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Numbers in Action A Naturalist Response to the Access Problem

Max Jones

A thesis submitted to the University of Bristol in accordance

with the requirements for the award of the degree of Doctor of

Philosophy in the Faculty of Arts, Department of Philosophy

December 2014

79,895 words

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Abstract

This thesis attempts to provide a response to the Access Problem by developing a

naturalist account of our access to mathematical knowledge. On the basis of recent

empirical research into the nature of mathematical cognition, it is argued that our

most basic access to arithmetical content is mediated by perceptual processes.

Moreover, in line with the theory of embodied cognition, arithmetical cognition is

grounded in the perceptual systems responsible for these processes, as well as other

perceptual and motor systems that are involved with our everyday interaction with

the world. This motivates a response to the Access Problem according to which

access to some mathematical content is on a par with our access to everyday objects

of perception. Whilst the picture that emerges on the basis of this response is

ontologically neutral, in the sense of being compatible with either a realist or anti-

realist approach to mathematics, it places significant constraints on a naturalistically

acceptable approach to the ontology of mathematics.

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Acknowledgements

First and foremost I would like to thank my amazing supervisor Richard Pettigrew

for his unending ingenuity, dedication, kindness and faith, which have allowed me to

produce this work. You have been there whenever I have needed you, and your

genuine interest in and encouragement of my sometimes somewhat strange ideas has

been wonderful. I must also thank my secondary supervisory Finn Spicer both for his

help in my developing these ideas and for inspiring my passion for questioning the

nature of knowledge and mind.

Huge thanks too to the many other philosophers who’ve helped me throughout my

time at the University of Bristol. In no particular order, thanks to Øystein Linnebo,

Neil Coleman, James Ladyman, Mark Pinder, Anthony Everett, Giulia Terzian, Leon

Horsten, Benedict Eastaugh, Megan Rose, Vincenzo Politi, Chris Burr, Marianna

Antonucci, Sorana Vieru, Kit Patrick, Kate Hodesdon, Alexander Bird, Irina

Starikova, Michelle Montague, Aaron Guthrie, Ollie Lean, Sam Pollock, Katy Monk,

Aadil Kurji, Prakhar Manas, Pavel Janda, Chris Gifford, Dagmar Wilhelm, Richard

Craven, Jason Konek, Cedric Paternotte, Stuart Presnell, Tom Richardson, Alex

Malpass, Elina Pechlivanida, Chris Clarke, Toby Meadows, Steve Horvath and

probably many more who I’ve forgotten to write down here. I must also thank all of

the great academics from other universities who have helped me with insightful

conversations and comments at various conferences and via email. In particular to

Helen De Cruz, Jesse Prinz, Edouard Machery, Hannes Leitgeb, Guy Dove, Bart Van

Kerkhove, Mario Santos Sousa, Jean-Charles Pelland and Yacin Hamami. Harry

Famer deserves a massive amount of credit for all of the great debates we have had

over the last nine years of both of us trying to understand the mind. Special thanks to

my wonderful philosophy teachers Mark Hogarth and Jon Phelan for inspiring me to

get involved in philosophy many years ago.

I have received a huge amount of support from all of my adopted families of friends

over the last few years. You have kept me sane and jolly through both tough and

joyous times. I cannot thank you all in person here but you should know who you are.

Special thanks go to Jimmy Maurice for teaching me that there aren’t any numbers

bigger than twelve.

Above all I’d like to thank my family. Your love, care and support throughout my life

has allowed me to fulfil my ambitions to try and understand the world a little bit

more. You are all an inspiration to me and always will be.

Thanks to all who made my actions possible.

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Author’s declaration I declare that the work in this dissertation was carried out in accordance with the requirements of the University's Regulations and Code of Practice for Research Degree Programmes and that it has not been submitted for any other academic award. Except where indicated by specific reference in the text, the work is the candidate's own work. Work done in collaboration with, or with the assistance of, others, is indicated as such. Any views expressed in the dissertation are those of the author.

SIGNED: ............................................................. DATE:..........................

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Contents

1. The Access Problem for Naturalism 7

2. Natural Numerical Perception 49

3. The Objects of Numerical Perception 73

4. Embodied Numerical Cognition 105

5. Perceptual Access and Ontological Parity 149

6. External Symbols and Arithmetical Cognition 173

7. Against All-or-Nothing Ontology 203

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1

The Access Problem for Naturalism

In his landmark 1973 paper, ‘Mathematical Truth’, Paul Benacerraf presented a

challenge to the world of philosophy of mathematics in the form of a dilemma, which has

become known as the Access Problem.1 To this day many still regard Benacerraf’s

challenge as ‘the philosophical problem of mathematical knowledge’.2 Over the last forty

years, many attempts have been made to address the challenge from a diverse array of

different positions in the philosophy of mathematics, yet the challenge remains very

much alive. Some may feel that the problem can easily be sidestepped or ignored.

However, I will argue that Benacerraf’s challenge remains particularly threatening to any

attempt to provide a naturalist account of mathematical knowledge. The challenge is

particularly threatening to the naturalist, as it exposes an underlying tension between

the two main strands of naturalist thought; Naturalist Epistemology and Naturalist

Ontology.3 Those who have attempted to provide a naturalist approach to the philosophy

of mathematics have either failed to adequately overcome Benacerraf’s challenge or have

done so in a manner that undermines their naturalist credentials.

Mathematical beliefs are a near ubiquitous feature of human thought. Whilst only

a small minority are familiar with sophisticated mathematical reasoning and concepts,

nearly everyone engages in some thought involving mathematical content. In particular,

nearly all of us think about numbers at some point in their lives. Given the prevalence of

arithmetical beliefs, it is somewhat surprising that in many ways they remain

mysterious. There is no consensus as to how exactly we are able to acquire these beliefs

or even as to what exactly these beliefs are about. Benacerraf’s challenge helps to

highlight the roots of this mystery.

In what follows, the aim is to shed new light on the challenge by adopting the

perspective of naturalised epistemology. As such, the primary focus will be on

1 Benacerraf, (1973) 2 Leng, (2007) pg. 1 3 Quine (1969), Quine (1948)

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understanding the psychological mechanisms that support our most fundamental

mathematical beliefs and concepts. The scope is restricted to an account of arithmetical

cognition for three reasons. Firstly, Benacerraf’s challenge is equally problematic with

respect to all mathematical content, so providing a response that alters our

understanding of just some of this content is sufficient to engender a reassessment of the

challenge as a whole. Secondly, numbers are often taken to be paradigmatic cases of

purely abstract entities and, as such, are clear culprits in giving rise to the challenge.

Other basic mathematical entities, such as geometrical objects, are less obviously

divorced from our experience of the world. Thus, in addressing the issue of number, the

aim is to develop a methodology that could potentially also be applied in these other,

perhaps less challenging, cases. Thirdly, arithmetical content is developmentally

fundamental in both an ontogenetic and a historical sense. Numerical content is the first

mathematical content that we encounter in our lives and forms the basis on which the

vast edifice of modern mathematics has been built. As such, by accounting for our

knowledge of arithmetic, the aim is to provide the foundations for an explanation of

more complex mathematical thought.

Benacerraf’s Challenge

Problems first arise when we begin to consider the meaning of sentences with

mathematical content. In the case of ordinary sentences devoid of mathematical content

it is usually assumed that the truth of a given sentence depends on the existence of the

entities and relations referred to in the sentence. For example, “the cat is black” is true

just in case there really exists a cat and that cat really possesses the property of being

black. Most agree that ordinary mundane sentences of this kind do not exhaust the range

of true sentences. In particular, most assume that mathematical claims, such as “3 is

prime”, are also true. Mathematical claims are often held up as paradigmatic cases of

true claims. If anything is true, then, surely, “2 + 2 = 4” is true. Applying the same

strategy to mathematical claims as we do to everyday more mundane claims, it seems as

if the truth of the claim “3 is prime” depends on the existence of some entity 3 that has

the property of being prime. Thus, just as ordinary claims commit us to the existence of

ordinary objects and properties, mathematical claims commit us to the existence of

mathematical objects and properties.

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It would be good if it were possible to develop a ‘homogeneous semantical theory

in which semantics for the propositions of mathematics parallel the semantics for the

rest of language’.4 We should have a method for determining the truth of a proposition

that works equally well in the case of mathematical claims as in the case of those devoid

of mathematical content. Tarski’s theory of truth provides just such a theory for

determining the truth-conditions of a sentence in a formal language from the

composition of its constituent parts.5 Furthermore, Montague provides a precise formal

theory of semantics that applies to both formal and natural languages.6 On both of these

accounts, use of singular terms commits one to the existence of the entities to which the

terms refer. As such, the truth of mathematical claims seems to entail the existence of

mathematical entities. Benacerraf points out that this issue is neither limited to our

intuitive conceptions of meaning nor to formal accounts of semantics presented by

Tarski and Montague. Any attempt to provide a unified semantics for both ordinary and

mathematical claims will entail an equal commitment to both ordinary and

mathematical entities. If one accepts that the truth of ordinary claims about, say, cats

commit one to the existence of cats then one must admit that mathematical claims

commit one to mathematical entities of some kind. Thus, in order to avoid commitment

to mathematical entities whilst maintaining a homogeneous semantics, one would have

to avoid ontological commitments for all claims. However, this is an approach that few

would find palatable.

An initial worry that arises at this stage is that the entities that serve as truth-

makers for mathematical claims seem mysterious and very different from the truth-

makers of ordinary everyday claims, such as cats, table and chairs. We encounter entities

of the latter kind on a daily basis. We can see cats or failing that bump into and trip over

them. However, mathematical entities, such as the number 3, don’t, at face value, seem

like the kind of things that we can encounter. ‘We do not bump up against’ them ‘nor do

we see or hear them’.7 As a result of considerations of this kind, most assume that

mathematical entities, if they exist at all, must be abstract as opposed to concrete

objects.

4 Benacerraf (1973) pg. 661 5 Tarski (1944) 6 Montague (1970) Montague argues that there are ‘no important theoretical differences’ (pg. 373) between formal and natural languages. 7 Shapiro (1997) pg. 109

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The nature of the distinction between abstract and concrete objects is a

convoluted issue, as it can be formulated in a number of distinct ways.8 Some define

abstract entities as entities that are causally inefficacious, whilst others define them as

entities that lack spatiotemporal location. In either case it should be clear that such

entities are not the kind of entities that occupy our physical realm. As a result, those who

are committed to the existence of mathematical entities are often taken to be committed

to their existence in an abstract realm, distinct from our physical realm, in some senses

akin to Plato’s realm of the Forms. As a result, realists about mathematical entities are

commonly dubbed Platonists.

A further motivation for consigning mathematical entities to the abstract realm

arises from the vast scale of some of the entities considered by mathematicians.

Mathematicians like to think big and a large swathe of mainstream mathematical

theorising deals with entities that far transcend the apparently finite bounds of the

universe that we live in, such as the natural numbers, the set-theoretic hierarchy or

Hilbert-spaces of infinite dimension. Assuming that there are only finitely many entities

in the universe, it becomes apparent that most mathematical entities transcend the scale

of the physical realm, even before one takes into account the kind of exotic infinite

mathematical entities just mentioned. Suppose there are n entities in the universe.9 It

seems clear that we can make true mathematical claims about the successor of n,

namely, n+1. However, there seems to be no collection of physical entities to which

claims about n+1 could possibly refer. As a result, it seems correct to infer that truths

about n+1 and beyond are truths about abstract rather than concrete entities. At this

stage it is tempting to suggest that, whilst claims about large and infinite mathematical

entities are claims about abstract entities, claims about entities smaller than n+1 are

truths about the concrete realm. However, this would involve introducing an arbitrary

divide to which mathematics itself is blind. If it even makes sense to talk about n, there is

nothing within mathematics that will inform us of what n is. As such it seems incorrect

to impose such a weighty metaphysical distinction on the basis of such an arbitrary

divide from the mathematical perspective. As a result, the fact that most mathematical

8 Rosen (2012) 9 It should be noted that the idea of a specific fixed number of entities in the universe may be somewhat unrealistic from the perspective of modern physics. For example, the notion of the number of entities in the universe may seem somewhat nonsensical from the perspective of quantum field theory. However, the consequences of this example are still valid, since for any limit that one places on the “size” of the universe, there are claims that mathematicians take to be true that seemingly transcend this limit.

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entities of interest to mathematicians transcend the limits of the physical realm can be

seen as a further motivation for taking all mathematical entities to be abstract.

An important consequence of this is that Benacerraf’s problem is equally

problematic in the case of the finite mathematical entities that people consider on a daily

basis as it is for the more exotic mathematical entities considered by professional

mathematicians. Much of the debate in the philosophy of mathematics over the last two

centuries has focussed on justifying the acceptance of exotic mathematical entities that

transcend the finite, such as the transfinite cardinals that occupy the higher reaches of

the set-theoretic hierarchy. Whilst this work is undoubtedly of immense value in terms

of establishing and consolidating the foundational justifications of our mathematical

knowledge, it does little to answer the problems raised by Benacerraf. These problems

are just as troubling for explaining our knowledge of the number 3, as they are for

explaining our knowledge of the further reaches of the set-theoretic hierarchy.

Benacerraf argues that the best theory of knowledge that we have available is a

causal theory of knowledge. Causal theories of knowledge were developed as a response

to Gettier’s famous challenge to the traditional definition of knowledge as true justified

belief.10 In certain cases, one can have true justified beliefs that most would intuitively

reject as being knowledge. In response, some argued that the problem in these cases is

that the given belief of the subject in question is not suitably causally related to that

which it is about.11 If one accepts a causal account of knowledge then mathematical

knowledge becomes problematic. Since mathematical entities are supposedly abstract,

they are, by definition, acausal and/or lacking in spatiotemporal location. Thus, on a

causal account of knowledge, mathematical knowledge seems to be rendered impossible.

There is no way that one could be causally related to an acausal entity and, since

causation is a physical relation, there is no way one could be causally related to an entity

that has no location in the physical realm. Benacerraf’s challenge has thus become

known as The Access Problem, since mathematical knowledge seems to be rendered

impossible by our lack of access to mathematical entities.12

We have thus reached the heart of Benacerraf’s dilemma. In order to provide a

universal and fully general theory of semantics, whilst also accepting the truth of

10 Gettier (1963) 11 E.g. Goldman (1967) 12 MacBride (2004)

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mathematical claims, it is necessary to posit mathematical entities, which, by their very

nature, are inaccessible and, thus, unknowable. If you want mathematical knowledge

then you cannot also have a standard notion of truth, and if you want a standard notion

of mathematical truth then you must accept that it is impossible to know such truths.

Naturalism and Realism

At this stage, from a naturalist perspective, it is tempting to take a step back and

try to resist positing a vast realm of mathematical entities. This would certainly fit with

Quine’s predilection for ontologically parsimonious ‘sparse desert landscapes’.13 Perhaps

giving up on a general theory of semantics for both ordinary and mathematical claims is

a price worth paying in order to avoid ontological extravagance. However, such a move

seems less palatable from a naturalist perspective, once one considers the role that

mathematical claims play in our scientific theories. A central aspect of the naturalist

approach to philosophy is to argue that questions of ontology are ultimately to be

decided on the basis of the commitments of our best scientific theories.14 We should

believe in just those things that our best scientific theories tell us exist.

The most common reason naturalists tend to adopt a realist stance to

mathematical entities is as a result of the supposed indispensability of mathematical

entities in our best scientific theories. This argument is widely known as the Quine-

Putnam indispensability argument. The use of mathematical methods pervades modern

scientific discourse and theorising, and for some can even be seen as the hallmark of

scientific enquiry. Thus, use of mathematics can be seen as indispensable to scientific

practice. However, in order to motivate the indispensability of mathematical entities one

must go further than this. This motivation stems from the fact that reference to

mathematical entities is indispensable from our best scientific theories. The Quine-

Putnam indispensability argument can be stated as follows:15

(P1) We ought to have ontological commitment to all and only the entities that

are indispensable to our best scientific theories.

13 Quine (1948) pg. 23 14 Ibid. pg. 36 15 Colyvan (2011) §2

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(P2) Mathematical entities are indispensable to our best scientific theories.

(C) We ought to have ontological commitment to mathematical entities.

The first premise can simply be seen as a statement of what is required from a naturalist

approach to ontology. The second premise is slightly more controversial. Whilst it is

clear that many scientific statements make reference to mathematical entities, it takes

more to show that such reference is indispensable. One way that one might try to

dispense with mathematical entities would be to separate out the mathematical and non-

mathematical parts of our scientific claims and try to show that we are only committed to

the latter.16 However, this is likely to prove difficult, since ‘there is no obvious way of

disentangling the purely mathematical propositions from the main body of our science.

Our empirical theories have the so-called empirical parts intimately intertwined with the

mathematical’.17 Furthermore, even if such a separation were possible with respect to our

scientific descriptions of the world, it seems impossible to eliminate reference to

mathematics when considering principles of statistical inference, which ‘is an essential

part of the scientific method’.18

Calling indispensability arguments of this kind the Quine-Putnam argument is

somewhat odd, since ‘neither Quine nor Putnam explicitly formulated the

indispensability argument that bears their names. And it is not clear that they would

have endorsed the particular versions that we find in the literature.’19 In the case of

Quine, the notion of indispensability can be seen as at odds with his commitment to

holism. Quine’s holism can be seen to derive from his rejection of the analytic-synthetic

distinction.20 As a result, no aspect of our web of knowledge is unrevisable in principle.21

Thus it seems strange to credit Quine with suggesting the absolute indispensability of

mathematical entities to our scientific ontology, since his holism would suggest that

there might, in principle, be circumstances where any aspect of our overall ontology

might be jettisoned. We are committed to mathematical entities, not because they are

16 Field (1980) attempts to do just this. Whether or not he is successful in doing so is a question that goes beyond the scope of the current work. However, it should suffice to say that there are prima-facie reasons for thinking that the task is difficult at best if not impossible. Furthermore, in attempting to do so, Field ends up being committed to odd metaphysical entities such as space-time points and space-time regions, which could arguably be seen to be as mysteriously unknowable as the mathematical entities that he is trying to replace. 17 Colyvan (2001) pg. 36-37 18 Resnik (1997) pg. 57 19 Pettigrew (2012) pg. 687 20 Quine (1951) 21 Resnik (2007)

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indispensable, in principle, but because the scientific theories that refer to them are

better scientific theories. For example, such theories are simpler, more unifying and

more explanatory.

As a result of these considerations, it might make more sense to see the Quine-

Putnam argument as an argument for parity rather than indispensability. The reason for

clinging to belief in mathematical entities is the fact that they are as dispensable as other

entities that we tend to take our scientific theories to commit us to. In particular,

mathematical entities are taken to be at least as dispensable as unobservable entities,

such as electrons or black holes. In both cases we have no direct perceptual contact with

the entities in question. However, we take the indispensable role that they play in our

best scientific theories as evidence for their existence.

(P1) We should grant the same ontological status to entities that play the same

role in our best scientific theories.

(P2) Mathematical entities play the same role as unobservable entities in our best

scientific theories.

(P3) The role that unobservable entities play in our best scientific theories

commits us to their existence.

(C) The role that mathematical entities play in our best scientific theories

commits us to the existence of mathematical entities.

On these grounds it should be clear that there are reasons for a naturalist to commit to

the existence of mathematical entities that are independent of indispensability per se. As

long as the reasons for believing in unobservable entities are on a par with the reasons

for believing mathematical entities and one has good reason for believing in the former

then one should believe in the latter. However, it is important to note that unlike the

indispensability argument, the parity argument depends on the potentially contentious

premise that our best scientific theories do commit us to unobservable entities. This

premise would be called into dispute by Constructive Empiricists, such as Van

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Fraassen.22 Thus, if we remove this somewhat controversial premise, all that parity

arguments can show is that we should either believe in unobservable and mathematical

entities or we should believe in neither.

At this stage it is worth considering whether our reasons for believing in

unobservable and mathematical entities are really on a par. Sober has questioned

whether the role of mathematical entities is really on a par with that of unobservable

entities.23 He argues that when testing scientific theories we are not testing for the

existence of mathematical entities in the same way that we are testing for the existence of

theoretical entities. One of the most important features of theoretical entities is that they

are dispensable in light of evidence that falsifies the theory that posits them.24 However,

there seem to be no conceivable observations that would falsify our belief in

mathematical truths. As such even if mathematical entities are indispensable, this

indispensability renders them as not being on a par with theoretical entities. Maddy has

also argued against the idea that mathematical entities should be seen as on a par with

theoretical entities and should thereby be seen as indispensable.25 She argues that

mathematical methods have featured to the same extent in false scientific theories as

they do in true scientific theories. Furthermore, much of the use of mathematics in

science involves abstractions and idealisation which form parts of a theory that do not

even aim to be strictly true of the actual world.26 Both of these approaches seem to

question ontological parity with theoretical entities as a basis for the indispensability of

mathematical entities. As such, the necessity of realism for naturalists may be

undermined.

However, Resnik has argued that mathematical realism might still be necessary

for naturalists. He argues that there are reasons for taking mathematics as indispensable

to science even though mathematical entities are not on a par with theoretical entities.27

Scientific practice involves drawing conclusions on the basis of taking mathematical

claims to be true. This is the only way that science can be done. Thus, we are justified in

taking mathematical claims to be true and thereby believing in the things that they refer

22 Monton & Mohler (2012) 23 Sober (1993) 24 Ibid. pg. 44 25 Maddy (1992) 26 Ibid. pg. 281 27 Resnik (1995)

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to.28 On these lines the indispensability of mathematics to science can be motivated even

if the role of mathematical entities is not on a par with that of theoretical entities.

From the Quinean holist perspective, one could even argue that our belief in

mathematical entities is in some sense sturdier than our belief in unobservables, since

the former are more deeply embedded in our web of knowledge than the latter.29 For

example, our beliefs about electrons only directly impact upon a relatively small portion

of our overall knowledge and, as such, if empirical evidence suggested the need to

jettison our commitment to electrons, only a small portion of our overall knowledge

would need revision. Mathematical beliefs, on the other hand, are of direct significance

to a wide range of other scientific beliefs, and so jettisoning commitment to

mathematical entities might lead to a much more widespread revision of overall

knowledge. In this sense mathematical beliefs can be seen as more central to our web of

knowledge and beliefs in unobservables as more peripheral. On these grounds, there

may be room to assert that we have more reason to believe in mathematical entities than

unobservable entities or alternatively that mathematical entities are less dispensable

than unobservables.

A naturalist approach to ontological questions advocates the view that we should

be committed to the existence of the entities that are posited by our best scientific

theories. However, the naturalist’s deference to the scientific worldview is not limited to

questions of ontology. Many also argue that we should take a naturalised approach to

epistemology. Traditional epistemology primarily focused on attempting to explain and

define knowledge by methods of conceptual analysis. On these lines, knowledge is to be

understood by introspecting and analysing our intuitions about the nature of knowledge.

Naturalists have argued that this approach is unscientific and thus insufficient. A

naturalist approach to ontology dictates that the mind is a purely physical entity, since

our best scientific theories make no reference to the mysterious non-physical mind

posited by dualists. As a result, the natural sciences, in particular psychology, cognitive

science and neuroscience, offer the best explanation of mental phenomena. Since

knowledge is, at least in part, a mental phenomenon, the best way to understand it is to

turn to the sciences that deal with the underlying mechanisms of the mind. Thus,

epistemology ‘simply falls into place as a chapter of psychology and hence of natural

28 Ibid. pg. 171 29 Resnik (2007) pg. 422-423

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science. It studies a natural phenomenon, viz., a physical human subject.’30 As a result,

traditional epistemology should either be replaced with or supplemented by the scientific

study of the physical mechanisms that allow creatures like us to come to know about the

world.31 ‘Any faculty that the knower has and can invoke in the pursuit of knowledge

must involve only natural processes amenable to ordinary scientific scrutiny’32

This combination of the need for realism and for scientific explanations of

knowledge renders Benacerraf’s challenge particularly challenging for the naturalist. The

first of these aims seems to demand the existence of abstract mathematical entities,

which have no spatiotemporal location or causal efficacy. However, science is necessarily

limited to only studying aspects of the physical realm with which we can have causal

contact. As a result, Benacerraf’s challenge seems to render the satisfaction of these two

naturalist desiderata as impossible. If mathematical knowledge really is knowledge, and

scientific practice seems to suggest that it is, then there can be no naturalistically

acceptable explanation of how we are able to acquire this knowledge.

Abandoning the Causal Theory of Knowledge

At first sight the most vulnerable part of Benacerraf’s argument appears to be his

insistence upon a causal theory of knowledge. The causal account of knowledge is only

one amongst many responses to Gettier’s challenge to the traditional theory of

knowledge. It is thus tempting to merely abandon such an approach and hope to escape

the challenge by adopting one of these alternatives. Steiner attempts to dissolve

Benacerraf’s challenge in just this manner, arguing that ‘the most plausible version of the

causal theory of knowledge admits Platonism, and the version most antagonistic to

Platonism is implausible’.33

Whether or not one buys Steiner’s argument against the causal theory of

knowledge in this context, the lack of causal access to mathematical entities can still be

seen as problematic from a naturalistic perspective. ‘It is a crime against the intellect to

try to mask the problem of naturalising the epistemology of mathematics with

30 Quine (1969) pg. 82 31 Feldman (2012) §3, §4 32 Shapiro (1997) pg. 110 33 Steiner (1975) pg. 116

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philosophical razzle-dazzle. Superficial worries about the intellectual hygiene of causal

theories of knowledge are irrelevant to and misleading from this problem, for the

problem is not so much about causality as about the very possibility of natural

knowledge of abstract objects’.34 To see why this is the case it is useful to consider how

one might attempt to solve the problem using an alternative strategy. For instance, one

might invoke a reliablist theory of knowledge in the hope that this might circumvent

Benacerraf’s challenge. According to this approach a justified true belief qualifies as

knowledge just in case it is the result of a reliable belief-forming process. Suppose we

grant that the methods employed by expert mathematicians are reliable belief-forming

processes. At face value, it would seem as though we have managed to circumvent

Benacerraf’s challenge in allowing for mathematical knowledge despite the causal

inertness of abstract entities. However, such an approach is unlikely to be satisfactory

from a naturalistic perspective, since the reliability of the processes in question still

requires explanation.35 From a naturalistic perspective it is unclear how one could

explain the reliability of such a process without invoking some kind of causal or

spatiotemporal relation to mathematical entities in any one instance of the given

process.

The issue is not with the causal theory of knowledge per se but with the idea that

there could be any adequate naturalistic explanation of knowledge that fails to invoke

some kind of causal or spatiotemporal relation between the knower and the thing that

they know about. ‘Benacerraf’s challenge… is to provide an account of the mechanisms

that explain how our beliefs about these remote entities can so well reflect the facts about

them’.36 One way of fleshing out this idea is in terms of information. It seems a minimal

requirement of knowledge about a given entity that the knower must have some way of

acquiring information about the entity in question. However, transfer of information

requires some kind of transfer of energy and given the law of the conservation of energy

it is clear that no energy could be transferred from the abstract realm into our physical

realm carrying information about abstract mathematical entities. Regardless of one’s

views about causation, it is still necessary to explain ‘how our beliefs could be about

energetically inert objects’.37 The problem that Benacerraf’s challenge provides for

naturalism is not merely limited to those that subscribe to a causal theory of knowledge.

34 Hart (1977) pg. 125-6 35 Maddy (1990) pg. 43 36 Field (1989) pg. 26 37 Hart (1977) pg. 125

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Any naturalist account of knowledge should aim to explain knowledge in terms of purely

physical processes and, as such, will always face problems in explaining knowledge of

nonphysical abstract entities.

The Problem of Reference

Whilst Benacerraf’s challenge was initially seen as problematic for explaining

mathematical knowledge, the difficulties that arise are arguably far wider in scope. The

notion of causation plays a prominent role in theories of reference as well as theories of

knowledge, in particular those theories of reference that attempt to provide naturalistic

explanations of reference. For instance, Kripke argues for a causal theory of reference,

whereby one’s ability to refer to a given object depends upon one standing in an

appropriate causal relation to an event where the object in question was first

encountered and given its name.38

As Lear and Hodes both point out, this is problematic in the case of mathematical

objects, since there is no way that one could be appropriately causally related to an

encounter with a causally inert entity lacking spatiotemporal location.39 Thus, as well as

rendering mathematical knowledge as mysterious, Benacerraf’s dilemma also highlights

difficulties in explaining how it is possible to successfully talk about mathematical

entities. Again, the problem is particularly acute from a naturalist perspective, since

naturalist approaches to reference are restricted to explaining reference in terms of some

kind of physical relationship between a word and its referent.

As with the case of naturalist accounts of knowledge, problems are still likely to

arise if one adopts an alternative to the causal theory of reference, since one still faces

the problem of explaining how a physical utterance or inscription of a word can refer to a

non-physical mathematical entity, entirely in physical terms. For example, Hodes argues

that the problem cannot be avoided by adopting a descriptivist theory of reference, such

as Frege’s. In order to illustrate this, he presents the example of Adam, who speaks a

language that is just like English apart from having non-standard meanings for “4”, “5”,

“successor”, “less than” and “the number of”, such that, despite these differences in

38 Kripke (1980) pg. 91, 96-97 39 Lear (1977) pg. 88, Hodes (1984) pg. 127

20

reference, ‘there is no systematic difference between the sentences that we accept and

those that Adam accepts’.40 Problems arise because there seem to be no ‘physical,

psychological and social facts’ that can explain why everyone happens to hit upon the

same standard referents for number words, when descriptions equally consistent with

the evidence are available.41 The only available options are to either resort to a

naturalistically unacceptable appeal to mathematical intuition or to fall back on a causal

theory of reference and face the problem just mentioned.

The Real Problem: The Problem of Mental Content

It is tempting to see the problems that Benacerraf’s dilemma raises for

mathematical knowledge and reference as two similar, yet distinct, issues. However, I

shall argue that both problems are symptomatic of a single deeper problem that

Benacerraf’s challenge reveals. This underlying mystery is not the problem of how

knowledge of mathematical objects is possible nor is it the problem of how successful

talk of such objects is possible. The real problem lies in explaining, in naturalistically

acceptable terms, how it is possible to think about mathematical objects.

Benacerraf’s access problem is essentially a problem of how it is possible to

acquire beliefs with mathematical content. Beliefs are generally taken to be constituted

by concepts, with the content of a belief being determined by its constituent concepts in

much the same way that the meaning of a sentence is determined by its constituent

words. As such, in order to explain how we are able to acquire beliefs with mathematical

content it is necessary to explain how we are able to acquire mathematical concepts. If

one accepts the existence of abstract mathematical entities then it becomes mysterious

as to how one could ever come to have concepts of these objects which, in principle,

cannot be encountered.

As with the case of theories of reference, most recent attempts to provide

naturalist accounts of mental content incorporate some kind of causal condition. For

example, Dretske’s Causal-Informational theory of content suggests that a mental state S

can be said to be about Xs if it has developed so as to reliably respond to signals that

40 Hodes (1984) pg. 134 41 Ibid. pg. 134

21

carry information about Xs or, in other words, if S is reliably caused by information

about Xs. Thus, a prima facie problem arises for mathematical concepts, since it seems

as nothing in the environment carries signals about the presence of mathematical

entities that occupy an entirely distinct abstract realm.

Dretske’s approach to mental content faces a number of criticisms and is widely

held to have been superseded by alternatives, such as the Asymmetric Dependency

theory advocated by Fodor or the Teleosemantic theory of Millikan.42 Fodor’s approach

was developed in order to deal with the problem that beliefs about Xs can often reliably

be caused by the presence of non-Xs. For example, a mental state with the content DOG

might reliably be caused by foxes on a foggy night and yet we would not want a theory of

content that entailed that our DOG concept was also a FOX ON A FOGGY NIGHT concept.

Thus, Fodor argues that S has content X iff Xs reliably cause S and, for all Ys that cause

S, the fact that Ys cause X is dependent on the fact that Xs cause S. Thus, although foxes

might cause S, they only do so in virtue of looking like dogs under certain conditions, so

since S ultimately depends on its causal relations to dogs it can be said to have the

content DOG. Again this attempt to naturalise mental content seems inapplicable to the

case of mathematical entities, since there can be no mental states that are reliably caused

by acausal abstract entities.

Millikan provides an alternative approach to naturalising mental content, which

places the theory of content in the context of our evolutionary history.43 She argues that

it isn’t sufficient for a mental state to merely be reliably caused by the presence of a

certain condition in order to represent that condition. As well as this, the mental state

must have developed in order to serve the function of responding to that particular

condition. Once again, the lack of causal contact with abstract entities is problematic,

since it seems to undermine the prospects of a naturalist account of concepts with

mathematical content. This problem can be seen as the root of both the challenge to a

theory of knowledge and the challenge to a theory of reference for mathematical claims,

as both theories of knowledge and theories of reference are dependent on a satisfactory

account of mental content.

42 Fodor (1990), Millikan (1984) 43 Millikan (1984)

22

Whilst many aspects of the theory of knowledge are debatable and up for grabs, it

is widely accepted that possession of a belief about the object of one’s knowledge is a

necessary condition for knowledge. As such, any account that fails to explain how we are

able to acquire the relevant beliefs will therefore fail to provide a satisfactory account of

knowledge in the given context. Since, the content of beliefs is derivative of the content

of the concepts that comprise them, it is only possible to explain acquisition of the

former by explaining acquisition of the latter. Thus, the original problem highlighted by

Benacerraf with respect to mathematical knowledge can be seen as a result of problems

in providing a naturalist account of how it is possible to acquire mental states with

mathematical content in the first place.

An important consequence of locating the root of Benacerraf’s challenge in the

mysteriousness of the acquisition of mathematical beliefs is that it rules out attempts to

escape the challenge by merely altering the definition of knowledge. It is tempting to

think that the problem can be remedied either by adding a further condition to the

traditional justified true belief account or by tweaking the notion of justification.

However, since the problem infects the belief condition of the tripartite definition, no

additional conditions or tweaks to the justification condition will impact upon the

problem. The only way to avoid the problem would be to argue for the counter-intuitive

position where knowledge is entirely independent of belief. Some have indeed

questioned whether knowledge always entails belief.44 However, this falls far short of

questioning whether belief is ever relevant to knowledge. Furthermore, even if one could

develop an account of knowledge that is entirely independent of belief, one would still

face the problem of explaining the fact that people clearly possess some mathematical

beliefs, regardless of their truth or falsehood and regardless of their relation to

mathematical knowledge. As such, it seems that no attempt to tweak the definition of

knowledge will allow an escape from Benacerraf’s challenge.

As with the case of Benacerraf’s challenge regarding mathematical knowledge.

The problem of explaining mathematical reference can also be seen to result from the

difficulties in explaining the acquisition of mathematical mental content. In the case of

naturalist theories of reference, it is generally accepted that an individual’s capacity for

successful reference to a given object depends on there being some kind of suitable

association between their mental representation of the object in question and the word

44 Radford (1966) Myers-Schulz & Schwitzgabel (2013)

23

used to refer to it. Thus, reference becomes problematic in the case of mathematical

objects, since if it isn’t possible to explain how we are able to acquire mental

representations of mathematical entities then it is hard to explain how these mental

representations could be associated with the words that refer to mathematical entities.

The real problem that is brought to the fore by Benacerraf’s challenge is that it

seems impossible to explain how we are able to acquire concepts with mathematical

content. Most accounts of conceptual content build some kind of causal condition in,

such that a concept must have some kind of causal contact with that which it represents.

However, in the case of concepts of abstract mathematical objects, any explanation of

this kind if precluded.

Access and Justification

It is important to note, at this stage, that Benacerraf’s challenge is primarily a

problem for explaining how we acquire mathematical content, rather than a problem of

how we justify mathematical knowledge. Even if there turned out to be no way of

justifying our mathematical knowledge it would still be necessary to explain where our

mathematical beliefs came from in the first place. This distinction can be seen as akin to

Reichenbach’s distinction between the context of discovery and the context of

justification.45 Traditionally within the philosophy of science, only the latter is taken to

be relevant to epistemological enquiry. The context of discovery was considered to be the

domain of psychology and a belief’s ‘psychological origin’ was taken to be ‘irrelevant to

epistemology’.46 How we access our beliefs was taken to be irrelevant to assessing

whether they qualify as knowledge.

This claim of the irrelevance of issues of access for epistemology can be called into

question once one adopts a naturalist approach to epistemology. Once one takes the

study of psychological processes to be central to the goals of epistemology, the context of

discovery suddenly plays a more significant role. ‘Questions about how we actually arrive

at our beliefs are thus relevant to questions about how we ought to arrive at our beliefs.

Descriptive questions about belief acquisition have an important bearing on normative

45 Reichenbach (1938) pg. 5-7 46 Siegel (1980) pg. 300

24

questions about belief acquisition’.47 If one adopts a naturalist approach to epistemology

then how one arrives at one’s beliefs is central to whether those beliefs can be seen as

justified. ‘Questions about a belief’s justification cannot be answered independently of

questions about a belief’s causal ancestry’.48 It is in this sense in which Benacerraf’s

challenge can be seen as a challenge to naturalised epistemology. The challenge itself is

primarily a challenge of explaining how we are able to acquire mathematical content.49

However, on a naturalised epistemological picture, any account of epistemic justification

will have to make reference to epistemic access. It is only possible to explain how certain

beliefs are good beliefs by showing that they have some connection with their content.

However, it is important to be clear that the problem that Benacerraf’s challenge raises

for naturalised epistemology is derivative of the more fundamental problem of

explaining how we are able to acquire psychological states that represent mathematics.

Non-Naturalist Responses to the Challenge

To be fair to Non-Naturalist Platonists, most acknowledge that some answer to

the problem is required. Even Plato saw that he needed to provide some account of how

we could acquire knowledge of abstract entities that transcend the concrete realm. As

such, he suggested that our knowledge of the abstract is not acquired through learning in

the physical realm but remembered from some mysterious prenatal stage where our

souls existed in contact with the abstract realm.50 It goes without saying that, in positing

the existence of an undetectable eternal soul that exists before birth in the realm of

Forms, Plato’s explanation of the origins of mathematical knowledge is not

naturalistically acceptable. However, it will be important to keep in mind Plato’s

important insight that, in order to explain the origins of mathematical knowledge, it

might be necessary to invoke more than our direct contact with the physical realm. In

47 Kornblith (1997) pg. 3 48 Kornblith (1982) pg. 238 49 This has interesting implications for those, such as Field (1980), who respond to Benacerraf’s challenge by simply denying that we have mathematical knowledge. By doing so they avoid the main threat of the challenge, in the sense that our concepts need not be about abstract entities and so may be possible to explain. However, in denying that our mathematical beliefs are justified they do not provide a full answer to the challenge, since they too must provide some explanation of how we acquire mathematical concepts and beliefs in the first place, which tallies with their denial of mathematical knowledge. 50 Plato (2005) 112-113

25

particular, it may be necessary to invoke some inherited mechanisms that, in a certain

sense, can be seen as acquired before our birth.51

Gödel, the most famous advocate of Platonism in recent times, also saw the need

to explain the possibility of acquiring mathematical knowledge, suggesting that we

possess a special faculty of mathematical intuition, akin to the faculty of perception but

directed at abstract rather than concrete entities. He argues that ‘despite their

remoteness from sense experience, we do have something like a perception also of the

objects of set theory, as is seen from the fact that the axioms force themselves upon us as

being true’ and goes on to claim that he doesn’t ‘see any reason why we should have less

confidence in this kind of perception, i.e., in mathematical intuition, than in sense

perception’.52

The problem with Gödel’s account is that the mechanism of mathematical

intuition is doomed to remain forever mysterious from a naturalist perspective. In the

case of perception, we are not merely certain in the usual veridicality of our sensory

experiences; we also have psychological and neurophysiological theories that at least

begin to explain how perception works. In the case of mathematical intuition, it is

difficult to even say what it is, let alone how it works. Furthermore, given the non-

physical nature of mathematical entities, any physical explanation of mathematical

intuition, of the kind we accept for perception, will be impossible. Thus, Gödel does not

answer Benacerraf’s challenge. He merely stipulates that there is some mysterious

mechanism that allows us contact with the abstract realm. ‘There is nothing wrong with

supposing that some facts about mathematical entities are just brute facts, but to accept

that facts about the relation between mathematical entities and human beings are brute

and inexplicable is another matter entirely’.53 Unless some such mechanism can be

discovered, Gödel’s account will remain unsatisfactory from a naturalistic perspective.

The main thrust of Benacerraf’s challenge is to suggest that the discovery of any such

mechanism is impossible, since our physical theories of the world do not tend to deal

with mechanisms involving both physical and nonphysical entities.

51 This issue will be addressed in Chapter 2, where it will be argued that at least some of our access to mathematical content is mediated by innate psychological mechanisms. 52 Gödel (1964) 272 53 Field (1990) 215

26

Whilst Gödel clearly fails to provide a naturalistically acceptable account, it is

possible to give him a more charitable reading. Instead of taking him to be arguing that

we possess a special intuition faculty akin to perception, one can take him to be making

the more negative claim that we have as much reason to doubt perception as we do

intuition. He notes that seemingly straightforward aspects of our perception of the world

remain mysterious on a traditional account of sensory perception. For instance, an

account of perception in terms of ‘sensations or mere combinations of sensations’ fails to

explain the origins of as simple a concept as ‘the idea of object’.54 Thus, Gödel can be

taken as highlighting the paucity of our theories of perception and suggesting that if we

are to understand the origins of either our concept of object or of mathematical concepts,

we may need to invoke ‘another kind of relationship between ourselves and reality’, that

goes beyond traditional sense-data theories of perception.55 Thus, as with Plato’s

account, whilst Gödel’s response to the Benaceraffian challenge may be unacceptable

from a naturalistic perspective, he highlights an important consideration, which

naturalists should take note of.56

Avoiding Objects: Structuralism

So far, the responses to Benacerraf’s challenge that have been considered have

primarily involved attempts to tinker with the definition of knowledge. However, an

alternative response lies in questioning whether a standard account of the truth of

mathematical claims necessarily leads to commitment to mathematical objects. In

response to an entirely different challenge to Platonism, also put forward by Benacerraf,

many philosophers of mathematics have adopted a Structuralist position, whereby

straightforward commitment to mathematical entities is avoided.57

In ‘What Numbers Could Not Be’ Benacerraf highlights a problem for Platonists

that arises from attempts to reduce numbers to their set-theoretic counterparts. Zermelo

and Von Neumann proposed differing set-theoretic accounts of the natural numbers,

where either one is equally as viable as the other.

54 Gödel (1964) pg. 271 55 Ibid. pg. 272 56 This issue will be addressed in Chapter 3, where it will be argued that our access to mathematical content can be explained in perceptual terms by looking to contemporary theories of perception, which address precisely the concerns that Gödel raises. 57 Benacerraf (1965)

27

Von Neumann: {Ø}, {Ø,{Ø}}, {Ø,{Ø},{{Ø}}}, …

Zermelo: {Ø}, {{Ø}}, {{{Ø}}}, …

Whilst both systems agree when it comes to the truths of arithmetic, they disagree as to

the exact nature of the numbers. For instance, for Von Neumann 2 is a member of 3,

whilst for Zermelo it is not and for Von Neumann 3 has a cardinality of 3 whilst for

Zermelo it has a cardinality of 1.58 Furthermore, Zermelo’s and Von Neumann’s are not

the only ways of reducing the natural numbers to set-theory. There are, in principle,

indefinitely many ways of conducting such a reduction, where each is equally viable and

succeeds in preserving the truths of arithmetic. This is problematic for the traditional

Platonist, as if, for instance, the number 3 is a real entity then there should be some fact

of the matter as to its identity and its properties. However, in the case of Von Neumann’s

and Zermelo’s differing accounts there seems to be no way to decide which of “{Ø, {Ø},

{{Ø}}}” and “{{{Ø}}}” is the real number 3. Both are equally suitable to play the role of

number 3 and no mathematical considerations allow us to favour one over the other.

Structuralists offer a solution to this problem by arguing that we are wrong to

interpret mathematical claims as straightforwardly referring to mathematical objects.

Apparent reference to mathematical objects is better construed as reference to positions

in mathematical structures. “3” does not refer to an independent abstract entity; it refers

to a position in an abstract structure, which could be occupied by any entity that bears

the right kind of relations to the other entities in the given structure. Thus, both “{Ø,

{Ø}, {{Ø}}}” and “{{{Ø}}}” can be seen as 3, since both of them occupy the 3 position in

the structures that they belong to. The true claims of mathematics need not be seen to

engender an ontological commitment to a distinct abstract object for each mathematical

term. Instead, one can just be committed to the existence of certain abstract structures,

with terms such as “3” merely referring to positions within such structures.

At first sight this might seem like a departure from a standard theory of semantics

and thus be in violation of Benacerraf’s call for a standard semantics for mathematical

and everyday claims. However, this worry can be assuaged, since reference to positions

in patterns is relatively commonplace in everyday language. For example, when

58 Ibid. pg. 55

28

explaining the rules of football, someone might say “the goalkeeper is allowed to use

their hands in the penalty area”. At face value this sentence seems to commit us to a

particular single entity, “the goalkeeper”. However, anyone who understands the

sentence properly can see that, rather than being about a particular entity, it is about any

entity that happens to occupy the role of goalkeeper in a football match. Thus, similarly

we should take claims involving “3” to refer to any entity that happens to occupy the role

of the number 3 in the natural number structure, where this role is defined purely in

terms of relations to other positions in the structure regardless of the intrinsic properties

of the entity in question.

Both Shapiro and Resnik, two of the most prominent advocates of Structuralism,

attempt to provide a naturalist account of the subject matter of mathematics and our

knowledge of it. As such, both argue that Structuralism provides a better account of

mathematical knowledge than traditional Platonism by avoiding the problems raised by

Benacerraf’s challenge. At first, this may seem somewhat mysterious, since, in affirming

the existence of abstract structures, the Structuralist could be seen to be positing entities

that are just as inaccessible as the Platonists’ abstract objects. However, whilst the

abstract mathematical objects of the Platonist bear no relation to the concrete objects of

the physical realm, abstract structures can be instantiated by patterns or systems of

concrete objects in the physical realm. Physical objects bear certain physical relations to

one another and, by focussing on these relations whilst ignoring irrelevant features of the

objects themselves, we can come to learn about the abstract structures that these

patterns of physical objects exemplify. ‘With processes much like – or even identical to –

ordinary sense perception, a subject comes to recognise and learn about patterns’.59

Whilst the exact mechanisms involved in pattern recognition ‘pose deep and interesting

problems for cognitive psychology’, the fact that humans possess a faculty of pattern

recognition is deemed ‘philosophically unproblematic’.60 Thus, Structuralism is able to

offer the beginnings of an escape from Benacerraf’s challenge, in that we are able to

acquire at least some knowledge of mathematical structures by encountering and

recognising patterns of concrete objects that exemplify these structures.

Whilst our capacity for pattern recognition offers the structuralist a route to an

explanation of our access to mathematical knowledge, it is clear that such a strategy can

59 Shapiro, (1997) pg. 111 60 Ibid. pg. 112

29

only take us so far. We only ever encounter relatively small finite patterns of concrete

objects and when we encounter larger patterns it is unclear that our capacity for pattern

recognition is adept enough to recognise the given pattern. For instance, upon

encountering a pattern of 9,427 objects, we lack both the capacity and the time or

inclination to recognise that this is the particular pattern that we have encountered.

Since most interesting mathematics involves our capacity to comprehend large finite and

often infinite structures, the structuralist must explain how we are able to go beyond

simple recognition of small finite patterns in order to comprehend these larger

structures. In order to explain this capacity, Shapiro again invokes our capacity for

pattern recognition. However, he argues that we are also able to notice higher-order

patterns that hold between small finite patterns and to project these patterns so as to

comprehend large and infinite structures. For instance, we are able to notice that the

sequence of patterns of strokes (below) exemplifies a higher-order pattern, where each

new pattern in the sequence has one more stroke than the last.

I , II , III , IIII , IIIII , …

We are then able to project this pattern and come to the realisation that for each pattern

in the sequence there is a subsequent pattern with one more stroke and that, as such, any

large finite pattern will occupy a determinate position in the overall structure.

Furthermore, we can come to realise that for any pattern in the sequence there will

always be a further pattern with one more stroke and, thus, come to realise that the

higher-order structure, of which the sequence exemplifies an initial segment, is infinite

in extent.

Numeral systems play a significant role in this ability to project from small finite

patterns to large finite and infinite patterns, since the numeral system can itself be seen

to exemplify the structure of the natural numbers. If one considers the relations between

the numerals in a numeral system, in isolation from the particular properties of the

numerals themselves, then this abstract structure is itself an instance of the natural

number structure. ‘The point is that… understanding how to work with numerals…

presupposes everything needed for arithmetic… In short, understanding how to use the

language of arithmetic is sufficient for understanding and referring to a system that

exemplifies the natural numbers.’61 The ability to understand and use the language of

61 Ibid. pg. 137

30

arithmetic may not be necessary for arithmetical knowledge but it is indicative of

possession of such knowledge.

Whilst our capacities for pattern recognition, abstraction and projection

supposedly allow us to acquire a significant amount of mathematical knowledge, Shapiro

acknowledges that these capacities are not sufficient to explain all of our mathematical

knowledge. In particular, they are not powerful enough to generate knowledge of the

kinds of transfinite structures considered in set-theory. In order to explain our belief in

these much larger structures, Shapiro invokes a further capacity for generating

mathematical knowledge; description. He argues that we can come to know of the

existence of a particular mathematical structure by providing a coherent and categorical

description of that entity.62 We can infer the existence of a particular structure if we have

a consistent set of axioms that uniquely picks out the given structure.

Whilst Resnik concurs with Shapiro in emphasising the significance of our faculty

of pattern recognition in providing a naturalist account of mathematical knowledge, he

takes a different approach when it comes to explaining knowledge of structures that go

beyond the patterns we encounter in everyday experience. He offers a Quinean account

of our justification for believing in mathematical structures, arguing that, since our

scientific knowledge is dependent upon mathematical knowledge, we are justified in

believing in mathematical structures in order to avoid unwanted revisions of our belief in

the physical objects studied by science. As has been mentioned earlier, this may be a

good way of justifying our mathematical beliefs, but, when it comes to Benacerraf’s

challenge, justification isn’t the issue. What is required is an account of the acquisition of

beliefs about mathematical structures.

In order to address this issue Resnik offers a quasi-historical account of the

development of mathematical beliefs.63 He begins with the observation that the

languages of aboriginal peoples and thus, presumably, prehistoric peoples too contain

words that refer to small collections of entities. He then argues that the next step in the

development of our knowledge of number was the development of ‘indefinitely

protractible systems of numerals for counting’.64 This step allowed members of ancient

societies, such as the Babylonians and Egyptians, to understand the relations between

62 Ibid. pg. 132 63 Resnik (1997) pg. 176-180 64 Ibid. pg. 178

31

mathematical entities that transcended the limits of immediate perceptual awareness by

appreciating the systematic relations in their numeral systems. They could, for example

appreciate that one-billion-and-three is less than one-billion-and-five without even

comprehending the meaning of “one billion” by understanding the rules that govern the

generation of number words. As a result, these ancient societies end up ‘implicitly

positing’ abstract structures, to which they have no direct access.65

Problems for Structuralism’s Naturalist Epistemology

By simply acknowledging the existence of pattern recognition as the process by

which we apprehend small finite mathematical structures, Shapiro and Resnik merely

name the process that, in light of Benacerraf’s challenge, requires explanation. It may

seem philosophically unproblematic to assert that we possess such a capacity but the

question of how such a capacity works is exactly the issue at hand in trying to provide a

naturalist response to the challenge. By invoking pattern recognition the structuralists

isolate the phenomenon that requires explanation but by relegating the issue of

explaining pattern recognition as only of interest to cognitive psychology they undermine

their own naturalist credentials. In a way all that is achieved is a restatement of the

original problem. In the case of mathematical knowledge, the central aspect of

Benacerraf’s challenge isn’t the question of whether such pattern recognition exists but

how such a process is possible. Similarly, in order for the invocation of pattern

recognition to do any useful work in explaining how we are able to acquire beliefs with

mathematical content, one must go further than merely acknowledging the existence of

such a capacity. One must explain how such a faculty works. In particular, it is necessary

to explain how our capacity for pattern recognition allows us to recognise distinctly

mathematical patterns. Whilst the fact that we are able to recognise patterns of one kind

or another is somewhat trivial, the fact that we are able to recognise mathematical

patterns is anything but. Shapiro and Resnik do nothing to explain how our general

capacity for pattern recognition includes this specific capacity, nor do they explain why

our capacity for recognising mathematical patterns should be seen as a specific case of a

broader recognitional capacity.

65 Ibid. pg. 180

32

The failure to adequately explain the nature of our capacity for pattern

recognition also causes problems for Shapiro’s explanation of our ability to project from

small finite patterns to larger and infinite ones. Our ability to project to these larger

patterns relies on applying our capacity for pattern recognition to higher-order patterns.

However, since the initial capacity for pattern recognition remains unexplained,

invoking it again in a different context can hardly be any more enlightening.

Furthermore, since we lack any grasp on the nature of pattern recognition, it is hard to

say what warrants the assertion that the same kind of process is going on at the level of

recognising higher-order patterns. In order to ascertain that these two processes are the

same kind of process rather than distinct and unrelated capacities, one would need to

have some idea about the underlying mechanics of our pattern recognition processes.

However, this is exactly the point at which Shapiro defers to the cognitive scientists

whilst also dismissing their interests as somewhat irrelevant. However, their interests

should not be seen as irrelevant, since, if cognitive scientists were to discover that basic

pattern recognition and the projection involved in recognising higher-order patterns are

supported by entirely distinct cognitive mechanisms then Shapiro’s account would be

undermined.

Even if one accepts that there are some patterns in the physical realm, the purely

structural aspects of these patterns seem to be as abstract as the mathematical entities

that structures are supposed to replace. Without an adequate account of the mechanisms

of pattern recognition, the introduction of mathematical structures merely pushes the

problem from one of accounting for our access to abstract objects to one of accounting

for our access to abstract structures. In both cases we lack an account of how and why it

is that we are able to acquire beliefs about such abstracta in the first place.

Shapiro’s further focus on our powers of description also fails to provide

illumination. Firstly, in making use of the set-theoretic notions of coherence and

categoricity, Shapiro’s argument can seem somewhat circular, in that he assumes some

knowledge of set-theory when knowledge of set-theory is one of the phenomena that he

is purporting to explain. 66 Shapiro might respond by suggesting that he is not intending

to invoke the formal notions of coherence and categoricity but is instead appealing to our

informal understanding of these notions. However, if this is the case, our belief in the

coherence and categoricity of the structures whose existence is in question depends on

66 MacBride (2008) pg. 162-3

33

mere intuitions that this is the case. As such, if Shapiro were to adopt this more informal

approach, it would be unclear how his approach to explaining our knowledge of the more

exotic entities of set-theory has any advantages over Gödel’s approach, since both

ultimately appeal to unexplained mathematical intuitions.

It is arguably the case that the circularity that MacBride highlights need not be

seen as vicious with respect to the justification of our belief in large infinite structures.

However, the justification for our belief in such structures is not what is at issue.

Benacerraf’s challenge is concerned with the question of how we acquire beliefs about

these structures in the first place. Appealing to our ability to describe such structures

fails to help in this context, since presumably the ability to describe a structure

presupposes the ability to think about the structure in question. ‘An account of how

concrete finite creatures can reliably access truths about the abstract and infinite is just

as wanting in the case of set theory as it is in any other branch of mathematics. We

simply have no idea… of how to account for the reliability of mathematicians’

judgements about sets whilst invoking “only natural relations”. So in relation to the goal

of providing a naturalistic account of our access to mathematical objects, Shapiro’s

epistemology is viciously circular’, even if with respect to justification the circularity is

acceptable.67

By providing a quasi-historical story, Resnik’s approach can be seen as an

improvement on Shapiro account of mathematical knowledge acquisition, in the sense

that Resnik appreciates the need to study the actual processes through which

mathematical beliefs are acquired. He rightly emphasises the significance of the

universal application of small number concepts and of the transformative power of

numeral systems when it comes to developing more complex mathematical beliefs.

However, his quasi-historical account is descriptive without succeeding in being

explanatory. He points to evidence that suggests that most people are able to form beliefs

with mathematical content and that numeral systems play an important role in the

formation of more complex mathematical beliefs. However, he fails to offer an

explanation of how and why this is the case. In order to gain insight into the way in

which people acquire mathematical beliefs it is necessary to do more than merely

describe cases in which they do. The heart of the problem lies in the fact that he is merely

providing a quasi-historical account. In order to provide a naturalistic account of

67 Ibid. pg. 163

34

knowledge of mathematical structures, it is necessary to go beyond speculation about the

history of mathematical thought based on a few arbitrary facts and delve into the details

of how our natural cognitive mechanisms and historical technological advancements

actually make the acquisition of mathematical knowledge possible.

With their structuralist approaches in the philosophy of mathematics, Shapiro

and Resnik arguably provide adequate naturalist accounts of the justification of

mathematical knowledge. This is without doubt a task of utmost importance. However,

they fail to provide an adequate response to the Access Problem. Answering this problem

requires one to explain how it is that we are able to acquire beliefs with mathematical

content. The nature of acquisition is acknowledged to be significant to justification,

hence Resnik and Shapiro’s emphasis of the significance of pattern recognition.

However, establishing the methods for justifying our beliefs is not the same as explaining

their origins. A naturalised epistemological account of the acquisition of mathematical

beliefs requires attention to the underlying psychological mechanisms that support such

processes. The structuralists tend to ignore these details at their peril, focussing, instead,

on introspective assessment of mathematical beliefs or analyses of the role of such

beliefs in scientific theorising. Whilst both of these things are clearly significant when it

comes to the justification of mathematical beliefs, their relevance to issues of acquisition

is far from obvious.

Sacrificing Universal Semantics: Modal Structuralism

Structuralism fails to overcome Benacerraf’s challenge, since the commitment to

abstract structures renders our access to these structures as no less mysterious than our

access to mathematical objects. We still lack an explanation of how it might be possible

to acquire beliefs about such structures. A potential solution to this problem may lie in

accepting the positive contributions of structuralism, whilst avoiding the troublesome

ontological commitments that allow Benacerraf’s challenge to resurface. Perhaps one can

accept that mathematics is the science of structure whilst remaining quiet as to whether

any such structures exist in the abstract realm.

The immediate problem with such an approach, with regards to Benacerraf’s

challenge, is that it seems to violate the requirement for a homogenous semantics for

35

mathematical and ordinary claims. We usually take such claims to be committed to the

actual existence of the entities that they refer to. Thus, in order to endorse an approach

where ontological commitments are avoided, it is necessary to motivate the idea that, in

the case of mathematical claims there may be more to them than is suggested by their

surface grammar. In doing so it may be possible to justify giving up on a homogenous

semantics.

Putnam provides a motivation for such an approach by advocating the construal

of mathematical claims in modal terms. ‘We can reformulate classical mathematics so

that instead of speaking of sets, numbers or other “objects”, we simply assert the

possibility or impossibility of certain structures’.68 Significantly, the former Platonist

view and the latter modal view are mathematically equivalent, to the extent that they

‘might as well be synonymous, as far as the mathematician is concerned’.69 For each

mathematical claim that seems to commit us to the existence of mathematical objects

there will be a corresponding claim that merely commits us to the possibility of

mathematical structures. Furthermore, the two seemingly very different formulations

are equally powerful; anything that can be proved about the existence of mathematical

structures can also be proved about the possible existence of mathematical structures.

Hellman adopts this Modal Structuralist approach, arguing that mathematical

claims are not claims about actually existing entities or structures but should instead be

construed as claims about possible structures. Despite their surface structure in natural

language, each mathematical claim is broken down into two distinct claims, when

reduced to its underlying logical form. The first type of claim, for example (H1), suggests

that it is necessarily the case that all structures of a given kind possess a particular

property.70

(H1) □∀X (X is an ω-sequence ⊃ S holds in X )

The second type of claim, for example (H2), suggests that it is possible for a structure of

the given kind to exist.

68 Putnam (1994) pg. 508 69 Putnam (1983) pg. 300 70 Hellman (1989) pg. 16

36

(H2) ◊∃X (X is an ω-sequence)

Thus, whilst mathematical statements might initially appear to commit to the existence

of mathematical entities, when their underlying logical structure is revealed they merely

commit to the possible existence of certain structures and the properties that those

structures would necessarily possess were they to exist. By sacrificing the desire for a

homogenous semantics for mathematical and non-mathematical claims, Hellman is thus

able to account for the truth of mathematics, whilst maintaining ontological innocence.

An initial worry that arises from the modal structuralist approach is that it merely

replaces Benacerraf’s initial problem of access to abstract mathematical objects with an

equally troubling problem of access to possible mathematical structures. As with abstract

objects, merely possible structures are causally isolated and spatiotemporally

disconnected from our physical realm.71 As such, one could argue that we are no better

off in explaining the mechanisms that allow us to form beliefs about possible structures

than we are in the case of abstract entities. However, this initial worry may be slightly

less troubling than it first seems. In the case of abstract objects there are no such objects

with which we have causal contact and no such objects that possess spatiotemporal

locations. The case for possible objects is different, since all actual entities are possible

entities. As such there are at least some possible entities with locations in our physical

realm and with which we can have causal contact. It might thus be possible to explain

our access to beliefs about merely possible structures with reference to our contact with

actualised possibilities.

The problem with this response on is that it only explains how we are able to

acquire knowledge of possibility but not how we acquire knowledge of possible

mathematical structures. This problem arises out of Hellman’s desire to maintain

ontological innocence. He is committed to providing a theory that can eliminate any talk

of real mathematical entities or structures. However, in order to explain how we can

acquire knowledge of possible mathematical structures from the world, it is necessary to

argue that we access some actual mathematical structures in the world. This is

problematic since, ‘if we had a satisfactory epistemology for the possibility of

mathematical objects, we would already have one for mathematical objects

71 Vaidya (2011) §4

37

themselves’.72 However, this undermines the need to treat mathematics in modal terms

in the first place. A further problem arises from the fact that Hellman must be

committed to the possibility of infinite concrete mathematical objects. However, there

are reasons to think that there is no way we could acquire knowledge of such a possibility

from our limited access to finite concrete non-mathematical objects.73 It thus seems as

though Hellman’s account cannot preserve ontological innocence whilst simultaneously

providing a satisfactory response to Benacerraf’s challenge. Our access to the realm of

possible structures is as remote from everyday experience as our access to the realm of

the abstract. In order to avoid such a problem it may be necessary to account for

knowledge of possibilities in more down to earth terms.

Sacrificing Universal Semantics: Kitcher’s Empiricism

Like Hellman, Kitcher offers an account of the subject matter of mathematics that

involves going beyond taking mathematical statements at face value. However, where

Hellman argues for the translation of mathematical statements into statements about

possible structures, Kitcher’s account suggests that we should translate mathematical

statements into statements about possible actions for an agent. An advantage of this

approach is that it is explicitly grounded in the actual practices through which we acquire

mathematical beliefs. He argues that, as children, we first acquire mathematical beliefs

by engaging in processes of manipulating collections of objects.74 For example, a child

playing with some blocks can gain an understanding of threeness by gathering three

blocks together. They can learn about the principles of addition and subtraction by

coming to appreciate that when they remove a block from a collection of three blocks

they end up with two blocks and when they add one they end up with four. Eventually,

through the practice of manipulating collections they can come to appreciate the

successor principle by acknowledging that it is always in principle possible to perform

the action of adding another block.

Kitcher cannot simply stop at this point, however, since it seems as though

thought processes of this kind will lead us to something other than knowledge. By

72 Resnik (1997) pg. 78 73 Hale (1996) 74 Kitcher (1984) pg. 107

38

manipulating objects we can learn that a vast range of further manipulations are

possible. However, this induction cannot be carried on indefinitely whilst still providing

knowledge of the actual world. This is simply because in actuality there are limits to the

kind of manipulation that it is possible to perform. As mortal beings in a finite universe

there are only so many blocks that we could pile together in a single lifetime. This seems

to conflict with our mathematical knowledge of the infinite nature of the natural number

structure and the possibility of carrying out addition operations indefinitely. In order to

accommodate worries such as this, Kitcher argues that ‘arithmetic owes its truth not to

the actual operations of actual human agents, but to the ideal operations performed by

ideal agents’.75

An initial worry about Kitcher’s approach is that it seems to ground the subject

matter of arithmetic in the seemingly insignificant practice of collecting and

manipulating medium sized macroscopic entities. However, this fails to do justice to the

myriad of different ways in which arithmetical content is employed, even by young

children. There is a sense in which Kitcher fails to take on board one of the main lessons

of structuralism, namely the astounding multiple realisability of mathematical

structures. Whilst it may be the case that gathering together medium sized objects is one

instantiation of possible arithmetical action, it is by no means the only one. By focussing

on this one form of action, Kitcher overly restricts his account of what arithmetical

actions might be. Furthermore, in doing so he leaves his account open to the challenge

that many of the ways in which we employ arithmetical concepts might not be

explainable in terms of object manipulation. For example, one can count the number of

beats in a song or the number of thoughts one had before breakfast this morning, despite

the fact that neither sounds nor ideas seem to be the kinds of thing that are, even in

principle manipulable.

Kitcher responds to this problem by suggesting that acts of collecting together

macroscopic entities by manipulation are merely the prototypical examples of a broader

class of ‘collective activity’.76 Manipulating objects into spatially bounded collections

might be the first type of collective activity that most children tend to engage in but it is

not the only kind of collective activity. Kitcher suggests that most of the typical instances

of our use of arithmetical thought, whether, for example, applied to concrete collections

75 Ibid. pg. 109 76 Ibid. pg. 111

39

or series of thoughts, all count as some form of collective activity. However, by

broadening his definition of collective activity he runs the risk of trivialising it to the

extent that it no longer plays the role of connecting our mathematical knowledge to

simple interactions with the physical world. It is not immediately clear what collecting

physical objects together in space has in common with collecting a number of ideas

together in thought. In order to provide an acceptable naturalist explanation of our

access to knowledge of collective activities, Kitcher needs to provide some explanation of

what it is that unites these seemingly diverse activities such that they deserve to be

understood as the single capacity for collective activity.77

A second problem with Kitcher’s account is that he fails to live up to his own

professed naturalist credentials, since his account of the way in which we acquire beliefs

with arithmetical content is at odds with the best available evidence from the relevant

sciences. There is a wealth of evidence to suggest that young infants and animals are

capable of behaviour that can only be explained by granting them some basic

arithmetical mental content.78 However, in the case of infants, this behaviour emerges

way before they have the opportunity or capacity to manipulate objects and arrange

them into collections.79 This is particularly problematic for Kitcher’s account because he

argues that our capacity for more abstract collective activity is developmentally

dependent on engaging in the prototypical collective activity of manipulating concrete

objects into collections. However, the fact that our access to arithmetical content seems

to emerge before engagement in such activities drastically undermines his

developmental story. As was the case with structuralists such as Shapiro and Resnik, it

does not suffice to merely tell a story about a possible means of acquiring mathematical

content. In order to provide an adequate naturalist account of our acquisition of

mathematical beliefs it is necessary to pay attention to the relevant data from the

cognitive sciences on how mathematical beliefs are, in actual fact, acquired.

A further problem with Kitcher’s account is that, in order to accommodate

knowledge of arithmetical actions that go beyond the possible actions of a limited human

agent, he has to introduce the notion of an ideal agent. As with the case of Hellman’s

approach, by positing ideal agents, Kitcher seems to be merely giving rise to a further

77 This issue will be revisited in Chapter 3. 78 This evidence will be addressed in much more detail in Chapter 2. 79 The case of other animal species is even more problematic, since basic arithmetical abilities have been demonstrated in animals that don’t manipulate objects by arranging them into collections.

40

version of Benacerraf’s problem. We have no more access to ideal agents than we have to

abstract objects. Furthermore, an ideal agent seems to be nothing other than a certain

kind of abstract entity. Kitcher’s comparison of ideal agents with ideal gases fails to help

here, since we are only able to comprehend the notion of an ideal gas as a result of

applying a particular mathematical description of gases that idealises away from

complicating features of reality. If anything an ideal gas is nothing other than a

particular kind of mathematical object. However, there is no use considering ideal agents

as a certain kind of mathematical entity, since knowledge of mathematical entities is

exactly what Kitcher is attempting to explain in appealing to ideal agents.

Despite these problems, Kitcher’s approach may fare better than Shapiro’s,

Resnik’s and Hellman’s, in the sense that we at least have some contact with the notion

of an actual agent. Knowledge of mathematics can be explained in terms of taking our

knowledge of actual agents’ capacities for action and stripping away certain features until

we arrive at an idealised version of our notion of an agent. Furthermore, Kitcher argues

that some of this knowledge may be derived from our basic perceptual faculties.80 This

strategy seems promising and motivates much of the picture that will be painted in

subsequent chapters. However, it is important to note that Kitcher falls far short of

providing a fully naturalised account of mathematical epistemology. It is all well and

good suggesting that we are able to think about mathematics in terms of actions of an

ideal agent but mere suggestion is not enough. A naturalised epistemology of

mathematics requires an explanation of how such thoughts are possible and which

mechanisms are responsible for them.

Bringing Mathematical Objects Down to Earth

Whilst most naturalist attempts to respond to Benacerraf’s challenge involve

either construing reference to mathematical entities as reference to structures or

translating mathematical statements into some other, perhaps more ontologically

innocent form, Maddy, in her early work, attempts to address the challenge head on.

Mathematical statements are taken at face-value as referring to real mathematical

entities. However, rather than assuming that mathematical entities are inherently

abstract, Maddy argues that they are concrete objects. In doing so she brings ‘them into

80 Kitcher (1984) pg. 103

41

the world we know and into contact with our familiar cognitive apparatus’.81 Benacerraf’s

challenge is thus undermined, since our access to mathematical entities need not be seen

as any more mysterious than our access to other everyday concrete entities. Instead of

positing some magical faculty of mathematical intuition to explain our cognitive access,

mathematical entities are taken to be accessed using our commonplace faculty of

perception.82 Maddy argues that the mathematical entities that exist in the world are sets

and ‘that we can and do perceive sets’.83

In much of what follows I shall argue that Maddy’s response to Benacerraf’s

challenge is close to being correct and that a similar response is supported by our best

scientific understanding of the origins of our mathematical knowledge. However, at this

stage it is important to highlight why her account fails to provide a satisfying naturalist

solution to Benacerraf’s challenge. A problem that Maddy herself acknowledges is that it

is unclear why we should take the mathematical objects that people perceive to be sets

rather than some other kind of mathematical entity. The concrete existence of sets is

invoked to explain the fact that we can perceive the numerical properties of collections of

objects. The problem with this is that there are ‘various candidates for the bearer of the

number property’.84 Sets are not the only type of entity that could be said to bear

numerical properties so the fact that we have perceptual access to numerical properties

does not give us warrant to believe in the existence of sets. In a way this problem can be

seen as a concrete incarnation of Benacerraf’s other challenge, according to which there

is no fact of the matter as to which sets should be identified with the numbers.85

However, in this case the problem is that there is no fact of the matter as to which

concrete mathematical entities should be seen to bear numerical properties, since the

numerical content of our perceptions seems unaffected by whether they are sets,

aggregates, concepts, classes, collections or whatever else. We only have perceptual

access to numerical properties and, as such, various different bearers of numerical

properties are perceptually indistinguishable.86

Maddy’s response is to claim that we are justified in taking the bearers of

perceptually accessible numerical properties to be sets due to the foundational role that

81 Maddy (1990) pg. 48 82 Ibid. pg. 50 83 Ibid. pg. 58 84 Ibid. pg. 61 85 Benacerraf (1965) 86 Carson (1996) pg. 7

42

such entities play in our more general mathematical theorising.87 However, it isn’t clear

how the role of sets in mathematics is relevant, since the issue at hand is the extent to

which our mathematical theorising is related to our understanding of the physical world.

Furthermore, it isn’t clear that sets play any fundamental role in our physical theorising.

It is the numerical properties themselves that play a role in our physical theories and, as

such, any entities that bear these properties are equally viable as candidates for physical

existence.

This problem is further exacerbated by the fact that knowledge of numerical

properties is both historically and ontogenetically prior to an understanding of sets. Set

theory is a relatively new branch of mathematics and arithmetic was developed way

before anyone had any clear notion of sets. Furthermore, the ability to perceive number

and form beliefs about arithmetic is nearly ubiquitous amongst humankind. However,

most people never gain any understanding of the notion of set. It thus seems very

strange to suggest that perceptual capacities allow one to acquire a concept of sets that at

the same time fails to give rise to an understanding of the notion of set. In response,

Maddy could claim that we are able to acquire concepts of water by perceiving bodies of

H2O, without thereby coming to understand that water is H2O. However, this case seems

to be significantly different. Our understanding of water as H2O has significant

implications for our physical theories about water. Identifying sets as the bearers of

number properties has no comparable implications for our understanding of the roles of

numbers in physical theory. The positing of sets seems to play no significant role in

explaining the processes through which we acquire knowledge of numerical properties.

Furthermore, it plays no significant role in explaining the way in which we utilise this

knowledge in order to explain our knowledge of the physical world. From the perspective

of naturalised epistemology, one should take the historical and ontogenetic priority of

knowledge of number properties seriously. As such, one should argue that we perceive

numerical properties and remain quiet as to exactly which kind of entity bears such

properties. Maddy’s argument may provide reasons for realism about some bearer of

numerical properties but it fails to justify the assertion that the bearers are sets and that

sets physically exist.

A further reason for doubting Maddy’s set-theoretic realism is that it is far from

clear that we take the physical bearers of numerical properties to conform to the axioms

87 Maddy (1990) pg. 62

43

of set theory. For example, it isn’t clear what the Null Set Axiom, which asserts the

existence of an empty set, could mean in the context of the physical world. Furthermore,

one could argue that the Axiom of Infinity is false, or at least far from easily

interpretable, when applied to the physical world. Many of the most important

properties of sets are seemingly absent from our perceptual access to the physical world.

For example, when perceiving the numerical properties of a collection we do not thereby

perceive that this collection is itself a set which is capable of being a member of a further

higher set.88 These considerations further emphasise the point that invoking sets as the

real concrete bearers of numerical properties contributes nothing to our understanding

of the physical world. The reasons for adopting set-theoretic approaches are entirely

contained within mathematical theory and as such have no bearing on our

understanding of physical reality. Invoking sets as the bearers of numerical properties

adds nothing to our understanding of arithmetical knowledge that cannot be gained by

simply admitting the existence of physical numerical properties.

A further problem with Maddy’s account is that it fails to meet the standards of

naturalist epistemology, by being based on an outdated and confused understanding of

both the science of perception and the science of arithmetical cognition.89 The attempt to

provide a naturalist account of perception of mathematical entities invokes two distinct

psychological theories. The notion that we are able to perceive mathematical objects, as

opposed to mere sensory properties from which we infer their existence, is motivated by

an adoption of a direct realist approach, as championed by Gibson.90 As will become

clear in the forthcoming chapters, there are good reasons for invoking direct realism in

this context. However, Maddy mistakenly assumes that this approach is widely accepted

by psychologists of perception, when at the time at which she was writing it was

considered a highly controversial approach. The theory of direct perception is then

combined with an account of Hebb’s theory of cell assemblies to account for how we are

able to form neural structures that represent number on the basis of experience.91 There

are a number of problems with invoking this theory. The first problem is that, at face-

value, Gibson’s and Hebb’s theories are incompatible. Gibson’s approach is often

88 Carson (1996) pg. 8 89 It may be somewhat mean spirited to criticise Maddy for failing to provide an up to date account, given that her account predates a large amount of significant work in these fields and that her specialist area is mathematics rather than cognitive science. However, the fact that the failings in her account were somewhat unavoidable given the context and her background does nothing to temper the fact that her own account fails to meet the naturalistic standards that it sets itself. 90 Maddy (1990) pg. 50, Gibson (1979) 91 Maddy (1990) pg. 55-56

44

characterised as an attempt to explain perception without invoking neural

representations, whereas Hebb’s theory is presented as a theory of how neural

representations form as a result of perceptual processes and then go on to influence

perceptual processes. The idea that these two theories might be made compatible is

extremely interesting and could lead the way to a novel theory of perception. However,

Maddy fails to provide such a theory and so the assumption that the two are compatible

is unwarranted.

Maddy’s reason for invoking Hebb’s theory of neural assemblies is to account for

how we might be able to form representations of numerical properties on the basis of

experience with physical collections. This theory is roughly based on the idea that

neurons that fire together tend to form connections with each other.92 When we are

repeatedly exposed to various perceptual stimuli that share a certain property, such as a

series of triangles, the similar low-level firing patterns on each exposure will lead to the

development of a neural assembly that corresponds to the given property, such as

triangularity.93 There are three clear problems with attempting to apply this account to

numerical properties. Firstly, it isn’t clear what perceptual properties various instances

of perceiving threeness have in common. One can perceive threeness in collections of

objects that are perceptually different from one another and even in collections that are

accessed through distinct sensory modalities. As a result, there is nothing to guarantee

similar firing patterns across different instances of threeness and therefore nothing to

motivate the idea that neural assemblies for numerical properties could be formed on the

basis of Hebbian learning. A second problem, is that our current best theories of the

nature of arithmetical cognition suggest that our capacity for perceiving numerical

properties is innate.94 As such, the notion that our mental representations of number

develop through a process of Hebbian learning from experience just seems false.

Both of the aforementioned problems with invoking Hebb’s theory of neural

assemblies might potentially be surmountable. However, the main issue with invoking

the formation of neural assemblies is that it fails to provide any justification for the

realist position that Maddy is trying to support. Hebbian theories of neural assembly

provide a general account of the formation of higher-level representations in the brain.

The theory itself is somewhat outdated and so cannot form the basis of a satisfactory

92 Hebb (1949) 93 Maddy (1990) pg. 56 94 Evidence for this claim will be addressed in detail in Chapter 2.

45

naturalist account. However, even if it were a complete theory of neural representation it

would still fail to provide justification for a realist account. This results from the simple

fact that we are able to represent both existent and non-existent entities. Given the

supposed generality of the Hebbian approach, it would presumably explain both how we

are able to form representations of real physical entities, such as tables and chairs, and

fictitious entities, such as unicorns. Thus, even if it were the case that we are able to form

neural assemblies that represent numerical properties, this gives us no reason to thereby

take such properties or their bearers to be real.

Despite the problems with Maddy’s account, the strategy of suggesting that our

access to mathematical knowledge is fundamentally perceptual is a promising one. In

what follows I will argue that Maddy is right to claim that we acquire knowledge of

numerical properties through the same kinds of perceptual mechanism that provide us

with knowledge of everyday objects. However, such an account need not be committed to

the existence of sets as the bearers of such numerical properties. Furthermore, such an

account will need to be motivated by the best contemporary theories of both perception

and arithmetical cognition, in order to fulfil the requirements of a truly naturalist

epistemology.

Desiderata for a Naturalist Response to the Access Problem

The broad aim of this work is to provide a naturalist account of the acquisition of

mathematical knowledge, which avoids the Access Problem; in doing so the aim is to

open the door to a novel way of understanding mathematical knowledge and its subject

matter. Various attempts to provide a naturalist solution to the challenge have thus far

been unsuccessful. Some of these attempts explicitly ignore the demands of naturalism,

whilst others aim to respect the naturalist’s demands but fail to fulfil them, either by

failing to adequately address the problem at hand or by paying too little attention to the

relevant empirical data on the nature of mathematical knowledge acquisition processes.

However, this diagnosis points to what is required from a naturalist response to the

challenge.

Naturalism can roughly be defined as the idea that we should give precedence to

scientific knowledge over pure philosophical theorising. If we want to understand a given

46

phenomenon we should first look to what relevant scientists have to say on the matter

rather than idly musing from the armchair. However, once one adopts this attitude, one

must still decide who the relevant scientists are. Naturalism can be categorised as falling

into two distinct camps with respect to this issue. Internalist Naturalism suggests that, in

order to understand the nature of a particular subject matter and our epistemic access to

it, we should defer to the scientists whose job it is to investigate that particular subject

matter.95 For example, philosophy of physics should defer to the views of the physicist,

philosophy of mind should defer to those working in the cognitive sciences and

philosophy of mathematics should defer to mathematicians. In the philosophy of

mathematics, Maddy’s later work is an example of Internalist Naturalism, in the sense

that she takes questions of mathematical ontology and epistemology to only be

ultimately answerable by appealing to mathematicians’ own views about the nature of

their subject matter.96 Externalist Naturalism, on the other hand, takes the practice and

practitioners of the given scientific subject as its subject matter. We can learn about what

a given subject is about and how knowledge of its subject matter is acquired by

scientifically studying the way practitioners go about their practice. As such, Externalist

Naturalism uses the tools of psychology, cognitive science, neuroscience and

anthropology. In the case of the philosophy of mathematics this means studying the

cognitive mechanisms that underlie processes of mathematical knowledge acquisition.

At first sight, it seems as though Internalist Naturalism is best suited to

ontological questions whilst Externalist Naturalism is best suited to epistemology. For

example, if you want to know what quarks are and whether they exist, you are best off

asking a particle physicist. However, if you want to know how they know this you are

best off studying their practices. Externalist Naturalism is particularly important with

regards to naturalised epistemology, since it is well established that knowers are not

always best placed to understand how they come to know things. It is well-established in

the psychological sciences that introspection is a poor guide to the nature of the mind.97

As such there is no guarantee that asking experts in a certain field about how they know

things will provide any useful guide to their actual processes of knowledge acquisition.

95 This terms “Internalist” and “Externalist” Naturalism are taken from (Van Kerkhove (2006) pg. 19), however, they are used in a slightly different manner here. 96 Maddy (1997, 2007) 97 Watson (1913), Schwitzgebel (2008)

47

However, this division of labour between Internalist and Externalist naturalism

might not be quite so simple, since the two are likely to influence one another. How we

know about something tells us something about what that thing is, and vice versa.

Furthermore, Internal Naturalism about epistemology implies External Naturalism

when explaining mathematical knowledge. Scientists of the mind are best placed to

understand the nature of knowledge and, as such, when the aim is to study mathematical

knowledge the best approach is to investigate the psychological processes that go on in

the heads of mathematicians, rather than to ask the mathematicians. The case of

mathematical knowledge also differs, in that the External Naturalist cannot restrict their

studies to expert practitioners. Unlike particle physics, almost all of us possess some

knowledge of mathematics. As such, it is necessary to study the processes of knowledge

acquisition in everyday subjects as well as those of experts. Henceforth, the aim will be to

adopt the Externalist Naturalist approach and attempt to provide an explanation of

mathematical knowledge acquisition by turning to the cognitive sciences and their study

of the nature of mathematical knowers.

The philosophy of mathematics can be seen as somewhat odd in that ontological

issues tend to take centre stage, whilst epistemological considerations are relegated to a

supporting role. Mathematical knowledge or the appearance thereof is often taken for

granted and the hard work is perceived to be in accounting for a compatible ontology.

Philosophers argue as to whether either all or no mathematical entities exist and then,

having settled on an answer to this question, try to explain an approach to mathematical

knowledge that is consistent with their position. ‘Nearly all philosophical concern

revolves around the ontological enigma of where on earth the numbers are, with

epistemological questions consequently moved to the background’.98 However, this is by

no means an orthodox approach within philosophy. For example, when one enquires as

to the reasons for being ontologically committed to everyday objects, such as tables and

chairs, one’s natural response is to appeal to one’s certainty about the processes that lead

to one’s belief in them. If somebody asks you why you believe in tables, the natural

response is to reply that you perceive them and that you have faith in your perceptual

capacities. Thus, in other areas of philosophy it is natural to take epistemology first and

ontology second. This is the methodology that will be followed in the forthcoming

chapters. The aim will be to take a naturalist approach to investigating our knowledge of

number in order to determine how we actually acquire mathematical knowledge without

98 Van Kerkhove (2006) pg. 17

48

being encumbered by any ontological assumptions. ‘A new epistemology of mathematics

is needed before confronting ontological questions’.99 Contrary to the methodology that

underlies Benacerraf’s challenge, we should address questions of the actual mechanisms

that support access to mathematical knowledge first and only then go on to investigate

how answers to these questions constrain our mathematical ontology.

Although I have argued that Hellman’s, Kitcher’s and Maddy’s attempts to escape

Benacerraf’s access problem in a naturalistically acceptable manner are all

unsatisfactory, I will go on to argue that, once relevant psychological data are taken into

account, a hybrid of certain aspects of these theories can provide a naturalist solution to

the problem of mathematical knowledge. Hellman’s insistence on the ontological

innocence of mathematical claims and his reframing of mathematics in terms of

modality, Kitcher’s emphasis on the significance of possible actions and Maddy’s claim

that some mathematical knowledge has perceptual origins will all have a part to play in

the approach that I will put forward. A hybrid of these apparently diverse theories may

seem like a somewhat odd creature. However, once the details of the cognitive

mechanisms through which we acquire mathematical knowledge are taken into account,

such an approach will hopefully appear more palatable.

99 Echeverría (1996) pg. 21

49

2

Natural Numerical Perception

In order to provide a satisfactory naturalist response to Benacerraf’s challenge it

is necessary to explain the origins of our capacity for mathematical thought. Whilst it is

valuable to provide a means of justifying our mathematical beliefs, the issue at hand is to

provide an account of where these beliefs come from in the first place. It will only be

possible to provide a satisfying naturalist account of justification once this primary aim

has been established. In order to achieve this aim it is necessary to eschew the

commonplace naturalist strategy of deferring to active practitioners of mathematics.

Whilst they may be the expert at practising mathematics, they are not experts in the

nature of the underlying cognitive processes that support their abilities. Instead, what is

needed is a naturalised epistemology approach, which treats the thinkers of

mathematical thoughts and the processes that underlie these thoughts as its primary

subject matter. Furthermore, given the ubiquity of mathematical thought, it will not

suffice to limit this form of investigation to the thought processes of expert mathematical

practitioners. What is required is an account of the cognitive mechanisms that allow

almost anyone and everyone to form mathematical beliefs.

Recently, the subject of mathematical cognition has received an increasing level

of attention amongst researchers in psychology, animal behavioural studies, cognitive

science and neuroscience. This provides the opportunity to investigate the actual

mechanisms that are responsible for our acquisition of arithmetical concepts. Thus,

rather than starting from the apparent impasse between naturalist approaches and

Benacerraf’s challenge, the aim is to start from an understanding of the actual

mechanisms that underlie arithmetical knowledge and, from this derive a response to

the challenge.

Numbers are not usually seen as things that can be encountered in the world and

perceived. For many, mathematics is solely a product of human culture. For some,

numbers are a mere creation of the human imagination, on a par with dragons and

unicorns.100 For others, they are more like social constructions, such as money or

100 Yablo (2005)

50

marriage, merely invented to serve a particular purpose.101 In either case, number is seen

as artificial and anthropocentric, with no basis in the natural world. This perspective is

appealing when one conceives of mathematical thought purely in terms of the kinds of

activity that take place in mathematics departments and school maths classes, such as

grappling with abstract mathematical structures remote from everyday experience or

engaging with sophisticated technological inventions such as numerals, graphs, abacuses

and calculators. There are, without doubt, many ways in which our concept of number

has been shaped and transformed by human inventiveness, culture and technology.

However, the main thrust of this chapter will be to argue that evidence about the nature

of numerical cognition points towards the idea that our access to number is rooted in

natural processes of perception. Far from being a mere artificial construct, number is

one of the fundamental aspects of the perceptual realm and we have evolved specific

systems dedicated to its perception.

Subitising and the Approximate Number System

There is now a wealth of behavioural and neurological evidence which

overwhelmingly suggests that basic forms of numerical cognition are dependent upon an

innate system for perceiving number. Humans and a wide range of other species possess

the capacity to rapidly apprehend the number of entities in a collection. This capacity is

known as subitising and has been known about since the late nineteenth century.102 The

capacity for subitising is somewhat counterintuitive. It is natural to think that

apprehending the number of entities in a collection requires perform some kind of

counting procedure, perhaps by attending to each entity in sequence and keeping a

mental tally. As such, numerical apprehension would seem to be a complex cognitive

activity that requires conscious attention to each and all of the objects in the collection to

be enumerated. In actual fact, our capacity for subitising is more automatic than this.

Priming experiments show that we can perceive number unconsciously. 103 Furthermore,

patients that lack the capacity for serial attention can still perceive number and there are

convincing models of numerical perception based on parallel as opposed to serial

processing.104

101 Ernest (1998) 102 Kauffman et al. (1949), Jevons (1871) 103 Naccache & Dehaene (2001), Nieder & Miller (2004) 104 Dehaene & Cohen (1994), Dehaene & Changeux (1993)

51

Accurate subitising is possible when dealing with collections of between one and

three or four entities. However, our ability to perceive the number of entities in a

collection becomes increasingly inaccurate as the size of the target collection increases.105

The level of inaccuracy increases roughly logarithmically in line with the so called

Weber-Fechner law. Similar results can also be found with respect to subjects’ reaction

times, with longer reaction times for larger collections.106 This limitation manifests itself

in the form of two well documented effects that arise in number comparison tasks; the

size and distance effects. In the case of the size effect, subjects’ inaccuracy and reaction

times increase as the number of entities in the target collections increases, even when

the absolute difference between the target collections remains fixed. For example,

subjects are slower and more error-prone when comparing collections of 18 and 20

objects than with collections of 8 and 10. In the case of the distance effect, subjects’

inaccuracy and reaction times increase as the numerical distance between the collections

decreases. For example, subjects are slower and more error-prone when comparing

collections of 8 and 10 objects than when comparing collections of 8 and 12. Variance in

accordance with the Weber-Fechner law, and thus manifestation of size and distance

effects, are a signature of this particular capacity for number apprehension. As such,

when these effects arise, one can detect that our system for numerical apprehension is

responsible. These results suggest that we possess an Approximate Number System

(ANS) dedicated to the apprehension of number. Apprehension of the number of entities

in a collection by the ANS is not limited to enumerating objects in the visual scene. There

is evidence that we can also rapidly perceive the number of sounds, touches, or

actions.107 In all such cases, subjects’ performance is similar, bearing the signature

limitations of the ANS, suggesting that a single system is responsible. Furthermore,

performance is even similar when subjects are given tasks where they must enumerate

objects using more than one sensory modality, such as if they count both dots and

beeps.108 The ANS is dedicated to apprehending number, regardless of the nature of

sensory input.

105 van Oeffelen & Vos (1982) 106 Moyer & Landauer (1967), Moyer & Bayer (1976) 107 Wynn (1996), Xu & Spelke (2000), Riggs et al. (2006) 108 Barth et al. (2005) As a result of these considerations, psychologists have argued that our capacity for

numerical apprehension is supported by “abstract” representations in the brain (Dehaene, Dehaene-Lambertz &

Cohen (1998)). From a philosophical perspective, however, it is important not to read too much into this

terminology. Philosophers and psychologists tend to use the term “abstract” in very different senses (Prinz

(2006b) pg. 438). For philosophers, “abstract” is the opposite of “concrete” or “physical” and, as such “abstract

representations” would not be the kinds of thing that one could find in the physical brain. For psychologists, on

the other hand, “abstract” simply means not reducible to a single sensory modality. When understood in these

52

The Approximate Number System is Innate

This capacity for rapidly perceiving number is shared by a remarkably diverse

range of species. Examples include chimpanzees, macaques, dolphins, lions, pigeons,

salamanders, some fish and even invertebrates such as beetles, bees and ants.109 This

suggests that it is either evolutionarily ancient or that a similar mechanism underlying

this ability has evolved independently in a number of different lineages. Either way, the

extraordinary prevalence of such a specific capacity suggests that the ANS is an innate

system.

It is easy to see why this capacity would be evolutionarily beneficial. When

foraging, our ancestors needed to be able to compare the number of food items in

different areas. They would have needed to be able to perceive the number of predators

in the vicinity, since the difference between two and three lions might be the difference

between life and death. It would have been important to keep track of the numbers of

one’s offspring, to prevent them from getting lost and going astray. All of these

important survival capacities could be greatly enhanced by possessing the capacity to

quickly perceive the number of entities in a collection without having to go through time-

consuming and cognitively costly serial counting procedures. It also makes evolutionary

sense for this capacity to become less accurate when dealing with larger collections.

Accurate representation of number presumably comes at a cost and in most ecologically

salient scenarios exact representation of larger collections might not be particularly

beneficial. For example, discriminating between two and three lions might be highly

salient for an organism’s survival, whereas discriminating between twenty and twenty-

one lions is somewhat unnecessary given that one should probably run away in either

case.

Further evidence that the ANS is an innate system comes from studies into the

numerical capacities of young infants. Infants as young as one week old are capable of

discriminating between collections of two and three objects.110 At the age of five months

infants are capable of simple arithmetical calculations, fixating longer on

terms, many aspects of the physical realm, such as space, time, shape and number, could all be represented using

abstract representations, without engendering any need for philosophical worries about the abstract realm.

109 Boysen & Berntson (1989), Brannon & Terrace (2000), Killian et al. (2003), McComb, Packer & Pusey (1994), Emmerton, Lohmann & Niemann (1997), Uller et al. (2003), Agrillo et al. (2008), Carazo et al. (2009), Gross et al. (2009), Reznikova & Ryabko (2011) 110 Antell & Keating (1983)

53

demonstrations that appear to violate the rules of addition and subtraction (see Fig.

2.1).111

Fig. 2.1

Six-month-old infants can also discriminate between larger collections of, for example, 8

and 16 objects.112 When results of this kind first appeared, some speculated that infants

might be using some non-numerical property, such as the overall volume of the target

collections, as a proxy for detecting number.113 However, in cases where such non-

numerical properties were manipulated, infants were primarily sensitive to the

numerical properties.114 Evidence also suggests the ANS is already responsible for the

111 Wynn (1992), McCrink & Wynn (2004) (Fig. 2.1 from Wynn (1992) pg. 749) 112 Xu & Spelke (2000) 113 Clearfield & Mix (1999), Feigenson, Carey & Spelke (2002) 114 Wynn, Bloom & Chiang (2002), Lipton & Spelke (2003), Brannon, Abbot & Lutz (2004)

54

apprehension of number across various sensory modalities in newborn infants, since

they recognise the equinumerosity of collections of visual and auditory stimuli.115 The

fact that our capacity for numerical apprehension develops at such an early stage lends

further support to the claim that the ANS is an innate system.116

This claim is bolstered by the existence of genetic disorders that lead to

developmental dyscalculia, a selective impairment to numerical cognitive capacities.

Evidence suggests that developmental dyscalculia is usually not a consequence of general

cognitive impairments but is instead the result of impairments to number-specific

capacities.117 One such example is Turner’s syndrome, which results from abnormalities

in the X chromosome and leads to selective deficits in number apprehension and

arithmetical reasoning, whilst leaving many other cognitive capacities intact.118 The

genetic basis of our numerical capacities is further supported by evidence for higher

rates of developmental dyscalculia in siblings of subjects with the condition.119

Evidence from animal studies, infant studies and genetic disorders all support the

idea that our basic numerical capacities are supported by an innate system dedicated to

number. As a result, the ANS can be understood to be an evolved system for numerical

apprehension. However, some might still question how significant the presence of such a

system is to our more sophisticated engagement in numerical reasoning as adults. Given

the limitations of the ANS, one might expect adults trained in sophisticated mathematics

to bypass using such a system in order to overcome these limitations. However, there is a

wealth of evidence to suggest that adult humans, like animals and infants, also use their

ANS for the apprehension of number.120 Adults also make errors that match the

signature performance limitations of the ANS and performance in formal mathematics is

correlated with the acuity of the ANS in infancy.121 The ANS can thus be seen as an

innate system for perceiving number, which remains active and central to our numerical

abilities throughout life.

115 Izard et al. (2009) 116 De Cruz & De Smedt (2010) 117 Butterworth (2008) 118 Temple & Marriott (1998), Butterworth et al. (1999), Braundet et al. (2004) 119 Shalev et al. (2001) 120 Moyer & Landauer (1967), Dehaene (1997) pg. 66-80 121 Halberda, Mazzocco & Feigenson (2008)

55

The Neural Basis of the Approximate Number System

There is also a burgeoning array of evidence to suggest that the ANS is supported

by a dedicated neural system. Neural imaging studies have consistently demonstrated

heightened activation in a region known as the intraparietal sulcus (IPS) during tasks

involving numerical cognition (see Fig. 2.2 & Fig. 2.3).122

Fig. 2.2

Fig. 2.3

122 Piazza et al. (2004), Nieder & Dehaene (2009), (Fig. 2.2 from http://en.wikipedia.org/wiki/Intraparietal_sulcus , Fig. 2.3 from Piazza et al. (2004))

56

Activity within the IPS during numerical tasks is strongest in a particular segment of the

IPS known as the horizontal intraparietal sulcus (hIPS).123 As such, this region has been

proposed as the site of the neural mechanisms that support the ANS. It has also been

shown that macaque homologs of the IPS contain neurons that are selectively tuned to

respond to specific numbers.124 These neurons show maximum levels of excitation when

presented with a specific number of entities and the level of excitation progressively

drops off as the number of entities in the presented collection becomes more remote

from this specific number.125 It has so far not been possible to demonstrate the existence

of number-specific neurons in humans, since the required techniques are too invasive for

use on human subjects. However, the presence of these neurons in a similar system in a

near evolutionary relative warrants the prediction that the human hIPS probably also

contains number-specific neurons that function similarly.

Further evidence for this neural basis comes from the development of neural

network models. The most successful of these models involves the parallel individuation

of the entities in a target collection to produce an approximate representation of number

(see Fig. 2.4).126

Fig. 2.4

123 Piazza et al. (2004) 124 Nieder & Miller (2004) 125 Nieder & Dehaene (2009) pg. 188 126 Dehaene & Changeux (1993)

57

Studies of macaque homologs of the hIPS have found neurons that seem to play similar

functional roles to those proposed in the neural network model.127 There is still some

debate as to exactly which neural network model best fits the activity in the hIPS.128

However, both of the leading models are compatible with the evidence from macaque

studies. Thus, attempts to construct neural network models of the ANS strongly support

the localisation of the ANS in the hIPS.

The ANS’s being located in the hIPS is further supported by clinical studies.

Patients with lesions to this area often suffer from severe impairments to their

arithmetical abilities despite most of their other cognitive capacities remaining intact. In

one such case the patient in question struggled when dealing with collections of 1-4

objects and found dealing with collections of any more objects impossible.129 Evidence

also suggests that cases of developmental dyscalculia with genetic origins also result

from structural anomalies in the hIPS.130 There are also cases of patients who lose a wide

range of cognitive functions as a result of severe neurodegenerative diseases but whose

arithmetical capacities are spared. In these cases the disease is often found to have

caused damage to prefrontal areas but not the IPS.131 This evidence of a double

dissociation strengthens the case for the hIPS as the basis of the ANS. Taken together,

the evidence from neural imaging, neural network modelling and clinical studies

provides such a compelling case for taking the ANS to be located in the hIPS that this is

now widely accepted throughout the field.132 The ANS can be understood to be an innate

system that occupies a specific location within a particular system in the human brain.

Object Tracking, Pattern Recognition and Subitising

It was originally hypothesised that a single mechanism, the ANS, can explain both

our capacity for accurately subitising the number of objects in collections of 1-4 objects

and our ability to estimate the number of objects in larger collections.133 The increased

accuracy for smaller collections was merely thought to result from the logarithmic

representation of number. However, recent experiments have demonstrated a

127 Roitman, Brannon & Platt (2007) 128 See Verguts & Fias (2004) for an alternative to Dehaene & Changeux (1993), similar findings are supported by a model based on the more contemporary approach of modelling neural systems in terms of hierarchical generative models (see Stoianov & Zorzi (2012)) 129 Cipolotti, Butterworth & Denes (1991), Butterworth (1999) pg. 165-167 130 Molko et al. (2003) 131 Rossor, Warrington & Cipolotti (1995) 132 Nieder & Dehaene (2009) 133 Dehaene & Changeux (1993)

58

discontinuity between responses to small collections as opposed to larger collections,

diverging from what one would expect if the ANS alone were responsible.134 Subjects

were far faster at responding to collections of 1-4 objects than predicted by the Weber-

Fechner model. The idea that the ANS is solely responsible is further challenged by

evidence that, in tasks involving 1-3 objects, infants often fail where one would expect

them to succeed if they were utilising ANS representations. Infants succeed in

discriminating between 1 and 2 objects and between 2 and 3 objects but fail to

discriminate between 2 versus 4 objects and even between 1 and 4 objects.135 If the ANS

were responsible, one would expect them to be far better at discriminating 1 versus 4

objects than 2 versus 3 objects, since the ratio in the case of the former is far more

favourable.

As a result of these findings a number of theorists postulate two distinct systems,

with one system responsible for subitising small collections, and another, the ANS, for

estimation of number for larger collections. Some argue that subitising is supported by

the Object Tracking System (OTS).136 Others argue that it is supported by the Pattern

Recognition System (PRS).137

The OTS, as the name suggests, is the system responsible for keeping track of the

objects in a perceptual scene. This is a particularly useful function, since, for example, it

allows us to recognise that an object that moves behind an occluding surface and

emerges at the other side is still one and the same object. The OTS achieves this function

by representing each object to be tracked with a single representation known as an object

file. The suggestion in the case of numerical tasks is that, when dealing with small

collections, rather than using representations from the ANS, infants use their OTS. The

OTS is a good candidate to explain infants’ surprising performance since the limitations

on the capacity of the OTS and the limitations in their performance coincide. It is well

established that the capacity of the OTS is limited. It can only deal with up to three

separate objects by forming three separate object files. Thus, infants’ surprising failure to

discriminate 1 and 4 objects can be put down to them having too few object files to keep

track of all four objects. As a result of these considerations, it has been argued that the

OTS rather than the ANS is responsible for performance on some numerical tasks. As

such, it is tempting to claim that the OTS constitutes another innate system dedicated to

134 Revkin et al. (2008) 135 Feigenson, Carey & Hauser (2002), Feigenson & Carey (2005) 136 Feigenson & Carey (2005), Carey (2009b) 137 Mandler & Shebo (1982), Krajcsi, Szabó & Mórocz (2013), Jansen et al. (2014)

59

the representation of number.138 However, this conclusion will be resisted for reasons

detailed below.

The OTS is not the only system invoked to explain subjects’ unexpected

performance in tasks involving small collections. Some have argued that our enhanced

capacity when dealing with small collections of objects might be, at least partially, the

result of our more general capacity for pattern recognition.139 Experiments have shown

that response times are faster and success rates higher when small collections of objects

are presented in canonical configurations, such as lines, regular shapes or dice patterns,

as opposed to randomised configurations.140 It makes sense that the PRS would be more

effective at apprehending the number of entities in smaller collections, since the fewer

entities there are in a collection, the more likely the objects will form a canonical

configuration. For instance, collections of two objects always form the pattern of a

straight line, whilst collections of three are usually likely to form a triangle.141 One could

thus either conclude that the PRS is responsible for the apprehension of number for

small collections or that it supplements and enhances the performance of other systems.

It is, again, tempting to include the PRS as another innate system dedicated to the

representation of number. However, this conclusion will, again, be resisted.

There are two reasons for resisting the move to include other systems as being

responsible for the apprehension of number. Firstly, it is not clear that one needs to

downplay the role of the ANS to explain the experimental data. Secondly, neither the

OTS nor the PRS explicitly represent number. They both implicitly represent number

and, as such, cannot accurately be described as systems dedicated to numerical

representation

Some interpret the experimental data as suggesting that there are two exclusive

systems for numerical representation. The OTS or PRS is taken to be responsible for the

representation of number in the case of small collections, whilst the ANS is taken to be

responsible for representation of number for larger collections. This approach therefore

posits at least two distinct systems dedicated to numerical representation. An alternative

approach is to argue that the ANS is involved in perceiving number for both small and

large collections but that the representations that it forms are not solely responsible for

driving behaviour. Even if the OTS or the PRS drive behaviour on certain occasions, this

138 Carey (2009), pg. 141 139 Mandler & Shebo (1982) 140 Mandler & Shebo (1982), Krajcsi, Szabó & Mórocz (2013), Jansen et al. (2014) 141 Mandler & Shebo (1982)

60

does not entail that the ANS is not also active in generating representations of number. It

just shows that on these particular occasions it isn’t ANS representations that are driving

behaviour.

This alternative approach is supported by evidence that, when subjects’ attention

is taken up by a different task, performance is as one would expect if ANS

representations were responsible.142 This suggests the ANS is always actively engaged in

the process of apprehending number but that its representational resources are only

deployed when other more costly strategies are unavailable. This hypothesis is also

consistent with neurological evidence, since exposure to collections leads to activation in

the hIPS regardless of whether the collections are small or large.143 Since neither the OTS

nor the PRS are located in the hIPS, this suggests that the ANS actively represents

number even in cases where these representations do not drive behaviour.

It is also worth noting that much of the evidence against the ANS representing

number in small collections comes from studies involving infants and young children

that are still developing.144 Whilst these studies are extremely significant with respect to

the developmental trajectories of the systems that we use to engage with collections, it is

dangerous to use these studies to draw strong conclusions about the functions of these

systems. Adults and relatively young children do not make the same mistakes as infants

and can discriminate 1 versus 4 objects with ease. Thus the fact that infants seem to

utilise the OTS rather than the ANS to drive their behaviour when dealing with small

collections may simply reflect the developmental immaturity of the ANS. The OTS may

be the best tool that infants have available at an early stage, with the ANS taking over

sole responsibility for numerical representation at a later stage.

It is worth noting that much of the evidence for OTS involvement comes from

experiments where objects in the target collection are hidden. These experiments often

involve presenting infants with collections of desirable food items before hiding them in

boxes and assessing their numerical discrimination capacity based on which box they

choose to search in.145 As such, these experiments do not necessarily test their capacity

for apprehending the number of entities in a collection, since the infants must also

remember how many objects are in each box. Thus, these results might merely indicate

that ANS representations are harder to hold in memory. The fact that the OTS is used to

142 Burr, Turi & Anobile (2010) 143 Nieder & Dehaene (2009) 144 Feigenson, Carey & Hauser (2002), Feigenson & Carey (2005) 145 Ibid.

61

keep track of these occluded objects is also unsurprising since keeping track of occluded

objects is one of the primary functions of this system. Thus, the ANS could be the sole

system dedicated to the apprehension of number even if it isn’t the sole driver of

behaviour in tasks that involve keeping track of small collections of objects.

When taken together, these considerations suggest that it is perfectly viable to

consider the ANS as responsible for the representation of number for both small and

larger collections. The fact that these representations are not the sole driver of behaviour

on all occasions is no reason to deny their existence. Furthermore, there are good

reasons to believe that the ANS is the only system that can properly be described as

being dedicated to the explicit representation of number.

It may seem strange to deny that the OTS is dedicated to numerical

representation. After all, the overall state of this system is dependent on the number of

entities that are being tracked. However, there are good reasons for thinking that the

OTS contains no explicit representations of number. The OTS functions by assigning one

object file to each object in the sensory array. As such the number of object file

representations is inevitably correlated with the number of objects. However, there is no

explicit representation of the numerical properties of the given collection of objects.146 A

representation of threeness is not the same thing as three representations of oneness. In

order for the OTS to explicitly represent number it would need to be comprised of a

further system that produces representations corresponding to the numerical properties

of the collections of object files. Thus, representing threeness would require four

representations, one file for each object and one representation for the collection of

objects. Such an auxiliary system is obviously superfluous, since we have a system, the

ANS, which is capable of directly representing the number of objects in a collection.

Furthermore, it isn’t clear that the OTS is capable of representing the correlation

between the number of objects and the number of object files. All that the OTS does is

provide one representation for each object. As such, even if the OTS can be used as a

proxy for dealing with number, it fails to qualify as a dedicated system for the explicit

representation of number.

The PRS can be ruled out as a system dedicated to the representation of number

for similar and more straightforward reasons. The PRS is clearly not a system dedicated

to the representation of number, since it is primarily responsive to spatial patterns.

146 Margolis & Laurence (2008) pg. 936

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These will often serve as good proxies for the number of entities in a collection but this

will by no means always be the case. Furthermore, it is unclear whether the PRS can

really be characterised as a single system at all. For instance, consider the fact that we

are able to subitise both collections of visually apprehended objects and auditorily

apprehended sounds. There is no immediate similarity between the patterns in these two

distinct modalities. In the case of vision a canonical configuration might be a shape and

in audition it might be a steady rhythm but it seems unclear how one system could be

responsible for the apprehension of both. The PRS is best understood as an array of

systems, which can contribute to our apprehension of number, but are not dedicated to

numerical representation. There may be a whole host of different systems that are

involved in our interaction with collections of entities, however the ANS is the only

system that we know of that is primarily dedicated to representing number.

The Approximate Number System and Numerical Perception

It is natural to think of our capacity for the apprehension of number to be the

product of complex cognitive processes. However, the remainder of this chapter is

dedicated to arguing that the ANS is best understood as a perceptual, rather than a

purely cognitive, system. We directly perceive the number of objects in a collection

rather than inferring it as a result of some deliberative counting process. It seems natural

to characterise number apprehension as a cognitive, rather than perceptual, process,

since we intuitively assume that it requires serial enumeration of some sort. One

naturally assumes that apprehending number requires focusing one’s attention on each

object in sequence and keeping some memory trace of each object. However,

experimental data, neural network models and introspection upon the experience of

exposure to collections all weigh against this intuitively plausible account. Experiments

suggest that our apprehension of number is too quick to be explainable in terms of

attending to each entity in sequence and that it can take place unconsciously, without

attention being paid to each distinct object.147 This fits with the neural network model of

the ANS, mentioned earlier, according to which numerical apprehension is accomplished

by parallel, as opposed to serial, individuation.148 Whilst the notion of direct number

perception might not be immediately intuitive, it is easy to convince people of its truth

via introspection. When subjects are presented with collections very briefly, they usually

147 Piazza et al. (2004), Bahrami et al. (2010) 148 Dehaene & Changeux (1993)

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report that they know the number of objects without having the subjective experience of

having counted them.

A further reason for understanding the ANS as a perceptual system is that it

arguably fits all of Fodor’s criteria for perceptual input modules.149 Numerical

representation in the ANS is mandatory, since subjects are susceptible to numerical

priming effects.150 As has been mentioned, it is fast, and is also shallow, in the sense that

it uses a computationally cheap mechanism to produce a simple output with consistent

limitations on its accuracy.151 The ANS is informationally encapsulated in the sense that

we cannot utilise higher cognitive capacities to improve our performance in subitising

tasks. This is clear from the fact that adults make similar performance errors to infants

and animals.152 It is also inaccessible, since we cannot use introspection to uncover its

underlying mechanics. As we have seen it is widely held to be localised in the hIPS and is

subject to characteristic breakdowns when this region is damaged.153 Furthermore,

evidence for the ANS’s innateness suggests that its development is ontogenetically

determined.154 It is also arguably the case that the ANS is domain specific in that it is

dedicated to dealing with a specific property of perceptual input, namely the number of

entities in a collection. Thus, the ANS seems to fit all of the Fodorean criteria for a

perceptual system.

The ANS meeting these criteria is unlikely to convince everyone that it is a

perceptual system. On the one hand, proponents of massive modularity are unlikely to

find it convincing, since they take both perceptual and cognitive systems to be

characterised by these criteria.155 On the other hand, some will be unconvinced, as they

take modularity to be a poor way of characterising perceptual systems.156 However, the

ANS meeting these criteria still lends support to it being a perceptual system. Even if one

rejects a modular view of the mind, Fodor’s criteria are still characteristic of the kinds of

low-level systems thought to be involved in perceptual processes.

Another reason for seeing the ANS as a perceptual system is that its limitations

are characteristic of perceptual systems. The Weber-Fechner law was originally

developed in order to describe capacities for perceptual discrimination of physical

149 Fodor (1983) 150 Bahrami et al. (2010) 151 Piazza et al. (2004) 152 Barth et al. (2006) 153 Ashkenazi et al. (2008) 154 Nieder, Freedman & Miller (2002), Molko et al. (2003) 155 Carruthers (2006) 156 Prinz (2006a)

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magnitudes such as weight and illumination.157 Variance in accuracy according to

Weber-Fechner laws is a signature feature of perceptual systems. This points to the idea

that numerical quantity is just another one of the physical magnitudes that our

perceptual systems are sensitive to. This conclusion is further supported by evidence that

our apprehension of number is ‘susceptible to adaptation’ effects in the same manner as

other perceptual properties, such as ‘colour, contrast, size and speed’.158 It is well

established that our perceptual systems are susceptible to adaptation, whereby repeated

or continuous exposure to a particular stimulus can lead the brain to adapt to these

stimulus conditions, thereby affecting future responses. For example, this kind of effect

is noticeable when one removes tinted sunglasses to find that the world appears to be

tinted with the opposite colour to that of the sunglasses. In the case of numerical stimuli,

it has been shown that adaptation to small collections of objects increased the apparent

number of objects in subsequent tests, whilst adaptation to larger collections of objects

decreased the apparent number of objects in subsequent tests.159

Although highly compelling, this evidence is not sufficient to establish that

subjects are directly perceiving number. It might be the case that some other perceivable

property is being used as a proxy. For example, numerical apprehension might be

accomplished by perceiving the density and spatial extent of the target collection and

inferring the approximate number. However, a number of experiments have ruled out

this alternative hypothesis. Subjects have been shown to be capable of apprehending the

number of entities of a particular colour in densely packed arrays of multicolour dots just

as well as when the dots are presented on their own. If number were computed as a

product of density one would expect the former condition to interfere with the task.160

Furthermore, lowering the number of objects whilst keeping the density fixed, by

grouping objects together with connections, leads to correct perceptions of fewer

objects.161 Number specific adaptation effects also still arise when potential confounding

features are systematically varied.162

Considerations regarding the location and character of hIPS also suggest the ANS

is a perceptual, as opposed to a purely cognitive, system. The hIPS forms a part of the

parietal cortex which is located towards the rear of the brain. One should always be

157 Hecht (1924) 158 Burr & Ross (2008), pg. 425 159 Ibid. 160 Halberda, Sires & Feigenson (2006) 161 Franconeri, Bemis & Alvarez (2009) 162 Ross & Burr (2011)

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cautious when drawing conclusions on the basis of crude coarse-grained neural

geography. However, if it makes any sense to separate neural regions out into those that

are responsible for perception and action and those responsible for higher cognition then

it is generally accepted that systems located in the hind brain are responsible for the

former, whilst the latter takes place in more frontal regions. Moving beyond this

simplistic characterisation of the brain, there are further reasons to think that the

parietal cortex is primarily involved in perceptual processes. The parietal cortex forms a

part of the dorsal stream of perceptual processing, which is also known as the “where”

stream, since its primary function is arguably the processing of the spatial location of

perceived objects, in particular in order to mediate the control of object-directed

actions.163 One of the primary functions of the parietal cortex is the control of spatial

attention.164 It is implicated in both the conscious deliberative attention associated with

gaze fixation and goal-directed action and also in more subconscious processes such as

the coordination of saccades. The significance of the attentional function of the parietal

cortex for numerical cognition will become clearer in the next chapter. However, for

now, it suffices to emphasise that coordination of spatial attention is clearly a perceptual

as opposed to a purely cognitive function.

For some the involvement of the parietal cortex in both perception and control of

action might not sit well with its characterisation as a perceptual system. On a traditional

picture, perception, cognition and action are three distinct kinds of process with the link

between perception and action always being mediated by some kind of cognitive

processing. The story is, roughly, that perception provides a representation of the

external world, which is passed on to cognitive systems and these systems form

intentions, which then engage the motor systems to accomplish the performance of a

relevant action. Thus on this picture, the parietal system seems like a prime candidate

for a cognitive system, since it sometimes mediates between perception and action.

However, there are good reasons to be suspicious of this ‘classical sandwich’ model of the

mind.165 Perception and action might not be so readily separable. For instance, visual

perception takes place in the context of ongoing ocular motion due to visual saccades

and frequent changes of gaze direction. As such, the perceptual system responsible for

vision might not be separable from the oculomotor system.166 This is not to deny that, at

times, higher cognitive systems are involved in the mediation between perception and

163 Mishkin, Ungerleider & Macko (1983), Goodale & Milner (1992), Culham & Kanwisher (2001) 164 Colby & Goldberg (1999), Culham & Kanwisher (2001), Grefkes & Fink (2005) 165 Hurley (1998) pg. 401-402 166 Gibson (1966)

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action. All that is being claimed here is that the function of the parietal cortex does not fit

this profile. Rather than mediating between perception and action, the parietal cortex’s

role in coordinating spatial attention is best understood as enabling perception and

spatially-directed action to happen in the first place.

The fact that the hIPS forms a part of a system dedicated to perceptual processes

supports characterising the ANS as a perceptual system. This conclusion is bolstered by

studies of the nature of the component parts of the hIPS. Macaque homologues of the

human hIPS contain neurons that respond selectively to specific numerosities.167 These

neurons arguably function in a similar manner to selective neurons in the visual system,

such as edge-detectors in the primary visual cortex or face-detectors in the fusiform

gyrus. In each case, neural activation is strongest when subjects are exposed to the

neuron’s preferred input and decreases as the input deviates from the input to which the

neuron is dedicated. The question of whether either number-specific or other feature-

specific neurons can truly be understood as feature-detectors is highly contentious.168

However, this controversy is tangential, since the similarity of function between number-

specific neurons and neurons in systems that are uncontroversially perceptual provides

reason for characterising the ANS as a perceptual system. The presence of number-

specific neurons supports the idea that we directly perceive the number of objects in

small collections.169 As such, it also supports the claim that the ANS is a perceptual

system.

As a result of these and other considerations, Dehaene refers to the capacity of the

ANS as “The Number Sense” and argues that

‘Number appears as one of the fundamental dimensions according to

which our nervous system parses the external world. Just as we cannot

avoid seeing objects in colour… and at definite locations in space… in the

same way numerical quantities are imposed on us effortlessly’170

Given the evidence, there is good reason to take Dehaene literally and view the ANS as a

system involved in the perception of numerical properties that utilises perceptual

representations of number.

167 Nieder, Freedman & Miller (2002) 168 Martin (1994) 169 Prinz (2006b) pg. 443 170 Dehaene (1997) pg. 71 (emphasis mine)

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The Problem of Approximate Representations

In much of the literature on the ANS it is claimed that it functions to detect

numerosity as opposed to number. This might be seen to threaten the idea that the ANS

is an innate system dedicated to the perception of number, since it seems to be dedicated

to the perception of numerosity instead. One reason for this distinction is to highlight

the fact that the ANS only provides approximations of the number of entities in a

collection when dealing with collections larger than three. Thus far I have avoided the

use of this terminology, arguing that the ANS is a system dedicated to the perception of

number. As a result of this disavowal of the terminological distinction, an immediate

objection arises. Whilst it might be the case that the ANS is best seen as a perceptual

system, this system could arguably be seen to be dedicated to detecting approximate

number or numerosity, rather than being dedicated to the perception of number. One

could argue that we cannot truly be said to be detecting numbers because we know that

fifty-seven and fifty-eight are different numbers, yet our ANS will often fail to allow us to

discriminate between collections of fifty-seven and fifty-eight objects.

The main problem with this objection is that it places far more exacting demands

on the capacities of the ANS than one would want to place on any other sensory

capacities. Unless one takes an unacceptably naïve view of the contents of our perceptual

experiences, one has to accept that our perceptual systems have limitations. In order to

make sense of the idea that our perceptual systems in some sense allow us to represent

features of the external environment, it is necessary to accommodate the possibility that

these systems will, under certain conditions, misrepresent exactly those features that

they are dedicated to representing. Furthermore, by understanding the way in which our

perceptual systems are limited in their capacity to represent the world we can learn a lot

more about how, when and why they are successful in representing the world.

Take the example of colour vision. Some would argue that the function of the

perceptual system responsible for colour vision is to allow us to detect variations in the

wavelength of light impinging on our retinas from the environment, so as to allow us to

successfully navigate and interact with our environment. However, there are obvious

limitations to our capacity to accurately detect wavelengths of light impinging on our

retinas. For one thing, we are only sensitive to a certain portion of the electromagnetic

spectrum. There are certain wavelengths of light that we are unable to detect.

Furthermore, our ability to discriminate between different wavelengths is not perfect.

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The underlying problem with suggesting the ANS represents numerosity rather

than number is that it makes the error of confusing representational vehicle with

representational content.171 It may be the case that the representations that the ANS

utilises are fuzzy and only approximate but this, in and of itself, tells us nothing about

the nature of what they represent. To argue that the ANS represents approximate

numerosities as opposed to precise numbers is akin to arguing that a blurry photograph

of a face is a photograph of a blurry face. In general, representations will tend to be

impoverished in comparison to the thing that they represent but this is no reason to

suggest that they represent something else altogether.

The perceptual limitations of the ANS are not merely arbitrary. They reflect a

careful balance between fulfilling the evolutionary demands that this system evolved to

satisfy and achieving an efficient cognitive system that did not put too high an energy

demand on our ancestors. It might be the case that a system, similar to the ANS but with

a far more fine-grained degree of accuracy could have evolved. However, such a system

would be likely to demand greater cognitive resources for very small benefits in terms of

survival prospects.172 The particular way in which the discriminatory capacities of the

ANS are limited adds further reason to see it as a system for perceiving number. The

limitations of the ANS are closely tied to the Weber-Fechner law which also applies to

systems for detecting objective properties such as mass and luminance. However, it

would be odd to argue that we have systems for detecting approximate weight or

approximate luminance, since there are no such properties in the world.

The fact that the ANS has specific limitations for detecting numerical properties,

far from being problematic for the notion of numerical perception, helps to support the

idea that this system evolved in order to detect numerical properties that are particularly

salient. This is no different from any other perceptual capacity. One of the hallmarks of

any system capable of representation is the fact there are limitations to its

representational capacity and circumstances in which it will malfunction and

misrepresent. Thus, the sometimes approximate nature of the representational

capacities of the ANS should not rule out its dedication to the perception of precise

number. The ANS is a system responsible for approximately representing number,

rather a system for representing approximate number or numerosity.

171 Millikan (1991) 172 Whether there are fifty-seven or fifty-eight lions chasing you is pretty insignificant when it comes to your survival prospects, your behavioural response is likely to be pretty similar in either case.

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The Problem of Multi-Sensory Perception

Another potential objection to the idea that the ANS is a perceptual system arises

from its performance being similar regardless of the sensory modality to which stimuli

are presented. Subjects are as adept at counting beeps as they are at counting dots and

the signature performance limitations of the ANS show up regardless of the particular

form of sensory input.173 Subjects’ performance is even similar in tasks which involve

counting multiple kinds of stimuli, for example counting beeps and flashes.174

Furthermore, neurological evidence suggests that similar areas of the hIPS are activated

in tasks involving different types of sensory stimuli.175 These findings are problematic for

the notion of numerical perception, as it is common to understand perceptual systems as

being divided up into distinct sensory modalities, with each dedicated to a single form of

sensory input. On a traditional picture, information from these distinct streams of

sensory input is only combined together in higher cognitive systems. The ANS’s

insensitivity to the form of sensory input could thus be taken as evidence that it is not a

perceptual system but rather a cognitive system responsible for integrating information

from lower level perceptual systems.

One way of responding to this problem is to argue that the available evidence is

compatible with the ANS being divided up into modally specific subsystems. If this were

the case then the criticism would fail, as the ANS would be a collection of distinct

systems, with one system responsible for each form of perceptual input. Each of these

subsystems could be functionally similar to one another, explaining the fact that

behaviour is similar regardless of specific input. However, each might involve a different

population of neurons, for example, one subset of hIPS neurons might be responsible for

visual number perception and a different subset for auditory number perception. At

present, the available neuroimaging data is not fine-grained enough to test such a

hypothesis in the case of humans. However there is some evidence from tests on

nonhuman primates to support the claim. Recent detailed neuroimaging studies found

that primate homologs of the hIPS contain some neurons that fire for both visual and

auditory numerical stimuli, as well as other modally specific number neurons, which

only fire for stimuli from a single form of input.176 Whether or not the ANS can be

173 Xu & Spelke (2000) 174 Barth et al. (2005) 175 Piazza et al. (2006) 176 Nieder (2012), Nieder (2013). At first sight, these findings could be interpreted as undermining the present hypothesis, since the existence of multimodal number-specific neurons seems to weigh against the idea of modally specificity. However, the claim that the hIPS is divided into modally specific subsystems need not imply that these systems are anatomically distinct from one another. The present hypothesis is consistent with there

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understood as being divided up into functionally distinct modally specific subsystems is

an open empirical question, particularly in the case of humans. However, if future

research supports this hypothesis then the problem of multisensory perception can be

avoided.

Even if this hypothesis isn’t vindicated, it may still be possible to avoid the

apparent problem of multi-sensory perception. The apparent problem can be

undermined by questioning the idea that all perceptual systems must be modally

specific. It makes sense to take modal specificity to be a sufficient condition for

perceptual systems. If a system is solely dedicated to processing a specific form of

sensory input then it makes sense to classify it as a perceptual system. However, modal

specificity need not be a necessary condition for being a perceptual system. As such, the

multimodal nature of the ANS may be compatible with its characterisation as a

perceptual system.

One reason for questioning modal specificity as a necessary condition for

perceptual systems is the recent emergence of evidence to suggest that systems that are

unequivocally accepted as being perceptual systems are nonetheless responsive to

multimodal sensory inputs. Thus, if the problem of multimodal perception is a problem

with respect to the ANS then it might be equally problematic for seemingly

uncontroversial perceptual systems such as the visual or auditory system. In recent years

a growing body of evidence has built up to challenge the notion that low-level perceptual

systems, such as the visual or auditory system, are solely responsive to inputs from the

sensory receptors from which they derive their names.177 For example, parts of the visual

cortex have been found to also be responsive to auditory and tactile stimuli.178 Parts of

the auditory cortex have been found to be responsive to visual and tactile stimuli.179

Furthermore, neuroanatomical studies have found a surprisingly large quantity of

interconnections between low-level perceptual systems.180 As such, it makes little sense

to characterise the distinction between perception and cognition in terms of modally

specific processing in the former and multimodal processing in the latter. If one were to

do so, one would end up with a picture of the brain whereby even the early visual and

being some overlap between the different subsystems. All that is required is that different subsets of neurons are activated for different forms of sensory input and this is vindicated by the discovery of some modally specific number neurons. 177 Ghazanfar & Schroeder (2006), Driver & Noesselt (2008) 178 Morrell (1972), Sathian & Zangaladze (2002) 179 Giard & Peronnet (1999), Foxe et al. (2000) 180 Cappe, Rouiller & Barone (2009)

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auditory systems count as cognitive systems. As such, the multimodal nature of the ANS

need not be a barrier to its characterisation as a perceptual system.

The idea that we possess perceptual systems that go beyond modally specific

information streams should not be altogether surprising. There is a lot more to

perception than the mere passive reception and filtering of information. In order to

perceive the world we must actively explore our environment, suggesting that action and

perception might not be easily distinguishable. We also need to integrate inputs from

different senses to produce a coherent picture of the world, suggesting that entirely

distinct information streams cannot provide the whole story with respect to perception.

On top of all this we also need to coordinate our attention so as to allow us to process the

information that is most salient and relevant to our ongoing activity. At any given time

we may need to attend to stimuli from any of our different senses and so one would

expect attentional mechanisms to go beyond the notion of distinct channels of sensory

input. All of these processes can be understood as perceptual processes undertaken by

perceptual systems and, yet, none of them can be easily explained with perception

characterised purely in terms of modal specificity. As will become clear, these are exactly

the kinds of processes that are relevant for the notion of numerical perception. As such,

the multimodal nature of the ANS is hardly surprising. The ANS can, thus, be

understood as a perceptual system despite the fact that it responds similarly to multiple

forms of sensory input. It can either be understood as a collection of subsystems each

responsible for the perception of number within a given modality or as a single

multimodal system responsible for the perception of number regardless of sensory input.

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3

The Objects of Numerical Perception

In the case of most of our perceptual capacities we seem to have a pretty good

idea about what properties of the world our perceptual systems are dedicated to

representing. For example, vision might be seen as detecting the properties of light

impinging on the receptive neurons in the retina or it might be seen to detect changes in

the ambient optic array as we move around our environment. Despite disagreements as

to exactly what vision represents, it has something to do with detecting certain physical

properties of light. The case seems to be similar for other perceptual capacities. For

example, audition detects vibrations in the environmental medium and olfaction detects

various chemical properties.

The fact that the ANS responds to a number of different forms of sensory stimuli

should not, in and of itself, be seen as problematic for the notion that it is a perceptual

system for detecting number. However, this immediately leads to the question of what

property of the physical world the ANS responds to. In the case of a perceptible property

such as colour, we seem to have some idea about its physical basis. However, in the case

of number it is hard to tell where to begin. One reason for this is the standard

philosophical assumption that, whatever number might be, it is definitely abstract and,

therefore, nonphysical in nature. In the current context of addressing Benacerraf’s

challenge, it makes good sense to withhold judgement on the abstract nature of

numerical properties, since this judgement is an important step in the argument that is

being assessed. Thus, by suspending the philosophical assumption that number must be

abstract, one can begin to inquire as to what kind of physical property number could be.

The notion that we acquire our knowledge of number through experience with

certain properties of the physical world is certainly not a new one. It formed the basis of

Mill’s attempt to provide an empiricist account of mathematical knowledge. Mill argues

that our knowledge of mathematics is the result of ‘observation and experience founded,

in short, on the evidence of the senses’.181 However, the issue of a perceptual basis for

181 Mill (2002) pg. 399

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mathematical knowledge has largely been neglected for more than a century. This is

probably due to the widespread belief that Mill’s approach had been thoroughly

demolished by Frege’s scathing critique.182 Although Frege highlighted significant

problems with Mill's account, they are not so severe as to rule out any attempt to account

for number as a property that is perceptually accessible.

For Mill, number is a property of ‘agglomerations’, namely ‘the characteristic

manner in which the agglomeration is made up of and may be separated into parts’.183

Thus, number is seen as an objective property of collections of objects which determines

the ways in which we are able to interact with such collections. We are able to acquire

knowledge of these properties from our experiences of manipulating and rearranging

collections. In light of these experiences we are able to make generalisations and come to

see that the claims of arithmetic are true of all collections of physical objects.

Problems with Mill’s Empiricism

There are clearly some aspects of Mill’s account of arithmetical knowledge

acquisition that do not fit with the psychological evidence that was presented in the

previous chapter. For Mill, as an Empiricist, our knowledge of number is entirely derived

from our experiences of manipulating collections and learning about the results of our

manipulative activity through perception. There are two ways in which this picture

clashes with the available evidence from the cognitive sciences. Firstly, there is a wealth

of evidence to suggest that our capacity to acquire arithmetical beliefs is to some extent

innate. We are able to apprehend a collection as having three entities without having to

have learned from any experiences of manipulating collections.184 A second related

problem is that, for Mill, arithmetical knowledge is derived from prolonged and repeated

interactions with the environment. However, our access to arithmetical content is far

more direct. We directly perceive the number of entities, without having to inductively

learn about the nature of collections through repeated manipulation experiments.

Despite the difficulties in reconciling Mill’s Empiricist account with the available

psychological data, the notion that number is a property that we apprehend perceptually

remains viable. Mill was wrong to emphasise an inductive basis for mathematical

182 Frege (1960) pg. 12-32 183 Mill, (2002) pg. 400 184 This should be clear from the presence of this capacity in infants that lack the dexterity for object manipulation and in animals that lack the kinds of limbs needed to manipulate collections.

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knowledge acquisition but he was right to highlight the importance of perception. As

such, it might be fruitful to try to maintain Mill’s account of number as the property

according to which collections are ‘made up of and may be separated into parts’185, whilst

departing from his account of how we acquire knowledge of such properties, in favour of

an account based on the direct perception of number.

Kitcher’s Millian Account

Despite the general consensus that Mill’s Empiricist account was roundly

defeated by Frege’s criticisms, some have tried to resurrect a Millian approach.186 The

most pertinent of these attempts, given current concerns, is that of Kitcher, who

explicitly reframes Mill’s account so as to emphasise our perceptual access to

arithmetical knowledge.187 In order to justify this, Kitcher appeals to Gibson’s ecological

theory of perception.188 This theory will be addressed in more detail soon. For now, the

important feature to highlight is that it suggests we are able to directly perceive

opportunities for possible action. Thus, whilst Mill argues that we acquire arithmetical

knowledge by engaging in the activity of manipulating collections, Kitcher argues that we

do so by perceiving opportunities to manipulate collections. As such he can explain the

fact that we seem to have perceptual access to arithmetical facts prior to engaging in any

actual manipulative activity. This reframing of Mill’s approach is best captured by the

claim that arithmetic is ‘true in virtue not of what we can do to the world but rather of

what the world will let us do to it’.189 In order to access these truths, we do not need to

generalise from experience of what we can do, we merely need to see what it is possible

to do.

An immediate problem for Kitcher’s account is that most of the truths of

arithmetic fail to correspond to any process of object manipulation that it would be

possible for a normal human subject to engage in. Our mortality means that, in practice,

we will never be able to segregate a vast number of objects and, given the apparent

finitude of the universe, we would eventually run out of objects to manipulate. In order

to address this problem, Kitcher argues that arithmetic is an idealised science whose

185 Mill (2002) pg. 400 186 Kessler (1980), Kitcher (1980, 1988), Irvine (2010) 187 Kitcher (1980) pg. 11 188 Gibson (1979) Michaels & Carello (1981) 189 Kitcher (1980) pg. 108

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subject matter is the possible actions of ideal, as opposed to actual human, agents.190 As

has already been mentioned, it is unclear how this move can help, since our access to

knowledge of ideal agents seems at least as problematic as our access to the abstract

Platonic entities that Kitcher is trying to avoid. As such, it would be good to develop an

account of numerical perception that can avoid this move. Despite this problem,

Kitcher’s appeal to Gibson’s theory of ecological perception represents a significant

advance from Mill’s crude Empiricism. Before addressing this theory in more detail, it is

important to address aspects of Frege’s critique that directly challenge the notion of

numerical perception.

Frege’s Criticism of Perceiving Numerical Properties

The idea that the number of objects in a collection is an objective property

pertaining to ‘the characteristic manner’ in which a collection is made up of and may be

separated into parts’ was roundly criticised by Frege.191 His main objection to this idea

was that there is no unique characteristic manner in which a given collection can be

separated. ‘There are very various manners in which an agglomeration can be separated

into parts, and we cannot say that one alone would be characteristic’.192 For any one

collection there are many ways of separating it and there is no fact of the matter as to

which of these many ways is characteristic. For example, a pack of playing cards could be

seen as fifty-two cards, four suits or one pack.193

This criticism of Mill’s account is, at face value, troubling for a perceptual account

of our access to arithmetical content. If any one collection can be understood as

possessing many different incompatible numerical properties then it is unclear how

perception can pick up on one property rather than another on any particular occasion.

One tends to assume that if the physical make-up of a collection is kept stable from one

occasion to the next then the perceptual input that one receives from that collection will

also be the same from one occasion to the next. Thus if there is any difference in our

judgements of the number properties of a stable physical collection from one occasion to

the next, it follows that this difference cannot be a result of differing perceptual input. If

Frege is correct that we are able to judge one and the same physical collection as

190 Ibid. pg. 118 191 Mill (2002) pg. 400 192 Frege (1960) pg. 30 193 Ibid. pg. 28

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possessing different numerical properties on different occasions then this difference

must arise from something other than perceptual content and the notion that

arithmetical content is derived from the perception of real physical properties is

undermined.194 If anything is to determine the numerical properties, it would seem to be

our own subjective cognitive judgements as opposed to the perception of external

properties.

Given the evidence from the preceding chapter, we should expect some response

to be forthcoming. A wide range of animals possess the innate ability to perceive the

number of entities in a collection. Furthermore, at least within a given species,

organisms tend to agree on the number of entities perceived, as long as one factors in the

inherent limitations on accuracy. For example, when confronted with an image of four

circles subjects will tend to immediately perceive them as being four, despite the fact that

they could in principle conceive of them as eight semi-circles. Thus, it would seem that

there is a characteristic manner in which the number of entities in a collection is

perceived. Frege is correct that a given agglomeration could be conceived in a different

way. However, this does not detract from the fact that in actual scenarios there is a single

characteristic manner in which numerical features are perceived. Whilst this gives cause

for optimism, explaining how such a characteristic manner can arise is harder than it

may first seem.

Frege’s argument for the indeterminacy of attributing numerical properties to

collections is similar in important ways to a more famous indeterminacy argument

proposed by Quine. In two related arguments, known as the indeterminacy of translation

thesis and the argument for the inscrutability of reference, Quine argues that the

information available to perception is insufficient to determine either the meaning or the

reference of words.195 To illustrate this point, he presents a thought experiment in which

a field linguist is attempting to determine the meaning of a native word “gavagai”, which

the native speakers tend to utter when they see a rabbit. Intuitively it seems correct to

conclude that “gavagai” means “rabbit”. However, there are many other interpretations

that are equally compatible with the available evidence from perception. For example,

one could instead conclude that “gavagai” means “undetached-rabbit-parts”, since it

seems to be the case that whenever one sees a rabbit one also sees a collection of

194 Another way of looking at this argument is in terms of indeterminacy. If we assume, with Frege, that there are many ways of separating an agglomeration into parts and also assume that for any given agglomeration the perceptual input remains fixed, then it follows that the perceptual input is insufficient to determine the numerical properties of a given agglomeration. 195 Quine (1960) pg. 29-40, (1969) pg. 30-38

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undetached-rabbit-parts. From these considerations, Quine concludes that there is no

determinate fact of the matter as to the meaning of words and that the reference of a

given utterance is behaviourally inscrutable.

At this stage, this problem may seem quite remote from issues concerning

numerical perception, being primarily about the meaning and reference of linguistic

items. However, in order to see the relevance to Frege’s challenge to numerical

perception, it is important to appreciate that these problems ‘begin at home’.196 Just as

the field linguist has no way of determining whether the native is referring to a rabbit or

some undetached-rabbit-parts, so too from the inside we have no way of determining

whether our perceptual content is of a rabbit or some undetached-rabbit-parts, since the

two are taken to be equivalent with respect to incoming stimuli. Once one has taken the

apparent indeterminacy of our own mental content on board, the similarity between

Fregean and Quinean indeterminacy becomes more apparent. Just as our perceptual

input seems to be indeterminate with respect to rabbits and undetached-rabbit-parts, it

is also indeterminate between seeing a single deck of cards or fifty-two distinct cards.

Thus, in order to explain the possibility of numerical perception, it is necessary to try to

find a way around this perceptual indeterminacy.

Shani has diagnosed the underlying problem behind Quinean indeterminacy as

stemming from the assumption that perceptual input is purely extensional.197 Since

rabbits and undetached-rabbit-parts share the same material extension, it is assumed

that the process of perceiving the former must be the same as that of perceiving the

latter.198 This assumption also seems to underlie Frege’s argument for the indeterminacy

of attributing numerical properties. They both assume that, for entities or collections

with a fixed material extension, the perceptual input that one receives from the given

entity or collection must also remain fixed. Perceptual content is thus entirely

determined by the material extension of a given object of perception. If one maintains

this assumption then it is clear that perception can never deliver content that pertains to

features of the world that are more fine-grained than material extension. It is certainly

natural and intuitive to assume that perceptual content is determined by material

extension. However, whether this is in actual fact the case is an empirical question about

196 Quine (1969) pg. 46, Shani (2009) pg. 744 197 Shani (2009) pg. 746 198 Following Shani (2009) pg. 745-746 I make reference to “material extension” as opposed to mere extension, since, technically, rabbits/undetached-rabbit-parts and deck-of-cards/32-cards have different logical extensions from each other since they differ with respect to cardinality. Given that the perception of numerical properties, such as cardinality, is exactly what is at issue here, the notion of material extension seems more apt.

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the nature of perceptual processes and, thus, should not be decided on the basis of

intuitions or a priori judgements about how we think our perceptual systems work. If it

turns out that our perceptual input is in fact more fine-grained than merely being

determined by the material extensions of the objects that we perceive then Frege’s

argument against the perceptibility of number properties might be undermined.199 In

order to assess whether Frege is justified in assuming the extensional nature of

perception it is thus necessary to look to prevailing theories of perception from the

cognitive sciences.

Perceiving More than Extension

According to the traditional computational view of perception, the primary role of

perception is to construct a rich model of the world on the basis of the relatively

impoverished stimuli that impinge upon our sensory receptors. For example, in the case

of vision, the goal is to provide a detailed 3-dimensional representation of the

environment on the basis of the 2-dimensional patterns of excitation on the retina.

Perception is thus a process that involves the production of a representation through the

process of computational inference purely based on the input data from sensory

receptors.200 This approach thus seems to support the idea that our perceptual systems

are only responsive to purely extensional features. For example, a single pack of cards

will produce the same pattern of excitation on the retina independently of whether one is

considering it as 52 cards or 4 suits. Thus, a traditional computational approach to

perception supports Frege’s claim that assigning numerical properties to a given

agglomeration is more a matter of internal subjective cognitive processes than a matter

of direct perception. Perception seems to provide us with a rich model of the extensional

properties of the 3-dimensional environment, to which we then apply cognitive

processes that allow us to divide the scene up into distinct objects and collections and

then to assign numerical properties to the collections thus divided. As such, the

possibility of numerical perception seems to be undermined both by the intuitive

199 In the case of Quinean indeterminacy the case is somewhat more complicated. If perception is more fine-grained than extension, i.e. if there is intensional perception, then one’s own mental content might be rendered determinate. However, the problem of interpreting the mental content of someone else from their behaviour alone remains. 200 In the case of this kind of approach, sensory modalities tend to be individuated in terms of distinct sensory receptors and their corresponding information channels. As such, the input data that perception deals with tends to be understood as sequences of patterns of excitation at the periphery of sensory systems.

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account of perception offered by Frege and by the prevailing computational account of

perception from mainstream cognitive science.

In order to support the idea that we are able to perceive number, it is thus

necessary to consider alternatives to the prevailing computational approach. In recent

years the computational account of perception has come under attack from alternative

approaches that emphasise the direct and active nature of perceptual processes. There

are a number of reasons why such alternative approaches might be seen as more

attractive. Firstly, support for the computational approach to perception has been

dwindling in recent years, to the extent that the “alternative” on offer here might be the

more widely accepted contemporary approach. Secondly, both Maddy and Kitcher

tentatively point to the potential applicability of Gibson’s theory of direct perception to

the problem of mathematical knowledge.201 Thus, by investigating its application to the

problem of perceiving numbers in a detailed manner it might be possible to flesh out the

ideas to which Maddy and Kitcher merely hinted. Furthermore, the theory of direct and

active perception is the best candidate to challenge the notion that perception is purely

extensional and thus to overcome Frege’s indeterminacy objection to the perceptibility of

numerical properties.

The first step in challenging the computational approach lies in questioning the

idea that our perceptual input data are mere sequences of patterns of excitation on

sensory receptors. The idea that our sensory inputs are richer than this stems from

Gibson’s theory of direct perception and has been further developed by his followers.202

The problem with the computational approach, according to Gibson, is that it mistakenly

considers the inputs to sensory systems to be series of isolated snapshots or patterns of

excitation. This assumption ignores the fact that perception is a dynamic process of

actively exploring the environment. Our sensory receptors do not merely sit there

passively receiving excitations; they actively sample dynamic properties of the unfolding

sensory array. For example, in visual perception, our eyes are never merely sitting still

and passively soaking up the light that impinges upon them. They constantly engage in

rapid movements, known as saccades, darting from one place to another so as to

maximise the reception of salient information. The importance of this dynamicity is best

demonstrated by experiments where subjects are presented with images that shift in line

with the movements of their eyes. In such cases, where sensory input is essentially kept

201 Kitcher (1988) pg. 11-12, 108, Maddy (1990) pg. 48 202 Gibson (1979), Michaels & Carello (1981)

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stable, the images will begin to disappear from sight in a matter of mere seconds.203

Thus, it seems as if active dynamic exploration is necessary, at least for visual perception.

Significantly, once one adopts this dynamic perspective on the nature of sensory input, it

becomes clear that the input is far richer than the computational approach assumes.

Perceptual systems are able to pick up on higher-order regularities of the dynamically

unfolding sensory array that cannot easily be discerned from snapshots of excitation.

The most significant of these higher-order properties are perceptual invariants.

In order to understand the notion of a perceptual invariant it will help to consider

the example of vision. The first significant thing to note is that the light emanating from

light sources and reflecting off surfaces in the environment does not merely structure the

excitations of an organism’s sensory receptors. These processes directly influence the

overall structure of the light in the environment, which Gibson calls the ‘ambient optic

array’.204 For example, a blue square of card might reflect a roughly trapezoid patch of

light onto the retina of an organism, whilst also reflecting different roughly trapezoid

patches of light in the direction of other parts of the environment.205 If one considers the

organism to be a stationary passive observer then at any one moment the only

information available to the organism will be the particular excitations on its sensory

receptors. However, once one appreciates that the organism is constantly engaging in

active exploration of its environment, it becomes clear that there is more information

available about environmental structure. For example, as the organism moves towards

the blue square, the shape of the trapezoid patch of light on the retina will change in a

structured and predictable way, dependent upon the direction and speed of motion and

the changing angle of orientation between the blue square and the organism’s sensory

receptors. Similarly, if the organism approaches the blue square head on, the size of the

trapezoid patches on the retinae will grow in proportion to the speed at which the object

is approached. The most significant fact about these processes is that certain features of

the sensory input will remain relatively invariant despite the dynamic variation in

sensory input caused by active exploration. For example, whilst the particular trapezoid

patches of light on the retinae will change shape from one perspective to the next, their

four-cornered-ness and four-sided-ness will be maintained from almost every

perspective. Invariant features of the dynamic sensory input over time are thus able to

203 Riggs et al. (1953) 204 Gibson (1979) pg. 65 205 Whilst the account here is restricted to visual perception, a similar account could be provided for other sensory mediums. For example, sound waves structure the air throughout the environment, not just the air impinging upon an organism’s ear drum.

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uniquely specify the structure of the environment. The blue square of card structures the

ambient optic array in a specific manner and exploration of the ambient optic array

allows organisms to pick up on the invariant properties that specify the given structure.

The nature of such dynamic perceptual invariants can be formalised and studied in a

rigorous manner using the tools of affine geometry, projective geometry and topology.206

Another important feature of Gibson’s approach is the interdependency of

perception and action. The kinds of invariant that an organism can detect will depend

upon the ways in which it can move about its environment. Thus, perception is

dependent on observer-relative actions. For example, human vision allows us to pick up

on the kinds of invariants that are revealed by typically human actions, such as features

that remain invariant as our eyes rapidly saccade, as we move our head from side to side

or as we move forwards by walking upright. What one can perceive depends on how one

can act and thus perception is always dependent on what kind of organism one is.

However, there is also an important sense in which action is dependent upon perception.

It is uncontroversial that our ongoing actions are continuously guided by the information

we receive from our senses. However, the Gibsonian approach goes further than this by

suggesting that what we perceive is directly relevant for action. In particular, through

perceiving invariant features of the sensory array we are able to directly perceive

affordances. The notion of affordance is central to the account of how we can perceive

number and so will require elucidation in some detail.

Affordances are opportunities for possible action in the environment. For

example, when one sees a table one might see that it affords climbing on, thereby

perceiving the affordance of climbability. A more traditional account would assume that

one perceives the various sensory attributes before inferring the shapes of objects in the

environment and then making the further inference that one can interact with them in

particular ways. However, on the Gibsonian approach one directly perceives the actions

that the environment makes available, since the invariant features of the perceptual

array directly specify information relevant for action. Affordances are always relative to a

particular organism. For example, a chair might afford sitting for an adult but not for a

toddler, or a drainpipe might afford sheltering for a mouse but not for a human. Despite

the organism-relative nature of affordances, they can still be thought of as fully objective

properties. There are facts of the matter about what a given organism can or can’t do in

its environment and the information that a given organism is able to detect perceptually

206 Michaels & Carello (1981) pg. 34-36

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pertain to just these facts. The objects of perception are affordances and affordances are

all that we perceive.

At first sight, the notion that affordances are the only objects of perception seems

to conflict with both our introspective access to our own perceptual states and with some

of the earlier details of the direct and active perception account. When we consider our

own perceptual states they seem to be far richer and more detailed than mere

specifications of possible actions. Furthermore, as it was introduced here, the direct and

active perception account seems to suggest that we perceive affordances by perceiving

perceptual invariants. Thus, it may seem that what we directly perceive are the

invariants and that the affordances are inferred on this basis. In order to fully appreciate

the fact that we directly perceive affordances it is necessary to pay more attention to the

fact that perception itself is an inherently active process.207 Since perception involves

active exploration, some of the affordances that we perceive will be opportunities for

further perceptual action. For example, when one encounters a table, as well as

perceiving that it is climbable, one perceives that it is possible to move in a certain

manner with respect to the table so as to acquire further information about the table’s

affordances. Some of these possible movements might be overt intentional acts of

exploration, such as walking around the table to reveal its various facets. However, the

notion of action at play here allows for much more minimal activities to qualify as action,

for example, the rapid saccadic motions of the eyes.208 One does not infer the

affordances of an object from perceptual invariants. Perceptual invariants are always

relative to a certain kind of motion and thus to a certain kind of possible action. When

one perceives an object, one perceives the perceptual affordances that specify how one’s

perception will change as a result of possible interaction with it. Thus, the notions of

invariant and affordance are inseparably bound together. Once one takes into account

the ideas that some of the affordances that we perceive are perceptual affordances and

that these affordances can be specified in terms of relatively minimal actions, such as

visual saccades, the apparent conflict between the Gibsonian account and our

introspective access to our perceptual experiences dissolves. If one only considers large-

scale deliberate actions such as climbing or walking then it is clear that our perceptual

experience is richer than an affordance based account would suggest. However, once one

207 O’Regan & Noë (2001) 208 Aloimonos, Weiss & Bandyopadhyay (1988), Findlay & Gilchrist (2003) pg. 178-180

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takes into account the possibility of perceiving far more fine-grained perceptual

affordances, it is clear that the discrepancy in detail and richness can be accounted for.209

The idea that we directly perceive affordances is supported by arguments about

the evolutionary origins of our perceptual systems. ‘Affordances and only the relative

availability (or nonavailability) of affordances create selection pressure on animals;

hence behaviour is regulated with respect to the affordances of the environment for a

given animal’.210 From an evolutionary perspective there is no need for perceptual

systems that have no impact on the way an organism can act in the world. Only an

organism’s behaviour is relevant to its survival and reproduction and, as such, natural

selection will favour perceptual systems that are geared to detecting information relevant

to action that serves these goals. The evolutionary function of perceptual systems is to

guide action by detecting opportunities for possible evolutionarily beneficial actions,

which is, in essence, to say that the function of perception is to detect affordances.

The Gibsonian approach to perception emphasises the direct nature of perceptual

processes. A core feature of the approach is eliminativism with respect to internal

representations or computational inferences.211 Whilst the question of whether or not

perceptual processes require representations is of great interest and significance in the

cognitive sciences, it is tangential to the issue currently at hand. In order to support the

claim that it is possible to perceive numerical properties, all that is required is that the

objects of perception are affordances or, in other words, that perception is action-

oriented. However, one can adopt an action-oriented view of perception from either an

anti-representationalist or a representationalist standpoint.212 In either case, numerical

properties are perceivable properties that are intimately related to the specific kinds of

action that a given organism is able to perform.

By adopting this active and action-oriented account of perception it is possible to

challenge the Fregean assumption that perceptual content is solely determined by

material extension. Instead perceptual content is taken to be determined by both the

state of the environment, including facts about material extension, and the actions

available to the organism doing the perceiving. Which actions are available to an

organism depends on a number of factors. Some of these factors will be related to

209 O’Regan & Noë (2001) 210 Reed (1996) pg. 18 211 Anti-representational accounts of this kind are sometimes referred to as theories of Radical Embodied Cognition (see Chemero (2009)) 212 Clark (1998), Mandik & Clark (2002), Mandik (2005)

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physiological constraints of the organism in question. However, others might be more

temporary and depend upon the context of the organism in question’s current goal-

directed behaviour. For example, a particular tree might be seen as affording sheltering

when trying to avoid rain whilst affording climbing when fleeing a predator. According to

the Gibsonian approach, one would thus have a case of differing perceptual content

despite material extension remaining the same in both cases. In the case of Frege’s

problem a similar response is available. In seeing a deck of cards as one deck of cards we

perceive different affordances to the case where we see it as thirty-two cards, in the

sense that we see the opportunity for different courses of action. By adopting a

Gibsonian account of perceptual content it is therefore possible to avoid the problems

that arise from assuming that perceptual content is solely determined by the material

extension of external objects. In doing so one needn’t throw away the objectivity of the

numerical properties that are perceptually represented. Affordances can be understood

as being real properties of environment-organism systems. The fact that they are

organism-relative does not thereby render them subjective, since facts about what an

organism can do are to some extent independent of the organism’s thoughts or beliefs

about the world.

Enumerative Affordances and Manipulation

Having established the notion of affordances it is time to return to the main task

by showing how the perception of numerical properties can be understood in these

terms. The central claim is that, in perceiving the numerical properties of a collection, we

perceive the enumerability of the collection. We directly perceive that the given

collection affords some sort of enumerative action. A lot more needs saying about the

notion of enumerability and of enumerative actions. However, it will be useful to first

contrast the notion of direct perception of enumerability with the kind of account of

enumeration that both an intuitive conception and the computational approach provide.

Both our intuitive conception and the computational theory of perception suggest

that enumeration is a complex process that requires a sequence of cognitive operations

on perceptual representations. We must seemingly begin by forming a representation of

the perceptual scene based on the data impinging on our sensory receptors, then go on to

recognise the objects in the representation, then group some of these objects together as

a collection before finally assessing the number of objects by performing some kind of

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sequential enumeration operation. On such an account, the number assigned to a given

collection will depend upon the kinds of cognitive operations that we use to recognise

objects, group them together and enumerate them. Numerical properties seem to be

inferred rather than perceived.

Once one takes into account the possibility of affordances being the objects of

perception, the necessity of performing this kind of cognitive process is undermined.

Instead one can argue that we directly perceive the enumerability of the collection. We

perceive that the collection affords a certain kind of interaction. Thus, rather than

perceiving a collection of entities and inferring their enumerability, we directly perceive

collections as collections by perceiving their enumerability. On this approach all that it is

to be a collection is to be the possible subject of an enumerative action.

This notion of direct perception of enumerability fits nicely with empirical

evidence about the ANS. Numerical perception provides us with a direct but sometimes

approximate specification of the kind of enumerative act that a given collection affords.

With small collections we can reliably perceive the exact enumerative action that a given

collection affords, whilst with increasingly larger collections our number sense provides

increasingly approximate specifications of the enumerative actions available. However,

even in the case of more approximate specifications we arguably directly perceive a

course-grained specification of the kind of enumerative action available.

By adopting an active and direct account of perception of this kind one can thus

argue that it is possible to directly perceive numerical properties in the environment,

where such numerical properties are affordances of enumerability. We directly see

opportunities for possible enumeration. However, at this stage, it remains to explain

exactly what is meant by an enumerative act. For both Mill and Kitcher the fundamental

kind of enumerative act lies in the manipulation of external macroscopic objects into

bounded spatial regions. For example, Kitcher argues that ‘arithmetic describes those

features of the world in virtue of which we are able to segregate and recombine

objects’.213 This approach is also at the heart of Lakoff & Núñez’s account of the cognitive

foundations of arithmetic.214 There are at least two obvious problems with this approach,

which were first suggested by Frege in his critique of Mill and are further supported by

recent findings in the cognitive sciences. Firstly perception of numerical properties does

not seem to depend on a capacity for object manipulation. Secondly, it is possible to

213 Kitcher (1984) pg. 108 214 Lakoff & Núñez (2000) pg. 54-65

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perceive the numerical properties of collections of entities that are non-manipulable in

principle.

The first of these criticisms is brought out nicely in a passage where Frege

ridicules Mill by exclaiming ‘what a mercy, then, that not everything in the world is

nailed down, for if it were,… 2 + 1 would not be 3!’215 The first point highlighted by

Frege’s sarcastic chide is that our ability to count is independent of our ability to move

things around. For example, we can count the number of mountains on the horizon or

the number of clouds in the sky. Furthermore, our ability to count objects that are

manipulable is independent of our capacity to move them around. Nailing objects down

has no effect on this ability. Kitcher argues that we are able to perform such counting

operations merely by imagining the possibility of manipulating the objects in the

collection under consideration. We are able to imagine such manipulative activities by

mentally drawing lines around the objects and this capacity arises as a result of our

capacity for actual object manipulation.216 The problem with this approach is that

counting objects that are “nailed down” seems to be our default capacity. Infants and

animals have some capacity for enumeration despite lacking the manual dexterity

required to manipulate objects into orderly piles. Furthermore, these capacities emerge

in the absence of any experience of manipulating objects. In perceiving a collection’s

enumerability we must perceive a far more basic affordance than the possibility of

specific kinds of orderly manipulation.

This problem is further exacerbated by the fact that we can also enumerate

entities that are not even manipulable in principle. We do not only count macroscopic

manipulable entities. We are equally able to enumerate sounds, flashes, events, fictional

entities and ideas. It would be strange if our ability to enumerate these kinds of entities

were dependent on imagining performing manipulations on them. It is unclear what

such a process would even involve. Furthermore, infants and animals are capable of

enumerating sounds and events as well as ordinary objects, suggesting that our capacity

for enumeration is developmentally independent of our capacity for perceiving

manipulability. We are able to enumerate nonmanipulable entities before we are even

able to manipulate entities, so it makes little sense to explain this capacity in terms of

applying our ability to perceive manipulability in an unorthodox manner. As such,

evidence again suggests that our capacity for perceiving enumerability of a collection

215 Frege (1960) pg. 9 216 Kitcher (1988) pg. 111

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involves far more basic forms of affordance than those involved with manipulating and

organising macroscopic objects into spatially segregated collections.

These problems arise because object manipulation provides a far too restricted

notion of an enumerative act. Certain kinds of object manipulations might involve

enumerative actions but enumerative actions themselves seem to be far more basic and

general. In terms of affordances, manipulable collections afford enumeration but so do

other nonmanipulable collections. In order to see how an account based on the

perception of affordances can help, it is necessary to take into account the hierarchical

nature of affordances and also to take on board a minimal notion of action. Having done

so, it will be possible to characterise enumerative affordances in terms of attention as

opposed to manipulation.

Enumerative Affordances and Attention

Affordances are organised in hierarchies. Complex affordances are built up out of

more basic affordances. For example, in perceiving the climbability of a staircase one

also perceives the stepability of each step. Similarly, in perceiving the manipulability of a

collection one also perceives the graspability of the objects that comprise it. Given that

affordances are organised hierarchically with more basic affordances being nested in

more complex ones, the question arises as to how basic affordances can be. As has

already been emphasised, Gibsonian approaches do not only claim that perception is

action-oriented but also claim that perception is a process of active exploration. In other

words, perception invariably involves perceptual actions. An upshot of this is that

perception involves detection of affordances for further perceptual acts. For example, in

seeing something as a cube, we see that its occluded faces afford being seen given certain

movements with respect to it.

The most significant type of perceptual affordance for current concerns is that of

attendability. In the process of perceiving our environment we perceive that it is

possible to direct our attention to certain aspects of the environment. Attention is thus a

basic form of perceptual action and, relatedly, attendability is amongst the most basic

forms of affordance. Many complex interactions with the world depend upon certain

aspects of the world being attendable to. For example, in order to manipulate a collection

in a certain way one must be able to see that the objects that comprise it are graspable

and to apprehend this it may be necessary to attend to the particular objects. As such,

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our ability to perceive manipulability can be seen to depend upon the attendability of the

relevant objects. Attendability is one of the basic affordances that we perceive. We

perceive opportunities for possible attention. There are objective facts of the matter as to

which features of the environment are attendable for a given organism and organisms

are perceptually sensitive to these affordances.

For an account of this kind to work, it is necessary to adopt a minimal sense of

what counts as an action, such that attending is considered as an action. For some this

may be somewhat controversial, so it will be important to guard against some potential

objections to conceiving of attention as a form of action. In many cases, attending to a

particular feature of the environment will involve certain overt actions. For example,

attending to a visual stimulus will often involve motor responses such as turning one’s

head to align the stimulus with the centre of one’s visual field or initiating characteristic

patterns of saccadic motion. Thus, it is tempting to understand attentional actions in

terms of overt motor responses. If one were to do this then perceiving attendability

would amount to perceiving opportunities for enacting particular motions in order to

coordinate sensory receptors with stimuli in the environment. The problem with such an

approach is that shifts in attention are not always accompanied by overt motions. There

is a wealth of evidence which suggests that we are also capable of covert attention, where

the locus of attention can shift without any change in overt behaviour, such as a change

in the eyes’ point of fixation.217 For some, the possibility of covert attention might be

problematic for the idea that attention is a basic form of action, since action tends to be

understood in terms of an agent’s overt movements and interactions with the world.

There are, however, at least two ways to respond to the apparent problem of

covert attention. Firstly, similarities in the mechanisms involved in overt and covert

attention suggest that they should either be seen as one and the same or two extremely

closely related processes. Secondly, there are good reasons for rejecting the idea that

action should be defined in terms of movement. Although covert attention is

characterised by a lack of overt motor behaviour, evidence suggests that the mechanisms

that support the control of covert attention are the same as those that support

oculomotor activity in overt attention.218 Thus, according to so-called premotor theories

of attention, overt and covert attention can be seen to utilise the same mechanism,

however, in the case of the latter, the movements of the eye are somehow suppressed.219

217 Posner & Cohen (1984) 218 Moore, Armstrong & Fallah (2003) 219 Rizzolatti et al. (1987) pg. 37

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Furthermore, there is some evidence to suggest that, even in cases of apparent covert

attention, the eyes undergo barely noticeable microsaccadic motion, which is determined

by the target of attention.220 Since both overt and covert attention seem to be supported

by the same mechanisms and since the former seems to uncontroversially count as a

form of action then the latter should also be seen as an acceptable form of basic action.

Some might still insist that the case of covert attention is problematic, as it fails to

involve the kind of overt movement that one generally associates with actions. However,

one can question whether this is a good way of characterising action. It is first important

to note that covert attention is usually only deployed in unusual circumstances. In most

cases shifts in attention are accompanied by eye movements and other movements such

as head turning. Subjects deploy covert attention when they are explicitly given a task

that requires them to maintain a point of fixation that conflicts with the target of their

attention. Thus the suppression of their eye movements are a part of the action itself. It

is easy to think of ecologically relevant scenarios where such a capacity might be of use.

For example, an animal might want to keep track on the location of a potential mate

without alerting the suspicions of their rivals. In order to see how such behaviour could

be classed as a form of action it is worth considering another example that seems like

action without overt movement. Consider an animal that is in an environment where it is

well camouflaged when it remains stationary. Upon spotting a predator the animal might

stay still in order to avoid detection. It seems right to say that the animal’s staying still

counts as an action, regardless of the fact that it involves no overt movement.

Furthermore, such a scenario can be described in Gibsonian terms. The animal in

question can perceive that it is in an environment that affords freezing so as not to be

spotted by a predator. However, if the animal were in a different environment, such as

out in the open, the same behaviour would have less fortuitous results. Thus, it seems

wrong to define action in terms of overt movements. In some cases lack of motion can

count as an action to the same extent as various kinds of motion. Thus, the fact that

attention needn’t be accompanied by overt motion need not invalidate it as a basic form

of action.

So far the account of attention as action has primarily focussed on visuospatial

attention. For some this may be seen as problematic, since, whilst some shifts in visual

attention are usually accompanied by overt movements, such as eye saccades or head

turns, this is less obviously the case when one considers shifts of attention pertaining to

220 Hafed & Clark (2002), Engbert & Kliegl (2003)

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other sensory modalities. For example, it seems as though we don’t need to carry out any

form of movement in order to orient ourselves to an auditory or tactile stimulus. As such,

one could argue that only visuospatial attention should be understood in terms of action,

undermining the general account of attention as action. However, there are a number of

reasons to resist this move. One reason is that attentional mechanisms in distinct

sensory modalities might be harder to separate than one would think. For example,

when one hears a loud noise one instinctively tends to take action to visually fixate upon

the source of the noise. Evidence suggests that subjects’ performance at processing

auditory and tactile stimuli is better when they are visually fixating on the location of the

source of the stimulus.221 Furthermore, this is even the case prior to saccadic motion and

when subjects only produce microsaccadic motion, suggesting a similar mechanism to

that responsible for covert visual attention in the cases of auditory and tactile

attention.222 Thus, even in the case of nonvisual attention, ‘attention is subserved by the

same mechanisms that program eye movements’.223 At first sight, the idea that

oculomotor mechanisms subserve all kinds of attention can seem somewhat strange.

However, the central idea of the premotor theory of attention is that attention is

mediated by the mechanisms responsible for goal-directed spatially oriented actions. It

just happens to be the case in humans and primates that spatially-oriented actions

usually involve oculomotor orientation. Thus, one might expect animals, such as bats or

moles, which have different dominant modalities, to have their attention governed by

different sensorimotor mechanisms. However, in the case of humans, the dominance of

visuospatial mechanisms in supporting attention is an innate feature, since even

congenitally blind subjects seem to utilise visuospatial mechanisms for orienting

auditory attention.224 Thus, attention can be understood as a form of visuospatial action

even in cases that do not involve attending to visual stimuli.

A further problem for understanding attention as action arises from cases of

nonspatial attention. Thus far, all the cases covered involve directing attention towards a

particular spatial location. However, attention can also be directed at specific features or

specific objects.225 For example, when presented with a green triangle it is possible to

attend to either its greenness or its triangular shape and it is also possible to attend to it

as a whole object extended across multiple locations. Since these cases seem to involve

221 Driver & Spence (1994, 2004) 222 Rorden & Driver (1999), Rorden et al. (2002), Rolfs, Engbert & Kliegl (2005) 223 Rizzolatti, Riggio & Sheliga (1994) pg. 245 224 Garg, Schwartz & Stevens (2007) 225 Duncan (1984) Mazer (2011)

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more than merely directing one’s attention towards a specific location they could be seen

as problematic. In the case of the green triangle example, it intuitively seems as though

an action-based account of attention fails, since the same action is required to orient

one’s attention to the triangle’s greenness as to its shape. In response, it is first

important to note that some spatial attention is required. Attention to the triangle or it’s

features still requires one to direct one’s attention towards the region of space that it

occupies. Furthermore, it isn’t clear that attending to different features of a single object

involves just a single kind of attentional process. It is well established that our eyes are

constantly engaged in ballistic saccadic motion and undergo such motion, on average,

three to four times every second.226 As a result, patterns of saccadic motion are far more

complex than is suggested by the notion of static fixation upon a particular location.

Thus, whilst attending to the triangle’s shape and colour might both involve attending to

the same region of space, the way in which one attends to this region could be different

in each case. Attending to an object’s shape might involve different fine-grained patterns

of saccadic motion than when attending to the object’s colour. Thus, attending to the

shape and attending to the colour of a single object could be different actions. A similar

story can be told with regards to object-based attention, where one would expect

different patterns of saccadic motion when attending to a whole object rather than a

specific location. It is unlikely that differences in saccadic motion tell the whole story

with regards to different kinds of feature based attention. In particular some aspects of

feature based attention seem to also involve top-down neural signals which modulate the

excitability of feature-detecting neurons in lower level perceptual systems.227 However,

this needn’t threaten the account of attention as action. As in the case of covert attention,

the lack of movement associated with the modulation of perceptual systems is no reason

to deny that such processes are a form of action. Furthermore, feature based attention is

still primarily supported by the same neural mechanisms as those responsible for active

spatial attention.228

So far, the focus has been on the role of attention in guiding perception and

action with respect to the immediately available environment. However, it intuitively

seems as though one can also attend to cognitive entities that aren’t present in the

environment and are unperceivable in principle. For example, one can focus one’s

attention on a particular thought or memory and one can switch one’s attention from

226 Findlay & Gilchrist (2003), pg. 25 227 Treue & Martinez Trujillo (1999), David et al. (2008), Mazer (2011) 228 Mazer (2011)

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one idea to another. This form of attention has received very little focus in the empirical

literature, where attention is primarily considered as being related to perception and

action.229 As such, many would be reluctant to call this phenomenon attention at all.

However, in the current context of explaining enumerability in terms of attention, it

seems necessary to address this issue, since it is clear that we are able to enumerate more

than just the things that are immediately present in our perceivable environment. The

case of attention to merely cognitive entities seems particularly problematic for an

account of attention as action, since these entities seem to lack spatial location altogether

and so it is unclear how one could actively direct one’s attention towards them.

Given the relative poverty of empirical literature on the subject the response to

this problem will be somewhat speculative. However, there are plausible responses

available. Firstly, in the case of attention to memories and imagined scenes and objects,

mechanisms for spatial attention are still likely to be relevant, as attending to particular

locations within remembered or imagined scenes may use the same mechanisms that are

responsible for attention to real locations. This idea is supported by experiments

involving patients with hemispatial neglect, who lack the capacity to attend to a region of

space, as a result of a particular kind of neural lesion. When asked to describe a

particular square in Milan from a certain vantage point, these patients were only able to

describe features from half of the remembered scene but when they were asked to

imagine turning so as to bring the previously neglected features into the region without

neglect, they were able to describe the previously neglected features.230 This suggests

that the mechanisms responsible for spatial attention are also involved in attention to

memories and imagined scenes, since deficits in spatial attention are still present in

memory and imagination. Whilst this suggests that spatial attentional mechanisms are

responsible for attending to memorised and imagined scenes, which involve memorised

or imagined spatial locations, it is less obvious how it could apply to cognitive entities

such as beliefs, which seem to lack any kind of spatial element. In these kinds of cases it

is hypothesised that mechanisms for spatial attention are still utilised. Although beliefs

lack locations, attention to distinct beliefs could still be accomplished by simulating

them as having particular locations. If this were the case then switching attention from

one belief to another could still involve switching attention from one simulated location

to another. For example, one might imagine one’s beliefs as a spatially organised

sequence and shift attention from one position in the sequence to the next. Thus, despite

229 Mole (2013) §3.4 230 Bisiach & Luzzatti (1978)

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the apparently nonspatial nature of cognitive entities, attention to these entities could

still be accomplished by the mechanisms responsible for spatial attention and, thus,

could still be understood as a form of action.

Despite not necessarily involving any overt movement, paying attention is a form

of action, supported by the sensorimotor mechanisms that are responsible for orienting

sensory apparatus. Thus within a Gibsonian framework, part of what we do when we

perceive the world is to register opportunities for attentional actions. These attentional

affordances can be understood as some of the most basic affordances, since many kinds

of action depend upon being able to actively direct attention. For example, the ability to

grasp and manipulate an object depends upon being able to attend to the object in

question. Significantly for current concerns, our capacity for enumeration seems to

depend upon a capacity for attending to objects in sequence and, thus, is intimately

linked to our ability to perceive attendability.

Perceiving Enumerability

Once one accepts that it is possible to perceive affordances as basic as

attendability, this opens up room for an account of enumerative acts that both fits with

the Gibsonian account of perception and avoids the problems associated with grounding

enumeration in object manipulation. The key lies in taking enumerative acts to be acts of

sequential attention. Just as there are facts of the matter as to what an organism is able

to attend to, so too there are facts of the matter as to what sequences of attentional acts

are available to an organism. Thus, one can understand the perception of enumerability

in terms of the perception of opportunities for sequential attention. All that it is for

something to be a collection is to afford a certain kind of sequence of attentional acts.

This account allows us to define the notion of perceiving the cardinalities of

collections in terms of perceiving differing affordances for sequential attention. For

example, when we perceive the threeness of a collection of three entities, we perceive the

opportunity to engage in three separate attentional acts in sequence. We can rephrase

this by saying that we perceive the affordance of 3-ability. Our ANS provides us with

direct access to the specific affordance of enumerability of a given collection. In other

words, it allows us to perceive the collection’s n-ability. This is a vital ability as it allows

us to then coordinate our attentional acts so as to carry out more complex actions. For

example, in order to manipulate three objects and collect them into a pile, we must first

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perceive their 3-ability, and then we must selectively attend to each object in sequence,

and this forms part of the process of picking each object up and moving it. These distinct

processes do not need to be seen as temporally separate, as they might be embedded in

each other. However, they can be differentiated as a result of occupying different levels

in the hierarchy of affordances. Perceiving an opportunity for a particular kind of

manipulative act is dependent on perceiving 3-ability, which is in turn dependent on

perceiving instances of attendability.

It should be clear that such an account is able to account for Frege’s problem of

the indeterminacy of attributions of number to external collections. If the perception of

number is perception of affordances then it is correct that there is no fact of the matter

as to the objective cardinality of a given agglomeration when considered in isolation

from any particular organism. As has already been mentioned, affordances are always

organism-relative. However, for a given organism-agglomeration system in a specific

context there will be a fact of the matter as to the ways in which the organism can

sequentially attend to parts of the agglomeration. A single agglomeration may afford

different sequences of attention for different organisms in different contexts. For

example, a rabbit might be 1-able for most of us but be 8-able for a butcher in the

business of cutting up rabbits and selling off their separate parts.231 Once one takes into

account the wide range of organisms and the wide range of behavioural contexts that a

given organism can find itself in, the fact that a given agglomeration with a fixed material

extension can possess different numerical properties in different contexts is far less

mysterious.

The ANS is only able to provide consistently accurate perceptual representations

of affordances of 1-ability, 2-ability and 3-ability. However, in providing approximate

representations of n-ability for collections larger than three it can still serve the purpose

of constraining sequential attention. Seeing that a given agglomeration affords a process

of sequential attention with roughly seven loci of attention can still play a significant role

in determining action. If and when the approximations of the ANS go awry, this will

become apparent when the actual action of sequential attention is carried out.

Furthermore, by construing numerical perception as perception of affordances for

potential sequential attention, it is possible to explain why the ANS is limited in the way

that it is. The ANS must provide some representation of affordances for sequential

231 Returning to the example from Quine (1969), a butcher may often see rabbits as sets of undetached rabbit parts when they are in the process of trying to turn the given rabbit into a set of detached rabbit parts!

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attention in order for actions of sequential attention to be carried out. However, it must

also operate within the constraints of its own capacity. The majority of acts of sequential

attention relevant to our evolutionary ancestors’ survival in their natural environments

would have been likely to involve relatively small collections. The ANS is thus able to

consistently provide sufficiently accurate perceptual representations to guide sequential

attention in the kinds of cases that are most salient to an organism’s natural behaviour.

The idea that our perception of number is essentially detection of affordances of

sequential attendability receives independent support from neurological data about the

region of the brain where the ANS is generally held to be located. The hIPS, where the

ANS is usually held to be situated, forms a part of the intraparietal sulcus. There is a

wide range of evidence suggesting that one of the primary functions of this region of the

brain is the direction and coordination of spatial attention, in particular in the context of

the control of eye and hand movements.232 Imaging studies have demonstrated increased

activation in this region during spatial attention tasks.233 Evidence from lesion studies

also suggests that this region of the brain is crucial for spatial attention.234 It is important

to be cautious in inferring functional similarity from crude considerations of similarity in

neural location. However, the fact that the hIPS is part of a system that is largely

dedicated to spatial attention adds further independent support to the idea that the

primary function of the ANS is the coordination of sequential attention. The fact that this

region is also implicated in the guidance of eye and hand movements further supports

the idea that affordances of attendability lie at the base of the hierarchy of affordances.

Our capacity for sequential attention is required in order to carry out these more

complex actions. Thus, by focussing on the role of perceiving affordances of sequential

attendability, it may be possible to explain the role of more complex actions, such as acts

of object manipulation, which Mill and Kitcher took to be so central.

Object Manipulation as Epistemic Engineering

The problem with both Mill and Kitcher’s accounts is that they take object

manipulation to be constitutive of our apprehension of number, when in actual fact it is

better understood as an important aid to help facilitate this capacity. Mill argues that

number is the property of how collections can be separated into parts, thereby placing

232 Grefkes & Fink (2005) 233 Coull & Frith (1998), Goldberg et al. (2006) 234 Gillebert et al. (2011)

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object manipulability at the centre of the story of our access to arithmetical content.

Kitcher, on the other hand, takes object manipulation to be the paradigm and

developmentally fundamental case of enumerative activity. Both are right to emphasise

the important role that object manipulation plays in our apprehension of number.

However, they are wrong to see such manipulations as the fundamental or original

source of our arithmetical knowledge.

Thus far, it has been argued that our access to numerical content is primarily

mediated by perceptual processes. In many cases it may be possible to perceptually

apprehend the number of entities in a collection without the need for manipulation of

the given collection. In light of this, rather than seeing object manipulation as essential

to our access to arithmetical content, it makes more sense to see the practice of

manipulating collections as an important tool for rendering perceptual access to

arithmetical content as cognitively tractable. Our capacity to perceptually apprehend the

number of entities in a collection is constrained by the limitations of our natural

cognitive mechanisms. The ANS is only capable of reliable apprehension of number in

the case of relatively small collections. Furthermore, our capacity for more precise

apprehension of number mediated by actually carrying out processes of sequential

attention is limited by constraints on our capacity for short term memory. Our limited

memory means that in the case of larger collections it will often be difficult to keep track

of which members of a collection we have already attended to and, as such may end up

with an incorrect assessment by mistakenly attending to the same member twice.

Manipulating the spatial arrangement of objects in a target collection allows one

to lighten the cognitive load and overcome these limitations, by rendering a previously

overly demanding task as achievable using the basic perceptual capacities already

mentioned. Object manipulation is thus best seen as a form of ‘epistemic action’,

whereby the environment is manipulated in order to render a previously tricky task as

cognitively tractable.235 By arranging objects into certain kinds of spatial configurations

we can lighten the cognitive loads on our memory by delegating some of the memory

tasks to the environment. For example, when attempting to apprehend the number of

entities in a collection through sequential attention, one can place the objects that have

already been attended to in a spatially separated region from those that are still to be

attended to, in order to provide a simple perceptual means for avoiding the problem of

attending to the same object twice without having to rely on memory. Thus, whilst object

235 Kirsh & Maglio (1994), Kirsh (1995), Clark (2008) pg. 68-73

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manipulation plays a significant role in rendering perceptual access to number possible,

it is perceptual access that is fundamental.

Given the fact that most attempts to apprehend number would be cognitively

intractable without this kind of epistemic engineering, it is easy to see why both Mill and

Kitcher give such a prominent role to object manipulation in their stories of our access to

arithmetical knowledge. This move makes even more sense once one appreciates that

processes of perceptual apprehension of number and of object manipulation are often

difficult to disentangle. Perceiving the sequential attendability of a collection is a

requirement for both apprehending the number of entities in a collection and

performing the kind of object manipulations that make this very task more tractable.

One must attend to an object in order to manipulate it. As such, the act of apprehending

number is often likely to be embedded in the act of manipulation. However, this should

not be allowed to obscure the fact that it is the former that is the fundamental source of

our precise apprehension of number and that the latter is merely a means for making

such a process tractable.

Spatially segregating objects in order to aid sequential attention is not the only

kind of epistemic action relevant to our acquisition of arithmetical content. We are also

able to engineer our environments so as to render overly challenging feats of numerical

comparison achievable using basic perceptual capacities. It is clear that the limitations

of the ANS and of our memory capacity mean that we will often lack the required

accuracy and precision to reliably discern the larger of two collections or to assess their

equinumerosity. However, by spatially arranging the collections to be compared by

pairing each member of one collection with one of the other and then arranging the

collections in a line, one can easily apprehend the larger of the two collection using basic

perceptual capacities. Furthermore, one can then go on to perceptually apprehend the

difference between the collections using the kinds of perceptual procedure already

described. In this case, as with the first, it is clear that object manipulation plays a key

role in making the tasks at hand cognitively tractable in the face of our limited capacities.

However in each case the ultimate goal is to facilitate the perceptual apprehension of

number.

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What is the Metaphysical Status of Numbers as Affordances?

Having argued that numerical perception involves the perception of affordances,

the immediate question arises as to what the metaphysical status of affordances is. The

most straightforward answer would seem to be that affordances are objective properties.

In perceiving affordances we detect some objective feature of the world. This would fit

with an orthodox account of the nature of perception. However, in the case of

affordances the issue is slightly trickier, due to the fact that affordances are always

organism-relative. For some, this might be seen to conflict with the idea that real

objective properties are mind-independent. Various proponents of a Gibsonian approach

have offered differing metaphysical accounts of affordances as objective properties, two

of which will be assessed below. If either of these approaches is viable then numerical

perception can be understood as the detection of real physical properties. However, even

if one denies the objective existence of affordances, it may be possible to put forward an

account whereby our arithmetical knowledge originates from perceiving affordances.

Before addressing these options it is worth addressing an immediate worry that might be

raised regarding the metaphysical status of affordances.

The idea that we can perceive affordances may seem to fly in the face of some

relatively widespread metaphysical assumptions. One of the central claims is that we are

able to perceive opportunities for possible actions. However, intuitively, it seems as if we

can only sense that which is actual and not that which is merely possible. For example,

McGinn has argued that ‘you do not sense modalities with your sense modalities. You do

not see what would obtain in certain counterfactual situations; you see only what

actually obtains’.236 At face value this intuition seems to present a problem for the idea

that we always and only perceive what actions we could do, prior to any such action

taking place and even in cases where the action in question never becomes actualised. It

would be somewhat worrying if an account of numerical perception required one to

explain how we are able to perceive the contents of merely possible worlds. In order to

avoid this worry, it is necessary to make an important distinction between the states of

affairs that our perceptual states respond to and the states of affairs that our perceptual

states represent.237 Whilst it might be the case that our perceptual states cannot respond

to or be caused by mere possibilities, this does not imply that these states are incapable

of representing possibilities. It is possible to perceptually represent something as being a

236 McGinn (1996) pg. 540 237 Nanay (2011) pg. 300-303

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certain way without representing what causes it to be that way. For example, ‘we can

perceive something as being cold, without perceiving it as having a certain kinetic

energy’.238 Thus, similarly we can represent an action as being possible without

representing that which grounds this possibility. Furthermore, our representation of the

possibility of a given action can be caused by features of the actual world. For example,

an organism’s perceptual representation of the climbability of a staircase might be

caused by physical features of the staircase and physical features of the organisms own

body. As such there is nothing particularly mysterious about perceptually representing

modality, since such perceptual representations need not be caused by aspects of merely

possible worlds.

One potential option for the realist about affordances is to conceive of them as

dispositional properties.239 On such an account the arithmetical affordance of a given

collection would be a dispositional property of the collection to elicit counting behaviour

in organisms with the right kind of behavioural capacities. Thus, the collection could be

seen to possess the dispositional property even if no animals with the capacity to engage

in such behaviour actually existed. Arithmetical affordances could then be seen as

objective mind-independent dispositional properties of collections. It might require the

presence of an animal with a mind to encounter the collection in suitable circumstances

for this disposition to become manifest but this needn’t threaten the mind-independence

of the dispositional property itself.

Whilst this kind of approach is appealing, in that it locates numerical properties

as objective dispositional properties of collections, it faces a major problem. On most

accounts of dispositional properties, a disposition will become manifest whenever it is in

the presence of the conditions for its manifestation, as a result of the laws of nature.240

For example, since salt possesses the dispositional property of solubility it will dissolve

whenever it is in contact with a solvent, such as water, as a result of the relevant

chemical laws. This aspect of dispositions is fully embraced by those who favour a

dispositional account of affordances. For example, Turvey argues that ‘dispositionals

never fail to be actualised when conjoined with suitable circumstances’.241 However, such

an approach seems to immediately lead to problems, since it seems as though an

organism could be in a situation that affords a certain action and yet fail to actually carry

238 Ibid. pg. 305 239 Turvey (1992) 240 Choi & Fara (2012) 241 Turvey (1992) pg. 178

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out the given action. At any one time an organism will encounter a vast range of different

affordances, some of which will be mutually exclusive.242 For example, an apple might

afford eating and afford throwing but cannot be both eaten and thrown. As a result, it

will be impossible for all such affordances to become actualised despite the apparent

presence of suitable circumstances for their actualisation. This is problematic for the

dispositional account, as it would seem that only those opportunities for action that end

up being acted upon deserve the status of affordance, despite the fact that before any

action takes place the various opportunities seemed to be on a par. One way of avoiding

this problem is to provide a far more detailed specification of the circumstances relevant

to an affordance’s actualisation. For example, an apple could be said to afford eating only

in the circumstances where the organism in question is sufficiently hungry and where

there are no other actions that are more important to the organism to engage in at the

given time. However, it is easy to see that such an account would quickly become

extremely complex and involve relations between circumstances and affordances that

most would be hesitant to call laws of nature. There are many good reasons to want an

account of affordances that allows for the possibility of unactualised affordances and,

since this is incompatible with the dispositional account, such an account should be

dismissed.

One of the main benefits of the dispositional approach was that it rendered

affordances as objective properties of the external environment, which could be said to

exist even in the absence of any organisms to perceive or fulfil them. However, the

objectivity of affordances need not rest upon their existence being independent of the

existence of organisms. An alternative approach is to conceive of affordances as relations

between a certain kind of organism and features of its environment.243 On such an

approach, arithmetical affordances would be relations between the physical properties of

aspects of the environment that underlie their attendability and the properties of

organisms that render them capable of engaging in acts of sequential attention. Thus,

affordances are not properties of the external environment but properties of organism-

environment systems.244

A somewhat counterintuitive upshot of this position is that the existence of

collections, construed as affordances, depends upon the existence of organisms capable

of sequential attention. If there were no organisms capable of perceiving number then, in

242 Stoffregen (2003) pg. 119 243 Stoffregen (2003) pg. 122, Chemero (2009) pg. 140 244 Stoffregen (2003) pg. 122

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a certain sense, there would be no such thing as number. Despite the seemingly

counterintuitive nature of this consequence, it is important to highlight that this does not

render number a merely subjective property. The existence of arithmetical affordances

depends upon the existence of certain physical feature and certain organisms but is

entirely independent of the subjective mental states of the organisms in question. For

example, a given collection might afford sequential attention for a given organism,

regardless of whether the organism has ever encountered the collection and regardless of

whether the organism will ever perceive the given affordance. Organism-dependence and

mind-dependence are very different, where only the latter is a sign of subjectivity.

Another somewhat counterintuitive consequence of construing affordances as organism-

environment relations is that it leads to a vast proliferation of relations. For any one

organism there will be a vast multitude of relations between it and every part of the

environment that affords some kind of action. Even if one only focuses on arithmetical

affordances, the quantity of relations between a given organism and a quite restricted

region of the environment will be unimaginably vast, since there are a huge variety of

ways in which an organism can selectively attend. In order to maintain a view of

affordances as objective properties, it is necessary to accept that they are extremely

unorthodox organism-dependent entities and that they do not leave one with a clean,

simple and parsimonious ontology. However, these might be prices worth paying in

order to allow for a simple explanation of the perception of numerical properties as

perception of objective features of the world.

For some, however, these prices will be far too high. Fortunately, however, it is

possible to give an account of perceptual access to numerical properties that does not

depend upon viewing affordances as objective properties. All that is required for the

current project of explaining our perceptual access to numerical properties is to explain

how perceptual processes give rise to mental states with affordances as their content.

Thus, one could maintain that arithmetical affordances are the content of perceptual

states without committing to their existence as objective properties of external objects or

as objective relations of organism-environment systems. To see how this could be fruitful

it is worth considering a comparison with the case of colour. As with affordances, some

have offered dispositional or relational accounts of colour.245 However, some have also

offered accounts of colour where it is explicitly seen as a subjective, mind-dependent

245 E.g. Johnston (1992) and Levin (2000) advocate dispositionalism, Averill (1992) and Cohen (2009) advocate relationalism.

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property.246 Despite the fact that such theorists do not take colour to be a mind-

independent property, they would likely agree that our beliefs about colour derive from

our perceptual processes. It would be strange to argue against the view that knowledge of

colour is in some sense dependent upon perception. Thus, a property’s mind-

independence and objectivity need not be seen as necessary conditions for it being

represented perceptually. As with the case of colour, the fact that we seem to perceive

affordances might merely reflect the way in which our minds work. We might represent

the world as being populated by opportunities for action despite the fact that such

opportunities cannot be understood as real objective features. As with the case of colour,

one could argue that arithmetical affordances are mind-dependent properties and yet

still subscribe to the idea that our knowledge of such properties is ultimately dependent

upon perceptual processes.

An important upshot of this is that one can provide an account of perceptual

access to arithmetical affordances that is, to some extent, independent of the ontological

status of affordances. If one is willing to accept the strange metaphysical consequences

of a realist view of affordances then one can see numerical properties as real features of

organism-environment systems and thereby understand numerical perception as the

perceptual detection of these features. Thus, this account of numerical perception is

compatible with an unorthodox form of realism, according to which at least some

arithmetical properties are features of the physical world. However, there is room for

seeing arithmetical affordances as merely subjective features of our experience that we

project onto the world, whilst maintaining that our knowledge of numerical properties

originates from perception. On this approach, numerical perception could be compatible

with a nominalist perspective that denies the existence of arithmetical properties. Thus,

the idea that our arithmetical knowledge originates from perceiving affordances, whilst

controversial and counterintuitive in a number of ways, can be seen as ontologically

neutral.

Perceiving Numerical Affordances

There is a sense in which Frege is right to ridicule Mill for offering a view on

which it seems impossible to acquire knowledge of the number in cases where

246 Wright (2003)

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‘everything in the world is nailed down’.247 There are many cases where we can

apprehend number regardless of the manipulability of the objects in the target

collection. However, there is an important sense in which, despite Frege’s intention to

convey an absurdity, arithmetical knowledge would be impossible in a world in which

everything is static. This is not because we cannot count immovable objects but because

we cannot apprehend number without moving ourselves. Since these capacities are

clearly central to the possibility of apprehending number, the dependence of our

numerical perception on our capacity for motion should not come as a surprise. If we

were not able to move and to interact with our environment, even in the most minimal

sense of interaction in terms of shifting attention, then arithmetical knowledge might

indeed be impossible.

Humans and other animals are endowed with an innate capacity to perceive

opportunities for sequential attention. This capacity plays a vital role in enabling our

interactions with the world. It is also this capacity that allows us to perceive the

numerical properties of collections. In essence all that it is for something to be a

collection is to afford sequential attention and all that it is for a collection to possess a

particular numerical property is for that collection to afford a certain kind of pattern of

sequential attention for a certain kind of organism. Thus numerical properties need not

be seen as mysterious abstract entities that only exist in a remote platonic realm. At the

same time, one need not thereby be forced into viewing numerical properties as mere

inventions or constructions of the subjective mind. Numerical properties can be

understood as objective features of organism-environment systems or as aspects of the

way that we perceive the world and, as such, our access to these properties can be

understood in terms of perception.

247 Frege (1960) pg. 9

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4

Embodied Numerical Cognition

Thus far the focus has been on our innate capacity for numerical perception. At

least some of our arithmetical knowledge originates from this basic capacity. However,

most of our engagement with arithmetic involves going beyond immediate perception.

When we engage in arithmetical reasoning, we are rarely in the presence of collections

that instantiate the numerical properties that we are reasoning about. Furthermore,

much of our mathematical reasoning transcends the capacities of our innate numerical

perceptual system. We can think about numerical values that transcend the practical

limitations of sequential attention. Furthermore, whilst the ANS is only able to provide

approximate representations, our capacity for arithmetical reasoning is paradigmatically

rigorous and precise. In order to understand the origins of arithmetical knowledge, it is

necessary to provide an account of the nature of number concepts and numerical

cognition.

Given the supposedly abstract nature of mathematical entities, it is natural to

suppose that number concepts are somewhat divorced from experience. Mathematical

reasoning, on such an account would involve manipulation of purely cognitive

representations according to abstract computational processes, untainted by the messy

practicalities of everyday perception and action. In what follows, this traditional view of

arithmetical reasoning as transcending everyday experience will be challenged, in favour

of a view according to which number concepts and numerical cognition are embodied.

Embodied Cognition

The embodied cognition movement is an emerging research program in the

cognitive sciences, encompassing a wide range of theoretical claims, which, whilst closely

related to one another, are yet to cohere as a single unified theory.248 The main thing that

248 Shapiro (2011) pg. 3

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unites these various strands is their opposition to the previously prevalent cognitivist

paradigm. In order to draw out the specific sense of embodied cognition relevant here, it

will be helpful to provide an account of traditional cognitive science with which to

contrast it.

Traditional cognitivism stems from the idea that cognition can be understood in

terms of computation and, hence, has often been dubbed the Computational Theory of

Mind (CTM). There are three commitments of this theory that are of particular

significance. Firstly, mental states, such as concepts, are taken to be symbols in some

kind of language of thought.249 Like words, these symbols possess both syntactic and

semantic properties and can be understood to represent states of affairs in the world.

Moreover, as in the case of words, the structure of the symbols need not bear any

relation to that which they represent. The symbols’ structures are arbitrary with respect

to their semantic content. Secondly, cognition can be defined in terms of computation, in

the sense that it involves the combination and manipulation of symbols according to

formal rules determined by the syntactic properties of the symbols. Thirdly, perception,

cognition and action are taken to be three distinct and separate kinds of process, each

with its own specific form of representation. On such a view, perception provides the

inputs to cognition, allowing for processing that leads to changes in cognitive states and

possibly also to action as output via the motor system.

At face value, mathematical concepts and mathematical cognition seem to fit very

nicely with the traditional picture of cognitive science. Turing’s pioneering work on the

notion of a Turing machine, which provided the impetus for the development of CTM,

was originally focussed on the issue of numerical computation.250 Furthermore, the kind

of proof and calculation conducted by mathematicians using pen and paper closely

mirrors the kind of inferential processes that CTM posits as going on inside the head.251

Operations are constrained by rules that are only sensitive to syntactic properties and

thereby allow for rigour and truth-preservation. The idea of concepts as arbitrary

symbols also fits nicely with a traditional view of mathematical entities. Contrary to the

claims of the last two chapters, mathematical entities have generally been considered as

249 Fodor (1975), Pylyshyn (1980) 250 Turing (1936). However, some (Wells (2002), Barrett (2011)) have argued that Turing’s original work was far closer to the picture suggested by embodied cognition than most people tend to assume, since Turing was attempting to formally describe how human “calculators” performed computations using their bodily interactions with external media, such as pens and paper. 251 It should be noted that very few mathematicians actually write down all the steps of their formal proofs (with notable exceptions being the foundational projects of Frege and Russell). Informal proofs are by far the norm. However, even informal proofs and elementary calculations can be understood in terms of the manipulation of symbols according to specific algorithmic rules of thumb.

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abstract and thus unperceivable, leading to questions as to how they could be

represented in the mind. The positing of symbols that bear no structural similarity to

that which they represent opens up the possibility that one could have symbols that

represent abstract entities.

Embodied cognition (EC) can be seen to challenge aspects of all three of the

central claims of the traditional approach. Firstly, EC challenges the idea that

perception, action and cognition are three separate and distinct kinds of process.252

Understanding perception and action as separate processes is problematic, since

perception is best understood as an inherently active and action-oriented process.

However, the embodied cognition approach also challenges the idea that cognition can

be understood as separate from perception and action. The idea that cognition uses its

own distinct form of symbolic representations is rejected and replaced with the idea that

cognition shares the same representational resources as perception and action.

Conceptual representations are taken to be modal as opposed to amodal symbols, since

they utilise the same representational resources as the sensory modalities. Concepts are,

in part, taken to be constituted by reactivation of the sensory and motor systems that are

activated by encounters with the referent of the given concept.253 This strand of the

embodied cognition movement is often referred to as Concept Empiricism.254 This name

derives from its endorsement of a position similar to that of traditional Empiricists

according to which thoughts and ideas are constituted by ‘less forcible and lively’ copies

of sensations, movements and emotions.255 Any talk of embodiment or EC, from here on,

can be taken as referring to the essential claim of Concept Empiricism, that cognition is

accomplished using only representational resources that are also involved in perception

and the guidance of action.256

The notion of perceptual representation used here is broader than an orthodox

view of perception might suggest. Modal representations go beyond the traditional

252 Hurley (1998) pg. 401-402 253 Barsalou (1999) 254 Prinz (2002) pg. 108 255 Hume (1999) pg. 97. However, where the traditional empiricists emphasised the phenomenological similarities between experience and thought, Concept Empiricists instead emphasise commonalities between the vehicles of representation utilised for perceptual, motor and cognitive processes. Furthermore, where traditional Empiricism was committed to the claims that all content is ultimately derived from experience, Concept Empiricism is compatible with some representations being innate, albeit innate perceptual or motor representations. 256 The claim that cognition is accomplished using perception and action based representations is endorsed by a wide range of proponents of the wider embodied cognition movement, some of whom might be reluctant to call themselves proponents of Concept Empiricism. Although the account on offer endorses the central claim of Concept Empiricism with respect to number concepts, there are other aspects of particular Concept Empiricist accounts, such as Prinz (2002), that are not endorsed, lending further reason to avoid reference to the theory on offer as Concept Empiricism.

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Aristotelian taxonomy of the five sensory modalities.257 For example, they also include

proprioceptive representations of the positions of bodily parts, interoceptive

representations of internal bodily states such as hunger or fatigue, representations of

emotional states and motor representations responsible for the coordination of actions.

Furthermore, given the significance of action for perception it may be necessary to blur

the boundary between what have traditionally been considered as separate perceptual

and motor systems.258 The question of how best to divide up and individuate distinct

sensorimotor systems is a tricky one. It may turn out that multiple viable divisions

suggest themselves, with each one being suited to a different level of analysis.

Thankfully, however, settling upon the correct taxonomy of sensorimotor systems is a

desirable but not necessary goal for EC. All that EC requires is that the representational

resources that are involved in perception and the guidance of action are also used to

support cognitive processes and that these are the only resources involved. In other

words, cognition involves no purely cognitive amodal representations.

A consequence of blurring the boundaries between perception and action on the

one hand and cognition on the other is that mental states, such as concepts, can no

longer be understood as merely arbitrary symbols. According to the embodied cognition

approach the structure of a mental representation is intimately related to the kinds of

perceptual and motor activities associated with its genesis and use. Due to the nature of

perceptual and motor processes, the representations that are used are by no means

arbitrary with respect to their content. Since conceptual representations are taken to

reuse the same representational resources as perceptual and motor systems, they inherit

this lack of arbitrariness. This reconstrual of the nature of cognitive representations also

has implications for the idea that cognition is simply computation according to formal

rules. In denying the arbitrary nature of mental symbols, proponents of embodied

cognition erase the clear divide between syntactic and semantic properties. Thus, one

would expect cognitive processing to be affected by the content of the representations

being processed, often in ways that go against what would be expected if cognition were

merely transformation according to formal rules.

It is important to differentiate the account on offer here from other claims of

embodied cognition. Some take embodied cognition to be the claim that cognition is

constituted by non-neural bodily processes. The issue of what constitutes the mind is

257 Barsalou (1999) pg. 585, Prinz (2002) pg. 120-122 258 Gibson (1966) pg. 56-57, Hurley (2001)

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somewhat tangential to the main issue at hand, since an account of numerical cognition

that diverges from the traditional cognitivist approach can be developed without

challenging traditional assumptions about the boundary of the mind. The main claim

here is that the neural processes that underlie numerical cognition are closely tied to

perceptual and motor processes. Some might wish to include the bodily manifestation of

such processes as part of the cognitive system. However, one can subscribe to the main

claim of EC endorsed here without making any revolutionary claims that involve

extending the boundaries of the mind. It is also important to distinguish EC from what

has become known as Radical Embodied Cognition.259 Proponents of this position agree

that cognitive processes use the same systems as perception and action. However, they

argue that, due to the dynamical nature of these systems, such an interpretation obviates

the need to posit mental representations.260 The question of whether the kinds of states

that are posited by the forthcoming account qualify as representations is a thorny issue.

It is clear that if the definition of representation is simply lifted from the traditional

approach to cognitive science then no entities of this kind will be found in the newer

embodied approach. However, this would have little to do with the concerns regarding

dynamical systems that motivate the rejection of representations. Moreover, there are

many reasons to think that the dynamical systems approach is compatible with the

notion of representation.261 In what follows reference to representations will be

maintained.262

The aim here is not to defend EC as a general theory of concepts and cognition.

There may be aspects of cognition for which this approach is explanatorily unsuitable.

The aim is to argue that EC provides the best framework for understanding the nature of

numerical cognition. Number concepts, in particular are argued to be constituted by

perceptual and motor representations and, thus, it is claimed that there are no purely

cognitive representations of number. However, given that concepts for supposedly

abstract entities, such as numbers, are taken to be particularly problematic for the EC

259 Clark (1997) pg. 148, Chemero (2009) 260 Freeman & Skarda (1990), Van Gelder (1995), Chemero (2009) 261 Bechtel (1998), Prinz & Barsalou (2014) 262 This is not because the relevant arguments depend on taking a particular side in the debate about whether

representations should be eliminated. For those who reject the notion of representation, the main point, that

cognition utilises the same resources as perception and action, should still be valid. If the reader is more inclined

towards an anti-representationalist approach, they are invited to substitute mention of number concepts or

representations with talk of the non-representational processes that underlie numerical cognition.

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approach, the arguments presented here could also serve the purpose of providing a

partial defence of EC as the right way to understand cognition in general.263

There are two main reasons for adopting an EC account of number concepts.

Firstly, it is arguably the case that the ANS plays a significant role in numerical

cognition. Given that the ANS is best understood as a perceptual system, this would

suggest that our number concepts are at least partly constituted by perceptual

representations. It is pretty clear that ANS representations alone are not sufficient to

explain our capacity for precise and rigorous numerical cognition. However, the second

reason for adopting an EC approach is that the resources required to augment these ANS

representations so as to allow for the development of sophisticated number concepts are

also drawn from perceptual and motor systems. Thus, in line with the predictions of EC,

number concepts are taken to be constituted by perceptual and motor representations

from the ANS and other perceptual and motor systems.

From Numerical Perception to Numerical Cognition

It should be clear from the arguments in chapter two that the ANS provides us

with some perceptual representations of number. However, it should also be clear that

these representations are not sufficient to explain our capacities for numerical cognition.

Mathematical reasoning is characteristically precise and rigorous. However, the

perceptual representations from the ANS are, in most cases, inherently fuzzy and

approximate. If our number concepts were solely constituted by these perceptual

representations then, for example, we would not be able to reliably distinguish FIFTY-SIX

from FIFTY-SEVEN. Thus, questions arise as to whether these perceptual representations

play a role in numerical cognition and, if they do, as to what more is required in order to

render numerical concepts sufficiently precise to allow for rigorous reasoning.

The issue of whether perceptual representations from the ANS play a role in

numerical cognition is a matter of contention. It is possible to distinguish three different

approaches to the question.264 The non-nativist approach suggests that innate number-

specific systems, such as the ANS, play no role in sophisticated number concepts and

that, instead, these concepts are acquired as a result of more general purpose learning

263 Machery (2007), Dove (2009) 264 Laurence & Margolis (2007) pg. 139

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mechanisms.265 For example, some argue that we acquire number concepts by

developing an implicit understanding of arithmetical axioms through the use of our

innate logical concepts and deductive capacities.266 Weak nativists, on the other hand,

suggest that innate number-specific systems, such as the ANS, play some role in the

acquisition and constitution of number concepts but that these innate systems are

insufficient for the possession of number concepts. As such, weak nativists argue that

some form of learning is required for the acquisition of any fully-fledged number

concepts.267 Finally, strong nativists argue that some of our number concepts are

innate.268 They argue that, as well as possessing innate systems for the perception of

number, we possess a distinct ‘innate number module’, which contains some innate

conceptual representations of number.269

In order to investigate the acquisition of number concepts, it is necessary to set an

agreed standard as to what capacities are required for attributing number concepts to an

organism. It is widely agreed that possession of number concepts entails some degree of

understanding about the nature of the positive integers. One might argue that this

requires some sort of appreciation of the axioms of Peano Arithmetic that characterise

the structure of the positive natural numbers.

1 is a number.

If a is a number, the successor of a is a number.

1 is not the successor of a number.

Two numbers of which the successors are equal are themselves equal.

If a set S of numbers contains 1 and also the successor of every number

in S, then every number is in S.270

However, to demand that organisms possess an explicit representation of these axioms

seems like a step too far. The axioms were formulated relatively recently, in the late 19th

265 E.g. Piaget (1952) and Rips, Bloomfield & Asmuth (2008). It is important to note that non-nativists do not deny that our number concepts arise as the result of innate mechanisms; they merely deny that they arise from innate mechanisms that are specific to number. Laurence & Margolis (2007) refer to this position as empiricism, however, this label is somewhat misleading, since these theorists do not believe that our number concepts are solely the result of experience and are also potentially opposed to the Concept Empiricist position according to which number concepts are partly constituted by perceptual representations. 266 Rips, Bloomfield & Asmuth (2008) pg. 638 267 E.g. Dehaene (1997), Spelke (2003), Carey (2009a, 2009b) 268 E.g. Gelman & Gallistel (1978), Laurence & Margolis (2007) 269 Ibid. pg. 145 270 Weisstein (2014) (Most modern formulations of Peano’s Axioms take zero to be the first number. However, taking one to be the first number has no detrimental effect and is in line with Peano’s original formulation (Peano (1973) pg. 113). Furthermore, it seems as though an appreciation of zero should not be a necessary condition for the possession of number concepts.)

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century, and it would be absurd to suggest that all of the great mathematicians that came

before Peano lacked number concepts.

Rather than explicit representations of the axioms, number concept possession

depends on possessing representational resources that implicitly conform to the axioms.

The best evidence that we have for an organism’s implicit mastery of these axioms is

their successful engagement in counting procedures. Engaging in these procedures

requires that subjects have a unique way of representing each cardinal value and that

they have a way of representing the relation characterised by the successor function. The

ability to engage in counting procedures depends upon the appreciation of five ‘counting

principles’.271

i. The One-One Principle:

There is a one-one correspondence between the items to be counted

and the distinct representations used to count them.

ii. The Stable-Order Principle:

The representations used for counting must be arranged in a stable

order.

iii. The Cardinal Principle

The final representation assigned in a count procedure corresponds to

the cardinality of the counted collection.

iv. The Abstraction Principle

Counting principles (i)-(iii) apply to any collection of entities.

v. The Order-Irrelevance Principle

The order in which representations are assigned to items in an array is

irrelevant.

Any organism that is able to carry out counting procedures in line with these principles

can thereby be understood as possessing number concepts. In practice, the best evidence

that we have for mastery of the counting principles comes from studying the

development of children’s ability to count using numerical language. However, it is

important to note that the ability to engage in linguistic counting procedures is merely

indicative of number concept possession and might not be necessary for number concept

possession. All that is required for number concepts is the possession of some system of

representations that can satisfy the counting principles, and this could potentially be

271 Gelman & Gallistel (1978) pg. 77-82

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achieved by an organism that lacks linguistic capacities. Neither Peano’s axioms nor the

counting principles are intended as an exhaustive analysis of number concepts. Number

concepts are likely to be far richer than axioms or principles of this kind could ever

capture. Instead, implicit capturing of the axioms through compatibility with the

counting principles is supposed to be taken as a minimum threshold for the attribution

of number concepts. The question of whether innate number-specific systems are

involved in numerical cognition can thus be framed in terms of whether these systems

provide sufficient representational resources to support principled counting procedures.

The EC approach to number concepts on offer here is a form of weak nativism. It

is opposed to the non-nativist approach in that it suggests that number-specific

representations from the ANS play a significant role in the constitution of number

concepts. However, it is also opposed to strong nativism, since it denies any purely

cognitive innate representations of number that reside in a number module distinct

from perceptual systems. Whilst many agree that weak nativism is the right approach,

the EC perspective provides a more detailed picture of exactly how learning can enable

the combination of ANS representations with other perceptual and motor

representations so as to produce fully-fledged number concepts. Before going into more

detail about the specifics of this EC approach, it is first important to present the general

case for weak nativism and to defend this approach against opposing views.

Weak Nativism: The “Bootstrapping” Hypothesis

The most detailed weak nativist account of the development of number concepts

has been proposed by Carey.272 According to this account, our innate number-specific

mechanisms are insufficient to explain our capacities for numerical cognition. However,

these innate systems are held to play a significant role in forming sophisticated number

concepts. Carey takes both the ANS and the OTS to be innate systems that support our

numerical capacities. Representations from the ANS are insufficient for numerical

cognition for two reasons. Firstly, they fail to capture small numerical differences

between large collections of objects. Secondly, they fail to capture a unique relation that

corresponds to the successor function. As such, ANS ‘representations are not powerful

enough to represent the natural numbers and their key property of discrete infinity’.273

272 Carey (2009b) 273 Ibid. pg. 295

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OTS representations are also held to be insufficient for two reasons. Firstly, they

represent number, at best, only implicitly. Secondly, they are only capable of implicitly

representing the number of objects in collections of 1-3 or 1-4 objects.

Since neither of the innate systems is sufficient for the possession of number

concepts, there must be a significant developmental stage which enables the

development of sophisticated number concepts. Furthermore, this stage cannot merely

involve the combination of representational resources from systems already present.

Instead it involves the creation of a new kind of representation from the basis of innate

representational resources that, at the same time, transcends the capacity of these

resources. Carey refers to this kind of process as Quinean “bootstrapping”, whereby ‘the

structure one builds consists of relations among the concepts one will eventually

attain’.274 A crucial stage in this “bootstrapping” process is the acquisition of numerical

language, since, for bootstrapping to take place, it is necessary that the subject has access

to explicit symbols that can serve as placeholders or scaffolds for as-yet-unformed

concepts to form around.

To see how this bootstrapping process might work in practice it will be helpful to

consider how number concepts might develop from the basis of ANS representations and

the acquisition of numerical language. It is hypothesised that children first learn the

arbitrary list of number words and the counting routine ‘without recognising the

numerical significance of these activities’.275 They then develop associations between

representations of number words and ANS representations, by noticing an analogy

between later steps in count lists and larger magnitude ANS representations. This then

allows the child to ‘come to the induction that each number word is associated with a

different numerical magnitude and that larger magnitudes correspond to words that

come later in the count list’.276 This goes some way in explaining how number concepts

could be “bootstrapped” from ANS and linguistic representations. However, more needs

to be done to show how children could come to represent the successor function. This

can potentially be achieved by the child carrying out a further induction. They first notice

that the ANS representation that corresponds to the word “two” is achieved when the

ANS representation corresponding to the word “one” is added to the ANS representation

corresponding to the word “one” and notice that the relation amongst corresponding

274 Ibid. pg. 306 275 Ibid. pg. 309 276 Ibid. pg. 312

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ANS representations is similar for other small numbers.277 They are then able to reason

inductively that there will be two ANS representations corresponding to n and n+1 for

any n, despite the fact that the ANS alone might not discriminate between larger

numbers.278 The acquisition of representations of numerical language can thus enable

the development of number concepts whose representational capacity transcends that of

the ANS on its own.

Carey rejects the idea that our number concepts result from “bootstrapping”

using ANS representations, since evidence suggests associations between ANS

representations and number words develop later than children’s successful deployment

of number concepts in line with the counting principles.279 Furthermore, she argues that

an account based on “bootstrapping” from ANS representations fails to explain a

‘striking discontinuity’ between performances when dealing with collections of 1-4

objects and when dealing with larger collections.280 As a result, an alternative is

proposed, whereby number concepts are initially “bootstrapped” from representations of

number words and OTS representations, which only later become associated with ANS

representations. According to this account children again start from learning the count

list without associating it with any numerical content. They then learn to distinguish

cases of singular and plural reference, which allows them to differentiate states of the

OTS with a single file from those with more and also to appreciate that plural words

might refer to collections that are too large to be represented by the OTS.281 They are

then able to notice that number words apply in cases where items in an array can be put

in one-one correspondence with the mental files that constitute particular states of the

OTS. This allows the child to recognise an analogy between the next item in a numeral

list and the state of the OTS that would result from adding another individual to the

array under consideration. The child is thus in a position to appreciate that the last word

on the count list indicates the cardinal value of the collection being counted and, since

the child has already mapped representations of individuals to the word “one”, has the

capacity to represent the successor function in terms of adding one individual to a

collection.282

277 Ibid. pg. 313 278 Le Corre & Carey (2007), Condry & Spelke (2008) 279 Carey (2009b) pg. 313-318 280 Ibid. pg. 318 281 Ibid. pg. 325 282 Ibid. pg. 327

116

There are a number of reasons to doubt whether this alternative approach is

viable. The main problems arise from the fact that the OTS fails to explicitly represent

number. The account rests on the idea that the child can notice the one-one

correspondence between items in an array and files in the OTS. However, it is unclear

how this process of “noticing” is supposed to take place. Each mental file in the OTS

represents a particular item in the array. However, the child lacks a representation of the

overall state of the OTS. ‘Having one, two or three such active [OTS] representations

does not amount to a representation of oneness, twoness or threeness’.283 The

representations in the OTS are the subpersonal vehicles that enable object tracking and,

as such, their structure is not available for the child to notice and reason about.284 In

order for innate representations to help in the process of bootstrapping, these

representations must have some explicit numerical content. Given that the ANS is the

only innate system that we know of which explicitly represents numerical properties,

Carey’s dismissal of its involvement in the formation of number concepts might be too

hasty. Without ANS representations or some other form of explicit number

representations, number concepts would have no numerical content.

The fact that mapping of ANS representations to number words seems to happen

later than childrens’ acquisition of number concepts, need not invalidate the role of the

ANS. Carey only shows that ANS representations and linguistic representations are not

jointly sufficient to enable the generation of number concepts. However, there may be

more representational resources available. Carey reasons from the insufficiency of

linguistic representations and ANS representations to the conclusion that the latter are

not partly necessary for concept acquisition. However, an alternative conclusion is to

suggest that further representational resources are required. For example, the EC

approach would point towards the involvement of ANS, OTS, linguistic and further

perceptual and motor representations in the generation of number concepts. Whilst it is

clear that the acquisition of numerical language plays a significant role, it is by no means

the only possible significant factor.

This overemphasis on the role of language also manifests itself in the kinds of

evidence that are taken to be significant. The lack of involvement of the ANS in number

concepts is supposedly entailed by a lack of mapping between ANS representations and

283 Rips, Bloomfield & Asmuth (2008) pg. 629 284 The fact that three OTS files instantiate the property of threeness does not entail that they represent threeness and thus they can serve no useful role unless we possess some further system that explicitly represents the states of the OTS.

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number words. However, mastery of linguistic counting is not constitutive but indicative

of the possession of number concepts. It is perfectly possible that an organism might

possess all of the representational apparatus required for number concepts and yet lack

the capacity to generate relevant verbal behaviour. By only considering the acquisition of

numerical language as a potentially decisive turning point and by only considering

linguistic performance in counting tasks as evidence, Carey unnecessarily constrains the

possible explanations of our capacity for numerical cognition. The EC approach can

overcome these problems by showing how there are far more representational resources

available for the development of number concepts. Furthermore, these extra resources

may make it possible to overcome some of the problems associated with “bootstrapping”

with only ANS and language representations to work with. However, before turning to

this alternative weak nativist account, it is important to defend the weak nativist position

against non-nativist and strong nativist challenges.

Defending Weak Nativism against Non-Nativism

The most famous non-nativist account of number concept acquisition is that of

Piaget, who argued that number concepts must be derived from experience and relatively

sophisticated reasoning.285 However, these findings have been widely discredited by a

host of evidence suggesting that children acquire a surprising degree of numerical

competence much earlier than Piaget had found.286 Unlike Piaget, recent proponents of

non-nativism are motivated by the negative claim that it is not possible to develop

number concepts from the starting point of either ANS or OTS representations. As a

result, they argue that number concepts must arise from other cognitive systems that are

not necessarily number-specific.287

The non-nativist account presented by Rips et al. first takes ANS or OTS

representations to be insufficient for the acquisition of number concepts for much the

same reasons as Carey. ANS representations are not precise enough and fail to represent

a successor function and ONS representations have too limited a range and fail to

explicitly represent number. However, the non-nativists depart from the weak nativist

approach by denying that the inductive steps required to “bootstrap” number concepts

are viable. They argue that innate resources when combined with language might fail to

285 Piaget (1952) 286 Dehaene (1997), Carey (2009b) 287 Rips, Bloomfield & Asmuth (2008) pg. 638

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give rise to number concepts, since they are entirely compatible with the generation of

non-standard number concepts. For example, even when children seem to have

mastered the counting principles, it could be that they take the number sequence to

repeat after the largest number word that they know, such that if they knew number

words up to “eight” they would count a collection of ten objects as follows: “one, two,

three, four, five, six, seven, eight, one, two”.288 One would think that such an inductive

step would be unlikely to be made once a child understands the association between

number words and the cardinalities of concrete collections. However, they argue that

possession of number concepts is independent of the way in which these concepts are

applied to concrete collections. Number concept possession is taken to depend upon

developing an appreciation of the abstract natural number structure, by implicitly

representing the axioms of Peano Arithmetic, and so considerations of how one applies

number concepts in concrete situations are deemed irrelevant.289

As a result of these considerations, Rips et al. argue that our innate numerical

systems play no role in the acquisition and development of number concepts. Whilst

these systems may play an important role in tasks such as assessing the number of

entities in a collection, these tasks need not necessarily involve number concepts.

Instead number concepts are formed from the top-down by developing abstract schemas

through the use of more general purpose systems for deductive reasoning. Schemas are

constructed purely on the basis of an innate grasp of logical notions such as uniqueness,

mapping and function, which employ no number-specific mechanisms. The non-nativist

approach can thus be seen as directly opposed to the EC account that follows, since

neither perceptual representations from the ANS nor representations from any other

perceptual or motor systems are involved in number concepts. On the non-nativist line

of reasoning, number concepts are, instead, purely cognitive representations,

constructed by the central cognitive systems that govern deductive reasoning.

The non-nativist account faces two main problems. Firstly, there is a wealth of

evidence suggesting that ANS representations play a significant role in processes of

numerical cognition. Secondly, there are good reasons to think that non-nativists set too

high a standard for number concepts by suggesting that they should fully capture all of

the relevant aspects of the natural number structure.

288 Rips, Asmuth & Bloomfield (2006) pg. B53-B54 289 Rips, Bloomfield & Asmuth (2008) pg. 625

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Contrary to the predictions of a non-nativist approach, there is a wealth of

evidence to suggest that perceptual representations from the ANS play a significant role

in cognitive tasks employing numerical concepts. Thus, in line with the predictions of

embodied cognition, perceptual representations from the ANS can be seen to partially

constitute the number concepts that are used for sophisticated numerical cognition.

Behavioural evidence suggests that characteristic performance effects of the ANS do not

only arise in perceptual tasks where subjects are dealing with concrete collections of

objects. They also arise in cases where subjects are presented with numerals and number

words. For example, size and distance effects arise in numerical comparison tasks using

numerals and number words.290 Furthermore, there is a wealth of neurophysiological

evidence from imaging studies, which suggests that patterns of activation in the hIPS are

similar regardless of whether subjects are engaging in tasks involving concrete

collections or tasks involving numerals or number words.291 At face value, there is no

reason why engaging with numerals or number words should elicit effects associated

with the perception of number in concrete collections, since the symbols used bear no

direct resemblance to concrete collections. Thus, the best explanation for discovering

performance and activation associated with the ANS in these scenarios is that perceptual

representations from the ANS play some role in constituting number concepts used in

cognition.

There is also evidence to suggest that perceptual representations from the ANS

are recruited when subjects engage in more complex arithmetical cognition, such as

conducting addition, subtraction, multiplication and division calculations. For example,

many common errors in learning multiplication tables can be explained when one takes

into account the typical performance limitations of the ANS.292 Furthermore, a number

of studies suggest that many of the same neural regions are activated during complex

arithmetical tasks such as addition, subtraction, multiplication and division as in tasks

that involve the perception of number.293 There is even evidence to suggest that the same

regions are involved when subjects engage in highly sophisticated mathematical

reasoning, such as solving integration problems.294 Further evidence implicating

perceptual ANS representations in cognitive tasks comes from developmental studies.

These studies suggest that the acuity of a preschool child’s numerical perception is a

290 Dehaene (1992) Dehaene & Akhavein (1995) 291 Pinel et al (2001), Piazza et al. (2007) 292 Dehaene (1997) pg. 126-133 293 Fehr, Code & Hermann (2007) Arsalidou & Taylor (2011) 294 Krueger et al. (2008)

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good predictor of their later success in formal school mathematics.295 This suggests that

perceptual representations from the ANS may play a significant role in more formal

arithmetical cognition.

A second problem with the non-nativist approach is that it sets the bar far too

high for attributing number concepts. Rips et al. adopt a structuralist position in the

philosophy of mathematics and set their standards for the nature of number concepts

accordingly. Thus, subjects are only said to possess number concepts when they have

developed an appreciation that numbers are merely positions in an abstract structure

and can be fully defined in terms of their relations to one another.296 This view is the

main motivation for their argument that systems responsive to the cardinalities of

concrete collections play no role in the formation of number concepts. However, this

approach is confused in at least three ways. Firstly, it blurs the line between common

sense understanding and scientific understanding.297 Secondly, it fails to distinguish

philosophical and psychological notions of concept. Thirdly, it conflates the concept THE

NATURAL NUMBERS with natural number concepts, such as THREE, TEN and ONE-HUNDRED.

The structuralist conception of the natural numbers is a relatively recent

development in the history of mathematics, widely held to have stemmed from the work

of Dedekind in the late nineteenth century.298 Thus, it would be strange to insist that

only those who possess an explicit understanding of the significance of purely structural

features of numbers can be said to possess number concepts. If this were the case then

most people engaging in numerical cognition must be said to lack number concepts,

along with all of the great mathematicians whose work preceded that of Dedekind. It is

clear that Rips et al. do not set the bar quite so absurdly high; however they still insist

that number concept possession requires an implicit appreciation of these structuralist

ideas. However, this is to confuse concept possession with possession of up to date

scientific knowledge.299 For example, ancient civilisations’ biological theories were less

developed than ours to the extent that many classified whales as fish. However, it would

be very odd to claim that members of these civilisations lacked the concept FISH. It would

be weirder still to insist that their possession of a FISH concept depends on their implicit

understanding that whales are not fish, despite the explicit behaviour to the contrary.

Similarly, just because infants’ and most adults’ concepts fail to capture all aspects of our

295 Gilmore, McCarthy & Spelke (2010), Libertus, Feigenson & Halberda (2011) 296 Rips, Bloomfield & Asmuth (2008) pg. 625 297 Barner (2008) pg. 643 298 Dedekind (1963) 299 Barner (2008) pg. 644

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developed theory of number this is no reason to deny that they have number concepts at

all.

This confusion may arise as a result of the non-nativists confusing philosophical

notions of concepts with psychological notions of concepts. For some philosophers, such

as Frege, concepts are abstract objects as opposed to mental entities.300 They are the

abstract meanings of words or thoughts. For others they are definitions, often spelled out

in terms of lists of necessary and sufficient conditions.301 However, neither of these

approaches sits well with the psychological notion of concepts, according to which

concepts are mental representations. By insisting that our concepts conform to the high

standards of a philosophical analysis of number, Rips et al. render number concepts

incapable of explaining most everyday cases of numerical cognition in terms of number

concepts.

Part of the problem with the non-nativist account may stem from their failing to

appreciate the important difference between the possession of number concepts such as

THREE, TEN and ONE-HUNDRED and the possession of the concept THE NATURAL NUMBERS.

Intuitively it seems as if it is possible to possess the former without possessing the latter.

Furthermore, it would be strange if the development of the latter was in no way related

to the former. Rips et al. might be correct in suggesting that possession of THE NATURAL

NUMBERS concept requires implicit representation of the axioms of Peano Arithmetic.

However, this needn’t also be the case for number concepts. It is without doubt

important to explain how we are able to generate THE NATURAL NUMBERS concept from the

basis of our number concepts and also to explain the development of these basic number

concepts. However, by setting the standards for possession of the latter in terms of the

former, the non-nativists kill this explanatory project before it gets off the ground.

The non-nativists fail to provide convincing arguments against the involvement of

innate number-specific systems in the development of our number concepts. There is a

wealth of evidence to suggest that perceptual representations from the ANS play a

significant role in numerical cognition, even in the case of quite sophisticated

mathematical reasoning. This supports a weak nativist EC approach since perceptual

representations are used in cognitive processes. However, before detailing this approach

it is necessary to defend weak nativism against the challenge from strong nativism.

300 Laurence & Margolis (1999) pg. 6 301 Ibid. pg. 8-9

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Defending Weak Nativism against Strong Nativism

Strong nativists agree with both weak nativists and non-nativists that innate

number-specific perceptual systems, such as the ANS, are insufficient for the

development of number concepts. They also agree with the non-nativists that

“bootstrapping” from ANS representations and linguistic representations isn’t a viable

route to number concepts. However, they differ from non-nativists in suggesting that our

number concepts do in fact develop from a number specific system. In order to maintain

this position, they argue that we possess an innate number-specific cognitive system, the

innate “number module”, which already includes number concepts for small numbers,

such as ONE, TWO and THREE.302 As such, some of our number concepts are innate.

However, the question then arises as to what extra role this mechanism is supposed to

play. Strong nativists argue that the innate number module allows us to employ precise

number concepts in a manner that neither the ANS nor the OTS are able to. We are only

able to develop a full gamut of precise number concepts by starting from the basis of

some innate number concepts and generalising from their properties or combining them

to form new concepts.

One problem with the strong nativist approach is that the innate number module

is posited in the absence of any behavioural or neurobiological evidence for its existence.

In the case of both the ANS and the OTS there is a long history of studies detailing the

way in which these systems behave and their implementation in the brain. However,

thus far, nobody has found any evidence for a distinct innate number module. This extra

mechanism is merely posited on the basis of arguments that the known mechanisms are

insufficient. Thus, if the ANS can be seen to be capable of achieving just as much as the

innate number module then there is no need to overcomplicate the theory by positing the

latter mechanism.

There are good reasons to think that the ANS may be able to do at least as much

as the innate number module. The strong nativists suggest that the innate number

module only supports concepts for small numbers, such as ONE, TWO, THREE and possibly

also FOUR. The question then arises as to what extra functions this system could achieve

that the ANS isn’t already capable of. The claim is that this extra cognitive mechanism

could allow for precise representation of number where the ANS cannot. However, the

302 Laurence & Margolis (2007), pg. 145-146. The account here focuses on Laurence & Margolis’ strong nativist account. However, an alternative form of strong nativism has been proposed by Leslie, Gallistel & Gelman (2008). The criticisms of strong nativism provided here can be seen to apply equally to both accounts.

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ANS is relatively accurate in the range of numbers for which strong nativists posit a

separate conceptual module. There are still clear incremental differences between ANS

representations of one to four objects, so these representations could presumably fulfil

exactly the same function as symbols in the innate number module.303 The redundancy

of the innate number module is further backed-up by the fact that strong nativists also

rely on the acquisition of number language to account for number concepts greater than

four.304 Furthermore, since the innate number module is posited to be a purely cognitive

system, the strong nativists must explain how our perception of number leads to

activation of representations in the innate number module. Presumably this would

involve mapping the fuzzy representations of the ANS to precise representations in the

innate number module. However, this is exactly the kind of operation that Carey saw as

problematic with respect to the relationship between fuzzy ANS representations and

precise linguistic representations.305 Thus, it is hard to see how positing an innate

number module does anything other than shift the problem from one of mapping ANS

representations to language to one of mapping ANS representations to innate internal

symbols.

Proponents of strong nativism might reply that a separate representational

system is required for developing a representation of the successor function. The idea

would be that we can only arrive at the notion of successor if we have a means of

representing the difference between successive numbers as always being precisely one

and that ANS representations ‘are by their nature approximate and hence incapable of

expressing a difference of exactly one’.306 However, Katz argues that one could arrive at

the notion of a unique successor function using the ANS alone. The first thing to note is

that, in the range of small numbers, the ANS reliably produces representations with clear

incremental differences.307 Whilst the difference in magnitude between the

representation of one and the representation of two might not be reliably the same as the

difference between two and three, it is still the case that there is a clear incremental step

between one and two and then between two and three. Thus, although completed ANS

representations are only approximate, they can be understood as being formed from a

precise number of increments.308 Despite the variability of the increments, the notion of

the successor function is implicit in the precise number of incremental steps that could

303 Katz (2013) pg. 692 304 Laurence & Margolis (2007) pg. 147 305 Carey (2009b) pg. 313-318 306 Margolis & Laurence (2008) pg. 935 307 Katz (2013) pg. 692 308 Ibid. pg. 701

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be taken to construct a collection of a certain number. Whilst, both Carey and the strong

nativists might be right to suggest that acquiring number concepts by mapping number

words to completed ANS representations is not viable, this is not the only option.

An immediate problem for this approach comes from the wide range of evidence

to suggest that ANS representations result from of directly perceiving collections by

individuating entities in parallel, whilst the appreciation of incremental steps between

ANS representations seems to rely on individuating entities in series.309 However, the

fact that the ANS acquires representations through parallel rather than serial

individuation does not render serial representation of collections impossible. It may be

possible to form a series of ANS representations by first focusing on one entity in a

collection and then two and then three and so on, and thereby come to an appreciation of

the precise number of incremental steps between the approximate representations at

each stage. At each stage, ANS representations are formed directly and are only

approximate but the notion of precise successor is implicit in the incremental differences

between representations at each stage. The process required for the appreciation of the

successor relation is thus more complex than processes of numerical perception and is

thus likely to require more than the representational resources of the ANS alone.

However, as with other weak nativist accounts, further representational resources, such

as linguistic representations, could enable the construction of the required sequence of

ANS representations. As a result of mapping number words and perhaps other

representational resources to the precise number of increments in a series of ANS

representations, subjects arguably have the resources to perform a “bootstrapping”

operation, by making the induction that number words correspond to the precise

number of incremental steps for numbers greater than four.

Since this response to the strong nativist suggests that number concepts can be

“bootstrapped” from ANS representations without the involvement of OTS

representations, one might expect it to be vulnerable to the criticisms that Carey levelled

at theories of this kind. In particular it might be vulnerable to the objection that children

seem to master the counting principles before they are able to map number words to

ANS representations.310 However, this evidence is consistent with the current approach,

since number words are not taken as being mapped to completed ANS representations.

Instead number words are taken to be mapped to the precise number of increments that

309 Dehaene & Changeux (1993) 310 Le Corre & Carey (2007)

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are compounded to form an ANS representation, when ANS representations are formed

in sequence.311 As such, children might have the required representational resources to

“bootstrap” from relations between ANS representations of small numbers to number

concepts that obey the counting principles for larger numbers without yet having the

capacity to map completed ANS representations of larger numbers to number words.

They can carry on the pattern of taking each new word in the number sequence to

represent a further incremental increase in magnitude of ANS representation without

being able to map these number words onto specific states of the ANS. Thus, it is

arguably the case that the ANS is capable of doing all of the things that the innate

number module is posited for.

The case for strong nativism is blunted since it is not clear what an innate number

module could achieve that could not already be achieved by the ANS. ANS

representations are reliably accurate enough to allow for the differentiation of numbers

within the range of the innate number module and it may be possible to develop an

appreciation of the successor relation by “bootstrapping” based on the incremental steps

between ANS representations. Since there is no independent evidence for the existence

of an innate number module, the strong nativist position can be rejected for reasons of

parsimony, as it posits a further theoretical entity without evidence and without thereby

gaining any further explanatory power.

The Development of Embodied Number Concepts

The evidence against non-nativism is widely suggestive of the idea that ANS

representations play a significant role in the constitution of number concepts. For better

or for worse, sophisticated mathematical cognition is affected by the limitations of the

ANS. This evidence can also be seen to support an EC account of number concepts, since

the ANS is best understood as a perceptual system and its representations seem to

partially constitute number concepts. In line with the predictions of EC perceptual

representations are also implicated in cognitive processes. However, it should be clear by

now that ANS representations alone are not sufficient to explain our numerical cognition

capacities and thus cannot be the only constituents of number concepts. As such the

important question arises as to what must be added to these representations to enable

the acquisition and development of number concepts. The arguments against strong

311 Katz (2013) pg. 700

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nativism also support an EC approach and help to answer this question, since they rule

out the need for purely cognitive innate number concepts and suggest that there is a

viable route to the formation of number concepts on the basis of ANS representations

being augmented with further representational resources. In what follows it will be

argued that these further representational resources can be accounted for on an EC

approach.

There are good reasons to believe that the acquisition of number language has an

important role to play. Furthermore, the role that number language is hypothesised to

play is arguably best explained within an EC framework. However, whilst Carey

considered the acquisition of representations of number language as the sole decisive

factor in enabling the development of number concepts, an EC approach would suggest

that there are far more relevant representational resources available. In particular,

number concepts might also be partially constituted by the perceptual and motor

representations associated with other numerical practices, such as finger counting.

Furthermore, cultural practices may play a role in shaping our number concepts, leading

to a degree of heterogeneity in the number concepts of different cultures. It will be

argued that, once one takes on board this much wider pool of representational resources,

the reasons for doubting that ANS representations form the basis of our number

concepts disperse. Number concepts, in line with the claims of EC are constituted by

number-specific perceptual representations from the ANS augmented by further

perceptual and motor representations from other systems.

The Role of Embodied Linguistic Representations

One thing almost universally agreed upon is that the acquisition of number

language plays a significant role in the development of number concepts. Evidence from

behavioural, anthropological and neurophysiological studies all supports this idea.

However, it is still possible to question whether numerical language is necessary for the

development of all number concepts. Furthermore, the precise role that number

language plays still needs explaining. After surveying some of the evidence for the

importance of number language, it will be argued that an EC account can provide an

explanation of its role. Representations of external linguistic entities by perceptual and

motor systems can be seen to partially constitute number concepts.

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It is clear that learning the sequence of number words plays an important role in

most children’s pre-school education. Parents devote a lot of time and energy to getting

their children to engage in learning the counting sequence. This is reflected in the

widespread popularity of childrens books that focus on number words and the

prevalence of numbers in nursery rhymes from a wide range of cultures (see Fig. 4.1).312

Fig. 4.1

Many with experience of young children will testify to their eagerness to show off their

counting abilities. However, merely learning the count list is not sufficient for using it in

mathematical cognition. Children might merely learn the count list as a sequence of

arbitrary sounds without assigning any numerical meaning to the words. Significantly

though, once children have developed an appreciation of the numerical significance of

number words, there is a qualitative change in their behavioural capacities.313

Further evidence for the significance of number language comes from

anthropological studies of societies whose language only contains a very limited array of

number words. One of the earliest accounts of such societies was provided by Locke, who

spoke of the Tououpinambos, a tribe with no words for numbers beyond five.314 The

Pirahã and the Mundurukú Amazonian tribes both use languages which only have

number words for the first few numbers.315 Studies of their numerical abilities have

312 Normanton (2011) (Fig. 4.1 from The Very Hungry Caterpillar, Carle (1969)). 313 Carey (2009b), Le Corre et al. (2006) 314 Locke (1975) pg. 207 315 Pica et al. (2004), Gordon (2004), Everett (2005) The Pirahã only have words for “one”, “two” and “many”, whilst the Mundurukú have words for “one”, “two”, “three”, “four”, “five” and “many

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found that their capacity for dealing with number approximately is similar to that of

animals, infants and adults that are not given enough time to explicitly count. However,

they lack the capacity to deal with numbers in a precise manner for numbers greater

than three or four.316 It, thus, appears as if they are relying solely upon ANS

representations. This suggests that the acquisition of number language is a necessary

step for developing the ability to deal with number precisely. Representations of number

words are needed to augment ANS representations so as to allow for precise numerical

cognition.

Further evidence for the importance of numerical language comes from the

recruitment of neural areas associated with verbal behaviour in numerical tasks. As well

as activation in the hIPS, numerical tasks consistently involve recruitment of the left

angular gyrus.317 This area of the brain is involved in a range of language-mediated

tasks.318 Furthermore, the level of activation of the angular gyrus during numerical tasks

varies in accordance with the linguistic demands of the task, with much stronger

activation in tasks involving precise number calculations and even more so in the case of

complex arithmetical calculations such as multiplication and division.319 Precise

calculations, multiplication and division are taken to rely on language to a greater extent

because they often involve recall of rote-learned phrases. Thus, the apparent role of the

angular gyrus in numerical cognition provides further evidence that linguistic

representation plays an important role in allowing us to go beyond approximate ANS

representations.

At first sight, the significance of linguistic representations for the acquisition of

number concepts looks at odds with an EC approach. Most external linguistic

representations are symbolic and arbitrary and proponents of cognitivism argue that this

is a reflection of the nature of the underlying cognitive states that natural language is

used to express. However, even if it were the case that natural language reflects the

language-like structure of cognitive states; it would still remain to explain how we are

able to engage in linguistic activities, such as speaking, listening, reading and writing. In

order to explain these capacities, one must make reference to the role of our perceptual

and motor systems. Speaking and listening must to some extent involve auditory

representations and motor representations that control the mouth muscles and voice

316 Ibid. 317 Dehaene et al. (2003) 318 Fiez & Petersen (1998), Price (1998) 319 Dehaene et al. (2003) pg. 495

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box. Reading and writing must to some extent involve visual representations and motor

representations that govern hand movements for writing or typing. There is even some

evidence to suggest that there is some overlap between the motor systems responsible

for the production of language and the perceptual systems responsible for its reception

and that this can help explain our capacity for linguistic social interaction.320 On an EC

account of linguistic representation, representing linguistic entities involves activation of

the same perceptual and motor representations that would be used in online linguistic

activity.321 Thinking about a word involves some of the same systems that would be used

to say, hear, read or write that word. On such an account, there is no need to posit an

extra system of internal amodal symbols that correspond to words, since perceptual and

motor representations of words are already required and these are sufficient to do all the

work for which amodal symbols are posited.322 External language is already a symbolic

and arbitrary system so perceptual and motor representations of external symbols can

inherit all of the benefits of a representational system of this kind without the need to

posit a further auxiliary amodal system to do the job.323

In the context of numerical cognition, EC can provide a detailed explanation of

the way in which the acquisition of numerical linguistic competence can enable the

development of number concepts. ANS representations can be augmented by becoming

associated with the perceptual and motor representations that underlie our ability to

perceive and produce number words and numerals. Thus, whilst most accounts merely

emphasise the significance of numerical language, an EC account provides claims about

the structure of the underlying representational vehicles which explain this significance.

Whilst ANS representations might not be accurate enough to distinguish proximate

larger numbers from one another, our perceptual and motor representations of the

corresponding number words or numerals are different enough so as to render them

distinguishable. For example, whilst the ANS might not be able to reliably distinguish

320 Fadiga et al. (2002), Gallese (2008) 321 It is important to distinguish the EC account of linguistic representation from a related strand of EC focused on linguistic comprehension. The claim on offer here is merely that EC can provide an account of how we represent external linguistic entities, such as symbols, words and sentences, in terms of perceptual and motor representations. Some proponents of EC also support an embodied account of linguistic comprehension (Glenberg & Kaschak (2002), Zwaan (2004), Pulvermüller (2008)). On such an account, comprehending a word involves the activation of perceptual and motor systems associated with the word’s content. For example, hearing the word “lick” activates motor representations responsible for the control of tongue muscles, whilst hearing the word “kick” activates motor representations that govern leg muscles. (Hauk & Pulvermüller (2004))The EC account of number concepts on offer here does not depend on the EC account of comprehension. The present account merely suggests that embodied representation of external linguistic entities supports the development of number concepts. An EC account of number language comprehension would thus be a prediction of an EC account of number concepts rather than a prerequisite for the possibility of number concept acquisition. 322 Dove (2014) 323 Clark (2006a, 2006b), Dove (2014)

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collections of sixteen and eighteen, our perceptual and motor representations of the

words and symbols for sixteen and eighteen are clearly distinct. As such, forming

perceptual representations of external number symbols enables us to transcend the

representational power of the ANS by allowing for distinct representations for each

cardinal value. ‘When we add number words to the more basic biological nexus… we

acquire an evolutionarily novel capacity to think about an unlimited set of exact

quantities. We gain this capacity not because we now have an encoding of 98-ness just

like our encoding of 2-ness. Rather, the new thoughts depend directly (but not

exhaustively) upon our tokening the numerical expressions themselves’.324 Number

concepts could thus be thought of as at least partially constituted by ANS representations

combined with embodied representations of external numerals and number words.

Activating the concept SIX involves partial reactivation of approximate ANS

representations of six as well as perceptual and motor representations of the word “six”

and the numeral “6”. Furthermore, since numerical language is primarily learned within

the context of a regimented counting routine, representations of numerical language can

play some role in grounding the relations between number concepts. For example,

perceptual and motor representations of “six” are likely to become closely associated

with perceptual representations of “five” and “seven” due to their repeated concatenation

during counting routines.

Some have challenged the notion that perceptual and motor representations of

external linguistic entities can help ground concepts on an EC approach.325 Perceptual

representations of words arguably do not help with the grounding of abstract concepts,

since these external symbols are often ambiguous, arbitrary and unsystematic.326 They

are ambiguous in the sense that many different meanings can be ascribed to the same

external symbol. As such, distinct internal amodal symbols may be required to

differentiate between ambiguous meanings of a single external symbol. They are

arbitrary in the sense that their physical form is unrelated to their content. It is thus

unclear how adding a perceptual representation of an arbitrary symbol can have any

significant effect on the nature of a given concept, since the perceptual character of the

added representation bears no relation to the content of the concept. External symbols

are also unsystematic in the sense that they have a wide range of associations, of which

only some are relevant to the content of the concept, yet there is usually no systematic

324 Clark (2006b) pg. 297 325 Dove (2009) 326 Ibid. pg. 420

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way in which to distinguish relevant from irrelevant associations.327 In most cases, it

would thus seem as though perceptual representations of external symbols are of no use

to the proponent of EC in explaining the nature of abstract concepts and, as such, one

would expect the same to be the case for number concepts.

However, there are also reasons to think that the case of number is special. Unlike

ordinary language, numerical language and numeral systems are relatively

unambiguous, to some extent iconic and also systematic. The meanings of number words

are rarely ambiguous. In many of the cases where number words are polysemous their

seemingly nonnumerical meanings can be explained in numerical terms.328

Furthermore, numbers are unique in having their own dedicated system of external

symbols, numerals. As such, the set of perceptual and motor representations of the

numeral for a given number is likely to be unique and, thus, unambiguous.

Numerical language also differs from ordinary language in the sense that the

structure of the symbols used is not purely arbitrary with respect to their meaning.

Numerals and number language are, to some extent, iconic. For examples, larger

numbers are, in general, represented using a larger quantity of digits than smaller

numbers. In some number systems, such as Roman numerals, some symbols, such as

“III”, are fully iconic, in that they instantiate the property that they represent.

Furthermore, even spoken number words, for the most part, involve more syllables for

larger numbers. There is obviously some degree of arbitrariness to numerical language

and notation. The symbol “6” bears no resemblance to a collection of six entities. Our use

of the base-10 counting system, as opposed to any other, is also, to some extent,

arbitrary. However, this arbitrariness is often explainable with respect to further aspects

of embodied representation. For example, use of the base-10 counting system could be

explained in terms of the significance of motor representations associated with counting

on our fingers. In cultures that use other counting bases the choice of counting base can

also be understood in terms of bodily constraints. For example, the Native American

Yuki use a base-8 system as a result of counting the spaces between fingers.329 In the

case of Roman numerals it is significant that the more arbitrary symbol “IV” is

327 Ibid. pg. 420-421 328 For example, in English, the word “one” is often used to refer to a hypothetical person but is still being used to refer to a single entity. In many languages the word for “five” and “hand” are the same but this is a reflection of the number of fingers on a hand (Dehaene (1997) pg. 93). One exception, in English, is the possible ambiguity between spoken external symbols “two”, “to” and “too”. 329 Ascher (1991) pg. 9

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introduced at exactly the stage where our ANS representations begin to become less than

completely reliable.

The most important feature that distinguishes numerical language from ordinary

language is its systematic nature. Number words and numerals are presented in the

context of well-defined systems with clear rules for “correct” use and for the generation

of novel number symbols through combination. Furthermore, unlike the case of ordinary

language, where direct tuition plays a lesser role, the systematic relations between

number words are directly regimented by the teaching of counting procedures and

explicit rules for symbol manipulation. In the case of most words there is no way of

determining which associations are relevant for content. However, in the case of number

words the relevant associations are both made explicit and exhausted by their

regimented contexts of use. We are explicitly taught the counting routine and the rules

for generating novel number words to extend this routine indefinitely. As a result of the

systematic nature of number language we are thereby both able to appreciate that each

cardinal value has a distinct number word associated with it and to appreciate that the

sequence of number words that can be generated is endless. Evidence for the significance

of the systematic nature of number language comes from comparisons in arithmetical

performance between speakers of languages that vary with respect to how systematic

they are. The Chinese system of number words is more systematic than the English,

since, when counting beyond ten, instead of using idiosyncratic words, such as, “eleven”

and “twelve”, Chinese speakers simply count “ten-one, ten-two, …”. This linguistic

difference provides an explanation for the more rapid development of arithmetical

competence amongst speakers of Chinese, which even manifests itself in nonlinguistic

tasks.330

It might be correct to suggest that, in most cases, representations of external

linguistic symbols are unable to significantly impact upon concepts due to their

ambiguity, arbitrariness and their lack of systematic associations.331 However, since the

absence of these three properties is a distinctive feature of number language, this

problem need not apply in this case. As such, number concepts may be a special case

where augmentation with perceptual representations of external linguistic symbols can

play a particularly significant role in conceptual development. Perceptual

330 Miura et al. (1988), Miller et al. (1995), Miller et al. (2000) This explanation has been argued to be far more compelling than alternatives that argue for the significance of more general differences in educational practices and culture, in that it can explain specific differences in competencies with two-digit numbers. 331 Dove (2009)

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representations of number from the ANS could be augmented by perceptual and motor

representations of numerals and number words, which, due their systematic and iconic

nature, could allow for the more fine-grained distinction required for the development of

precise and sophisticated number concepts.

Are Embodied Linguistic Representations Necessary?

It is clear that the acquisition of number language plays a very significant role in

the development of number concepts and that this can be accommodated and explained

by an EC approach. However, one can still question whether number language

representations are necessary for number concepts or whether they just happen to play a

significant role in most cases. One can also question whether ANS representations

combined with these number language representations are sufficient for the

development of number concepts and, if not, what else is needed.

The absence of precise number concepts in subjects that lack sophisticated

number language is not enough to support the claim that language is necessary for

number concept acquisition. In cases such as the Pirahã and the Mundurukú, impaired

precise arithmetic might result from a different deficiency, which might also explain the

absence of number language. For example, both their lack of numerical language and

their lack of precise arithmetic abilities might merely reflect the fact that their particular

environment and culture render precise number concepts less important. This is

supported by the finding that bilingual Mundurukú subjects with knowledge of

Portugese number words still perform badly in precise arithmetic tasks.332 Furthermore,

Mundurukú subjects only engage in rudimentary finger-counting strategies, providing

equal reason to see sophisticated finger-counting as necessary for number concepts.333

There is also evidence to suggest that Pirahã subjects perform relatively well at tasks that

seemingly require precise number concepts when tasks involve collections of entities

that are immediately perceivable and manipulable. Only tasks that involve keeping track

of a precise number of objects in memory elicit significant behavioural differences.334 As

such, it could be argued that number language is not necessary for the development of

332 Pica et al. (2004), Gelman & Butterworth (2005) pg. 9 333 Andres, Di Luca & Pesenti (2008) pg. 643 334 Frank et al. (2008)

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number concepts but instead enables ‘the ability to remember the cardinalities of large

sets’.335

Neurological evidence is also indecisive as to whether number language is

necessary for the development of number concepts. The involvement of the left angular

gyrus in numerical cognition is inconclusive, since it is also involved in various non-

linguistic functions that might also be significant for numerical cognition. For example,

the angular gyrus might be involved in representing numerical magnitudes in spatial

terms.336 Furthermore, evidence from patients with lesions and genetic disorders

suggests that linguistic and numerical cognition are dissociated at the neural level. For

example, a patient with global aphasia so profound that he had no viable capacity to

comprehend or express language was found to have perfectly functional capacities for

arithmetical calculation.337 Another patient suffering from semantic dementia was found

to have lost all memory for the meanings of words yet maintained nearly all aspects of

numerical knowledge.338 Furthermore, patients with lesions that cause severe

grammatical impairments have also been shown to have intact abilities for carrying out

calculations.339 Thus neither semantic nor grammatical aspects of our linguistic ability

seem essential for the successful deployment of number concepts. There are also cases

where subjects’ numerical abilities are severely impaired whilst cognitive functions

associated with language remain intact, such as in cases of Gertsmann’s syndrome.340

Whilst in most cases representation of numerical language plays a significant role in the

development of number concepts, these representations need not be seen as a necessary

prerequisite for the possession of all sophisticated and precise number concepts. It may

be possible to acquire some precise number concepts via a different route.

As well as questioning the necessity of numerical language, it is possible to

question whether number language representations combined with ANS representations

are jointly sufficient for the acquisition of number concepts. Representations of number

language enable the distinguishability of proximate large number concepts and the

appreciation that each cardinal value has a distinct unique representation. By

appreciating the systematic nature of number language it is also possible to come to the

appreciation that there is an endless sequence of numbers for which novel number

335 Ibid. pg. 823 336 Göbel, Walsh & Rushworth (2001), Price & Ansari (2011), Krause et al. (2014) 337 Rossor, Warrington & Cipolotti (1995) 338 Cappelletti, Butterworth & Kopelman (2001) 339 Varley et al. (2005) 340 Gerstmann (1940), Cipolotti & van Harskamp (2001)

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words can be systematically generated. However, one can argue that a key ingredient of

number concepts is still missing. There is still no notion of a unique successor function.

The problem is that a potentially endless list of external labels says nothing about the

nature of the relation between the labels. Language provides no clues to the fact that the

relationship between three and four is the same as that between four and five. This is

further exacerbated by the fact that relationships between successive ANS

representations vary as the numbers in question get higher. Thus, whilst linguistic

representations may play a significant role in the development and constitution of

number concepts, their combination with innate ANS representations is arguably

insufficient for the development of fully-fledged number concepts.

Finger-Counting and the Role of Action Representations

As well as evidence for the significance of linguistic representations in the

development of number concepts, there is a growing body of evidence to suggest that

perceptual and motor representations of action also have a significant role to play. A

particularly important type of action for the development of number concepts is that of

finger counting. Finger-counting clearly plays a significant role in the development of

our arithmetical abilities. Whilst specific finger-counting strategies vary from culture to

culture and from individual to individual, it is found, in some form, in all cultures across

the world and throughout known history.341 Furthermore, children engage in finger-

counting strategies spontaneously without explicit instruction from adults.342 It is easy to

see how finger counting could aid our counting practices. Members of collections to be

counted might not be very stable or might only appear very briefly, making it hard to

keep track of them. By using a finger to represent each object counted we create a much

more stable external record of the entities in the collection, which remains readily

observable and under our control. In this way we use our fingers as an external memory

aid which allows us to go beyond the limitations of our internal working memory.

Furthermore, canonical hand postures associated with finger-counting provide us with a

useful tool for communicating numerical content to others.

Whilst finger-counting strategies clearly provide a useful aid to our arithmetical

cognition and communication, the EC approach makes a far stronger claim. The EC

341 Ifrah (1985) pg. 26, 38, 55-80 342 Brissaud (1992)

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approach suggests that finger-counting is not a mere heuristic aid. Number concepts are

partially constituted by activation in the sensory and motor systems that are involved in

actively engaging in finger-counting. This hypothesis, which has become known as the

“manumerical cognition” stance, is supported by a wide range of behavioural and

neurological evidence.343

One of the most compelling pieces of evidence for the significance of finger-

counting is the prevalence of base-10 counting systems and their independent

emergence in isolated cultures. This prevalence is significant because the choice of such

a counting base is somewhat arbitrary and by no means optimal. A base-12 system would

arguably be superior from a purely mathematical perspective.344 The best explanation for

the base-10 systems’ emergence is the role of our fingers in arithmetical cognition.345

This idea is supported by linguistic evidence, for example, the word digit refers to both

fingers and numerals and the word for five and for hand have similar roots in a wide

range of languages.346 Children have been found to make a disproportionate number of

split-five errors in relatively simple mental addition or subtraction tasks, suggesting that

children represent calculation problems in terms fingers and hands and fail to appreciate

that more than one hand may be necessary for the task.347

The role of the fingers in numerical cognition is further supported by

developmental evidence. A child’s ability to discriminate and recognise its fingers is a

better predictor of later mathematical ability than standard tests of intellectual

capacity.348 Furthermore, training children in these abilities can increase arithmetical

performance.349 In line with this behavioural evidence, children who are born unable to

use their fingers as a result of congenital hemiplegia also show hampered numerical

abilities.350 Furthermore, in numerical tasks where subjects were required to respond by

pressing buttons with certain fingers while their hands were flat on a table, responses

were quicker when the finger used to respond matched the corresponding position in a

343 Wood & Fischer (2008) 344 Andrews (1936) 345 It should be noted that there are examples of cultures where different bases are preferred. For example, some Mayan and Aztec cultures used a base-20 system (Ifrah (1985) pg. 38-39) and the Oksapmin of Papua-New-Guinea use a system with base-27 or even higher (Saxe (1981)). However, it is significant that, in cultures with different bases, further body parts other than fingers are often used. Thus, rather than undermining the EC account, the presence of body-centric alternatives to finger-counting only strengthens this approach. 346 Menninger (1969) 347 Domahs, Krinzinger & Willmes (2008). (Split-five errors are where the answer that the children give is either +5 or -5 away from the correct result.) 348 Noël (2005) 349 Gracia-Ballafuy & Noël (2008) 350 Thevenot et al. (2014)

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prototypical finger counting strategy.351 Subjects also respond faster to numbers after

being unconsciously primed with images of hands in congruent positions from

prototypical finger-counting strategies.352 As well as behavioural evidence from humans,

evidence from robotics suggests that simulated robots develop number concepts and

arithmetical capacities quicker and more efficiently when they are able to use bodily

motions such as finger movements when learning the meanings of counting words.353

The manumerical cognition stance is also supported by a wide range of

neurological evidence. The lesion-based neurological disorder, known as Gertsmann’s

syndrome, impairs both arithmetical abilities and the ability to recognise and

discriminate fingers.354 Furthermore, experiments where similar lesions are simulated

using rTMS also lead to the impairment of both finger abilities and numerical abilities,

suggesting they involve the same underlying neural mechanism.355 Evidence also

suggests that engaging in numerical tasks results in increased excitability of cortical

regions responsible for the control of hand muscles, even in cases where subjects are

prevented from using explicit finger-counting strategies.356 Furthermore, neural

activation is correlated with a subject’s particular finger-counting strategies, with, for

instance, left-hand-starters showing more activation in the neural regions that control

left-hand movements for smaller numbers.357 Taken together, this evidence suggests a

significant role for the motor system responsible for finger counting in the processes that

support our numerical abilities. In line with the predictions of an EC approach, our

number concepts are partly constituted by activation of the sensory and motor systems

that are involved in one of the key elements of our arithmetical engagement, finger

counting.

Although the combination of ANS and linguistic representations may not be

sufficient for the acquisition of sophisticated number concepts, it is arguably the case

that the addition of finger-movement representations provides the ‘missing tool’.358

Finger counting allows subjects to generate their own associations between number

words and ANS representations. Whenever a subject engages in finger counting whilst

also practicing the linguistic counting routine they bring into being a collection of raised

351 Di Luca et al. (2006) 352 Di Luca & Pesenti (2008) 353 De La Cruz et al. (2014) 354 Gerstmann (1940), Cipolotti & van Harskamp (2001) 355 Rusconi, Walsh & Butterworth (2005) 356 Andres, Seron & Olivier (2007), Sato et al. (2007) 357 Tschentscher et al. (2012) 358 Andres, Di Luca & Pesenti (2008)

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fingers that will naturally engage ANS representations. This allows subjects to create

sequences of ANS representations that are correlated with the sequence of the first ten

count words. ANS representations approximately capture the cardinalities of collections

without capturing their place within an ordered sequence, whilst number language

representations capture the notion of an indefinitely long ordered sequence of distinct

places without capturing their link to cardinalities. Finger counting can provide a link

between the two by allowing subjects to generate ordered sequences of symbols and ANS

representations simultaneously. The motor representations that support finger-counting

may also provide the necessary representational apparatus to capture the successor

function. ANS representations fail to capture the notion that each successive step in the

counting routine involves adding exactly one, whilst linguistic representations indicate

nothing about the similarity between steps between symbols. Finger counting can

overcome this deficit, since for any particular finger-counting strategy there is a natural

next step in the sequence that is dictated by spatial and morphological constraints.

In line with an EC approach our number concepts are partly constituted by the

perceptual and motor systems that underlie the process of finger counting. They are

likely to be partly constituted by the visual and proprioceptive perceptual

representations that result from engaging in finger counting as well as being partly

constituted by the motor representations that govern the action of finger counting. There

is thus no need for distinct amodal representations of number, since number concepts

are constituted by combinations of ANS, linguistic and action based representations, all

of which can be understood entirely in terms of perceptual and motor systems.

It is unlikely that finger counting is strictly necessary for the development of

number concepts. For example, subjects that went blind early in life tend not to use

finger counting strategies but still perform as well as others on certain numerical

tasks.359 There may thus be many other action representations that could play a similar

role. The most significant aspect of finger counting in numerical cognition is not the

finger movements themselves but the fact that there is a stable and repeatable sequence

of actions that correspond to the sequence of numbers. ANS representations are

arguably representations of attendability. In line with this approach, finger counting is of

particular use because it allows us to generate our own stable sequences of attendability.

However, one can generate stable sequences of attendability without the use of fingers.

All that is required is some means of naturally generating an ordered sequence of actions

359 Crollen et al. (2014)

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in space. Finger counting can thus be seen as a specific consequence of our natural

ability to structure space into ordered sequences. However, the question of how we are

able to naturally comprehend the notion of an ordered sequence remains to be

explained.

Spatial-Numerical Associations and Ordinality

Up until this point, the main concern has been establishing the cognitive basis of

our cardinal conception of number. It has been argued that the ANS provides us with the

capacity to perceive the number of entities in a collection and that this system is

augmented to enable precise conceptual representations of cardinality. However,

sophisticated numerical cognition is about more than mere assignments of quantity. A

complete picture must also explain how we represent ordinality. As well applying

number to collections’ cardinalities we also apply it to positions in sequences. For

example, we conceive of “c” as the 3rd letter in the alphabet or of “June” as the 6th month

of the year. In some senses the two different conceptions of number are very closely

connected. After all, in naming the ordinal position of an item in a sequence we also

implicitly refer to the quantity of items in the sequence up to and including it. However,

important questions remain regarding how our cognitive mechanisms allow us to

connect these two conceptions of number together.

Neurological and behavioural evidence suggests that the capacities to engage with

these two conceptions of number are closely linked. The hIPS has been shown to be

equally activated by non-numerical stimuli associated with ordered sequences, such as

letters of the alphabet, as it is by numerical stimuli.360 There is also evidence to suggest

that deficits in assessing the cardinality of collections are also often accompanied by

deficits in processing ordinal sequences.361 Whilst this evidence backs up the notion that

the two conceptions are closely linked, it fails to show that they are supported by exactly

the same neural mechanisms.362 Unlike the case of the ANS where the underlying

mechanisms are well understood, the cognitive basis of our ordinal conception is

unclear.

360 Fias et al. (2007) 361 Cipolotti et al. (1991) 362 Jacob & Nieder (2008)

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However, recent discoveries of associations between numerical and spatial

cognition potentially reveal the basis of our conception of ordinality. In the previous

section the role of finger counting was emphasised. However, it was argued that this is

merely a manifestation of a more fundamental capacity to generate ordered sequences of

actions in space. This capacity is essential in going from ANS and linguistic

representations to sophisticated number concepts. It will be argued that the way in

which we naturally represent space provides all the representational tools that we need

to ground the notion of an ordinal sequence. Since the representation of space is a

fundamental aspect of our sensorimotor engagement with the world, this account lends

further support to the idea that our number concepts are constituted by embodied

representations.

Spatial-Numerical Associations

Interest in Spatial-Numerical Associations (SNAs) took off with the discovery of

the so-called SNARC effect.363 The Spatial-Numerical Association of Response Codes

effect (SNARC) came to light in experiments where subjects were asked to compare the

magnitudes of two numbers and indicate whether the second number is larger or smaller

by pressing a button on either their left or their right hand side.364 Subjects responded

faster when they had to press a button on the left to indicate that the number was

smaller and on the right to indicate that it was larger and slower when the button

assignment was reversed. This was taken as evidence that we have a tendency to

associate smaller numbers with the left-hand-side of space and larger numbers with the

right-hand-side of space. This finding was bolstered by experiments which showed that

the SNARC effect was unaffected when subjects had to cross their hands, ruling out the

possibility that the effect had something to do with the hand that was used to make the

response.365 The effect also emerges in tasks not directly related to assessing numerical

magnitude, such as assessing whether a presented digit is odd or even, and also in tasks

where numerical properties are irrelevant, such as judging whether the word for the

number presented contains an “e” sound, suggesting that activation of the SNARC effect

is automatic.366 The SNARC effect is not restricted to numbers presented as visual

stimuli, as there is also evidence for an auditory SNARC effect, nor is it restricted to

363 Dehaene, Bossini & Giraux (1993) 364 Dehaene, Dupoux & Mehler (1990) 365 Dehaene, Bossini & Giraux (1993) 366 Ibid., Fias et al. (1996)

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manual responses, as the effect arises when subjects are required to move their eyes left

or right.367

Whilst the SNARC effect is by far the most famous and well replicated

demonstration of SNAs, in recent years many more have been discovered. For instance,

priming subjects with numerical stimuli can lead to systematic errors in line-bisection

tasks.368 Subjects were presented with lines made up from numerical digits and asked to

bisect them. When the lines were made up of lower numbers subjects tended to err by

bisecting the line too far to the left, whereas when the lines were made up of higher

numbers they tended to err to the right (see Fig. 4.2).

Fig. 4.2

Although this result is interesting in its own right, it becomes even more significant in

the context of certain other experiments involving patients with hemi-spatial neglect.

Hemi-spatial neglect arises as a result of lesions to the parietal lobe and results in

patients’ inability to attend to the region of space contralateral to their lesion.369 Patients

suffering from the condition often err in standard line-bisection tasks and tend to only

bisect the portion of the line that is not in the region of neglect and so end up dividing

lines into one-quarter and three-quarter portions. Remarkably, these patients make

similar errors when asked to pick the mid-point of numerical intervals.370 For example,

they are likely to respond that five is half way between two and six.371 Similar findings

have been replicated in subjects where hemi-spatial neglect is simulated using

367 Nuerk, Wood & Willmes (2005), Fischer et al. (2003), Fischer et al. (2004) 368 Fischer (2001) 369 Heilman, Watson & Valenstein (1979) 370 Zorzi, Priftis & Umiltà (2002) 371 Umiltà, Priftis & Zorzi (2009)

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transcranial magnetic stimulation.372 These findings are particularly significant, as they

suggest that spatial and numerical processing systems share a common neural substrate.

SNAs are also revealed by the discovery of unconscious influences between motor

activity and numerical processing. In one experiment, subjects were asked to turn their

heads from side-to-side and pick a number “at random” from a specific number interval.

Subjects were more likely to pick a lower number when their head happened to be facing

left and a higher number when facing right.373 Thus, the area of space to which subjects

are attending unconsciously influences what they take to be a free choice as to which

number they name. In a further experiment, researchers were able to predict the

numbers that subjects would “randomly” generate, by analysing their gaze-direction.374

These findings both suggest that spatial attention and the motor activity that

accompanies it directly and unconsciously influence numerical cognition. It has also

been found that priming subjects with numerical stimuli can influence their subsequent

gaze direction when given a free choice to look at a face presented on either the left or

the right.375 Similarly, numerical priming can influence where subjects choose to position

an object.376 This suggests that numerical cognition can also have direct and unconscious

effects on spatial attention. The presence of these involuntary and unconscious effects

suggests that the association between space and number is deeply ingrained.

The Origins of Spatial-Numerical Associations

Whilst the existence of SNAs is now widely accepted, the question as to their

origins is somewhat more controversial. The initial hypothesis was that they are

primarily culturally determined, resulting from reading and writing habits and from

encounters with cultural artefacts, such as rulers, that display numbers ascending from

left to right.377 This hypothesis is bolstered by evidence that the SNARC effect is reversed

in subjects whose language is written and read from right-to-left, such as Arabic.378

372 Göbel et al. (2006) 373 Loetscher et al. (2008) 374 Loetscher et al. (2011) 375 Ruiz Fernandez et al. (2011) 376 Gianelli et al. (2012) 377 Dehaene (1997) pg. 82 378 Zebian (2005)

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There is also evidence to suggest that the directions of reading for both words and

numbers contribute to the SNARC effect.379

Despite the early promise of this hypothesis, it now seems unlikely that the appeal

to reading habits can provide a full explanation of the origins of SNAs. The association

between SNAs and reading habits seems to be highly context dependent. Bilingual

speakers, whose two languages are read in opposite directions, can have their SNARC

greatly weakened or even reversed by being primed with just a single word from one of

their two languages.380 Furthermore, the effect is quite fragile and can be disrupted by

engaging in tasks involving incompatible spatial-numerical orientations, such as on a

clock face.381 This fragility is further emphasised by results suggesting that SNARC

effects can be suppressed or reversed by priming subjects with fictional recipes in which

the placement of numbers on the page conflicts with the subject’s usual SNAs.382

Amongst the majority of people, direction of reading and writing is a relatively stable

feature. Thus, if this was the sole determinant of SNAs, one would expect a similar

degree of stability. The context-dependence and fragility of SNAs suggests that, whilst

partly shaped by reading and writing, their ultimate origin may lie elsewhere.

Further evidence against reading and writing being the sole determinant of SNAs

comes from studies that point to the existence of a vertical SNARC and a near-far

SNARC effect.383 Although we tend to read and write from top to bottom and often

encounter lists, where numbers increase further down the page, in the case of the vertical

SNARC effect small numbers are associated with lower regions of space whilst large

numbers are associated with higher regions of space. In the case of the near-far SNARC

effect, where smaller numbers are associated with near space and larger numbers with

space further away, there is no obvious relation to reading and writing practices. As such,

alternative hypotheses must be sought for the origins of these SNAs. One potential

hypothesis is that these effects are grounded in invariant features of our physical

interactions with the world.384 For example, due to the invariant influence of gravity,

piles of objects tend to get larger in the vertical direction as we add more objects and,

due to the nature of our method of locomotion, positions further away tend to take more

379 Shaki, Fischer & Petrusic (2009). Whilst English and Arabic speakers possessed left-right and right-left SNARCs respectively, Hebrew speakers, who read words from right-to-left but numbers from left-to-right, merely showed a weakened SNARC effect. 380 Fischer, Shaki & Cruise (2009) 381 Bachtold, Baumüller & Brugger (1998) 382 Fischer, Mills & Shaki (2010) 383 Schwarz & Keus (2004), Ito & Hatta (2004), Shaki & Fischer (2012) 384 Fischer (2012)

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steps to reach. Alternatively these effects could be seen to result from linguistic

conventions, whereby larger numbers are described as higher and smaller numbers as

lower.385

As a result of the failure of the reading and writing hypothesis to fully explain the

emergence of SNAs, some have argued that finger-counting practices may also have a

significant role to play.386 The association of small numbers with the left side of space

could result from subjects tending to begin counting on their leftmost digit on their left

hands. This hypothesis is to some extent supported by the finding that subjects that start

counting on their right hand show negligible SNARC effects.387 However, if finger-

counting were the primary determinant of SNAs one would expect these subjects to

exhibit a reversed SNARC effect. Furthermore, in experiments where subjects are made

to keep their hands flat on a table or to cross their hands, smaller numbers are found to

be associated with whichever digit is furthest to the left, suggesting that spatial position

is more important than orthodox finger counting strategies in determining SNAs.388

Thus, whilst finger-counting, like language, may play some role in shaping SNAs it seems

unlikely that it is the primary determinant. One could even argue that the ability to

engage in finger-counting strategies is an effect of already possessing an association

between numbers and space, rather than the cause.

Given that neither language nor finger-counting can fully explain the origins of

SNAs, some have suggested the presence of SNAs might be, in a certain sense, innate. At

first sight, this may seem like a strange suggestion, since SNAs are a misrepresentation

of reality. Asymmetry between small quantities appearing in the left hand side of space

and larger ones on the right certainly isn’t a feature of our evolutionary environment. As

such one might wonder how such an arbitrary spatial bias could be the result of

evolution. Despite this there is mounting evidence for some kind of innate asymmetry

with regards to the association of space and number. For instance, newborn chicks have

been found to show a leftward bias when given the task of locating an object based purely

on its ordinal position in a line of identical objects.389 Rhesus monkeys have also been

shown to map number onto space and to show SNARC-like effects.390 Furthermore,

385 However, this merely begs the question as to the origins of these linguistic conventions. The linguistic convention could equally be taken as evidence for the existence of an underlying cognitive association between number and space (see Lakoff & Nuñez (2000)). 386 Fischer & Brugger (2011) 387 Fischer (2008) 388 Plaisier & Smets (2011) 389 Rugani et al. (2010) 390 Drucker & Brannon (2014)

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human infants as young as 7-months-old, who clearly lack the capacity for reading or

finger-counting, show a preference for left-to-right oriented numerical sequences.391

Taken together, this evidence from nonhuman species and human infants suggests that

the basis of SNAs may indeed be innate. Whilst the existence this arbitrary bias may at

first seem odd, it could potentially be explained in terms of the wider framework of

attentional biases. In order to achieve exploration of space it is necessary to make some

choice as to which side of space to direct attention to first. In the absence of particularly

salient stimuli it is necessary to make some arbitrary choice as to which side of space to

choose to attend to first. As such, there are good reasons to have a predetermined yet

arbitrary attentional bias in exploring visual space. It is likely that this kind of attentional

bias could then also form the basis of the asymmetry that enables the formation of

SNAs.392

There are three distinct questions concerning the origins of SNAs. Firstly one can

question the origin of the association between numbers and space. Secondly, one can

question the origin of the asymmetry in our spatial representation that allows for the

possibility of ordinal relations being tied to a particular direction in space. Thirdly, one

can ask about the origins of a particular direction in space being associated with

numbers. It seems as though the capacity for relating numbers and space is innate, as is

the tendency to represent space asymmetrically. However, the exact orientation of our

SNAs is determined by a number of distinct factors and is fragile and context-dependent.

It is likely that physical invariants, reading and writing habits, finger-counting strategies

and contextual factors all contribute to the orientation of a given SNA on a given

occasion. However, despite their cultural dependency and context-sensitivity, the basic

mechanisms that underlie SNAs are best understood in terms of natural features of the

way in which we represent and interact with space. Our natural tendency to impose an

order onto space enables the generation of ordered sequences of actions, such as finger

counting routines. As a result ANS representations and linguistic representations can be

augmented with the required representational apparatus to form fully-fledged number

concepts.

391 de Hevia et al. (2014) 392 de Hevia, Girelli & Macchi Cassia (2012)

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Ingredients of Embodied Number Concepts

It is now possible to outline the EC account of number concepts. All of the

ingredients required for sophisticated and precise number concepts can be accounted for

in terms of representations from sensory and motor systems. The ANS can be

understood as a perceptual system which provides approximate representations of the

number of objects in a collection. This capacity can be spelled out in terms of perceiving

affordances for attendability. Perceptual representations from the ANS seem to partially

constitute our number concepts, even when we are not directly engaged with perceiving

a collection of entities.

These representations are insufficient to account for number concepts on their

own due to their inherently approximate nature. In order to go beyond these

approximate representations we need some means of representing number such that

each distinct cardinality has a precise and distinguishable representation.

Representations of external number words and symbols are able to fulfil this role due to

the iconic and systematic nature of numerical language. The sensory and motor systems

involved in speaking, hearing, reading and writing number words and numerals support

distinct representations for each number. As such, embodied representations of

numerical language can also be seen to partly constitute number concepts.

However, combining ANS representations with linguistic representations is still

not enough to yield sophisticated number concepts. Linguistic representations fail to

capture the link between ordered sequences and cardinalities. In particular they fail to

capture the notion of a unique successor function. In order to capture this notion we

need to engage in some form of stable ordered sequential action. Finger counting is one

example of the kind of action that can fulfil this role. As such, our number concepts are

also likely to be partly constituted by the sensory and motor representations involved in

finger counting. Ordered sequences of actions of this kind and their associations with

number concepts are made possible by our natural tendency to associate number with

space. This natural sense of ordinality emerges out of a tendency to represent space

asymmetrically, which is then shaped and manipulated by cultural practices and

contextual factors. Our number concepts are, thus, entirely built from sensory and motor

representations. There is no need to posit purely cognitive amodal symbols for number

or to insist that our number concepts must be constructed from the top down using

general logical principles. Number concepts are embodied.

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Embodied Cognition and Conceptual Heterogeneity

An upshot of an EC account of number concepts, which distinguishes it from

traditional amodal approaches, is that one would expect a degree of heterogeneity with

respect to the vehicles that support numerical cognition. There are clear differences in

the kinds of sensory and motor processes that different individuals and different cultures

employ in learning and developing their number concepts. As a result of these

differences, an EC account would predict differences in the kinds of sensory and motor

representations that constitute number concepts. In some cases one would expect these

differences to manifest themselves in terms of differences in performance. This

prediction is vindicated by studies comparing the numerical cognitive capacities of

subjects whose arithmetical training was predominantly linguistic with subjects whose

arithmetical training focussed more heavily on the use of motor capacities, such as

students trained in the use of abacuses. Studies have shown that there are significant

differences between linguistic and abacus trained subjects. Abacus trained subjects are

able to perform calculations much faster and are less likely to be influenced by irrelevant

information regarding the physical size of the target objects when engaging in numerical

tasks.393 Many will have encountered an annoying friend who attempts to distract them

from a numerical task by shouting out irrelevant numbers.394 However, abacus users are

also less susceptible to this form of verbal interference.395 Neural activation during

numerical tasks also differs between linguistic and abacus trained subjects, with the

latter showing stronger activation in visual and motor areas associated with the

manipulation of abacuses.396 Remarkably, one abacus-trained patient, whose

arithmetical abilities were hampered by a stroke that caused a lesion to her language

system, was able to recover her abilities by actively focusing on using mental abacus

strategies. As EC would predict, the recovery of her arithmetical capacities was

accompanied by a shift in neural activation to more visuospatial areas when engaging in

arithmetical tasks.397 In line with the predictions of EC, the nature of the representations

that end up augmenting the ANS depends upon the particular experiences of the

individual during arithmetical training.

The inherent heterogeneity of embodied number concepts implies that there is no

simple recipe for building a number concept. Different individuals might use different

393 Hatano, Miyake & Binks (1977), Wang et al. (2013) 394 Although, I’d like to hope that your friends are not quite so annoying! 395 Frank & Barner (2011) 396 Chen et al. (2006) 397 Tanaka et al. (2012)

148

combinations of neural systems to accomplish the same goal of representing number. As

such, the question of whether or not someone possesses sophisticated number concepts

cannot be decided on the basis of whether they have acquired a particular array of

representations but instead must be decided on the basis of whether the particular array

of representations that they happen to use are capable of fulfilling the required role. We

should not assess the presence of number concepts in terms of what these concepts are

made of but instead in terms of what they can do.

Another consequence of this conceptual heterogeneity is that the nature of

number concepts can be expected to change over time as new numerical technologies

develop. An EC approach implies that our number concepts are partly constituted by the

way in which we engage with external numerical technologies and so as new technologies

develop we should expect our number concepts to change. As has been emphasised

earlier, the orthodox view of mathematics is that it is a purely cultural development with

no natural basis. Up until this point much emphasis has been placed on countering this

viewpoint by highlighting the role of natural mechanisms in numerical cognition.

However, by emphasising the significance of sensory and motor engagement in building

our number concepts, an EC account can explain the transformative effects of numerical

technologies, whilst maintaining that our number concepts are fundamentally rooted in

our natural capacities for perception and action. Furthermore, the way in which

numerical technologies have developed over time can be seen to be intimately linked to

the natural capacities such technologies augment. Thus, in order to develop a deeper

understanding of the roots of our knowledge of number, it will be necessary to

investigate the interactions between our innate numerical capacities and the numerical

technologies that enable sophisticated arithmetical reasoning. However, before

addressing this task, it is time to take stock and return to Benacerraf’s epistemological

challenge in order to assess the impact of the various findings from the cognitive sciences

that have been covered so far.

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5

Perceptual Access and Ontological Parity

The processes through which we acquire arithmetical beliefs are in no way as

mysterious as Benacerraf’s initial challenge might have suggested. These beliefs are

fundamentally rooted in the way in which we perceive and interact with our everyday

environments. They are beliefs about the things that we can do. Our best theories of

numerical cognition suggest that at least some of our mathematical beliefs are accessed

via perceptual processes. Thus, our scientific understanding of the mind seems to

undermine the conclusions of Benacerraf’s challenge. Our basic forms of numerical

cognition are primarily accomplished using the same basic mechanisms that we use to

interact with the physical world. Our number concepts are constituted by activation in

systems responsible for numerical and spatial perception.

Benacerraf’s challenge arises as a result of giving ontological concerns priority

and only then going on to address epistemological issues. However, when one first

attempts to provide a naturalist account of the epistemology of arithmetic, a potential

solution to the challenge becomes available. The mechanisms through which we access

mathematical knowledge are no different in kind from those with which we access

knowledge of ordinary physical objects and, as such, one’s choice of ontological stance

with respect to mathematical entities should be constrained so as to reflect this fact.

As was argued in the first chapter, Benacerraf’s challenge seems to be a valid

argument. However, our best empirical evidence about the nature of the mind seems to

challenge one of the possible conclusions of the argument. Given that the challenge

comes in the form of a dilemma, we are thus left with two courses of action. The first is to

drop one of the assumptions that motivated the first horn of the dilemma. The second is

to either bite the bullet by accepting the problems of the second horn or to find a way

around them. The strategy here will be to argue that the intuitions that motivate

Benacerraf’s challenge in the first place, when combined with the evidence for numerical

perception, directly undermine at least one of the assumptions that drive the first horn of

the dilemma. As a result one should accept the unproblematic nature of our access to

mathematical content. Furthermore, adopting such a position puts significant

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constraints on the nature of one’s overall ontological position. However, avoiding the

first horn of the dilemma need not force one to bite the bullet and fall on the second

horn. The way in which the assumptions of the first horn are undermined suggests a new

way of interpreting the challenge posed by the second horn which renders it avoidable.

The Access Parity-Ontological Parity Principle

One of the central assumptions of Benacerraf’s challenge is that the abstract

nature of mathematical entities renders them inaccessible. The intuition behind this

seems reasonable. If the entities in question occupy an entirely separate realm of

existence from the agents in question then there is no way that the agents could have

epistemic access to the entities. We have epistemic access to everyday objects of

perception, such as tables and chairs, because we are physical, they are physical and a

physical process, perception, links us with them. The same cannot seemingly be said for

abstract objects, since, if such entities have such a different kind of ontological status

from our own status as physical entities, there is seemingly no physical process that

could link their realm with ours. Thus, one of the central assumptions of Benacerraf’s

challenge is that, if a type of entity has a suitably different ontological status from the

entities of our physical realm then it is cognitively inaccessible or, at the very least,

inaccessible via any naturally explainable means of access. This implies that if a type of

entity is cognitively accessible via a naturally explainable means of access then the

entities in question must be ontologically on a par with the entities of our physical realm.

If we tell the same story about how we acquire beliefs about one kind of entity as we do

with another type of entity then we should take the ontological status of both kinds of

entity to be the same. These considerations motivate the following general Access Parity-

Ontological Parity (APOP) principle.

APOP: Cognitive access parity implies ontological parity.

At first sight, this principle may seem somewhat suspect in the sense that an ontological

conclusion is entailed by a fact about epistemic origins. However, this worry should be

dampened by the widespread acceptance of Benacerraf’s use of the contrapositive

according to which ontological disparity implies cognitive inaccessibility. It is generally

regarded as uncontroversial that epistemic claims can be justified on the basis of

ontological claims. However, this acceptance is enough to guarantee the validity of some

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derivations of ontological claims from epistemic claims, on pain of violating the logically

valid principle of contraposition.

Furthermore, the intuition behind the principle seems consistent with the

common philosophical practice of using the everyday objects of perception as

metaphysical yardsticks. When engaging in metaphysical enquiry, many philosophers

will simply assume the existence of the medium sized dry goods that we perceive, such as

tables and chairs, and assess the ontological status of a more exotic category of objects by

asking whether they are as real as the familiar objects. The only obvious reason for giving

medium dry goods such a privileged role in our ontology in the first place is that we have

a pretty good idea about our cognitive access to them. Thus anything that we know to be

accessed in a similar way should be awarded a share in this privileged role.

The main conclusion thus far has been that our access to some mathematical

entities is via ordinary processes of perception. As such, in line with the original

reasoning behind Benacerraf’s challenge and motivated by the APOP principle, we

should infer that some mathematical entities are ontologically on a par with the everyday

objects of perception. Our access to some numbers is the same as our access to tables

and chairs and as such our attitude to both categories of entity should be the same. These

considerations suggest that ‘the problem of access to mind-independent mathematical

objects is misconceived. The mystery is not in the ability to perceive mathematical

objects, but in the ability to perceive any “object” whatsoever’.398 Thus, in many ways the

position advocated here echoes Gödel’s claim that access to mathematical objects via

intuition is no more mysterious than access to ordinary objects via perception.399

However, where Gödel adopts Platonism and insists that all mathematical objects can be

accessed using intuition, the findings of cognitive science suggest that we have

perceptual access to some mathematical entities as well as to ordinary objects.

At this stage it is important to be clear as to exactly what perceivable

mathematical facts are taken to be on a par with. So far all that has been mentioned are

the ordinary everyday objects of perception. This relatively imprecise and vague

terminology has been chosen for a reason. Numbers are to be taken as ontologically on a

par with whichever objects we have access to using our perceptual systems, where our

belief in the existence of such objects is justified on this very basis. For example, most

accept that tables and chairs exist and justify their acceptance of the existence of tables

398 Voorhees (2004) pg. 88 399 Gödel (1947)

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and chairs on the basis of their immediate perceptual contact with them and so for such

people there are some mathematical entities that should have the same status as tables

and chairs. There may also be some disagreement as to whether we perceive objects

themselves or the properties of objects. The consequence of the APOP principle is that

some arithmetical facts achieve parity with whatever kinds of entity one takes to be the

primary content of perception.400 Depending on one’s particular philosophical position,

exactly which objects are taken to be the objects of perception and the specific

ontological status one bestows on such objects will vary. However, regardless of one’s

specific position on these issues, everybody agrees that, in some sense, we see objects

and, as such, everybody has something to regard numbers as on a par with.401

It is also important to try to clarify exactly which mathematical entities are taken

to be perceivable and, as such, on a par with the objects of perception. However, this is a

far from easy task. If one were being extremely strict one could limit the perceivable

numbers to those that we are able to reliably perceive directly. On such lines one might

restrict perception of mathematical entities to just the first three or four numbers. We

can perceive oneness, twoness, threeness and fourness but not fiveness. However, this

seems overly restrictive. On more occasions than not we are able to accurately perceive

the number of entities in relatively small collections greater than four. Furthermore, our

less than perfect accuracy seems no reason to deny our capacity for perception, since we

do not set such high standards for perception in the case of ordinary objects. The fact

that one might mistake a cow for a horse on a foggy night is no reason to suggest that we

have no perceptual access to cows or horses. The possibility of misrepresentation is a

precondition for representation, so any account of our perceptual representation of

number must account for cases where we sometimes go wrong. As such, it makes sense

to see numbers that we only perceive approximately as also on a par with the objects of

perception.

Direct perception isn’t the only means we have for perceptually accessing

numerical properties of collections. We are able to verify the number of entities in a

collection by engaging in counting procedures. In principle, these could extend to any

400 The fact that there may thus be disagreement as to whether these arithmetical entities are on a par with objects or properties may to some extent be reflected in the fact that number words function both as nouns and as adjectives. 401 It should be noted that some might argue that we do not perceive objects at all but rather that we perceive sense-data (see Huemer (2011) for a review of various arguments of this sort). However, in cases like this and other similar cases where it is claimed that we do not perceive objects, it is possible to argue that numbers should be on a par with sense-data or whatever else is invoked. Thus, “objects of perception” is to be taken to loosely refer to whatever it is that one’s favourite theory of perception claims that we have perceptual access to, rather than committing to any specific theory that entails the perception of objects.

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arbitrarily large number and in practice the numerical properties that are accessible in

this manner go way beyond those that can be directly perceived. Some might argue that

counting procedures do not really count as perceptual access and involve a different form

of access to our usual access to everyday objects and so do not provide any support to the

APOP principle. For example, one might argue that these procedures fail to qualify as

perceptual, since they tend to involve active manipulation of members of the collection

under consideration, as well as some internal memory based representations. However,

it seems unfounded to limit our conception of perceptual access in this manner. For

instance, in order to be sure that an object is a cube it may be necessary to manipulate

the object and view it from different angles, whilst keeping past perspectives on it in

mind, but this seems to be an absurd reason to deny that we access the relevant

information perceptually. A further reason for extending the notion of perceptually

accessed number beyond the first few numbers is that we are able to access certain

numerical properties of very large collections perceptually. For instance, when

confronted with a collection of 500,000 entities and a collection of 1,000,000 entities,

we are able to perceptually discern that the latter collection contains more entities than

the former, even if we are unable to discern the exact number of entities in each

collection. In other words we are able to perceive that these collections differ numerically

even if we are unable to discern the precise numerical properties that are exemplified by

each collection.

Given these considerations, there may be no definitive answer as to which

arithmetical entities are perceivable. This is unsurprising given that numbers are, on the

current framework, interpreted as opportunities for a certain kind of action. Exactly

which numbers are perceivable will be determined in part by which such actions are

possible and in part by which possible actions we are able to perceive. Both of these

factors are contentious issues. The main issue for current concerns is that, at the very

least, the first four numerical properties can be understood as being perceivable.

Furthermore, it seems reasonable to assume that the perceivable numerical properties

extend far beyond this, even if it is not possible to be clear or certain regarding how far.

The fact that any numerical properties are perceivable is enough to motivate a response

to Benacerraf’s challenge. This guarantees that there are at least some mathematical

entities that should be seen as on a par with the everyday objects of perception.

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Rescuing Universal Semantics

Despite seemingly overcoming the problem of inaccessibility by bringing

arithmetical properties down to earth, it might seem at this stage as if the horns of

Benacerraf’s dilemma have not been avoided. The claim that mathematical knowledge

can only be attained by impossibly accessing the realm of abstract entities may have been

brought into doubt. However, in doing so the worry remains that all that has been

achieved is an elaborate gymnastic manoeuvre ending in gory impalement on the other

horn of the dilemma, namely, the problem of failing to provide a universal semantics.

The other horn remains a threat, since the objects of our arithmetical beliefs have been

reconceived as affordances or possibilities for action. As a result, this would seem to

suggest that statements involving apparent reference to mathematical entities require a

special kind of translation, whilst other ordinary nonmathematical statements do not,

thereby threatening universal semantics. Statements about cats, rocks and trees seem to

be statements about the entities that exist out there in the world, whereas statements

about sets, odds and threes seem to be statements about possible actions.

One option is to simply bite the bullet and accept that the goal of a unified

account of semantics for natural languages is unattainable, even in principle. The surface

grammatical structure of our sentences has no immediate bearing on the underlying

meaning. Instead in order to analyse the commitments of a particular sentence it would

be necessary to determine whether that sentence contained any mathematical content

and then to translate it accordingly. For example, sentence (1) could be translated as

something like sentence (1*)

(1) “There are three birds in the garden”

(1*) “It is possible to carry out a three-stage sequential attention procedure with respect

to the birds in the garden.”

In principle, it seems as though any statement involving numbers could be translated in

this manner. Failing to tackle the second horn of Benacerraf’s dilemma need not be seen

as disastrous. There is no particularly good reason to assume that the surface grammar

of our everyday language is transparent with respect to its content and there are many

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innocent examples where it is natural to engage in translational practices of this kind.402

For example, most are happy to translate sentences like (2) into sentences like (2*)

(2) “Nobody turned up to the meeting”

(2*) “It is not the case that somebody turned up to the meeting”

Furthermore, acceptance of the second horn of the dilemma has been advocated by

some, such as Hellman, who also advocates translating mathematical statements into

statements about modality.403 As will be addressed in more detail below, the account on

offer here may be more promising than accounts such as Hellman’s when it comes to

answering Benacerraf’s dilemma. Hellman’s translation forces us to take mathematical

claims to be about possibilities that are no more accessible than the abstract entities that

they are supposed to replace. However, the possibilities invoked here are supposed to be

directly perceivable and hence epistemologically innocent.

Despite providing a more satisfying answer than other accounts which bite the

bullet and fall on the second horn, an account which accepts heterogeneous semantics

can still be seen as less than satisfactory. It would be far better if there were a way in

which sentences that refer to mathematical entities could be understood in the same

manner as sentences that don’t. This is particularly important given the problem of

interpreting statements where arithmetical and ordinary content is so intermingled as to

be inseparable. Furthermore, the intuitions that motivate the call for ontological parity

could be brought to bear on the issue of semantic parity. In a similar vein, one could

argue that the fact that we access ordinary and mathematical content in the same

manner supports our interpretation of sentences pertaining to each type of content in a

single unified manner.

Thankfully, a way out is suggested by considering the action-oriented approach to

perception that motivated the reconstrual of arithmetical statements in the first place.

The first step lies in considering our reasons for believing in everyday objects. We believe

in these objects because they are the immediate objects of perception. When we refer to

these ordinary objects we take ourselves to be referring to the very same things that we

402 Mathematics need not be the only domain in which translations are required to access a sentences true content. For example, quasi-realists in meta-ethics argue that a similar kind of translation procedure is required when dealing with moral claims in order to reveal the true content behind the surface grammatical structure (Blackburn, 1993). 403 Hellman (1989) pg. 16-18

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perceive and act upon. However, according to the action-oriented approach to

perception, the objects of perception are affordances. Thus we are left with three options

1. Eliminate reference to ordinary objects of perception and replace it with reference

to affordances.

2. Understand reference to ordinary objects as on a par with reference to

affordances.

3. Take reference to objects to imply reference to affordances and reference to

affordances to imply reference to objects (broadly construed).

The first option would allow the preservation of universal semantics by

advocating a similar translation procedure for all sentences. Just as sentences that

appear to refer to mathematical entities should be translated into sentences about

possible enumerative actions so too sentences about ordinary entities should be

translated into sentences about other types of possible action. For example, the sentence

“there is a chair in the room” could be translated as “the state of the room is such as to

make possible the action of sitting” or “a certain region of the room affords sitting”. Such

translations are without doubt unwieldy and somewhat unnatural. However, this would

explain why we tend to favour far simpler sentences that make reference to objects and

properties. Following the first option involves asserting that this advantage in referential

simplicity is the only reason that we talk about the ordinary objects of perception. Deep

down we are really referring to possible actions for an agent. To say that something

exists is just to say that there are some possible actions. When we are apparently

referring to ordinary entities what we are actually referring to are organism-relative

action-oriented affordances.

The first of these options will be considered far too radical by many. Surely if

anything exists, the ordinary objects of perception, such as tables and chairs, rocks and

trees, exist. Thankfully, this radical form of eliminativism is not the only way to preserve

universal semantics. Rather than taking affordances to be more fundamental than the

existence of everyday entities, it is possible to argue that the two are on a par. ‘The bare

and obvious reality of affordances – opportunities and dangers in the environments of

an organism – really ought not to be denied, any more than the reality of rocks and trees

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ought to be denied’.404 It isn’t necessary to choose which of objects or affordances are

ontologically more fundamental, since object-talk and affordance-talk might just be two

ways of talking about the same thing. It seems intuitive to understand the claim that

there is a chair in the room as saying much the same thing as that there is an affordance

of sit-ability present.

One could object that the apparently intuitive nature of the parity in this

particular case is merely the result of the idiosyncratic artefactual nature of chairs.

Chairs are designed and created with a particular action in mind and so it is unsurprising

that they can be described in action terms.405 When one considers sentences about less

artificial entities the parity between objects and affordances becomes less clear. For

example, it is unclear what actions clouds, trees or tigers afford or whether talk of

affordances can capture all that we mean when we mention such entities. At face value,

the actions afforded by a tiger fail to capture all we mean by tiger, since, for instance,

they don’t seem to capture anything of the tiger’s stripes.406 However, this problem need

not be insurmountable. As has been mentioned earlier, we are prone to only considering

affordances that relate to relatively large-scale deliberative actions, such as sitting,

picking up or running away. However, the action-oriented account of perception is

committed to a far richer realm of affordances, including low-level perceptual

affordances. As such, it may be possible to explain how construing sentences about, for

example, tigers, can preserve the rich content that we associate with objects. Talk of a

tiger is not simply talk of the possibility for running away but also talk of the possibility

of moving our heads so as to get a different glimpse of its particular stripy pattern.

In essence the move being made here is the same as the move that Putnam

advocates for mathematical entities but writ large for all cases of ontological

commitment.407 Paraphrasing Putnam, one could say that ‘we can reformulate’ our

metaphysics ‘so that instead of speaking of’ tables, rocks, trees ‘or other “objects”, we

simply assert the possibility or impossibility of certain’ actions.408 Any statement about

404 Sanders (1997) pg. 100 405 It is notable that philosophers are prone to using artefacts, such as tables and chairs, as paradigmatic examples of objects whose existence is uncontroversial. This tendency could be seen as indicative of a tacit acceptance of the significance of affordances for ontology. Tables and chairs can be seen as paradigmatically real objects because their affordances are relatively apparent and easy to comprehend, whilst clouds and tigers are less obvious as examples, since the actions that they afford are far more complex. This point is merely speculative. However, if some reason for philosophers’ preponderance for talking about tables and chairs can be provided, which goes beyond citing their tendency for a sedentary lifestyle, then this is one more reason for philosophers to favour an affordance-based account! 406 Presumably a dangerous cat this big would afford running away regardless of whether it has stripes or not 407 Putnam (1983) 408 Putnam (1994) pg. 508

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the existence of an entity can be translated into a sentence about the opportunities for

action that it affords. Furthermore, any statement about possibilities for action could be

rephrased as one that commits us to the existence of a certain state of the world. In

practice, the particular mode of talking one chooses shouldn’t have any implications for

the consequences of accepting the sentence in question. One can sit on a chair whether

one sees it simply as a chair or as an opportunity for sitting. If one adopts this approach

it is possible to maintain a universal semantics for both mathematical and ordinary

sentences whilst preserving the possibility of differentiating between the two. Sentences

about numbers pertain to a certain kind of affordance, namely, possibilities for

sequential action. Sentences about other entities pertain to other, perhaps more complex

affordances or sets thereof. Talk of tigers involves commitment to a rich complex of

affordances that may involve run-away-ability, move-your-eyes-to-see-stripes-ability

and many more affordances that are at least as hard to express in normal linguistic

terms. However, talk of both numbers and tigers can, at heart, be interpreted as talk of

possibilities for interaction with the world.

This second option may still be unpalatable for some who may insist that in

positing the existence of an object we are doing something very different from

committing to the possibility of certain actions. It is certainly intuitive to insist that

sentences starting “there is a…” have a very different underlying meaning from those

that start “it is possible to…”. On an orthodox philosophical reading, the former only

commits us to the existence of certain states of the actual world, whilst the latter

commits us to certain states of merely possible worlds. Without getting into too many of

the messy details of the ontology of modality, it may be possible to preserve something

close to a universal semantics without undermining this fairly central philosophical

intuition. Rather than seeing affordances as fundamental or seeing claims about

affordances as on a par with claims about objects, it may be possible to preserve some

semblance of a universal semantics by arguing that each type of claim implies the other.

When we say that a particular entity exists in the actual world we thereby commit

ourselves to the possibility of some kind of action with respect to that entity. To see why

this approach is compelling it helps to consider what it would mean for this not to be the

case. Committing to the existence of an entity that affords no possible actions would

seem to involve accepting the existence of things that could in principle play no role in

our understanding of the world. Given the action-oriented approach to perception, an

entity that affords no action would be unobservable, in principle, since any evidence we

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would have for such an entity would presumably involve some kind of interaction with

the entity in question. An entity without affordances would therefore be unknowable and

thus theoretically superfluous. If this seems like too bold a step, it is worth considering

the minimal nature of action involved with affordances. Even changing one’s position

with respect to an object can count as an action with respect to that object. Thus, in

committing to the existence of an actual entity without affordances one would have to be

committed to a physical entity without location. Given these considerations it seems

reasonable to assume that any sentence of the form “there is an x” implies a sentence of

the form “actions a1, … an with respect to x are possible for an agent S”.

The idea that talk of possible actions can be understood to imply talk of entities is

perhaps less controversial. The reason for this is that reification is relatively cheap. It is

easy to see that any sentence of the form “actions a1, … an with respect to x are possible

for an agent S” can be taken to imply a sentence of the form “there is a y such that y

affords actions a1, … an”. In other words it is always easy to generate a sentence which

treats “affordances” or “opportunities” or even “possibilities for action” as entities in

their own right. At first sight, this move may seem like sleight of hand. The implied

sentence only contains reference to dubious entities, such as affordances or possibilities,

whereas in the converse case it was argued that statements about more commonplace

entities imply the possibility of action. Some might thus be suspicious of this asymmetry

in the implications of the two kinds of sentences. However, it important to keep in mind

that we are only looking for an analysis that yields something close to a universal

approach to semantics. Thus all that is needed is some method for deriving a sentence

that makes an ontological commitment in the manner of a usual existential claim from a

sentence about possibilities for action. As long as some such sentence is implied, the

supposed dubiousness of the entity quantified over is by the by. Those that doubt the

existence of entities such as “opportunities” or “affordances” are welcome to evaluate the

implied sentences as false. All that we are trying to establish is a unified framework for

evaluation.

This third option wouldn’t strictly involve preserving a universal semantics.

Statements about numbers could strictly be seen as statements about affordances, whilst

statements about ordinary entities could be taken at face-value. However, since each

statement of one form implies a corresponding statement of the other form, the

motivation behind the desire for universal semantics could be saved. Since any

statement about objects will imply a corresponding statement about affordances and any

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statement about affordances will imply a corresponding statement about entities any

pair of these corresponding statements will stand or fall together.

The APOP principle only suggests that we should adopt the same attitude to some

mathematical entities as we adopt towards the ordinary objects of perception. This

needn’t seriously threaten the need for universal semantics, since we can interpret

claims about perceivable objects either as claims about affordances or claims that imply

claims about affordances. However, a potential upshot of this is to undermine the

possibility of a universal semantics for observable and unobservable entities. It is less

than immediately obvious how claims about quarks or entities that exist beyond the

light-cone of any agent could be understood in terms of affordances. It isn’t possible to

delve into this issue in a satisfying level of detail here. However, it is important to

highlight that there are options available. One option would be to adopt a position like

Hacking’s Entity Realism, according to which we are justified in adopting a realist

position with respect to some unobservable entity as long as it is possible to carry out

some interaction with the world using that entity.409 ‘The final arbitrator in philosophy is

not how we think but what we do’.410 Another option for understanding unobservable

entities in terms of affordances is to follow Sanders and consider the perspectives of

possible but non-actual agents.411 Along these lines, electrons could be seen to afford

possible actions for miniscule possible agents. Whilst this may seem fanciful at first, it

need not be seen as any more controversial than when physicists consider the

perspective of “observers” moving at close to the speed of light despite the fact that no

physically possible biological system could do so.412 It may turn out that neither account

is viable, threatening the notion of a universal semantics for both observable and

unobservable entities. Even so, the fact that it is possible to provide something like a

homogenous semantics for mathematical claims and claims about ordinary objects

remains.

Much more could be said about the relationship between ordinary objects and

affordances, since this is a relatively underexplored topic with potentially revolutionary

implications for many traditional philosophical assumptions. However, such an

exploration would take us too far off course. It suffices to say that adopting an

affordance-based view of numerical perception need not imply a failure to overcome the

409 Hacking (1983) pg. 22 410 Ibid pg. 31 (This compatibility between Hacking’s position and an affordance based ontology is noted by Chemero (2009) pg. 192) 411 Sanders (1997) pg. 109 412 Sanders (1999)

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second horn of Benacerraf’s challenge. An action-oriented view of perception implies

that affordances are relevant for all of the things that we ordinarily perceive and not just

for explaining numerical perception. As such any translation or implication that is

required to make sense of reference to number can be mirrored in the case of explaining

our reference to the objects of perception.

Ontological Neutrality

At this stage it may seem as though we are forced into accepting some kind of

realist conception of numbers. After all, it has been argued that numbers are amongst

the things that we perceive in the same way that we perceive everyday objects. However,

it is important to note that the need for ontological parity is neutral with respect to the

ontological status of mathematical entities. All that it requires is that some mathematical

entities are treated the same as the everyday objects of perception. However, this leaves

it wide open as to the correct ontological attitude to both.

It is generally assumed that, if mathematical entities exist at all, then they are

abstract entities. Secondly, it is usual to assume that abstract entities are not the kinds of

things that we can perceive. Thirdly, the everyday entities of perception are usually taken

to be both real and concrete. However, given the evidence surveyed so far it seems that at

least one of these three widespread assumptions will need to be dropped. As such, one’s

choice of ontological position will be influenced by which assumption one is willing to

drop.

One option is to drop the first assumption that all mathematical entities are

abstract. Instead one could take some mathematical entities to be constituents of the

physical world. We perceive some mathematical entities in the same manner that we

perceive some concrete physical entities and, thus, we should take these perceivable

mathematical entities to be concrete too. Taking this line with respect to perceivable

mathematical entities may have interesting consequences for our understanding of the

mathematical entities that we don’t perceive. Once one has admitted that mathematical

entities can be physically realised this opens up the possibility of dividing up our

mathematical ontology in a similar manner as we do our physical ontology. For example,

there may be real mathematical entities that are perceivable in principle though not in

practice. Large numerals could be taken to refer to real possibilities for enumerative

action that are only rendered practically impossible by the finiteness of our lifespans and

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the need to engage in practical activities other than counting. Furthermore, it may be

possible to accept the existence of mathematical entities that are unobservable in

principle. For example, one could admit the physical existence of the actual infinite,

despite the fact that the completion of an infinite enumerative act is impossible. At first

sight, this may sound like a very strange idea. However, it need not be considered any

stranger than admitting the existence of particles, such as photons, which are by their

very nature impossible to perceive using our visual systems. Our acceptance of ordinary

objects is based upon of ability to perceive them but this does not necessarily prevent us

from accepting other more exotic entities with respect to which we lack this ability.413

The fact that some mathematical entities can be posited on perceptual grounds in no way

rules out that others might be posited for other reasons. At the same time, admitting the

existence of unobservable mathematical entities is, by no means, mandatory once one

has admitted some mathematical entities as concrete. One could easily combine a

concrete view of mathematical entities with a finitism motivated by the desire to only

admit perceivable entities.414 The important point is that any decision on such a matter

should be motivated by considerations about the nature of the physical universe rather

than by considerations of mathematics in isolation. If mathematical entities exist in a

concrete sense then any decision about which such entities exist should be a contingent

matter depending on our best empirical accounts of the nature of the physical world.

A second option involves dropping the assumption that abstract entities are

necessarily unperceivable. Along these lines one could maintain that the subject matter

of mathematics is abstract but suggest that this is not an obstacle to it being perceptually

accessible. Furthermore, in order to comply with the APOP principle one would have to

also assert that our ordinary perception involves perception of abstract entities. This may

seem like a very strange and counterintuitive position, particularly to those versed in

philosophical orthodoxy. However, such an approach can be motivated independently.

Firstly, it is important to note that the claim on offer would not be that all abstract

entities are perceivable. Only some mathematical abstracta need to be seen as

perceivable. Similarly only those properties that we normally take to be involved in our

everyday perception need to be reconstrued as abstract. To see how this could be made

to make sense it is worth considering two distinct notions of the relationship between

abstractness and perception. Certain properties, such as colours, are seen as abstract

413 Maxwell (1962) 414 For example, one might be motivated to only admit observables in the manner of Van Fraassen (1980) pg. 14-19

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because they are multiply instantiated. However, it seems very odd to suggest that we do

not perceive colour properties. Their abstractness is not a result of their imperceptibility

but a result of their being perceivable in many different instances. Other properties, such

as truth or justice, are seen as abstract due to the fact that they are not the kinds of thing

that could be perceived. Numerical properties could thus be seen as perceivable in much

the same sense as colour properties could be seen as perceivable.

This line of argument will seem extremely odd if abstract entities are defined as

those that do not exist in space-time. However, an alternative option is to define

‘abstract objects as those which need not exist in space-time’.415 For example, Hume’s

missing shade of blue, by definition, does not exist in the physical universe but the blue

of the lid of my pen does but might not have. Furthermore, when we consider the nature

of the perceivable properties in a bit more detail, they begin to look a lot more like

abstract entities than concrete ones. When one looks at the surface of a table one sees it

as a flat Euclidean plane of certain dimensions. However, modern science tells us that

this isn’t the true nature of the surface of a table. Viewed at a different scale the same

table would have an extremely complex structure and would be anything but flat. In

normal conditions, there is a sense in which we don’t see the true structure of the table.

Instead we see a Euclidean plane. However, a Euclidean plane is a paradigm example of

an abstract entity. The surface of the table that we perceive is obviously more than

merely a Euclidean plane. It has certain colour features and visible deviations from

perfect flatness. However, as has already been mentioned, colour properties can be

understood as abstract themselves. Furthermore, any deviation from a flat plane could

presumably be modelled by a more complex geometrical entity. The important thing to

note is that there is no reason to distinguish this complex geometrical entity from the

surface that we perceive since they would in principle share every single feature with

each other. There is no need to posit two structurally identical entities, one abstract and

non-physical, which we don’t perceive, and the other physical that we do. Instead one

could say that what we perceive is an abstract object.416

These two options of taking mathematical objects to be concrete or taking

ordinary objects to be abstract have received relatively little attention in the philosophy

of mathematics. Notable exceptions include Maddy (in her early work), Bigelow,

Franklin, Tegmark, Tymoczko and Nicholas Goodman, all of whom maintain that some

415 Tymoczko (1991) pg. 208 (emphasis mine) 416 Ibid. pg. 218

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mathematical entities can be thought of as real constituents of the physical realm.417

Maddy’s early views resonate with the picture on offer here in the sense that she takes

some mathematical entities to be directly accessed through perception. Her physicalist

realism is clearly compatible with the notion that some mathematical entities are

ontologically on a par with ordinary objects.418 However, Maddy goes beyond what is

being argued for here by claiming that we have perceptual access to sets as opposed to

numerical facts. Given the possibility of understanding numerical properties in set-

theoretic terms this approach is clearly compatible with the APOP principle. However,

more empirical evidence would be needed to support the claim that we have perceptual

access to sets, rather than to other kinds of entities with numerical properties.

Bigelow and Franklin both take a slightly different stance by endorsing an

Aristotelian form of realism with respect to number, whereby mathematical universals

are taken to be real constituents of the physical realm.419 As such both could arguably be

seen as Pythagoreans in the sense that they take physical reality to be at least partly

constituted by mathematical entities or structures. It is clear that both of their views are

compatible with the current framework in the sense that Franklin explicitly endorses

perceptual access to some mathematical entities that is on a par with access to ordinary

physical objects, whilst Bigelow argues that mathematics is ‘the theory of universals’ and

so all of our perceptual access to universals can be conceived of access to the

mathematics.420 Tegmark takes Pythagoreanism much further by arguing that our

physical universe is just an abstract mathematical structure. Perception of ordinary

objects is by definition on a par with perception of mathematical entities, since all

constituents of physical reality are nothing other than mathematical entities.421

Tymoczko and Goodman take a slightly different line by arguing for the

abandonment of the distinction between abstract and concrete in the case of

mathematical entities.422 This position is perhaps the most reasonable in light of the

current considerations, since there seems to be little difference between the two

positions of arguing that mathematical entities are concrete and of arguing that abstract

entities are perceivable. Abstractness is often defined in terms of nonphysicality or lack

of perceivability and so the fact that supposedly paradigmatic abstract entities flout these

417 Goodman (1979), Bigelow (1988), Maddy (1990), Tymoczko (1991), Franklin (2014) 418 Maddy (1990) pg. 50 419 Bigelow (1988) pg. 1, Franklin (2014) pg. 11-12 420 Franklin (2014) pg. 19-20 165-179, Bigelow (1988) pg. 16 421 Tegmark (2008) 422 Goodman (1979), Tymoczko (1991)

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criteria adds further weight to calls for the distinction between abstract and concrete to

be abandoned or heavily revised.

Both of the options considered so far line up with a form of mathematical realism,

according to which at least some mathematical entities are considered to exist in just the

same way that the ordinary objects of perception can be seen to exist. However, neither

the fact that we seem to perceive some mathematical objects nor the fact that they are on

a par with the ordinary objects of perception forces us into adopting a realist approach.

Instead, one could simply argue that our perceptions are, in some sense, systematically

misleading and adopt a fictionalist or subjectivist account of the mathematical entities

that we perceive.423 The idea of our perceptions being systematically misleading is a

slightly odd one, particularly given that “perceive” is often understood as a success term.

However, this idea is by no means novel. In order to see this, it is again useful to

consider a comparison with the case of colour. Despite the fact that the existence of

colour experiences is indisputable, one can dispute the fact as to whether colours

themselves exist independent of our experience of them. For instance one could argue

that a mature scientific theory could describe all relevant aspects of the world whilst

eliminating any reference to colours.424 Alternatively, one could argue that colours are

inherently subjective and mind-dependent properties and, as such, cannot be said to

exist in a robust sense.425 The existence of these approaches shows that anti-realism is a

viable option even with respect to entities that we seem to be immediately perceptually

aware of.

Furthermore, the fact that the mathematical entities that we perceive are here

defined in terms of affordances may lend itself to a form of anti-realism. For some, the

modal aspects of affordances may be enough to motivate anti-realism. If one were a

strict determinist, one would have good reason to doubt that we can really be perceiving

real possibilities for action, since at any one time there is only one predetermined course

of action. Along these lines, when we perceptually represent an action as possible we are

misrepresenting reality unless that action actually goes on to be fulfilled, since, strictly

speaking, only actual courses of action are possible. Even without committing to such a

strict deterministic view it may be possible to motivate a form of anti-realism with

423 In the case of the former position one could argue that we are systematically misled in believing that mathematical entities exist at all (e.g. Field, 1980, Yablo, 2005). In the case of the latter position one could argue that we are misled in believing that mathematical entities exist mind-independently (e.g. Ernest, 1998, Lakoff & Núñez, 2000). 424 Boghossian & Velleman (1989), Hardin (2003) (This would be on a par with Field’s, 1980, treatment of mathematical entities as eliminable). 425 Johnston (1992)

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respect to mathematical entities as affordances. As with the case of colour, one could

argue that affordances are inherently subjective and mind-dependent. What one

perceives as a possible action in the environment is arguably determined not just by the

state of the organism and the environment but also by the organism’s own subjective

characteristics. Along these lines, affordances could be seen as features that we project

out onto the environment rather than features inherent to it. For example, one could

argue that there is nothing objectively sit-able-on about a chair. The chair only acquires

this property with respect to the subjective experiences it causes in us.

The anti-realist option with respect to perceivable mathematical entities may still

seem unpalatable to some, since in order to comply with the APOP principle, it would

entail taking an anti-realist stance with respect to all of the objects of perception. In

other words, being anti-realist about small numbers would also involve being anti-realist

about chairs, tables, trees and tigers. There are two ways in which an anti-realist may be

becalmed about these potential worries. Firstly, the kind of anti-realism about everyday

objects on offer here is only anti-realism about them as perceptual objects. Thus, this

view is compatible with a scientific realism that justifies our belief in some entities on

theoretical grounds. Secondly, global anti-realist approaches of this sort, whilst radical,

are by no means new. For example, on certain interpretations, Kant’s transcendental

idealism can be seen as endorsing a form of global anti-realism with respect to the

objects of perception, at least in the sense of denying that they are objective and mind-

independent.426 When it comes to mathematical entities, the most common form of anti-

realism is nominalism. Nominalism is sometimes characterised as the rejection of

abstract objects and would thus be consistent with a realist approach to mathematical

entities that characterised them in concrete terms. However, nominalism can also be

characterised as the rejection of universals. When characterised in this sense,

nominalism is an anti-realist position compatible with the APOP principle. Along these

lines, one could reject that there is anything real that all instances of threeness have in

common. Furthermore, one could take a similar attitude to other perceptual objects, for

example, by denying that there is anything real that different instances of the property of

chairhood or tigerness have in common. As such, a global form of austere nominalism of

the kind advocated by Goodman and by Quine could be seen as a consistent way of

426 Rohlf (2014)

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maintaining anti-realism about perceivable mathematical entities whilst conforming to

the constraints of the APOP principle.427

It is not possible to survey all of the metaphysical options that comply with the

APOP principle here. It should suffice to say that, even once one takes on board the

constraints on ontology that a naturalistic response to Benacerraf’s dilemma entails,

there are still a wide range of ontological positions available. All that is important is to

maintain a consistent ontological attitude towards entities that are accessed in the same

manner as each other. However, it is also important to highlight that the APOP principle

puts pressure on certain theories which advocate different ontological attitudes to

mathematical objects and observables. One such theory is Constructive Empiricism. This

approach advocates a realist attitude towards observables whilst adopting anti-realism

with respect to the unobservable theoretical entities of science. As such, the Constructive

Empiricist is able to provide a response to indispensability arguments, by accepting the

parity between unobservable entities and mathematical entities but rejecting the

existence of both. Van Fraassen, the leading advocate of Constructive Empiricism,

admits to having no clearly worked out philosophy of mathematics but insists that it

would need to be an anti-realist one with respect to mathematical entities.428 However, if

one were to accept the APOP principle, this approach would not be viable since some

mathematical entities should be awarded the same ontological status as observables. As

such, the Constructive Empiricist should accept at least those mathematical entities that

we directly perceive.

Numerical Perception and Knowledge of Mathematical Modality

One of the benefits of adopting an account of numerical perception as perception

of affordances is that it allows for a vindication of modal accounts of mathematical

entities, whilst both preserving a uniform semantics and avoiding encountering a new

version of the access problem for modal entities. Putnam and Hellman can both be seen

as correct in asserting that a modal reading of mathematical statements is on a par with

an object-based reading.429 However, they need not have restricted this analysis to

mathematical statements. All perception is understood in terms of the detection of

possibilities for action. Therefore, our access to the modal content that is central to our

427 Goodman & Quine (1947) 428 Van Fraassen (1985) pg. 283 429 Putnam (1983), Hellman (1989)

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understanding of the nature of mathematical entities need not be any more mysterious

than our perception of ordinary entities. As we explore the world we perceive what it is

possible for us to do. Furthermore, we couldn’t perceive the world in the way that we do

if things were otherwise. Our perceptual capacities are in part explained by our ability to

represent and predict the outcomes of possible perceptual actions that may never

actually be fulfilled.430 Our perceptual representation of the actual depends on our

representation of the counterfactual. An upshot of this approach is that, for at least some

mathematical entities, it is possible to help oneself to Kitcher’s notion of mathematics as

the science of possible action without needing to commit to the notion of ideal agents.431

At least for small numbers, it is possible to explain our access to mathematical content in

terms of actions that are possible for real agents.

Things get more complicated when one tries to pin down exactly what possibility

is supposed to mean in this context. Even if one accepts that our perceptual capacities

give us access to possibilities for enumeration, it remains unclear as to what one should

say about our perception of collections that are difficult or impossible to enumerate

either in practice or in principle. As has already been mentioned, from a strictly

determinist point of view, only those enumerative actions that actually take place can be

understood as being possible. For most, this would be too strict a notion of possibility to

employ. However, it, at least, provides a determinate answer as to which enumerative

acts we perceive as being possible and, therefore as to which numbers qualify for

ontological parity with objects of perception. Once one goes beyond actual acts of

enumeration things get a bit messier. For example, one could choose to limit the possible

enumerative acts to those that a normal human being could feasibly achieve. There is

presumably some limit to the length of sequences of serial attention that a human could

perform in a lifetime. As such, one could argue that only these possibilities for action

should be on a par with ordinary objects. However, the problem arises as to which

person and which lifetime we should be taking into account. For example, humans need

to eat too, so should this kind of fact be taken into account when considering the limits of

possible enumeration? Taking such seemingly irrelevant features into account seems

somewhat odd. However, if one is to discount constraints of this kind then there seems

no reason to take our own mortality into account. As such, others may want to define the

notion of possibility in a manner that ignores contingent limitations, such as our own

430 Noë (2004), Friston et al. (2012), Seth (2014) 431 Kitcher (1984) pg. 110

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mortality, and consider possibilities for enumerative action in principle rather than in

practice.

Whilst the issue of how we should interpret possibility is a vexed one, the fact that

many competing interpretations are viable should be seen as a boon for preserving

ontological neutrality. The wide range of different ways of interpreting the notion of

possibility relevant to the perception of affordances gives rise to a wide range of

positions with respect to the ontological status of numbers. Some may simply reject the

existence of both numbers and ordinary objects of perception. Others might suggest that

only the numbers that we are able to directly perceive with a certain degree of reliability

should be understood as being on a par with ordinary objects. Others still may argue that

numbers that could be perceivable in principle should be admitted to one’s ontology.

What is clear is that our perceptual capacities do not involve built in

representations of our own contingent limitations. For example, we do not perceive that

some of our actions are limited by our inevitable but contingent mortality or by the fact

that we will eventually get hungry or bored and give up. These are empirical discoveries

that are not built in to our innate cognitive architecture and, although most of us learn

them at a very young age, they require learning. As such there is nothing prohibiting the

seemingly counterintuitive notion that we acquire beliefs about the potentially infinite

perceptually. The fact that we are finite beings with finite capacities provides no reason

for thinking that we are unable to represent the possibility of an infinite process

perceptually. Furthermore, studies of childrens’ understanding of the notion of infinity

suggest that their concept of infinity is primarily shaped by perceptual notions of

unending processes.432 Thus one of the benefits of taking our access to mathematical

beliefs to be primarily mediated by perceptions of the possibility for action is that it

allows for an explanation of how we form beliefs about the potentially infinite. However,

whether we take such beliefs to be true is another matter altogether. When dealing with

our perceptual access to possibilities for action that go beyond the humanly possible and

extend to the potentially infinite, we are left to choose between two options. On the one

hand, one could argue that our numerical perceptual systems systematically

misrepresent actually impossible enumerative actions as being possible. On the other

hand, one could argue that our perceptual systems accurately represent forms of

possibility that go beyond our contingent constraints. Significantly, each option which

takes knowledge of arithmetic as pertaining to possible action is compatible with the

432 Singer & Voica (2008)

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notion that such knowledge ultimately arises from perceptual processes of interaction

with the physical world.

The Limitations of Perceptual Access

At this stage it seems as though we have arrived at a tentative answer to

Benacerraf’s challenge. Our access to some mathematical entities is achieved in much the

same manner as our access to the other ordinary entities of perception. However, an

obvious objection at this point is that this is an insignificantly small gain, since the

majority of our mathematical beliefs do not seem to arise in this manner. Even if we stick

to arithmetical knowledge alone, it seems as though most of our arithmetical beliefs do

not arise as a result of perceptual processes. Arithmetical practice is not usually

characterised as the process of looking around the world and counting things. Instead

the majority of our arithmetical beliefs seem to be the result of complex reasoning that,

first and foremost, involves the manipulation of symbols according to formal practices.

This is particularly problematic since the arithmetical facts that we access through

perception seem to be of the same kind as those that we access through our engagement

with arithmetical practices involving symbols. However, given the APOP principle, we do

not seem entitled to grant the same ontological status to the entities that we access

through perception and the entities that we access through this alternative method. As a

result this threatens to limit our mathematical knowledge in a manner that nobody

would want to take seriously. Surely, if we can be said to have mathematical knowledge

at all then this knowledge should extend beyond the basic arithmetical facts associated

with the kinds of small numbers that we have perceptual access to. Even if one takes

numerical perception to provide us with beliefs beyond the small numbers, problems still

arise. Intuitively, our practices of manipulating symbols when engaging in arithmetical

reasoning gives us access to knowledge of the very same facts that we perceive when we

perceive number.

In order to overcome this problem, it is necessary to give an account of how our

arithmetical reasoning involving symbol manipulation is related to our capacity to

perceive number. Furthermore, to preserve the parity of access between arithmetical

facts and facts about ordinary objects it is necessary to provide an account of

arithmetical reasoning which does not invoke any novel cognitive mechanisms that differ

from those that govern our access to ordinary objects. In this way our further

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arithmetical knowledge could be seen as building upon such access, as opposed to

arising from a different source altogether. In the next chapter it will be argued that an

account of our arithmetical reasoning practices of this kind can be provided.

Before moving on to this account it is worth highlighting that it is not a necessary

component for the undermining of Benacerraf’s challenge. Our perceptual access to

number alone is enough to show that the challenge does not succeed. An essential

feature of Benacerraf’s challenge is that all mathematical entities are inaccessible and

that, as such, all mathematical content is impossible. Thus, even the miniscule portion of

arithmetic that can be readily accepted as being accessed through perception alone is

enough to cause the challenge problems. However, the unnerving upshot of this solution

is that only a very small proportion of our mathematical knowledge gets explained. It is

possible to preserve a universal semantics, in the sense that what it takes for an

arithmetical claim to be true is consistent with an understanding of what it takes for an

ordinary claim to be true. However, from perception alone we have very little

justification in believing that the vast majority of mathematical claims are true. Thus, the

account on offer so far provides an explanation of some arithmetical content that leaves

us in the dark regarding the majority of what is usually taken to be arithmetical

knowledge. The aim of the next chapter, therefore, is to move from a response to

Benacerraf’s challenge to a satisfying response to the challenge. This is important, since

providing an account of such a small portion of our mathematical knowledge as to clash

with our intuitions about what we normally take to be mathematical knowledge could be

seen as a somewhat pyrrhic victory. To avoid this it is necessary to explore how our

arithmetical symbol systems allow us to go beyond immediate perception without

invoking mechanisms that involve anything other than basic perceptual access to the

world.

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6

External Symbols and Arithmetical Cognition

So far the focus has been on our most basic perceptual access to numerical

content and the ways in which this shapes our capacity for numerical cognition. For

some, this focus may seem to ignore some of the most basic features of the development

of mathematical abilities. When learning about arithmetic children do much more than

simply perceiving and cognising the number of entities in concrete collections. The

development of arithmetical competence involves a lot of work carrying out calculations

with pens and paper by manipulating symbols. Furthermore, the emphasis on

manipulating symbols on paper is not restricted to school maths classes. Practising

mathematicians in university departments do a large proportion of their work using

pens, paper, blackboards and chalk.

According to a traditional view of the role of formal procedures, mathematicians’

workings using pen and paper serve as external records of the processes that are going

on inside their heads. This is generally taken to serve two purposes. Firstly it allows the

mathematician to avoid having to remember all the steps taken in solving a particular

problem. Secondly, it allows the mathematician to communicate the steps taken in

solving the problem to other mathematicians. However, it is generally assumed that

mathematical cognition is primarily conducted using an internal purely cognitive code

and that, in principle, mathematical proofs and calculations could all be conducted in the

absence of any external activities using pen and paper.

The main aim of this chapter will be to challenge this assumption. It will be

argued that our capacity for arithmetical cognition is in significant ways dependent on

the nature of the external systems that we use to express our mathematical ideas. At the

same time, the nature of these external symbol systems is to some extent determined by

our natural capacities for numerical cognition. In this way, our capacity for engaging in

arithmetical practices, such as calculation, can be seen to emerge from a reciprocal

relationship between our innate cognitive systems and the external symbol systems that

have developed over many centuries of cultural engagement with mathematics. When we

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do engage in purely internal mental arithmetic, we do not do so in some internal mental

code. Instead we carry out internal simulations of manipulations of external symbols.

It has already been mentioned that numeral systems differ from ordinary

language in important respects. Numeral systems are, to some extent, iconic, in the

sense that the structure of the symbols bears a systematic relation to their content. They

are also systematic, in the sense that there are clear rules for combining and

manipulating the symbols, so as to produce symbols with determinate content. These

features make representations of number language suitable for enabling the

development of fully-fledged number concepts. However, they also open up new ways of

reasoning about number that are grounded in our ability to manipulate external

symbols. In order to see how this is possible, it is worth looking at the iconic and

systematic nature of numeral systems in more detail. It will be argued that these features

of numeral systems emerge from the exploitation of aspects of our natural capacity for

numerical cognition. There are many different ways in which one could use iconic

symbols to represent number but we use ones that are shaped to fit with our innate

number systems.

The Development of Numeral Systems

To appreciate how numeral systems have come to have these features, it helps to

pay attention to their historical development. Archaeological evidence suggests that

humans have been using external representations of number for at least 50,000 years.433

As such, external representations of number can be seen as the earliest form of external

symbolic representation and, until around 2700 BC, all external symbol systems were

dedicated to numerical representation.434 The earliest external numerical

representations come in the form of tallies (see Fig 5.1).435

433 Cain (2006) 434 Dutilh Novaes (2012) pg. 42 435 Fig. 5.1 is an image of the Ishango Bone, one of the oldest recorded uses of tally marks as external representations of number.

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Fig 5.1

Tallies are clearly iconic forms of representation, since the structure of the symbols

shares some property with that which they represent. In the case of tallies, the number of

tally marks is the same as the number of entities represented. Tallies are extremely

useful in that they allow a subject to keep track of the number of entities without keeping

an internal memory trace.436 However, the nature of the ANS places limitations on the

usefulness of tallies. The ANS fails to support reliable numerical perception of collections

of more than three or four entities and, as such, tallies of larger numbers might not

provide much of a cognitive advantage. It is thus, no surprise that tally systems often

include conventions to group marks together when tallies exceed three of four marks

(See Fig 5.2).

Fig. 5.2

It is important to note that use of tally systems already involves a relatively specific form

of iconic representation. There are a huge variety of ways in which one could arrange

marks such that they are equinumerous with the entities that they represent. However,

tally systems tend to conform to the convention of using equidistant parallel vertical

dashes. Some aspects of such conventions are likely to be determined by practical

considerations.437 However, certain aspects of notational conventions seem to be

determined by features of our natural systems for representing number. We have a

436 This may allow them to keep track of the number of entities when the entities are not immediately present or when such a task would exceed the capacity of our memory. 437 For example, it is easier to carve a dash into a bone than a more complex symbol.

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natural cognitive tendency to represent number in terms of positions along a spatial axis

and external representations such as tallies exploit this capacity.

As numbers get larger, there comes a point at which tally systems lose their

practical value. Even with conventions for grouping tally marks, there will come a point

at which the number of groups exceeds the capacity of the ANS to be reliable.

Circumventing these problems involves introducing new symbols, such that a single digit

can be used to represent a quantity greater than one (see Fig. 5.3).438

I, II, III, IIII, V, VI, VII

Fig. 5.3

The introduction of new numerals of this kind renders the symbols less straightforwardly

iconic. For example, the Roman numeral for six is made up of two separate digits or

three dashes but represents six rather than two or three. Some cases even seem to

undermine the iconic nature of the representations. For example, the numeral for five is

made up of fewer digits than the numeral for three. However, in the majority of cases,

larger numbers will still be represented by numerals that are made up of a greater

number of digits.439 Thus, whilst the structural similarity between symbol and referent is

not perfect, numeral systems like the Roman numerals still involve a degree of iconicity.

Again it is important to note that the conventions for introducing new symbols

are closely related to the nature and limitations of our natural systems for representing

number. From an objective point of view, there is no particular advantage to introducing

a new symbol at three or four as opposed to at seven or eight. However, it is no surprise

that in the Roman numeral system a new symbol is introduced at four or five, exactly at

the point at which ANS representations of the number become less reliable (see Fig. 5.3

and Fig. 5.4).

I, II, III, IV, V, VI, VII

Fig. 5.4

438 For example, in one version of the Roman numeral system the first four numerals are akin to tally marks before a new symbol, “V”, is introduced for five. 439 Furthermore, exceptions to this tendency become less prevalent as one moves to considering higher and higher numbers. For example, whilst it may be the case that a large number, such as 1,000 is represented by a single symbol, “M”, most numbers beyond 1,000 will be represented by a larger number of symbols than are used to represent numbers less than 1,000.

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Furthermore, the exploitation of our natural tendency to represent number in terms of

spatial positions is, to some extent maintained. Systems like the Roman numerals also

exploit the way in which our bodies shape numerical cognition. In particular, the

introduction of a new symbol at five seems to result from the fact that we have five

fingers on each hand and, thus, the “V” can be seen to represent one hand’s worth of

fingers. Given the significance of finger counting for the development of numerical

concepts, this provides further evidence that the conventional aspects of our numeral

systems are to a large extent shaped so as to conform with and exploit our natural

cognitive capacities.

Although the Roman numeral system has clear advantages over simple tallies, it

too faces serious limitations. As a result, systems like the Roman numeral system have

largely been usurped by systems like the Arabic numeral system. Whilst the main reason

for the transition from Roman to Arabic numerals may relate to the systematic

advantage of the latter, it is still important to consider the issue of iconicity in the case of

Arabic numerals. It is clear that in the case of the first nine numerals Arabic numerals

are less straightforwardly iconic than Roman numerals, in that a single digit is used to

represent each number. However, when it comes to numbers larger than nine, the Arabic

numeral system can be seen to restore some of the iconicity that was lost in the shift

from tally marks to Roman numerals. Whereas in the case of Roman numerals the link

between number of digits and number represented was only a rough generalisation, in

the case of Arabic numerals there is a systematic link between the two. For Arabic

numerals, it is always the case that a numeral with more digits represents a higher

number than one with less digits.

The conventional aspects of Arabic numerals are also closely tied to our natural

capacities for numerical cognition. As with both tallies and Roman numerals, Arabic

numerals exploit our natural tendency to associate numerical and spatial representation.

Arabic numerals also pay heed to the limitations of the ANS, albeit in a slightly different

manner to the previous cases. For instance, when writing down numbers from one-

thousand upwards, it is conventional to separate the digits into groups of three using

commas. For example, one writes one million as “1,000,000” rather than “1000000”.

This convention allows us to parse the numerals far more easily, by rendering them as

more amenable to the limitations of the ANS. The Arabic numeral system allows us to

exploit the natural capacity of the ANS for accurately representing the number of entities

in small collections for the task of representing differences between large collections. In

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a certain sense, our ability to think about one hundred using the symbol “100” is

grounded in our ability to perceptually represent threeness and our ability to represent

one million using the symbol “1,000,000” is grounded in our perceptual ability to

represent two groups of three. As well as exploiting the natural capacities of the ANS, the

Arabic numeral system, like the Roman numeral system, builds upon the role of finger

counting in the development of number concepts. Although, the use of a base-10 system

can seem somewhat arbitrary and even detrimental from a purely mathematical

perspective, once one takes the role of embodied representations of finger counting on

board, it becomes clear that the base-10 system is more suited to our natural cognitive

apparatus. 440

Our external symbols for representing number differ from ordinary linguistic

representations in being to some extent iconic. However, of all of the many ways in

which one could represent number using icons, the numeral systems that we have

developed use icons that are specifically tailored to our natural cognitive capacities. The

numerals that we use activate the very same systems as the content that they represent

and the conventional aspects of our external representations of number are shaped by

the character and limitations of our natural systems for numerical cognition. When

engaging with numerical symbols the ANS plays a dual role. For example, when we

encounter a multi-digit numeral, such as “758”, ANS representations firstly play a role in

constituting the number concepts that are activated by each digit and secondly play a

role in representing the fact that the numeral itself has three digits.

Numeral Systems in the Brain

It shouldn’t be surprising that the forms of external representation that we have

developed are suited to human brains. However, it is important to pay attention to the

manner in which our external notational systems are constrained by our own cognitive

capacities. Our capacity for using external symbol systems is highly unlikely to be

accomplished by a dedicated innate system. Archaeological evidence suggests that

written language emerged far too recently in our evolutionary history for a specialised

innate system for reading and writing to have emerged.441 As such, our capacity to

engage with external symbol systems must involve recruiting parts of our brains that

440 Andrews (1936) 441 Dehaene (2009) pg. 4

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initially possessed some alternative evolutionary function.442 An interesting feature of

written symbols is that the symbols that appear most frequently in a wide range of

different languages closely resemble configurations of lines that appear most frequently

in natural scenes.443 Thus, it seems as though the development of symbols is shaped by

the constraints of our visual system. We create symbols that are naturally easy to detect

by parts of the visual cortex that already have the function of detecting common features

of the environment. Given the fact that our capacity for reading and writing is culturally

determined combined with the fact that there is wide variation in reading and writing

practices from culture to culture, one might expect the neural systems responsible for

engaging with external linguistic symbols to be highly variable. However, the left lateral

occipito-temporal sulcus has found to be consistently activated by letters and words,

regardless of language or culture.444

At face value, numerals and letters are extremely similar. One would expect that

they could be interchangeable, in the sense that swapping the symbols for the first nine

letters of the alphabet for the symbols for the first nine numerals would seemingly make

little difference. As a result, one would expect the capacity for recognising numerals to be

intimately linked to the capacity for recognising letters. Similarly one would expect the

visual word form area to be equally activated by both numerals and letters. Surprisingly,

however, patients with lesions that disrupt their ability to read letters and words

maintain the ability to read numerals.445 Furthermore, processing letters and processing

numerals leads to activation in different parts of the brain.446 Perhaps most surprisingly

of all, the differences between processing letters and numerals are not determined by the

form of the symbols alone. The dissociation of linguistic and numerical processing even

extends to cases of algebraic problem solving, where letters are used to represent

numbers. For example, one patient with severe aphasia, who had lost the ability to

recognise and comprehend letters and words, was still able to solve algebraic problems

where letters were used to stand in for unknown numbers.447 Furthermore, processing

letters in the context of algebraic problem solving leads to activation of different regions

442 Dehaene & Cohen (2007) 443 Changizi et al. (2006) 444 Dehaene (2009) pg. 69-71. The lateral occipito-temporal sulcus, often known as the human visual word form area, is sandwiched between a region dedicated to the visual detection of faces and a region dedicated to the detection of certain specific objects (Puce et al. 1996). In macaque brains a similar area is still primarily dedicated to the detection of faces and objects (Tanaka, 1996). These findings suggest exposure to written language alters the function of the left lateral occipito-temporal sulcus from dedication to faces and objects to dedication to letters and words (Dehaene, 2004). 445 Anderson, Damasio & Damasio (1990) 446 Park et al. (2012) 447 Klessinger, Szczerbinski & Varley (2007)

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of the brain to processing letters in the context of natural language.448 This shows that

regardless of the form of the symbol under consideration, we recruit different cognitive

mechanisms when dealing with mathematical as opposed to linguistic contexts.

In the case of both letters and numerals, reading and writing capacities are

supported by systems that are not innately specialised for this purpose. However, there

is an important difference between the kinds of systems that are recruited in each case.

In the case of letter processing, the brain recruits a system that would otherwise be

dedicated to processing a different kind of content, namely, faces or objects.449 However,

in the case of numerals, the brain utilises the very same system that is already innately

responsible for processing numerical content. Our external symbol systems utilise

numerical and spatial properties of symbols as a means for representing numerical

properties, which are in turn naturally represented in terms of perceptual

representations of number and space. As such our external symbol systems are doubly

grounded in our natural systems for the representation of number. The nature of

numerical notation is at least partially determined by the nature of our innate cognitive

capacities for perceiving and representing number. Our ‘internal cognitive processes

constrain the development and cultural transmission of external numerical

representations’.450

Spatially Systematic Symbols

Another significant difference between linguistic symbol systems and

sophisticated numerical symbol systems, such as the Arabic numeral system, is that the

latter are systematic. In particular, the spatial position of numerals in relation to one

another always has some semantic significance. As long as one aligns the digits along a

horizontal axis and avoids putting a “0” as the left-most symbol, any arrangement of

digits will result in a numeral with a determinate meaning.451 Numeral systems with this

property are known as “place-coding” systems. The base-10 system provides rules for

generating symbols for indefinitely many numbers through the systematic use of spatial

448 Lee et al. (2007), Monti, Parsons & Osherson (2012) 449 Puce et al. (1996) 450 De Cruz (2012) pg. 138 451 This is in stark contrast with language, where arbitrarily jumbling letters or words around tends to result in nonsense. In the case of numerals, relative spatial position of the constituent parts always has some bearing on the meaning of the whole. This is in contrast to ordinary language where, for example, “tender is the night” can mean the same as “the night is tender”.

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positions. Once one has mastered the sequence of the first nine numerals it is possible to

use a simple rule to generate all the rest.452

One of the major benefits of place-coding numeral systems is to enable simple

algorithmic procedures for carrying out complex calculations. For example, consider the

way in which children are taught to carry out addition calculations. The numbers to be

added are written so that the right-most digits of both numbers are vertically aligned and

then each vertical column is added starting from the right-hand-side and if the result is a

two-digit number the left-most digit of the result is added to the next column to the left.

Calculations that are way beyond our natural perceptual capacities can be achieved by

combining simple additions with simple rules for spatial manipulations. As a result of

the determinate and systematic relationship between the spatial positions of numerals

and their content, the relationships between numbers can be captured by rules that

determine which manipulations are permissible. As such, the place-coding ‘system is

both a medium for representing numbers and a tool for operating with numbers’.453

It is important to note that the capacities that allow us to engage in complex

calculation procedures are primarily sensorimotor capacities. We are able to use

perceivable properties, such as a digit’s spatial position or a reliably perceivable quantity

of digits, to stand for more complex numerical properties. Furthermore, engaging in a

calculation procedure involves carrying out concrete actions, by writing digits in relevant

spatial locations and “moving” symbols around.454 It had previously been assumed that

the development of place-coding systems was motivated by the impossibility of carrying

out certain forms of calculation, such as multiplication, using systems like Roman

numerals. In actual fact multiplication is possible using Roman numerals, however,

place-coding systems allow for shorter and simpler procedures that are far more

intuitive and cognitively tractable.455 In particular, procedures using Roman numerals

require a far higher number of ‘perceptual steps, attention shifts and motor actions’.456

452 In order to generate the next symbol, all that one needs to do is change the right-most symbol to the next symbol in the initial numeral sequence or, if the right most symbol is a “9” change that symbol to a “0” and change the symbol that is next to the left to the next symbol in the sequence. 453 Krämer (2003) pg. 531 454 We often talk of “moving” a symbol when what we really mean is writing the same symbol in a new spatial position. Actually moving the symbols would obviously be difficult with inscriptions on paper, since one would need some scissors and glue and things would soon get rather messy! Whilst reference to “moving” symbols may be somewhat metaphorical, this metaphor may reveal some interesting features of the way we think about numerals (see below). Furthermore, there are forms of external calculation systems, such as abacuses or Chinese rod calculus, in which the possibility of moving symbols is essential. 455 Schlimm & Neth (2008) 456 Ibid. pg. 2101

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The difference between calculation using Roman and Arabic numerals is that the latter

involves less demanding perceptual and motor processes.

It is important to note that the properties of spatial systematicity and iconicity are

at times incompatible. For example, Arabic numerals are able to achieve the former to a

greater degree by giving up on the latter for smaller numbers. Different numeral systems

can be seen to vary according to how they manage the trade-off between these two

properties. Different balances between spatial systematicity and iconicity lead to

differences in the cognitive demands on the user. For example, Arabic numerals require

the user to utilise internal associations for interpreting single digits such as “2” or “3”,

where with Roman numerals they can simply use perception, but Arabic numerals allow

for the perceptual representation of power values, where Roman numerals require the

user to remember the meaning of, for instance, “X” or “M”.457 Arabic numerals allow for

simple calculations to be achieved using relatively simple perceptual and motor

processes, due to the possibility of spatially aligning digits of the same power, whereas

dealing with Roman numerals either involves cognitive processes that are intractable or

spatial strategies that are too complex.

Fig 5.5

457 Zhang & Norman (1995) pg. 282-283

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Numeral systems thus exhibit so-called representational effects, whereby ‘different

isomorphic representations of a common formal structure can cause dramatically

different cognitive behaviours’.458 It has only been possible to consider three different

systems of numerical notation here, however there are many other systems that balance

the trade-off between systematicity and iconicity in different ways (see Fig 5.5).459 Each

system can be classified according to the particular kinds of cognitive demands and

advantages it bestows on users, and, when considered in the context of cultural and

social factors, these cognitive factors can explain the historical development of numeral

systems and the eventual convergence on place-coding systems, such as Arabic

numerals.460

The idea that numeral systems have developed so as to enable tractable

calculations using sensorimotor mechanisms also highlights further distinctions between

ordinary language and numerical symbol systems. In the case of ordinary language, its

primary functions are to enable communication and recording of information. As such

its use is primarily confined to multi-agent interactions.461 However, the case of numeral

systems seems different, since the function of these systems is often within-agent.462

Writing down numerals when engaging in arithmetical calculations goes beyond merely

trying to communicate our thought processes to others. It enables some of these thought

processes to take place in the first place. Furthermore, the ways in which such practices

aid cognition seem to go beyond the role of mere memory aids. Some practices, such as

“carrying over” a digit and writing it in the next-left column when carrying out long

addition, may seem to serve this purpose. However, the use of such techniques goes

further, since the spatial configurations of the symbols make appropriate actions

conspicuous to us in a way that they otherwise wouldn’t be. The spatially systematic

nature of our numeral systems allows them to go beyond mere communicative devices or

memory aids, such that they play a significant role in determining the nature of the

cognitive processes that support our capacity for arithmetical calculation.

458 Zhang & Norman (1994) pg. 88 459 (image from Zhang & Norman (1995) pg. 272) 460 Zhang & Norman (1995), Chrisomalis (2004) 461 When its use is confined to a single agent, as is sometimes the case with recording information for retrieval at a later date, such as when using a shopping list or writing a diary, this can still be construed in terms of the communication of information from an agent’s past state to their present state. 462 Dutilh-Novaes (2012) pg. 56-58. This may also apply to other formal notation systems, such as those used in logic.

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Arithmetical Calculations, Space and Motion

The spatially systematic nature of external numerical representations has

interesting consequences when considered in the context of our natural capacity for

representing number in terms of space. Since number is naturally represented in terms

of spatial position, arithmetical operations involving transformations of numerical

magnitudes, such as addition and subtraction, are likely to activate cognitive systems

dedicated to representing transformations of spatial position. In short, our capacity for

arithmetical reasoning is likely to be, at least partially, grounded in our capacity for

representing motion. A range of recent empirical evidence supports this hypothesis. This

evidence also supports the idea that our systems for actively engaging with external

representations of number recruit the very same systems that are naturally dedicated to

numerical cognition.

One of the first demonstrations of a cognitive association between arithmetical

operations and motion came from the discovery of the operational momentum effect.463

When subjects were shown videos of addition and subtraction operations on concrete

collections of objects and asked to choose the correct resulting collection, they

systematically tended to pick larger than correct results for addition and smaller than

correct results for subtraction. Representational momentum effects arise in many

aspects of spatial cognition, for example, subjects tend to overestimate the final position

of a moving target.464 Thus, the findings were explained in terms of similar

representational momentum effects arising from arithmetical operations being

represented in terms of movement in space. This operational momentum effect has also

been demonstrated in cases where subjects are engaging in symbolic arithmetic.465

Further evidence comes from motion-arithmetic compatibility effects. When

asked to carry out arithmetical calculations whilst also carrying out upward, downward,

leftward or rightward arm movements, subjects’ performance was better when the

movements undertaken were compatible with their SNAs.466 Addition performance was

better when arms were moved rightwards or upwards, whilst subtraction performance

was better when arms were moved leftwards or downwards. Similar effects were found

when the subjects themselves were in motion. Subjects were tasked with performing

arithmetical calculations whilst sitting on a platform that was either ascending or

463 McCrink, Dehaene & Dehaene-Lambertz (2007), Fischer & Shaki (2014) 464 Hubbard (2005) 465 Knops, Viarouge & Dehaene (2009) 466 Wiemers, Bekkering & Lindemann (2014)

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descending and, somewhat remarkably, performed addition better when ascending and

subtraction better when descending.467

As well as motion affecting arithmetical performance, there is evidence that

engaging in arithmetical tasks influences active motion. In one set of experiments,

subjects were asked to solve arithmetical problems and indicate their answers by using a

mouse to move a cursor from the bottom of the screen to the correct answer at the top of

the screen. By monitoring mouse movements, it was found that subjects’ hand

movements were significantly deflected to the left during subtraction and to the right

during addition.468 In a similar vein, engaging in arithmetical tasks has been shown to

induce shifts in attention, with addition causing subjects to shift attention to the right

and subtraction causing them to shift attention to the left.469 Furthermore, the signs for

addition and subtraction, “+” and “−”, are able to induce spatial response effects. An

effect similar to the SNARC effect was found whereby subjects respond faster to a “+”

when responding on the right-hand-side and faster to a “−” when responding on the left-

hand-side.470

Beyond this behavioural evidence there is also neurological evidence to back up

the idea that arithmetic calculations are tied to our representations of movement in

space. Engaging in addition and subtraction tasks has been shown to engage parts of the

brain that are responsible for the control of eye movements, with addition activating

parts of the brain responsible for rightwards eye movement and subtraction engaging

parts of the brain responsible for leftward eye movement.471 Furthermore, patients

suffering from hemispatial neglect on the left side of their visual fields, show impairment

when engaging in subtraction tasks but not when engaging in addition tasks, suggesting

that arithmetical calculation capacities are closely tied to the neural systems responsible

for spatial attention.472

The EC account suggests that number concepts are, at least partially, constituted

by spatial representations. These results suggest that operations with these concepts are

similarly grounded in our capacity for spatial cognition. We think about arithmetical

operations in terms of motion. Our external numeral systems exploit our natural

tendency to represent number in spatial terms, by reusing spatial representation as a

467 Lugli et al. (2013) 468 Marghetis, Núñez & Bergen (2014) 469 Masson & Pesenti (2014) 470 Pinhas, Shaki & Fischer (2014) 471 Knops et al. (2009) 472 Dormal et al. (2014)

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means for representing number. As such, one would expect our capacity for calculation

using external symbol systems to be influenced by spatial factors. This expectation is

vindicated by a number of recent empirical studies.

When carrying out complex arithmetical calculations it is important to pay

attention to the order in which the more basic calculations that comprise it are

undertaken. For instance, when presented with the problem “2 + 3 × 5 = ?” one could

give two answers, seventeen or twenty-five, depending on whether one carries out the

multiplication or the addition first, with the former being considered correct. One might

expect this to simply be a matter of following the rule, “always carry out multiplications

before additions”. However, surprisingly, subjects that are experienced at mathematics

go awry when spatial properties of the notation are manipulated. For example, subjects

performance is worse when the problem is presented with the numbers to be added

closer together, for instance, “2+3 × 5 = ?”, mistakenly carrying out the addition

operation first.473 Thus, it seems as though the order in which operations are undertaken

is influenced by what may have seemed like irrelevant spatial features of the external

notation. These findings support the idea that the cognitive processes that support

arithmetical calculation are to some extent determined by our capacity for spatial

cognition.

This conclusion is further supported by evidence that our capacity for

representing motion is also recruited when engaging with mathematical problems. In

particular, when engaging in algebraic problem solving, a significant part of the problem

solving strategy involves “moving” symbols from one side of an equation to the other.474

When subjects were presented with algebraic problems accompanied by moving patterns

in the background, their performance was improved when the patterns’ movement was

compatible with the required symbol manipulation and impaired when it was

incompatible.475 This suggests that we should understand talk of “moving” symbols to be

more than a mere metaphor. Our capacity to engage in formal reasoning about numbers

is supported by systems responsible for representing motion.

473 Landy & Goldstone (2007), Landy & Goldstone (2010), Jiang, Cooper & Alibali (2014). Similar results were found when the spacing was kept constant but the problems were presented along with visual cues that primed subjects to perceptually group the addition operands together (Landy & Goldstone, 2007). 474 For instance, when presented with the equation, “3x + 2 = 8”, it helps to move the “2” to the other side and swap its sign from a plus to a minus to get, “3x = 8 − 2”, so that one can then carry out the simple calculation of the right hand side of the equation. 475 Landy & Goldstone (2009)

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As with the case of numeral systems, our capacity for carrying out arithmetical

calculations using these numeral systems exploits our natural tendency to represent

numbers in terms of space. We represent addition and subtraction in terms of spatial

motion. However, as with the case of numeral systems, our practices for engaging in

arithmetical calculation by manipulating symbols are also grounded in the very same

systems for representing space and motion. We naturally utilise space to represent

arithmetical operations and our capacity to engage in such calculations is extended by

utilising external notation systems and practices that are themselves essentially spatial.

Symbol Translation or Symbol Manipulation

Once one takes the iconic and spatially systematic nature of numeral systems into

account, a new approach to our understanding of the nature of arithmetical cognition

becomes available. In particular, it is possible to develop an approach in which

perceptual and motor processes play a far more significant role than has been assumed.

Furthermore, on such an approach, the role of the external symbols themselves is

transformed from mere records or heuristic aids to active contributors to or even

constituents of our arithmetical cognitive processes. In order to appreciate the

significance of this new approach it will help to consider the established approaches to

arithmetical cognition on which it puts pressure.

The main point of contention in the established literature is whether to

understand arithmetical reasoning in syntactic or semantic terms.476 The former

approach is associated with computationalist approaches to the mind, and suggests that

arithmetical reasoning is carried out by translating external symbolic representations

into mental symbols in a language of thought.477 Reasoning then takes place by

manipulating these mental symbols according to formal rules that instantiate logical

principles and that are only sensitive to the syntactic properties of the symbols.478 The

result of such reasoning processes is the generation of a mental symbol that can then be

translated back into an external medium. The main alternatives to the computationalist

approach are various approaches that emphasise the semantic as opposed to syntactic

properties of reasoning problems. According to such accounts, arithmetical reasoning is

carried out by interpreting external symbolic representations and then constructing a

476 Landy, Allen & Zednik (2014) 477 Fodor (1975) 478 Fodor (1975), Pylyshyn (1980)

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mental model or simulation of a situation that instantiates the relevant properties

referred to by the symbols.479 Reasoning then proceeds by carrying out operations on

this internal model or simulation in order to arrive at a model or simulation that can

then be interpreted to yield an answer to the problem in terms of external symbolic

representations.

An important feature of the two rival traditional accounts is that they are

essentially translational.480 In both cases, arithmetical reasoning involves translating the

perceptual representations of external symbols into a different form of internal

representation. As such, the external symbols and perceptual representations thereof

play no significant role in the reasoning process. These approaches seem incompatible

with the evidence considered so far which suggests that our arithmetical reasoning is in

large part mediated by actively manipulating external symbols. For example, neither of

the traditional approaches is able to explain how manipulating the spatial properties of

external symbols could lead to alterations in performance. Considerations such as these

have led to the development of an alternative approach, known as Perceptual

Manipulation Theory.481 This approach suggests that the manipulation of external

representations plays a central role in our arithmetical reasoning. Arithmetical reasoning

is primarily achieved by following simple algorithms for manipulating the spatial

positions of external symbols so as to yield problems that can be solved using our natural

cognitive mechanisms. We are obviously able to engage in complex arithmetical

reasoning without always actually carrying out manipulations of external symbols using

pen and paper. However, in these cases arithmetical reasoning is accomplished by

producing mental simulations of manipulations using pen and paper. Arithmetical

reasoning is thus carried out either by directly manipulating external symbols or by

simulating such manipulations. Thus, the account is similar in some ways to the

semantic accounts of arithmetical reasoning, in the sense that it posits a major role for

mental simulations of perceptual and motor processes. However, it differs in that what is

simulated is the external symbols themselves rather than their content.

A benefit of this approach over semantic approaches is that it can explain our

ability to reason about number without being distracted by superfluous features of the

particular situation that we use as a model. This is illustrated by the case of a

479 Johnson-Laird (1983), Barsalou (1999) 480 Landy, Allen & Zednik (2014) 481 Landy, Allen & Zednik (2014), see also Clark (2006a, 2006b), Menary (2007), Dutilh Novaes (2012) for articulations of similar approaches.

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chimpanzee, named Sheba, who was taught to use numerals.482 Sheba was given a task

where she was presented with two piles of food and whichever pile of food she chose

would be given to another chimp, whilst she would receive the pile of food that she didn’t

choose. When presented with this task she would usually choose the larger pile of food

despite the fact that this would be detrimental. However, when presented with the same

task but with numerals replacing concrete collections of food items, Sheba was able to

successfully choose the symbol representing the smaller collection most of the time.

Thus, the use of symbols allowed her to ‘sidestep the capture of [her] own behaviour by

ecologically-specific fast-and-frugal subroutines’.483 By reasoning in terms of external

symbols rather than concrete collections she is able to reason about the mathematical

properties of number rather than being distracted by the natural drive for more food.

This goes against the semantic model approach, which would predict the same effects in

both scenarios, since, presumably an internal model of more food would be as attractive

as the external collections. This case also goes against the predictions of the purely

syntactic translational approach, since it would predict that dealing with both concrete

collections and with external symbols would involve translation into the same amodal

inner code.

Although the Perceptual Manipulation approach supports a significant role for

sensorimotor engagement with external symbols, it might be going too far to dispense

with translational accounts altogether. Evidence presented earlier suggests that there is

more to arithmetical reasoning than symbol manipulation alone. When we engage with

numerals, part of this process involves the activation of ANS representations and spatial

representations. In a certain sense we translate the external symbols into a different

form of internal representation. However, this form of translation is significantly

different from the other forms of translation on offer. It differs from the syntactic

approach in the sense that perceptual representations of symbols are translated into

perceptual representations of number, which are inherently contentful and whose

properties transcend the merely syntactical. However, it also differs from the various

semantic approaches, since rather than being translated into models or simulations that

exemplify given numerical properties, external symbols are represented directly in terms

of perceptual representations of those properties. Given the power of our natural

systems for representing number and the systematic spatial properties of our external

482 Boysen, Mukobi & Berntson (1999) 483 Clark (2006b) pg. 293-294

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numeral systems, there is no need to posit further forms of internal representation in

order to explain our capacity for arithmetical reasoning.

Arithmetical Cognition: Extended or Embodied

These considerations engender a reassessment of the role that external symbols

play in our processes of arithmetical reasoning. Rather than seeing external

representations of number as mere heuristic aids or means of communication, we should

take them to be active participants in arithmetical cognitive processes, either by partly

constituting such processes or determining the nature of the perceptual and motor

processes that support them. It is possible to distinguish four different positions with

respect to the relationship between external symbols and arithmetical reasoning.484

1. Arithmetical reasoning is independent of external symbols.

2. Arithmetical reasoning is dependent on natural language; symbolic notation is

mere shorthand.

3. Arithmetical reasoning is dependent on external symbol systems. It is constituted

by perceptual and motor representations of these symbols and our interactions

with them.

4. Arithmetical reasoning is partly constituted by external symbols.

It is also possible to distinguish three different time-scales that are relevant to

understanding the dependency relations between external symbols and arithmetical

reasoning.485 Firstly, arithmetical reasoning could be said to be synchronically

dependent on external symbols if it could only take place in the presence of external

symbols. Secondly, it could be said to be diachronically dependent in an ontogenetic

sense if development of arithmetical reasoning capacities depends on the presence of

external symbol systems in an individual’s environment. Finally, it could be said to be

diachronically dependent in a historical sense if the nature of arithmetical reasoning is

dependent on the particular historical development of our external symbol systems.

According to the first position, arithmetical reasoning could, in principle, take

place in the absence of external symbols. External symbols are best understood as

484 Dutilh Novaes (2013) pg. 46-49 (Dutilh Novaes distinguishes three distinct approaches. However, she fails to distinguish between embodied (3) and extended (4) cognition versions of the claim that cognition is constituted by external symbols.) 485 Ibid. pg. 49-55

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external records of the reasoning processes that go on inside our heads. They are merely

useful memory aids or means for communicating reasoning processes to others. On this

approach, arithmetical reasoning is independent of external symbols with respect to all

three time-scales. It should be clear from the evidence presented so far that this position

is untenable. Some arithmetical problems are too long and complex to be solved without

the use of external symbols and, as such, can only be solved by utilising the simple

spatial algorithms that certain external symbol systems make available. As a result,

arithmetical reasoning is synchronically dependent on external symbol systems. It seems

as if arithmetical reasoning is also diachronically dependent on external symbols in the

ontogenetic sense. Some form of external symbols are required to enable the

development of sophisticated number concepts.486 The immediate presence of external

symbols might not be necessary for arithmetical reasoning, in the sense that one can

clearly engage in mental arithmetic. However, when we engage in purely mental

arithmetic we are arguably simulating perception and manipulation of external symbols,

suggesting that mental arithmetic depends on prior exposure to them. Finally,

arithmetical reasoning seems to be diachronically dependent on external symbol systems

in a historical sense. Certain arithmetical reasoning processes are only available in the

context of a particular external symbol system. The development of new symbol systems

make new forms of arithmetical reasoning available.

According to the second position, arithmetical reasoning depends on natural

language but not on features specific to number language. Arithmetical reasoning takes

place in a particular language and the role of idiosyncratic mathematical symbols is as

mere abbreviations of statements in natural language. At first sight, this account seems

to fare better than the last in that it can provide an explanation for the apparent

synchronic and diachronic dependency of arithmetical reasoning on external symbols.

Arithmetical reasoning can be seen to synchronically depend on external symbols

because we are unable to keep long sentences of natural language in mind, so external

symbols help to abbreviate these sentences. Furthermore, this approach can seemingly

explain aspects of ontogenetic dependence, since number words could play the necessary

role in enabling the development of sophisticated number concepts. However, neither of

these explanations is satisfactory. In both cases it is the special features of numeral

systems that are of particular significance. The role that external symbol systems play in

synchronically enabling arithmetical reasoning is possible due to the features of iconicity

486 De Cruz (2008) and see Chapter 4

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and spatial systematicity and these features distinguish numerical symbol systems from

ordinary natural language. Similarly for the case of ontogenetic development, whilst

number words may play a significant role in the development of sophisticated number

concepts, it is the features that distinguish them from ordinary natural language that

allow them to play this role.

As a result of these considerations, a number of theorists have adopted the fourth

option and argued that external number symbols are constitutive of arithmetical

reasoning and that, as such, arithmetical reasoning is an example of extended

cognition.487 To the uninitiated the idea that external symbols, such as ink marks on

paper, can be parts of a cognitive process may seem a bit odd. As such, a brief

explanation of the roots of such an approach is in order. The extended cognition

approach has been a prominent issue of debate within the philosophy of mind since the

publication of Clark and Chalmers’ landmark paper ‘The Extended Mind’.488 In this

paper they argue that cognitive processes can extend into the world, utilising a

functionalist argument based on the Parity Principle.

Parity Principle: ‘if, as we confront some task, a part of the world functions as a process

which, were it to go on in the head, we would have no hesitation in accepting as part of

the cognitive process, then that part of the world is (for that time) part of the cognitive

process.’489

They then go on to provide examples of cases where engagement with external media

seems to play a suitably similar role to internal processes and conclude that, in such

cases, external media should be taken to be part of the mind. Returning to the case of

arithmetical reasoning, manipulations of external symbols using pen and paper seem, at

first sight, to be a good example of extended cognition. For example, when solving an

addition problem, the manipulation of external symbols seems to play the same

functional role as manipulations of internal representations play when engaging in

mental arithmetic.

However, once one pays closer attention to the nature of arithmetical reasoning,

this argument for cognitive extension falls apart. An important feature of arithmetical

reasoning is that it is dependent on manipulations of external symbols. For the majority

of cases of arithmetical reasoning it isn’t possible to carry out the given processes in a

487 Menary (2007), De Cruz (2008), Dutilh Novaes (2012), Dutilh Novaes (2013) 488 Clark & Chalmers (1998) 489 Ibid. pg. 8

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purely internal medium. When we do carry out processes of mental arithmetic we

simulate engagement with external symbols. As such, the Parity Principle fails to apply,

since there is no form of purely internal cognitive process with which to compare

processes involving external symbols. External symbol systems are significant precisely

because they enable cognitive processes that could not take place using only internal

resources. As a result of considerations such as these, a second wave of extended

cognition has been developed, which argues for the inclusion of external media in

cognitive processes without relying on the Parity Principle.490 Along these lines, Menary

develops a theory of cognitive integration, by defining cognition independently of the

nature of internal processes and then arguing that processes that incorporate parts of the

environment fit the definition.491 A process is defined as cognitive ‘when it aims at

completing a cognitive task; and it is constituted by manipulating a vehicle’, regardless of

whether the vehicles in question are external symbols or internal representations.492 In

the case of arithmetical cognition, the task can be seen as cognitive because it involves

transition from one epistemic state to another and also involves the manipulation of

representational vehicles in the form of external symbols. The processes of manipulating

external numerical symbols are best understood as partially constituting arithmetical

cognitive processes because there is a reciprocal dynamic causal coupling between

internal and external factors and both aspects of this coupling are essential to the

realisation of these processes.493 Internal processes and external symbol systems

mutually and interdependently contribute to bringing forth processes that we refer to as

arithmetical cognition.

At this stage it may seem strange that what started as an attempt to account for

our arithmetical reasoning capacities has developed into a discussion about the

metaphysical boundaries of the mind. Some may understandably balk at the idea that

explaining arithmetical reasoning requires us to commit to the mind leaking out into the

world. Thankfully, it is possible to explain the dependence of arithmetical reasoning on

external symbol systems without committing to external symbols as constituents of the

mind. Instead one can argue that arithmetical reasoning is embodied as opposed to

extended. Arithmetical reasoning is not constituted by external symbols on pen and

paper but by the perceptual and motor representations of our interactions with external

symbols. In order to manipulate external symbols, we need to perceive them and their

490 Menary (2007), Sutton (2010) 491 Menary (2007) pg. 57 492 Ibid. pg. 57 493 Ibid. pg. 52

194

spatial relations and to carry out actions, such as writing a new symbol, guided by our

motor system. Thus, there is no need to include the external symbols themselves and our

manipulations of them as components of our cognitive processes, since the perceptual

and motor representations of these features can play the same role. Although our

arithmetical reasoning is ‘genuinely hybrid’, in the sense of incorporating the

representational features of external symbol systems, the vehicles and the cognitive

process itself can be understood to be ‘fully internal to the biological agent’.494

The most remarkable feature of arithmetical cognition is not the fact that we are

able to carry out cognitive processes using external media. It is that our engagement with

external symbol systems can become internalised, such that we are able to carry out

simulations of external calculations and achieve feats of arithmetical reasoning that

would not be possible if we hadn’t encountered external symbol systems in the first

place. Arithmetical reasoning is not an example of our cognitive processes leaking out

into the world, but of features of the world leaking into the mind.495 We are able to do

this by utilising perceptual representations of the symbols and motor representations of

the manipulations. Thus, arithmetical reasoning can be seen as a prime example of

embodied cognition. Our number concepts are partially constituted by embodied

representations of external symbols and arithmetical reasoning is at least partially

constituted by embodied representations of the physical operations that we carry out on

such symbols. The external symbol systems are tailored so as to allow for complex

problems to be solved by relatively simple perceptual and motor processes. Once such

processes have been established, they can then be simulated offline. Thus, arithmetical

reasoning depends on external symbols synchronically, ontogenetically and historically,

without the need for extending the mind out into the world. By adopting an embodied

cognition framework with respect to arithmetical cognition, it is possible to acknowledge

the important insights from the cognitive integration approach. The nature of our

external representations and the nature of our cognitive processes can be seen as

interdependent. However, one need not commit to the view that written symbols in ink

on paper or movements of the pen in our hand are constituents of the mind.

494 Clark (2006b) pg. 299 495 Dartnall (2005)

195

Cognitive Practices and Representational Affordances

Our external systems for representing number allow us to far transcend our

innate capacities for numerical cognition and carry out complex calculations that would

be impossible in the absence of external representations. These novel capacities arise out

of the interdependency of our innate capacities for numerical and spatial perception and

the symbol systems that we use to enhance them. However, it remains to be explained

how manipulation of external systems is able to preserve the content of our natural

representation of numerical properties. In order to see how this is possible it is necessary

to invoke the notion of cognitive practices. Cognitive practices are ‘manipulations of

external representational and notational systems regulated by cognitive norms’.496

Arithmetical cognition thus provides a paradigmatic example of a cognitive practice. The

two most significant features of cognitive practices are their exploitation of physical

features of the environment and the fact that they are governed by normative

constraints. In the case of arithmetical cognitive practices, it is these two features that

enable cognitive processes involving external representations to faithfully represent

numerical content and to allow for operations that reliably preserve this content.

We are so used to working with external systems of representation that it is

sometimes easy to ignore the physical properties that render them useful for the tasks in

which we deploy them. Many of the benefits of using pen and paper to carry out

arithmetical calculations arise from the physical properties of the system involved. For

example, once a symbol has been inscribed on paper, its position remains fixed. Barring

unfortunate circumstances, such as dropping one’s arithmetical workings in a puddle,

written symbols do not tend to move about of their own accord. Using dark coloured ink

on light coloured paper or light coloured chalk on a dark blackboard leads to a high

degree of contrast between a symbol and its background, making the symbol easily

perceivable. Furthermore, use of flat surfaces such as paper or blackboards allows users

to treat their workspaces as if they were two-dimensional planes, thereby simplifying the

perceptual task and constraining the kinds of symbol manipulation that are available.

Similar things can be said for forms of external numerical representation that don’t

involve inscriptions. Abacuses utilise beads strung on solid bars in order to constrain the

ways in which beads can move, such that any possible movement of a bead corresponds

to a meaningful arithmetical operation. Furthermore, they exploit the forces of gravity

and friction that guarantee that beads don’t move from their intended positions. The

496 Menary (2007) pg. 84

196

significance of these ways in which we exploit the physical properties of external

representations becomes clearer when one takes into account the relationship between

physical and normative constraints.

Fig. 5.6

An illustration of the significance of the physical properties comes from

experiments involving the Tower of Hanoi problem (see Fig 5.6).497 Subjects were given

three different versions of the same abstract problem, one involving manipulating

oranges of different sizes, one involving manipulating doughnut shaped rings of different

sizes and one involving manipulating different sized cups of coffee. In the case of the task

involving oranges, the subjects had to keep all of the rules in mind, since many

impermissible manipulations were possible. In the cases of the doughnuts or the coffee

cups, fewer rules needed to be kept in mind because the physical properties of the

objects prevented subjects from breaking the rules.498 Subjects performed much better in

the doughnut and coffee cup cases, when the normative constraints were enforced by the

physical properties of the system rather than needing to be memorised.499 This case thus

illustrates the way in which the physical properties of external systems can be exploited

to simplify cognitive tasks. Furthermore, it demonstrates the relationship between

physical and normative constraints. When aspects of an abstract problem are encoded by

the physical properties of the external representation system, less cognitive resources

need to be dedicated to representing and following normative constraints. Furthermore,

each different form of external representation system will require its own set of

497 Zhang (2001) (Fig. 5.6 from Zhang (2001) pg. 5) 498 For example, in the case of the coffee cups, rules 2 and 3 were captured by the properties of the coffee cups themselves and the actions these made available, since placing a smaller cup in a larger cup would cause the coffee to spill and a cup could not be moved if there was another cup on top of it. 499 Ibid.

197

normative constraints, to ensure that permissible manipulations of the external system

conform to permissible operations with respect to the subject matter that the system

represents.

It should be clear that our engagement in arithmetical calculations with external

symbols is governed by rigid normative constraints. Learning how to solve arithmetical

or algebraic problems involves internalising strict rules for which kinds of manipulation

and symbol transformation are permissible. Manipulation of ‘these notations is

normative, in the sense that we learn or acquire a practice that is an established method

of manipulating notations to produce an end’.500 The norms for manipulating numerical

notation are specifically adopted so that the results of our manipulations maintain

contact with the subject matter that our symbols represent. A successful norm for

manipulating numerical notation is one which reflects the physical properties of the

systems that are represented. We use numerical notation to represent the numerical

affordances that we perceive in our environment. Our symbols represent the possibilities

for enumerative action that are the content of our numerical perceptions. As such, our

cognitive practices for manipulating symbols must be governed by norms that ensure

preservation of this content. Given a particular means for encoding numerical

affordances in terms of physical symbols, we require norms that prohibit

transformations that specify impossible actions. Our symbolic conventions acquire their

normative force by specifying that we should use representations that represent possible

rather than impossible actions.

An important feature of external representation systems is that they too give rise

to their own affordances. The physical features of pen and paper or beads on an abacus

make certain actions available to an agent. Obviously, these external media provide us

with a multitude of affordances, the majority of which will be irrelevant or detrimental to

the task of faithfully representing numerical content. The benefit of developing strict

rules for the manipulation of symbols is that it allows us to select from the vast range of

possible actions, which ones are relevant and beneficial for the task of arithmetical

cognition. When one first learns to deal with a particular external notation system,

following rules to manipulate the symbols in the appropriate way may require a large

degree of cognitive effort. For example, when carrying out a long multiplication problem

one might memorise a description of the actions required to correctly follow the

procedure. However, once one is practiced in following norms for carrying out such

500 Menary (2007) pg. 143 (emphasis mine)

198

procedures, ‘overt rule-following emerges from the fine-tuned interactions between the

perceptual and sensorimotor systems with well-designed physical notations’.501 There is

no need to rely on memorised rules, since one can directly perceive the actions that a

given collection of external symbols affords.

By developing external symbol systems and the norms for manipulating them we

engineer our environments so as to make new forms of representation possible. The

presence of these external representations contributes to the development of our

sophisticated number concepts, which are partially constituted by perceptual

representations of numerical symbols and motor representations of the things that we

can do with them. However, there is a sense in which the representational power of

external symbols is independent of the effects that they have on our internal

representations. It is natural to understand representation as being essentially

dependent on our own internal mental representations. For example, linguistic

representation is usually explained in terms of internal mental associations between

words and our experiences of the things that they represent. In the case of the

relationship between numerical notation and numerical content the situation seems to

be somewhat different. Numerical symbol systems represent numerical affordances by

being governed by physical and normative constraints that ensure that their structure

preserves the status of the represented affordances. The actions that we take in

inscribing and manipulating symbols directly represent possible acts of enumeration,

since the normative and physical constraints on symbol manipulation mirror the

constraints on enumerative acts. This allows us to reason about acts of enumeration that

are, in practice, unfeasible but, in principle, possible by engaging with and manipulating

the symbols that represent them.

Symbol Manipulation and Ontology

Direct perception is not the only means we have for acquiring arithmetical beliefs.

We are also able to access arithmetical content through the systematic manipulation of

numerical symbols. Although this provides an alternative route to accessing arithmetical

content, it does not involve invoking any different forms of cognitive mechanism. Our

capacity to manipulate numerical symbols to carry out arithmetical calculations is

primarily governed by the same kinds of perceptual and motor system that underpin our

501 Landy, Allen & Zednik (2014)

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perception of number. ‘When a mathematician sees the truth of a theorem, the neural

activity that allows physical seeing coupled to neural activity underlying basic

combinatorial operations on visually perceived symbols, allows a perception that a

certain sequence of symbol manipulations is valid’.502 However, the question then arises

as to what the consequences of these symbol manipulation practices are for our

understanding of the ontological status of mathematical entities.

There are two different questions to take into consideration in order to determine

the ontological impact of this account. The first thing to consider is whether our

encodings of arithmetical content in terms of numerical symbol systems are faithful to

the perceptual content that they encode. In other words, do our symbol systems and the

norms that govern their manipulation preserve some notion of the possibility of the

actions that they represent? A second important consideration is whether this form of

access can still be seen as on a par with the way we access content regarding ordinary

entities.

First it is worth addressing how successful one takes our symbol systems to be in

encoding the relevant features of our interactions with the world. This issue centres on

whether one takes permitted symbolic manipulations to reflect some sense of possible

actions. For example, if one took the encoding to be perfect, then any possible symbol

construction within the constraints of the given symbol system could be seen to refer to

an enumerative act that is, in principle, possible. However, if one doubted the fidelity of

such encodings then one might want to restrict endorsement of possibility claims to just

those that refer to acts of enumeration that are possible in practice. The idea that our

symbol systems provide faithful encodings of possible actions is supported by the iconic

nature of numerical symbols. We utilise numerical and spatial properties of external

symbols to represent numerical and spatial properties of possible actions. As a result

there is little reason to think that there will be a loss of content in encoding the latter

with the former. We can in a sense exploit our natural perceptual acquaintance with

smaller numerical properties as a means for representing numerical properties that

transcend immediate perception. Issues with the fidelity of the encoding do not really

arise with respect to whether the correct content is preserved. The area where significant

issues do arise is regarding the nature of possibility as encoded by symbols. There is a

clear sense in which the kind of operations that are possible to carry out through the

502 Voorhees (2004) pg. 87

200

manipulation of symbols are not practically possible as actions. For example,

determining the following…

123,456,789,987,654,321

+ 987,654,321,123,456,789

= 1,111,111,111,111,111,110

is relatively easy using simple spatially systematic procedures for carrying out long

additions. However, it is unclear in what sense, if any, the concrete operation

represented is a possible action for a normal human agent. There are two ways of

responding to this issue. The first is to suggest that we can know that the action in

question is in some sense possible precisely due to the fidelity of our symbol system in

encoding which actions are possible. The second is to respond that our symbol

manipulations provide us with access to something other than humanly possible action

and, as such we need not take arithmetical facts derived by symbol manipulation to be

akin to those derived from perception. This latter position is to some extent problematic,

since facts about small numbers can be accessed via either method. The pressure is thus

on those that deny that our symbol systems faithfully encode the content of numerical

perception to specify at what point and why this encoding breaks down. It seems

intuitive to take arithmetical calculations to have important consequences for our

understanding of concrete reality, even in cases where the numbers involved may

transcend the possible actions of a real agent or even the number of known entities in the

universe. This intuition is supported by the idea that our access to mathematical content

through symbol manipulation is access to content about what kinds of action are

possible.

The issue is further complicated when symbols are introduced for mathematical

entities that are unobservable in principle. For example, large swathes of advanced

mathematical reasoning involves manipulation of symbols that refer to infinite

collections, utilising similar manipulative norms as are associated with the manipulation

of symbols for finite collections. It is unclear whether such manipulations should strictly

be seen as legitimate or not and, if they are taken to be legitimate, how we should

interpret the notion of possibility in these contexts. As will be addressed in the next

chapter, there may be good reasons to think that our manipulation of symbols for the

infinite as if they were representations of finite collections may simply be metaphorical.

201

As such one might shy away from providing a realist interpretation in these contexts.503

However, in line with the aims of ontological neutrality, there is room to allow for even

infinite mathematical possibilities into one’s ontology without going beyond the idea that

our mathematical beliefs are formed on the basis of simple perceptual and motor

processes.

A further question arises as to whether the APOP principle still applies in the

context of mathematical content accessed through the manipulation of symbols. There is

a sense in which this seems to be a case where access parity does not apply. Numerical

symbol systems and our norms for manipulating them are highly idiosyncratic. As such,

it seems as though there is nothing on a par with epistemic access of this kind. This could

be seen to vindicate the idea that, for numerical properties that go beyond our

immediate perception, we should endorse a different ontological attitude to ordinary

objects of perception. However, this question is again complicated by the fact that we

access some of the very same arithmetical facts through both perceptual and symbolic

methods. For example, we can determine the sum of 14 and 15 either by engaging in a

counting procedure involving two concrete collections of the relevant sizes or by

engaging in a symbol manipulation procedure. Thus, whilst our symbolic access to

mathematical content may, at face value, seem different from any other form of cognitive

access, it is, in many ways, directly related to our means for perceptually accessing

numerical content. If the wider theory of embodied cognition is correct then there are

also striking similarities between nonperceptual access to arithmetical content and

nonperceptual access to ordinary content. When we think about either type of content in

the absence of direct perceptual contact, our cognitive access is mediated by simulation

of perceptual access.

Extending our arithmetical concepts beyond the immediately perceptually

available need not involve a sudden leap into an intangible abstract realm. Instead it

merely involves encoding the kinds of possible actions that we perceive in terms of other

possible actions within a framework of normative and physical constraints. It is an open

question as to whether and to what extent one takes this encoding to be faithful and

realistically interpretable. However, this is in keeping with the ontological neutrality of

the current approach to explaining our epistemic access to arithmetical content.

Regardless of the particular ontological attitude one takes to the entities of arithmetic,

503 Lakoff & Núñez (2000)

202

our beliefs in such entities can be explained by only invoking the kinds of basic

perceptual and motor mechanisms that we use to interact with the world on a daily basis.

203

7

Against All-or-Nothing Ontology

Most positions in the philosophy of mathematics tend to adopt one of two

extreme positions. Either all the theorems of mathematics are true or they are all, strictly

speaking, false. One either adopts Platonism by accepting the existence of all

mathematical entities or one adopts a position like nominalism or fictionalism by

denying that any mathematical entities exist. One of the consequences of buying into the

epistemological story on offer here is that a variety of intermediate views become

available. There is no need to be tied to an all-or-nothing attitude in the philosophy of

mathematics.

The availability of perspectives that lie between these two extremes is primarily

enabled by the notion that mathematical claims are claims about what is possible. There

are two ways in which this allows for a more diverse range of views. Firstly, questions of

ontology are divorced from questions of mathematical truth, in a manner similar to that

suggested by Putnam and Hellman.504 All mathematical claims about the possibility of

certain actions might turn out true even if no such actions ever actually take place. Given

that many of the relevant kinds of actions do seem to take place, this view might be seen

as a bit too extreme. However, there are infinitely many intermediate views available. If

one accepts that all mathematical claims make claims about possible actions then it

remains an open question as to which such actions are actualised. Furthermore, this

question no longer seems to be the kind of question that can be analysed entirely from

within the domain of the philosophy of mathematics. Questions about which of the

relevant kinds of actions are actually carried out are to a large extent empirical

questions. As such, one can accept the truth of all of mathematics whilst leaving the

question of which mathematical possibilities are actualised to those that study the actual

world. Mathematics can remain autonomous from more straightforwardly empirical

sciences, in the sense of dealing with claims about possibility, and yet still influence the

work of natural scientists who must decide which of these possibilities are realised.

504 Putnam (1983, 1994), Hellman (1989)

204

A second way in which the current perspective overcomes the hegemony of all-or-

nothing perspectives is in opening up the possibility of a divergence of views on the truth

or falsehood of accepted mathematical claims. It is usually assumed that if one accepts

the truth of a single mathematical claim then one is thereby committed to the truth of all

of them (and similarly for falsehood). However, once one understands mathematical

claims in terms of possibility the situation is more nuanced. Which mathematical claims

one takes to be true will depend upon which notion of possibility one takes to be

relevant. If one adopts a relatively lenient notion of possibility, for example, by arguing

that the relevant notion of possibility is that of logical possibility, then one can preserve

the idea that all mathematical claims are true, since they all correspond to logically

possible actions. On the other hand, if one adopts an extremely strict form of

determinism then very few, if any, mathematical claims will turn out to be true, since one

must either only accept the possibility of actual actions or go further and reject all

mathematical claims as false on the basis of their reference to possibility.505 However,

between these two positions lie a number of other interpretations of possibility that

assign truth or falsehood to mathematical facts in a less wholesale manner.

There are thus three central consequences of the epistemological picture on offer

here. The first is the parity claim motivated by the APOP principle. We should

understand mathematical entities as being ontologically on a par with entities to which

we have similar access. The second claim is the ontological neutrality claim. The

epistemological story on offer here does not provide the means for deciding between

realism and anti-realism or, in more positive terms, both realism and anti-realism are

compatible with the approach. The third consequence is the rejection of all-or-nothing

attitudes. There are a wide range of positions available that lie between full-blown

realism and full-blown anti-realism. These claims are somewhat controversial and as

such will require defending on two fronts.

Firstly, it will be necessary to defend against the view that the kind of account on

offer here entails a form of full-blown anti-realism. Others have offered a similarly

embodied account of mathematical cognition and argued for anti-realism about

mathematics on this basis.506 As such, it will be necessary to defend both the claim that

an embodied account is ontologically neutral and the claim that it is compatible with

505 It should be noted that most determinists will still accept the instrumental value of modal talk and, as such, might accept some kind of fictionalist account of talk of possibility, whilst still being anti-realist with regards to possibility as a metaphysical feature of the world. 506 Lakoff & Núñez (2000)

205

various different versions of realism from the argument that embodiment entails anti-

realism. Secondly, one could argue that full-blown realism is the only form of realism

that is true to the subject matter of mathematics. Any position that stands between anti-

realism and full-blown realism is committed to recognising a divide in the mathematical

facts that is mathematically arbitrary. As such, it is necessary to explain how an

intermediate position could be tenable and how if any arbitrariness arises this need not

threaten the viability of the approach.

Avoiding Embodied Anti-Realism

So far it has been argued that, by paying attention to the embodied nature of our

access to mathematical content, it is possible to provide a response to Benacerraf’s

challenge. This response constrains the nature of our ontological commitments but is

ontologically neutral, in the sense that it is compatible with both a realist and an anti-

realist approach. However, the account on offer is not the only account of mathematical

content based upon embodied cognition. Lakoff & Núñez (L&N) also offer an embodied

account of the nature of mathematical cognition.507 Furthermore, they take their account

to imply an anti-realist interpretation of mathematics, on the basis that it renders our

mathematical concepts as being fundamentally anthropocentric. In order to preserve

ontological neutrality, it is necessary to explain how the account on offer here differs

from that of L&N and to argue that embodied accounts of mathematical cognition need

not necessarily be committed to anti-realism.

Whilst L&N’s approach might be the correct account of more advanced

mathematical reasoning, it is not the right approach to arithmetical cognition. Even if

one were to buy wholesale into their approach, the kind of anthropocentricity that

emerges from their picture need not necessarily lead to anti-realism. Furthermore, if one

were to follow them in moving from anthropocentricity to anti-realism then one would

need to endorse a quite global form of anti-realism. Thus, despite appearances, their

arguments in favour of an anti-realist approach to mathematics can be seen to support

the APOP principle.

507 Ibid.

206

Embodied Mathematics and Conceptual Metaphor

L&N’s account is primarily based upon a specific version of embodied cognition

that takes linguistic behaviour as the primary source of data. The central idea is that the

majority of our capacity for mathematical cognition is based upon our ability to deploy

conceptual metaphors.508 The central idea of conceptual metaphor is that we utilise

representations of basic aspects of our experience, such as spatial representations, in

order to think about target domains that are more remote from experience.509 For

example, we often talk about time in terms of space, saying things like “I’m looking

forward to the concert tomorrow” or “back in the old days things were better”. Similarly

we talk about our affective states in terms of spatial metaphors, for example, we say

things like “I’m feeling low today” or “Things have been really up and down recently”.

Another significant example is the domain of temperature, where it is natural to talk of

temperatures rising or falling.

The most significant aspect of Lakoff’s approach to conceptual metaphor is that

these kinds of cases are taken to be more than mere features of language. They reveal the

nature of the cognitive mechanisms that underpin our reasoning about the target

domains. Thus, on the basis of analysing linguistic behaviour it is possible to develop

hypotheses about the nature of our cognitive systems. This is taken to be a particularly

useful methodological tool, since the nature of our underlying cognitive mechanisms is

primarily unconscious and, therefore, inaccessible to introspective analysis. As a result,

linguistic data can provide an indirect way of analysing our cognitive mechanisms.

Furthermore, many of the predictions that have emerged from employing this

methodology of cognitive linguistics have been vindicated by experimental and

neurological evidence. For example, there is both behavioural and neurological evidence

to suggest that we utilise systems primarily responsible for spatial cognition in order to

think about time.510 The fact that these conceptual metaphors invariably seem to be

based upon spatial cognition lends support to the more general embodied cognition

approach. It seems to suggest that much of our cognition employs the perceptual and

motor systems that are involved with our everyday representation of and interaction

with space.

508 Ibid. pg. 39-45 509 Lakoff & Johnson (1980) (1) pg. 195 510 Walsh (2003) Casasanto & Boroditsky (2008)

207

L&N concur with the account on offer here, to a certain extent, in that they agree

that at least some of our access to mathematical content is mediated by innate systems.511

However, they argue that most mathematical cognition arises from a different source,

namely, our engagement with conceptual metaphors. Since the main topic of the current

work is the nature of arithmetical cognition, it makes sense to focus on the role of

conceptual metaphors in this particular area. In a similar manner to weak nativists such

as Carey, L&N argue that our innate capacities alone are insufficient for number

concepts.512 However, unlike Carey they argue that the capacity for conceptual metaphor

is a necessary addition to move from these innate capacities to sophisticated number

concepts.513 They argue that we move from basic innate capacities to sophisticated

number concepts by grounding our number concepts in ‘extremely commonplace

physical activities’, such as object collection, object construction and motion along a

path.514 These conceptual metaphors are not merely linguistic devices, they are taken to

reflect the neural systems that support our arithmetical capacities. Thus, they infer from

the linguistic data that we talk about numbers as if we were talking about activities of

collecting to the hypothesis that our capacity to think about numbers is grounded in the

neural systems that govern such collecting activities. It is in this sense that the account is

best understood in the framework of embodied cognition, since this claim is consistent

with the idea that cognition involves activation of systems that are primarily devoted to

perception and action.

L&N appreciate that these grounding metaphors alone are not sufficient to enable

sophisticated arithmetical cognition and suggest that an important role is also played by

our ability to manipulate symbols. Their account is similar to that offered in the previous

chapter in the sense that they take contingent features of our numeral systems to be

determined by features of our bodies and the way that we naturally interact with the

world.515 However, they take our arithmetical calculation capacities to involve no more

than learning the rules for manipulating symbols ‘freed from meaning and

understanding’.516 Furthermore, the purpose of symbol manipulation is taken to be

merely to lighten the cognitive load rather than to play any constitutive role in cognitive

processes.517 Our practices of symbol manipulation are shaped by embodied constraints,

511 Lakoff & Núñez (2000) pg. 15-26 512 Carey (2009a, 2009b) 513 Lakoff & Núñez (2000) pg. 52 514 Ibid. pg. 54 515 Ibid. pg. 86 516 Ibid. pg. 86 517 Ibid. pg. 85

208

such as the fact that we have ten fingers or the fact that it is natural to write in horizontal

lines, however, these constraints are taken to be entirely devoid of mathematical

meaning or basis.

L&N argue that appreciating the embodied nature of our mathematical concepts

undermines what they call ‘the romance of mathematics’.518 ‘Mathematics is not about

objectively existing, external mathematical entities or mathematical truths’.519

Mathematics is neither true of abstract entities nor is it true of the physical world. The

reason than L&N take this stance is that they see mathematics as purely about

mathematical ideas. These ideas are mere products of human imagination and are

developed on the basis of distinctly human ways of interacting with the world. As a result

of such considerations they adopt an anti-realist conception of mathematics.

Mathematics cannot be seen as objectively real in that it is entirely dependent on human

minds. Without humans there would be no mathematics as we know it. However, it is

important to highlight that in taking mathematics to be mind-dependent, they do not

adopt a form of social constructivism. Mathematics is ‘not purely subjective’ and is ‘not a

mere matter of social convention’.520 Mathematics is dependent on the natural and

distinctively human ways that we interact with the world and on the distinctively human

capacity to apply concepts rooted in these basic bodily processes in an imaginative and

metaphorical manner.

Is Embodied Arithmetic Metaphorical?

Firstly, there are some general problems with L&N’s methodology. They aim to

uncover the metaphors on which our mathematical ideas are based. However, this makes

the assumption that such metaphors are unique. If anything is to be garnered from

mathematical structuralism, it is that mathematical structures can be instantiated by a

diverse range of different systems.521 Thus, whilst it could be the case that our

arithmetical ideas are grounded in metaphors of object collection, object construction

and sequential motion, this does not imply that these are the only possible metaphors.

Many other forms of interaction with the world may have the right kind of structure to

ground our arithmetical concepts. Thus, L&N are wrong to think that their conceptual

518 Ibid. pg. XV 519 Ibid pg. 365 520 Ibid. pg. 365 521 Benacerraf (1965), Shapiro (1997), Resnik (1997)

209

metaphors are the unique basis of mathematical thought.522 Furthermore, there are

likely to be differences in the ways that individual subjects ground their mathematical

thought, so L&N also fail to provide an account of mathematical cognition that is

universal. Differences in a subjects’ individual experiences may lead to the development

of different metaphors.523 For example, ‘a young child who spends hours playing with

LEGO pieces may develop’ different conceptual metaphors to one who doesn’t.524 As well

as varying from person to person, an individual’s conceptual metaphors may vary over

time, and as such are unstable.525 All of these considerations call in to question whether

L&N’s methodology is able to reveal the underlying metaphors that ground mathematical

thought and whether searching for such a unique, unified and stable cognitive structure

is a worthwhile endeavour in the first place.

The problems that arise for L&N can in part be seen to result from the limitations

of their methodology. By paying attention to linguistic data alone, they are inherently

blind to differences in cognition that lack any linguistic consequences. This is

problematic, since the nature of mathematical cognition may be underdetermined with

respect to the linguistic content we use to express our mathematical ideas. The technique

inherently misses out on cognitive differences that have no linguistic consequences.

However, thought may be more fine-grained than language and, as such, many different

underlying cognitive processes could be expressed by a single form of linguistic

utterance.

A result of these considerations is that L&N’s methodology can be seen as

somewhat ad hoc.526 They settle on explanations of metaphor that fit their particular

story when many other alternative explanations are available. As such, their method of

‘mathematical idea analysis’ can be seen to closely resemble the kind of traditional

‘armchair’ conceptual analysis that they are aiming to supersede.527 They choose to

analyse the way that mathematical ideas are linguistically presented in certain

mathematical textbooks and then try to uncover the specific metaphors that ground the

particular ideas. However, they neglect to acknowledge either the fact that these

textbooks might not reflect a universal picture of mathematical ideas or the fact that

their decisions are in large part influenced by their own introspective understanding of

522 Schiralli & Sinclair (2003) pg. 82 523 Ibid. pg. 85 524 Ibid. pg. 86 525 Ibid. pg. 85 526 Van Kerkhove & Myin (2004) pg. 360 527 Núñez (2000)

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the kinds of metaphors that could do the job of grounding. As such one can question

whether ‘the metaphors that are offered… really form a natural basis for our thinking’ or

whether, instead they are merely ‘the logical creations of the authors, who are trying to

develop a consistent epistemology’.528

Perhaps the most damning criticism of L&N’s methodology is that many of the

problems associated with the traditional Platonist approach to mathematics that they

reject remain for their project. By insisting that mathematical concepts are entirely

unconscious and purely metaphorical they render them as inaccessible as the kinds of

Platonic entities that they are supposed to replace. ‘Mathematics now becomes

determined by a fixed realm of entities, no longer situated in Plato’s heaven, but

constituted by the mechanics of the mind: mathematical structure has been moved from

heaven into our heads’.529 As such, we are still ‘out of touch with the world of

mathematics, now not because it’s up above in Plato’s heaven, but instead because it is

buried deep down in ourselves’.530 As such, we can only ever explain mathematical

cognition indirectly, by analysing the way we express mathematical thought in language.

However, L&N may be forced to accept that they are merely engaging in their own form

of traditional linguistic conceptual analysis, a route often taken by Platonists. As has

been emphasised throughout earlier chapters, a thorough understanding of

mathematical cognition requires direct study of the cognitive mechanisms that underpin

it. By embracing the possibility of such direct study of mathematical cognition, it is

possible to avoid the kind of inaccessibility that is entailed by either Platonism or L&N’s

view that we can only access mathematical cognition indirectly, through linguistic

analysis.

In highlighting the limitations of L&N’s methodology the aim is not to discredit

their approach. Analysing language to reveal conceptual metaphors is a powerful tool to

understand the mind. However, it is by no means the only tool available. Where they go

wrong is in considering the results of using this single methodology and then

extrapolating consequences from the fact that this methodology only reveals a certain

kind of mechanism. Their methodology is designed for the investigation of conceptual

metaphors and reveals them in a powerful and illuminating way. However, it is wrong to

infer from the results of applying such a methodology that conceptual metaphors are all

there are to cognition. There are many more aspects of cognition and many more

528 Dubinsky (1999) pg. 557 529 Van Kerkhove & Myin (2004) pg. 361 530 Ibid. pg. 361

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perspectives from which to investigate it and, as such, any conclusions that are drawn

without considering data from these alternative sources are bound to be impoverished.

By combining the methods that L&N employ with the wide range of other methods for

studying mathematical cognition we can arrive at a more comprehensive picture of the

way in which we acquire mathematical knowledge.

Leaving methodological issues aside, there are also reasons to think that

arithmetical cognition in particular cannot be fully captured in terms of conceptual

metaphor. We have an innate system for perceiving number and this system plays a

significant role in arithmetical cognitive processes. Thus, even if our concepts are

partially constituted by conceptual metaphors, there remains a significant part of our

representations of number which is not metaphorical in nature. L&N claim that ‘the

neural circuitry we have evolved for other purposes is an inherent part of mathematics,

which suggests that embodied mathematics does not exist independently of other

embodied concepts used in everyday life’.531 However, it is possible to question whether

the neural circuitry that supports our arithmetical capacities really evolved for other

purposes. A significant portion of the neural circuitry that supports our arithmetical

cognition evolved precisely for the perceptual representation of number. As such, there

is no reason to construe our numerical concepts as being purely metaphorical.

L&N take arithmetical cognition to be grounded in terms of representations of

object collection, object construction and sequential motion activities. However, the fact

that these activities seem to ground our arithmetical concepts may result from their

dependence upon the more basic activity of sequential spatial attention. L&N can be seen

as making the same mistake as Mill and Kitcher in taking object manipulation to be the

fundamental basis of arithmetical cognition.532 Numerical perception is arguably

perception of affordances for sequential attention. Even if we do conceptualise number

in terms of the activity of collecting objects, this is precisely because such activities

inherently involve the perception of number. The grounding metaphors for arithmetical

cognition can be explained as themselves being grounded in basic arithmetical content

and so, again, their metaphorical nature can be called into doubt. As such, it is hard to

make sense of the notion that in conceptualising number in terms of, for example, object

collection, we are ‘conflating’ our experience of perceiving number with our experiences

of object collection.533 The very same basic capacity is involved in both cases so this does

531 Lakoff & Núñez (2000) pg. 33 532 Kitcher (1988) pg. 108, Mill (2002) pg. 399, see Chapter 3 533 Lakoff & Núñez (2000) pg. 77

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not seem like a case of conflation. L&N argue that the reason grounding metaphors are

able to extend our innate arithmetical capacities is that they share structural similarities

that capture the basis of our abstract innate arithmetic.534 However, a more plausible

and explanatory account suggests that the reason that the grounding metaphors share a

structure is because they are all based on the more basic capacity for concrete numerical

perception.

It may be possible to argue that conceptual metaphor creeps in at the stage at

which symbols are introduced. However, if this is the case then conceptual metaphor

alone does not seem enough to motivate anti-realism, since we can use symbol

manipulation to derive both facts that are directly perceivable and those that aren’t.

Furthermore, if symbolic reasoning about number involves conceptual metaphor then it

is a particularly strange kind of conceptual metaphor, in the sense that the target domain

is partially cognised in terms of the same systems that we naturally use to process this

domain. In other words, once one accepts that our access to numerical content involves

numerical and spatial perception, it is odd to describe the use of spatial and numerical

systems for the symbolic representation of number as being metaphorical. There is, of

course a sense in which certain numerical features are utilised to represent other

numerical features. For example, the threeness of the digits in “100” is used as a means

for representing the numerical power. However, this looks less like a case of metaphor

and more like a case of simple encoding. L&N take our engagement with numerals to be

primarily algorithmic and involving no explicit understanding, arguing that ‘we can

manipulate the numerals correctly without having contact with numbers and without

necessarily knowing much about numbers’.535 However, this seems to be plainly false

once one notes the importance of numerical perception in interpreting numerals. Thus,

although there might be some motivation for suggesting that our arithmetical cognition

becomes metaphorical at the stage at which numeral systems are introduced, this is not a

route that L&N take.

It is important to emphasise that this rejection of construing arithmetical

cognition in terms of conceptual metaphor is not, thereby, a rejection of L&N’s general

approach to mathematical cognition. There may be valuable reasons for thinking that

conceptual metaphor plays a significant role in other areas of more advanced

mathematics. One particularly significant example is their treatment of the concept of

534 Ibid. pg. 78 535 Ibid. pg. 86

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actual infinity and the kinds of reasoning that mathematicians engage in with this

concept. Conceiving of the notion of a completed infinite collection may require the

deployment of metaphorical resources, in much the way that L&N suggest.536 However,

taking our understanding of arithmetic to be a non-metaphorical form of embodied

cognition opens up new ways for understanding the nature of the conceptual metaphors

that support our concept of actual infinity. Numerical perception can be understood as

one of our basic forms of interaction with the world and, as such, number is part of the

repertoire of basic perceptual concepts in which conceptual metaphors can be grounded.

Thus, reasoning in terms completed infinity can be understood as being grounded in the

metaphor of treating infinity as a (finite) number. We treat the infinite as if the kinds of

physical interactions that make sense in terms of finite concrete collections also make

sense in terms of completed infinite collections.

The idea that most of advanced mathematical cognition is based on deployment

of conceptual metaphors is extremely valuable. Despite the criticisms of the methodology

put forward here, it has a huge amount in its favour and may be the best means available

for addressing the nature of advanced mathematical cognition. As such, it seems like a

fruitful way in which to build upon the project of the current work in going beyond

arithmetic and analysing more advanced areas of mathematical thought. However, even

if this approach is the correct way to understand advanced mathematical cognition, it is

still possible to question the idea that adopting such an approach forces one to adopt

anti-realism.

Embodiment without Anthropocentric Anti-Realism

The move from the claim that mathematics is about ‘ideas that are ultimately

grounded in human experience’, in the sense of being based upon perceptual and action-

based concepts, to the claim that ‘there is no mathematics out there in the physical

world’ is an odd one.537 After all, there is a certain sense in which Maddy felt the need to

justify something like the former claim, in order to make sense of the negation of the

latter claim.538 Human actions are physical and, as such, there is a clear sense in which

knowledge grounded in ideas about action can be understood as knowledge about the

world. When we find out that certain actions are possible for ourselves we thereby find

536 Ibid. pg. 155-180 537 Ibid. pg. 366 & pg. 365 538 Maddy (1990) pg. 50-61

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out something about the physical world. Furthermore, if, as has been argued, we have

perceptual access to numerical features through perception of a certain type of possible

action, then any explanation of the basic nature of ‘human experience’ will arguably have

to make reference to our perception of some ‘mathematics out there in the world’.539 The

majority of more sophisticated mathematics may be quite hard to relate back to basic

interactions with the world. However, this is no reason to rule out a realist interpretation

of its content.

There is a sense in which L&N’s argument for anti-realism could be seen to rest

on a simple category error. They argue that ‘mathematics is primarily a matter of

mathematical ideas’ and that, as such, ‘mathematical objects are embodied concepts’.540

However, this seems to be a case of failing to distinguish vehicle from content. It is

somewhat trivial that mathematics is primarily accomplished by engaging with

mathematical ideas but this is true for any discipline. Science is primarily a matter of

scientific ideas, but few would take this to imply that the objects of science are merely

concepts and, thereby advocate scientific anti-realism. Such a position is obviously

available, however, it requires a lot more argument than L&N provide.

Núñez does provide some justification for why mathematics is unique, in the

sense that its subject matter is the very ideas that constitute it. He argues that

‘mathematics is a unique body of knowledge’, since ‘the very entities that constitute what

it is are idealised mental abstractions, which cannot be perceived directly through the

senses’.541 He cites mathematical entities, such as Euclidean points, the empty set and

infinity, as examples of such unperceivable abstractions. However, there are problems

with this line of argument towards anti-realism. Firstly, some of our mathematical ideas

are the result of direct perception and as such need not be seen as idealised abstractions.

Furthermore, science is replete with reference to unperceivable entities, such as

subatomic particles, and idealisations, such as perfect gases or frictionless planes.

However, we do not thereby suggest that science is about our concepts of subatomic

particles and ideal gases rather than being about the world. Some of the strongest

advocates of anti-realism about mathematics, such as Field, are happy to admit the

existence of space-time points and space-time regions, despite their lack of direct

observability.542 If lack of direct observability is enough to warrant anthropocentric anti-

539 Lakoff & Núñez (2000) pg. 366 & pg. 365 540 Ibid. pg. 365 & pg. 366 541 Núñez (2008) pg. 335 542 Field (1982) pg. 51

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realism then far more than mathematical entities may need to go and L&N might be

better off accepting something akin to constructive empiricism.543 However, if the

arguments from earlier chapters are accepted, this strategy will not work for all

mathematical entities anyway, since some of these are directly perceivable.

One of the main reasons for L&N taking an anti-realist stance with respect to

mathematics is the essentially anthropocentric nature of our mathematical concepts.

They argue that our mathematical ideas are uniquely human in that they are based on

concepts derived from uniquely human forms of interaction with the world. Thus,

mathematical concepts, although not social constructions, can still be seen to be entirely

dependent upon human minds.544 The claim that mathematical ideas are inherently

anthropocentric can be challenged. Our most basic capacities for perceiving the number

of entities in a collection are shared with a surprisingly wide range of other species,

including birds, amphibians, fish, insects and many other mammals.545 Since the system

responsible for this capacity in humans is also involved in our more complex

mathematical reasoning, our mathematical ideas seem less anthropocentric. Our

mathematical concepts are grounded in a basic perceptual system and many other

species possess either a homologous or analogous system. As such, our mathematical

concepts are grounded in a pretty ubiquitous form of interaction with the world. One

could argue that the ubiquity of numerical perception provides a counter to L&N’s

argument for anti-realism, since the best explanation for the evolution of a system

dedicated to the perception of number might be the existence of physical mathematical

facts.546 Even if one does not buy into this argument for realism on the basis of

evolutionary ubiquity, the fact that the basic mathematical capacities that support our

more complex mathematical reasoning are supported by systems that are near universal

amongst the animals diminishes L&N’s charge of anthropocentricity.

A more serious problem for L&N’s argument for anti-realism is that it seems to

lead to a more serious global anti-realism. If it is the conceptual metaphorical nature of

mathematical ideas that engenders anti-realism then it seems as if many of our concepts

fail to represent the world. When one looks at the long list of concepts that Lakoff has

argued are based in conceptual metaphors the number of things one would have to give

543 Van Fraassen (1980) 544 Lakoff & Núñez (2000) pg. 364-366 545 Boysen & Berntson (1989), Brannon & Terrace (2000), Killian et al. (2003), McComb, Packer & Pusey (1994), Emmerton, Lohmann & Niemann (1997), Uller et al. (2003), Agrillo et al. (2008), Carazo et al. (2009), Gross et al. (2009), Reznikova & Ryabko (2011) 546 De Cruz (personal correspondence)

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up on realism towards is worrying.547 For example, it is well established that we

conceptualise time in terms of space but this seems like a strange reason in and of itself

to take an anti-realist position with respect to time. Similarly, we conceptualise

temperature in terms of space but that seems like a poor reason to be anti-realists about

temperature. L&N need to explain why the use of conceptual metaphor in the case of

mathematics provides a special reason for anti-realism.

One possible reason for singling out mathematics may be that, unlike in the case

of time or temperature, L&N believe that we have no direct perception of mathematics. It

is conceived of as based on conceptual metaphor and conceptual metaphor alone,

whereas other concepts, such as of time or temperature, are partially based on

conceptual metaphor and partially based on perception. However, one of the main points

of the current work has been to argue that number, like time or temperature or any other

perceptual feature, is accessed by direct perception. As such, there is no reason to think

that the presence of conceptual metaphor in our mathematical cognition provides any

motivation for anti-realism in this specific case alone.

Perhaps L&N would bite the bullet at this stage and embrace a form of global

anti-realism. This would certainly be consistent with views expressed in some of Lakoff’s

earlier work, where he denies ‘that there is such a thing as objective truth’ and argues

that the idea of an objective mind-independent world is a mere ‘myth’.548 However, if

this were the case then their views would be entirely consistent with the APOP principle.

Their argument for anti-realism with respect to mathematics could be seen as nothing

more than a particular case of a more general argument against the existence of a mind-

independent reality or an objective way of carving up the world. However, whether or

not one would be willing to accept such a perspective is likely to depend on issues that go

way beyond mere considerations of the nature of mathematical cognition and our access

to mathematical beliefs. It may be that adopting an embodied approach leads to global

anti-realism but it certainly doesn’t entail anti-realism about mathematics alone.

L&N provide an extremely ambitious account of advanced mathematical

reasoning in terms of embodied cognition, which has many strengths and which is, in

many ways, compatible with the account on offer here. However, they are wrong to think

that an account of the cognitive basis of our thoughts about a domain is able to decide

either way as to the ontological status of that domain. Understanding the epistemological

547 Lakoff & Johnson (1980) (2) pg. 11-51 548 Ibid. pg. 159, 186-188

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picture with regards to a particular subject matter can constrain the ontological position

that one adopts but it cannot dictate it. The APOP principle supports ontological parity

but inevitably remains ontologically neutral. In general, understanding the nature of

representations shouldn’t provide definitive answers regarding the existence of the

things that they represent.

There are also reasons to be suspicious of the argument from embodiment to

anthropocentricity and from anthropocentricity to anti-realism. All of our concepts are

in some sense uniquely human concepts and, so, this line of argument should be avoided

unless one is happy with a form of global anti-realism. If one takes mathematical

knowledge to be knowledge of affordances then it will inevitably be organism-centric.

However, the fact that affordances must be defined relative to a particular organism with

a particular repertoire of possible actions need not be seen as any more reason for anti-

realism than the fact that in general relativity space-time structure must always be

defined relative to a particular observer’s perspective or frame of reference.549 Facts

about the actions that are possible for an organism are entirely objective and mind-

independent. As such, concepts that are grounded in representations of such facts can

still be seen as representations of the world. Thus, knowledge of possible human actions

can still potentially be understood to be knowledge of the world. Furthermore, there is

room to argue that by investigating possible human actions we are able to arrive at

knowledge about the physical world that far transcends the actual limitations of human

actions.

The Problem of Ontological Arbitrariness

It seems as though it is possible to provide an embodied account of our access to

mathematical knowledge without thereby being forced into anti-realism. However, more

work needs to be done in order to preserve ontological neutrality and maintain the

possibility of an ontology somewhere between the two extremes of all or nothing. The

current picture suggests that if some mathematical entities exist, they exist in the same

manner as the ordinary physical objects that we perceive. However, this leads to a

problem that besets any theory that attempts to locate mathematics in the physical

world. This problem will be referred to as the problem of ontological arbitrariness and

549 Sanders (1997) pg. 101

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can be seen as one of the main motivations for the tendency for all-or-nothing positions

in the philosophy of mathematics.

The problem can be roughly stated as follows. The physical universe is finite in

nature. As such, it only contains a finite number of entities. Let’s call this number n. If

arithmetical facts are taken to be facts about the physical universe then it seems as

though only mathematical facts that pertain to numbers less than or equal to n can be

taken to be true. For example, suppose 2 + 3 = 5 is true on the basis of the fact that when

one adds a collection of two entities to a collection of three entities one yields a collection

of five entities. The same does not seem to be the case for cases involving numbers

greater than n. If one takes a collection of n-1 entities and attempts to add two further

entities, one will not yield a collection of n+1 entities, since there simply aren’t enough

entities to form the collection with. At this stage one could simply bite the bullet and

claim that only arithmetical claims that involve numbers smaller than n are, strictly

speaking, true. However, this seems problematic from a mathematical perspective. There

is unlikely to be anything mathematically significant about n. From a mathematical

perspective n seems entirely arbitrary. Given this apparent arbitrariness it seems

unacceptable to suggest that n should play the significant role of distinguishing between

true and false mathematical claims.

In some ways this presentation of the argument is far too simplistic. From the

perspective of modern physics, it isn’t clear that the notion of the finite number of

entities in the universe makes much sense. Furthermore, it is still an open question as to

whether the universe is finite or infinite.550 However, even when one takes these

complications into consideration, the problem of arbitrary ontology can still be

motivated. The problem is that whatever physical limitations one places on

mathematical ontology, the mathematician seems to be able to make true claims about

structures that transcend those limitations. As such, any such limitations, whether

limiting mathematics to particular finite or particular infinite structures, will be arbitrary

from a mathematical point of view.

As a result of these considerations, if one wants to respect mathematical practice,

it seems necessary to adopt either an all or a nothing ontology. If one wants any

mathematical entities at all then one is forced to accept all of them in order to avoid

arbitrary limitations on ontology. If one wants to restrict one’s mathematical ontology

550 Tegmark (2004), Aguirre & Tegmark (2011)

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one should reject all mathematical objects and explain the apparent truth of

mathematical claims in terms of something other than the existence of mathematical

entities. If one takes the first of these options one seems to be forced into denying

mathematical entities physical reality, since, whatever the “size” of the physical universe,

it is too small to fit them all in, and as such one must invoke a platonic abstract realm in

order to find somewhere big enough to fit them all in.

Mathematical Modality and Mathematical Ontology

The picture on offer here is able to sidestep the initial challenge posed by the

problem of ontological arbitrariness, since mathematical claims are interpreted as claims

about what is possible rather than what actually exists. Mathematics is about possible

actions and, as such, the fact that there may be in actual fact only finitely many objects to

enumerate says nothing about the limits of possible enumeration, as long as one is

willing to permit a lenient enough interpretation of possibility. A limited physical

universe is consistent with truths about possibilities that transcend these limits. Even if

the number of entities in the physical universe is finite, possible acts of counting need

not be.

However, this manoeuvre may not be enough to escape the challenge of

arbitrariness altogether. All accepted mathematical claims can be seen as true, in the

sense of making true claims about which actions are possible, so no arbitrary divide

between true and false mathematical claims arises. However, it seems as though the way

in which the notion of possibility is interpreted within these claims will have to vary.

Basic arithmetical claims might be claims about the kinds of enumerative acts that are

physically possible for real humans. Arithmetical facts involving much larger numbers

might instead be claims about what is physically possible for merely possible or idealised

agents. Mathematical facts about the infinite and about transfinite cardinals might go

beyond anything physically possible and merely be seen as pertaining to actions that are

metaphysically or logically possible. As such, the problem of arbitrariness seemingly re-

emerges, since the transitions from one sense of possibility to another seem again to be

arbitrary with respect to mathematics. There might be some number m, such that

mathematical claims about m involve reference to physically possible actions but where

claims involving m+1 involve reference to metaphysical possibility. Thus, it seems as

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though another metaphysically significant set of distinctions arises despite the fact that

mathematics itself is blind to the basis of such distinctions.

At this stage it may be necessary to merely bite the bullet and argue that

mathematics pertains to a specific form of possibility, even if the motivation for making

such a move cannot be solely motivated from within mathematics. Despite the fact that

mathematical claims seem to be claims about what is possible, there seems to be nothing

from within mathematics that can tell us about the nature of this possibility and how it

relates to the actual world.

A means of accommodating this worry emerges from considering the contingent

nature of the limits to mathematical ontology. If the universe is indeed finite then there

may be limits to the kinds of action that can be interpreted as being possible in the actual

world. However, the finiteness of the universe is a contingent fact about our universe

that has only achieved widespread acceptance through the relatively recent development

of the Big Bang theory. Furthermore, there are alternatives to this theory and it is too

soon to say for sure that we know that the universe is finite. It is not possible to know a

priori what the limits to the application of mathematics to the actual universe will be.

The same goes for other limitations like that of our own mortality. It may be true that

certain possible acts of enumeration go way beyond anything possible in a human

lifetime. However, the fact that we are mortal is also a contingent fact that we can only

find out about by studying the world. Admittedly it is more certain and easier to discover

than the facts about the finiteness of the universe or lack thereof, however, it is a

contingent empirical fact nonetheless.

Furthermore, it is unclear which aspects of mathematics we will ultimately take to

apply to the physical universe. In the past, mathematical entities that were thought to

have no possible application to the physical universe, from fractals to non-Euclidean

geometries to infinite Hilbert spaces, have all turned out to be applicable in surprising

ways. Given that we do not know what the limits on the applicability of maths in science

are, it makes sense to explore the range of mathematical possibility in full, since it may

turn out that mathematics that we thought transcended the physical has an unforeseen

physical application. Once one takes the contingency of limits on actual instantiations of

mathematics into account the second form of arbitrariness becomes less threatening.

Mathematics is the science of possible action but, as such, we wouldn’t expect it to tell us

about what is actually the case. When we want to know which actions are possible in the

actual world it makes sense that we turn to science. If science tells us that there are a

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finite number of entities in the universe then this may change the way that we apply

mathematics to the physical world. However, it needn’t impact upon the practice of

mathematics, which is primarily concerned with what actions are possible and not which

are actual.

Embodied Numbers and Mathematics in the World

In order to understand how we know about mathematics, it is necessary to look to

the cognitive sciences and their study of mathematical cognition. They tell us that our

access to mathematics is less mysterious than the intuitions behind Benacerraf’s

challenge lead one to believe. Our basic access to arithmetic is mediated by perceptual

processes and our numerical concepts are based upon our everyday embodied

interaction with the world. The power of these concepts is greatly enhanced by the fact

that we have sculpted our environment, through the development of symbol systems, so

as to allow us to use these very same forms of embodied interaction to take our

mathematical content far beyond that provided by our innate capacities for numerical

perception.

However, looking to the cognitive sciences alone can never tell us definitively

whether mathematical entities exist or not. L&N come close to admitting as much when

they suggest that from the perspective of the cognitive sciences ‘there is no way to know

whether there are objectively existing, external, mathematical entities or mathematical

truths’.551 No story of how we acquire a particular form of content from the world can

provide a definitive answer as to whether that content is accurate or not, without also

telling some story about the nature of the world. Despite this, understanding the nature

of our access to some mathematical facts can constrain the nature of our ontology. We

should take a similar ontological attitude to entities that are accessed in a similar way,

even if the correct attitude to both requires much more work to determine.

By looking at contemporary theories of perception, it becomes clear that the

notion that mathematical content can be seen as inherently modal need not be

incompatible with our access to such content being perceptual.552 Our mathematical

percepts and mathematical beliefs are about possibilities for action but, again, this does

not provide us with an answer about the ontology of the actual world. In order to know

551 Lakoff & Núñez (2000) pg. 365 552 Putnam (1983), Gibson (1979), Nanay (2011)

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what kinds of action are actually possible, it is necessary to go beyond cognitive science’s

story of epistemic access and look to the sciences that study the nature of the world. Only

by studying the nature of the world can we tell the extent to which our mathematical

content, framed in terms of possible action, latches on to reality.

There may be a sense in which our scientific description of the world in

mathematical terms is metaphorical. By describing the world in terms of mathematics

we describe the world in terms of possible human action, despite the fact that much of

what we describe goes beyond any intuitive sense of what is humanly possible. Thus the

extent to which we take such uses of mathematics to be metaphorical will depend upon

what notion of possibility we take mathematical claims to entail and to what extent such

a notion of possibility can still be taken to be relevant to reality. Exactly how possibility is

to be understood in these contexts is a vexed issue. However, the effectiveness of

mathematics in the natural sciences is an empirical fact that needs explaining, whether

or not one takes it to be mysterious.553 It may turn out that the only way to explain this is

to accept that our mathematics based on possible human action is able to tell us truths

about the world. Even if use of mathematics does involve metaphor, this is not sufficient

for anti-realism. Metaphorical reasoning is central to a large proportion of our scientific

knowledge acquisition.554

The picture that emerges from the scientific study of mathematical cognition

suggests that various positions in the philosophy of mathematics are on the right lines.

Maddy is right to suggest that our mathematical beliefs have perceptual origins.555 Mill

and Kitcher are right to suggest that our mathematical beliefs are ultimately about our

interactions with the world.556 Putnam and Hellman are right to suggest that our

mathematical beliefs are about what is possible rather than what is actual.557 Tymoczko,

Goodman, Franklin and others are right to question the traditional divisions between

abstract and concrete, allowing room for views that lie between the extremes of

Platonism and Nominalism.558

Mathematics is not about a mystical realm where all mathematical entities exist.

Neither is it a pure fiction that, strictly speaking, is about nothing that exists. Far from

being aimed at intangible abstract mathematical entities, our mathematical knowledge is

553 Wigner (1960) 554 Hoffman (1980), Brown (2003) 555 Maddy (1990) 556 Mill (2002), Kitcher (1984) 557 Putnam (1983), Hellman (1989) 558 Goodman (1979), Tymoczko (1991), Franklin (2014)

223

generated from the everyday physical embodied capacity to perceive tangible facts about

possible bodily actions. We believe in numbers because we can see them but whether or

not they really exist is a whole different question.

224

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