Post on 24-Jan-2023
1
Revista de Filosofie
CIRCULARITIES IN THE CONTEMPORARY PHILOSOPHICAL
ACCOUNTS OF THE APPLICABILITY OF MATHEMATICS
IN THE PHYSICAL UNIVERSE
Cătălin Bărboianu
CIRCULARITIES IN THE CONTEMPORARY PHILOSOPHICAL ACCOUNTS OF
THE APPLICABILITY OF MATHEMATICS IN THE PHYSICAL UNIVERSE
Abstract. Contemporary philosophical accounts of the applicability of mathematics in physical
sciences and the empirical world are based on formalized relations between the mathematical
structures and the physical systems they are supposed to represent within the models. Such
relations were constructed both to ensure an adequate representation and to allow a justification
of the validity of the mathematical models as means of scientific inference. This article puts in
evidence the various circularities (logical, epistemic, and of definition) that are present in these
formal constructions and discusses them as an argument for the alternative semantic and
propositional-structure accounts of the applicability of mathematics.
Keywords: philosophy of mathematics; applicability of mathematics; mathematical entities;
mapping accounts; semantic accounts; circular definition; epistemic circularity; Frege; formal
language; second-order logic; first-order logic; contingent truth; isomorphisms
2
I. INTRODUCTION
I.I CONTEMPORARY ACCOUNTS OF THE APPLICABILITY OF MATHEMATICS
Until the 1990s, philosophy of mathematics focused on pure mathematics, especially on the
ontological aspects of mathematical objects and structures, generating the nominalism-platonism
duality, to which any contemporary theory still reports, and granting a central role to the so-called
indispensability arguments1, as well as on the aspects of mathematical creation, generating the
traditional philosophical views of logicism, formalism, and intuitionism. The notable exception that
moved the focus toward the applicability of mathematics in the physical universe was Frege’s work
from the end of 19th century which, although centered on the notion of number but generalizable to
the mathematical objects and structures in their wide diversity, has captured the essential problems
of the philosophy of applicability. Of contemporary philosophers in this field, Mark Steiner has
declared that Frege solved part of the problems of the philosophy of applicability. Others (C.
Pincock, O. Bueno, M. Colyvan) have developed more recent theories based on some primary
principles of Frege’s theory.
The basic problems of the philosophy of applicability of mathematics are grouped around
the central metaphysical problem of establishing an optimal connection/relation between the
abstract mathematical structures and the physical systems of the empirical universe, by which to
ensure on one hand an adequate representation of the physical system within the mathematical
model, so as to have a rigorous semantic interpretation of the mixed statements (with physical and
mathematical terms) as well as of their contingent truth, and on the other hand to rationally justify
and provide a “certificate of validity” for the application of pure mathematics in the empirical
universe, in the usual way it is applied by the mathematician. This need for a rational justification
of the application derives from the act of mathematical modeling itself, in which the mathematician
– in the last phase of the modeling – interprets the derived mathematical results in terms of the
modeled physical system with an amazing effectiveness and accuracy, confirmed more than
sufficiently throughout the history of science. This general effectiveness, not justified satisfactorily
until now by any philosophical theory of applicability, has been synthesized by physicist Eugene
Wigner through his famous syntagm “the unreasonable effectiveness of mathematics” [Wigner,
1960]. Because of that effectiveness of mathematics, the focus of the philosophers has moved
1 Called the Quine-Putnam argument after the authors who stated and sustained it for the first time (independently of each other, in 1980 and 1970 respectively) and developed in various forms in more recent works (see [Baker, 2009]), the argument consists of two premises and a conclusion, namely: (P1) We ought rationally to believe in the existence of any entities that are indispensable to our best scientific theories; (P2) Mathematical entities are indispensable in our best scientific theories; (C) We ought rationally to believe in the existence of mathematical entities.
3
toward the nature of the relation that connects the abstract structures with the physical ones, under
the idea that the simple representation – even of a mathematical nature – is not enough for
explaining the success of the application of mathematics in physical sciences, and mathematics
must represent the physical universe in the right way for reaching the intended justification. M.
Steiner – who declared Frege’s solutions for the semantic applicability of mathematics and the
internal conceptual relation between the physical object and the mathematical one as complete and
unobjectionable – still admitted as open and unsolved the rhetorical question
“How2 does the mathematician – closer to the artist than to the explorer – by turning away from nature, arrive at its most appropriate descriptions?” [Steiner, 1995].
This query summarizes one of the greatest unsolved problems of the philosophy of applicability,
namely the justification of the inference based on mathematical models.
More recent contemporary accounts (C. Pincock’s and O. Bueno & M. Colyvan’s),
following the goal of justification of applicability through going deeper into the nature and
properties of the relation between the mathematical structures and the physical universe, have
developed formal systems for representing and describing this relation, still in a mathematical
context as a premise considered indispensable for rational justification. Within these formal
systems, the explanatory function of the mathematical model has been followed more than before,
and employing explanation, generated new debates on the balance between the explanatory role and
the representational role of mathematics in describing physical phenomena, as well as on the criteria
that qualify an explanation as genuinely mathematical.
A primary classification – belonging to Pincock – of the contemporary accounts of the
applicability of mathematics is based on the relation between the mathematical object and the
physical one, but also on the criterion of accepting (the truth of) mixed statements. Thus, he
identified two types of accounts, namely internal-relation accounts and external-relation accounts,
the latter also called structural accounts3 [Pincock, 2004] which will be treated further with respect
to the subject of this paper.
The purpose of this paper is not to discuss the weaknesses of these accounts with respect to
their aim and accordingly to suggest formal constructive alternatives, but to put in evidence the
circularities that are present in their constructions. Even though I shall discuss aspects of
2 my emphasis 3 Actually, Pincock’s complete classification has three types, including a so-called no-relation account type, which denies the existence of any relation between the mathematical domain and the physical one. I have ignored this type as trivial in the sense that from the time when that classification was made until the present, no concrete theory into which to fit that type has been developed, with the degenerate (in sense of the definition of the type) exception of Hartry Field’s fictionalism, which denies the existence of mathematical objects and considers that mathematics is dispensable in science. There is also a practical reason for ignoring the no-relation account type, in the sense that this article’s topic – circularity – does not apply in that case.
4
malignity/benignity of the circularities in the specific context of those accounts, I will not label
those circularities with one of the two attributes, arguing for this undecidability (in the next section)
through the fact that there exist no objective universal criteria necessary for this labeling. In turn,
the existence of the circularities represents an additional argument for reconsidering the semantic-
type account (initiated and partially developed by Frege) and the propositional-structure account
(not developed yet) as non-circular alternatives.
I.II EPISTEMIC CIRCULARITIES, LOGICAL CIRCULARITIES, AND DEFINITION
CIRCULARITIES. ISSUES WITH MALIGNITY
In the literature, argumentative circularities are categorized as epistemic and logical, and the
definitions of both types involve the concepts of argumentation and truth. In addition, epistemic
circularity (EC) involves the concept of trustworthiness (reliability) of a source and of belief in the
truth derived through argumentation; thus, EC holds an anthropocentric component. More precisely,
epistemic circularity characterizes the formation of a rational agent’s own belief (either inferentially
or non-inferentially) in the trustworthiness of one of the agent’s belief sources X, if the formation of
that belief depends upon X, where to depend upon a belief source X in forming a belief B is for B
either to be an output of X or to be held on the basis of an actually employed inference chain leading
back to an output of X. The term “source” from this definition must not limit in any way the
existence of epistemic circularities to argumentations and theories using knowledge provided by
other agents (or by other theories created by other agents), because it can be extended naturally to
any kind of logical or mixed statement (in Frege’s sense, described previously) to which a truth-
value is assigned, or moreover, to a scientific method of investigation or principle. Thus, “source”
can represent not only information and knowledge, but also principle, conjecture, and theory4.
Through this extension, we can identify two types of epistemic circularity – one in which the
agent’s belief in one of the premises of the argumentation depends upon X and another one in which
the act of inferring the conclusion from the premises is an instance of depending upon X. A classic
example of the latter type of epistemic circularity – the type which I call methodological circularity
– is an inductive argument for induction: The agent cites many examples of inductive arguments
(with true premises) whose conclusions have been independently confirmed and then inductively
infers that induction is trustworthy. In such an argument, none of the premises depends on induction
(the source whose trustworthiness is the conclusion), but the inference from the premises to the
conclusion does.
4 For a complete characterization of epistemic circularity, see [Alston, 1986].
5
Logical circularity characterizes a logical derivation in which the conclusion is embedded
directly or indirectly in one of the premises; such characterization shows some primary structural
analogy with epistemic circularity. However, the main distinction between the two types is made by
the employment of the concept of trustworthiness of the premise for the epistemic circularity.
Logical circularity applies only to the argumentation process itself as logical derivation, generating
even a general logical truth with possibly false premises (proposition p p→ is logically true, but it
does not prove p), while the epistemic circularity applies to both argumentation and their premises,
the derived truth being dependent on the trustworthiness of the premises and necessary for
“warranting” a premise.
An important philosophical problem with circularities was (and has remained) their
malignity/benignity within a scientific theory, the degree of malignity (if it exists), and the criteria
that qualify a circularity as having one of the two attributes (if such criteria exist) – in other words
the complete answer to the question “Is circularity something bad in science?” I shall not insist in
this paper on these general problems, because its purpose is just to put in evidence the circularities
that are present in the current theories on the applicability of mathematics in physical sciences. I
will limit myself to mentioning that the general tendency is to not characterize the epistemic
circularities as malignant – appearances to the contrary notwithstanding – in large part in
themselves, but also in their effect, and to characterize the logical circularities as malignant.
In the sense of the definition of malignity of EC given by Bergmann [2004], according to
which EC is malignant if it prevents beliefs infected by it from being justified (definition adopted
by contemporary theories), the first proponents of benignity were the reliabilists, as being
committed to accept the track-record5arguments; this thesis is also confirmed by Alston [1993].
M. Bergmann, an important proponent of the benignity of EC, proposed a contextualization
of EC by identifying contexts and large categories of EC malignant in itself, arguing that for the rest
of the categories – the wide majority – EC is benign. His argument in favor of benignity is built on
the fundamentalist thesis that there can be noninferentially justified beliefs, and the basic principle
of his argument is that one who accepts this thesis must accept that track-record arguments are not
something bad6. His thesis is also supported by the primary Reidian principle that human faculties
are reliable, and this belief is non-inferential; hence the primary principle is a self-evident
contingent truth [Reid, 1969]. Bergmann identifies malignant EC in what he calls questioned source
context, namely that context in which the agent begins by doubting or being unsure of his/her
source’s trustworthiness and looks for a second opinion (independent of his/her perception) on that
source. The explanation offered by Bergmann of the labeling of this type of EC as malignant, is, in 5 A track-record argument is an argument in which one infers from the past success of a belief source in producing true beliefs that it is a trustworthy source. Such an argument tends to be infected with EC [Alston, 1993]. 6 still in the sense of Bergmann’s definition of malignity.
6
brief, the following: In a questioned source context, the EC-belief in a source X’s trustworthiness is
produced, at least in part, by a source (X) whose trustworthiness the agent questions. Because the
agent questions X’s trustworthiness, the agent has an undercutting defeater7 for all his or her beliefs
produced by source X, including the belief that X is a trustworthy source. This defeater keeps the
belief in X’s trustworthiness being justified. In addition, there is no reason for thinking EC-beliefs
in unquestioned source contexts are defeated in this way.
Other approaches of EC – pro benignity – worth mentioning are in brief: Goldman’s [2003],
for whom the evaluation of arguments and argumentation must be done exclusively
epistemologically and not syntactically (in the vein of Sorenson’s [1991] thesis) and who shows that
EC is not formally defective, but it may be epistemologically objectionable, and the arguments in
favor of benignity do not hold for the interpersonal argumentation; Brown’s [2004], who states that
we might have to accept EC reasoning if it were shown to follow from an epistemic commitment
which is unavoidable; Alexander’s [2008], who denies the No Self-Support Principle (NSS), on the
reason that NSS has the skeptical consequence that the trustworthiness of all our sources ultimately
depends upon the trustworthiness of certain fundamental sources that we cannot justifiably believe
to be reliable, and so we should not trust any of our sources at all.
Among the proponents of malignity of EC, emerge Fumerton, Vogel, and Cohen. Fumerton
and Vogel do not argue in favor of malignity of EC in itself, but postulate it, and the result is the
rejection of reliabilism on the basis of the principle that we cannot gain knowledge and justified
beliefs through EC. The core argument of Fumerton [1995] is the following: If the mere reliability
of a process is sufficient for giving us justification, as reliabilism entails, then we can use it to
obtain a justified belief even about its own reliability. Vogel [2000] proposes the rejection of what
he calls neighborhood reliabilism8 via his “gauge argument,” consisting of the following relevant
example: One is checking a gauge (for example, indicating the fill level of a hidden recipient, such
as the fuel tank of a car) about whose reliability s/he does not know, although in fact it is reliable.
For gaining knowledge on the gauge’s reliability, s/he undertakes successive readings of the gauge.
Through these readings, s/he forms beliefs not only about the level of fuel in the tank, but also about
what the gauge displays, combining them as follows: (1) On this occasion, the gauge reads ‘F’ and
the tank is filled in proportion F. From a representative set of such beliefs, she concludes
inductively that (2) The gauge is reliable. The perceptual process by which the agent forms his/her
7 We are not speaking here of the sort of defeater specific to the defeasibility accounts (such as Klein’s [1976]), namely in propositional form, but of a defeater in the form of a mental state. 8that reliabilism in which knowledge is defined as follows: Agent S knows that p if and only if, in the actual and neighboring worlds, the belief or the process that produced that belief is reliable. Neighborhood reliabilism contrasts with ‘counterfactual reliabilism,’ according to which, S’s belief that p is knowledge only if, in the nearest not-p world, S would not believe that p, or would not believe it via the method by which s/he formed the belief that p in the actual world.
7
belief about the reading of the gauge is reliable, but by the reliabilist hypothesis, so is the process
through which s/he reaches the belief that the tank is filled as the gauge indicates. Because
induction is also a reliable process, the whole process by which the agent reaches his/her conclusion
is reliable. Thus reliabilism allows gaining knowledge about the reliability of the gauge by knowing
the premises of the argumentation, which is EC reasoning.
Cohen [2002] argues that any theory that rejects the following principle allows knowledge
about reliability too easily:
(KR) A potential knowledge source K can yield knowledge for agent S, only if S knows K is
reliable.
Theories that reject this KR principle allow that a belief source can deliver knowledge prior
to one’s knowing that the source is reliable, called basic knowledge9; these theories can appeal to
our basic knowledge for justifying the reliability of the sources, and so in obtaining such knowledge
of reliability, we reason in a way that is epistemically circular; the problem posed by Cohen is that
in this way we gain knowledge too easily.
Proposed directions for research on the malignity of EC
From this overview we can see that research on EC with respect to malignity/benignity has
been limited to the process itself of gaining knowledge, specific to each theory infected with EC,
including both arguments and argumentation. The approach and research of malignity have not
studied in depth (and sometimes have not even made reference to) the possibility of providing
decidability criteria for malignity/benignity of EC, even though the problem is of the first degree of
interest, tracing back from the period of ancient skeptical Greeks – namely, that of resolving the
philosophers’ disagreements on various theories about knowledge and attaining any knowledge
through criteria that distinguish beliefs that are true from those that are false. Still, this generates
new disagreements as to the appropriate criteria of truth. In order to resolve these disagreements and
to know what the appropriate criteria are, we need to know first which beliefs are true – the ones the
criteria are supposed to select – and thus we encounter circularity. If we understand the right criteria
of truth as reliable sources of belief – sources that mostly produce true beliefs – we arrive at the
following formulation of the problem of the criterion:
(1) We can know that a belief based on source K is true only if we first know that K is reliable.
(2) We can know that K is reliable only if we first know that some beliefs based on source K are
true.
Assumption (1) is a formulation of the KR principle, and together with assumption (2), it leads to
skepticism: we cannot know which sources are reliable nor which beliefs are true. Assumption (2)
9in a sense other than the standard Reidian sense.
8
does not require us to know that beliefs based on K are true through K itself; we can rely on some
other source. However, (1) posits that this other source can deliver knowledge only if we first know
that it is reliable, and (2) that, in order to know this, we need to know that some beliefs based on it
are true. In order to know this, in turn, we once again have to rely on some third source, and so on.
Because we cannot have an infinite number of sources, sooner or later we have to rely on sources
already relied on at some earlier point. Thus, the argumentation is circular logically and not
epistemically.
Coming back to the problem of research on the decidability criteria for the malignity of EC,
a first possible explanation for its not going deeply would be the unanimous acceptance of the
existence of the logical circularity exposed above, which would, in turn, inhibit the tendency of
going deeper. I also identify a conceptual cause which manifests once the problem is posed: in what
sense should we see the malignity of EC? As an internal property characterizing circularity in itself
or as a potential of producing effects in the theory to which it belongs – or further, as a mental state
of the scientist prior to finishing the theory in progress?
Accepting malignity of EC in sense of Bergmann’s definition limits all that means effect of a
malignant EC strictly to the argumentation to which it belongs, and thereby makes harder the effort
toward obtaining a decidability criterion. In a natural semantic mode, something cannot be “bad” in
itself, but only as it relates to the effects it produces in an ensemble or system to which it belongs or
with which it is in relation and more, in the aims of that ensemble or system.
My proposal is to externalize the definition of the malignity of EC toward the theory to
which it belongs and even toward the ensemble of possible alternative theories assigned for the
same concrete scientific goal. For having an objective approach toward the problem of decidability,
it is necessary to transfer the malignity attribute to the effects EC might have within the (complete)
theory that it contains, with respect to theory’s goals, but also to the existence of the alternatives not
infected with EC (considered either benign or malignant). In other words, an EC won’t be
malignant in itself, but only if it produces “negative” effects in the theory that contains it, and
malignity may not even be brought into discussion (but just avoided) if there are acceptable non-
infected alternatives to the infected theory, which to serve the same goal10. Thus, by relating the
decidability criteria to the link between EC and the ensemble theory-goal-alternatives, the
previously mentioned logical circularity will be avoided.
Because such a partially utilitarist direction will strengthen the link between the
malignity/benignity of EC and the concrete theory that contains it, decidability will be granted a
relative character that will transcend the general types of EC as stand-alone objects, and thus we can
10 Formally, if for a EC theory there exists a non-EC alternative (theory) serving the same goal, and the two theories are not comparable through other objections (except EC), we shall choose as “better” the non-EC theory. In this sense, EC can be seen as “malignant” with respect to the theory to which it belongs.
9
assume there will not be universal criteria of decidability, but at most, criteria relative to the theory
in question and its goals. This assumption is also sustained by the anthropocentric character of EC,
as it is defined, through the notions of belief, reliability, and argumentation specific to a rational
agent, as well as part of a complex rational process creating a theory, also specific to humans.
If we accept non-universality for the decidability criteria (in case these criteria exist), the
principles expressed previously can best be verified and developed in concrete situations of
scientific practice, and the problem of applicability of mathematics in the physical universe is a
situation relevant for this endeavor, as we shall see in the next sections, where I shall refer to
aspects of “malignity” of the circularities that will be put in evidence in the light of these
principles11.
I shall mention in this section one more type of circularity, this time non-argumentative,
namely the circularity of definition, because this type of circularity is also present – as I will show
– in the existent accounts of applicability of mathematics.
Circular definitions
A definition is circular if it uses the term(s) being defined in the definiens or assumes a prior
understanding of the term being defined. The concept of malignity of a circular definition is usually
expressed through the term non-legitimacy of a definition, which semantically suggests the fact that
possible decidability criteria (if they exist) will relate more to the construction itself of the
definition and less to its effects within a theory, because the effects depend strictly on the function
of the definition.
In the traditional account of definition, there are postulated two intuitive criteria through
which is established an internal legitimacy of a definition, namely the Conservativeness (the
definition should not be able to establish, by means of itself, new knowledge) and the Use (the
definition should fix the use of the defined expression, and it should be the only definition available
to guide us in the use of the defined expression; in other words, it should fix the meaning of the
definiendum). Starting from these intuitions, a theory of definition has been developed on the basis
of three principles: 1) definitions are generalized identities, 2) their structure is sentential, and 3) the
reduction principle: the use of any formula containing the defined term is explained by
reducing it to a formula in the ground language12. The reduction principle conjoined with the
sentential one leads to a strong version of the Use criterion, called the Eliminability criterion: the
11 The reason I have insisted in this introductory section on some general aspects and previous research on EC is exactly the relevance of the accounts of applicability of mathematics as examples for the proposed direction of research on malignity. 12 In the terminology of theory of definition, the ground language L is that used in stating the definition, less the defined
term (L is applicable to definiens), and the expanded language L+ is obtained by adding the defined term to the ground language L; in this context, “formula” is the term for sentences and sentence-like things with free variables.
10
definition must reduce each formula containing the defined term to a formula in the ground
language, which is the distinctive thesis of the traditional account.
The conservativeness and eliminability criteria, which can be expressed formally in model-
theoretic terms [Urbaniak & Hämäri, 2012] are not an absolute property of a legitimate (in the
traditional sense) definition, because satisfaction is relative to the ground language. The traditional
account does not impose for the legitimacy of a definition a certain syntactic structural form of it,
but three requirements, two logical canonical and one criterial: i) definiendum must contain the
term being defined; ii) definiendum and definiens must belong to the same logical category; iii)
definition must obey the conservativeness and eliminability criteria. In this form, the group of the
three requirements expresses more clearly a general definition of definition and less
legitimacy/malignity criteria for objectionable definitions (such as the circular definition).
Weakening the conditions of legitimacy, we can find circular definitions that provide some
guidance in the use of the defined term13 and therefore, they have semantic value, being also
logically valid. Moreover, many of the inductive-recursive definitions from mathematics and logic,
while apparently circular, do obey all the requirements of the traditional account because they can
be expressed in normal form14 [Moschovakis, 1974], which grants the legitimacy of a circular
definition a relativity with respect to the type of circularity with which it is infected.
Gupta [1988/1989] shows that circular definitions do not obey the eliminability criterion. On
the other hand, the strong parallelism between the logical behavior of the concept of truth and that
of concepts defined by circular definitions suggests that since truth is manifestly a legitimate
concept, so also are concepts defined by circular definitions. Following the same parallelism, Gupta
and Belnap [1993] developed a revision theory of definitions in which a circular definition imparts
to the defined term a meaning that is hypothetical in character; the semantic value of the defined
term is a rule of revision; not so with non-circular definitions, which hold a rule of application. The
revision process consists of repeated applications of the revision rule to each hypothetical
interpretation of the defined term (generating successive revision sequences) and can yield
a categorical verdict15 on the objects (non-logical constants) from definiens. Objects for which the
process does not yield a categorical verdict are said to be pathological (relative to the revision rule, 13 Gupta [1988/1989] proposes as a particular example the following definition of a one-place predicate G: Gx = (def) = (x = Socrates) ˅ (x = Plato & Gx) ˅ (x = Aristotle & ̴ Gx); then he argues that although the definition (circular with respect to G) leaves unsettled the status of the two objects Plato and Aristotle, and assuming a certain status of Aristotle with respect to G, we reach a paradoxical situation in either case (a behavior similar to that of the concept of truth in some examples of classical paradoxes), the definition provides substantial guidance on the use of predicate G. 14 The normal form of a definition is: (def ) ( )a x x= = Ψ , for names a; ( ) ( )1 1, ..., (def ) , ...,n nH x x x x= = Φ , for n-
ary predicates H and ( ) ( )1 1, ..., (def ) , ...,n nf x x y x x= = χ , for n-ary functional symbols f. The conditions are: the
definiens can be separated in a specific part and the definition must not contain the defined term or any free variables other than those in the definiendum. In a normal-form definition, the defined term is the only non-logical constant in definiendum. 15 in sense of belonging to a category, in that context the category of the defined concept.
11
the definition, or the defined concept) and when an object has a logically contradictory behavior in
all revision sequences, it is said to be paradoxical.
The fundamental idea of the revision theory is that the derived interpretation is better than
the hypothetical one, and the semantic value that the definition confers on the defined term is not an
extension16, but a revision rule. The revision rule explains the behavior, both ordinary and
extraordinary, of a circular concept, and the revision processes help provide a semantics for circular
definitions. Under this theory, the logic and semantics of non-circular definitions remain the same
as in the traditional account, and revision stages are dispensable. Circular definitions do not disturb
the logic of the ground language; conservativeness holds,17 but eliminability fails to hold, even
though the weaker use criterion holds [Gupta & Belnap, 1993].
Among the attempts to establish legitimacy criteria for definitions it is also worth
mentioning Russell’s Vicious-Circle principle,18 whose primary motivation is the elimination of
logical and semantic paradoxes. Still, the principle is not universal and serves as an effective
motivation for a particular account of legitimate concepts and definitions, namely that embodied in
Russell’s Ramified Type Theory.19
In conclusion, qualifying the circular definitions as non-legitimate with respect to the
criteria of the traditional account does not grant them a malignant character, in the external sense
(as I described it in the case of EC), but establishes their membership to a narrowed class defined
through internal properties, one of which is that of not obeying the eliminability criterion. In
addition, the existence and use of circular definitions in well-established theories suggest – as in the
case of EC – the employment of an external-utility criterion (if such criterion exists). Mathematics
is abundant in circular definitions, whose circularity is more or less visible. I limit myself to
mentioning inductive and recursive definitions, the systems of geometric definitions with
interdependent axioms, and the concept of classical mathematical probability, which is grounded on
the primary concept of equally-possible elementary events. Rejecting such definitions as non-
legitimate within well-established theories would be possible only as a characterization through a
certain internal status and not a character of malignity with respect to the ensemble, as long as we
do not have objective criteria by which to establish in what way those theories are affected or
whether there exist valid non-infected alternatives.
For answering the question of whether externalizing the concept of malignity of a circular
definition is possible, we must – as in the case of EC – approach the problem of the theoretical
16 a demarcation of the universe of discourse into objects that fall under the defined term, and those that do not. 17 No definition, no matter how vicious the circularity in it, entails anything new in the ground language. 18 “If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total” [Russell, 1908]. 19 notions like “truth,” “proposition,” “class.”
12
alternative20. The problem is far more difficult than in case of EC, where the sequentiality of the
argumentative process and the existence of belief sources offer a certain degree of flexibility in
finding and constructing alternative theories with the same goal; this flexibility is contrary to the
case of circular definition, which represents one single object (concept) that is present in the
theory’s goal. The Gupta-Belnap revision of circular definition does not resolve the problem of the
external alternative, but only that of semantics, the revised definition still not obeying the
eliminability criterion.
In the traditional account there are mentioned types of non-legitimate definitions (including
circular definitions) which are recognized as having some value with respect to their function21 and
this value is strictly related to utility. It is remarkable that definitions with subtle circularities22 have
a greater functional value than definitions with obvious circularities, which suggests the idea of
different malignity degrees for the various types of circularities.
Obviously, a possible external alternative of a circular definition can be only conceptual,
and I leave as a research theme the problem of externalizing the concept of and the malignity
criteria for circular definitions, with the proposal of employing utility and alternative in scientific
practice. As in the case of EC, but with a different motivation, it can be assumed that these
decidability criteria – if they exist – will not be universal, first because of the fairly wide typology
of definitions23. In addition, even though circular definition has a general anthropocentric character
(as being created by a rational agent), this character is much more limited in comparison with EC,
because we do not have here the conceptual correlation agent-trustworthiness in defining
circularity.
Further research might establish whether EC or definition circularities can be assigned labels
(or degrees) of malignity in the sense explained in this section or they will remain, under traditional
accounts, at their status of (just) objectionable objects of scientific research. But even this latter
status raises the problem at least of the theoretical alternative, which thus must be associated with
circularity.
An alternative to this research proposal is evaluating circularity in specific theoretical
context, with no appeal to general criteria of decidability on malignity/benignity, but still related to
theoretical alternative and utilitarian value, criteria that comply with the principle of choosing the
best theory. Such a scientific practice will be exemplified through the theme of the current paper, in
20 It is not about an alternative of the infected definition, but a procedural one within the theory or even a complete alternative theory. 21 semantic, nominative, stipulative, descriptive, explicative, etc. 22 Circularity of the definition, as it is generally defined, can refer to the presence of the defined term in the definiens not only as distinct visible object, but also as its manifestations that assume knowledge of the defined concept prior to its definition; both the object and its manifestations can be present on various conceptual levels of the objects from the definiens, which grant the circularity a certain degree of subtleness. 23 real, nominal, lexicographic, ostensive, stipulative, descriptive, explicative, implicit, etc.
13
which circularities that are present in the contemporary accounts of applicability of mathematics in
the physical universe can stand for relevant examples and a starting base for eventual research on
the malignity of circularities, in the previously proposed direction.
II. GENERIC CIRCULARITY
What I call generic circularity is that type of circularity occurring when stating both the
problem and the goal of research prior to creation of the theory; in our case, this is about finding a
formal (logico-mathematical) means through which to represent and rationally justify the
application of mathematics and to provide an explanation for its so-called “unreasonable
effectiveness” [Wigner, 1960]. In this formulation, in fact equivalent to the problem of finding a
mathematical (meta)model of the mathematical modeling, the terminological circularity is visible
even from the first level and is an epistemic one, more precisely, methodological, as I will argue
further.
The requirement that the rational justification of the applicability of mathematics should be
done in a logico-mathematical framework is natural and at the same time limited by the bounds of
human reason themselves, because the critical analysis of a scientific method of investigation can
be undergone only through methods of at least the same precision as the method analyzed; this
requirement is actually the element generating the methodological circularity. Among the two
functional components of the sought theoretical model, namely representation and justification, the
former does not raise epistemic circularities in itself, although a certain construction type of
representation within a specific theory may raise more or less subtle definition circularities, as we
shall see further. The justificative component is that which holds EC. Through the created
metamodel, we do not aim to justify the inference based on an arbitrary mathematical model,
namely the mathematical application24, but the general use of a mathematical model in inference,
namely the universal applicability of mathematics to the investigation of physical systems. In
contemporary accounts of the applicability of mathematics, this justification is done by establishing
the existence and functionality of a relation of a certain mathematical type, which fulfill the
representation of the mathematical structures in the physical systems, and the arguments of the
justification are based on the nature and properties of that relation.
In terms of the traditional account of EC, the justification obtained through the created
metamodel is the argumentation of the fact that applicability of mathematics is a trustworthy source.
Thus, the reliability of the created metamodel is based upon the reliability of the source from the
24 This justification is made within pure mathematics in the derivation step (in terms of Bueno & Colyvan, [2011]), combined with the justification given by the representation function of an arbitrary model.
14
conclusion (as being also a mathematical model), which qualifies such argumentation as circular, in
the sense of the traditional definition, more precisely as a methodological circularity, as strictly
related to the method of investigation. As regards the malignity of such circularity, in Bergmann’s
sense, the discussion remains open as to whether the circularity is one of the questioned-source type
or not. Although the standard formulation of the problem does not express a source questioning
from the agent, an implicit questioning may be considered as embedded in the generic character of
the circularity, and this idea is confirmed if we see the problem as a general-theoretic-constructive
one25. I shall not insist on this categorization of Bergmann type, because generic circularity has an
immediate generalization that transcends any possible malignity detected through particular
theories. It is about the fundamental question of whether we are allowed to analyze and critique
human reason through even this reason, a question posed but undeliberated by most of the major
classical philosophers26. Thus, if we reject the theory of a mathematical metamodel on the basis of
its generic circularity, we should by the same criterion reject the whole analytical and critical
philosophy, of which the problem of the mathematical metamodel is part. Still, this contradiction is
not an argument pro benignity, but at least a principle of practical reason. Obviously, the problem of
the malignity of the generic circularity cannot be resolved in the sense of the external malignity
either27, because any alternative would assume the use of a different type of reason than the logico-
mathematical one. In favor of the benignity of the generic circularity as scientific utility, it is worth
mentioning here the example of metamathematics as a well-established theory studying the
mathematical theories though merely mathematical methods.
Characterization of the metamodel remains open, specific to each theory in part as
exclusively mathematical, partially mathematical, or pseudo-mathematical, involving the study of
the existence of non-mathematical methods of inference within the metamodel.
The generic circularity will also manifest through other types of circularities in each specific
account of the applicability of mathematics, as I will show in the next sections, for which a possible
verdict on malignity can be reached by the application of some criteria of alternative and utilitism.
25 The problem of finding a rational theory which will justify the inference through mathematical models, seen as a general philosophic endeavor and not as an analysis of a particular solution (theory), assumes both analysis and critics of the existent theories, as well as construction of possible alternatives. In this general context, the analysis has also a dubitative character with respect to the rational agent, which may entail the questioning of a certain source. 26 Through the distinction made between theoretical and practical reason, Kant denied the absolute skepticism consisting of the rejection of critical philosophy on the basis of this circularity, but his distinction still left room for circularity issues regarding the interdependent descriptive definitions of the two concepts. 27 in section Proposed directions of research on the malignity of EC.
15
III. INTERNAL-RELATION ACCOUNTS
The accounts of the applicability of mathematics in the physical universe originated in
Frege’s logicist view on arithmetic in particular, centered on the idea of number as concept-property
applied to classes of physical objects and consequently the need of a constant interpretation of the
mixed statements in all contexts, as an indispensable step toward approaching the truth of these
sentences. Thus, from the beginning28, the research acquired a strong semantic influence,
abandoned (somehow inexplicably) over time to the detriment of formalizing relations between the
mathematical structures and the physical structures in a pure-mathematics context.
For representing applicability, it was necessary to establish a relation between the universe
of pure mathematics and the physical universe, such that the mixed statements would benefit by not
only a rigorous semantics, but also an interpretation of their contingent truth on the basis of the
necessary mathematical truth. The accounts that are the subject of this section were developed
based on the principle that such a relation – through which to ensure the interpretation of mixed
statements and their contingent truth – must be of an internal type in the following sense: An
internal relation relating an object A with other objects is a kind of relation in which A must stand
for being that particular object A – that is, the internal relation is an identity criterion29. The most
relevant example of internal relation is that between a set and its elements: Given a set M = {a, b, c,
...}, any set having other elements will be different from M, and thus we can say that M stands in an
internal relation (the membership relation) with a, as well as with b or c.
Contemporary internal-relation accounts are based on Frege’s theory of the (mathematical)
concept of number, defined as an extension of a second-level concept containing first-level
concepts, which in turn contain physical objects. Frege defined the natural numbers in terms of their
paradigm application of counting30, starting from the idea that the number is applicable31 to a
concept and not to specific object or objects. For example, the natural number two is the extension
that contains all the first-level concepts that fall under (or apply to) the concept of counting two
objects.
In its essence, this stands for the definition of cardinality as an equivalence class in a pure
mathematics context, which catches the quality and not quantity for a number, formulated in
conceptual terms that may also include objects of the physical universe. The Fregean definition of
28 beginning with Frege’s work [1884] 29 the term belongs to C. Pincock [2004], being adapted from Bradley, Russell, and Moore. 30 The real numbers are defined by Frege in terms of their paradigm application of measuring. The formal-conceptual construction is considerably more complicated than for the natural numbers, but it is still based on an internal relation. 31 in the sense of logical predicate
16
number is based on the notion of equinumerosity32 and suggests (for the first time) the principle that
sets (classes) can be considered as having physical objects as their elements (members).
With this concept of number at hand, Frege resolved the semantic problem of mixed
statements with numerals. For instance, in the mixed statement “There are 3 apples and 4 pears, so 7
fruits, on the table,” the numerals, taken individually, are naming the mathematical objects “number
3, number 4, number 7,” but logico-syntactically they are predicates characterizing those fruits. This
difference of status invalidates the following argumentation, which is specific to the application of
pure mathematics in a physical context:
(1) 3 + 4 = 7
(2) There are 3 apples on the table.
(3) There are 4 pears on the table.
(4) No apple is a pear.
(5) Apples and pears are the only fruits on the table.
“(1) & (2) & (3) & (4) & (5)” implies:
(6) There are exactly 7 fruits on the table.
The philosophical problem is to find an interpretation for all six sentences, by which to
explain the validity of the argument, and by this reasoning, the relation with applicability of
mathematics becomes obvious. Frege solved this semantic problem, and his solution is independent
of his logicism applied to arithmetics. In his view, having the concept of number defined as a
second-level concept, any mixed statement in arithmetics is of the form “The number of Fs is m”
(NxFx = m), where F is the first-level concept (in our previous example, the fruits of a specific
kind), and “is” means identity. Numerical attributions (m) are predicates, not to physical objects, but
to the concept F itself; thus, numerical predication is (at least) second-order predication. Formally,
the argument of the previous example is based on the following theorem:
∀F∀G(NxFx = m ˄ NxGx = n ˄ ¬∃x(Fx˄Gx) → Nx(Fx˅Gx) = m + n) (T)
which expresses a connection between addition of natural numbers and disjoint set union.
32 Frege defined the notion of equinumerous concepts in terms of first-order logic. For example, if concepts P and Q satisfy the following condition “exactly two objects fall under F”, in logical terms ∃x∃y(x≠y & Fx & Fy & ∀z(Fz → z=x ∨ z=y)), where F is a variable ranging over concepts, then there is a one-to-one correspondence between the objects which fall under P and the objects which fall under Q, and we say that P and Q are equinumerous; the class of all concepts that are equinumerous with P or Q is called number 2. Once all possible numbers are defined in this way, Frege defined the predecessor of a number within a relation series (aRb, bRc, cRd, etc., where R is a two-place relation) with the help of the second-order predicate calculus, as corresponding to the mathematical notion of succession from number theory [Frege 1889, 1884]. With the notions of number and predecessor, Frege built the set of natural numbers in logical terms. Although Frege’s initial goal was the logical formalization of the whole of arithmetics (in his Die Grundlagen, 1884), his project failed because his system was inconsistent [Steiner, 1998]. Still, stand-alone systems like that dedicated to natural numbers have remained valid conceptual alternatives until nowadays, and Frege’s basic ideas were appealed in contemporary research on the applicability of mathematics.
17
For Frege, (T) is a theorem of pure logic, because the objects m and n are logical. For
moving to application, the next step is instantiating concepts in (T) for F and G, in case of our
example, apples on the table and pears on the table. Although these instantiated concepts are not
mathematical, the result is logico-mathematical, which, once the theorem is proved, leads to a
necessary truth.
This conceptual instantiation is in fact the essential process which, in virtue of the definition
of number as an extension of the class of concepts under which it falls, fulfills the semantic
application of arithmetics through mixed statements. Starting from the concept of number, Frege
came to solve the semantic problem of mixed statements (not necessarily limited to arithmetics) and
implicitly, to a solution with regard to the representation of the applicability of mathematics on the
basis of an internal relation (“falling under the concept”) of semantic nature. Thus, in his logicist-
platonist view, mathematical entities are not in any direct relation with the physical universe, but
with concepts, and concepts are applicable to physical universe semantically. This is a solution that
seems to resolve as well the central metaphysical problem of applicability of mathematics, namely
that of filling the “gap” between the mathematical universe and the physical one, which was also
confirmed by Steiner [1998].
Extending the semantic applicability from arithmetics to the whole of pure mathematics
does not exclusively depend on the Fregean logicism. If we regard set theory, rather than second-
order logic, as the foundation of all mathematics33, every mathematical concept can be defined in
set-theoretic terms, and for applying mathematics to the physical universe, one needs to identify
some functions (by which to ensure correspondence) from the physical objects to the mathematical
ones; since a function can be regarded as a set (actually, an ordered pair34), it follows that the only
type of relation necessary for the representation of the applicability of mathematics is set
membership [Steiner, 1998].
Pincock [2004], by endorsing this principle, retakes the Fregean representation of the
applicability of mathematics on the basis of the internal semantic relation, trying to accommodate it
to set theory enough to allow elimination of the semantic component. One motivation (which I
assume) would be that appealing to semantics for the representation prevents the completely
rational justification of applicability35, as the semantic relations cannot be formally modeled so as
to become operable within a logico-mathematical argumentation.
33 according to Zermelo-Fraenkel axiomatic system 34 ordered pair (a, b) is set {a, {a, b}} or {{a}, {a, b}} 35 Frege’s semantic applicability solves the interpretation of mixed statements, but a rigorous interpretation only does not also ensure a justified transformation of the necessary mathematical truth in contingent truth (of the mixed statement) because such a justification would employ the nature and properties themselves of this transformation. A “more mathematical” nature of the internal relation and implicitly of this transformation is supposed to take the scientific endeavor closer to the desiderata of a complete justification.
18
Pincock treats Frege’s conceptual extensions as sets, and appealing to the notion of
transitive closure of a set36 and to the principle – suggested by Frege and motivated by Steiner
[1998] – that a set can have physical objects as its elements, establishes the existence of a direct
internal relation between the modeled physical objects and mathematical objects, definable in terms
of set membership. Thus, in set-theoretic terms, number two is a set (class) having as members all
the sets with two elements (objects), namely the objects number two is applied to in counting:
{ }{ }1 22 (def ) ,i ii
O O= = (D)
The physical objects to which the natural number two is applied in counting will be
members of sets that are in turn members of the natural number two.
Pincock generalizes this representation to all the mathematical objects, which then can be
viewed as sets containing the physical objects they apply to. Thus, every mathematical object will
have in its transitive closure the physical objects to which it applies, and since a set stands in an
internal relation with all the elements of its transitive closure, we can establish a direct relation
between the mathematical objects and the physical ones37.
In Pincock’s approach, where we already talk about a mathematical metamodel, the
representational function of the applicability remains valid, as in Frege’s approach, and gains in
addition a (pretended) purely mathematical nature and a character of direct connection. In
exchange, the justification function, although it gains in capacity even through the nature of the
representation, cannot completely explain (yet) the applicability, because the much too general
character of the definition of the relation between the mathematical system and the physical one
does not allow the representation of structures in their complexity, the representation being limited
to objects and groups of objects as independent entities.
What counterbalances this apparent gain of Pincock’s model is just circularity, which I will
put in evidence in what follows.
36 The transitive closure of a set S is a set S' that has as elements all the elements of S, all the elements of the elements of S (if S contains other sets), and so on. If S = {a, {b, c}}, then S'= {a, b, c}. 37 Pincock’s [2004] argumentation is made around the illustrative example of natural number two and appeals to the transitive closure as generating the internal relation between number two and the physical objects standing in the right-hand member of relation (D). However, a physical object that number two applies to actually consists of two objects,
namely the physical system (or set, if we ignore other structural aspects) formed by two isolated objects 1iO and 2
iO .
Since the transitive closure has as elements singular instances 1iO and 2
iO , we obtain a relation between number two
and each singular object, which is not what was intended. In other notation (without {...}), the physical object to which
number two applies is actually 1 2( , )i i iO O O= , which would resolve the confusion. Thus, employing the transitive
closure for the considered particular example was not necessary, and I take this inadvertence more as a degenerate example than a circumstantial error, admitting that in the case of more complex objects and mathematical applications the extended set could contain as elements sets containing other sets. Moreover, the relation (transitively obtained) between the mathematical object and the singular instances of the physical objects is one of the main objections to the internal-relation account, also expressed by Pincock [2004], another objection being that it is committed to a modal realism about possible physical objects that may stand in the same internal relation with the mathematical object.
19
Relation (D) is not circular, even though apparently number two shows in the indexation of
each set-element of the right-hand member. In the analyzed theory, (D) is not a definition38, but a
denotation of a set defined through the mathematical object “number two”, whose symbol it uses.
For catching the circularity, we have to follow not the symbolic, but the intentional aspect of the
scientific endeavor, as well as the goal of the theory. In this view, relation (D) must be interpreted
in a certain chronological mode, as follows:
1. (posing the problem) The mathematical object is given and defined, while the extension-set of the
physical objects is not, being the object to be created (defined, put in evidence). The goal is to find
(to prove that it exists) a direct relation39 between the mathematical object and each of the physical
objects of the extension-set. Therefore, the extension-set is the central object of the argumentation,
which must start from what is given (the mathematical object), this direction being the only
logically valid.
2. (constructing the extension-set) In the creation of the extension-set, the property “the
mathematical object applies to”40 is used as being common to the elements, and essential for the
definition, of that set. The mathematical object is given; however, “applicable to” already assumes
the existence of a direct relation between the mathematical and the physical object41, which is just
what we follow as the goal.
3. (putting in evidence the relation) Once the extension-set is created, we identify it with the
mathematical object. Identification does not mean identity, but still a relation, defined through the
common property of the elements of the set (if this property exists) and symbolized in (D) by sign
“= (def) =.” But the existence of the direct relation (as internal relation of set membership) depends
on identity in Pincock’s argumentation and not on another different kind of relation42. Denoting by
MO the mathematical object, by E the extension-set, by fO a physical object of E, by aR the two-
place membership relation, and by pR the two-place relation between MO and E based on the
(mathematical) property common to the elements of E, Pincock’s account puts in evidence the
composed relation a pR R , as M p a fO R ER O .
38 It would become definition if we switched between goal and premises: we start from characteristics of the physical objects and define each mathematical object through the common characteristics; however, such an endeavor is specific to the creation of pure mathematics on empirical grounds and stands for a matter different from that of the current paper. 39 Although assigning the attribute “direct” to a relation seems pleonastic and circular, the examples presented in this paper put in evidence a clear distinction between the two-place relations connecting objects from unconnected domains, in the sense of different logical categories. A pertinent analysis of the direct character can be done only by going deep into the concept of relation and will be subject to forthcoming research. 40 The Fregean correspondent is “[concept F] falls under [mathematical object]”. 41 Moreover, it assumes an external relation (mapping), in the sense of correspondence between two distinct domains. 42 A set A stands in an internal relation of membership type with an object only if A is identical to a set containing that object or containing a set that contains that object.
20
However, relation pR is neither of identity and nor of membership, and, moreover, it is not a
direct relation, but one of conceptual-semantic type in the Fregean sense, and so the relation
a pR R that makes the connection between the mathematical and the physical object is not a direct
one, as it was claimed through the goal of the theory43.
In the unfolding 1-3, circularity is present at step 2. If we see the argumentation of step 2 as
a construction process, the circularity is methodological-epistemic, because the agent relies on the
trustworthiness of the source “applicability” in the construction of an object through which s/he will
finally argue for the applicability as a relation. With this status of the circularity, we cannot decide
on its malignity in the traditional sense unless we accept Bergmann’s categorization, in favor of
benignity in this particular case and sustained by the somewhat similar example of mathematical
induction.
However, if we see the argumentation of step 2 as a premise included in a merely deductive
process, and we admit its logico-mathematical context, namely:
Hypothesis:
P1: It is given mathematical object MO , which is defined through property p.
P2: There exists a set E of physical objects with which MO stands in a relation44 in virtue of
property p45.
P3: Set membership establishes a relation between a set and any of its elements.
Conclusion:
C: For any fO E∈ , there exists a relation between MO and fO ,
then the deduction P1 ˄ P2 ˄ P3 → C is logically circular, because both the relation to which the
conclusion refers and that from premise P2, not only that depend on p, but they are defined
through p. Thus, premise P2 essentially uses the existence of the relation expressed through C.
Seen as such, the circularity becomes logically vicious (malignant).
The argumentation process formed by steps 2-3 can also generate definitions, for which we
can pose the problem of circularity. Two terms may be the object of a definition: the mathematical
object and the extension-set. The definition { }( ) applies toM f M fO def O O O= = is obviously circular,
as it contains the defined term at the first level of the definiens; however, it is not one actually used
in the unfolded 1-3 theory, but one possible within a possible inverse theory with respect to the
proposed goal, namely starting from the extension-set as being provided, toward a judgment
43 In my view, the approach of the internal relation in set-theoretic terms is not complete, and hence the justification of the applicability does not gain in the accuracy of the method of investigation in Pincock’s approach, being in essence still an instance of the Fregean semantic representation. This decompensation adds to circularity in my final argument of moving the focus toward propositional and semantic accounts. 44 of identity, in Pincock’s approach 45 Set E can even be empty.
21
regarding the mathematical object. A second definition would be { }( ) applies tof M fE def O O O= = ,
which has a merely referential function and is not circular. Still, it generates an EC-infected
argumentation when employed in the process 1-3. The circular definition of the mathematical object
offers a contribution of utilitarian value (the extension-set concept being central), and the lack of a
non-circular alternative would incline the balance toward benignity of this definition circularity, in
an externalized sense. However,, there exists an alternative in Frege’s semantic applicability
account, as I will show further.
Both the argumentation and definition circularities exposed above are effects of the generic
circularity treated in the previous section, which originates in the “forcing” of a model as
mathematical as possible through which to represent the applicability of mathematics in general. An
objective attribution of the malignant character of these specific circularities in not possible in the
traditional sense, and one of the reasons is that circularity types can switch for the same instance46,
as I showed before (epistemic and logical). The analysis of the non-circular alternative may be
decisive for the characterization of these circularities as malignant in an externalized sense, and in
what follows I will argue that Frege’s corresponding internal-relation account does not show
circularities.
In Frege’s conceptual approach, the goal is to establish not a direct relation between the
mathematical and the physical object, but a semantic47 relation connecting two domains of different
categories, that of the mathematical concepts and the physical one. Keeping as possible the analogy
(of the denotations and argumentation process) with Pincock’s approach, we have:
1. Also here the mathematical object MO is given, while extension E (this time conceptual and not
as a set in sense of collection of stand-alone objects) is created.
2. Extension E uses as a variable the first-level concept F, which is also given as an interpretation of
the physical object fO 48: concept F is the concept “being the physical object fO in the given
situation”, and E is the second-level concept “being a concept F falling under MO ”. The
formulation of E is logical, because MO is also a mathematical concept. E is not a set, but a
conceptual extension in Frege’s terms, which is in turn a concept.
46 This concrete example in which a circularity can switch between epistemic and logical suggests the need for a revision of the theories on malignity of circularities, which are still in an incipient stage. 47 This wording is made with the reservation that the semantic connection may not be considered a relation because the related objects do not belong to an universal set, but to distinct domains of different categories. 48 The physical object must be seen not in general context, but belonging to the physical situation specific to the application, that is object fO in physical situation fS . In the previous example of counting the fruits on the table, if we
refer to apples, the physical object to which number three as mathematical object applies is not “apples” (in general), but “apples on the given table.” This point applies also to the previous discussion regarding Pincock’s approach.
22
3. The identification of extension E with mathematical object MO is still of a semantic nature, being
a particularization through instantiation of the generality of object MO , with the purpose of
interpretation: concept F generated by the given physical situation belongs to E, but so do other
concepts 1 2, , ...F F falling under MO generated by similar physical situations49. At this step is made
an attribution of an interpretation to extension E, namely the mathematical concept MO . This
attribution still is not an identity, but a relation of the semantic type, of the same nature as that
fulfilling the interpretation of the physical object fO as concept F.
Let us observe that process 1-3 is neither argumentative nor deductive, as is the one in the
case of the set-theoretic approach, but one of a composition of two semantic interpretations. As a
result, at step 2 we cannot pose the problem of circularity any more (neither epistemic or logical),
especially because there is no longer the goal of putting in evidence a direct relation. The problem
of the definition circularity vanishes as well. At step 3 (the identification of E with MO ), the
presence of the term MO in the description of E exposes a circularity only apparently, because MO
stands for the essence captured by the process of conceptualization, without which the concept
cannot be established, and in this sense circularity is allowed in any semantic interpretation. In
addition, the identification (def )MO E= = does not stand for a definition in the traditional sense (as
non-legitimate), because the definiendum and the definiens are not of the same logical category, and
hence, we cannot pose the problem of definition circularity as well.
In summary, Frege’s semantic model puts in evidence the connection between the physical
and the mathematical object as a composition of two semantic interpretations, as follows:
1
2
...
if
iM
F O
FO E
F
←⎯⎯←⎯⎯
The two relations composed together aR and pR showing in Pincock’s approach become
one and the same “relation” at Frege, namely the conceptual-semantic interpretation (i).
Interpreted as a concept about concepts regarding physical objects50, the mathematical
concept applies semantically in the physical universe, ensuring a rigorous interpretation of any
mixed statement, with no circularities. Within this model, the application of a mathematical theorem
consists of the instantiation of a higher-degree logical truth, through higher-degree concepts and
quantification. The alternative of this model under set theory is finally reducible (step 3) to Frege’s
model. In addition, the elimination of the set-theoretic model’s circularities can be done only 49 Particularizing to the application of the natural number in a mixed context, 1 2, , , ...F F F would be equinumerous
concepts, in Frege’s terms. 50 In other Fregean wording, the mathematical laws are laws of the laws of the physical universe.
23
through the unemployment of the concept of “direct relation.” Even though the study of the non-
circular alternatives assumes going deeper in the general concept of relation, at this moment I
suggest that such an alternative still converges toward the semantic model.
For Frege, it is necessary that a mathematical formula be applied only if it expresses a
thought, and it is senseless to infer from something that never been thought. This position, as well
as the Fregean model, even though they remain open to possible objections regarding the
representation and optimal justification of the applicability of mathematics, have the merit of being
natural, unifying for any type of internal relation, non-circular, as well as being able to solve – at
least at present – the main metaphysical problem of the applicability, namely filling the ideatic
“gap” between the physical and the mathematical universes. Frege’s answer on the gap matter was
straightforward: such a direct relation between the two universes (by which to represent and justify
the applicability) simply does not exist; in exchange, there exists a (semantic) relation between
mathematical objects and concepts of the physical universe, just as there is a relation between these
concepts and physical objects.
III. EXTERNAL-RELATION ACCOUNTS
The internal-relation accounts ensure the representation function of the mathematical
metamodel and partially – more in principle than explicatively – the justification function.
However, the explicative justification is based on representation, on its nature and its epistemic
potential, and the representation type yielded by internal-relation accounts does not capture the
internal structures of the two domains – mathematical and physical – in their essential details so as
to make any application of mathematics workable. This is the reason for which researchers further
focused on the aspects of structural representation, aiming to put in evidence a new type of relation
between the two models, in order to preserve its logico-mathematical character, and meanwhile, to
be able to capture in as much detail as possible the actual mathematical application. This orientation
gave the name “structural” to the newly created theories .
Pincock [2004], still starting from the mixed statements and extending the theoretical goal to
cover not only their interpretation, but also their truth as derived truth (from the necessary
mathematical truth), declared as necessary the existence of an external relation51 between the
mathematical domain and the modeled physical situation. Such an approach seemed very
promising, because it would have resolved two major desiderata: a) If the relation is merely
external, then it would not affect the necessity of the mathematical truth, since mathematical objects
51 An external relation is a relation that is not internal, that is, a relation that does not involve criteria of identity of the objects standing in that relation.
24
and their properties would exist independently of the physical universe, and b) a rigorous solution
for the construction of such a relation would guarantee the applicability of mathematics in specific
situations, which means an explicative justification for the metamodel.
Pincock is basing on the idea of analogy between mathematical structures and certain
structures of the physical universe obtained through idealization, which can be mathematically
represented through the notions of homomorphism or isomorphism52, as a mapping that preserves
structures between two different domains53. Thus, the sought external relation is a structure-
preserving function. Pincock does not develop this account further, limiting himself to the general
status of this external relation, without providing a formalism of the structures that have been put in
homo/isomorphic correspondence. The structural formalism is approached further by Bueno &
Colyvan [2011] in their theoretic model based on the homo/isomorphic function, called by the
authors “the inferential conception of the applicability of mathematics” (I shall abbreviate it
ICAM). Even though it is an extension of Pincock’s external-relation account, ICAM is not merely
structural, leaving room to some pragmatic and context-dependent features of the process of
application of mathematics. The core principle of ICAM is that the fundamental role of applied
mathematics is inferential (even though the functions of a mathematical model may be several), and
this role ultimately depends on the ability of the model to establish inferential relations between the
empirical phenomena and the mathematical structures. In Bueno & Colyvan’s terms, ICAM consists
of a three-step scheme:
1. (Immersion) Establishing a homo/isomorphic mapping from the empirical context to convenient
mathematical structures, by which to connect the relevant aspects of the empirical situation to the
mathematical context appropriate for the application54.
52 depending on the case of the mathematical application 53 Although the isomorphic model is attributed to Pincock, there are references to it in previous works of some proponents of the principles of this model, such as Baker [2003], Balaguer [1998], or Leng [2002]. 54 The mapping is not unique, and choosing the right one is a contextual problem in the mathematician’s responsibility, depending on the specificity of the application.
25
2. (Derivation) Derivation of the consequences through mathematical formalism, within a specific
mathematical theory, using the structures evidentiated through immersion.
3. (Interpretation) Interpretation of the consequences obtained at the derivation step in terms of the
empirical context, by establishing a homo/isomorphic mapping from the mathematical structures to
the initial empirical context55.
In this theoretical framework, the concept of structure is the “static” one, consisting of a set
of objects (nodes, positions, etc.) along with a set of relations between them [Resnik, 1997]. The
epistemological problem of the structure of the physical universe as falling under this concept of
structure56 reverts to the problem of the universality of the created metamodel and I will not discuss
that here, just as I will not discuss the strengths and weaknesses of this account of the applicability,
limiting myself to circularities. The issues with the structure surplus – physical or mathematical –
remaining outside the process of mathematical modeling57 are accommodated through the
introduction of the notions of partial structures, then partial isomorphism/homomorphism58:
An n-place partial relation R over a non-empty set D is a triple 1 2 3, ,R R R , where
1 2 3, ,R R R are mutually exclusive sets, with 1 2 3nR R R D∪ ∪ = and: 1R is the set of n-tuples of
elements from D that we know that stand in relation R, 2R is the set of n-tuples of elements from D
that we know that do not stand in relation R, and 3R is the set of n-tuples of elements from D that
we do not know whether they stand or not in relation R 59. A partial structure is an ordered pair
, i i ID R
∈, where D is a non-empty set and ( )i i I
R∈
is a family of partial relations over D.
Further, the notions of partial isomorphism and partial homomorphism are defined in terms
of partial relations and structures, as follows:
Let , i i IS D R
∈= and ' ', 'i i I
S D R∈
= be partial structures (so they are of the form
1 2 3, ,R R R and 1 2 3' , ' , 'R R R respectively). We say that a function : 'f D D→ is a partial
isomorphism between S and S' if (i) f is bijective, and (ii) for every x and y from D,
55 This mapping is not necessarily the inverse of the immersion mapping, although in many concrete situations it may be. 56 in which causal relations, as explanans for most of the phenomena, cannot be accommodated 57 the former, from the idealization of the physical system, and the latter within the derivation step, which assumes the selection of the convenient mathematical structure from a wider mathematical context 58 The authors of ICAM advanced this formalism of partial structures also as an argument in defense of their theoretical model against some objections from Batterman [2010], who – among others – claims that ICAM does not adequately accommodate singularities and divergences coming from limited mathematical operations, for which physical analogies do not exist. Batterman also raises the question “How can idealizations [as false assumptions, N.A.] play an explanatory role?” Bueno and Colyvan answer his questions by appealing to their partial-structures model and advancing an iterated ICAM account, consisting of two successive composed modelings, the last starting from the partial structure to the total mathematical structure. Their complete answers to Batterman’s objections can be analyzed in [Bueno & French, 2012]. 59 If 3R is empty, the notion reverts to the usual notion of n-place relation R, which becomes identical to 1R .
26
1 1( ) ' ( )xR y f x R f y⇔ and 2 2( ) ' ( )xR y f x R f y⇔ . We say that a function : 'f D D→ is a partial
homomorphism between S and S' if only (ii) holds60.
The partiality of the relations and structures reflects formally the incompleteness of our
information about the investigated physical domain, having more an epistemic than ontological
character. However, let us observe that this formalism of the partial structures captures only a
“quantitative” aspect of the idealization, and does not accommodate aspects with a strong
anthropocentric character of the idealization process itself, as the process preceding the immersion
step. The idealization process is essential in identifying the (partial) structures that will be mapped
and is a primary test in labeling a mathematical model as good or bad61. Idealization (as well as the
entire process of immersion) is part of what Steiner [1995] called “the art of the mathematician,” as
opposed to a process with clear criteria of objectivity, and this “art” has not been formally
accommodated yet within a theoretical model of the applicability of mathematics.
Next I will move to evidentiating the circularities present in external-relation accounts, with
a focus on ICAM, as an extended account of Pincock’s primary model. The circularities can be
caught – as in the case of the internal-relation accounts – if we see the process 1-3 as an intentional
one with respect to its epistemic goal.
I shall begin with the domain of the external function (the homo/isomorphism) which makes
the immersion, namely the (partial) physical structure from the empirical context. In the formulation
of ICAM’s authors, the physical structure is “embedded” in the investigated physical system.
Advancing the nature of the external function as homo/isomorphic (a mathematical object necessary
for the justification function) implicitly assumes a mathematical nature of the two domains that are
mapped one within another, that of mathematical structure, which entails granting the physical
structure a mathematical nature. The structural-ontological options for the status of the physical
domain with respect to homo/isomorphism are to consider it either as a (purely) mathematical
structure or a physical structure. The former option assumes the identification of the physical
(sub)system with a mathematical structure given a priori, but the identification is in turn equivalent
to establishing an internal relation between the physical and the mathematical domains, necessary
as identity criterion. Thus, we deal in this case with an internal relation – not immediately visible in
the ICAM reduced scheme – which, formalized under set theory, generates circularities of three
types, as I showed in the section dedicated to internal-relation accounts. The representation of the
physical domain into the mathematical one within the ICAM model thus becomes a composition of
two relations, one internal and the other external. The external relation is no longer necessary for 60These definitions are particularized for two-place relations, for the sake of simplicity. Observe that if 3R and 3'R are
empty, we revert to the standard notions of isomorphism and homomorphism. 61 For an objective analysis of the quality of the mathematical models in an epistemological view, see [Cartwright, 1983].
27
the representation, which is fulfilled by the internal relation, but contributes with explicative surplus
to the justification function of the metamodel; the essence of representation (and part of the
justification) derives from the internal relation, and hence in this case ICAM loses a significant part
of its potential of modeling the applicability. The latter option – keeping the physical status of the
(sub)system of the function’s domain – commits us to the compromise of requiring the existence of
the structure at least in the format objects-sets-relations, by accepting the existence of physical
objects as elements of the sets and as relata, but preserving the mathematical nature of the concepts
of set and relation62, so as to be consistent with the definition of homo/isomorphism. Evidentiating
the structure in the format objects-sets-relations as mathematical object assumes the reorganization
of the physical system and identification of parts of it (as part of the idealization process, included
in the ICAM scheme at immersion step) with sets and relations, which again represents an internal
relation between the physical and mathematical domains, generating circularities.
I shall move now to the external function, the homo/isomorphism. This function is not
provided prior to modeling, and its mere existence (if proved) does not indicate a sufficient
condition for the correct functioning of a mathematical model63. Thus, the mathematician not only
commits to establishing the existence of an arbitrary homo/isomorphism between the two domains,
but s/he chooses the right one. This attribute of the external function also derives from the
formulation of the immersion step, in which the mathematical structure representing the co-domain
of the external function is “convenient” (conveniently chosen64). This convenience refers not only
to the co-domain, but implicitly to the entire mapping ensemble “domain - external function - co-
domain.” Since we cannot establish convenience except with respect to the final goal, the criterion
of choosing must be strictly related to the result of the last step, the interpretation.
In other words, provided a function makes the correspondence between the physical and
mathematical domains as sets of objects, we cannot decide whether this function is a
homo/isomorphism based only on that correspondence. It follows, then, that the structures of the
two domains that need to be preserved must be known completely, hence including the relations
from the physical structure that are unknown prior to modeling (i.e. those relations belonging to set
3R from the formalism of the partial structures). Those unknown relationships represent the target
of the inference and are discovered through interpretation.
In conclusion, the method of reasoning leading to interpretation is based on this
interpretation, which is an instance of methodological-epistemic circularity.
62 One type can be mentioned instead of two, keeping in mind that a relation is a set as an ordered n-tuple. 63 A mathematical model may provide incorrect results not only as a result of an inappropriate choice for the external function, but also as a result of a too restrictive idealization, and the errors may represent not only false statements in empirical context, but also approximated numerical results with an unacceptable error range. 64 my rephrasing
28
The main question and an essential research direction becomes: Given that the
homo/isomorphism is not determined through proof, but chosen by the mathematician65, and
consequently the homo/isomorphic nature of the chosen function is actually postulated, how can we
explain the very high rate of success66 of the choices in general? The question is a reformulation of
the rhetorical question regarding the “art of the mathematician” used in applications, and a possible
answer that can be analyzed and developed is the following: The mathematician chooses the
correspondence between the objects of the two domains on the basis of an intuition which is
supported on one hand by the natural conceptual interpretation (of the internal type) itself of these
objects67, which leads to a certain way of conceptual interpretation of relations either (even though
the relations are not put into correspondence), and – on the other hand – by the experience of
observing the “partial” functioning of the homo/isomorphism, for a large enough set of observable
relations or non-relations. In this sense, the explanation would have a Bayesian probabilistic
character, but also one of asymptotic reasoning.
I shall exemplify the arguments above by presenting a trivial example68 of mathematical
modeling-interpretation of an empirical context, in which we can capture the method of choosing
the external function and the effects of the choice made. The example is adapted from an example
presented by Lange [2013]69, who discussed it in the context of the possibility of genuine
mathematical explanations. As Lange does, we shall retain explanation as the main goal of the
model.
Empirical context: Mother aims to distribute evenly twenty-one strawberries to her two
children (without cutting any), after which to distribute evenly another twenty-one strawberries to
the three friends of her children. She tries to distribute evenly the strawberries to her two children
several times and – of course – fails every time. The goal of the modeling: To find a (mathematical)
explanation of this (physical) failure. The mathematical model appeals to the notion of divisibility
within number theory (at the derivation step in ICAM scheme), the results being interpreted back in
the empirical context through a homomorphism that puts in correspondence the following
structures:
The physical structure of the physical (sub)system that participates in the immersion consists
of:
65 Obviously, a homo/isomorphic correspondence might not even be possible in a specific application. 66 The success means that a certain chosen function proves to be a homo/isomorphism with respect to the two structures, through post-modeling empirical confirmation that will empirically discover the unknown relations from the physical structure. 67 For example, the groups of physical objects with attributed numerals are put in a natural correspondence with those natural numbers respectively, in Fregean mode. 68 In the field of the philosophy of applicability of mathematics, the trivial examples proved to be the most relevant with respect to the constructive and structural aspects, also confirmed by the literature in this field. 69 and inspired from Braine [1978]
29
Physical objects: C – the group of the mother’s children, P – the group of the three friends,
F – the first batch of twenty-one strawberries, G – the second batch of twenty-one strawberries (4
objects)
(Two-place) relations: R – not being possible to be evenly distributed; T– being possible to
be evenly distributed to (2 relations); at this moment, we do not know whether (or why) FRC.
The entire physical system has a richer structure, as it includes as object also the mother
(and the table where the strawberries lie, etc.), as well as the relations between mother and children,
mother and fruits, or children and their friends. The chosen subsystem represents the first
idealization done, the second being the ignoring as irrelevant of other possible relations in empirical
context between the four objects.
The mathematical structure (conveniently chosen) standing for the co-domain of the external
function is { } ( )2, 3,21 , ,D D , namely the set consisting of natural numbers 2, 3, and 21, and the
relations D, defined as “being divisible by” and D , as “not being divisible by”. The mathematical
suprastructure (which is used at the derivation step) is the set of natural numbers together with
relations D and D , and the theory used (the governing theory) is number theory (the remainder
theorem, etc.). We choose the external function as { } { }: , , , 2, 3, 21f C P F G → , with f(C) = 2, f(P)
= 3, f(F) = f(G) = 21; f is surjective and non-injective. This choice is a natural one, suggested by the
conceptual interpretation of the numerals in the given mixed context. Observe that:
in mathematical structure in physical structure
f(F) D f(C) we do not know whether FRC
f(G)D f(P) GTP
f(C) and f(P) do not stand in relation C and P do not stand in relation
The intuition that f is a homomorphism between the two structures is based on:
a) the correspondence made by f (“physical groups with numerals” to “natural numbers”), which
induces a conceptual correspondence of the relations ( D with R and D with T) in virtue of the
natural interpretation “being possible to be evenly distributed” through the concept of divisibility
between numbers and
b) the experience that the relation T between G and P is preserved through f (becoming D) and that
C and P do not stand in any relation, as well as f(C) and f(P)70; the knowledge GTP can be gained,
70 In its simplistic formulation with illustrative purpose and for space considerations, the example does not hold a “numerous” set of observable relations and non-relations on the basis of which is intuited the unknown relation necessary for the homomorphic quality of f (actually it holds only one relation and one non-relation). The example may
30
for instance, by actually imagining how the twenty-one strawberries are distributed to three children
through any valid empirical procedure; the knowledge that C and P do not stand in any relation is
derived from the definition of the relations of the physical structure – they (the relations) make
sense only as connecting groups of strawberries with groups of children.
Admitting that f is a homomorphism between the two structures, we infer (through interpretation,
that is through the isomorphism { } { }: 2, 3, 21 , ,g C P F→ , with g(2) = C, g(3) = P, g(21) = F, and
the same structural relations as those preserved through f ) the fact that FRC, which is the
explanandum.
Choosing another external function might vitiate the inferred result. For example, choosing
f(F) = f(G) = 22 and keeping the rest of the correspondence as it is, as well as the previous relations,
by using intuition that f is a homomorphism (obtained this time only on the basis of b), we would
infer that FTC, contrary to the previous result and reality. Completing the mathematical structure by
putting 2 and 3 in relation ( D ) would cancel the homomorphic character of function f, and so the
inference through the model would be compromised. Completing the physical structure with other
objects and/or relations would have the same effect, as long as we cannot find analogous objects
and relations in the mathematical structure. In exchange, ignoring objects and relations from the
physical domain considered (erroneously) as irrelevant for the described situation would also vitiate
the result of the inference. For example, if the mother has a certain medical condition that affects or
limits her action of distributing strawberries in certain conditions, and these physical facts are
considered as irrelevant and not represented in the structure, this would be a modeling error, and
any further inference would no longer be relevant, even though it can be communicated on the basis
of the same isomorphic principle.
I conclude that the presented example captures very well the circularity generated by the
process of choosing the external function and the interpretation in terms of the empirical context, as
well as the actual way of modeling and interpretation as based on mathematician’s “intuition” and
the assumed conceptual-experimental motivation of this intuition.
Finally, I will analyze the idealization, as the process that determines the domain of the
external function. This process consists in fact of three idealizations: The first one is the reduction
of an extended physical system to a subsystem specific to the investigated empirical context, from
which an object-sets-relations structure will be extracted through a second idealization. This partial
physical structure comes from a suprastructure from which a surplus is eliminated through criteria
of relevance of the participation to the physical phenomenon that is the target of the model. A
similar idealization (the third) takes place in the mathematical domain. Although we work formally
be extended by introducing other objects, for instance another group of seven children the strawberries will be distributed to, which would increase the number of evidential observations.
31
within a unique mathematical theory at the derivation step, we retain just a certain mathematical
structure, by not employing the surplus, through criteria of convenience. The difference between the
two types of structural idealizations – in empirical and mathematical context – is the fact that while
the latter does not pose problems of logical truth to the act of formal inference, the former could be
epistemically vicious and affect the empirical truth. (In the previous example, the hypothesis that
the mother has a medical issue is not represented structurally.)
Let us observe that relevance and convenience of the idealizations are interdependent via the
construction of the external function as an homo/isomorphism, which follows chronologically the
idealization at immersion step and implies the determination of the domain and co-domain of this
function, as well as its homo/isomorphic character, which would assume knowing the function and
the corresponding structures prior to idealization. We encounter again an epistemic circularity
somewhat like that detected in the previous analysis of the homo/isomorphism with respect to the
interpretation step, and the two circularities are mutually dependent. The establishing of
convenience and relevance is based on mathematician’s intuition, and this latter concept was not
formalized within ICAM. Even though idealizations can be accommodated in Bueno & Colyvan’s
[2011] formalism of the partial structures, the volition act of construction and argumentation based
on relevance and convenience is impossible to be accommodated prima facie, due to its
anthropocentric component.
The circularities of the external-relation accounts are also the effect of the generic circularity
of the mathematical metamodel, because they are created around the mathematical object
homo/isomorphism, as central object through which the justification of the model is rationally
obtained. By labeling them as benign through the argument of their derivation from the generic
circularity or that of not belonging to a questioned-source context per Bergmann’s categorization71
would not exclude considering the alternatives, also in the light of other objections to external-
relation accounts, among which I mention here only the problem of the derivation of the contingent
truth as sentential truth in mixed context from the necessary truth. Any non-EC-infected alternative
theory would need either to eliminate the implicit chronology of the ICAM formalism (also present
in Pincock’s reduced model), or to propose other structural types of the domains, or to base on
another nature of the external function, or simply dismiss the mediation of the external function,
which would lead to the degeneration of the theoretical model into one of internal-relation based.
71 This second argument needs further analysis, especially because we deal with an anthropocentric character of the argumentative process more intensely than in the case of the other circularities detected in previous sections.
32
IV. CONCLUSIONS
In contemporary philosophical accounts of the applicability of mathematics in the physical
universe, we identified circularities of all kinds – epistemic, logical, and of definition – detectable
especially when putting in evidence the intentional aspect of the scientific endeavor and the goal of
the theory, except Frege’s semantic applicability account. The issue of malignity of these
circularities was treated in both the traditional sense and in the externalized sense I proposed, which
involves theoretic alternative and utilitarian value.
In absence of a well-established theory on the malignity of circularities and of universal
criteria of decidability on malignity, we cannot reject a theory on the reason of the existence of
circularities. In this vein, I analyzed the detected circularities in their specific theoretical context,
and established the following: a) regardless of their type, the circularities are effects of the generic
circularity of creating a mathematical model for representing and justifying the mathematical model
in general; b) in the case of the accounts based on an internal relation of set membership kind, the
epistemic and logical types can switch when characterizing the same circularity; c) all circularities
expose a certain constructive-epistemic analogy. The analysis of the non-circular alternatives for
each analyzed account shows that they are possible only considering either the dismissing of the
structural type objects-sets-relations, or another nature of the direct relation between the physical
and the mathematical domains, or simply dismissing this relation as a whole. This fact together with
characterizations a) - c) of the detected circularities suggests the reconsideration of the Fregean
semantic model. Other weaknesses of the contemporary accounts of the applicability of
mathematics in the physical universe – among which I mention only the problem of the derivation
of the contingent truth in an empirical context from the necessary mathematical truth – again
suggest the reconsideration of the semantic models. Thus, on one hand we come to the necessity of
employing the unformalizable psychological component that is specific to mathematical modeling
through its anthropocentric character, and on the other hand the need of a formal model within
which a transfer of truth between two domains of different natures will be possible and more
justified than through interpretation. A proposal for such a theoretical model – and the matter of
further research – is to consider propositional structures instead of the objects-sets-relations ones,
where the transfer and status of truth can be better accommodated. Such an alternative – if it proves
valid – is acceptable to both platonist and nominalist.
33
References
Alexander, D. - In Defense of Epistemic Circularity. “Acta Analytica”, Vol. 26, Nr. 3, 2011, pp. 223-241.
Alston, W.P. - Epistemic circularity; “Philosophy and Phenomenological Research”, Vol. 47, Nr. 1, 1986, pp.1-30.
Alston, W.P. - The Reliability of Sense Perception; Ithaca: Cornell Univesity Press, 1993. Baker, A. - The Indispensability Argument and Multiple Foundations for Mathematics; “Philosophical
Quarterly”, Vol. 53, 2003, pp. 49-67. Baker, A. - Mathematical Explanation in Science. “The British Journal for the Philosophy of Science”, Vol.
60, Nr. 3, 2009, pp. 611-633. Balaguer, M. - Platonism and Anti-Platonism in Mathematics. New York: Oxford University Press, 1998. Batterman, R. W. - On the Explanatory Role of Mathematics in Empirical Science. “The British Journal for
the Philosophy of Science”, Vol. 61, Nr. 1, 2010, pp. 1-25. Bergmann, M. – Epistemic Circularity: Malignant and Benign. “Philosophy and Phenomenological
Research”, Vol. 69, Nr. 3, 2004, pp.708-725. Braine, D. - Varieties of Necessity. “Supplementary Proceedings of the Aristotelian Society”, Nr. 46, 1972,
pp. 139–70. Brown, J. - Non-inferential justification and epistemic circularity. “Analysis”, Vol. 64, Nr. 4, 2004, pp. 339-
348. Bueno, O. & Colyvan, M. - An Inferential Conception of the Application of Mathematics. “Noûs”, Vol. 45,
Nr. 2, 2011, pp. 345-374. Bueno, O. & French, S. - Can Mathematics Explain Physical Phenomena? “The British Journal for the
Philosophy of Science”, Vol. 63, Nr. 1, 2012, pp. 85-113. Cartwright, N. - How the Laws of Physics Lie. Oxford: Clarendon Press, 1983. Cohen, S. - Basic knowledge and the problem of easy knowledge. “Philosophy and Phenomenological
Research”, Vol. 65, Nr.2, 2002, pp. 309–329. Frege, G. - Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der
Zahl, Breslau: W. Koebner, 1884. Translated as The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number, by J.L. Austin, Oxford: Blackwell, second revised edition, 1974.
Frege, G. - Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle a. S.: Louis Nebert, 1889. Translated as Concept Script, a formal language of pure thought modelled upon that of arithmetic, by S. Bauer-Mengelberg in J. vanHeijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Cambridge, MA: Harvard University Press, 1967.
Fumerton, R. - Metaepistemology and Scepticism. Maryland: Rowman and Littlefield1, 1995. Goldman, A. I. – An Epistemological Approach to Argumentation. “Informal Logic”, Vol. 23, Nr. 1, 2003,
pp. 51-63. Gupta, A. - Remarks on Definitions and the Concept of Truth. “Proceedings of the Aristotelian Society”,
Vol. 89, 1988/89, pp. 227–246. Gupta, A. & Belnap, N. - The Revision Theory of Truth. Cambridge MA: MIT Press, 1993 Klein, P. - Knowledge, causality, and Defeasibility. “Journal of Philosophy”, 73, 1976, pp.792-812 Lange, M. - What Makes a Scientific Explanation Distinctively Mathematical?. “The British Journal for the
Philosophy of Science”, Vol. 64, Nr. 3, 2013, pp. 485-511. Leng, M. - What’s Wrong with Indispensability? (Or the Case for Recreational Mathematics). “Synthese”,
Vol. 131, Nr. 3, 2002, pp. 395-417. Moschovakis, Y. - Elementary Induction on Abstract Structures, Amsterdam: North-Holland, 1974. Pincock, C. - A New Perspective on the Problem of Applying Mathematics. “Philosophia Mathematica”,
Vol. 12, Nr. 3, 2004, pp. 135-161. Reid, T. - Essays on the Intellectual Powers. Cambridge: MIT Press, 1969 [1785]. Resnik, M. D. - Mathematics as a Science of Patterns, Oxford: Clarendon Press, 1997. Russell, B. - Mathematical Logic as Based on the Theory of Types, 1908, reprinted in his “Logic and Knowledge:
Essays 1901–1950”, London: George Allen & Unwin (1956), pp. 59–102. Sorenson, R - “P, therefore, P” without Circularity. “Journal of Philosophy”, Vol. 88, Nr. 5, 1991, pp. 245-
266.
34
Steiner, M. - The Applicabilities of Mathematics. “Philosophia Mathematica”, Vol. 3, Nr. 3, 1995, pp. 129-156.
Steiner, M. - The Applicability of Mathematics as a Philosophical Problem. Cambridge, MA: Harvard University Press, 1998.
Urbaniak, R. & Hämäri, K. S. - Busting a Myth about Leśniewski and Definitions. “History and Philosophy of Logic”, Vol. 33, Nr. 2, 2012, pp. 159-189.
Vogel, J. - Reliabilism levelled. “Journal of Philosophy”, Vol. 97, Nr. 11, 2000, 602-623. Wigner, E. P. - The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications on
Pure and Applied Mathematics, Vol. 13, Nr. 1, 1960, pp.1-14.