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)

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)

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)

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)

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)

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

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.

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 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)

ABSTRACT This paper explores the role of aesthetic judgements in mathematics by focussing on the relationship between the epistemic and aesthetic criteria employed in such judgements, and on the nature of the psychological experiences underpinning them. I claim that aesthetic judgements in mathematics are plausibly understood as expressions of what I will call ‘aesthetic-epistemic feelings’ that serve a genuine cognitive and epistemic function. I will then propose a naturalistic account of these feelings in terms of sub-personal processes of (...) representing and assessing the relation between cognitive processes and certain properties of the stimuli at which they are directed. (shrink)

To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept from (...) a wide variety of philosophical positions. Once the axioms are generally accepted, mathematicians can expend their energies on proving theorems instead of arguing philosophy. Given this account of the role of axioms, I give four criteria that axioms must meet in order to be accepted. Penelope Maddy has proposed a similar view in Naturalism in Mathematics, but she suggests that the philosophical questions bracketed by adopting the axioms can in fact be ignored forever. I contend that these philosophical arguments are in fact important, and should ideally be resolved at some point, but I concede that their resolution is unlikely to affect the ordinary practice of mathematics. However, they may have effects in the margins of mathematics, including with regards to the controversial “large cardinal axioms” Maddy would like to support. (shrink)

Husserl's contributions to the nature of mathematical knowledge are opposed to the naturalist, empiricist and pragmatist tendences that are nowadays dominant. It is claimed that mainstream tendences fail to distinguish the historical problem of the origin and evolution of mathematical knowledge from the epistemological problem of how is it that we have access to mathematical knowledge.

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)

In this paper we consider the major development of mathematical analysis during the mid-nineteenth century. On the basis of Jahnke’s (Hist Math 20(3):265–284, 1993 ) distinction between considering mathematics as an empirical science based on time and space and considering mathematics as a purely conceptual science we discuss the Swedish nineteenth century mathematician E.G. Björling’s general view of real- and complexvalued functions. We argue that Björling had a tendency to sometimes consider mathematical objects in a naturalistic way. One (...) example is how Björling interprets Cauchy’s definition of the logarithm function with respect to complex variables, which is investigated in the paper. Furthermore, in view of an article written by Björling (Kongl Vetens Akad Förh Stockholm 166–228, 1852 ) we consider Cauchy’s theorem on power series expansions of complex valued functions. We investigate Björling’s, Cauchy’s and the Belgian mathematician Lamarle’s different conditions for expanding a complex function of a complex variable in a power series. We argue that one reason why Cauchy’s theorem was controversial could be the ambiguities of fundamental concepts in analysis that existed during the mid-nineteenth century. This problem is demonstrated with examples from Björling, Cauchy and Lamarle. (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 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)

The world's leading authorities in the sciences and humanities—dozens of top scholars, including three Nobel laureates—join a cultural and intellectual battle that leaves no human life untouched. Is the universe self-existent, self-sufficient, and self-organizing, or is it grounded instead in a reality that transcends space, time, matter, and energy?

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)

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)

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)

Naturalism is the label for the thesis that the tools we should use in answering philosophical problems are the methods and findings of the mature sciences—from physics across to biology and increasingly neuroscience. It enables us to rule out answers to philosophical questions that are incompatible with scientific findings. It enables us to rule out epistemological pluralism—that the house of knowledge has many mansions, as well as skepticism about the reach of science. It bids us doubt that there are (...) facts about reality that science cannot grasp. It gives us confidence to assert that by now in the development of science, absence of evidence is prima facie good grounds for evidence of absence: this goes for God, and a great deal else. (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)

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)

This article describes .a course in the philosophy of mathematics that compares various metaphysical and epistemological theories of mathematics with portions of the history of the development of mathematics, in particular, the history of calculus.

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 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)

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)

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.

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)

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)

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)

Mathematics as a Science of Patterns . By Michael D. Resnik. (Oxford: Clarendon Press, 1997. Pp. xiii + 285. Price £35.00.) Naturalism in Mathematics . By Penelope Maddy. (Oxford: Clarendon Press, 1998. Pp. viii + 254. Price £32.50.) Realistic Rationalism . By Jerrold J. Katz. ( MIT Press, 1998. Pp. xxxiv + 226. Price £22.50.) The Principles of Mathematics Revisited . By Jaakko Hintikka. ( Cambridge UP, 1996. Pp. xii + 288. Price £40.00.).