This paper aims to reassess the philosophical puzzle of the “applicability of mathematics to physical sciences” as a misunderstood disciplinary interplay. If the border isolating mathematics from the empirical world is based on appropriate criteria, how does one explain the fruitfulness of its systematic crossings in recent centuries? An analysis of the evolution of the criteria used to separate mathematics from experimental sciences will shed some light on this question. In this respect, we will highlight the historical influence of three (...) major disciplinary paradigms. According to the Aristotelian classification of the sciences, the separation of mathematics from physics is based on their respective objects of study. The Baconian system distinguishes these sciences by the type of knowledge involved in each field. Finally, the Whewellian disciplinary layout categorises these disciplines by their respective methods. In this paper, we argue that the cascading effect of such disciplinary categorisations—based successively on ontological, epistemological, and heuristic criteria—resulted in a profound redefinition of the border between mathematics and physics, with the consequence of obscuring the foundations of the interplay between these fields. Our approach intends to put forward such a mechanism as a constitutive piece of the puzzle of the applicability of mathematics. (shrink)

This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of (...) the physical world. The no-miracles argument is the primary motivation for scientific realism. It is a presupposition of this argument that unobservable entities are explanatory only when they determine the empirical phenomena they explain. I argue that mathematical entities should also be seen as explanatory only when they determine the empirical facts they explain, namely, the modal structure of the physical world. Thus, scientific realism commits us to a metaphysical determination relation between mathematics and physical modality that has not been previously recognized. The requirement to account for the metaphysical dependence of modal physical structure on mathematics limits the class of acceptable solutions to the applicability problem that are available to the scientific realist. (shrink)

According to Steiner (1998), in contemporary physics new important discoveries are often obtained by means of strategies which rely on purely formal mathematical considerations. In such discoveries, mathematics seems to have a peculiar and controversial role, which apparently cannot be accounted for by means of standard methodological criteria. M. Gell-Mann and Y. Ne׳eman׳s prediction of the Ω− particle is usually considered a typical example of application of this kind of strategy. According to Bangu (2008), this prediction is apparently based on (...) the employment of a highly controversial principle—what he calls the “reification principle”. Bangu himself takes this principle to be methodologically unjustifiable, but still indispensable to make the prediction logically sound. In the present paper I will offer a new reconstruction of the reasoning that led to this prediction. By means of this reconstruction, I will show that we do not need to postulate any “reificatory” role of mathematics in contemporary physics and I will contextually clarify the representative and heuristic role of mathematics in science. (shrink)

In the last decades two different and apparently unrelated lines of research have increasingly connected mathematics and evolutionism. Indeed, on the one hand different attempts to formalize darwinism have been made, while, on the other hand, different attempts to naturalize logic and mathematics have been put forward. Those researches may appear either to be completely distinct or at least in some way convergent. They may in fact both be seen as supporting a naturalistic stance. Evolutionism is indeed crucial for a (...) naturalistic perspective, and formalizing it seems to be a way to strengthen its scientificity. The paper shows that, on the contrary, those directions of research may be seen as conflicting, since the conception of knowledge on which they rest may be undermined by the consequences of accepting an evolutionary perspective. (shrink)

We outline a framework for analyzing episodes from the history of science in which the application of mathematics plays a constitutive role in the conceptual development of empirical sciences. Our starting point is the inferential conception of the application of mathematics, recently advanced by Bueno and Colyvan. We identify and discuss some systematic problems of this approach. We propose refinements of the inferential conception based on theoretical considerations and on the basis of a historical case study. We demonstrate the usefulness (...) of the refined, dynamical inferential conception using the well-researched example of the genesis of general relativity. Specifically, we look at the collaboration of the physicist Einstein and the mathematician Grossmann in the years 1912--1913, which resulted in the jointly published ``Outline of a Generalized Theory of Relativity and a Theory of Gravitation,'' a precursor theory of the final theory of general relativity. In this episode, an independently developed mathematical theory, the theory of differential invariants and the absolute differential calculus, was applied in the process of physical theorizing aiming at finding a relativistic theory of gravitation. We argue that the dynamical inferential conception not only provides a natural framework to describe and analyze this episode, but it also generates new questions and insights. We comment on the mathematical tradition on which Grossmann drew, and on his own contributions to mathematical theorizing. We argue that the dynamical inferential conception allows us to identify both the role of heuristics and of mathematical resources as well as the systematic role of problems and mistakes in the reconstruction of episodes of conceptual innovation and theory change. (shrink)

This paper distinguishes between 3 meanings of reversal, all of which are mathematically equivalent in classical mechanics: velocity reversal, retrodiction, and time reversal. It then concludes that in order to have well defined velocities a primitive arrow of time must be included in every time slice. The paper briefly mentions that this arrow cannot come from the Second Law of thermodynamics, but this point is developed in more details elsewhere.

In Grundgesetze, Vol. II, §91, Frege argues that ‘it is applicability alone which elevates arithmetic from a game to the rank of a science’. Many view this as an in nuce statement of the indispensability argument later championed by Quine. Garavaso has questioned this attribution. I argue that even though Frege's applicability argument is not a version of ia, it facilitates acceptance of suitable formulations of ia. The prospects for making the empiricist ia compatible with a rationalist Fregean framework appear (...) thus much less dim than expected. Nonetheless, those arguing for such compatibility eventually face an hardly surmountable dilemma. (shrink)

This thesis offers a naturalist revision of Alain Badiou’s philosophy. This goal is pursued through an encounter of Badiou’s mathematical ontology and theory of truth with contemporary trends in philosophy of mathematics and philosophy of science. I take issue with Badiou’s inability to elucidate the link between the empirical and the ontological, and his residual reliance on a Heideggerian project of fundamental ontology, which undermines his own immanentist principles. I will argue for both a bottom-up naturalisation of Badiou’s philosophical approach (...) to mathematics and a top-down naturalisation. Articulating my particular understanding of what realism and naturalism should commit us to, I propose a creative fusion of Badiou’s attention to metamathematical results with a structural-informational metaphysics, proposing a ‘matherialism’ uniting the more daring speculative insights of the former with the naturalist and empiricist commitments motivating the latter. (shrink)

One of the important challenges in the philosophy of mathematics is to account for the semantics of sentences that express mathematical propositions while simultaneously explaining our access to their contents. This is Benacerraf’s Dilemma. In this dissertation, I argue that cognitive science furnishes new tools by means of which we can make progress on this problem. The foundation of the solution, I argue, must be an ontologically realist, albeit non-platonist, conception of mathematical reality. The semantic portion of the problem can (...) be addressed by accepting a Chomskyan conception of natural languages and a matching internalist, mentalist and nativist view of semantics. A helpful perspective on the epistemic aspect of the puzzle can be gained by translating Kurt G ̈odel’s neo-Kantian conception of the nature of mathematics and its objects into modern, cognitive terms. (shrink)

This paper explores the question of what logic is not. It argues against the wide spread assumptions that logic is: a model of reason; a model of correct reason; the laws of thought, or indeed is related to reason at all such that the essential nature of the two are crucially or essentially co-illustrative. I note that due to such assumptions, our current understanding of the nature of logic itself is thoroughly entangled with the nature of reason. I show that (...) most arguments for the presence of any sort of essential re- lationship between logic and reason face intractable problems and demands, and fall well short of addressing them. These arguments include those for the notion that logic is normative for reason (or that logic and correct reason are in some way the same thing), that logic is some sort of description of correct reason and that logic is an abstracted or idealised version of correct reason. A strong version of logical realism is put forward as an alternative view, and is briefly explored. (shrink)

In his classic work The Mind and its Place in Nature published in 1925 at the height of the development of quantum mechanics but several years after the chemists Lewis and Langmuir had already laid the foundations of the modern theory of valence with the introduction of the covalent bond, the analytic philosopher C. D. Broad argued for the emancipation of chemistry from the crass physicalism that led physicists then and later—with support from a rabblement of philosophers who knew as (...) much about chemistry as etymologists—to believe that chemistry reduced to physics. Here Broad’s thesis is recast in terms more familiar to chemists. In the hard sell of particle physics, several prominent figures in chemistry—Hoffmann, Primas, and Pauling—have had their views interpreted to imply that they were sympathetic to greedy reductionism when in fact they were not. Indeed, being chemists without physicists as alter egos, they could not but side with Broad’s contention that chemistry, as a science that deals primarily in emergent phenomena which are beyond the purview of physicalism, owes no acquiescence to particle physics and its ethereal wares. Historically, among the most widely used expediencies in chemistry and materials science are additivity or mixture rules and their cohort transferability, all of which are devised and used under the mantle of naive reductionism. Here it is argued that while the transfer of functional groups between molecules works empirically to an extent, it is strictly outlawed by the no-cloning theorem of quantum mechanics. Several illustrative examples related to chemistry’s irreducibility to physics are presented and discussed. The failure of naive reductionism exhibited by the deep-inelastic scattering of leptons by A > 2 nuclei is traced to the same flawed reasoning that was the original basis of Moffitt’s ‘atoms in molecules’ hypothesis, the neglect of context, nuclei in the case of high-energy physics and molecules in the case of chemistry. A non-exhaustive list of other contexts from physics, chemistry, and molecular biology evidencing similar departures from the ideal of additivity or reductionism is provided for the perusal of philosophers. Had the call by the mathematician J. T. Schwartz for developments in mathematical linguistics possessed of a less single, less literal, and less simple-minded nature been met, perhaps it might have persuaded scientists to abandon their regressive fixation with unphysical reductionism and to adapt to new methodologies that engender a more nuanced handling of ubiquitous emergent phenomena as they arise in Nature than is the case today. (shrink)

Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That series (...) rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology. (shrink)

Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions.

In recent years it has been convincingly argued that the Church-Turing thesis concerns the bounds of human computability: The thesis was presented and justified as formally delineating the class of functions that can be computed by a human carrying out an algorithm. Thus the Thesis needs to be distinguished from the so-called Physical Church-Turing thesis, according to which all physically computable functions are Turing computable. The latter is often claimed to be false, or, if true, contingently so. On all accounts, (...) though, thesis M is not easy to give counterexamples to, but it is never asked why—how come that a thesis that transfers a notion from the strictly human domain to the general physical domain just happens to be so difficult to falsify. In this paper I articulate this question and consider several tentative answers to it. (shrink)

The standard mathematical models in population ecology assume that a population's growth rate is a function of its environment. In this paper we investigate an alternative proposal according to which the rate of change of the growth rate is a function of the environment and of environmental change. We focus on the philosophical issues involved in such a fundamental shift in theoretical assumptions, as well as on the explanations the two theories offer for some of the key data such as (...) cyclic populations. We also discuss the relationship between this move in population ecology and a similar move from first-order to second-order differential equations championed by Galileo and Newton in celestial mechanics. (shrink)

I analyze the importance of parts in the style of biological theorizing that I call compositional biology. I do this by investigating various aspects, including partitioning frames and explanatory accounts, of the theoretical perspectives that fall under and are guided by compositional biology. I ground this general examination in a comparative analysis of three different disciplines with their associated compositional theoretical perspectives: comparative morphology, functional morphology, and developmental biology. I glean data for this analysis from canonical textbooks and defend the (...) use of such texts for the philosophy of science. I end with a discussion of the importance of recognizing formal and compositional biology as two genuinely different ways of doing biology – the differences arising more from their distinct methodologies than from scientific discipline included or natural domain studied. Ultimately, developing a translation manual between the two styles would be desirable as they currently are, at times, in conflict. (shrink)

This paper sets out a new framework for discussing a long-standing problem in the philosophy of mathematics, namely the connection between the physical world and a mathematical domain when the mathematics is applied in science. I argue that considering counterfactual situations raises some interesting challenges for some approaches to applications, and consider an approach that avoids these challenges.

Although the recent emphasis on models in philosophy of science has been an important development, the consequence has been a shift away from more traditional notions of theory. Because the semantic view defines theories as families of models and because much of the literature on “scientific” modeling has emphasized various degrees of independence from theory, little attention has been paid to the role that theory has in articulating scientific knowledge. This paper is the beginning of what I hope will be (...) a redress of the imbalance. I begin with a discussion of some of the difficulties faced by various formulations of the semantic view not only with respect to their account of models but also with their definition of a theory. From there I go on to articulate reasons why a notion of theory is necessary for capturing the structure of scientific knowledge and how one might go about formulating such a notion in terms of different levels of representation and explanation. The context for my discussion is the BCS account of superconductivity, a `theory' that was, and still is, sometimes referred to as a `model'. BCS provides a nice focus for the discussion because it illuminates various features of the theory/model relationship that seem to require a robust notion of theory that is not easily captured by the semantic account. (shrink)

This paper presents a novel account of applied mathematics. It shows how we can distinguish the physical content from the mathematical form of a scientific theory even in cases where the mathematics applied is indispensable and cannot be eliminated by paraphrase.

Wigner found unreasonable the "effectiveness of mathematics in the natural sciences". But if the mathematics we use to describe nature is simply a carefully coded expression of our experience then its effectiveness is quite reasonable. Its effectiveness is built into its design. We consider group theory, the logic of symmetry. We examine the premise that symmetry is identity; that group theory encodes our experience of identification. To decide whether group theory describes the world in such an elemental way we catalogue (...) the detailed correspondence between elements of the physical world and elements of the formalism. Providing an unequivocal match between concept and mathematical statement completes the case. It makes effectiveness appear reasonable. The case that symmetry is identity is a strong one but it is not complete. The further validation required suggests that unexpected entities might be describable by the irreducible representations of group theory. (shrink)

One central aim of science is to provide explanations of natural phenomena. What role(s) does mathematics play in achieving this aim? How does mathematics contribute to the explanatory power of science? Rules to Infinity defends the thesis, common though perhaps inchoate among many members of the Vienna Circle, that mathematics contributes to the explanatory power of science by expressing conceptual rules, rules which allow the transformation of empirical descriptions. Mathematics should not be thought of as describing, in any substantive sense, (...) an abstract realm of eternal mathematical objects, as traditional platonists have thought. In Rules to Infinity, this view, which I call mathematical normativism, is updated with contemporary philosophical tools, and it is argued that normativism is compatible with mainstream semantic theory. This allows the normativist to accept that there are mathematical truths, while resisting the platonistic idea that there exist abstract mathematical objects that explain such truths. Furthermore, Rules to Infinity defends a particular account of the distinction between scientific explanations that are in some sense distinctively mathematical – those that explain natural phenomena in some uniquely mathematical way – and those that are only standardly mathematical, and it lays out desiderata for any account of this distinction. Normativism is compared with other prominent views in the philosophy of mathematics such as neo-Fregeanism, fictionalism, conventionalism, and structuralism. Rules to Infinity serves as an entry point into debates at the forefront of philosophy of science and mathematics, and it defends novel positions in these debates. (shrink)

I argue that mathematical representations can have heuristic power since their construction can be ampliative. To this end, I examine how a representation introduces elements and properties into the represented object that it does not contain at the beginning of its construction, and how it guides the manipulations of the represented object in ways that restructure its components by gradually adding new pieces of information to produce a hypothesis in order to solve a problem.In addition, I defend an ‘inferential’ approach (...) to the heuristic power of representations by arguing that these representations draw on ampliative inferences such as analogies and inductions. In effect, in order to construct a representation, we have to ‘assimilate’ diverse things, and this requires identifying similarities between them. These similarities form the basis for ampliative inferences that gradually build hypotheses to solve a problem.To support my thesis, I analyse two examples. The first one is intra-field, that is, the construction of an algebraic representation of 3-manifolds; the second is inter-fields, that is, the construction of a topological representation of DNA supercoiling. (shrink)

In a paper titled, “The Unreasonable Effectiveness of Mathematics”, published 20 years after Wigner’s seminal paper, the mathematician Richard W. Hamming discussed what he took to be Wigner’s problem of Unreasonable Effectiveness and offered some partial explanations for this phenomenon. Whether Hamming succeeds in his explanations as answers to Wigner’s puzzle is addressed by other scholars in recent years I, on the other hand, raise a more fundamental question: does Hamming succeed in raising the same question as Wigner? The answer (...) is no. My goal is to show that Hamming’s reading misses Wigner’s highly original formulation of the problem.Through a close and contextual reading of Wigner’s work, as I will show, we are led in new directions in addressing and solving the applicability problem. (shrink)

Mi objetivo es discutir las principales dificultades que Frank P. Ramsey encontró en Principia Mathematica y la solución que, vía el Tractatus Logico-Philosophicus, propuso al respecto. Sostengo que las principales dificultades que Ramsey encontró en Principia Mathematica están, todas, relacionadas con que Russell y Whitehead desatendieron la forma lógica de las proposiciones matemáticas, las cuales, según Ramsey, deben ser tautológicas.

In this excellent book Sebastien Gandon focuses mainly on Russell's two major texts, Principa Mathematica and Principle of Mathematics, meticulously unpicking the details of these texts and bringing a new interpretation of both the mathematical and the philosophical content. Winner of The Bertrand Russell Society Book Award 2013.

This article presents a challenge that those philosophers who deny the causal interpretation of explanations provided by population genetics might have to address. Indeed, some philosophers, known as statisticalists, claim that the concept of natural selection is statistical in character and cannot be construed in causal terms. On the contrary, other philosophers, known as causalists, argue against the statistical view and support the causal interpretation of natural selection. The problem I am concerned with here arises for the statisticalists because the (...) debate on the nature of natural selection intersects the debate on whether mathematical explanations of empirical facts are genuine scientific explanations. I argue that if the explanations provided by population genetics are regarded by the statisticalists as non-causal explanations of that kind, then statisticalism risks being incompatible with a naturalist stance. The statisticalist faces a dilemma: either she maintains statisticalism but has to renounce naturalism; or she maintains naturalism but has to content herself with an account of the explanations provided by population genetics that she deems unsatisfactory. This challenge is relevant to the statisticalists because many of them see themselves as naturalists. (shrink)

Arguments from non-causal analogy form a distinctive class of analogical arguments in science not recognized in authoritative classifications by, e.g., Hesse and Bartha. In this paper, I illustrate this novel class of scientific analogies by means of historical examples from physics, biology and economics, at the same time emphasizing their broader significance for contemporary debates in epistemology.

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these (...) options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

In his famous article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” Eugen Wigner argues for a unique tie between mathematics and physics, invoking even religious language: “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve”. The possible existence of such a unique match between mathematics and physics has been extensively discussed by philosophers and historians of mathematics. Whatever the merits (...) of this claim are, a further question can be posed with regard to mathematization in science more generally: What happens when we leave the area of theories and laws of physics and move over to the realm of mathematical modeling in interdisciplinary contexts? Namely, in modeling the phenomena specific to biology or economics, for instance, scientists often use methods that have their origin in physics. How is this kind of mathematical modeling justified? (shrink)

Many mathematicians, physicists, and philosophers have suggested that the fact that mathematics—an a priori discipline informed substantially by aesthetic considerations—can be applied to natural science is mysterious. This paper sharpens and responds to a challenge to this effect. I argue that the aesthetic considerations used to evaluate and motivate mathematics are much more closely connected with the physical world than one might presume, and (with reference to case-studies within Galois theory and probabilistic number theory) show that they are correlated with (...) generally recognized theoretical virtues, such as explanatory depth, unifying power, fruitfulness, and importance. (shrink)

In this paper I argue that constructivism in mathematics faces a dilemma. In particular, I maintain that constructivism is unable to explain (i) the application of mathematics to nature and (ii) the intersubjectivity of mathematics unless (iii) it is conjoined with two theses that reduce it to a form of mathematical Platonism. The paper is divided into five sections. In the first section of the paper, I explain the difference between mathematical constructivism and mathematical Platonism and I outline my argument. (...) In the second, I argue that the best explanation of how mathematics applies to nature for a constructivist is a thesis I call Copernicanism. In the third, I argue that the best explanation of how mathematics can be intersubjective for a constructivist is a thesis I call Ideality. In the fourth, I argue that once constructivism is conjoined with these two theses, it collapses into a form of mathematical Platonism. In the fifth, I confront some objections. (shrink)

This paper formulates generalized versions of the general principle of relativity and of the principle of equivalence that can be applied to general abstract spaces. It is shown that when the principles are applied to the Hilbert space of a quantum particle, its law of coupling to electromagnetic fields is obtained. It is suggested to understand the Aharonov-Bohm effect in light of these principles, and the implications for some related foundational controversies are discussed.

Otávio Bueno* * and Steven French.** ** Applying Mathematics: Immersion, Inference, Interpretation. Oxford University Press, 2018. ISBN: 978-0-19-881504-4 978-0-19-185286-2. doi:10.1093/oso/9780198815044. 001.0001. Pp. xvii + 257.

Analogies between classical statistical mechanics and quantum field theory played a pivotal role in the development of renormalization group methods for application in the two theories. This paper focuses on the analogies that informed the application of RG methods in QFT by Kenneth Wilson and collaborators in the early 1970's. The central task that is accomplished is the identification and analysis of the analogical mappings employed. The conclusion is that the analogies in this case study are formal analogies, and not (...) physical analogies. That is, the analogical mappings relate elements of the models that play formally analogous roles and that have substantially different physical interpretations. Unlike other cases of the use of analogies in physics, the analogical mappings do not preserve causal structure. The conclusion that the analogies in this case are purely formal carries important implications for the interpretation of QFT, and poses challenges for philosophical accounts of analogical reasoning and arguments in defence of scientific realism. Analysis of the interpretation of the cutoffs is presented as an illustrative example of how physical disanalogies block the exportation of physical interpretations from from statistical mechanics to QFT. A final implication is that the application of RG methods in QFT supports non-causal explanations, but in a different manner than in statistical mechanics. (shrink)

ABSTRACT Historical structuralist views have been ontological. They either deny that there are any mathematical objects or they maintain that mathematical objects are structures or positions in them. Non-ontological structuralism offers no account of the nature of mathematical objects. My own structuralism has evolved from an early sui generis version to a non-ontological version that embraces Quine’s doctrine of ontological relativity. In this paper I further develop and explain this view.

The aim of this article is to explore the impact of Darwinism in metaethics and dispel some of the confusion surrounding it. While the prospects for a Darwinian metaethics appear to be improving, some underlying epistemological issues remain unclear. We will focus on the so-called Evolutionary Debunking Arguments (EDAs) which, when applied in metaethics, are defined as arguments that appeal to the evolutionary origins of moral beliefs so as to undermine their epistemic justification. The point is that an epistemic disanalogy (...) can be identified in the debate on EDAs between moral beliefs and other kinds of beliefs, insofar as only the former are regarded as vulnerable to EDAs. First, we will analyze some significant debunking positions in metaethics in order to show that they do not provide adequate justification for such an epistemic disanalogy. Then, we will assess whether they can avoid the accusation of being epistemically incoherent by adopting the same evolutionary account for all kinds of beliefs. In other words, once it is argued that Darwinism has a corrosive impact on metaethics, what if its universal acid cannot be contained? (shrink)

It is argued that mathematics is unreasonably effective in fundamental physics, that this is genuinely mysterious, and that it is best explained by a version of Pythagorean metaphysics. It is shown how this can be reconciled with the fact that mathematics is not always effective in real world applications. The thesis is that physical structure approaches isomorphism with a highly symmetric mathematical structure at very high energy levels, such as would have existed in the early universe. As the universe cooled, (...) its underlying symmetry was broken in a sequence of stages. At each stage, more forces and particles were differentiated, leading to the complexity of the observed world. Remnant structure makes mathematics effective in some real world applications, but not all. (shrink)

I develop a non-representationalist account of mathematical thought, on which the point of mathematical theorizing is to provide us with the conceptual capacity to structure and articulate information about the physical world in an epistemically useful way. On my view, accepting a mathematical theory is not a matter of having a belief about some subject matter; it is rather a matter of structuring logical space, in a sense to be made precise. This provides an elegant account of the cognitive utility (...) of mathematics. Further, it makes explicit how the brand of non-representationalism I develop is compatible with there being substantive rationality constraints on our mathematical theorizing. (shrink)

In this paper I argue for the view that structuralism offers the best perspective for an acceptable account of the applicability of mathematics in the empirical sciences. Structuralism, as I understand it, is the view that mathematics is not the science of a particular type of objects, but of structural properties of arbitrary domains of entities, regardless of whether they are actually existing, merely presupposed or only intentionally intended.

I discuss the applicability of mathematics via a detailed case study involving a family of mathematical concepts known as ‘fractional derivatives.’ Certain formulations of the mystery of applied mathematics would lead one to believe that there ought to be a mystery about the applicability of fractional derivatives. I argue, however, that there is no clear mystery about their applicability. Thus, via this case study, I think that one can come to see more clearly why certain formulations of the mystery of (...) applied mathematics are not convincing. (shrink)

Lately, philosophers of mathematics have been exploring the notion of mathematical explanation within mathematics. This project is supposed to be analogous to the search for the correct analysis of scientific explanation. I argue here that given the way philosophers have been using “ explanation,” the term is not applicable to mathematics as it is in science.

In this paper I defend mathematical nominalism by arguing that any reasonable account of scientific theories and scientific practice must make explicit the empirical non-mathematical grounds on which the application of mathematics is based. Once this is done, references to mathematical entities may be eliminated or explained away in terms of underlying empirical conditions. I provide evidence for this conclusion by presenting a detailed study of the applicability of mathematics to measurement. This study shows that mathematical nominalism may be regarded (...) as a methodological approach to applicability, illuminating the use of mathematics in science. (shrink)

This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply (...) the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the 'no miracles' argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific 'realists' should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism. (shrink)

I identify and characterize a type of noncausal explanation in physics. I first introduce a distinction, between the physical properties of a system, and the representational properties of the mathematical expressions of the system’s physical properties. Then I introduce a novel kind of property, which I shall call a dual property. This is a special kind of representational property, one for which there is an interpretation as a physical property. It is these dual properties that, I claim, are amenable to (...) noncausal (mathematical, in fact) explanations. I discuss a typical example of such a dual property, and an example of an explanation as to why that dual property holds (the explanation of the quantization of the linear momentum). (shrink)

Recent case studies have revealed that purely formal analogies have been successfully used as a heuristic in physics. This is at odds with most general philosophical accounts of analogies, which require analogies to be physical in order to be justifiably used. The main goal of this paper is to supply a philosophical account that justifies the use of purely formal analogies in physics. Using Bartha’s (2010) articulation model as a starting point, I offer precise definitions of formal and physical analogies (...) and propose a new submodel of analogical reasoning that accounts for the successful use of purely formal analogies in the development of renormalization group methods for use in particle physics in the early 1970’s (Fraser 2020). Two distinctive features of this new applied mathematics submodel for analogical reasoning are that the conclusion of the argument from analogy includes both an entire model (and not only a hypothesis or a prediction) and the construction procedure for this model. A third important difference from arguments from physical analogy is that only the prima facie plausibility of the conclusion is established, and not stronger types of plausibility associated with confirmation. The use of purely formal analogies is justified because they are suited to supporting conclusions of this sort. Formulating a general philosophical account of analogies that covers purely formal analogies also serves two additional purposes: (1) to highlight the features of this case that are novel in the context of examples of analogies traditionally considered by philosophers and (2) to establish that physicists should not automatically dismiss purely formal analogies when evaluating heuristics for the development of new models. (shrink)

Gauge symmetries provide one of the most puzzling examples of the applicability of mathematics in physics. The presented work focuses on the role of analogical reasoning in the gauge argument, motivated by Mark Steiner's claim that the application of the gauge principle relies on a Pythagorean analogy whose success undermines naturalist philosophy. In this paper, we present two different views concerning the analogy between gravity, electromagnetism, and nuclear interactions, each providing a different philosophical response to the problem of the applicability (...) of mathematics in the natural sciences. The first is based on an account of Weyl's original work, which first gave rise to the gauge principle. Drawing on his later philosophical writings, we develop an idelaist reading of the mathematical analogies in the gauge argument. On this view, mathematical analogies serve to ensure a conceptual harmony in our scientific account of nature. We further discuss the construction of Yang and Mills's gauge theory in light of this idealist reading. The second account presents a naturalist alternative, formulated in terms of John Norton's account of a material analogy, according to which the analogy succeeds in virtue of a physical similarity between the different interactions. This account is based on the methodological equivalence principle, a simple conceptual extension of the gauge principle that allows us to understand the relation between coordinate transformations and gravity as a manifestation of the same method. The physical similarity between the different cases is based on attributing the success of this method to the dependence of the coupling on relational physical quantities. We conclude by reflecting on the advantages and limits of the idealist, naturalist, and anthropocentric Pythagorean views, as three alternative ways to understand the puzzling relation between mathematics and physics. -/- . (shrink)

This paper aims to study the foundations of applied mathematics, using a formalized base theory for applied mathematics: ZFCAσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathsf {ZFCA}_{\sigma }$$\end{document} with atoms, where the subscript used refers to a signature specific to the application. Examples are given, illustrating the following five features of applied mathematics: comprehension principles, application conditionals, representation hypotheses, transfer principles and abstract equivalents.