Putnam rejects having argued in the terms of the argument known in the literature as “the Quine-Putnam indispensability argument”. He considers that mathematics contribution to physics does not have to be interpreted in platonist terms but in his favorite modal variety. The purpose of this paper is to consider Putnam’s acknowledged argument and philosophical position against contemporary so called in the literature ‘fictionalist’ views about applied mathematics. The conclusion will be that the account of the applicability of mathematics that stems (...) from Putnam‘s acknowledged argument can be assimilated to so-called ‘fictionalist’ views about applied mathematics. (shrink)

For quite some time the indispensability arguments of Quine and Putnam were considered a formidable obstacle to anyone who would reject the existence of mathematical objects. Various attempts to respond to the indispensability arguments were developed, most notably by Chihara and Field. Field tried to defend mathematical fictionalism, according to which the existential assertions of mathematics are false, by showing that the mathematics used in applications is in fact dispensable. Chihara suggested, on the other hand, that mathema­tics makes (...) true existential assertions, but that these can be interpreted so as to remove the commitment to abstract objects. More recently, there have been various attempts to show that the indispensability arguments contain assumptions that are conceptually misguided in ways having little to do with mathematical content. All of this work is of considerable interest, and the result has been a gathering consensus that the indispensability arguments, as put forth by Quine and Putnam, do not provide convincing reason to accept mathematical realism. The focus here will be on the ways of responding to the indispensability arguments, and in particular on the obstacles to fictionalism that remain after the versions of Quine and Putnam are undercut. (shrink)

Our best scientific theories explain a wide range of empirical phenomena, make accurate predictions, and are widely believed. Since many of these theories make ample use of mathematics, it is natural to see them as confirming its truth. Perhaps the use of mathematics in science even gives us reason to believe in the existence of abstract mathematical objects such as numbers and sets. These issues lie at the heart of the Indispensability Argument, to which this Element is devoted. The Element's (...) first half traces the evolution of the Indispensability Argument from its origins in Quine and Putnam's works, taking in naturalism, confirmational holism, Field's program, and the use of idealisations in science along the way. Its second half examines the explanatory version of the Indispensability Argument, and focuses on several more recent versions of easy-road and hard-road fictionalism respectively. (shrink)

Numbers without Science opposes the Quine-Putnam indispensability argument, seeking to undermine the argument and reduce its profound influence. Philosophers rely on indispensability to justify mathematical knowledge using only empiricist epistemology. I argue that we need an independent account of our knowledge of mathematics. The indispensability argument, in broad form, consists of two premises. The major premise alleges that we are committed to mathematical objects if science requires them. The minor premise alleges that science in fact requires mathematical objects. The most (...) common rejection of the argument denies its minor premise by introducing scientific theories which do not refer to mathematical objects. Hartry Field has shown how we can reformulate some physical theories without mathematical commitments. I argue that Field’s preference for intrinsic explanation, which underlies his reformulation, is ill-motivated, and that his resultant fictionalism suffers unacceptable consequences. I attack the major premise instead. I argue that Quine provides a mistaken criterion for ontic commitment. Our uses of mathematics in scientific theory are instrumental and do not commit us to mathematical objects. Furthermore, even if we accept Quine’s criterion for ontic commitment, the indispensability argument justifies only an anemic version of mathematics, and does not yield traditional mathematical objects. The first two chapters of the dissertation develop these results for Quine’s indispensability argument. In the third chapter, I apply my findings to other contemporary indispensabilists, specifically the structuralists Michael Resnik and Stewart Shapiro. In the fourth chapter, I show that indispensability arguments which do not rely on Quine’s holism, like that of Putnam, are even less successful. Also in Chapter 4, I show how Putnam’s work in the philosophy of mathematics is unified around the indispensability argument. In the last chapter of the dissertation, I conclude that we need an account of mathematical knowledge which does not appeal to empirical science and which does not succumb to mysticism and speculation. Briefly, my strategy is to argue that any defensible solution to the demarcation problem of separating good scientific theories from bad ones will find mathematics to be good, if not empirical, science. (shrink)

In his Meditations, René Descartes asks, "what am I?" His initial answer is "a man." But he soon discards it: "But what is a man? Shall I say 'a rational animal'? No: for then I should inquire what an animal is, what rationality is, and in this way one question would lead down the slope to harder ones." Instead of understanding what a man is, Descartes shifts to two new questions: "What is Mind?" and "What is Body?" These questions develop (...) into Descartes's main philosophical preoccupation: the Mind-Body distinction. How can Mind and Body be independent entities, yet joined--essentially so--within a single human being? If Mind and Body are really distinct, are human beings merely a "construction"? On the other hand, if we respect the integrity of humans, are Mind and Body merely aspects of a human being and not subjects in and of themselves? For centuries, philosophers have considered this classic philosophical puzzle. Now, in this compact, engaging, and long-awaited work, UCLA philosopher Joseph Almog closely decodes the French philosopher's argument for distinguishing between the human mind and body while maintaining simultaneously their essential integration in a human being. He argues that Descartes constructed a solution whereby the trio of Human Mind, Body, and Being are essentially interdependent yet remain each a genuine individual subject. Almog's reading not only steers away from the most popular interpretations of Descartes, but also represents a scholar coming to grips directly with Descartes himself. In doing so, Almog creates a work that Cartesian scholars will value, and that will also prove indispensable to philosophers of language, ontology, and the metaphysics of mind. (shrink)

Identity, Essence, and Indiscernibility - Intrinsicness - Cause and Essence - Advertisement for a Sketch of an Outline of a Prototheory of Causation - Does Ontology Rest on a Mistake? - Apriority and Existence - Go Figure: A Path through Fictionalism - Abstract Objects: A Case Study - The Myth of the Seven - Carving Content at the Joints - Non-Catastrophic Presupposition Failure - Must Existence-Questions Have Answers?

The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, (...) and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field. (shrink)

Professor Hilary Putnam has been one of the most influential and sharply original of recent American philosophers in a whole range of fields. His most important published work is collected here, together with several new and substantial studies, in two volumes. The first deals with the philosophy of mathematics and of science and the nature of philosophical and scientific enquiry; the second deals with the philosophy of language and mind. Volume one is now issued in a new edition, including an (...) essay on the philosophy of logic first published in 1971. (shrink)

The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox, a challenge to 'classical' mathematics from a world-famous mathematician, a new foundational school, and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection (...) brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field. (shrink)

"The first two lectures place the alternative I defend -- a kind of pragmatic realism -- in a historical and metaphysical context. Part of that context is provided by Husserl's remark that the history of modern philosophy begins with Galileo -- that is, modern philosophy has been hypnotized by the idea that scientific facts are all the facts there are. Another part is provided by the analysis of a very simple example of what I call 'contextual relativity'. The position I (...) defend holds that truth depends on conceptual scheme and it is nonetheless 'real truth'. "In my third lecture I turn to the Kantian antecedents of this view, explaining what I think should be retained of the Kantian idea of autonomy as the central theme of morality, and extracting from Kant's work a 'moral image of the world' that connects the ideals of equality and intellectual liberty. In this lecture I defend the idea that moral images are an indispensible part of our moral and cultural heritage. "In the final lecture I defend the idea of moral objectivity. I compare our epistemological positions in ethics, history, analysis of human character, and science, and I argue that in no area can we hope for a 'foundation' which is more ultimate than the beliefs that actually, at a given time, function as foundational in the area, the beliefs concerning which one has to say 'this is where my spade is turned'. In ethics such beliefs are represented in moral images of the world.". (shrink)

First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general.

First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general.

Michela Bella & Anna Boncompagni – This conversation will focus on your role and your position with respect to pragmatism; those who you consider allies and enemies in the field; and then finally your ideas about the future of philosophy and the future of pragmatism. You worked with famous philosophers like Carnap, Reichenbach and many others. But let us start from the beginning. Hilary Putnam – My alma mater was the University of Pennsylvania. The first teacher who really influenced (...) me there... (shrink)

Professor Hilary Putnam has been one of the most influential and sharply original of recent American philosophers in a whole range of fields. His most important published work is collected here, together with several new and substantial studies, in two volumes. The first deals with the philosophy of mathematics and of science and the nature of philosophical and scientific enquiry; the second deals with the philosophy of language and mind. Volume one is now issued in a new edition, including an (...) essay on the philosophy of logic first published in 1971. (shrink)

It has been maintained by such philosophers as Quine and Goodman that purely ‘extensional’ language suffices for all the purposes of properly formalized scientific discourse. Those entities that were traditionally called ‘universals’ — properties, concepts, forms, etc. — are rejected by these extensionalist philosophers on the ground that ‘the principle of individuation is not clear’. It is conceded that science requires that we allow something tantamount to quantification over non-particulars (or, anyway, over things that are not material objects, not space-time (...) points, not physical fields, etc.), but, the extensionalists contend, quantification over sets serves the purposes nicely. The ‘ontology’ of modern science, at least as Quine formalizes it, comprises material objects (or, alternatively, space-time points), sets of material objects, sets of sets of material objects,... but no properties, concepts,or forms. Let us thus examine the question: Can the principle of individuation for properties ever be made clear? (shrink)

ABSTRACT:This essay describes three commitments that have become central to the author's philosophical outlook, namely, to liberal naturalism, to metaphysical realism, and to the epistemic and ontological objectivity of normative judgments.Liberal naturalismis contrasted with familiar scientistic versions of naturalism and their project of forcing explanations in every field into models derived from one or another particular science. The form ofmetaphysical realismthat the author endorses rejects every form of verificationism, including the author's one-time ‘internal realism’, and insists that our claims (...) about the world are true or false and not just epistemically successful or unsuccessful and that the terms they contain typically refer to real entities. ‘Representationalism is no sin’. The central part of the essay is an account oftruthbased on a detailed analysis of Tarski's theory of truth and of the insights we can get from it as well as of the respects in which Tarski is misleading. The account ofthe objectivity of the normativein this essay draws on insights from Dewey as well as Scanlon. (shrink)

Professor Hilary Putnam has been one of the most influential and sharply original of recent American philosophers in a whole range of fields. His most important published work is collected here, together with several new and substantial studies, in two volumes. The first deals with the philosophy of mathematics and of science and the nature of philosophical and scientific enquiry; the second deals with the philosophy of language and mind. Volume one is now issued in a new edition, including an (...) essay on the philosophy of logic first published in 1971. (shrink)

A laboratory-testable, solid-state Maxwell demon is proposed that utilizes the electric field energy of an open-gap p-n junction. Numerical results from a commercial semiconductor device simulator (Silvaco International–Atlas) verify primary results from a 1-D analytic model. Present day fabrication techniques appear adequate for laboratory tests of principle.

A fictionalist attitude towards an area of discourse encourages us to assent to certain sentences of that discourse without believing that they are true. Prima facie, this amounts to a suggestion that we should also assent to sentences of the form 'S but I don't believe that S'. Traditional versions of fictionalism have an answer to this challenge, but I argue that the answer is unavailable for a currently popular type of fictionalism. This is bad news for fictionalism in general (...) because the currently popular variety is the one that deals best with the Quine-Putnam indispensability argument. (shrink)

There are old and convincing arguments for intellectual liberty in all of its forms — freedom to think, to speak, to publish — based on assumptions that we who have been brought up in Western democratic countries take for granted. Two major arguments are particularly powerful. The first I shall call the Utilitarian argument which, in its simplest form, says that without intellectual liberty any Party and any government will harden into an exploiting class, a tyranny. The Kantian argument is (...) that, quite apart from its value to society, intellectual liberty — Kant calls it autonomy — is absolutely indispensable to the integrity of the person. In this paper I defend the Kantian approach. The philosophical-epistemological question "How do you know autonomy is a good thing?" remains unanswered. No further foundation can be given. (shrink)

There are old and convincing arguments for intellectual liberty in all of its forms — freedom to think, to speak, to publish — based on assumptions that we who have been brought up in Western democratic countries take for granted. Two major arguments are particularly powerful. The first I shall call the Utilitarian argument which, in its simplest form, says that without intellectual liberty any Party and any government will harden into an exploiting class, a tyranny. The Kantian argument is (...) that, quite apart from its value to society, intellectual liberty — Kant calls it autonomy — is absolutely indispensable to the integrity of the person. In this paper I defend the Kantian approach. The philosophical-epistemological question "How do you know autonomy is a good thing?" remains unanswered. No further foundation can be given. (shrink)

Most science requires applied mathematics. This truism underlies the Quine-Putnam indispensability argument: we cannot be mathematical nominalists without rejecting whole swaths of good science that are seamlessly linked with mathematics. One style of response (e.g. Field’s program) accepts the challenge head-on and attempts to show how to do science without mathematics. There is some consensus that the response fails because the nominalistic apparatus deployed either is not extendible to all of mathematical physics or is merely a deft reconstrual equivalent (...) to standard mathematics. A second style of response (suggested, e.g., by Balaguer and Maddy) denies that indispensability entails realism: when we mathematize a physical problem we treat its physical content as if it were the mathematical representation; provided the two are sufficiently similar, we can use the mathematics to draw conclusions about the physics; even if we cannot represent physical facts without mathematical tools, as-if-fictionalism is reasonable. In this paper I argue that uses of mathematics in science reach deeper than is appreciated by this second response and, indeed, in the more general literature. More specifically, our confidence that we can use the mathematics to draw conclusions about the physics itself depends on mathematics. If the mathematical premises we employ in concluding that a certain application is trustworthy are false, we may lack a justification for supposing that the application will reliably lead us from correct input to correct output. For example, solutions to many physical problems require the determination of a function satisfying a differential equation. Sometimes (e.g., if the differential equation is linear) the existence of a solution for initial value problems can be established directly; where direct methods fail, the existence of a solution must be established indirectly, generally by constructing a sequence of functions that converges to a limit function that satisfies the initial value problem. Moreover, the solution often cannot be evaluated by analytic methods, and scientists must rely on finite element numerical methods to approximate the solution. Mathematical analysis of errors provides further useful information governing the choice of approximation method and of the step size and number of elements needed for the approximation to reach a desired precision. Mathematical physicists rely on the background mathematical theories (e.g., theory of differential equations) presupposed in proving the existence of the solutions and approximating them. It is difficult to see how they could do this while adding the fictionalist disclaimer, “But, you know, I don’t believe any of the mathematics I’m using”. It is difficult to see how a fictionalist pursuing the second strategy can account for the soundness of mathematical reasoning in mathematical physics and elsewhere in the sciences. The paper will fill out this argument by appeal to examples and attempt to make clear (a) how mathematics is indispensable to understanding – and thus underwriting our confidence in – applications that would otherwise be shaky approximations and idealizations and (b) how this role is difficult to square with fictionalism. (shrink)

The term ‘social capital’ is a way of defining the intangible resources of community, shared values and trust upon which we draw in daily life. It has achieved considerable international currency across the social sciences through the very different work of Pierre Bourdieu in France and James Coleman and Robert Putnam in the United States, and has been widely taken up within politics and sociology as an explanation for the decline in social cohesion and community values in western societies. It (...) has also been adopted by policy makers, particularly in international governmental bodies such as the World Bank. (shrink)

Conventionalism about mathematics has much in common with two other views: fictionalism and the multiverse view (aka plenitudinous platonism). The three views may differ over the existence of mathematical objects, but they agree in rejecting a certain kind of objectivity claim about mathematics, advocating instead an extreme pluralism. The early parts of the paper will try to elucidate this anti‐objectivist position, and question whether conventionalism really offers a third form of it distinct from fictionalism and the multiverse view. The paper (...) then turns to anti‐objectivism about logic, and suggests that here too conventionalism offers no distinct alternative, and that there are limits on the extent of pluralism/anti‐objectivism. It also argues that these limits can't be used to argue for more extensive objectivity within mathematics, and also that they do allow for more limited pluralism within logic, e.g. as regards resolution of semantic and property‐theoretic paradoxes. (shrink)

This paper elaborates on the fictionalist conception of mathematics, and on how it accommodates the obvious fact that mathematical claims are important in application to the physical world. It also replies to Maddy's argument that fictionalism does not have the epistemological advantage over Platonism that it appears to have; the reply involves a discussion of whether mathematics should be regarded as conservative over second order physical theories as well as first order ones.

There has been much discussion of the indispensability argument for the existence of mathematical objects. In this paper I reconsider the debate by using the notion of grounding, or non-causal dependence. First of all, I investigate what proponents of the indispensability argument should say about the grounding of relations between physical objects and mathematical ones. This reveals some resources which nominalists are entitled to use. Making use of these resources, I present a neglected but promising response to the indispensability argument—a (...) liberalized version of Field’s response—and I discuss its significance. I argue that if it succeeds, it provides a new refutation of the indispensability argument; and that, even if it fails, its failure may bolster some of the fictionalist responses to the indispensability argument already under discussion. In addition, I use grounding to reply to a recent challenge to these responses. (shrink)

The Quine-Putnam indispensability argument runs as follows: We have reason to believe in Fs if Fs are indispensable to our best available science. Mathematical entities are indispensable to our best available science. Therefore, we have reason to believe in mathematical entities.According to the standard understanding, in order to refute the argument the nominalist has to show that mathematical entities are dispensable by providing an at least as good theory of the same phenomena that is not ontologically committed to mathematical entities. (...) Most philosophers who write in this area, including John Burgess, Mark Colyvan, Hartry Field, Penelope Maddy, and Gideon Rosen, accept the standard understanding. Many nominalists who accept the standard understanding propose nominalistic paraphrases or alternatives, claiming that these are either equally good or better than our current scientific theories. Platonists deny that they are either equally good or better. (shrink)

I defend a unified fictionalism about modality and mathematics. First, I defend each view separately against internal objections. Then, I attempt a unified fictionalism by giving an analysis of truth in fiction which is neither modal nor platonistic. Finally, I explore the prospects for nominalistic unified fictionalism. ;In the first chapter, I defend modal fictionalism: the view that statements about possible worlds are best understood as claims about the content of a fiction, the 'many-worlds story'. I address the Brock-Rosen objection (...) that the fictionalist is a modal realist despite himself. I argue that Noonan's solution, the best response so far, is inadequate, and then offer an alternative fictionalist translation scheme for modal statements. ;In the second chapter, I defend mathematical fictionalism. Mathematical fictionalism is the view that mathematics is best understood as a fiction whose value is quite independent of its truth-value. Field argues for fictionalism in this sense by constructing nominalistic alternatives to standard physical theories. The aim is to establish that mathematical entities are 'dispensable for the purposes of science', and so to undermine the only reason we might have for believing in the literal truth of standard mathematics. I address an objection that the nominalization of current science would not suffice to establish the 'dispensability' of mathematical entities in the epistemologically relevant sense. I argue that there is a plausible interpretation of the indispensability argument according to which successful nominalization would establish that it is unreasonable to believe in mathematical entities. ;The most important challenge for any attempt at a unified fictionalism is to provide an account of truth in fiction that is neither implicitly modal nor implicitly platonistic. To this end, I offer in the third chapter an analysis of truth in a story which is based on a modified version of Walton's account of truth in fiction. ;A second challenge is to show that the timelessness and necessity of modal truth are compatible with the view that fictions are contingent and temporally restricted entities. In the final chapter, I argue this problem constitutes a genuine obstacle for a unified nominalistic fictionalism. (shrink)

Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that this attribution is mistaken. Quine's argument for the existence of abstract mathematical objects differs from the argument which many philosophers of mathematics ascribe to him. Contrary to (...) appearances, Putnam did not argue for the existence of abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam's writings reveals some neglected arguments for platonism which may be superior to the indispensability argument. (shrink)

The ramifications are explored of taking physical theories to commit their advocates only to ‘physically real’ entities, where ‘physically real’ means ‘causally efficacious’ (e.g., actual particles moving through space, such as dust motes), the ‘physically significant’ (e.g., centers of mass), and the merely mathematical—despite the fact that, in ordinary physical theory, all three sorts of posits are quantified over. It's argued that when such theories are regimented, existential quantification, even when interpreted ‘objectually’ (that is, in terms of satisfaction via variables, (...) rather than by substitution-instances) need not imply any ontological commitments. (shrink)

In this paper I do two things: (1) I support the claim that there is still some confusion about just what the Quine-Putnam indispensability argument is and the way it employs Quinean meta-ontology and (2) I try to dispel some of this confusion by presenting the argument in a way which reveals its important meta-ontological features, and include these features explicitly as premises. As a means to these ends, I compare Peter van Inwagen’s argument for the existence of properties with (...) Putnam’s presentation of the indispensability argument. Van Inwagen’s argument is a classic exercise in Quinean meta-ontology and yet he claims – despite his argument’s conspicuous similarities to the Quine-Putnam argument – that his own has a substantially different form. I argue, however, that there is no such difference between these two arguments even at a very high level of specificity; I show that there is a detailed generic indispensability argument that captures the single form of both. The arguments are identical in every way except for the kind of objects they argue for – an irrelevant difference for my purposes. Furthermore, Putnam’s and van Inwagen’s presentations make an assumption that is often mistakenly taken to be an important feature of the Quine-Putnam argument. Yet this assumption is only the implicit backdrop against which the argument is typically presented. This last point is brought into sharper relief by the fact that van Inwagen’s list of the four nominalistic responses to his argument is too short. His list is missing an important – and historically popular – fifth option. (shrink)

The main task of this paper is to defend anti-platonism by providing an anti-platonist (in particular, a fictionalist) account of the indispensable applications of mathematics to empirical science.

The Quine-Putnam Indispensability argument is the argument for treating mathematical entities on a par with other theoretical entities of our best scientific theories. This argument is usually taken to be an argument for mathematical realism. In this chapter I will argue that the proper way to understand this argument is as putting pressure on the viability of the marriage of scientific realism and mathematical nominalism. Although such a marriage is a popular option amongst philosophers of science and mathematics, in light (...) of the indispensability argument, the marriage is seen to be very unstable. Unless one is careful about how the Quine-Putnam argument is disarmed, one can be forced to either mathematical realism or, alternatively, scientific instrumentalism. I will explore the various options: (i) finding a way to reconcile the two partners in the marriage by disarming the indispensability argument (Jody Azzouni [2], Hartry Field [13, 14], Alan Musgrave [18, 19], David Papineau [21]); (ii) embracing mathematical realism (W.V.O. Quine [23], Michael Resnik [25], J.J.C. Smart [27]); and (iii) embracing some form of scientific instrumentalism (Ot´ avio Bueno [7, 8], Bas van Fraassen [30]). Elsewhere [11], I have argued for option (ii) and I won’t repeat those arguments here. Instead, I will consider the difficulties for each of the three options just mentioned, with special attention to option (i). In relation to the latter, I will discuss an argument due to Alan Musgrave [19] for why option (i) is a plausible and promising approach. From the discussion of Musgrave’s argument, it will emerge that the issue of holist versus separatist theories of confirmation plays a curious role in the realism–antirealism debate in the philosophy of mathematics. I will argue that if you take confirmation to be an holistic matter—it’s whole theories (or significant parts thereof) that are confirmed in any experiment—then there’s an inclination to opt for (ii) in order to resolve the marital tension outlined above.. (shrink)

In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument (...) for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct. (shrink)

This book does three main things. First, it defends mathematical platonism against the main objections to that view (most notably, the epistemological objection and the multiple-reductions objection). Second, it defends anti-platonism (in particular, fictionalism) against the main objections to that view (most notably, the Quine-Putnam indispensability objection and the objection from objectivity). Third, it argues that there is no fact of the matter whether abstract mathematical objects exist and, hence, no fact of the matter whether platonism or anti-platonism is true.

Crucial to Hilary Putnam’s realism in the philosophy of mathematics is to maintain the objectivity of mathematics without the commitment to the existence of mathematical objects. Putnam’s indispensability argument was devised as part of this conception. In this paper, I reconstruct and reassess Putnam’s argument for the indispensability of mathematics, and distinguish it from the more familiar, Quinean version of the argument. Although I argue that Putnam’s approach ultimately fails, I develop an alternative way of implementing his form of realism (...) about mathematics that, by using different resources than those Putnam invokes, avoids the difficulties faced by his view. (shrink)

The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their (...) dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. (shrink)

Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.

In this paper, I examine Putnam’s nuanced views in the philosophy of mathematics, distinguishing three proposals: modalism, quasi-empirical realism, and an indispensability view. I argue that, as he shifted through these views, Putnam aimed to preserve a semantic realist account of mathematics that avoids platonism. In the end, however, each of the proposals faces significant difficulties. A form of skepticism then emerges.

Most science requires applied mathematics. This truism underlies the Quine-Putnam indispensability argument: we cannot be mathematical nominalists without rejecting whole swaths of good science that are seamlessly linked with mathematics. One style of response accepts the challenge head-on and attempts to show how to do science without mathematics. There is some consensus that the response fails because the nominalistic apparatus deployed either is not extendible to all of mathematical physics or is merely a deft reconstrual equivalent to standard mathematics. A (...) second style of response denies that indispensability entails realism: when we mathematize a physical problem we treat its physical content as if it were the mathematical representation; provided the two are sufficiently similar, we can use the mathematics to draw conclusions about the physics; even if we cannot represent physical facts without mathematical tools, as-if-fictionalism is reasonable. In this paper I argue that uses of mathematics in science reach deeper than is appreciated by this second response and, indeed, in the more general literature. More specifically, our confidence that we can use the mathematics to draw conclusions about the physics itself depends on mathematics. If the mathematical premises we employ in concluding that a certain application is trustworthy are false, we may lack a justification for supposing that the application will reliably lead us from correct input to correct output. For example, solutions to many physical problems require the determination of a function satisfying a differential equation. Sometimes the existence of a solution for initial value problems can be established directly; where direct methods fail, the existence of a solution must be established indirectly, generally by constructing a sequence of functions that converges to a limit function that satisfies the initial value problem. Moreover, the solution often cannot be evaluated by analytic methods, and scientists must rely on finite element numerical methods to approximate the solution. Mathematical analysis of errors provides further useful information governing the choice of approximation method and of the step size and number of elements needed for the approximation to reach a desired precision. Mathematical physicists rely on the background mathematical theories presupposed in proving the existence of the solutions and approximating them. It is difficult to see how they could do this while adding the fictionalist disclaimer, “But, you know, I don’t believe any of the mathematics I’m using”. It is difficult to see how a fictionalist pursuing the second strategy can account for the soundness of mathematical reasoning in mathematical physics and elsewhere in the sciences. The paper will fill out this argument by appeal to examples and attempt to make clear how mathematics is indispensable to understanding – and thus underwriting our confidence in – applications that would otherwise be shaky approximations and idealizations and how this role is difficult to square with fictionalism. (shrink)