Markus PantsarPost-doctoral researcherUniversity of Helsinkimarkus.pantsar@helsinki.�Please do not circulate without permission!

An empirically feasible approach to the epistemol-

ogy of arithmetic

1 Introduction: a wish-list for an epistemological

theory

In this paper I wish to propose a framework for an empirically feasible epistemo-logical theory of arithmetic. The bulk of the paper focuses on explaining whatthat empirical feasibility entails, but we should �rst establish context by specify-ing what we generally desire of an epistemological theory of mathematics. Whilethere is bound to be little consensus over any comprehensive and detailed listof criteria, it seems that at least the following �ve conditions should be ful�lledby an epistemological theory of mathematics:

(1) It should not require any unreasonable ontological assumptions.

(2) It should be epistemologically feasible as a part of a generally empiricistphilosophy.

(3) It should be able to explain the apparent objectivity of at least somemathematical truths.

(4) It should not make the applications of mathematical theories in empiricalsciences a miracle.

(5) It should not rid mathematics of its special character.

A short description of each criterion is in place, and the condition (1) isparticularly important. It may seem trivial since there are such wide di�erencesbetween the views that di�erent philosophers consider ontologically reasonable.But this lack of consensus actually gives us a useful criterion for an epistemolog-ical theory of mathematics. Taking into account all the ontological di�culties,it makes sense to avoid ontological assumptions as far as possible when we areprimarily concerned with epistemology. For many philosophers, a Platonist on-tology would be unreasonable. But for some, denying the objective existenceof mathematical objects would be equally unacceptable. Having such ontologi-cal assumptions play key roles in an epistemological theory could be needlessly

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limiting, and thus the approach here is to formulate as ontologically neutral anepistemological theory as possible.

The purpose of criterion (2) is to include the epistemology of mathematics asa part of general epistemology. This does not mean that mathematical knowl-edge could not be essentially non-empirical. Rather, the main idea is that suchnon-empirical character can never be simply assumed. The study of mathemat-ical knowledge should not include assumptions we would not be ready to acceptin the study of other modes of knowledge. And while there is much disagreementin epistemology, it should be safe to say that it is now widely accepted that ageneral philosophical theory of knowledge should be fundamentally empiricist.

Criterion (3) deals with the simple fact that mathematicians as well as lay-men often get the impression that mathematics deals with objective truths.Whether or not this is an illusion - and here di�erent areas of mathematicsmust be dealt with independently - it is something that an epistemological the-ory should be able to explain.

With criterion (4) we move on to the subject of applicability. The variousscienti�c applications of mathematical theories are often (e.g. Field 1980) con-sidered to be the strongest argument for mathematical realism. Whether ornot we agree with that, applications are an important problem in epistemology.There is obviously some way in which our knowledge of mathematics enhancesour knowledge of the world, yet there are common �ctionalist and convention-alist understandings, which state that mathematical knowledge is not about theworld. This connection has to be somehow included in a satisfactory epistemo-logical theory.

Condition (5) focuses on mathematics as an intellectual discipline, and assuch it clearly is special. Regardless of its ultimate subject matter, mathematicalresearch appears to have an a priori character that must be explained by anepistemological theory.

These �ve conditions are not meant to be a comprehensive list based on thephilosophical literature, although they can be recognized in the goals of mostepistemological accounts. Further criteria may be required, but at the very leastthey should provide us with a good starting point. The conditions also point outdi�culties in particular epistemological theories. It is clear that, for example,Platonist epistemology faces serious issues with criteria (1) and (2). Criterion(5) is a challenge for empiricist approaches, whereas a conventionalist will needto explain questions concerning criteria (3) and (4) with particular care.

However, rather than go into such considerations in detail, the purpose ofthe criteria here is to test the outline for an epistemological theory presented inthis paper: the contextual a priori. But that theory is based on ful�lling an ad-ditional criterion, one that is starting to be increasingly accepted in philosophyof mathematics, although still often ignored: empirical feasibility.

Empirical feasibility must not be confused with empiricist epistemology. Ifthe best empirical data on mathematical cognition implies that we should pur-sue a Platonist epistemology, for example, that is what an empirically feasibleepistemological theory should be like. The motivation for including empirical

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feasibility as a criterion is thus not connected to any particular epistemologicaltheory. It depends only on whether we consider the empirical data on math-ematical cognition to be strong and reliable enough to warrant including it asa criterion. I believe that point has been reached. In the next section I willpresent an overview of some of the philosophically relevant empirical results onarithmetical cognition. But from now on, we work with a sixth criterion for anepistemological theory of mathematical knowledge:

(6) It should be empirically feasible: the best scienti�c data about mathe-matical cognition cannot con�ict with philosophy.

Before we move on to the scienti�c data, a short disclaimer is in place. Ido not believe that the current set of data establishes conclusively the resultsI will later use as the criteria of empirical feasibility. While some of the datais indeed very strong and the theories used to explain them widely supported,at the moment we are still in a phase when new and surprising discoveries aremade constantly.

If one is highly skeptical about the empirical data, perhaps this paper isnot best understood as arguing for the correctness of a certain epistemologicaltheory. Instead, it should be seen as a fundamentally hypothetical pursuit: ifthe best-supported theories in psychology, cognitive science and neurobiologyare correct, what kind of an epistemological theory of arithmetic do we need?This by itself should be worth studying, especially since we have so little otherdata to base epistemology of mathematics on.

2 Empirical data

The state of the art in psychology, cognitive science and neurobiology is thathuman beings are not the only animals able to deal with numerosities.1 In thewords of the neurobiologist Andreas Nieder (2011, 107):

Basic numerical competence does not depend on language; it isrooted in biological primitives that can already be found in animals.Animals possess impressive numerical capabilities and are able tononverbally and approximately grasp the numerical properties ofobjects and events. Such a numerical estimation system for repre-senting number as language-independent mental magnitudes (ana-log magnitude system) is thought to be a precursor on which verbalnumerical representations build, and its neural foundations can bestudied in animal models.

1Here is a case where the empirical scientists often are less than careful about equivocatingterminology. To talk about natural numbers is premature at this primitive stage, as manyscientists note. Hence the term �numerosity�, referring to a primitive non-linguistic conceptionof a discrete quantity. In this paper, numerosities are also referred to as cardinalities. Anothercommon confusion is to call the primitive processing of numerosities �arithmetic�. In thispaper, arithmetic means a system with explicit rules and number symbols (or number words).For the more primitive processing of numerosities, it is better to talk of proto-arithmetic.

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This passage captures well the current situation in empirical study of proto-mathematical thinking. It has been established time and again that nonhumananimals and human infants have a basic ability to recognize numerosities. In-stead of simply focusing on duration, size or other magnitudes, experimentshave shown a common propensity toward processing observations based on thequantity of objects. This ability was detected long ago in primates like chim-panzees, but also in rats and even small �sh (Rumbaugh, Savage-Rumbaugh,& Hegel (1987)). The ability of chimpanzees may not be that surprising giventheir relatively developed linguistic abilities, but one would not expect rodentsor gold�sh to be able to deal with numerosities. However, rats can learn to usesmall quantities quite impressively. They can learn to press a lever a certainamount of times to get food, also when the time that the presses take is varied.They can distinguish the cardinality of tones from the total duration of tones.The experiments have been controlled for other variables and there is littledoubt that rats not only have the ability, but also a natural tendency to processobservations in terms of numerosities (Mechner (1958), Mechner & Guevrekian(1962) and Church & Meck (1984)). The case of �sh is even more surprising,yet the evidence is strong also for their ability to deal with numerosities. Small�sh can learn to choose the right hole to go through based on the number ofobjects drawn above the hole. Even when the combined surface area and theillumination of the objects was the same, mosquito�sh were able to make thedistinction, clearly pointing toward ability to process observations in terms ofquantities (Agrillo et al (2009)).

The ability to recognize numerosities that many nonhuman animals haveis an intriguing phenomenon, but it has also been found that newborn babiesat very early stages show a similar ability. The groundbreaking work when itcomes to infants was Wynn (1992), in which she showed that babies reactedto the unnatural arithmetic of 1 + 1 = 1 in experiment settings. When theinfants saw one plus one dolls placed behind a screen, they expected to see twodolls and were puzzled when there was only one. This tendency has later beenestablished to occur in many variations of the experiment: including ones wherethe size, shape and location of the dolls are changed. The infants still foundthe changing quantity most surprising (Simon et al (1995) and Dehaene (2011,40-48)).2

There are many more experiments which point toward the primitive abilityto deal with numerosities, and I will not present further examples at this point.The recent second edition of Stanislas Dehaene's classic Number Sense is a goodsource to the early experimental material, also providing updated referencesto subsequent studies. Dehaene & Brannon (eds.) (2011) is a comprehensivecollection of recent advances in the �eld. But the empirical study of proto-

2I believe that the many replications of the Wynn experiment quite clearly show that theinfants acted based on the quantity of objects they saw. But at the same time, it seemsthat Wynn and others are hasty in making the conclusion that the infants are adding andsubtracting. They could equally well just be keeping track of the quantity of the objects theyexpect to see, thus holding only one numerosity in their minds. Postulating the ability to doarithmetical operations is not necessary, and in fact quite problematic.

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arithmetical thinking is a growing and highly active area of research. Newrelevant data is being published constantly and it is not possible to give here acomprehensive account of the current state of the art in the �eld. It is beyonddoubt, however, that during the past decade the amount of research on subjectshas developed in immense steps. Recently, researches have even been able tolocate speci�c neurons in the monkey brain that activate when the animal isprocessing objects in terms of a certain numerosity. The same neurons activateindependently of the sensory, i.e. regardless of whether the monkey sees or hearsthree things, or indeed sees the number symbol 3 (Nieder, Freedman & Miller(2002), Nieder & Miller (2003), Nieder & Dehaene (2009)). Such supramodalityof numerosities is a highly interesting result and it could be at the very root ofwhat we understand as abstractness of natural numbers.

At this point it seems hardly worthwhile to contest these �ndings. Rather,we should accept that processing observations in terms of numerosities is a muchwider phenomenon than scientists and philosophers used to believe. Previously,even primitive forms of mathematics were thought to be an exclusively humana�air, and one that takes years to develop in children.3

In hindsight, such �ndings should not have been particularly surprising. Themodern history of biology reminds us constantly that human beings are muchless special than originally thought. If dealing with numerosities is useful for us,it makes sense that it is bene�cial for other animals, as well. Is there any betterway, for example, to keep track of one's o�spring than somehow recognizing thecardinality of them? And if some human ability is present in animals, it is likelyto be present in some form already in human infants.

Looking at such empirical results, we can immediately see potential philo-sophical relevance. If there are ways of dealing with quantities language-independently,in the philosophy of mathematics we should be interested in such �ndings. Af-ter all, one of the most prominent philosophical theories concerning mathe-matics is conventionalism, which takes mathematical statements to consist ulti-mately of human conventions and as such not about the world in any substantial

3While the nonhuman and infant abilities with numerosities are wide accepted in moderncognitive science and psychology, not everybody subscribes to it. Kelly Mix, in particular, hasbeen a prominent critic of the ability to deal with numerosities in infants, claiming that it israther continuous quantities than discrete cardinalities that the infants respond to. I cannotgo here into the details (for one part of them, see Mix et al (2002)) but it should be notedthat many experiments seem extremely unlikely to fall into such equivocations. For example,experiments are often controlled so that the total visible surface area of objects remains thesame when their quantity is changed. One criticism by Mix (2002), however, is philosophicallyinteresting and should be addressed here:

Just as animal researchers are at risk for anthropomorphizing their non-human subject, infant researchers are at risk for overlaying adult reasoning onbasic perceptual processes. Longer looking times are commonly interpreted asevidence that infants have formed an abstract representation, compared it to atest stimulus and e�ectively said to themselves, `Hey, that's di�erent!' or `Howsurprising!'

But this is the exact thing that proponents of numerical ability in infants and animals deny.They do not think that infants make abstract representations which they discuss within theirminds. Rather, the process happens automatically.

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sense. But any strong form of conventionalism must have a tight connection tolanguage-dependency of mathematics. The nature of conventions is a complexissue, but in no reasonable account can rats and infants be understood to processnumerosities based on a convention.

However, the basic ability to estimate and process quantities is of coursenothing like the arithmetic of natural numbers as we understand it. The re-lation between results of psychology and neurobiology and the philosophy ofmathematics is not a straight-forward one. In the above quotation of Nieder,we can locate perhaps the most important question in importing such empiricalresults into philosophy:

Such a numerical estimation system for representing number aslanguage-independent mental magnitudes [. . . ] is thought to be aprecursor on which verbal numerical representations build

It is important to remember that we are indeed dealing with an estimationsystem. In the jargon of cognitive science it is often called the approximate

number system, or the ANS. The ANS consists of two main parts. First isthe ability to subitize, that is, to immediately determine, without counting, thecardinality of objects in one's �eld of vision. Second is the �analog magnitudesystem�, a way of keeping track of quantities in the working memory (Dehaene(2011), Brannon & Merritt (2011) and Nieder & Dehaene (2009)).

While the existence of such abilities in nonhuman animals and infants (aswell as in adult human beings) has been an important discovery, it should notbe confused with discovering mathematical thinking in animals. The estimationsystem is very limited: when dealing with more than three or four objects, theability quickly starts to lose accuracy (Dehaene (2011), 17-20). In addition,this ability is best described as quasi-analog: the representation of quantities isonly discrete with small numerosities and gets more and more continuous as thequantities get bigger, thus making the primitive ability to deal with numerositiesfundamentally distinct from developed arithmetic. All this is clearly di�erentfrom actual arithmetical thinking, which deals (at least for a very large part)with the verbal numerical representations that Nieder mentions. And whileNieder is no doubt correct in stating that the primitive ability is among theempirical researchers generally thought to be a precursor to the actual mathe-matical ability, at this point this is still in considerable part a conjecture. Philo-sophically, this conjecture can be absolutely crucial. Take the above sketch ofan argument against conventionalism. Clearly that argument would be refutedif it turned out that our developed mathematical ability was independent ofthe primitive non-linguistic numerosity system. This way, the question here isnot whether ANS exists or whether it can be described as proto-mathematicalability. The question is whether it is plausible that our arithmetical thinkingactually develops from the ANS.

How could such a question be studied? The �rst idea is of course to trackthe development of numerical thinking and examine whether it is continuous inthe way that Nieder assumes. This, however, includes an important problem.

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As it happens, the primitive ability does not disappear when our mathematicalthinking develops. This is best seen in the phenomenon of subitizing. Adulthuman beings subitize and their ability in that closely resembles the primitiveability of animals and infants. Thus it appears that adult subjects have twosystems of dealing with numerosities: one an approximate estimation tool ofmagnitudes, the other a language-dependent system that we use in learningmathematics.

That, however, cannot be thought to be a refutation of the argument againstconventionalism. The existence of the two systems should not be confused withthe latter not developing out of the former. If the verbal numerical presentationsare indeed built on the primitive system of mental magnitudes, it is irrelevantto the conventionalist debate that both systems continue to exist. In fact thiswould hardly be surprising: people continue to have all kinds of intuitions andprimitive conceptions even when they develop a better knowledge about thesubject. Optical illusions are one case in point. Even though we perfectly wellknow that two lines are of equal length, in some circumstances we cannot helpseeing one as shorter. Yet we would never claim that our ability to establishthe length of the lines was not built on seeing the lines. We simply acceptthat our impressions of objects can be di�erent when the circumstances change.Obviously the same thing can happen with numerosities: we can continue tohave rough estimates of quantities even when we have developed tools to processthem in an exact manner.

Fortunately, the questions presented above have been studied empirically �and the results are philosophically highly relevant. What happens in the brainwhen we observe objects? The full story is too long to be told here, but thephilosophically important main idea is that the brain is full of di�erent typesof �neural �lters�. When we see something, an enormous amount of activitygoes on in the brain so that we can gather the relevant information from ourobservation. This is why it has been tremendously di�cult to develop visualrecognition in robots. In order to separate the relevant parts of the visual �eldfrom the irrelevant, the robot has to be programmed with excruciating detail.Our brain � and for that matter, the rat or even the �y brain � does suchthings automatically because it is accommodated to recognize the aspects thatare important. Crucially to the matter at hand, part of this activity has to dowith quantities (Nieder (2011)).

A lot is known about the primitive ability to deal with numerosities. It isknown, for example, that it is much more accurately modeled logarithmicallythan linearly. It is easy to tell the di�erence between 1 and 2, but harderbetween 4 and 5. At 19 and 20 monkeys (and us) are hopeless at recognizing thedi�erence without using more sophisticated tools, i.e., counting. The di�erencebetween 10 and 20, however, is easily established. Thus the ratio of cardinalitiesmodels our natural ability better than the di�erence between them and hence

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the ability is better described as roughly logarithmical.4

That much we know from observing the behavior of subjects in the exper-iments. Now the interesting question is whether that is mirrored also in theway the brain processes numerosities. As it turns out, there is a remarkable re-semblance. We can know that because, as described in Nieder (2011), it is nowknown that there are not only distinct areas of the brain where quantities areprocessed, but also speci�c neurons which represent certain quantities. Whenthe monkey is presented with two objects, a speci�c group of neurons activate.When the cardinality of objects is three, a di�erent group is activated. Theexperiments have been controlled for other variables, and the scientists havebeen able to tease out the e�ect of a particular quantity in the monkey brain.The brain, however, is a complex organ, and while there are speci�c neuronsfor each small numerosity, those neurons do not activate completely discretely.When the neurons for the numerosity �two� are activated, so is a small partof the neurons for �one� and �three�. And just like the behavior of monkeyspredicts, as the numerosities become larger, the bigger the �noise� is betweenthe di�erent groups of neurons. Distinguishing between four and �ve is muchmore di�cult than between one and two because in the former case more of thesame neurons activate. Our natural capacity to deal with numerosities is oneof approximate estimations that loses accuracy as the quantities become larger.What happens in the brain mirrors this.5

That is not all. Unlike many other animals, monkeys have the ability to(through extensive learning) understand symbols assigned to concepts, includ-ing numerosities. Diester and Nieder (2007) established that to large part thesame neurons in the pre-frontal cortex were activated regardless of whether themonkey saw two objects or the symbol 2.6

So far we have been dealing with primitive forms of dealing with numerositiesbased on subitizing and the analog magnitude system. But what happens whenwe are counting? Undoubtedly this is fundamental to our developed capacity todeal with numerosities and enables us to formulate the exact nature of numbers.In Nieder et al (2006), monkeys were presented objects one by one, to simulate anon-verbal account of counting. As expected, there were di�erences in the partsof the brain that activated compared to the task of seeing a group of objectsat once. However, the study found that at the end of the enumeration, a largepart of the activated neurons were the same as with the ANS. In short, whencounting, the monkey was dealing (partly) with the same representations fornumerosities as with subitizing.

What do these results suggest our ability with numerosities to be? The besthypothesis is that our basic experiences about small quantities are given by the

4The ability is not properly logarithmical, however, as distinguishing between two and fourobjects is easier than between eight and sixteen objects. See Dehaene (2011, chapter 4).

5In addition to cardinalities, similar results have been acquired in the study for proportions(Nieder (2011)).

6Interestingly, in the intraparietal sulcus, which is another part of the brain associated withnumerosity, the number signs triggered much less activity. This suggests that while there areclear connections between the number symbols and numerosities of ANS, these are di�erentin the two areas of the brain.

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approximate number system. As we develop the linguistic ability to count, weno longer require the ANS for our numerical ability. Indeed, even though wenever completely lose ANS, it could be said that the primitive ability is replacedby a language-based one. This gives us a lot of added expressive power, whichmakes arithmetic as we know it possible.

Of course these subjects need a lot of further study. But the data we cur-rently have is already relevant for the philosophy of mathematics. Let us con-sider the di�culty of refuting conventionalism on the basis of empirical sciencebecause our mathematical ability may be distinct from the primitive numerositysystem. We now know that there are important connections between that sys-tem, the symbolic presentation of numerosities, and counting. There is startingto be way too much correlation to be explained away simply as a coincidence.The data clearly points to the direction that our verbal ability to deal withcardinalities was built to accommodate the primitive non-verbal system. Thatis of course nothing extraordinary. Although there are di�erent areas of thebrain involved in subitizing, counting and recognizing symbols for numerosities,it would be surprising if there were no connections between them at all. Thebrain in general is built to facilitate learning and the existing information is usedto assimilate new data. If we have one mode for dealing with numerosities, aswe initially do, it would seem very unlikely that the brain starts to build anotherone completely from scratch instead of utilizing the existing connections.

In conclusion, the current set of empirical data points to philosophicallyinteresting theories about our capacity to deal with numerosities. At this laststage, however, we skipped past an important point. One crucial question is ofcourse whether these results concerning the monkey brain can be applied to thehuman brain. There is evidence also for this. In Piazza et al. (2007), it wasfound that when observing numerosities, the same areas of brain are activated inhumans as in monkeys. Given the similarity between monkey and human brains,such results are hardly surprising. Where di�erences are to be expected is ofcourse when the primitive numerosity system is little by little supported and�nally largely replaced by the verbal capacity to deal with cardinalities. But theprimitive approximate number system does not vanish. There have been manyexperiments that show college students to have similar patterns as monkeys andrats when solving number-ordering and quantity estimation tasks. It seems thateven people educated with discrete arithmetic often cannot help reverting to theprimitive approximate estimation system (Cantlon et al (2006)).7

With all the above data, it seems highly plausible that our capacity to pro-cess quantities has a shared origin with many nonhuman animals, and that itis connected to an ability that we have already as newborn infants. Of coursethere are many questions that still need to be answered about these originsand their development, and the answers should not be treated as a foregoneconclusion. Still, I believe it is de�nitely worthwhile to consider the philosoph-ical consequences of the modern developments in psychology, neurobiology and

7See also Brannon & Merritt (2011) for a good overview of the subject.

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cognitive science. The questions that interest philosophers most in this contextconcern the development from the approximate number system to axiomatictheories of arithmetic. At this stage, aside from the empirical work, we muststart to ask fundamentally philosophical questions. No amount of work on eventhe most brilliant monkeys or infants will present us with anything close to thehuman adult ability in mathematics. We now have a good idea what arithmeti-cal knowledge is initially based on. What we need to answer accordingly is whatarithmetical knowledge is.

3 Empirically feasible epistemology

Based on the data presented in the previous section, what kind of an episte-mological theory of arithmetic is empirically feasible? The only clear-cut re-quirement would seem to be that the theory cannot contradict with the proto-mathematical ability of ANS. If we accept that ANS is indeed the origin of ourarithmetic, then arithmetic knowledge cannot be completely conventional. Ipropose here that it also follows that arithmetical knowledge cannot be com-pletely conceptual in the language-dependent sense relevant to this context.

There is much debate on what we should understand by the concept of�concept�. It is not possible to go to that debate here, but for the purpose ofthis paper it makes sense to distinguish between concepts in a primitive senseand a linguistic sense. The simple argument for that position is that whatevercapacity the gold�sh, for example, uses in its ability to distinguish between smallnumerosities, it is not the same capacity that mathematicians use in practicingarithmetic. The primitive proto-arithmetical numerosity concepts given by theANS should not be confused with properly mathematical concepts, which aremore well-de�ned and precise, something that require a language to process.8

With this understanding of mathematical concepts as language-dependent,an empirically feasible epistemology requires that arithmetical knowledge is notonly about those concepts. The natural alternative is of course that it is aboutobjects. Along these lines, Burge (2007, 2010) has recently evoked empirical datain support of his argument that the objects of arithmetic can be represented ina de re fashion, i.e., the reference for numbers in arithmetical sentences is nottotally conceptual, de dicto. He uses forms of subitizing as an example of theway we use numbers without using numerals (2007, 74-75):

. . . these types of de re representation are associated with specialimmediate, non-descriptive powers � understanding and immediateperceptual applicability.

8It is a matter of debate whether we should accept there being non-linguistic conceptsin the �rst place. In any case, the de�nition of �concept� does not change the essentialargument here. If one is prepared to see the gold�sh's cognitive processing as conceptual inthe mathematically relevant sense, some of the terminology must be adjusted. But few woulddeny that there is a relevant di�erence between the gold�sh's processing of numerosities andour language-based understanding of number. This matter will be revisited in footnotes 15and 19.

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I believe Burge's argument uses a mistaken interpretation of the empirical data.Whereas I do agree that the empirical data presents a serious challenge for anyphilosophical account that takes mathematics to be purely language-dependent,this does not imply that in subitizing we refer to numbers as objects. Surely thereare various ways in which processes like subitizing can be explained withoutthere being such objects as numbers. Azzouni (2010, 33) suggests, for example,that we may have a natural tendency to put objects into one-to-one comparisonand thus be able to subitize between small quantities. There is some empiricalevidence that such a process indeed forms at least part of our ability to representobjects in the working memory (Leslie et al. (2008) and Feigenson (2011)). Buteven if this were false, it is easy to propose other plausible explanations. Whatif instead of numbers, whether de dicto or de re, we think of the primitivenumerosities as dealing with simpler concepts like equal, larger and smaller?

In this mode of explanation, when we subitize a group of four objects, wedo not arrive at an abstract object, the number four. Rather, we form a kindof baseline numerosity of �many.� When then presented with a group of threeobjects, we naturally subitize the concept �fewer�. With two objects, �halfas many�, and with one object �a lot fewer�. This is one very hypotheticalexplanation and I do not claim that these concepts correspond to anything thatactually happens in the process of subitizing, although there is evidence thatsomething like that is actually going on (Lourenco & Longo (2011)). The pointis that such simple comparative concepts would be enough to explain the abilityto subitize, remembering that it starts to radically weaken after groups of fourobjects. There is no need to evoke numbers as objects.

Such arguments, however, do not imply that there is no objective basisfor our ability to deal with numerosities - only that the objects that we areconcerned with in arithmetic (or proto-arithmetic) are not necessarily numbers.In philosophy of mathematics, this kind of argumentation is of course nothingnew, as there is a well-established distinction between the objectivity of truth-value and objective existence. The former, endorsed by, e.g., Kreisel9 is whatI mean by arithmetical objectivity here and ANS would seem to be a good �twith such a truth-value-based conception of objectivity. Even if we do not evokenumbers as objects, the research on ANS clearly points toward an objective basisfor arithmetic, one based on how our brains process observations. 10

This question is related to various basic epistemological problems. Objectiv-ity comes in many forms and it is not clear that all philosophers would accept

9Dummett (e.g. 1973, 228) has made this �Kreisel's dictum� famous: �The point is notthe existence of mathematical objects, but the objectivity of mathematical truth,� althoughthe quote is not known to be found in Kreisel's own writings. The quotation as attributed toKreisel can only be found in Putnam (2004, 67).

10In the philosophy of mathematics, such objectivism without objects is not a new suggestionin mathematical ontology. If we take a structuralist approach to mathematics, the focus can beturned from objects to characteristics that arithmetical structures have. Among structuralists,however, there is great disagreement about the ontological status of mathematical structures,ranging from the objectively existing ante rem structures of Resnik (1981) and Shapiro (1997)to the modal constructions of Hellman (1989).

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such physiological propensities toward processing quantities as enough to givearithmetic an objective basis � assuming all along that the ANS is indeed theprecursor to arithmetic. Such thinking, however, seems quite problematic. Ifthere is a basis for arithmetical knowledge that we already have as infants andshare with many nonhuman animals, it is hard to see what more we couldhope for in terms of objectivity from an epistemological theory of mathematics.Many mathematicians are convinced that they reach truths about Platonic ob-jects with their research. However, they will always have a hard time persuadingthe formalists and �ctionalists of that, especially since the formal part of mathe-matics, i.e. proving theorems from axioms, can be explained also from a strictlyconventionalist basis.11 But if there are features in our brain structure thatmake us process observations in certain proto-mathematical ways, and mathe-matics is based on those processes, the strict conventionalist case is suddenlymuch weaker. Infants and animals cannot be said to process quantities basedon conventions. If the ANS is behind our arithmetical ability, we should thinkof arithmetic as having an objective basis, one determined by our physiologicalstructure which we share with many animals.12

But in the case ANS is indeed the foundation for arithmetical thinking, howdoes arithmetical knowledge develop? In order to get a conclusive answer wewould need to have a much better empirical understanding of the development ofmathematical thinking than there currently is. However, there are some empiri-cal results which suggest an answer. The hypothesis that ANS is the foundationfor arithmetical thinking includes several predictions which have received cor-roboration from experiments. Studies have shown that we do not lose ANSwhen we develop symbolic means to deal with quantities (Butterworth 2010,Spelke 2011). In both older children and adults there is a strong correlationbetween improved performance in symbolic mathematics and the non-symbolicprocessing of quantities. Furthermore, it is known that the same brain areasactivate when dealing with symbolic and non-symbolic quantities and that theseareas activate in the brains of non-human primates (Piazza (2010)). Much ofthe current data suggest a fundamentally simple and coherent picture. Startingvery soon after our birth, we have a non-symbolic ability to deal with smallquantities. This ability is then developed to actual arithmetical ability, in aprocess where the development of language is likely to play an important role(Spelke (2011)).

Exactly how this happens is still largely unknown, but there is evidencethat grasping the idea of successor is central in this development. Based onsubitizing we have the ability to distinguish between small quantities, usuallyfrom one to four. This means that we have di�erent neural representations inour brain for those small cardinalities. Learning number words for those smallcardinalities comes naturally under such circumstances, as they correspond to

11Here by strict conventionalism I understand the position that mathematical statementsare ultimately mere human conventions not based on any objective basis.

12I acknowledge that this understanding of objectivity is somewhat non-standard, but itseems to be the relevant one when it comes to the present arithmetical context. See Footnote20 for more.

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to those representations. For understanding larger numbers words, a highlyplausible hypothesis is that children grasp the idea that these numbers form aprogression that can be continued (Butterworth (2010)). When one is addedto three, even an infant can tell that the numerosities are di�erent. But thisability comes from the approximate number system and it is only later thatthe child learns that adding one to four is similar; that there also is a distinctnumber for the end product of that process (Feigenson (2011)). From that thereis a short way to addition of two numbers, which in turn enables the process ofmultiplication and so on.13

Now the question is: how relevant should such hypotheses be in the philoso-phy of mathematics? We should be very careful at this point, because buildingepistemological theories of arithmetic based on empirical data is an extremelytricky endeavor. Carey (2009a), for example, has proposed such a theory inwhich she distinguishes between the �logical� and �ontogenetic� programs of ac-counting for the origin of arithmetic. Roughly put, the distinction is that thereare two kinds of building blocks. The logical building blocks are the logicallynecessary prerequisites to have some arithmetical capacity. The ontogeneticbuilding blocks are the actual capacities used in the historical development ofarithmetic. Importantly, Carey holds that the ontogenetic blocks depend onthe logical ones on characterizing the capacity. For example, we can use theDedekind-Peano axioms to characterize what is meant by natural number inthe ontogenetic building blocks, presumably also the hypothesis based on theANS.

This kind of argumentation, however, is extremely problematic since it seemsto simply assume what is the most important matter that an epistemologicaltheory should argue for: that we are speaking of natural numbers in the samesense in the ANS-based developmental theories as we do in the developed arith-metical theories. Clearly the primitive capacity to deal with numerosities isvery di�erent from our developed arithmetical ability, and the key question inthe whole matter of empirically feasible epistemology of arithmetic is whetherthere is a continuous development from the former to the latter. I do not claimthat we know the answer, and as long as we do not, we should be careful aboutmaking the kind of assumptions that Carey seems to make.14

Moreover, I believe that we should as a methodological guideline refrainfrom mixing the logical and ontogenetic aspects. Even if we were convinced

13This is of course nothing more than an outline of a sketch of the process, but anythingmore than that would still be speculative at this point. Although I believe that a theory likethat of Lako� and Núñez (2000) is generally plausible, we should be careful about confusingpossible developments with actual ones. I do not want to make speci�c claims about the stepsthat the development of our arithmetical ability has taken, but I do claim that this processwill be revealed empirically � if at all.

14Carey (2009b, Ch. 8) also presents an account of children learning the inductive natureof natural numbers that is based on the successor function. While I believe such an account ishighly plausible, we must be careful not to postulate the inductive ability before we know whenit is actually acquired. Interestingly, Carey recounts experiments with nonhuman animalswhich fail to make the inductive step, thus giving further evidence that while ANS may bethe basis for our arithmetical ability, it is not enough by itself.

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that ANS is the basis for our arithmetical ability, we should be interested inhow it develops into accepting systems like the Dedekind-Peano axioms. Whenwe simply draw properties from the latter to describe the former, we mightbe seriously distorting the ontogenetic explanations. Since, presumably, manypeople are not convinced that there is a continuous development from ANS toDedekind-Peano axioms, this is all the more important. What we should do istake the best empirical knowledge we have and try to �gure out what kind of anepistemological theory of arithmetic can be plausibly developed based on it. Ifit resembles closely our developed theories, we are likely to be on the right trackwith our ontogenetic explanations. As I have argued in this part of the paper,I believe this is quite feasible with the current set of data. I could be wrong: itneeds to be stressed that there is no conclusive evidence yet. But even with theexisting evidence, we should be interested in what kind of an epistemologicaltheory we need in order to account for the empirical �ndings.

This is not just a philosophical exercise: as long as the empirical evidenceunderdetermines the epistemological conclusions, we need philosophical theoriesto �ll out the picture. Of course these should not be confused with hard evidence.But if we can draw a plausible epistemological outline of an ANS-based theoryof arithmetical knowledge, at the very least we know that such accounts arepossible. In philosophy we should hardly wait for a complete neurobiologicaldescription of the development of arithmetical thinking.

4 Contextual a priori

In the previous section I have tried to argue that an empirically feasible epis-temological theory of arithmetic does not need to evoke independently existingmathematical objects, but can rather reach objectivity based on the physiolog-ical similarity of brain structures. It is on this similarity that arithmetic as weknow it can be developed upon. This is a fact so trivial that it almost goeswithout saying. When children are �rst introduced to number words and basicarithmetic, we simply assume that they observe the world as distinct objects, aswell as have the ability to process these observations in terms of discrete quan-tities. This assumption has been made as long as arithmetic has been taught,but only in the past few decades have we found an explanation for its success.In arithmetic, we do not introduce children to a new way of thinking. We givethem the necessary conceptual knowledge to expand and make exact a way ofthinking they already possessed in a primitive form.

The question is, if this account based on ANS is correct, what type of knowl-edge do we reach this way - and does this type correspond to our understandingof mathematical knowledge? Is it, for example, a priori in character, as mathe-matical knowledge is often understood to be? Or do the primitive origins meanthat it is essentially a posteriori, tied to empirical aspects?

I propose that mathematical knowledge is best understood outside of thisKantian dichotomy, as it includes characteristics from both but as a whole �tsneither concept. My rejection of the dichotomy is not Quinean in nature: I

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believe that there are fruitful distinctions to be made between a priori andposteriori modes of knowledge. Rather, I believe that the Kantian dichotomydoes not necessarily exhaust the �eld of such fruitful distinctions.

The question of a priori knowledge is of course actively studied and givenremarkably di�erent treatments among philosophers.15 Since Quine (1951), ithas been by no means obvious that the dichotomy between a priori and posteriorishould even exist. In the Quinean web of beliefs, traditional candidates fora priori knowledge, such as arithmetic and logic, are taken to be empiricallyfalsi�able due to their connections to scienti�c theories. The Quinean theory isoften understood to take no beliefs as separate from the web. In case of logic,for example, Putnam (1968) famously argued that a revision is already at handwith the quantum logic.

It is not possible to go here into such general questions about the possi-bility of a priori knowledge. In any case, the characterization of arithmeticalknowledge I defend in this paper is not a strictly a priori one. Whether otherforms of knowledge may be purely a priori is another question. Here I claimthat arithmetical knowledge has characteristics that do not �t what I see as afeasible notion of priori.

Let us consider the classic Kantian characterization of a priori as knowl-edge that can be gained essentially independently of sense experience. Underwhat kind of interpretation would the present ANS-based theory of arithmeticalknowledge count as a priori? A thorough analysis would open many philosoph-ical cans of worms, so let me focus here on just one aspect of the question. Anarithmetical statement like 2 + 1 = 3 undoubtedly has a strong appearanceof being an a priori truth. The way we de�ne the symbol �3� in axiomaticarithmetic is with the help of a successor function S. In a standard de�nition,we de�ne 0 as the �rst natural number S(0) as the next number, then S(S(0)),S(S(S(0)))) and so on. Then we name S(0) as 1, S(S(0)) as 2, S(S(S(0)))) as3, etc. The operation of addition we also de�ne with the help of the succes-sor function, in that the result of addition of X + Y is applying the successorfunction to the number X for Y times. In the addition 2 + 1 = 3 this simplymeans that we �rst use the function S two times on the number 0 to get thenumber X and then once more to add Y to it. Thus constructed, it is easy toget the impression that the symbol 3 means the same thing as 2 + 1, becauseboth mean applying the successor function S on the number 0 three times.

This, however, does not mean that addition is necessarily a priori knowledge.We must remember that axiomatization was a very late development in thehistory of arithmetic. When we have such sophisticated systems in place, it isnatural to understand arithmetic as a priori. Indeed, I will argue that once thecontext for arithmetical knowledge is set, it is best described as a priori. Wecan develop a context in which arithmetical truths are derived independently ofexperience and cannot be falsi�ed by empirical results. But in order to be strictlya priori, arithmetical knowledge would have to be independent of experience in

15See Boghossian and Peacocke (eds.) (2000) for a nice collection of the kind of problemsthat engage modern philosophers in the study of a priori.

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a much stronger sense. It could not depend on the empirical context set byANS.

Earlier I distinguished between the primitive ANS-based numerosity con-cepts and the linguistic arithmetical number concepts. If arithmetic were onlyabout the latter, there would be a strong case for arithmetical knowledge beinga priori. But since the content of arithmetic is at least partly determined bythe primitive ability to deal with numerosities, we must include them in a fullaccount of arithmetical knowledge. Can we have a priori knowledge of the prim-itive numerosity concepts? It seems that if we can, we are opening up too manypossibilities for a priori knowledge. There is strong evidence that the abilityto acquire primitive numerosity concepts is innate and thus completely under-written by our brain structures. If we consider knowledge about those conceptsto be a priori, what is there to stop us from considering knowledge about anyability underwritten by brain structures as a priori? Under such interpretation,arithmetical knowledge could indeed be classi�ed as strictly a priori. But itwould seem to follow that so could knowledge about many basic facts about ourlinguistic and cognitive abilities, which is unacceptable. Through introspectionwe may have privileged access to some insights about language and cognition,but it would be hopelessly antiquated to insist that these subjects can be stud-ied without empirical means and that our knowledge of them consists primarilyof empirical facts.

Arithmetical knowledge is of course a special case and it deserves an indepen-dent treatment. We should remember that the de�nition of a priori knowledgeis not that it is gained independent of experience, but that it can be gained so.But while it may seem plausible that we could gain arithmetical knowledge out-side of the empirical context of ANS, this is due to the traditional philosophicalunderstanding of the subject, rather than any evidence or argument. The mod-ern evidence suggests a di�erent picture where ANS is fundamentally linked tothe ability to develop the linguistic concept of natural number. When we learnarithmetic, we do it after years of experience in dealing with numerosities in aprimitive way, based on an ability we share to a large extent with other animals.For all the appearance of being a priori, arithmetical knowledge seems to be apriori - if at all - only within that empirical context.

It should be noted that I do not claim to refute the possibility of strictlya priori arithmetical knowledge, just like the epistemological account based onANS is not meant to refute Platonism. Even if the way we acquire mathematicalknowledge is not purely about the linguistic concepts, there remains the possi-bility that we have a special mathematical a priori faculty for gaining knowledgeof non-linguistic conceptual truths.16 However, the purpose of this paper is topropose an explanation of arithmetical knowledge on ontologically and episte-mologically minimal grounds. Special mathematical faculties for knowledge areno more acceptable than platonistic objects in such a pursuit.

16Such a faculty is most often associated with Gödel in the literature, but a version of itwould seem to be at the root of intuitionism, as well.

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There is a possible argument against strict apriority based on the di�erentarithmetical and systems that exist. This naturally concerns the di�erent ax-iomatizations of arithmetic - if they are not arithmetically equivalent - but morerelevantly, the di�erent ways in which people have developed arithmetic. In thephilosophy of mathematics, more attention is usually given to mathematicallyrevisionist approaches like ultra-�nitism (Essenin-Volpin 1961), which deniesthe possibility of an in�nity of natural numbers.17 But equally relevant arecultures which have considerably di�erent arithmetical systems, sometimes notdeveloped much beyond the proto-arithmetical experiences given to us by ANS.The Pirahã people, for example, have the number words corresponding to �one�,�two� and �many�. They can do the corresponding arithmetic in which one plusone is two and one plus two is �many�. However, two plus two is also �many�,even though one is di�erent from two (Gordon (2004)). The resulting system ofarithmetic is clearly very di�erent from ours. In order to have our knowledgeof arithmetic, we need to have developed the idea that we can add one to anynumber and the result is always (for �nite numbers) another number.

Most people would not want to consider the Pirahã system to be arithmeticat all, but there has also been at least one culture which was arithmetically con-siderably developed but yet their arithmetic took a considerably di�erent routefrom ours. The Mayans were highly sophisticated mathematicians and couldcalculate numbers up to billions (Ifrah 1998). They could also use arithmeticto great success in predicting astronomical events. In many ways, their abilitywith natural numbers was comparable to ours, and it would not make senseto claim that they were dealing with a fundamentally di�erent conception ofnatural numbers. However, their arithmetic seems to have been remarkably dif-ferent from ours. While they could calculate with extremely high precision, theydid not prove general truths about numbers. Moreover, they did not seem tohave a concept of in�nity of the numbers.18 All in all, the Mayan arithmetic atthe same time closely resembled our arithmetic and was fundamentally di�erentfrom it.

But should we consider the Mayan �arithmetic� to be arithmetic in the �rstplace? In the literature, there are many di�erent ways to understand �arith-metic�. To take a common understanding, the natural numbers form a discretesequence that is linearly ordered, each number has �nitely many predecessorsand exactly one successor, and the sequence does not have a largest element.Under such understanding, neither the Mayan system nor the ultra-�nitist ap-proaches - not to mention the Pirahã system - can be considered to be arithmetic.In this context, however, that seems needlessly limiting. There is an extremelyrelevant di�erence to be made between proto-arithmetic and arithmetic thatis more relevant here. When we develop our theory of cardinalities to includenumber symbols (or words) and explicit rules for operations on them, as well

17It should be mentioned that there are also �nitist approaches which aim to be non-revisionist mathematically. See Lavine 1994 for an example of non-revisionist set-theoreticalultra-�nitism.

18Ifrah (1998, 298) writes that Mayans had some notion of in�nity, but in the arithmeticalwritings that remain, in�nity is not to be found.

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expand it beyond the small numerosities of ANS to apply to large quantities, Ibelieve it makes sense to speak of an arithmetic. This makes sense particularlywhen we consider developmental aspects. While there is an important movefrom school arithmetic to formal systems like Peano arithmetic, developmen-tally the more crucial jump is from ANS to language-based arithmetic. Thisway, the use of the term �arithmetic� here may be somewhat non-standard inthe philosophy of mathematics. But that is mostly because in philosophy thedevelopmental angle is largely neglected in favor of logical and formal aspects.

Thus understood, it is possible that arithmetic can develop considerablydi�erently in di�erent cultures. Instead of proving things like the in�nity ofnatural numbers, a culture can use arithmetic exclusively as a tool of explain-ing the physical world. While the more radically di�erent cases like the Pirahãsystem may be explained away as unarithmetical, di�erences in the more de-veloped theories suggest that the shared initial concept of discrete cardinalityunderdetermines the development of arithmetic. It seems that we shared theconcept of number with the Mayans when it came to calculations, but in theend developed arithmetic di�erently.

Such di�erent ways of dealing with cardinalities do suggest that the de-velopment from the proto-arithmetical experiences given to us by ANS to ouraxiomatic systems of arithmetic was hardly inevitable. Does that suggest thatarithmetical knowledge is not objective? Is ANS enough of a reason to speak ofan objective basis for mathematics? Indeed, is it a reason at all to believe thatmathematical knowledge can be objective in any relevant sense?19

Interestingly, there is a related debate concerning moral realism. Ever sinceRuse (1986), evolutionary explanations for moral values have been considered tobe a problematic �t with moral realism. Street (2006), for example, argues thatwe cannot reconcile the position that there are objective moral truths with thefact that evolutionary forces have deeply in�uenced our values. Clarke-Doane(2012) has made the connection to mathematical objectivism explicit by arguingon epistemological basis that it may not be possible to be a moral anti-realistbut a mathematical realist.

19This matter was already mentioned in Footnote 11 and should now be dealt with in moredetail. The sense I have understood �objectivity� in this paper is obviously much weaker thansome standard understandings of objectivity in the philosophy of mathematics, including theobjective existence of natural numbers or the natural number structure. In the light of theempirical data reviewed in this paper, that kind of objectivity seems simply too much toask. If there are general physiological features responsible for at least some of the content ofarithmetical theories, it seems clear that arithmetical statements have content that is in somesense objective. Although it is not possible here to go deeper into the philosophical questionof objectivity, I believe that any arithmetically relevant understanding of �objectivity� shouldinclude this sense. That is why many important writings about mathematical objects andobjectivity, e.g. Putnam 1980 and Field 1998, do not touch the argument given here. Whilequestions treated in those writings, like whether the continuum hypothesis has an objectivetruth-value, clearly are central to the question of mathematical objectivity, so is the questionwhether that is the case for 1 + 1 = 2. To discuss the objectivity of this latter questionwithout taking into account the empirical data reviewed in this paper would seem to be takinga needlessly limiting view of objectivity and thus ending up ignoring important evidence.

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The evolutionary arguments against moral realism are quite forceful becausethere is a clear possibility of an epistemological gap between moral truths andthe moral beliefs natural selection favors. It Clarke-Doane's example, even ifkilling our o�spring were morally good, it would still be evolutionary advanta-geous to believe it is bad. Is the case with ANS-based arithmetic similar? Ifthere were objective arithmetical truths, do we have any reason to believe thatevolution has made it possible for ANS - and the arithmetic developed on it - tocapture them? Clarke-Doane argues for a strong similarity between moral andmathematical realism in this sense, even contending that 1 + 1 = 0 could, real-istically construed, be a mathematical truth while the �rst-order logical truthequivalent to 1 + 1 = 2 would be the evolutionary advantageous one. Even ifthat example seems far-fetched, the above examples of ultra-�nitism and Mayanmathematics surely give relevant examples of alternative arithmetics. Clearlynot all arithmetical theories can give us accurate knowledge of objective truths.How can we tell which one does? This is the �rst important question. Thesecond one is whether any arithmetical theory can give us objective truths.

How does the present ANS-based theory of arithmetical knowledge fare withsuch epistemological challenges? The �rst question seems less problematic.Compared to moral disagreements, the arithmetical ones seem quite minor. Themodern axiomatizations agree for the parts and the di�erent characteristics ofMaya and Pirahã arithmetics are more likely due to an underdevelopment ofthe arithmetical ideas than any substantial disagreement.

The second question is a more interesting one, but while the epistemolog-ical gap may indeed arise with a platonistic understanding of mathematicalobjectivism, there is no such problem with the present account. Clarke-Doane'sexample of 1 + 1 = 0 is inconceivable if we consider arithmetical truths to arisefrom the ANS. If arithmetical knowledge is in some way based on evolution-ary processes, there is naturally no epistemological gap involved in gaining thatknowledge. But now the question is whether such a basis for arithmetic can beconsidered objective under any relevant understanding of objectivity? What isthe di�erence to the problematic notion of moral objectivity?

I hope to have answered this latter question by the round-up of the empiricaldata in Section 2. ANS is something we share with �sh and rats, and althoughfurther analysis is not possible here, it should be clear that the roots of arith-metic go much deeper, and are much more speci�c, than those of human moralvalues. Indeed, that is where the analogue from moral values to mathematicsno longer applies. The arguments of Street and Clarke-Doane may be power-ful against moral realism and some of that power may apply to the argumentagainst mathematical realism. But we do not need to assume the existenceof mathematical objects in order to support the objectivity of mathematicaltruths. There may or may not be such things as natural numbers, but the em-pirical data strongly suggests that some of the basic rules concerning them areobjective in a very relevant sense, that is, they are not mere human conventions.

It is clear from the di�erent arithmetical systems that ANS underdeterminesour arithmetical theories. There are conventional and cultural factors involved

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which are independent of the content determined by ANS. That raises an impor-tant question: to what level do conventional aspects determine our arithmeti-cal conceptualizations? With the di�erence in Pirahã and Peano arithmetics,for example, it seems that the conventional part is indeed huge. Indeed, thatshould be expected based on the very primitive nature of ANS. Just like the �rstattempts at explaining empirical data can give completely di�erent results, ex-plaining and expanding on our proto-mathematical ability is hardly somethingthat happens uniformly.

The ratio of conventional and biological elements in particular theories ofnumerosities is an intriguing question, but based on the empirical data it seemsthat arithmetical knowledge is never completely conventional. There is an ob-jective element to our arithmetic, one determined by our biological structure.Arithmetical knowledge concerns the conventional factors, but never only them.Thus arithmetical knowledge is not only about the meanings of number con-cepts.

The case with nonhuman animals and infants should make this clear. Howcould arithmetical knowledge be totally conceptual if subjects that have littleor no ability for conceptual thinking can have proto-arithmetical abilities? If weunderstand that conceptual knowledge deals with relations within languages -and is not �about the world� - this seems untenable. Arithmetical knowledge, ifthe above empirical considerations are valid, is about the world in a substantialsense, i.e., about the way many animals observe the world.

�All bachelors are unmarried� is often presented as a standard case for con-ceptual knowledge. �1 + 1 = 2� is another one often suggested. But there is animportant di�erence between the two. The latter formulates a way of observingthe world we have as infants and share with many animals. The content of �1+ 1 = 2� is used constantly by agents who do not have number concepts. Theformer deals exclusively with the meanings of certain words. While arithmeticcan be presented as conceptual knowledge, as in our example with the successorfunction, that does not mean it is only about concepts.

If arithmetical knowledge is not strictly a priori, would it not make sense tocategorize it as a posteriori, captured by empiricist theories? In the account ofKitcher (1983), mathematical knowledge concerns generalizations of operationsthat we do in our environment. For instance, we play around with collectionsof pebbles and learn that the operations obey certain rules. We can then gen-eralize on these rules and end up with rules of arithmetic applicable insteadof only small collections of pebbles to an in�nite number of objects. Withoutgoing to the details, I believe this is indeed how we essentially �rst learn aboutnumbers. Empirical aspects play an important role in most people's learning ofarithmetic. It is also plausible that the development of arithmetic was initiallytightly connected to empirical aspects. However, there are two di�erences thatmake such an empirical account problematic as an explanation of arithmeticalknowledge. First, after the context is set, the empirical methods are not indis-pensable, which makes arithmetical knowledge di�erent in character. I do notwant to go too deep into the realm of thought experiments here, but simply

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by understanding the axioms and rules of PA, it is possible to build a (withthe Gödelian restrictions) complete knowledge of arithmetic. In actuality, mostchildren use their �ngers to learn to count as well as add, but this should by nomeans be thought to be necessary.

Second, there is the rather obvious point that while arithmetical statementsmay not be purely conceptual knowledge, they simply are not empirically falsi-�able under any relevant reading of �empirical�. Arithmetical calculations areneither corroborated nor refuted by direct experiment. Mathematical proof hasits own special character and an epistemological theory must include it.

There is, however, a more sophisticated argument for the empirical natureof arithmetic. If ANS is indeed the basis for our arithmetical ability, clearly wemust experience the world as discrete objects. But although ANS gives us onlythe ability to determine the quantity of objects up to small numerosities, clearlywe also experience pluralities of objects much greater than that. This may beseen as putting into question my earlier contention that we need language-dependent concepts to develop genuine arithmetical cognition. Isn't there thepossibility that we can non-linguistically grasp the successor relation and thusground the epistemology of arithmetic on a much stronger empirical contextthan what the ANS gives us?20

While that is indeed a possibility, there is empirical evidence against it.The underdetermination of arithmetical theories is one argument to that e�ect,but there is the possible counter-argument that only developed systems like PAshould be considered arithmetic. However, a much stronger argument can befound in the empirical data about the ANS. The rough logarithmic nature ofANS suggests that we do not observe large pluralities in the same way as wedo small ones. As described in Dehaene 2011 (Chapter 4), our doing mentalarithmetic is a constant duel between two systems in the brain. Dehaene locatesour �number sense� (that is, the non-linguistic ability to deal with quantities) topart of the intraparietal sulcus, what he and his colleagues call �hIPS� (horizontalpart of the intraparietal sulcus). Language-processing, on the other hand, islocated in regions of the left hemisphere. Now the question is, what happens inthe brain when we are given simple arithmetical tasks? What Dehaene and hiscolleagues found out was that the more exact we need to in our calculations,the more we rely on the regions of the left hemisphere and less on hIPS. Whenasked, for example, whether 4 + 5 = 7 or 9, we need to process the arithmeticin language. But when asked whether 4 + 5 = 8 or 3, when approximation ismore important, the hIPS activates more.

Such data is extremely important because it is strong evidence for the po-sition that the development of language-dependent concepts is the key factorin moving from the proto-arithmetical numerosity estimation of ANS to actualarithmetical thinking. While it does not prove that developed arithmetic islanguage-dependent, it clearly points to that. Two main areas of activity in thebrain have been found when given arithmetical tasks. One, the hIPS, deals withsmall cardinalities after which it becomes increasingly approximate. The other,

20I would like to thank an anonymous reviewer for pointing out this question.

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the left hemisphere regions, deals with exact operations on larger cardinalities -what we usually understand by actual arithmetical cognition. So far there is noevidence of there also being a non-linguistic capacity for those latter operations.Based on the current data, the best hypothesis is that while ANS is at the rootof our arithmetical ability, it takes language-dependent concepts to develop itinto actual arithmetical ability.

If arithmetic is neither a priori nor empirical, then what is it? I proposeit is actually a combination of the two, best described as contextual a priori.Contextual or relativized notions of a priori are not new in philosophy. Themost famous such conceptions in the literature are perhaps by Putnam andKuhn. The famous concept of paradigm in Kuhn can be interpreted as being anotion of contextual a priori. In his later work, Kuhn (1993, 331-332) describesthis connection:

[the concept of paradigm resembles] Kant's a priori when the lat-ter is taken in its second, relativized sense. Both are constitutive ofpossible experience of the world, but neither dictates what that ex-perience must be. Rather, they are constitutive of the in�nite rangeof possible experiences that might conceivably occur in the actualworld to which they give access. Which of these conceivable experi-ences occurs in that actual world is something that must be learned,both from everyday experience and from the more systematic andre�ned experience that characterizes scienti�c practice.

Putnam's (1976) characterization of his concept of contextual a priori alsosounds Kuhnian in spirit:

there are statements in science which can only be overthrown bya new theory�sometimes by a revolutionary new theory�and not byobservation alone. Such statements have a sort of `apriority' prior tothe invention of the new theory which challenges or replaces them:they are contextually a priori.�

Putnam's favored example is Euclidean geometry, which he holds to be a falsetheory about the world, but nevertheless contextual a priori knowledge. Theconcept of contextual a priori that I have in mind, however, is fundamentallydi�erent. I do not want to propose an epistemological theory in which falsebeliefs can be considered to be knowledge. Instead, I de�ne contextual a prioriknowledge to mean a priori in a context set by empirical facts. I believe thatarithmetical statements have objective truth-values. But they are not necessar-ily true about an objective world with no agents observing it. Rather, the bestexplanation is that simple arithmetical sentences are true when they correspondto the experiences that our primitive ability to deal with quantities gives us.

These experiences, usually dealing with numerosities from one to four, arequite simple. For those numerosities, the conception that our brain structureforces upon us is that there are four discrete quantities, forming a structure in

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which one succeeds the other. Even without knowing the rules of addition andsubtraction, we can tell when something goes wrong in the arithmetic of smallquantities. When we move to actual arithmetical thinking, we generalize onthese conceptions. This is not only a plausible philosophical picture, but as wehave seen, also supported by empirical evidence.

This characterization of arithmetical knowledge as contextual a priori mayresemble the above description by Kuhn, but in my view arithmetic is not justanother paradigm. Arithmetic certainly gives us a possible way of experiencingthe world, but based on the empirical data the proto-arithmetical ability seemsto give us much more than that: an e�ectively inevitable way of experiencingit. We cannot help categorizing observations in terms of quantities. Not onlyis this inevitable for adult human beings, but also for infants and various non-human animals. This way arithmetic is not relativized a priori knowledge inKuhn's sense. It is not a priori within some paradigm that could be completelyoverthrown by another paradigm. Of course a new paradigm may prevail theolder one, but for us to use it as arithmetic, it cannot con�ict with the proto-arithmetic that ANS gives us. How the proto-arithmetic develops based on theprimitive origins is an interesting question, and it is plausible that a considerablepart of arithmetic is essentially conventional. But since arithmetical knowledgeis always based on something non-conventional - the ANS - arithmetic is morethan simply another paradigm of categorizing our observations. It is a paradigmwe can never totally abandon, nor alter to a degree where it con�icts with theexperiences given by ANS.

In this way, Arithmetical knowledge can be true, but it is true as a theoryof what follows from one characterization (based on the successor function) ofour primitive ability to deal with numerosities. The primitive ability sets thecontext, the characterization sets the theory. And after this is done, arithmeticis not empirical as a theory of that process. It is essentially a priori in nature.As a theory of the world it could be false: for example, it could turn out thatthere is not an in�nite amount of things in the universe, and any ontologicaltheory postulating arithmetical objects as things in the universe would thusrender arithmetic false. But that does not need to be what arithmetic is: ratherthan a theory about the world � whatever we mean by that � we can think ofarithmetic merely as a theory of experience or categorization of observations.As such it seems to be best described as contextually a priori. Of course thisdoes not mean that arithmetic could not be a theory about the world, as well.But that is not a necessary step to take in order to save the epistemology ofarithmetic.

Above is of course only a very rough outline of an epistemological theory ofarithmetic, and the mere idea of generalizing on the application of the successorfunction leaves many arithmetical details open. However, in philosophy, I be-lieve we should be careful about �lling out these details. There is no doubt thatwe can build a detailed picture in the manner of Lako� & Núñez (2000), andfor the most part that work seems highly plausible. Their purpose is to formu-late a scenario of rigorous development of mathematics, starting from primitive

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arithmetic. The main idea (p. 53) is that mathematical ideas are metaphor-ical. Basic arithmetical operations are what Lako� & Núñez call groundingmetaphors, corresponding to simple operations on physical objects. Addition isgrounded on object collection, subtraction on taking objects away from a col-lection, etc. On top of the simple grounded ideas there are linking metaphors,which form abstract ideas. Examples are numbers as points on a line, algebraictreatment of geometrical �gures, etc. This way, they propose a cumulative de-velopment that our mathematics has had from its primitive origins. For themost part, it is highly plausible.

But what Lako� & Núñez draw is a possible picture. Such work is of coursevery important: it shows that from very humble origins we can step by stepbuild sophisticated mathematical theories. No doubt this is what actually hashappened with the development of mathematics. Nevertheless, there is limitedvalue in such explanations as long as they are not backed by hard empiricaldata. After all, from the fact that we have sophisticated mathematical theo-ries, we know that they can be developed from extremely simple ideas. It ishow this process actually happens in individuals (as well as historically withincultures) that interests us. We now have a very good idea about the �rst stepsin arithmetical knowledge. For the rest, empirical researches, philosophers andmathematicians should work together to build as strong theories as possible.

At the beginning, I composed a wish list for an epistemological theory ofmathematics. It is now time to see how the outlined theory of contextual apriori epistemology of arithmetic fares.

(1) It should not require any unreasonable ontological assumptions.

Regardless of what one's ontological leanings may be, the assumptions neededfor the present approach are hardly unreasonable. No mind-independent exis-tence of mathematical objects is presupposed, but neither is that denied. Onto-logically, the contextual a priori model is highly versatile. What are often seenas basic arithmetical intuitions are based on experience and hypothesized toarise from the structure of our brain. Whether that structure has developed tomirror some feature of the world is another question, and one de�nitely worthasking. One proposed explanation is that evolution will favor developmentsthat correspond to the structure of the world, and thus the proto-arithmeticalstructure of our observations can mirror the arithmetical structure of the world.I �nd such speculations unsatisfactory in many ways, but they are compatiblewith the epistemological theory proposed here. Indeed, the same goes for formsof Platonism: nothing in the current approach prevents the possibility that thebrain structure has developed to get information about numbers as indepen-dently existing objects. But the strength of the contextual a priori model isthat such assumptions are not needed. There are evolutionary advantages in�ltering observations in terms of quantities and that by itself can be enoughfor the proto-arithmetical structures to arise. The only ontological assumptionsneeded for that is the existence of the required biological organisms.

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(2) It should be epistemologically feasible as a part of a generally empiricistphilosophy.

The main problem many philosophers have with Platonism is that it seemsto require an epistemic access unlike all the other known psychological processes.The approach here leads to no such problems. Modern psychology and cognitivescience have shown that the brain plays a tremendously important role in con-structing our immediate experiences. That some of the brain �lters deal withquantities is perfectly feasible as part of a modern empiricist epistemology andgets solid support from the empirical data.

(3) It should be able to explain the apparent objectivity of at least somemathematical truths.

Since the empirical data strongly suggests that we have our proto-arithmeticalability to process quantities already as infants and share it with many nonhumananimals, it should be safe to say that there is very little that is subjective in theability. Whether we want to call the basic proto-arithmetical processes objectiveor something along the lines of �maximally intersubjective� is a moot point: theimportant matter is that they are not completely culture-dependent and con-ventional. If arithmetic is developed to accommodate these proto-arithmeticalprocesses, there is no great mystery that it appears to be objective. In therelevant sense, it is. Of course there are also culture-dependent aspects to arith-metic, and in other �elds of mathematics they are more prominent. It shouldcertainly be interesting philosophical work to consider the culture-dependencyof each area of mathematics based on the current model, but that will have towait for another occasion.

(4) It should not make the applications of mathematical theories in empiricalsciences a miracle.

This is a very tricky issue that would require (at least) another paper totackle. Generally speaking, it should be plausible that some mathematical the-ories have scienti�c applications since they are at the very core of our expla-nations of the world. If our understanding of the world is build in part onproto-arithmetical experiences, it is no wonder that we bene�t from applyingthem. And if there is a continuous development from arithmetic to, say, com-plex analysis or probability theory, it is plausible that these more specialized�elds will also have some applications. But this is considering the problem on avery general level. The fact is that many times the applications are surprisingand demand independent explanations. Moreover, there is also the questionwhy physical objects conform to such complex mathematical laws at all. Unlesswe accept the theory-dependency of observations in its extreme formulations,this is a bona �de question to ask. Here I do not want to propose an answer.Rather, I will have to be content with remarking that the problem is not any

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worse with the present approach than in competing epistemological theories.Indeed, I believe it is much less damaging than, say, in strict conventionalismor classical Platonism. In the former we assume that ultimately arbitrary con-ventions can help explain the world, while in the latter we assume that someabstract world of objects helps us explain the physical world. In Platonism thisjust raises a new question about the connection between these two worlds. Inconventionalism we are left with no explanation.

(5) It should not rid mathematics of its special character.

Since the model here is a priori in the empirical context, after the proto-mathematical beginning all arithmetical knowledge can be (in principle) ac-quired in a purely a priori manner. The method of proving theorems is retainedexactly as it is in traditional a priori explanations and none of the special char-acter of mathematics is lost.

(6) It should be empirically feasible: the best scienti�c data about mathe-matical cognition cannot con�ict with philosophy.

If the representation of the empirical data here is correct, and the episte-mological theory has been developed to accommodate the data as planned, thislast point is achieved by default.

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