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# Wigner's Puzzle on Applicability of Mathematics: On What Table to Assemble It?

Cătălin Bărboianu

► To cite this version:

Cătălin Bărboianu. Wigner's Puzzle on Applicability of Mathematics: On What Table to Assemble It?. Axiomathes, 2019, 30, pp.423 - 452. 10.1007/s10516-019-09465-8. hal-04089385

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Open licence - etalab
# Wigner's Puzzle on Applicability of Mathematics: On What Table to Assemble It?

Cătălin Bărboianu<fnref n="1" />

Received: 16 July 2019 / Accepted: 2 November 2019

© Springer Nature B.V. 2019

## Abstract

Attempts at solving what has been labeled as Eugene Wigner's puzzle of applicability of mathematics are still far from arriving at an acceptable solution. The accounts developed to explain the “miracle” of applied mathematics vary in nature, foundation, and solution, from denying the existence of a genuine problem to designing structural theories based on mathematical formalism. Despite this variation, all investigations treated the problem in a unitary way with respect to the target, pointing to one or two ‘why’ or ‘how’ questions to be answered. In this paper, I argue that two analyses, a semantic analysis ab initio and a metatheoretical analysis starting from the types of unreasonableness involved in this problem, will establish the interdisciplinary character of the problem and reveal many more targets, which may be addressed with different methodologies. In order to address objectively the philosophical problem of applicability of mathematics, a foundational revision of the problem is needed.

**Keywords** Applicability of mathematics · Unreasonableness · Wigner's puzzle · Applied mathematics

## 1 Introduction

In an influential paper, Wigner (1960) suggested that the overall success of the application of mathematics in the natural sciences is something unexplained and perhaps unexplainable. As Wigner put it in general terms, both the applicability of mathematics and the very high rate of success of applied mathematics require an explanation; if no explanation were given, the effectiveness of mathematics might be called “unreasonable.” Although the applicability of mathematics as a philosophical problem can be traced back to the ancient Greeks’ views on the relation between the abstract and the concrete, Wigner’s ‘unreasonable effectiveness’ can be the starting

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<footnote>Cătălin Bărboianu
cb@infarom.ro</footnote>

<footnote n="1">¹ Department of Philosophy, University of Bucharest, Bucharest, Romania</footnote>
point of a well-established discussion relating more closely the philosophy of math-
ematics and that of science. The general inquiry of whether the effectiveness of
mathematics is reasonable or not, (or whether this is even a genuine problem at all),
has benefited from the interest of philosophers, mathematicians and physicists; the
related catchword is ‘Wigner’s puzzle’.

Although Wigner points to several facts<fnref n="1" /> or “unexplained” aspects of apply-
ing mathematics successfully (briefly presented and named at the beginning of
Sect. 2), researchers have treated this problem within a philosophical framework as
just one *general* problem with respect to target, thereby adopting one of the follow-
ing approaches: (1) explain the success of mathematics, whatever this success may
mean; (2) show that nothing is unreasonable, whatever unreasonableness may mean;
(3) dissolve the puzzle as trivial or not genuine. Researchers have distinguished
between some kinds of applicability of mathematics which is defined through the
*roles* of mathematics, starting with Steiner (1995, 1998), who suggested that Wign-
er’s problem is actually a family of distinct puzzles, since there is more than one type
of applicability. This distinction was endorsed by Colyvan (2001), who also argued
that different philosophies of mathematics yield different hypotheses for Wigner’s
problem, which again results in different problems that have to be resolved. All clas-
sical views in the philosophy of mathematics have offered responses to Wigner’s
problem as interpreted in their particular conceptual frameworks, and some philoso-
phers have claimed to have found solutions<fnref n="2" /> for it.

What we find problematic ab initio about these solutions<fnref n="3" /> is that they assume
*one* general target of the problem, while there are rather several of them. When one
argues that “It is not unreasonable that mathematics is successfully applied because
A”, then the argument A employs all the notions used to frame the problem, includ-
ing the concepts of application, of mathematics, of applicability, of success, and that
of ‘unreasonableness’. The first four concepts were denoted through terms that ordi-
nary language also uses in a very wide palette of senses and meanings, but the con-
ceptual framework of any theoretical account of applicability of mathematics should
depend on a precise interpretation of these terms, even if such an interpretation may
either transcend the historical context of the problem or generalize Wigner’s original
formulation of the puzzle. The latter concept—unreasonableness—did not receive
a precise definition in this context. Since different conceptual frameworks can yield
different methodologies for solving the problem, it is clear that not only do we have
to find the way to assemble the pieces of the puzzle, but also the right table to sup-
port this assembling.

In the discussion, it is widely accepted that Wigner’s puzzle has received thus
far no complete and unobjectionable (dis)solution. Recent overviews revisiting

<footnote n="1">¹ In a methodological sense.</footnote>

<footnote n="2">² See (Bangu, 2012, pp. 133–139) for a well-organized presentation of these solutions.</footnote>

<footnote n="3">³ It is important to note that in this context of a philosophical puzzle, a ‘solution’ actually means the deductive chain or argument linking a set of concepts, hypotheses, and principles to the “unexplained” facts, even if those hypotheses and principles are not entirely tested and established or those concepts are not entirely tested for adequacy. With this definition, a solution is in fact a rational method available for making the unreasonable reasonable, but it needs a further, deeper analysis of the elements linked.</footnote>
this subject and commenting on an adequate research design for Wigner's puzzle<fnref n="4" /> brought into question the conceptual scheme underlying the problem and the necessity of reconstructing the puzzle, not only in its formulation, but also by adapting it to the contemporary philosophical accounts of applicability of mathematics. This approach is a way of admitting that in neither its original form nor subsequent versions, did the puzzle receive any solution and that the prospects of a solution are dim. We suggest that one reason for that idea is that the target or targets of the designed research are unclear, and the target(s) can only become clear if the conceptual framework of the puzzle is clear. This paper aims to ease the way to such clarification. After that, we suggest that the family of puzzles that comprises Wigner's puzzle is far larger than Steiner noted and that the methodologies required to deal with those puzzles are distinct, spanning several disciplines.

In Sect. 2, we will formulate the Wignerian facts of the successful application of mathematics, so as to reveal for each fact both the role of mathematics in scientific practice and the “strange” element that makes the success of that role appear ‘unreasonable’.

In Sect. 3, we will show how a semantic ab initio analysis of the key terms of the puzzle can split one target of the problem into several. The main distinction will be made between several notions of applicability that should be considered and a plurality of types of applicabilities qua roles. We will show that the relation between the concept of applicability and the success of the application is essential for both establishing a general target and the methodology of addressing that target.

In Sect. 4, we will take a closer look at the concept of ‘unreasonableness’ in Wigner’s problem, which is the main concern of this paper, and which has received too little attention so far. We will discuss the sense and meaning of the word ‘unreasonable’ and constituents of the unreasonableness as it takes shape as a concept from Wigner’s facts. We will identify six distinct types of unreasonableness involved in the facts that Wigner mentions, distinctions which are shown to enlarge the number of targets of the problem.

A subsection is devoted to the mathematically-driven discoveries in physics (not mentioned by Wigner in his paper) in which mathematics exhibits non-standard roles that have yet to be considered in the context of the distinction above. Types of unreasonableness already defined are detected in the success of these roles.

In Sect. 5, we will sketch in formal epistemological terms a definition of unreasonableness and a scheme of what it would mean to resolve an unreasonableness of a Wignerian type. In light of all the conceptual distinctions we have made, this scheme will be used to illustrate how to approach unreasonableness through changing its components.

In the last section we draw conclusions. The first conclusion drawn will be that the problem itself is interdisciplinary and the targets of the philosophical problem of applicability of mathematics stemming from Wigner’s puzzle are several. Both this new complexity and our formal account of unreasonableness suggest that in

<footnote n="4">⁴ See, for instance Ginammi (2011), Bangu (2016), Maudlin (2016) and McDonnell (2017), to name just a few of the most relevant.</footnote>
resolving the puzzle, conceptual rearrangement and reformulation are preferred
rather than continuing the search for a missing or impossible explanation to an
unclear ‘why’ target.

## 2 Empirical Hypothesis of Wigner's Problem and the Applicabilities of Mathematics

In his paper, Wigner (1960) presents the facts that make the successful application of mathematics appear as ‘unreasonable’ in a holistic and somehow contemplative mode, and in regard mostly to the practice of modern theoretical physics.<fnref n="5" /> In what follows, we will extract and synthesize these facts individually while formulating them in the general terms that philosophy of science and that of mathematics have addressed the problem of the ‘miracle of applied mathematics’ over the last half century. I take these formulations to be adequate for a theoretical-philosophical inquiry with regard to the applicability and application of mathematics to the natural sciences and non-dependent upon a certain ontology of mathematics<fnref n="6" />:

a (descriptive power). Mathematics has descriptive power that has allowed scientists to describe the phenomena they investigated so that they are amenable to a theoretical account (Wigner 1960, pp. 6–8, 10).

b (provision of language). Mathematics provides an appropriate (in fact, the best) language for the formulation of the laws of physics (Wigner 1960, pp. 6–8).

c (prior suitability). Mathematical concepts and theories created and developed independently of any empirical contexts prove to apply to certain empirical contexts (Wigner 1960, p. 3, 7).

d (cross suitability). Mathematical concepts that describe certain empirical contexts and whose creation was motivated or influenced by experience prove to apply to new contexts that are not related to the original ones (Wigner 1960, p. 1, 2).

e (accuracy). There is a remarkable accuracy in the results of the mathematical applications, and in general, the applications of concepts of type *c* or *d* have a very high rate of success; even false scientific theories gave amazingly accurate results, provided that they were mathematized (Wigner 1960, pp. 8–14).

<footnote n="5">⁵ In the interpretation of Islami (2017), this particularity turns, for Wigner, the metaphysical question into an epistemological one which Wigner actually solves for the specific case of theoretical physics by invoking the division between events, laws of nature, and invariance principles. As concerns the quality of ‘unreasonable’ granted by a physicist rather than philosopher, Ferreirós (2017) argues that the main reason for such qualification is even Wigner’s formalist views on mathematics and on its relation to theoretical physics.</footnote>

<footnote n="6">⁶ I do not take these formulations of Wigner's facts as a generalization, nor as interpretation, but rather as a conceptual unification with the philosophical framework of the problem. The acronyms in the list are mine.</footnote>
The facts *a* through *e* (apparently independent of each other with respect to any
possible explanation for the success of the application of mathematics), seem to be
connected to each other by some specific features of the discipline of mathematics
essentially responsible for its success. We hereafter refer to the facts *a* to *e* (as a
whole) as the *empirical hypothesis* of Wigner's puzzle.

Thus formulated, the empirical hypothesis has an immediate double effect: one is
that of shaping a primary concept of applicability of mathematics, and the other is a
surprise arising when *pairing* this applicability with some specific features of pure
mathematics regarding its nature, structure, content, and the process of its creation.

From the empirical hypothesis, one can shape a concept of a general applicabil-
ity of mathematics of a strong empirical character, given by the general success of
mathematics in the scientific practice, confirmed continuously during the history
of science. This general applicability consists of more than one type of applicabil-
ity, just as the empirical hypothesis is based on more than one role of mathematics.
However, such a concept may serve only the semantics of our discourse in ordi-
nary language, as this applicability has not yet assigned a definition in a theoretical
account employing the inner features of mathematics.

On the other hand, on the basis of this hypothesis, one can claim on merely inductive grounds a (kind of) *general-universal* applicability of mathematics. That is, we can expect that mathematics will be applied successfully to *any* future empirical context or problem of natural sciences during any time of observation. Of course, this concept of perpetual applicability is based on a dynamic concept of mathematics, which continually adds new concepts and theories to address the needs and problems of the sciences.

The conclusion that mathematics is generally applicable (in a sense adequately
defined) arises not simply as a result of an inductive inference, but rather, from a
philosophical motivation. The motivation is that we need a *metatheoretical* *justification* for an essential scientific method that was only empirically found successful thus far. Although the applicability expressed through the empirical hypothesis exhibits different kinds of unreasonableness (as will be argued in a subsequent section), the general target of any research following Wigner's paper has been formulated and has remained as *one* general 'why' question: *Why is mathematics so effective in the natural sciences?* This is a central question for the philosophy of applicability of mathematics. But before deciding whether we have one or more 'why' questions to answer as targets of any version of Wigner's puzzle, we should clarify what kind of applicability we are talking about when asking those questions.

## 2.1 Multiple Applicabilities

In the practice of science, mathematics has played some major roles due to its vir-
tues. This collaboration began even before Galileo noted that the great book of
nature is written in the language of mathematics, and the mathematical truths are
the essential truths for the inferences of science. The precision of mathematical lan-
guage, its capacity to formulate statements operable under propositional logic, and
the invariance of their necessary truths to any interpretations were and are the main
virtues of mathematics. These virtues assigned the classical roles of mathematics after it was engaged effectively in science: a role in describing and a role in inferring. Once it was acknowledged that mathematics played those roles very well, mathematics has been called *applicable* in different senses corresponding to those roles; these types of applicability are covered in the Wigner's empirical hypothesis.

First, we have *semantic applicability*, which is actually the referential feature of the mathematical concepts either created with empirical influence, having an empirical correspondent, or being applied directly to empirical concepts. We can unproblematically associate this type of applicability with that described in Frege's account of semantic applicability. Second, we have *descriptive applicability*, which is obviously related to the semantic applicability, yet based also on other features of mathematics such as those of language, logic, and set-theoretic structural constitution. Through the descriptive role of mathematics, we are able to formulate the laws of physics and describe the investigated empirical contexts, as idealizations. Both semantic and descriptive applicability are supported by the descriptive power and the provision of a language. Third, we have the *inferential applicability*, which is the ability of the mathematical models to allow inferences on the empirical context that otherwise would be hard or impossible to obtain (Bueno and Colyvan 2011, p. 352). This latter type of applicability is sustained by prior-suitability, cross-suitability, and accuracy; it is based on the two other types of applicability and the inferential character of a large class of mathematical applications. In his influential work on the applicability of mathematics as a philosophical problem, Steiner (1998) distinguished between these three types of applicability as inner features of mathematics and endorsed Frege, who solved the semantic-applicability problem through his second-order logicist model. Inferential applicability was addressed further through the so-called 'mapping accounts' and these accounts (Pincock 2004; Bueno and Colyvan 2011; Bueno and French 2012) have become the focus of philosophers studying applicability in the last decade.

Hence, the three classical types of applicability mentioned above are identified through the *roles* that mathematics played in the scientific practice.<fnref n="7" /> Mathematics played those roles successfully, and so it proved to be applicable (in the three senses above) on the basis of the standard scientific practice of using mathematics and applied mathematics throughout the history of science; in epistemological terms, that is a track-record kind of reliability based on success reached by the applied mathematician or scientist. However, for a metatheoretical justification of the application of mathematics, another concept of applicability (beyond the *assigned* roles of mathematics and their fulfillment) should be employed, transcending the empirical hypothesis and other particularities of the mathematical modeling. Such a concept of “no-assigned-role” applicability would be also expected to account for

<figure><img src="image_4.png" /></figure>

<footnote n="7">⁷ The distinction between semantic and descriptive applicability has already been made in the literature and is justified through the fact that the semantic applicability has more of a logical nature; however, this is not the logic of a descriptive language, but rather the logic of the application. These features are precisely represented in the Fregean account of semantic applicability. For a succinct presentation of this account in relation with the philosophical problem of applicability, see (Steiner 1998, pp. 13–23).</footnote>
Wigner's puzzle, since the prior- and cross-suitability facts are those reflecting an inner user-independent virtue of mathematics responsible for attaining the goals of some applications (including future ones).

The question arising is whether we can define a notion of applicability beyond the roles of mathematics, and if so, what its relevance would be for the resolving of Wigner's puzzle. We will investigate this question in another paper; however, a new question concerns both qua-roles and no-role applicability: Do we regard the concept of applicability as dependent on the concept of success or on the inner features of the applicable discipline? And what constitutes "success"?

### 3 Applicability and Success: Semantic Distinctions and Targets

The now-paradigmatic syntagma "unreasonable effectiveness of mathematics" has been equated with the question "Why is mathematics so effective in the natural sciences?" as the only target of the research, despite its various available formulations. The main contention of this paper is that reducing Wigner's puzzle to one 'why' question is problematic, because doing so could render a framework or methodology inadequate once we employ distinctions for the key terms of the problem.

Some researchers<fnref n="8" /> viewed the question “What is mathematics?” as essential (sometimes along with the complementary question “What is physics?”) determining priority in the investigation of the problem of applicability, but the semantic analysis can be extended further to other key terms such as ‘application’ and ‘applicability’, and ‘success’ of an application.

In ordinary language, ‘to apply’ has a wide palette of senses and meanings: to import (something in/from something), to map (something to something), to translate (something into something), to interpret (something as something), to superpose (something over something), to adjust (something to something), to use (something at something), to obey (something when something), to refer (to something), to measure (something with something) and so on, as well as combinations of those meanings. Then, the applicability (of something to something) would be either the *possibility* of the action chosen as application or the *success* of that action—that is, an internal or external applicability to refer to.

We apply mathematics to various empirical contexts,<fnref n="9" /> either in daily life or within the framework of a particular science. But what does it mean for mathematics to be applied to a target such as an empirical context or science? Roughly, it is about a triple import and usage: a conceptual import, a methodological import, and the import of truths. Indeed mathematical concepts describing empirical contexts are used as either referents or references of scientific discourse (note that some terms of pure

<footnote n="8">⁸ For instance Wigner (1960), Hamming (1980) and Sarukkai (2005).</footnote>

<footnote n="9">⁹ We refer here to ‘empirical context’ as a target of application in the sense of the standard model of application of Bueno and Colyvan (2011) that is characterized through a structural idealized description facilitating the application.</footnote>
mathematics have become terms of the mixed<sup>10</sup> language of science and even of the
ordinary language); science uses the axiomatic method and the analytical rules of
mathematical proof and inference; the necessary truths of mathematics become the
truths of sciences via the inferences through mathematical models. Of course, there
is no unique recipe for combining that triple import for any target of application.
Mathematics plays different roles for different targets: Besides the standard math-
ematical modeling of empirical contexts (an import of truths), mathematics plays for
instance a constitutive role in theoretical physics (a conceptual and methodological
import), which it does not play in the same way in biology where the role is instead
either directly inferential or played indirectly by physics.<sup>11</sup>

Now, note that all the verbs enumerated above for the ordinary-language concept of application are reflected somehow in the actions of what we just described as the application of mathematics. However, we can expect that applicability—either internal or external—may be something different for each particular action. Despite this variation, what is common to all applicabilities is their epistemic nature of being a *property* of the thing being applied. Therefore, a complete theoretical account of applicability of mathematics should be founded on a conceptual framework allowing a definition for that property, and this assumes a predication ranging over a class of variables, which would be the class of the disciplines, including that of mathematics. In other words, it is not enough to investigate mathematics in and of itself to account for its applicability, but also to establish what it would mean for a discipline to be applicable to an empirical context or to another discipline. However, a general theory of applicability is not the concern of our paper. For the moment, let us stick with the two options we have for defining applicability of mathematics—internal and external (of pure mathematics), that is, independent or dependent upon the success (of applied mathematics).

Is mathematics (or an arbitrary discipline) applicable because it has proved over-
all (acceptable) success in past applications to date, or is it applicable as a perpetual
inner feature of itself despite the failures of application? Of course, the first part of
this question can be posed only with the assumption that a discipline whose applica-
bility we do not know can still be applied to a different one in a somehow "forced"
manner.

Answers do not come easily for the concept of success either. We can distinguish
between empirical success (expressed in the accuracy fact, with only the individual
successful applications qualifying as such) and methodological success<sup>12</sup> (expressed
in all facts from *a* to *e*, retaining the *rate* of successful applications or the success
of the most relevant ones). This distinction raises the consequent question: Which
type of success does applicability depends on, if it is in fact dependent? Moreover,
the general concept of *success* can be referred to as either “accidental” or “pre-
formulated”: an “accidental” success is acquired through scientific methods but for

<figure><img src="image_5.png" /></figure>

<footnote n="10">¹⁰ A language with both mathematical and physical terms.</footnote>

<footnote n="11">¹¹ For an insightful analysis of the application of mathematics in biology, see Islami and Longo (2017).</footnote>

<footnote n="12">¹² The methodological success also has an empirical nature, as its ultimate indicators rely on the obser-
vational base consisting of the recorded individual empirical successes.</footnote>
which we do not have the certitude that they are adequate; a “pre-formulated” success is followed and acquired through a well-established and accepted methodology in which the successful result is the final target of a deductive-inferential chain or constructive process.<fnref n="13" /> Additionally, methodological success may be confirmed or not by the future evolution of sciences, so it is time-dependent and goal-dependent, which suggests that applicability may also be defined in such a dynamic view. This suggestion is supported by the “metabolic” feature of mathematics,<fnref n="14" /> which feeds from the problems of science, adds new concepts and develops new theories for addressing these problems, and serves the sciences with the result, thus influencing their evolution. This mutual-exchange process is perpetual. Note also that pure mathematics itself is time-dependent regarding its *content*, as an auto-generative content in expansion, where all that is created remains (unlike the sciences, where theories compete with each other and revolutions lead to the collapse of established theories). At this point, it worth noting that a dynamic evolutionary view of mathematics in relation to its success draws on the distinction between the mathematical content and mathematical knowledge, where the latter concept is dependent upon that of mathematical practice and successful application. Such dependency was recently characterized by Molinini (2019) through the notion of ‘weak objectivity’ of mathematical knowledge, which accommodates well with what we have called ‘methodological success’ and generally fits into a dynamic account of applicability.

This discussion on the distinctions for the concepts of applicability and success does not exhaust the typology, nor the subtleties of the concepts. Its only goal is to argue that the initial formulation of Wigner’s puzzle is simplistic to the extent that it suggests only one target, that of answering whether mathematics is reasonably or unreasonably effective and why it is so. Before discussing unreasonableness, observe that ‘effectiveness’ means *both* applicability and success, and the distinctions we have already made yield a major split for the single target into several targets: showing that mathematics is applicable and then accounting for its success in relation with the type of applicability defined. Of course, investigation on the adequacy of each type of concept of applicability (*qua*-roles/no-role, internal/external, empirical-/methodological-success dependent, static/dynamic) is itself a target prior to the applicability/success split.

The split is also visible when we take into account methodology. First, answering a mere ‘why’ question assumes a causal treatment, as long as the *empirical* hypothesis stands for the *explanandum*. However, employing certain types of concepts of applicability (as suggested earlier) makes room for formal acausal treatment, and thus various kinds of explanations are involved. Hence, we have methodologies of different natures required to solve the problem and the ‘why’ target splits into several ‘whys’ and ‘hows’ (or ‘how-is-it-thats’). However, the ‘hows’ may not assume

<footnote n="13">¹³ In ordinary language, the term *success* seems to be used preponderantly with the accidental (lucky) component, being attributed to a human endeavor that by its nature is subject to incertitude, subjectivism, imperfect preparation, and contingency, and it is exactly this incertitude that justifies the emphasis put on the result of the “successful” endeavor. Of course, the accidental/non-accidental distinction is related to the empirical/methodological one.</footnote>

<footnote n="14">¹⁴ In a sense that involve both scientific and mathematical practice.</footnote>
causal methodologies. Despite their multitude, let us group these general targets into two categories, which we will call *general applicability* and *explanation of success*.

Different frameworks and methodologies are also assigned when distinguishing between the meanings of other key terms of the problem, for instance that of *mathematics*. Working with a static concept of mathematics (as a stand-alone body of knowledge at a given time) versus a dynamic concept (time- and context-dependent) assumes theoretical models of different natures: discrete versus continuous, finitary versus infinitary, stochastic versus determinist, etc. Not only are methodologies different, but one methodology can invalidate another. Consider for example the so-called ‘many failures solution’ to Wigner’s problem.

Briefly, this solution is actually a criticism that tends to deny the problem by exploiting the fact there are so many failures in applying mathematical concepts that Wigner and his followers failed to address along with the surprising success of applied mathematics. The argument followed a line of thought that the number of failures (and not of successes) should be called “significant,” and moreover, only a small number of all the mathematical concepts are used in physics; therefore, quantitatively speaking, applicability might actually be simply a marginal phenomenon. Such arguments were developed around the ideas of Wilson (2000) and assume a hypothetical statistical estimation of the (performed, recorded, and unrecorded) applications over a timeline, in which the number of successes is claimed to have been much lower than the number of failures (and we have no reason to doubt that this is the case). Now combine that with a dynamic concept of mathematics, and such an account of the successes versus failures is no longer valid. Failure would then become time dependent (and hence non-constant as either the value of a random variable or the property/attribute of an application) with the time dependence of mathematics, and thus another type of estimation is required.

Or, as another example, take the ‘fudging’ solution, which argues from the premise that before any mathematical application, the physical context is stripped of its layers of complexity and made structural through idealization; hence it is prepared especially for facilitating the application.; as such, there is nothing surprising that mathematics applies.<fnref n="15" /> Observe that this argument at most counts to explain the methodological success. If we aim to explain the empirical success, the argument must somehow link pure mathematics not only to the idealized physical context, but directly to the real (physical) context. In addition, the argument seems to address only the descriptive applicability while empirical success is also dependent on the semantic and inferential applicability, at least for the predictive models.

The semantic distinctions in question are those that are necessary for posing the right problem or problems and for delimiting the frameworks specific to the different targets of the research. The semantic distinctions are closely related to the metatheoretical approach, since they also suggest directions for research, foundations, and

<footnote n="15">¹⁵ Idealization as a determinant procedure that makes application of mathematics possible is an idea that has been stressed by most of the philosophers dealing with applied mathematics, so the ‘fudging’ solution cannot be attributed to a certain author. This solution is named as such in Bangu (2012).</footnote>
methodology, including what is to be proved and what is to be explained and within
which framework.

Things seem to be clearer with the concept of unreasonableness in the context of Wigner's puzzle, at least for an exhaustive typology. Unreasonableness is not only the main object but also the target of Wigner's problem, with three possible options: investigating whether we have an ingenuous problem here, showing that nothing is unreasonable with the success of mathematics, or pursuing the direction that unreasonable can be turned into reasonableness. While these options—formulated as above—can stand for a general target, distinguishing between the different natures of such unreasonableness again yields different targets (or sub-targets) of an adequate research to the extent that different methodologies are required to deal with them. This is the subject of the next section.

4 Various Types of Unreasonableness

Let us see exactly where and how the *unreasonableness* manifests itself in the facts
from *a* to *e*, in Wigner's and his followers' view. Wigner emphasizes unreasonable-
ness especially in the prior suitability, cross suitability, and accuracy facts, but we
can detect various kinds of unreasonableness in all five facts, as we shall show in
this section, though not all are explicitly mentioned in Wigner's paper.

First, let us notice that the descriptive power, provision of language, and accu-
racy facts exhibit a connection or collaboration between objects, domains, or kinds
characterized by paradigmatic dichotomies based on the *extreme* difference of their
natures: abstract/concrete, mental/physical, ideal/non-ideal, and necessity/contin-
gency. Such collaboration occurs via mathematical modeling, and the unreasona-
beness detected is actually the association of this extremeness with the success of
this scientific practice. In brief, the unreasonableness sounds like this unanswered
question: *How is it that* a domain such as mathematics can be applied successfully
to another domain (the physical), *given* their different natures, ontologies, episte-
mologies, and logical categories? The “how is it that” (which is not equivalent to a
‘why’ in this context) expresses an absence of successful rational investigation and
the “given” expresses the surprise. These two epistemic elements are the exhaustive
ingredients of any labeling as unreasonable.<fnref n="16" /> This is a *background unreasonable*
*ness* influencing the particular unreasonableness coming from the prior suitability
and cross suitability facts, but Wigner did not emphasize it [though other researchers
of this puzzle did, such as Steiner (1995, 1998), Bangu (2012), or Morrison (2015)].
This unreasonableness has the immediate generalization to a philosophical prob-
lem thousands of years old, that of the general success of the human mind over the
empirical universe, including the so-called miracle of science. Such problems are
dealt with within the philosophy of mind, epistemology, philosophy of science, and
philosophy of language as *locus classicus*; however, newer approaches can appeal to
other disciplines, including non-philosophical ones, to be involved.

<footnote n="16">¹⁶ We will sketch a formal definition of the unreasonableness in Sect. 5.</footnote>
Second, exploring the particular unreasonableness from the prior suitability, cross suitability and accuracy facts takes us deeper into the specificity of pure and applied mathematics, which is called upon to account for the ‘miracle of mathematics ‘independently of the’ miracle of science’.

The prior suitability fact seems to be for Wigner the strongest evidence supporting the sensation of unreasonableness. The range of examples where mathematical concepts and theories were necessary and used successfully in physical theories is wide, and this is not the place to review them. Staying with the classical example of complex numbers, we are sure that they have been conceived independently of any methodology in physics and with no empirical correspondent, interpretation, or counterpart, proving them necessary for the formulation of quantum mechanics based on Hilbert spaces. The applicability of complex analysis in physics can be made evident in particular through the use of analytic functions, which turn out to be applicable to and essential for fluid dynamics, relativistic field theory, and thermodynamics. The creation of the concept of complex numbers in the 16th century, according to historians, has its purpose in the generalization of equation-solving techniques. This generalization included a unification of the natures of the solutions—a complex number becomes real if its imaginary part is made zero—and a simplification—the number of solutions is equal to the degree of the equation. The concept itself, as created, has nothing to do with any empirical interpretation and cannot even stand for a quantity or magnitude. Note that the idea of generalization, unification, and simplification is more a *mental impulse* than a scientific proposal, and that impulse is based on the criteria of aesthetics and symmetry, which are so specific to the pure mathematical creation. Now, formulated so as to catch the surprise and the lack of acceptable results of a rational investigation, what seems unreasonable is clear: *How is it that* mathematical concepts and theories conceived on criteria of aesthetics and symmetry—their conception having more an artistic freedom than a rational character—prove the ability to apply successfully in science—a rational domain—*given* that the application in science was not intended as the target by their creators? Or, as Steiner (1998, p. 46) put it in a syllogistic form when interpreting Wigner’s argument:

1. Mathematical concepts arise from the aesthetic impulse in humans.

2. It is unreasonable to expect that what arises from the aesthetic impulse in humans<fnref n="17" /> should be significantly effective in physics.

3. Nevertheless, a significant number of these concepts are significantly effective in physics.

4. Hence, mathematical concepts are unreasonably effective in physics.

<figure><img src="image_6.png" /></figure>

<footnote n="17">¹⁷ My emphasis.</footnote>
## 4.1 Standard Applications and “Classical” Types of Unreasonableness

Previously, we assigned the abstract-concrete background unreasonableness to the descriptive power, provision of language, cross suitability, and accuracy facts. All these facts fall within what we call the standard application of mathematics; such an application involves establishing correspondences between empirical contexts and mathematical structures and interpreting the mathematical results with physical reference so as to make inferences in the physical domain. With such characterization of the standard application and with respect to our aim of following the unreasonableness, we do not need a sharp distinction between investigating physical reality through a mathematized physical theory and investigating it directly through applied mathematics. When formulating their physical theories in mathematical language, physicists make the same kind of correspondences and interpretations as does someone within a problem of applied mathematics, in that both are structural and linguistic. If we limit our analysis just to the descriptive power of mathematics, *prima facie* nothing appears unreasonable: Mathematics is formulated into a formal language based on predictive logic and conformative syntax. Syntax is constituted by relations and implicitly structure. Ordinary language also has logic and structure, and therefore it is not so surprising that a correspondence can be established between mathematics and the empirical contexts described in ordinary language, scientific language or mixed language.<fnref n="18" /> Put as simply as possible, there is no surprise that physical objects can be mapped onto mathematical concepts, and actions (or phenomena) onto predicates (mathematical properties) at the level of language.<fnref n="19" /> However, what contributes decisively to the unreasonableness of the standard applications is the high accuracy of the results (the accuracy fact). That is, once the empirical context or physical theory is described in mathematical language, the model is put to work, and the inferred results are amazingly accurate. This result—an empirical success—acquires the quality of being unreasonable when paired with the specific premises of the standard modeling, of which two can be identified besides the general abstract-concrete dichotomy:

Firstly, mathematical models are widely seen to be *false* representations of the empirical contexts, and the first type of unreasonableness would sound like this: “*How is it that* applied mathematics yields such accurate results, *given* that they are inferred on the basis of false representations?”<fnref n="20" /> The apparent tension between

<footnote n="18">¹⁸ Either in sense of having both mathematical and physical terms, or of a combination of common and scientific language.</footnote>

<footnote n="19">¹⁹ This view accommodates well with what we call the *inseparability thesis* as a solution to Wigner’s puzzle through dissolving. The main premise of this solution is that the mathematical content cannot be separated within a physical theory and so mathematics is not something that can be added externally to the theory as a whole. In this way, the mere concept of *application* of mathematics in physics is in contradiction with this internally holistic view of physical theories. Proponents of this thesis are Boniolo and Budinich (2005, pp. 78–86).</footnote>

<footnote n="20">²⁰ Any analysis of the applied mathematics practice can account for the failures (or inaccurate results) as a result of inaccurate representation through either inadequate mappings or idealizations. Moreover, failures are the main argument for the proponents of a solution of dissolving Wigner’s puzzle (claiming that the number of failures, not the number of successes, is significant). However, if we stay with the concept</footnote>
false representations and accurate results can also be formulated as analogy-break-
ing: “How is it that applied mathematics yields such accurate results, given that
the results are inferred on the basis of an analogy within a false representation of
the empirical context which, by stretching the analogy further, should be expected
to yield inaccurate results instead?” Call this the analogy-breaking unreasonable-
ness. Of course, the relationship between false theories and accurate results is the
subject of an ongoing debate in the philosophy of science, and the “scientific” fal-
sity lies more in the postulation of ‘fictional’ entities rather than the precision of
a representation. However, in the case of mathematics, there are specificities that
could account for that relation independently of the general case of science. Among
those specificities, consider two in particular: firstly, the competition between scien-
tific theories and the subsequent collapse of the “losers,” which does not happen in
pure mathematics; secondly, the difference of epistemic status between a scientific
hypothesis and a set of axioms. The competition between hypotheses in science has
no counterpart in mathematics, where new sets of axioms create non-eliminable new
mathematical concepts. There are of course other specificities waiting to be discov-
ered. A deep analysis of the relationship between the success of science and the
success of mathematics with respect to Wigner’s puzzle deserves utmost attention,
as similarities could count as well toward finding an adequate solution to the puzzle.

The second specific premise is given by specific actions of the applied mathe-
matician within the methodology of building the models and applying mathematics
through these models. It has to do with the *choices* made throughout this multi-step
process when establishing the idealizations, structural arrangement, assignments,
and correspondences operated over the two domains. Such choices seem to be intu-
ition-based, arbitrary<fnref n="21" /> and subjective and are not governed by any criteria estab-
lished within a theory or metatheory of applying mathematics. The applied math-
ematician follows criteria of relevance (of establishing the mathematical structures
for the empirical context investigated) and convenience (with respect to both struc-
tural correspondence and the scientific goal of the modeling) when applying math-
ematics, and the overall results show that in most of the situations the right choices
are made. A first arising question is why the applied mathematician makes a par-
ticular choice since no normative disciplines apply for this selection, and a second is
how or why such choices prove to be the right ones. From this methodological point
of view, it seems that we come again to two general targets: that of a user-dependent
general applicability and that of explaining a success. Testing for the right or wrong
choice is done not only against confirmation of the predictions of the model, but also
against the preservation of a systematic coherence within the background results of
the science, which include all past successful applications of mathematics and the
well-established scientific theories. Steiner perfectly describes this premise as the

<figure><img src="image_7.png" /></figure>

<footnote>Footnote 20 (continued)

of a general (perpetual) applicability instead of a partial one, their argument loses power, and the accu-
rate results stands as a premise for this kind of unreasonableness.</footnote>

<footnote n="21">²¹ In sense of non-normative.</footnote>
"art of the mathematician": "How"<fnref n="22" /> does the mathematician—closer to the artist than to the explorer—by turning away from nature, arrive at its most appropriate descriptions?" (Steiner 1995).

Relevance and convenience, it would seem, are obtained through an artistic ability that apparently has nothing to do with the criteria of rigor specific to mathematics or science. The applied mathematician may employ an already created mathematics or develop a new mathematical theory just to serve the needs of the current problem. Such a new creation is—again—driven by criteria of aesthetics, symmetry and freedom. Again, we have an artistic-type unreasonableness stemming this time from the methodological process of applying mathematics.

Finally, let us now identify the specific unreasonableness in the cross suitability fact, where Wigner’s favorite example is the presence of π in an application engaged in the study of population statistics by means of normal distribution. The “mystery” comes from the common-sense belief that population has nothing to do with the circumference of the circle (Wigner 1960, p. 1).

Wigner does not analyze further his π example, staying with the philosophical aspect of the case. If we go to the mathematical insight of the normal distribution, it is easy to see how the concepts connect to each other and transports π and the equation of a circle to the expression of the probability density function. However, if this qualifies as an explanation, this is an explanation *within mathematics* of how circle and π are involved in the normal distribution, or how conceptual insight is transported within pure mathematics. It is widely known that mathematical concepts turn up in unexpected connections across different mathematical theories. This π example is just illustrative, as mathematics exhibits plenty of such connections and all can be explained within mathematics.<fnref n="23" /> However, Wigner’s concern was not for such explanation, but for the lack of one explaining why π, having its origins in Euclidian geometry and characterizing circle as a mathematical concept created with a clear empirical correspondent (the round objects or shapes), proved effective not in the theory of distributions within statistics, but when this theory was *applied* to an empirical context not apparently connected with the one the circle was created for. This is how the concern “the population has nothing to do with the circumference of the circle” (Wigner 1960, p. 1) translates, and the question is whether the mathematical explanation within mathematics is a necessary part of the missing explanation.

In this picture, the unreasonableness seems to stem from an apparent contradiction: The concept of circle (with its circumference or π) is present in (or at least connected with) the concept of normal distribution (1); normal distribution is an essential concept for modeling population trend (2); circle as a mathematical concept was defined to represent planar roundness as a metric property (3); population has nothing to do with roundness (4). The conjunction (1) ∧ (2) yields that the mathematical investigation of population trends uses indirectly the concept of circle, which seems to contradict the conjunction (3) ∧ (4). Of course, sensible issues like the structural ontology in (1), necessity in (2), and empirical meaning of roundness in (3) and (4)

<footnote n="22">²² My emphasis.</footnote>

<footnote n="23">²³ Take as a label for this general connection the famous Euler’s formula.</footnote>
are debatable within analysis of this example. However, the apparent contradiction within the argument is not logical, but rather conceptual, as related to the epistemic connection between the concepts involved.

These classical problems generate discussions for those arguing for the reasonableness or for dissolving Wigner's puzzle as ingenuous. Let us keep the scheme above when defining the (apparent) unreasonableness of the cross-suitability fact. That is, if there is something unreasonable in the cross suitability fact, it can be expressed in general in our 'how-is-it-that...given' form like this: "How is it that a mathematical theory proves effective in modeling an empirical context, given that this theory uses a concept having an empirical correspondent (or referent) that is not connected in any way to the modeled empirical context?", or better: "How is it that a mathematical concept created to represent a certain empirical object or context appears in a mathematical theory that proves effective in modeling a different empirical context, given that the two empirical contexts are not connected in any way (structurally, causally, etc.)?" Call this the contradiction unreasonableness.

## 4.2 Unreasonableness in Mathematically Driven Discoveries

A particular type of application of mathematics in physics is discovery via the manipulation of mathematical formalism. Wigner pointed (through the prior suitability fact) only to the discovery of laws of nature as related to unreasonableness. However, there is another type of discovery of a different nature, made just by pure observation of that formalism and not by manipulation; it has to do with discovering some physical particles—that is, stating their existence, followed by validation through empirical experiment—as a result of some tacit “suggestion” coming from the mathematical concepts, entities, and axioms. The mathematically driven discovery of physical particles occurred at the time Wigner’s paper was being developed, yet had not come to the spectacular contemporary results, and this would explain Wigner not giving more attention to such discovery. However, it certainly deserved such attention, as it potentiates the unreasonableness coming from the prior suitability fact.

To reference the most notable achievements in particle discovery with a focus on the way their discoverers came to state their existence, consider Dirac’s postulation of the positron in 1931 and the prediction of the omega-minus particle by Gell-Mann and Ne’eman in 1962. The cases are well known in the history of science, and this is not the place to reiterate them.<fnref n="24" />

What characterizes these two discoveries (positron and omega-minus particles) is the “unusual” way in which they were discovered: these discoveries were driven

<figure><img src="image_8.png" /></figure>

<footnote n="24">²⁴ Two excellent succinct histories of this discovery and its background are presented in Bangu (2012, pp. 91–95) and Steiner (1998, pp. 90–94).</footnote>
by a merely theoretical constraint that fits perfectly with an analogy<fnref n="25" /> applied to the natures (or significances) of the entities involved, both physical and mathematical.

We must make the distinction between this means of discovery and a somewhat similar one, that leading to the discovery of the neutrino by Pauli in 1931 or the planet Neptune by Leverrier and Adams in 1845, also well known. For these two discoveries, the scientist eliminates an observable *anomaly* in the behavior of a known system (which would violate a well-established law), by postulating the existence of a new element of the system and changing its initial conditions.

Simplifying as much as possible the two types of discoveries<fnref n="26" /> so as to see the resemblances and differences between them with respect to the role of mathematics in the discovering methodology, their schemes are as follows:

Scheme 1. Discovery via assigning referents to mathematical formalism: We have a physical system described by a mathematical formalism, within which some parts or entities have physical referents within a scientific theory using that formalism. We *observe* a group/set of mathematical entities of the same nature, status and role with respect to a mathematical concept essential for that theory (1a). We observe that some of these entities have physical referents within the theory and this assignment contributes to both the formulation and coherence of the theory (1b). Optionally, we might observe that the theory would be enhanced or questions left open might be answered within the same theory or there is a possibility of unification or simplification if the other entities also had physical referents (1b'). Then, physical referents are assigned to those entities by postulating/predicting them, and the theory is revised accordingly (1c).

Scheme 2. Discovery via preserving systematic coherence against dramatic change: We have a physical system described by a mathematical formalism and governed by well-established physical laws formulated mathematically. At some point, we detect experimentally an anomaly within the system that would violate one of these laws; such an anomaly is detected through measurement against calculation within the mathematical formalism which describes the interactions between the entities forming the system (2a). We *choose* not to revise the well-established laws (which also apply to other systems), but to eliminate the anomaly in a different way (2b). We decide or infer that the only way remaining is to modify the initial conditions for that system by adding (postulating/

<footnote n="25">²⁵ Steiner qualifies such an analogy as Pythagorean, arguing with several examples on the ground that “[T]he use of Pythagorean analogies is typical of twentieth-century physical thinking for two main reasons: the rich development of modern mathematics and the lack of any alternatives” (Steiner 1998, pp. 76–94).</footnote>

<footnote n="26">²⁶ In the philosophical literature dedicated to the subject of new particle discovery and also of the role of mathematics in natural sciences, such discoveries are usually called *predictions* (or *e-predictions*). Although the distinction prediction (on a par with *explanation*)/discovery certainly has significance with respect to the epistemology of scientific practice and its results, we must limit the discussion to what concerns the unreasonableness in Wigner’s empirical hypothesis. This is why we will stay with the term *discovery*, especially for catching the differences between the various aspects of facts *a* to *e*. Note also that prediction—among others—is also one of the functions/goals of a mathematical model, which is the main concern of facts *d* and *e*.</footnote>
predicting) a new physical entity that would change the mutual interactions within that system (2c). We revise the formal calculations for the new system by keeping all the parameters associated with the known entities and deduce the parameters of the postulated entity—as its physical characteristics—so that they fit the experimental data, including the anomalous data (2d).

Obviously, there are more differences than similarities between the two schemas, either with respect to scientific practice in general or the mere use of mathematics. As for similarities, note that the anomaly which essentially causes the discovery along scheme 2, is also dealt with in scheme 1, but there it has a different nature. While in scheme 2, the anomaly, empirically acknowledged, is a certain break in the systematic coherence, in scheme 1 it appears as merely theoretical and paired with a psychological tendency, or a non-equilibrium state of the rational thought searching for perfection. Firstly, it appears in process 1b, where we observe a possible break in the analogy and symmetry, and then, optionally, in process 1b', where the epistemic virtues of the new revised theory are not yet acquired. Seen as a resemblance between the two schemes, the anomalous component also qualifies scheme 1 as coherence-preserving, but this coherence is more local, while in scheme 2 it is more global. An important difference is the effect of the discovery for the scientific theory: while in scheme 1, the theory would not be invalidated in the absence of the postulation, but simply left open just for enhancement, in scheme 2 a well-established law or theory would be invalidated.

We cannot cover here all the subtleties of the differences between the two types of discovery,<fnref n="27" /> but we must focus on the arguments used to categorize such discoveries as belonging to the prior suitability fact of Wigner’s empirical hypothesis. Thus, what concerns us more are the similarities and differences with respect to any role of mathematics, to help us identify and characterize the unreasonableness in the prior suitability fact.

Let us notice first that in either scheme 1 or 2 mathematics has already served the purpose for the formulation of the scientific theories referred to in both schemes, in the vein of the descriptive power and provision of language facts by using its descriptive role. What mathematics does in scheme 1 is an *indirect* contribution to the enhancement of the theory via the intellectual impulse of the scientist. The mathematical formalism *itself* triggers this impulse by suggesting the assignments of physical referents. Such assignment, sometimes independent of the coherence of the theory in question (see Dirac’s particle case) not only leads to revision of the theory, but has ontological implications. Due to such implications (even if they are not direct, but mediated by empirical confirmation), we may qualify this role of mathematics as *predictive*; however, this attribute is controversial. Bueno and French (2018, pp. 145–150) advanced the view that such a role is at most heuristic and not predictive, based on the argument that the border between the formalism of mathematics and the domain of physics is still significant in such scientific practice; they

<figure><img src="image_9.png" /></figure>

<footnote n="27">²⁷ A detailed analysis of and comparison between the two types, on epistemological grounds, is made by Bangu (2012, pp. 97–108).</footnote>
go on to argue that there are actually the physical theory and the empirical realm, not mathematics, within which the prediction is made and respectively confirmed. They also supported their arguments with a historical study regarding the role of group theory in quantum physics (Bueno and French 2018, Ch. 4). By not entering the details of their arguments nor even asking for the exact concept of prediction to which they refer, we would not be so radical in rejecting a predictive role. The reason concerns the standard application of mathematics: A large class of mathematical models are considered predictive just because their goal is prediction, and hence the *inferences* made through them are equated unproblematically with predictions. The counterpart of this standard inferential role, in physical discoveries, is not a predictive role according to Bueno and French. If the same argument against the predictive role applies to the standard application (where the empirical/physical domain is even more independent of the mathematical domain and is the domain where the inferred relation is confirmed), this would undermine the inferential role of (applied) mathematics, which would be merely heuristic as well; however, interpretation is just a step in the inferential application scheme. To this extent, their argument must also explain why this is not the case or why prediction and inference would have different natures for different targets (postulated entities and relations between existing entities). While not discarding the heuristic role, henceforth we shall call the role predictive-heuristic.

In scheme 2, the scientist acts in a classically scientific way, as he has as the main objective the preservation of the systematic coherence within larger systems governed by well-established laws and theories, which is actually characteristic of the common rationale. At some point he will make the main choice (process 2b in scheme 2). The mathematical theory used within the model allows the introduction of a new variable as a mathematical correspondent for the postulated physical entity, due to its generality and flexibility. However, the scientist does not yet have the confirmation that his choice was right with respect to his objective. Next, through calculation, both formal and numeric, mathematics provides the scientist with that confirmation, and thus it exhibits a role, the *confirmation* role, which collaborates with its instrumental role. This is a *direct* contribution to the physical discovery along scheme 2, however it is not the mathematics which indicates that a physical entity is out there waiting to be discovered, but rather, the detected *physical* anomaly which triggers the whole process. This line of thinking seems a reason for discarding this type of discovery as an example for the prior suitability fact; instead, we must choose to keep it because the comparison is important for distinguishing between the various *roles* of mathematics.

We are now in a good position to identify the types of unreasonableness in such mathematically driven discoveries and see how this unreasonableness manifests itself. Firstly, let us notice that these types are specific for this kind of effectiveness of mathematics (that of discovery), as their characteristics go beyond the background abstract-concrete unreasonableness exhibited through the descriptive power, provision of language and accuracy facts. In these latter facts, we deal with a tight correspondence between the two different domains, along which we infer or predict *relations* in the physical domain on the epistemic basis of this very correspondence. In the case of discovery, we predict *objects* and moreover, we make ontological
commitments to them. However, this kind of prediction is not based on the estab-
lished or postulated correspondence between the abstract and the concrete, but
simply on the observation of a piece of the abstract, namely the mathematical for-
malism. What we do in processes 1a and 1b is a *total* equalization of the epistemic
natures of the mathematical objects, an equalization that goes beyond the mathemat-
ical theory in the physical domain. Granting solutions of the same equation the same
mathematical nature and status seems perfectly justified within pure mathematics,
but assigning physical referents to all of them only because some of them already
have such referents within a scientific theory is extending the epistemic equalization
beyond the mathematical domain. Such an action would be entirely justified only
by some *prevalence* of the abstract over the concrete, if this latter is indeed justi-
fied. The background unreasonableness stems from a sort of epistemic equivalence
between the abstract and the concrete, while the unreasonableness from the discov-
ery case stems from a non-justified higher status of the abstract over the concrete,
whereby the main characteristic responsible for this order resides in the ability of the
abstract to make ontological predictions for the concrete.

Such a specific unreasonableness can be expressed in our ‘how-is-it-that…given’ form as the question: “How is it that mathematics (or mathematical formalism) can successfully<sup>28</sup> suggest physical predictions for entities that successfully enhance a scientific theory, *given* that we have no justification or authority in granting the abstract a higher epistemic status than concrete<sup>29</sup> and *given* that the mathematical formalism was created independently of that theory?” (Call this the *no-justification unreasonableness*).<sup>30</sup>

However, this question does not entirely characterize unreasonableness in the
effectiveness of a predictive-heuristic role of mathematics. If we take a deeper look
into the process of equalizing the epistemic statuses of the mathematical objects, we
find that it is guided by—again—criteria of symmetry, aesthetics, unification and
simplicity. These together with freedom are the same criteria that have driven math-
ematical creation. Some of these criteria (at least symmetry and unification) stand
as a guide for further assignment through analogy between the abstract and the con-
crete in process 1c. Thus, the characterization of this unreasonableness falls within
Steiner’s syllogistic scheme—call it the *art unreasonableness*—and potentiates the
no-justification unreasonableness.

The fact that artistic features characterize each mathematical *creation*, nature as *we see* it, and the *intellectual* action of investigating nature with the help of mathematics does not present an argument against the unreasonableness, nor for a solution

<footnote n="28"><sup>28</sup> Here the success is the empirical confirmation of the discovered entity.</footnote>

<footnote n="29"><sup>29</sup> This assumption may be challenged by recent works in the field of theory of explanation that employ
mathematical explanations and abstract explanations; see, for instance, Pincock (2015).</footnote>

<footnote n="30"><sup>30</sup> Note that the lack of epistemic justification that characterizes this type of unreasonableness has nothing to do with the general concept of discovery, which assumes non-justification as almost a necessity. The circumstances of a discovery (walking, taking a bath, dreaming, guessing, etc.) need not bear any epistemic value for the discovery; otherwise, it would instead be an inference. That is, in the case of the mathematically driven discovery, the observation of the formalism is not such a circumstance, since it has the epistemic value of the *equal* status.</footnote>
to Wigner’s puzzle, nor even for a suggestion with respect to a certain methodology leading to a solution. Instead, this paradigmatic resemblance moves the focus to the role of the human mind in both formulating the problems of applicability of mathematics and the actual action of applying mathematics. Consequently, it raises metatheoretical questions about the adequate representation of this role and suggests that objective research of the puzzle has more than one target. We will come back to this point later.

We have identified thus far a no-justification unreasonableness and an art unreasonableness for the effectiveness of the role of mathematics in physical discoveries. The two unreasonablenesses are different and seemingly related since they serve the same role. But there is one more unreasonableness, this time more metaphysical, described in what follows:

Before using the predictive-heuristic role of mathematics, the scientific theory was already formulated in mathematical language, and inferences have been made through use of the standard mathematical modeling method. Thus, mathematics was used successfully in its descriptive and inferential roles. By analyzing the theory against the mathematical formalism used, that new role is activated especially when the theory is enhanced (process 1b'). By predicting physical entities, mathematics—via the scientist's action—can serve for the revision of a theory, but it also changes the *physical system* under investigation by addition of entities into such a system. The new revised theory applies to the new physical system by appealing to the same mathematics, which at any time may suggest new changes for the physical system, but also may itself change (through the addition of new concepts and deductions) as a result of the needs of the theory. As it appears, we deal with a *mutual exchange* between the source (mathematical domain) and the target (physical domain) of the investigation, mediated by the scientist or applied mathematician, in which mathematics seems to “know<fnref n="31" /> what is there” before the mediator knows. So we now have a new instance exhibiting the collaboration between the various roles of mathematics. What seems unreasonable here? Here is one answer:

“How is it that mathematics has the ability or power to change the investigated empirical context, *given* that it is used as an inert and non-causal tool or method of investigation?” or “How is it that mathematics—which proved effective—has the ability to become *more* effective for the same problems, *given* that the object of investigation should be fixed in order that the formulation not be self-contradictory?”<fnref n="32" /> Call this the *system-change unreasonableness* and observe that that mutual exchange leading to the system change is a feature of application which accommodates well with the concept of dynamic applicability of mathematics, as mentioned in a previous section.

With the distinctions expressed for the nature of the unreasonableness, the examples of mathematically driven discovery in physics given in this section strongly support the claim that Wigner’s puzzle does, as this paper maintains, have multiple targets. Unlike the other applicabilities of mathematics exhibited by the descriptive

<footnote n="31">³¹ Of course, the personification is metaphoric.</footnote>

<footnote n="32">³² The two questions are not equivalent, but share the same subject and are related to the same issue.</footnote>
power, provision of language, cross suitability and accuracy facts, which can be categorized as methodological, the applicability through the role of mathematics in physical discovery has an unavoidable ontological implication that potentiates the philosophical character of Wigner’s problem. Of course, there remain further debatable issues, such as whether the empirical hypothesis of the discovery case is sufficiently relevant for drawing conclusions on its unreasonableness and even on such a predictive-heuristic role of mathematics. Could the successful predictions be just lucky coincidences or isolated successes in a long series of failures? How can failure be tested? It is worth noting here that the Higgs boson had to wait 40 years before being empirically confirmed through measurements and simulations;<fnref n="33" /> during this period, at any time this prediction could have been declared a failure. These questions are legitimate as long as such kind of applicability is more isolated from the other applicabilities which are in a sense more continuous in scientific practice and the problem of their failure can be accounted for with different methodologies.

Let us summarize. We have identified six types of unreasonableness and formulated them: (1) the background (abstract-concrete) unreasonableness, (2) the no-justification unreasonableness, (3) the art unreasonableness, (4) the system-change unreasonableness, (5) the analogy-breaking unreasonableness and (6) the contradiction unreasonableness. To see more clearly how this differentiation influences the number of potential targets for research of Wigner’s puzzle, let us now analyze what exactly an *unreasonableness* is beyond these particular formulations and what would be a means for solving, dissolving or making such an unreasonableness reasonable.

# 5 How to Define ‘Unreasonableness’ and What It Means to Solve It

The “puzzling” unreasonableness can be defined only in epistemological terms, since the concept is directly derived from that of reason and acknowledgment through such reason. The following definition is an abstract schematic one open to improvement and its only role here is to point to the components of the unreasonableness that influence the targets of any attempt to solve Wigner’s puzzle. With this role, the character of this definition is both epistemic and metatheoretic.

**Definition** An unreasonableness is a conventional association of two elements I (the epistemic hypothesis) and F (the fact), (I ... F), for which the following holds true:

<figure><img src="image_10.png" /></figure>

<footnote n="33">³³ For a detailed history of the discovery of the Higgs boson as a study case for the problem of the relation between experimental data, simulation and theoretical mathematical modeling, see Morrison (2015, pp. 287–316).</footnote>
(a) An inferential relation (causally-explanatory, abstract-explanatory or abstract-deductive) or an identity relation<fnref n="34" /> linking *I* and *F* has not been established and

(b) Any attempt of establishing such a relation has failed due to contradictory elements of *I* and *F* that cannot stand simultaneously.

The epistemic hypothesis *I* may contain empirical facts and epistemic facts such as justified beliefs, accepted theories and/or well-established local or universal theories. The fact *F* may also contain both empirical and epistemic facts. Epistemic facts include beliefs and/or conjectures. In brief, an unreasonableness would be a “contra-diction” association between an epistemic hypothesis and a fact, where the contra-diction has both a logical and an epistemic character and is grounded on objective intellectual practice.

In the case of Wigner’s puzzle, for each of our formulations of a specific unreasonableness, *I* is (or includes) what is expressed after the “given,” and *F* is what is expressed after the “how is it that.” The empirical hypothesis is contained in *F*. The formulation “How is it that *F*, given *I*” is clearly conventional in the absence of an inferential relation linking *I* and *F*. The association (*I*, *F*), the unreasonableness per our definition above, creates the psychological effect of surprise not only through an analysis of the mere content of *I* and *F*, but also based on the prior intellectual practice in dealing with the problem. Time dependence via historic reference is specifically included in the abstract definition above and is exhibited by both conditions (a) and (b): the past attempts to establish an inferential relation within the history of this intellectual practice and a characterization of these attempts are not only to be referred to, but also constitutive for the definition. As such, an association of (*I*, *F*) type can be qualified as unreasonable only at a given moment, which is consistent with a common sense recognition of the unreasonableness. By this definition, a fact cannot be unreasonable by itself, but only in relation to an epistemic hypothesis, which is the entire universe of objective beliefs having as a subject that fact. In particular, an empirical fact per se cannot be unreasonable; what is unreasonable is the recorded failure of explaining that fact within the sphere of knowledge we uphold at a particular time.

Let us also observe that the strengthening of condition (a) by condition (b) is necessary for making room for the logical character of the unreasonableness, which cannot be left exclusively in the epistemic field. Without (b), any conjecture would also fall within the definition. Note that an unsolved conjecture as a *fact* holds nothing unreasonable, but on the contrary, its epistemic hypothesis is based on inductive knowledge, which remains reasonable even in the face of an assumption that the conjecture proves latter to be false. The unsolved conjecture expresses insufficient analytical effort to deduce the conclusion (fact) from the hypothesis or to deny the former, while the unreasonableness expresses sufficient intellectual effort to declare an association of hypothesis with fact as unreasonable.

<footnote n="34">³⁴ Such an identity relation is actually a criterion of identity; that is, without such a relation, one of the *relata* would not be what it is. For instance, the relation of set membership is an identity relation between the set and any of its elements, since without it, that set would not be what it is.</footnote>
This distinction itself suggests that any further focus on ‘why’ or ‘how’ ques-
tions—for denying condition (a)—does not have a good prospect for being a worth-
while intellectual endeavor in hopes of solving an unreasonableness, given that the
historical basis of results obtained during intellectual practice dealing with it is large
enough. So, how can we turn something unreasonable into something reasonable if
it has been declared that no inferential relation linking *I* and *F* is possible?

Logically, there are three ways to solve an unreasonableness: either by denying *I*, or by denying *F*, or by changing (revising) either *I* or *F* or both so as to come to new pairs (*I*', *F*') where condition (b) can be negated. The first two ways are termed by dissolving or establishing the ingenuineness of the unreasonableness. They seem to be easier (if possible) than the third way, which assumes focusing on all subjects *I*, *F*, the changed *I*' and *F*', and the inferential relation between them. Note that such schemes do not differ much from the usual basic scientific methodology dealing with an unsolved scientific problem. What is different is that a certain metatheoretic analysis has a primary role here: while in scientific practice only epistemic hypotheses can be changed<fnref n="35" /> and the target is fixed, solving an unreasonableness may also assume changing the target, such as the epistemic facts that *F* contains. It is important to note that the main part of such changes of *I* and *F* can be made through *formulation* only. Both the epistemic hypothesis *I* and the fact *F* are formulated in a language and changing them actually means changing the way they are described, expressed and the conceptual framework under which such description runs. This linguistic character of the unreasonableness points to the importance of the *semantic analysis* within any conceptual framework chosen for the solving of the unreasonableness, thus receiving the highest priority (Fig. 1).

What would it mean for Wigner’s puzzle to be solved in light of the definition
and scheme above? Recall this paper’s initial observation that any past research
toward this problem treated it unitarily with respect to target and kind of the
unreasonableness, by focusing on giving a solution for *one* or more ‘why’ ques-
tions as related to *one* unreasonableness. For one kind of unreasonableness, argu-
ments for dissolving the puzzle as ingenuine have been given (by denying *I* or *F*),
as well as solutions claimed for establishing an inferential relation linking *I* and *F*
that would make the puzzle reasonable rather than unreasonable. The arguments
for dissolving the puzzle were usually on the side of nominalists,<fnref n="36" /> associated
with their classical views on the philosophy of mathematics and its application,
but also on other sides (for instance, those sustaining the inseparability thesis<fnref n="37" />).
Solutions for the reasonableness that have been given vary in nature and argu-
ments, but keep the same unitary approach mentioned before.<fnref n="38" /> None of them
have been considered as clear, non-controversial, or unobjectionable in the litera-
ture, except perhaps the Fregean theoretical account of the semantic applicability

<footnote n="35">³⁵ From conceptual and theoretical frameworks, to local theories and even to well-established universal theories (the case of revolutions).</footnote>

<footnote n="36">³⁶ See Field (1980), Hellman (1989), Balaguer (1996), and Azzouni (2004, 2006).</footnote>

<footnote n="37">³⁷ See Boniolo and Budinich (2005).</footnote>

<footnote n="38">³⁸ See footnotes 2 and 3.</footnote>
<figure><img src="image_11.png" /><figcaption>Fig. 1 Solving an unreasonable by changing its components</figcaption></figure>

of mathematics; but this latter account only has one kind of applicability as its
target.

Accepting Wigner's puzzle as unreasonable and focusing on changing its *I* and/or *F*, the methodology employed would be expected to span several disciplines, and theoretical philosophy is certainly included since it is concerned with changing objective beliefs acquired through both scientific and philosophical knowledge. This maintains the philosophical character of the problem of applicability of mathematics.

For instance, working on solving the *background unreasonable* through
changing *I* would involve vanishing or blurring the extreme difference between
the natures of the abstract and the concrete, at least for the mathematical concepts
as the abstract. This change would also apply to the *no-justification unreasonable*,
which is specific to scientific methodology and not to concepts.

For the *art unreasonable*, one should for instance investigate (in order to
change *I*) whether the rationality and rigor specific to the scientific scenario is
actually unrelated to the artistic features (that drive the mathematical creation).
Perhaps such criteria are not arbitrary after all, and the biological laws would
explain such a behavior of the human mind, the same mind that formulates the
problems of science, solves them, and sees the investigated reality through all
perceptual means, including the mind. In literary words, assuming that the book
of nature is *written* in the language of mathematics, as Galileo said, is the genre
of this book fiction or non-fiction, or perhaps a third genre of which we are not
aware yet? Or, expressing the unreasonable literarily, it is not only that we
found that Galileo was right, but now we are told that we can recreate the unseen
content of this book by ourselves as mathematicians, from our table, merely with
our minds and not from observing out there in nature! Neurophysiology, psychol-
ogy and anthropocentrism would have a tough interdisciplinary project here.

For the *system-change unreasonable*, could a more dynamic account of
mathematics and its application, change *I* enough for us to be able to infer some-
thing about a certain kind of applicability, which might prove perpetual?

For the *analogy-breaking unreasonable*, where the empirical component
of fact *F* is the most striking, are mathematical models false representations?
If we take them to be representations of *formulated* empirical contexts rather
than mere reality, they are not false. If this is not the case, can we find a way
to accommodate structuralism with the problems of truth raised by conventional structures and structure-preserving, for the case of applied mathematics?

Finally, for the *contradiction unreasonableness*, are the two seemingly different empirical contexts not connected in any way? Is it possible that again the mind and its mathematical behavior are responsible for a connection, considering the empirical context as something formulated and not merely physical?

For all types of unreasonableness, fact *F* can be changed by adapting the conceptual framework we assign for *applicability, role* (of pure mathematics), and *effective-ness* or *success* (of applied mathematics). There are also specific changes available:

For instance, fact *F* of the *no-justification unreasonableness* can be changed (or denied) by claiming that it is not the mathematical formalism suggesting physical referents, but rather the *intellectual need* of completing a structure for preserving coherence and prior structures. Such completion does not employ any epistemic prevalence of abstract over concrete, but rather an equalization of these two. Such need could also be subject to a biological treatment, where the energy-saving feature of the brain might be employed. Further, the fact that this completion has proven effective and been confirmed remains to be dealt with.

For the *system-change unreasonableness*, the fact that mathematics has proved more effective than previously assumed—and exceeds its assumed roles—may be reformulated in terms of a dynamic, universal, and evolutionary applicability so as not to have a fixed mathematics applied to a fixed target.

For the *contradiction unreasonableness*, the fact can be reformulated so as to weaken effectiveness: how is it that a mathematical concept created for representing a certain empirical object or context appears in a mathematical theory *chosen* to model a different empirical context, a theory that proved effective? This formulation points to the non-uniqueness of mathematical models for a given empirical context, though with different degrees of effectiveness.

As for the *background unreasonableness, art unreasonableness, and analogy-breaking unreasonableness*, the question may be asked: is the accuracy of the empirical results something that we should qualify as true, approximately true, or false? If false is a valid option, the attribute would be turned into *inaccurate*. It should be investigated whether the last two options, approximately true and false, could accommodate with the asymptotic reasoning employed in the empirical hypothesis, for instance, instead of reasoning through analogy.

By presenting the above instances of changing *I* and *F*, we do not draw any emergent directions for further research, nor do we suggest possible solutions for Wigner's problem. Such particularizations are intended only as illustrative examples of how to approach an unreasonableness methodologically through changing its components *I* and *F*. Nevertheless, researches and advances have already been pursued in many of the directions depicted, some of them independent of Wigner's puzzle and their results await further development and employment in newer accounts on this problem.<fnref n="39" />

<figure><img src="image_12.png" /></figure>

<footnote n="39">³⁹ It is not the place here to make a review of these (partial) results, but they definitely deserve a further paper. We would just mention as significant the research on perceptual mathematics (Teissier 2005; Ye 2009; Mujumdar and Singh 2016), the constructivist views of the act of mathematical modeling and the constructivist representations of reality (Hennig 2010), the dissolution of abstract structures and the</footnote>
# 6 Targets of Wigner's Problem and Conclusions

The aim of this paper was to show that the problem of applicability of mathematics stemming from Wigner's puzzle cannot be approached and treated in the unitary—almost simplistic—way it has been thus far, from formulation to drawing the target and to designing research toward a solution. Regarding target, we argued that instead of a single 'why', the problem involves several 'whys' and 'hows'. Although this goal was technically accomplished after laying down the main semantic distinctions for the key terms, the metatheoretic analysis that included the types of unreasonable-ness involved in the problem revealed a variety of potential methodologies ostensi-bly able to deal with it.

Wigner's empirical hypothesis and the distinct roles mathematics does play in standard and non-standard applications were shown to exhibit distinct types of unreasonableness. Each distinct type of unreasonableness generates a different target of research, since changing the epistemic hypothesis also involves changing beliefs and this means creating a new system of beliefs whose coherence and adequacy need investigation. Moreover, once *I* and/or *F* are changed, we observe that linking *I'* and *F'* through a certain methodology is itself a target, even if it is partially addressed by the change. Thus, Wigner's puzzle actually has a double general target: not that of explaining the success of mathematics, but the 'unreasonable' success of mathemat-ics, which means solving an unreasonableness and then accounting for the success with a methodology specific to *that kind* of unreasonableness. Therefore, each type of unreasonableness yields two targets: the strategy of solving the unreasonableness and the methodology of linking the new *I* and *F*.

Staying with six different types of unreasonableness, only two semantic distinc-tions for the concept of applicability (say, internal and external), two for the concept of success (empirical and methodological), and no other distinctions for the rest of the key terms, a rough numerical estimation for the number of general targets of a complete research would be as follows:

6 (types of unreasonableness)×2 (particular unreasonableness strategy and further methodology)×2 (semantical distinctions for ‘applicability’)×2 (seman-tical distinctions for ‘success’) = 48 general targets (or targets and subtargets, if preferred).

Even assuming one type of unreasonableness (whichever it would be)—if an argument is posed for denying these distinctions—and accepting that a methodo-logical unification is possible, a multitude of targets still remain. Some of these tar-gets will eventually be seen as subtargets of one or more main targets, with the goal

<footnote>Footnote 39 (continued)</footnote>

<footnote>‘blurring problem’ in ontic structuralism (French and Ladyman 2003, 2011; Cao 2003), the works of Batterman (2002, 2009) on the asymptotic modeling and non-idealization modeling, and the dynamical account of the inferential conception of applied mathematics (Räz and Sauer 2015). They are just path-finders toward a non-classical (in sense of dealing with the unreasonable per our (*I*, *F*) scheme) account of Wigner’s problem from which solutions can be developed.</footnote>
of making the unreasonableness reasonable (including the solution of dissolving the problem).

Both the semantic analysis and the distinction between the various types of unreasonableness reveal aspects of the empirical hypothesis as well as of potential methodologies which are not specific to philosophy, but to scientific disciplines clearly specialized, such as psychology, neurosciences, evolutionary biology and history of science.

Accepting this interdisciplinary character of the problem apparently increases its complexity, as does accepting the multitudes of targets. This should not pose a reason to quit research, nor for a criterion for absolute rejection of the solutions that have been proposed to the puzzle. For the former, the increase in complexity opens a metatheoretical inquiry about how some conceptual frameworks from philosophy of science and of mathematics related to Wigner's puzzle may be unified in order to reduce or collapse targets. For the latter, the same inquiry on unification may be pursued; we may find components and arguments completing each other across the solutions, or waiving objections specific to another solution.

To state Wigner's problem economically (not enumerating all its targets) in the fewest possible 'whys' and 'hows', a crisp formulation would be: "Why is mathematics applicable in sciences and physical reality, *how* do we rationally justify the use of mathematical models in the investigation of physical phenomena and *how* do we explain their high rate of success, given the philosophical concerns about ontology, epistemology and the different natures of those entities involved in this scientific practice?"

Empirically and historically speaking, Wigner's problem has been left to the exclusive expertise of philosophy with no conclusive results. This situation has continued long enough to consider a shift, and the only possible shift is to science. This interdisciplinary perspective complies ab initio with our formal account of unreasonableness, where conceptual rearrangement and reformulation are represented in the scheme of solving an unreasonableness. Therefore, we claim that no 'why' for the applicability of mathematics should be further treated before deciding metatheoretically on the adequate conceptual and theoretical framework and methodology to deal with those questions.

Wigner's problem can only be resolved or degenerated (toward dissolving, simplifying, or trivializing<sup>40</sup>). Assume a third variant in which the problem is genuine, non-trivial, and non-resolvable. While we search for an acceptable solution, mathematics has long been applied successfully in science and everyday life. This leads to continuous credibility of the use of mathematical models in science based only on its previous record and acknowledging the success of applied mathematics in the physical universe as an observable regularity, perhaps formulated as a law of nature postulating the applicability of mathematics. On these grounds, applied mathematics would somehow work like science, through inductive means and/or through the laws of nature. This feature would diminish the epistemic "purity" of the truths of applied

<figure><img src="image_13.png" /></figure>

<footnote n="40"><sup>40</sup> Dissolving means that there is not a genuine problem with a well determined sense, and triviality means that the problem is trivially resolvable, having the solution visible from a subtle perspective.</footnote>
mathematics when composed<fnref n="41" /> with the truth of applicability, while the necessary truths of mathematics have already lost in purity with the application. But this would undermine the main epistemic reason for which we apply mathematics, since science itself cannot offer that kind of pure necessity. This suggests a new metatheoretical target for the problem of applicability of mathematics: Is Wigner's puzzle something that we *must* solve since we did not pose a similar problem for explaining the success of science<fnref n="42" /> or, more generally, the success of human rationality?

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