We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content to key (...) intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as explanatory generality is concerned. (shrink)

Mathematics appears to play a genuine explanatory role in science. But how do mathematical explanations work? Recently, a counterfactual approach to mathematical explanation has been suggested. I argue that such a view fails to differentiate the explanatory uses of mathematics within science from the non-explanatory uses. I go on to offer a solution to this problem by combining elements of the counterfactual theory of explanation with elements of a unification theory of explanation. The result is a theory according to which (...) a counterfactual is explanatory when it is an instance of a generalized counterfactual scheme. (shrink)

Some scientific explanations appear to turn on pure mathematical claims. The enhanced indispensability argument appeals to these ‘mathematical explanations’ in support of mathematical platonism. I argue that the success of this argument rests on the claim that mathematical explanations locate pure mathematical facts on which their physical explananda depend, and that any account of mathematical explanation that supports this claim fails to provide an adequate understanding of mathematical explanation.

Several philosophers have suggested that some scientific explanations work not by virtue of describing aspects of the world’s causal history and relations, but rather by citing mathematical facts. This paper investigates what mathematical facts could be in order for them to figure in such “distinctively mathematical” scientific explanations. For “distinctively mathematical explanations” to be explanations by constraint, mathematical language cannot operate in science as representationalism or platonism describes. It can operate as Aristotelian realism describes. That is because Aristotelian realism enables (...) mathematics to apply to the physical world without a morphism’s mediation. (shrink)

Easy-road mathematical fictionalists grant for the sake of argument that quantification over mathematical entities is indispensable to some of our best scientific theories and explanations. Even so they maintain we can accept those theories and explanations, without believing their mathematical components, provided we believe the concrete world is intrinsically as it needs to be for those components to be true. Those I refer to as “mathematical surrealists” by contrast appeal to facts about the intrinsic character of the concrete world, not (...) to explain why our best mathematically imbued scientific theories and explanations are acceptable in spite of having false components, but in order to replace those theories and explanations with parasitic, nominalistically acceptable alternatives. I argue that easy-road fictionalism is viable only if mathematical surrealism is and that the latter constitutes a superior nominalist strategy. Two advantages of mathematical surrealism are that it neither begs the question concerning the explanatory role of mathematics in science nor requires rejecting the cogency of inference to the best explanation. (shrink)

Pure mathematics can play an indispensable role explaining empirical phenomena if recent accounts of insect evolution are correct. In particular, the prime life cycles of cicadas and the geometric structure of honeycombs are taken to undergird an inference to the best explanation about mathematical entities. Neither example supports this inference or the mathematical realism it is intended to establish. Both incorrectly assume that facts about mathematical optimality drove selection for the respective traits and explain why they exist. We show how (...) this problem can be avoided, identify limitations of explanatory indispensability arguments, and attempt to clarify the nature of mathematical explanation. (shrink)

Aesthetics in philosophy of mathematics is too narrowly construed. Beauty is not the only feature in mathematics that is arguably aesthetic. While not the highest aesthetic value, being interesting is a sine qua non for publishability. Of the many ways to be interesting, being explanatory has recently been discussed. The motivational power of what is interesting is important for both directing research and stimulating education. The scientific satisfaction of curiosity and the artistic desire for beautiful results are complementary but both (...) aesthetic. (shrink)

Are there genuine mathematical explanations of physical phenomena, and if so, how can mathematical theories, which are typically thought to concern abstract mathematical objects, explain contingent empirical matters? The answer, I argue, is in seeing an important range of mathematical explanations as structural explanations, where structural explanations explain a phenomenon by showing it to have been an inevitable consequence of the structural features instantiated in the physical system under consideration. Such explanations are best cast as deductive arguments which, by virtue (...) of their form, establish that, given the mathematical structure instantiated in the physical system under consideration, the explanandum had to occur. Against the claims of platonists such as Alan Baker and Mark Colyvan, I argue that formulating mathematical explanations as structural explanations in this way shows that we can accept that mathematics can play an indispensable explanatory role in empirical science without committing to the existence of any abstract mathematical objects. (shrink)

There is growing debate about what is the correct methodology for research in the ontology of artworks. In the first part of this essay, I introduce my view: I argue that semantic descriptivism is a semantic approach that has an impact on meta-ontological views and can be linked with a hermeneutic fictionalist proposal on the meta-ontology of artworks such as works of music. In the second part, I offer a synthetic presentation of the four main positive meta-ontological views that have (...) been defended in philosophical literature about artworks and of some criticisms that can be lodged against them: Amie Thomasson’s global descriptivism, Andrew Kania’s local descriptivism, Julian Dodd’s folk-theoretic modesty, and David Davies’ rational accountability view. In the conclusion, I show the advantages of my view. (shrink)

Within the context of the Quine–Putnam indispensability argument, one discussion about the status of mathematics is concerned with the ‘Enhanced Indispensability Argument’, which makes explicit in what way mathematics is supposed to be indispensable in science, namely explanatory. If there are genuine mathematical explanations of empirical phenomena, an argument for mathematical platonism could be extracted by using inference to the best explanation. The best explanation of the primeness of the life cycles of Periodical Cicadas is genuinely mathematical, according to Baker (...) :223–238, 2005; Br J Philos Sci 60:611–633, 2009). Furthermore, the result is then also used to strengthen the platonist position :779–793, 2017a). We pick up the circularity problem brought up by Leng Mathematical reasoning, heuristics and the development of mathematics, King’s College Publications, London, pp 167–189, 2005) and Bangu :13–20, 2008). We will argue that Baker’s attempt to solve this problem fails, if Hume’s Principle is analytic. We will also provide the opponent of the Enhanced Indispensability Argument with the so-called ‘interpretability strategy’, which can be used to come up with alternative explanations in case Hume’s Principle is non-analytic. (shrink)

I show how mathematical platonism combined with belief in the God of classical theism can respond to Field's epistemological objection. I defend an account of divine mathematical knowledge by showing that it falls out of an independently motivated general account of divine knowledge. I use this to explain the accuracy of God's mathematical beliefs, which in turn explains the accuracy of our own. My arguments provide good news for theistic platonists, while also shedding new light on Field's influential objection.

In this paper I present a case study of mathematical explanation in science that is new to the philosophical literature, and that arises in the context of estimating the energetic costs of running in bipedal animals. I refer to this as the Bipedal Gait Costs explanation. I argue that it is important for examples of applied mathematics to be driven not just by philosophical and mathematical concerns but also by scientific concerns. After a detailed presentation of the BGC case study, (...) I discuss ways in which it is different from standard examples of MES. One distinctive aspect of BGC is that substantive mathematics occurs at two different levels. I argue for a distinction to be drawn between the mathematical modeling that occurs at one level, which is non-explanatory, and the mathematical theorem which is applied to results gleaned from these models, which is explanatory. I conclude by drawing some connections with broader indispensability-based arguments for platonism in the philosophy of mathematics. (shrink)

Science and mathematics continually change in their tools, methods, and concepts. Many of these changes are not just modifications but progress---steps to be admired. But what constitutes progress? This dissertation addresses one central source of intellectual advancement in both disciplines: reformulating a problem-solving plan into a new, logically compatible one. For short, I call these cases of compatible problem-solving plans "reformulations." Two aspects of reformulations are puzzling. First, reformulating is often unnecessary. Given that we could already solve a problem using (...) an older formulation, what do we gain by reformulating? Second, some reformulations are genuinely trivial or insignificant. Merely replacing one symbol with another does not lead to intellectual progress. What distinguishes significant reformulations from trivial ones? According to what I call "conceptualism" (or "conceptual empiricism"), reformulations are intellectually significant when they provide a different plan for solving problems. Significant reformulations provide inferentially different routes to the same solution. In contrast, trivial reformulations provide exactly the same problem-solving plans, and hence they do not change our understanding. This answers the second question about what distinguishes trivial from significant reformulations. However, the first question remains: what makes a new way of solving an old problem valuable? Here, a bevy of practical considerations come to mind: one formulation might be faster, less complicated, or use more familiar concepts. According to "instrumentalism," these practical benefits are all there is to reformulating. Some reformulations are simply more instrumentally valuable for meeting the aims of science than others. At another extreme, "fundamentalism" contends that a reformulation is valuable when it provides a more fundamental description of reality. According to this view, some reformulations directly contribute to the metaphysical aim of carving reality at its joints. Conceptualism develops a middle ground between instrumentalism and fundamentalism, preserving their benefits without their costs. I argue that the epistemic value of significant reformulations does not reduce to either practical or metaphysical value. Reformulations are valuable because they are a constitutive part of problem-solving. Both science and mathematics aim at solving all possible problems within their respective domains. Meeting this aim requires being able to plan for any possible problem-solving context, and this requires reformulating. By reformulating, we clarify what we need to know to solve problems. Still, one might wonder whether the value of reformulations requires underlying differences in explanatory power. According to "explanationism," a reformulation is valuable only when it provides a better explanation. Explanationism stands as a rival middle ground position to my own. However, it faces numerous counterexamples. In many cases, two reformulations provide the same explanation while nonetheless providing different ways of understanding a phenomenon. Hence, reformulating can be valuable even when neither formulation is more explanatory. Methodologically, I draw on a variety of case studies to support my account of reformulation. These range from classical mechanics to quantum chemistry, along with examples from mathematics. Symmetry arguments provide a paradigmatic example: the mathematics of symmetry groups radically recasts quantum mechanics and quantum chemistry. Nevertheless, elementary approaches exist that eschew this additional mathematical apparatus, solving problems in a more tedious but less mathematically-demanding manner. Further examples include reformulations of quantum field theory, Arabic vs. Roman numerals, and Fermat's little theorem in number theory. In each case, my account identifies how reformulations change and improve our understanding of science and mathematics. (shrink)

A perspective in the philosophy of mathematics is developed from a consideration of the strengths and limitations of both logicism and platonism, with an early focus on Frege’s work. Importantly, although many set-theoretic structures may be developed each of which offers limited isomorphism with the system of natural numbers, no one of them may be identified with it. Furthermore, the timeless, ever present nature of mathematical concepts and results itself offers direct access, in the face of a platonist account which (...) generates a supposed problem of access. Crucially too, pure mathematics has its own distinctive method of confirming or validating results - mathematical proof - which supplies a higher level of confidence and objectivity than that available elsewhere. The dichotomy of invention and discovery is too jejune a framework for analysing creative mathematical activity. The Gödelian platonist perspective is evaluated and queried through scrutiny of the part played by mathematical resources and constraints in relation to human activity. It appears that there can be non-causal mathematical explanations and mathematical constraint on purely natural processes. Valuable implications of Quine’s naturalism are explored, but one must be cautious of his thesis of confirmational holism. The distinction between algebraic and non-algebraic mathematical theories usefully contributes to our understanding of the internally differentiated nature of the subject. (shrink)

Mathematics clearly plays an important role in scientific explanation. Debate continues, however, over the kind of role that mathematics plays. I argue that if pure mathematical explananda and physical explananda are unified under a common explanation within science, then we have good reason to believe that mathematics is explanatory in its own right. The argument motivates the search for a new kind of scientific case study, a case in which pure mathematical facts and physical facts are explanatorily unified. I argue (...) that it is possible for there to be such cases, and provide some toy examples to demonstrate this. I then identify a potential source of scientific case studies as a guide for future work. (shrink)

Recent debates about mathematical ontology are guided by the view that Platonism's prospects depend on mathematics' explanatory role in science. If mathematics plays an explanatory role, and in the right kind of way, this carries ontological commitment to mathematical objects. Conversely, the assumption goes, if mathematics merely plays a representational role then our world-oriented uses of mathematics fail to commit us to mathematical objects. I argue that it is a mistake to think that mathematical representation is necessarily ontologically innocent and (...) that there is an argument from mathematics' representational capacity to Platonism. Given that it is common ground between the Platonist and nominalist that mathematics plays a representational role in science, this representationalist argument is to be preferred over the explanatory, or enhanced, indispensability argument. (shrink)

Defenders of the enhanced indispensability argument argue that the most effective route to platonism is via the explanatory role of mathematical posits in science. Various compelling cases of mathematical explanation in science have been proposed, but a satisfactory general philosophical account of such explanations is lacking. In this paper, I lay out the framework for such an account based on the notion of “the mathematical stance.” This is developed by analogy with Dennett’s well-known concept of “the intentional stance.” Roughly, adopting (...) the mathematical stance towards a particular physical phenomenon involves treating it as an abstract mathematical structure for the purposes of prediction and explanation. Interestingly, Dennett himself frequently draws analogies between his intentional stance towards beliefs and desires and scientists’ stance towards centers of gravity. I explore the theoretical role played by centers of gravity within science and discuss how an indispensabilist platonist ought to categorize the ontological status of this type of posit. I conclude with some thoughts on how an approach based on the mathematical stance might be developed into a more general philosophical account of the application of mathematics in science. (shrink)

This paper provides a sorely-needed evaluation of the view that mathematical explanations in science explain by unifying. Illustrating with some novel examples, I argue that the view is misguided. For believers in mathematical explanations in science, my discussion rules out one way of spelling out how they work, bringing us one step closer to the right way. For non-believers, it contributes to a divide-and-conquer strategy for showing that there are no such explanations in science. My discussion also undermines the appeal (...) to unifying power in support of the enhanced indispensability argument. (shrink)

The Königsberg bridge problem has played a central role in recent philosophical discussions of mathematical explanation. In this paper I look at it from a novel perspective, which is independent of explanatory concerns. Instead of restricting attention to the solved Königsberg bridge problem, I consider Euler’s construction of a solution method for the problem and discuss two later integrations of Euler’s approach into a more structured methodology, arisen in operations research and genetics respectively. By examining Euler’s work and its later (...) developments, I achieve two main goals. First, I offer an analysis of the role played by mathematics as a problem-solving instrument within scientific enquiry. Second, I shed light on the broader significance of well known contributions to the debate on mathematical explanation. I suggest that these contributions, which are tied to a localised explanatory context, achieve a greater relevance and attain a sharper formulation when they are referred to scientific enquiry at large, as opposed to its possible explanatory outcomes alone. (shrink)

Several philosophers have argued that to capture the generality of certain scientific explanations, we must count mathematical facts among their explanantia. I argue that we can better understand these explanations by adopting a more nuanced stance toward mathematical representations, recognizing the role of mathematical representation schemata in representing highly abstract features of physical systems. It is by picking out these abstract but nonmathematical features that explanations appealing to mathematics achieve a high degree of generality. The result is a rich conception (...) of the role of mathematics in scientific explanations that does not require us to treat mathematical facts as explanantia. (shrink)

Indispensability arguments occupy a prominent role in discussions of mathematical realism. While different versions of these arguments are discussed in the literature, their general structure remains the same. These arguments contend that insofar as reference to mathematical objects is indispensable to science, we are committed to the existence of these ‘objects’. Unsurprisingly, much of the debate concerning indispensability arguments focuses on the crucial contention that mathematical objects are indispensable to science. For these arguments to provide support for mathematical realism, what (...) we require are some compelling examples of mathematics playing an indispensable role in science. Towards this end, Alan Baker has supplied an example in which mathematics appears to be indispensable to a scientific explanation, what has come to be called the “Cicada MES”. This example has been the subject of much criticism, most notably that it depends on an unsupported empirical supposition. Recently, Baker proposed a number of revisions to the Cicada MES that aim to address some of these criticisms. In this paper, I argue that while Baker’s revisions go some way towards mitigating prominent criticisms of the Cicada MES, the underlying problem with the explanation remains. I argue that his revisions actually go in the wrong direction mathematically and empirically. In contrast, I propose a new Cicada MES. This new version accomplishes Baker’s goals and responds to the central objection to the explanation. It invokes a mathematical principle that plays a unifying role in the explanation, which makes providing adequate nominalist surrogates more difficult. And, more importantly, it accommodates an empirically grounded explanation of the phenomenon to be explained. Consequently, this new Cicada MES provides what proponents of indispensability arguments require, a compelling example of mathematics playing an indispensable role in science. (shrink)

This paper discusses Baker’s Enhanced Indispensability Argument for mathematical realism on the basis of the indispensable role mathematics plays in scientific explanations of physical facts, along with various responses to it. I argue that there is an analogue of causal explanation for mathematics which, of several basic types of explanation, holds the most promise for use in the EIA. I consider a plausible case where mathematics plays an explanatory role in this sense, but argue that such use still does not (...) support realism about mathematical objects. (shrink)