During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)

Advances in modern mathematics indicate that progress in this field of knowledge depends mainly on culturally inflected imaginative intuitions, or intuitive imaginings—which mysteriously result in the growth of systems of symbolism that are often efficacious, although fallible and very likely evolutionary. Thus the idea that a trouble-free epistemology can be constructed out of an intuition-free mathematical naturalism would seem to be question begging of a very high order. I illustrate the point by examining Philip Kitcher’s attempt to frame (...) an empiricist philosophy of mathematics, which he calls “mathematical naturalism,” wherein he proposes to explain novelty in mathematics by means of the notion of ‘rational interpractice transitions,’ only to end with an appeal to science to supply a meaning for rationality. A more promising naturalistic approach is adumbrated by Noam Chomsky who begins with a straightforward acceptance of mind and language as ‘natural’ or concrete facts which bespeak the need for a linguistic faculty. This indicates in turn that there may also be a mathematical faculty capable of generating and exploiting the powers of mathematical symbolisms in a manner analogous to the linguistic faculty. (shrink)

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)

The paper aims to capture a form of naturalism that can be found “built-in” in phenomenology, namely the idea to take science or mathematics on its own, without postulating extraneous normative “molds” on it. The paper offers a detailed comparison of Penelope Maddy’s naturalism about mathematics and Husserl’s approach to mathematics in Formal and Transcendental Logic. It argues that Maddy’s naturalized methodology is similar to the approach in the first part of the book. However, in the second part (...) Husserl enters into a transcendental clarification of the evidences and presuppositions of the mathematicians’ work, thus “transcendentalizing” his otherwise naturalist approach to mathematics. The result is a moderately revisionist view that takes the existing mathematical practices seriously, calls for reflection on them, and eventually gives suggestions for revisions if needed. (shrink)

I argue that a recent version of the doctrine of mathematical naturalism faces difficulties arising in connection with Wigner's old puzzle about the applicability of mathematics to natural science. I discuss the strategies to solve the puzzle and I show that they may not be available to the naturalist.

Penelope Maddy advances a purportedly naturalistic account of mathematical methodology which might be taken to answer the question 'What justifies axioms of set theory?' I argue that her account fails both to adequately answer this question and to be naturalistic. Further, the way in which it fails to answer the question deprives it of an analog to one of the chief attractions of naturalism. Naturalism is attractive to naturalists and nonnaturalists alike because it explains the reliability of (...) scientific practice. Maddy's account, on the other hand, appears to be unable to similarly explain the reliability of mathematical practice without violating one of its central tenets. (shrink)

Recent years have seen a number of naturalist accounts of mathematics. Philip Kitcher’s version is one of the most important and influential. This paper includes a critical exposition of Kitcher’s views and a discussion of several issues including: mathematical epistemology, practice, history, the nature of applied mathematics. It argues that naturalism is an inadequate account and compares it with mathematical Platonism, to the advantage of the latter.

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both (...) disciplines. (shrink)

What should a Quinean naturalist say about moral and mathematical truth? If Quine’s naturalism is understood as the view that we should look to natural science as the ultimate ‘arbiter of truth’, this leads rather quickly to what Huw Price has called ‘placement problems’ of placing moral and mathematical truth in an empirical scientific world-view. Against this understanding of the demands of naturalism, I argue that a proper understanding of the reasons Quine gives for privileging ‘natural (...) science’ as authoritative when it comes to questions of truth and existence also apply to other stable and considered elements of our inherited world-view, including, arguably, our firmly held mathematical and moral beliefs. If so, then the ‘thin’ mathematical and moral realisms of Penelope Maddy and T. M. Scanlon, respectively, are vindicated. We do not need to shoehorn mathematical and moral truths into the pushings and pullings of our empirical scientific world-view; for the busy sailor adrift on Neurath’s boat, mathematical and moral truths already have their place. (shrink)

This study addresses a central theme in current philosophy: Platonism vs Naturalism and provides accounts of both approaches to mathematics, crucially discussing Quine, Maddy, Kitcher, Lakoff, Colyvan, and many others. Beginning with accounts of both approaches, Brown defends Platonism by arguing that only a Platonistic approach can account for concept acquisition in a number of special cases in the sciences. He also argues for a particular view of applied mathematics, a view that supports Platonism against Naturalist alternatives. Not only (...) does this engaging book present the Platonist-Naturalist debate over mathematics in a comprehensive fashion, but it also sheds considerable light on non-mathematical aspects of a dispute that is central to contemporary philosophy. (shrink)

This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors tried (...) to naturalize mathematics by relying on evolutionism. But several difficulties arise when we try to do this. This chapter suggests that, in order to naturalize mathematics, it is better to take the method of mathematics to be the analytic method, rather than the axiomatic method, and thus conceive of mathematical knowledge as plausible knowledge. (shrink)

Naturalism in the philosophy of mathematics is the view that philosophy cannot legitimately gainsay mathematics. I distinguish between reinterpretation and reconstruction naturalism: the former states that philosophy cannot legitimately sanction a reinterpretation of mathematics (i.e. an interpretation different from the standard one); the latter that philosophy cannot legitimately change standard mathematics (as opposed to its interpretation). I begin by showing that neither form of naturalism is self-refuting. I then focus on reinterpretation naturalism, which comes in two (...) forms, and examine the only available argument for it. I argue that this argument, the so-called Failure Argument, itself fails. My overall conclusion is that although there is no self-refutation argument against reinterpretation naturalism, there are as yet no good reasons to accept it. Naturalism in mathematics The consistency of mathematical naturalism The failure argument Objections to the failure argument Philosophy as the default. (shrink)

persuasive argument for the claim that we ought to evaluate mathematics from a mathematical point of view and reject extra-mathematical standards. Maddy considers the objection that her arguments leave it open for an ‘astrological naturalist’ to make an analogous claim: that we ought to reject extra-astrological standards in the evaluation of astrology. In this paper, I attempt to show that Maddy's response to this objection is insufficient, for it ultimately either (1) undermines mathematical naturalism itself, leaving (...) us with only scientific naturalism, or (2) leaves open the possibility of other unpalatable naturalisms. (shrink)

Contemporary philosophy’s three main naturalisms are methodological, ontological and epistemological. Methodological naturalism states that the only authoritative standards are those of science. Ontological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural. In philosophy of mathematics of the past few decades methodological naturalism has received the lion’s share of the attention, so we concentrate on this. Ontological and epistemological naturalism in the philosophy of mathematics are discussed (...) more briefly in section 6. (shrink)

In this chapter, I consider some problems with naturalizing mathematics. More specifically, I consider how the two leading kinds of approach to naturalizing mathematics, to wit, Quinean indispensability‐based approaches and Maddy's Second Philosophical approach, seem to run afoul of constraints that any satisfactory naturalistic mathematics must meet. I then suggest that the failure of these kinds of approach to meet the relevant constraints indicates a general problem with naturalistic mathematics meeting these constraints, and thus with the project of naturalizing mathematics (...) itself. (shrink)

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)

The instability inherent in the historical inventory of mathematical objects challenges philosophers. Naturalism suggests we can construct enduring answers to ontological questions through an investigation of the processes whereby mathematical objects come into existence. Patterns of historical development suggest that mathematical objects undergo an intelligible process of reification in tandem with notational innovation. Investigating changes in mathematical languages is a necessary first step towards a viable ontology. For this reason, scholars should not modernize historical texts (...) without caution, as the use of anachronistic notation tends to impede, rather than enhance, our ability to recognize the emergent nature of mathematical objects. (shrink)

In current philosophical research, there is a rather one-sided focus on the foundations of proof. A full picture of mathematical practice should however additionally involve considerations about various methodological aspects. A number of these is identified, from large-scale to small-scale ones. After that, naturalism, a philosophical school concerned with scientific practice, is looked at, as far as the translations of its epistemic principles to mathematics is concerned. Finally, we call for intensifying the interaction between both dimensions of practice (...) and epistemology. (shrink)

In current philosophical research, there is a rather one-sided focus on the foundations of proof. A full picture of mathematical practice should however additionally involve considerations about various methodological aspects. A number of these is identified, from large-scale to small-scale ones. After that, naturalism, a philosophical school concerned with scientific practice, is looked at, as far as the translations of its epistemic principles to mathematics is concerned. Finally, we call for intensifying the interaction between both dimensions of practice (...) and epistemology. (shrink)

In this paper we shall concentrate on the issue of those ways of knowing in mathematics that have traditionally been taken to support apriorism. We shall do it by critizing pragmatic naturalism in the philosophy of mathematics, and in particular its historical approach in denying any role to apriority in mathematical epistemology. The version of pragmatic naturalism we shall be analyzing is Kitcher’s. In the paper we shall first set out a brief survey of the relevant features (...) of Kitcher’s pragmatic naturalism in the philosophy of mathematics and then indicate the points that provoke our disagreement.U ovome radu ćemo se koncentrirati na pitanje onih načina spoznaje u matematici za koje se tradicionalno smatralo da podržavaju apriorizam. To ćemo učiniti kritizirajući pragmatički naturalizam u filozofiji matematike, posebice njegov povijesni pristup u negiranju ikakve uloge apriorizma u matematičkoj epistemologiji. Verzija pragmatičkog naturalizma koju ćemo razmatrati je Kitcherova. U radu ćemo prvo iznijeti sažeti pregled relevantnih obilježja Kitcherovog pragmatičkog naturalizma u filozofiji matematike te potom naznačiti točke koje izazivaju naše neslaganje.Dans cet article, nous nous concentrerons sur la question des modes de la connaissance en mathématiques qui ont traditionnellement été considérés comme soutenant l’apriorisme. Nous le ferons en critiquant le naturalisme pragmatique dans la philosophie des mathématiques, notamment son approche historique de la négation de tout rôle de l’apriorité dans l’épistémologie mathématique. La version du naturalisme pragmatique que nous analyserons est celle de Philip Kitcher. Dans l’article, nous présenterons d’abord un court examen des aspects pertinents du naturalisme pragmatique de Philip Kitcher dans la philosophie des mathématiques, puis indiquerons les points qui suscitent notre désaccord.Im vorliegenden Paper konzentrieren wir uns auf die Problematik jener Wege der Erkenntnis innerhalb der Mathematik, denen traditionell Unterstützung des Apriorismus beigemessen wird. Wir verwirklichen dies, indem wir den pragmatischen Naturalismus in der Philosophie der Mathematik kritisieren, namentlich dessen historische Herangehensweise bei der Verneinung jeglicher Rolle der Apriorität in der mathematischen Epistemologie. Die Version des pragmatischen Naturalismus, der wir auf den Grund gehen, ist die von Kitcher. In der Abhandlung machen wir erstens einen Streifzug durch die relevanten Wesenszüge von Kitchers pragmatischem Naturalismus in der Philosophie der Mathematik und geben anschließend Anhaltspunkte an, die unseren Dissens hervorrufen. (shrink)

Recently, there has been a growing awareness that Russell’s post–1918 writings call into question the sort of picture that Rorty presents of the relation of Russell’s philosophy to the views of subsequent figures such as the later Wittgenstein, Quine, and Sellars. As I will argue in this paper, those writings show that by the early 1920’s Russell himself was advocating views—including an anti-foundationalist naturalized epistemology, and a behaviorist–inspired account of what is involved in understanding language—that are more typically associated with (...) philosophers from later decades whom are mistakingly often rpesented as dismantling Russell’s philosophy. (shrink)

This paper argues that Philip Kitcher's epistemology of mathematics, codified in his Naturalistic Constructivism, is not naturalistic on Kitcher's own conception of naturalism. Kitcher's conception of naturalism is committed to (i) explaining the correctness of belief-regulating norms and (ii) a realist notion of truth. Naturalistic Constructivism is unable to simultaneously meet both of these commitments.

Can a scientific naturalist be a mathematical realist? I review some arguments, derived largely from the writings of Penelope Maddy, for a negative answer. The rejoinder from the realist side is that the irrealist cannot explain, as well as the realist can, why a naturalist should grant the mathematician the degree of methodological autonomy that the irrealist's own arguments require. Thus a naturalist, as such, has at least as much reason to embrace mathematical realism as to embrace irrealism.

How should one conceive of the method of mathematics, if one takes a naturalist stance? Mathematical knowledge is regarded as the paradigm of certain knowledge, since mathematics is based on the axiomatic method. Natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some naturalists try to naturalize mathematics relying on Darwinism. But several difficulties arise when one tries (...) to naturalize in this way the traditional view of mathematics, according to which mathematical knowledge is certain and the method of mathematics is the axiomatic method. This paper suggests that, in order to naturalize mathematics through Darwinism, it is better to take the method of mathematics not to be the axiomatic method. (shrink)

Maddy’s book is an examination of an important question for the philosophy of mathematics: what justifies the axioms of set theory? In part 1, entitled “The Problem,” Maddy provides a summary of the philosophical and mathematical beginnings of set theory and highlights the importance that certain questions play in current debates about the foundations of the theory. Part 2, “Realism,” reviews three versions of mathematical realism and gives reasons for abandoning these views. Part 3, “Naturalism,” furnishes a (...) look at Maddy’s new philosophy of mathematics, Mathematical Naturalism, and explains the way it diagnoses the problem of justifying new axioms. (shrink)

In current philosophical research, there is a rather one-sided focus on the foundations of proof. A full picture of mathematical practice should however additionally involve considerations about various methodological aspects. A number of these is identified, from large-scale to small-scale ones. After that, naturalism, a philosophical school concerned with scientific practice, is looked at, as far as the translations of its epistemic principles to mathematics is concerned. Finally, we call for intensifying the interaction between both dimensions of practice (...) and epistemology. (shrink)

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both (...) disciplines. (shrink)

Review of PENELOPE MADDY. Naturalism in Mathematics. Oxford: Clarendon Press, 1997.

My thesis discusses the unique challenge that platonistic mathematics poses to philosophical naturalism. It has two main parts. ;The first part discusses the three most important approaches to my problem found in the literature: First, W. V. Quine's holistic empiricist defense of mathematical platonism; then, the nominalists' argument that mathematical platonism is naturalistically unacceptable; and finally, a radical form of naturalism, due to John Burgess and Penelope Maddy, which dismisses any philosophical criticism of a successful science (...) such as mathematics. I find faults with all of these approaches. ;The second part attempts to do better. First, I develop an improved epistemological challenge to mathematical platonism. Roughly, this challenge asks for an account of what mathematicians' justification for believing in platonistic mathematics consists in. I argue this challenge is immune to the criticism I leveled against the traditional challenges. To show that it has bite, I apply this challenge to Quine's philosophy of mathematics. I argue this shows that Quine must invoke, in addition to his confirmational holism, a semantic indeterminacy thesis that is so radical as to be implausible. Finally, I attempt to develop a better response to the improved challenge. I defend an unorthodox view of the concept of an object and argue this makes room for a neo-Dedekindian view of mathematical objects which identifies mathematical existence with the logical coherence of the theory describing the structure in question. This view transforms metaphysical and epistemological questions about mathematical objects into corresponding, but more tractable questions about the logical notion of coherence. (shrink)

With his book The Nature of Mathematical Knowledge (1983), Ph. Kitcher, that had been doing extensive research in the history of the subject and in the contemporary debates on epistemology, saw clearly the need for a change in philosophy of mathematics. His goal was to replace the dominant, apriorist philosophy of mathematics with an empiricist philosophy. The current philosophies of mathematics all appeared, according to his analysis, not to fit well with how mathematicians actually do mathematics. A shift in (...) orientation should invoke the more general reflection that causal, genetic factors are as significant for epistemology as logical structure. As I am to a large extent sympathetic with Kitcher's proposal, my aims here will be simple: first I start presenting Kitcher's argument, and then I try to raise some doubts about his contribution. These doubts come probably from my unskilfulness to follow the torrential flow of Kitcher's ideas. (shrink)

Many philosophers these days consider themselves naturalists, but it's doubtful any two of them intend the same position by the term. In Second Philosophy, Penelope Maddy describes and practices a particularly austere form of naturalism called "Second Philosophy". Without a definitive criterion for what counts as "science" and what doesn't, Second Philosophy can't be specified directly ("trust only the methods of science" for example), so Maddy proceeds instead by illustrating the behaviors of an idealized inquirer she calls the "Second (...) Philosopher". mhis Second Philosopher begins from perceptual common sense experimentation, theory formation and testing, working all the while to asses, correct and improve her methods as she goes. Second Philosophy is then the result of the Second Philosopher's investigations. Maddy delineates the Second Philosopher's approach by tracing her reactions to various familiar skeptical and transcendental views (Descartes, Kant, Carnap, late Putnam, van Fraassen), comparing her methods to those of other self-described naturalists (especially Quine), and examining a prominent contemporary debate (between disquotationalists and correspondence theorists in the theory of truth) to extract a properly second-philosophical line of thought. She then undertakes to practice Second Philosophy in her reflections on the ground of logical truth, the methodology, ontology and epistemology of mathematics, and the general prospects for metaphysics naturalized. (shrink)

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)

Leon Horsten has recently claimed that the class of mathematical truths coincides with the class of theorems of ZFC. I argue that the naturalistic character of Horsten’s proposal undermines his contention that this claim constitutes an analogue of a thesis that Daniel Isaacson has advanced for PA. I argue, moreover, that Horsten’s defence of his claim against an obvious objection makes use of a distinction which is not available to him given his naturalistic approach. I suggest a way out (...) of the objection which is in line with the naturalistic spirit of Horsten’s proposal but which further weakens the analogy with Isaacson’s Thesis. I conclude by evaluating the prospects for providing an analogue of Isaacson’s Thesis for ZFC. (shrink)

I argue that the most popular versions of naturalism imply nominalism in philosophy of mathematics. In particular, there is a conflict in Quine's philosophy between naturalism and realism in mathematics. The argument starts from a consequence of naturalism on the nature of human cognitive subjects, physicalism about cognitive subjects, and concludes that this implies a version of nominalism, which I will carefully characterize. The indispensability of classical mathematics for the sciences and semantic/confirmation holism does not affect the (...) argument. The disquotational theory of reference and truth is discussed but rejected. This argument differs from the Benacerrafian arguments against realism, because it does not rely on any specific assumption about the nature of knowledge or reference. It differs from the popular objections to the indispensability argument for realism as well, because it can admit both indispensability and holism. This argument motivates a new, radically naturalistic and nominalistic approach to philosophy of mathematics. (shrink)

Naturalism in philosophy is sometimes thought to imply both scientific realism and a brand of mathematical realism that has methodological consequences for the practice of mathematics. I suggest that naturalism does not yield such a brand of mathematical realism, that naturalism views ontology as irrelevant to mathematical methodology, and that approaching methodological questions from this naturalistic perspective illuminates issues and considerations previously overshadowed by (irrelevant) ontological concerns.

Steiner defines naturalism in opposition to anthropocentrism, the doctrine that the human mind holds a privileged place in the universe. He assumes the anthropocentric nature of mathematics and argues that physicists' employment of mathematically guided strategies in the discovery of quantum mechanics challenges scientists' naturalism. In this paper I show that Steiner's assumption about the anthropocentric character of mathematics is questionable. I draw attention to mathematicians' rejection of what Maddy calls ‘definabilism’, a methodological maxim governing the development of (...) mathematics. I contend that because definabilism is anthropocentric, its rejection casts doubts on Steiner's assumption. (shrink)

Mathematics poses a difficult problem for methodological naturalists, those who embrace scientific method, and also for ontological naturalists who eschew non-physical entities such as Cartesian souls. Mathematics seems both essential to science but also committed to abstract non-physical entities while methodologically it seems to have no place for experiment or empirical confirmation. The chapter critically reviews a number of responses naturalists have made including logicism, Quinean radical empiricism, and Penelope Maddy’s variant thereof and suggests some further problems both for ontological (...) and for methodological naturalists. (shrink)