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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

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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.

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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.

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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].

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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.

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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.

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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.

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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.

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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.

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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.

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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.”

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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

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