Does science justify any part of mathematics and, if so, what part? These questions are related to the so-called indispensability arguments propounded, among others, by Quine and Putnam; moreover, both were led to accept significant portions of set theory on that basis. However, set theory rests on a strong form of Platonic realism which has been variously criticized as a foundation of mathematics and is at odds with scientific realism. Recent logical results show that it is possible to directly (...) formalize almost all, if not all, scientifically applicable mathematics in a formal system that is justified simply by Peano Arithmetic (via a proof-theoretical reduction). It is argued that this substantially vitiates the indispensability arguments. (shrink)

Indispensability arguments for realism about mathematical entities have come under serious attack in recent years. To my mind the most profound attack has come from Penelope Maddy, who argues that scientific/mathematical practice doesn't support the key premise of the indispensability argument, that is, that we ought to have ontological commitment to those entities that are indispensable to our best scientific theories. In this paper I defend the Quine/Putnam indispensability argument against Maddy's objections.

The Quine-Putnam indispensability argument urges us to place mathematical entities on the same ontological footing as (other) theoretical entities of empirical science. Recently this argument has attracted much criticism, and in this paper I address one criticism due to Elliott Sober. Sober argues that mathematical theories cannot share the empirical support accrued by our best scientific theories, since mathematical propositions are not being tested in the same way as the clearly empirical propositions of science. In this (...) paper I defend the Quine-Putnam argument against Sober's objections. (shrink)

According to the indispensability argument, the fact that we quantify over numbers, sets and functions in our best scientific theories gives us reason for believing that such objects exist. I examine a strategy to dispense with such quantification by simply replacing any given platonistic theory by the set of sentences in the nominalist vocabulary it logically entails. I argue that, as a strategy, this response fails: for there is no guarantee that the nominalist world that go beyond the set (...) of sentences in the nominalist language such theories entail. However, I argue that what such theories show is that mathematics can enable us to express possibilities about the concrete world that may not be expressible in nominalistically acceptable language. While I grant that this may make quantification over abstracta indispensable, I deny that such indispensability is a reason for accepting them into our ontology. I urge that the nominalist should be allowed to quantify over abstracta whilst denying their existence and I explain how this apparently contradictory practice (a practice I call 'weaseling') is in fact coherent, unproblematic and rational. Finally, I examine the view that platonistic theories are simpler or more attractive than their nominalistic reformulations, and thus that abstract ought to be accepted into our ontology for the same sorts of reasons as other theoretical objects. I argue that, at least in the case of numbers, functions and sets, such arguments misunderstand the kind of simplicity and attractiveness we seek. (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)

One of the most influential arguments for realism about mathematical objects is the indispensability argument. Simply put, this is the argument that insofar as we are committed to the existence of the physical objects existentially quantified over in our best scientific theories, we are also committed to the mathematical objects existentially quantified over in these theories. Following the Quine–Putnam formulation of the indispensability argument, some proponents of the indispensability argument have made the mistake of taking confirmational holism (...) to be an essential premise of the argument. In this paper, I consider the reasons philosophers have taken confirmational holism to be essential to the argument and argue that, contrary to the traditional view, confirmational holism is dispensable. (shrink)

This paper considers Maddy’s strategy for naturalising mathematics in the context of Quine’s scientific naturalism. The aim of this proposal is to account for the acceptability of mathematics on scientific grounds without committing to revisionism about mathematical practice entailed by the Quine-Putnam indispensability argument. It has been argued that Maddy’s mathematical naturalism makes inconsistent assumptions on the role of mathematics in scientific explanations to the effect that it cannot distinguish mathematics from pseudo-science. I shall (...) clarify Maddy’s arguments and show that the objection can be overcome. I shall then reformulate a novel version of the objection and consider a possible answer, and I shall conclude that mathematical naturalism does not ultimately provide a viable strategy for accommodating an anti-revisionary stance on mathematics within a Quinean naturalist framework. (shrink)

I discuss Putnam’s conception of logical truth as grounded in his picture of mathematical practice and ontology. i begin by comparing Putnam’s 1971 Philosophy of Logic with Quine’s homonymous book. Next, Putnam’s changing views on modality are surveyed, moving from the modal pre-formal to the de-modalized formal characterization of logical validity. Section three suggests a complementary view of Platonism and modalism underlying different stages of a dynamic mathematical practice. The final section argues for the pervasive platonistic (...) conception of the working mathematician. (shrink)

This paper defends the Quine-Putnam mathematical indispensability argument against two objections raised by Penelope Maddy. The objections concern scientific practices regarding the development of the atomic theory and the role of applied mathematics in the continuum and infinity. I present two alternative accounts by Stephen Brush and Alan Chalmers on the atomic theory. I argue that these two theories are consistent with Quine’s theory of scientific confirmation. I advance some novel versions of the indispensability argument. I argue (...) that these new versions accommodate Maddy’s history of the atomic theory. Counter-examples are provided regarding the role of the mathematical continuum and mathematical infinity in science. (shrink)

Published originally as “Loogisen kielentutkimuksen näköaloja”, Ajatus 19,, pp. 81–96, the following piece by Jaakko Hintikka is the first essay he published in his mother tongue of Finnish. It is seen to provide both a state-of-the-art review of current topics emerging in the philosophy of language in the mid-1950, as well as outlines of Hintikka’s own evaluation of major theses of that era, in particular those of Quine’s and Wittgenstein’s concerning language use. Hintikka evaluates contributions that the logical study of (...) language use can make to the solving of philosophical questions. Published in 1956 in Ajatus: The Yearbook of the Philosophical Society of Finland, Hintikka’s essay served to introduce these topical issues to the Finnish-speaking audience soon after the original works had appeared in print. It also puts the theses into the perspectives of Hintikka’s own fledgling logical philosophy. One can see the germs of game-theoretical semantics is his comments on Wittgenstein’s notion of a language-game, and his remarks on general habits of action anticipate the importance of strategic rules in Hintikka’s mature, inquiry-led and ‘action-first’ epistemology. This is manifest in Hintikka’s characteristic approach to the methodology of philosophy of science and mathematics as an attempt to understand the nature of scientific and mathematical practices and operations through the logical analysis of their central concepts. Topics indicated below thus became the hallmarks of Hintikka’s next sixty years of work, which saw the development of a range of logical methods by which language and its use can be studied, and in particular in such a way that progress can be made in solving problems of genuinely philosophical nature that one encounters across various fields of science. This piece was very briefly summarized in English in 1963 by Arto Salomaa, 1963, p. 165), and it is translated from the original Finnish by Jukka Nikulainen and Ahti-Veikko Pietarinen. [A.-V. P, J. N.]. (shrink)

In this lively look at current debates in American philosophy, leading philosophers talk candidly about the changing character of their discipline. In the spirit of Emerson's The American Scholar , this book explores the identity of the American philosopher. Through informal conversations, the participants discuss the rise of post-analytic philosophy in America and its relations to European thought and to the American pragmatist tradition. They comment on their own intellectual development as well as each others' work, charting the course of (...) American philosophy over the past few decades. Giovanna Borradori, in her substantial introduction, explains the history of the analytic movement in America and the home-grown reaction against it. In the late nineteenth and early twentieth centuries, American philosophy was a socially engaged interdisciplinary enterprise. In transcendentalism and pragmatism, then the dominant currents in American thought, philosophy was connected to history, psychology, and public issues. But in the 1930s, the imported European movement of logical positivism redefined philosophical discourse in terms of mathematical logic and theory of language. Under the influence of this analytic view, American philosophy became a professionalized discipline, divorced from public debate and intellectual history and antagonistic to the other, more humanistic tradition of continental thought. The American Philosopher explores the opposition between analytic and continental thought and shows how recent American work has begun to bridge the gap between the two traditions. Through a reexamination of pragmatism, and through an attempt to understand philosophy in a more hermeneutical way, the participants narrow the distance between America's distinctly scientific philosophy and Europe's more literary approach. Moving beyond classical analytic philosophy, the participants confront each other on a number of topics. The logico-linguistic orientations of Quine and Davidson come up against the more discursive, interdisciplinary agendas of Rorty, Putnam, and Cavell. Nozick's theory of pluralist anarchism goes face-to-face with the aesthetic neo-foundationalism of Danto. And Kuhn's hypothesis of paradigm shifts is measured against MacIntyre's ethics of "virtues." Borradori's conversations offer an unconventional portrait of the way philosophers think about their work scholars and students will not be its only beneficiaries, so will everyone who wonders about the current state of American philosophy. (shrink)

In this lively look at current debates in American philosophy, leading philosophers talk candidly about the changing character of their discipline. In the spirit of Emerson's _The American Scholar_, this book explores the identity of the American philosopher. Through informal conversations, the participants discuss the rise of post-analytic philosophy in America and its relations to European thought and to the American pragmatist tradition. They comment on their own intellectual development as well as each others' work, charting the course of American (...) philosophy over the past few decades. Giovanna Borradori, in her substantial introduction, explains the history of the analytic movement in America and the home-grown reaction against it. In the late nineteenth and early twentieth centuries, American philosophy was a socially engaged interdisciplinary enterprise. In transcendentalism and pragmatism, then the dominant currents in American thought, philosophy was connected to history, psychology, and public issues. But in the 1930s, the imported European movement of logical positivism redefined philosophical discourse in terms of mathematical logic and theory of language. Under the influence of this analytic view, American philosophy became a professionalized discipline, divorced from public debate and intellectual history and antagonistic to the other, more humanistic tradition of continental thought. _The American Philosopher_ explores the opposition between analytic and continental thought and shows how recent American work has begun to bridge the gap between the two traditions. Through a reexamination of pragmatism, and through an attempt to understand philosophy in a more hermeneutical way, the participants narrow the distance between America's distinctly scientific philosophy and Europe's more literary approach. Moving beyond classical analytic philosophy, the participants confront each other on a number of topics. The logico-linguistic orientations of Quine and Davidson come up against the more discursive, interdisciplinary agendas of Rorty, Putnam, and Cavell. Nozick's theory of pluralist anarchism goes face-to-face with the aesthetic neo-foundationalism of Danto. And Kuhn's hypothesis of paradigm shifts is measured against MacIntyre's ethics of "virtues." Borradori's conversations offer an unconventional portrait of the way philosophers think about their work; scholars and students will not be its only beneficiaries, so will everyone who wonders about the current state of American philosophy. (shrink)

role in scientific theories, and their relation to time. ;Chapter 1, "Why Apply Mathematics?" argues that scientific theories are not about the mathematics that is applied in them, and defends this thesis against the Quine-Putnam Indispensability Argument. ;Chapter 2, "Scientific Ontology," is a critical study of W. V. Quine's claim that metaphysics and mathematics are epistemologically on a par with natural science. It is argued that Quine's view relies on a unacceptable account of empirical confirmation. ;Chapter 3, (...) "Prior and the Platonist," demonstrates the incompatibility of two popular views about time: the "Platonist" thesis that some objects exist "outside" time, and A. N. Prior's proposal for treating tense on the model of modality. ;Chapter 4, "What has Eternity Ever Done for You?" argues time), and presents a defense of the rival view that they exist sempiternally. (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)

In the philosophy of mathematics, indispensability arguments aim to show that we are justified in believing that abstract mathematical objects exist. I wish to defend a particular objection to such arguments that has become increasingly popular recently. It is called instrumental nominalism. I consider the recent versions of this view and conclude that it has yet to be given an adequate formulation. I provide such a formulation and show that it can be used to answer the indispensability arguments. -/- (...) There are two main indispensability arguments in the literature, though one has received nearly all of the attention. They correspond to two ways in which we use mathematics in science and in everyday life. We use mathematical language to help us describe non-mathematical reality; and we use mathematical reasoning to help us perform inferences concerning non-mathematical reality using only a feasible amount of cognitive power. The former use is the starting point of the Quine-Putnam indispensability argument ([Quine, 1980a], [Quine, 1980b], [Quine, 1981a], [Quine, 1981b], [Putnam, 1979a], [Putnam, 1979b]); the latter provides the basis for Ketland’s more recent argument ([Ketland, 2005]). I begin by considering the Quine-Putnam argument and introduce instrumental nominalism to defuse it. Then I show that Ketland’s argument can be defused in a similar way. (shrink)

Quine, taking the molecular constitution of matter as a paradigmatic example, offers an account of the relation between theory confirmation and ontology. Elsewhere, he deploys a similar ontological methodology to argue for the existence of mathematical objects. Penelope Maddy considers the atomic/molecular theory in more historical detail. She argues that the actual ontological practices of science display a positivistic demand for “direct observation,” and that fulfillment of this demand allows us to distinguish molecules and other physical objects from (...) class='Hi'>mathematical abstracta. However, the confirmation of the atomic/molecular theory and the development of scientists’ ontological attitudes towards atoms was more complicated and subtle than even Maddy supposes. The present paper argues that the history of the theory in fact supports neither Quine’s and Maddy’s accounts of scientific ontology. There was no general demand from scientists to “see” atoms before they were reckoned to be real; but neither did the indispensable appearance of atoms in the best theory of chemical combination suffice to convince scientists of their reality. (shrink)

This is a dissertation of philosophy of mathematics, in the analytical tradition, about the Quine-Putnam mathematical indispensability argument, that we ought to have ontological commitment to mathematical entities that are indispensable to our best scientific theories. It is an argument for the metaphysical mathematical realism supported by Quinean doctrines such as naturalism and holism. My overall aim is to make a discussion of the argument. The argument will be defended against generic objections or some of its (...) detractors such as Azzouni, Maddy, Cheyne and Balaguer. Matters connected with the indispensability of mathematics, such as the epistemic problem of Benacerraf and the agnosticist view of Balaguer, will be discussed. Primarily, the discussion is ontological; secondarily, the discussion is epistemological and metaphysical. (shrink)

W.V. Quine is a well-known proponent of naturalism, the view on which reality is described only in science. He is also well-known for arguing that our current scientific theories commit us to the existence of abstract objects. It is tempting to believe that the naturalistic philosopher should think scientists outside of philosophy are in the best position to assess the merits of revising our current commitment to abstract objects. But Quine rejects this deferential view. On the reading of Quine’s (...) philosophical methodology that I defend in this paper, the naturalistic philosopher not only may assess the merits of revising the commitments of our scientific theories, but also will recommend we make such revisions if doing so simplifies and clarifies our science. To develop my reading, I will examine John Burgess and Gideon Rosen’s naturalism in ontology that includes the deferential view Quine rejects. By explaining how Quine’s naturalism differs from the anti-revisionist, deferential naturalism in philosophy of mathematics that Burgess and Rosen advance, I seek to clarify and advance contemporary debates on naturalism. (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 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)

This dissertation makes two primary contributions. The first three chapters develop an interpretation of Carnap's Meta-Philosophical Program which places stress upon his methodological analysis of the sciences over and above the Principle of Tolerance. Most importantly, I suggest, is that Carnap sees philosophy as contiguous with science—as a part of the scientific enterprise—so utilizing the very same methods and subject to the same limitations. I argue that the methodological reforms he suggests for philosophy amount to philosophy as the explication (...) of the concepts of science through the construction and use of suitably robust meta-logical languages. My primary interpretive claim is that Carnap's understanding of logic and mathematics as a set of formal auxiliaries is premised upon this prior analysis of the character of logico-mathematical knowledge, his understanding of its role in the language of science, and the methods used by practicing mathematicians. Thus the Principle of Tolerance, and so Carnap's logical pluralism, is licensed and justified by these methodological insights. This interpretation of Carnap's program contrasts with the popular Deflationary reading as proposed in Goldfarb & Ricketts. The leading idea they attribute to Carnap is a Logocentrism: That philosophical assertions are always made relative to some particular language, and that our choice of syntactical rules for a language are constitutive of its inferential structure and methods of possible justification. Consequently Tolerance is considered the foundation of Carnap's entire program. My third chapter argues that this reading makes Carnap's program philosophically inert, and I present significant evidence that such a reading is misguided. The final chapter attempts to extend the methodological ideals of Carnap's program to the analysis of the ongoing debate between category- and set-theoretic foundations for mathematics. Recent criticism of category theory as a foundation charges that it is neither autonomous from set theory, nor offers a suitable ontological grounding for mathematics. I argue that an analysis of concepts can be foundationally informative without requiring the construction of those concepts from first principles, and that ontological worries can be seen as methodologically unfruitful. (shrink)

According to the Quine-Putnam indispensability argument (QPIA), we should be realists about mathematics because mathematics is indispensable to science. QPIA’s reasoning can be understood in two ways. Under the confirmational analysis, QPIA argues that mathematics is confirmed as part of our best scientific theories. Under the pragmatic analysis, QPIA argues that our scientific practices implicitly assume the truth of mathematics. The usual reasons given in favour of the pragmatic analysis are that it affords advantages to proponents of QPIA (...) by avoiding some dependencies of the confirmational analysis. This paper argues that these reasons are indecisive, because the pragmatic analysis introduces some dependencies of its own. Nevertheless, the paper also argues, there are other reasons to prefer the pragmatic analysis. QPIA is an instance of a wider class of realist arguments—indispensability arguments—and the pragmatic analysis yields a preferable overall understanding of indispensability arguments than does the confirmational analysis. (shrink)

W. V. Quine was one of the most influential figures of twentieth-century American analytic philosophy. Although he wrote predominantly in English, in Brazil in 1942 he gave a series of lectures on logic and its philosophy in Portuguese, subsequently published as the book O Sentido da Nova Lógica. The book has never before been fully translated into English, and this volume is the first to make its content accessible to Anglophone philosophers. Quine would go on to develop revolutionary ideas about (...) semantic holism and ontology, and this book provides a snapshot of his views on logic and language at a pivotal stage of his intellectual development. The volume also includes an essay on logic which Quine also published in Portuguese, together with an extensive historical-philosophical essay by Frederique Janssen-Lauret. The valuable and previously neglected works first translated in this volume will be essential for scholars of twentieth-century philosophy. (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)

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)

Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (Putnam, J Symbol Logic 45:464–482, 1980 ). (...) Therefore, it is not clear how this logic should be interpreted. (3) Second-order logic also has strong ontological commitments: (a) it is ontologically committed to classes (Resnik, J Phil 85:75–87, 1988 ), and (b) according to Quine (Philosophy of logic, Prentice-Hall: Englewood Cliffs, 1970 ), it is nothing more than “set theory in sheep’s clothing”. (4) It is also not better than its first-order counterpart, in the following sense: if first-order logic does not characterize adequately mathematical systems, given the existence of non - isomorphic first-order interpretations, second-order logic does not characterize them either, given the existence of different interpretations of second-order theories (Melia, Analysis 55:127–134, 1995 ). (5) Finally, as opposed to what is claimed by defenders of second-order logic [such as Shapiro (J Symbol Logic 50:714–742, 1985 )], this logic does not solve the problem of referential access to mathematical objects (Azzouni, Metaphysical myths, mathematical practice: the logic and epistemology of the exact sciences, Cambridge University Press, Cambridge, 1994 ). In this paper, I argue that the second-order theorist can solve each of these difficulties. As a result, second-order logic provides the benefits of a rich framework without the associated costs. (shrink)

In this dissertation on Hilary Putnam's philosophy, I investigate his development regarding meaning and necessity, in particular mathematical necessity. Putnam has been a leading American philosopher since the end of the 1950s, becoming famous in the 1960s within the school of analytic philosophy, associated in particular with the philosophy of science and the philosophy of language. Under the influence of W.V. Quine, Putnam challenged the logical positivism/empiricism that had become strong in America after World War II, with influential exponents (...) such as Rudolf Carnap and Hans Reichenbach. Putnam agreed with Quine that there are no absolute a priori truths. In particular, he was critical of the notion of truth by convention. Instead he developed a notion of relative a priori truth, that is, a notion of necessary truth with respect to a body of knowledge, or a conceptual scheme. Putnam's position on necessity has developed over the years and has always been connected to his important contributions to the philosophy of meaning. I study Hilary Putnam's development through an early phase of scientific realism, a middle phase of internal realism, and his later position of a natural or commonsense realism. I challenge some of Putnam’s ideas on mathematical necessity, although I have largely defended his views against some other contemporary major philosophers; for instance, I defend his conceptual relativism, his conceptual pluralism, as well as his analysis of the realism/anti-realism debate. (shrink)

Philosophy of mathematics is moving in a new direction: away from a foundationalism in terms of formal logic and traditional ontology, and towards a broader range of approaches that are united by a focus on mathematical practice. The scientific research network PhiMSAMP (Philosophy of Mathematics: Sociological Aspects and Mathematical Practice) consisted of researchers from a variety of backgrounds and fields, brought together by their common interest in the shift of philosophy of mathematics towards mathematical (...) practice. Hosted by the Rheinische Friedrich- Wilhelms-Universitat Bonn and funded by the Deutsche Forschungsgemeinschaft (DFG) from 2006-2010, the network organized and contributed to a number of workshops and conferences on the topic of mathematical practice. The refereed contributions in this volume represent the research results of the network and consists of contributions of the network members as well as selected paper versions of presentations at the network's mid-term conference, "Is mathematics special?" (PhiMSAMP-3) held in Vienna 2008. (shrink)

Deflationsism about truth is a pot-pourri, variously claiming that truth is redundant, or is constituted by the totality of 'T-sentences', or is a purely logical device (required solely for disquotational purposes or for re-expressing finitarily infinite conjunctions and/or disjunctions). In 1980, Hartry Field proposed what might be called a 'deflationary theory of mathematics', in which it is alleged that all uses of mathematics within science are dispensable. Field's criterion for the dispensability of mathematics turns on a property of theories, called (...) conservativeness. I present some technical results, some of which may be found in Tarski (1936), concerning the logical properties of truth theories; in particular, concerning the conservativeness of adding a truth theory for an object level language to any theory expressed in it. It transpires that various deflationary truth theories behave somewhat differently from the standard Tarskian truth theory. These results suggest that Tarskian theories of truth are not redundant or dispensable. Finally, I hint at an analogy between the behaviour of mathematical theories and of standard (Tarskian) theories of truth with respect to their indispensability to, as Quine would put, our 'scientific world-view'. (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)

Much of the current thought concerning mathematical ontology and epistemology follows Quine and Putnam in looking to the indispensable application of mathematics in science. A standard assumption of the indispensability approach is some version of confirmational holism, i.e., that only "sufficiently large" sets of beliefs "face the tribunal of experience." In this paper I develop and defend a distinction between a pure mathematical theory and a mathematized scientific theory in which it is applied. This distinction allows for (...) the possibility that pure mathematical theories are systematically insulated from such confirmation in virtue of their being distinct from the "sufficiently large" blocks of scientific theory that are empirically confirmed. (shrink)

The Quine/Putnam indispensability approach to the confirmation of mathematical theories in recent times has been the subject of significant criticism. In this paper I explore an alternative to the Quine/Putnam indispensability approach. I begin with a van Fraassen-like distinction between accepting the adequacy of a mathematical theory and believing in the truth of a mathematical theory. Finally, I consider the problem of moving from the adequacy of a mathematical theory to its truth. I argue that the (...) prospects for justifying this move are qualitatively worse in mathematics than they are in science. (shrink)

According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this criterion with our own brand of (...) metaphysics, 'Aristotelian realism', in order to identify the truthmakers of mathematics. We consider in particular as a case study the indispensability to physics of real analysis (the theory of the real numbers). We conclude that it is possible to run an indispensability argument without Quinean baggage. (shrink)

Chihara introduces this book as a response to critics of his last book, which gave an account of mathematical objects in terms of possible constructions of open sentences. Several reviewers charged him with exchanging an ontology of platonistic mathematical objects for an equally extravagant ontology of possible entities. In this book Chihara replies with an extended account how one can use modal logic, and even the notions of possible worlds semantics, without accepting merely possible worlds or objects. A (...) final chapter extends his anti-platonist arguments about philosophy of mathematics, directed primarily against Penelope Maddy and the “Indispensibility” argument associated with Quine and Putnam. (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.

Both sides in the debate about scientific realism have argued that their view provides a better account of actual scientific practice. For example, it has been claimed that the practice of theory conjunction presupposes realism, and that scientists' use of multiple and incompatible models presupposes some form of instrumentalism. Assuming that the practices of science are rational, these conclusions cannot both be right. I argue that neither of them is right, and that, in fact, all (...) class='Hi'>scientific practices are compatible with both realism and instrumentalism. I also repudiate van Fraassen's argument to the effect that the instrumentalist account of scientific practice is logically weaker, hence better, than the realist account. In the end, there are no scientific practice arguments on the table that support either side of the debate. It is also noted that the deficiencies of van Fraassen's argument are recapitulated in Putnam's miracle argument for realism. My pessimistic assessment of the state of the debate is reminiscent of Arthur Fine's. However, Fine's argument for the ‘natural ontological attitude’ once again repeats the problems of van Fraassen's and Putnam's arguments. (shrink)

According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this criterion with our own brand of (...) metaphysics, 'Aristotelian (...) realism', in order to identify the truthmakers of mathematics. We consider in particular as a case study the indispensability to physics of real analysis (the theory of the real numbers). We conclude that it is possible to run an indispensability argument without Quinean baggage. (shrink)

Contents: Introduction. NATURALIZED EPISTEMOLOGY. Ton DERKSEN: Naturalistic Epistemology, Murder and Suicide? But what about the Promises! Christopher HOOKWAY: Naturalism and Rationality. Mia GOSSELIN: Quine's Hypothetical Theory of Language Learning. A Comparison of Different Conceptual Schemes of Their Logic. THE NATURE OF PERCEPTUAL KNOWLEDGE. Jaap van BRAKEL: Quine and Innate Similarity Spaces. Dirk KOPPELBERG: Quine and Davidson on the Structure of Empirical Knowledge. Eva PICARDI: Empathy and Charity. ONTOLOGY. Sandra LAUGIER: Quine: Indeterminacy, ‘Robust Realism', and Truth. Roger VERGAUWEN: Quine and Putnam (...) on Conceptual Relativity and Reference: Theft or Honest Toil? Igor DOUVEN: Empiricist Semantics and Indeterminism of Reference. Lieven DECOCK: Domestic Ontology and Ideology. Paul GOCHET: Canonical Notation, Predication and Ontology. About the Authors. (shrink)

First published in 1978, this reissue presents a seminal philosophical work by professor Putnam, in which he puts forward a conception of knowledge which makes ethics, practical knowledge and non-mathematic parts of the social sciences just as much parts of 'knowledge' as the sciences themselves. He also rejects the idea that knowledge can be demarcated from non-knowledge by the fact that the former alone adheres to 'the scientific method'. The first part of the book consists of Professor Putnam's John (...) Locke lectures, delivered at the University of Oxford in 1976, offering a detailed examination of a 'physicalist' theory of reference against a background of the works of Tarski, Carnap, Popper, Hempel and Kant. The analysis then extends to notions of truth, the character of linguistic enquiry and social scientific enquiry in general, interconnecting with the great metaphysical problem of realism, the nature of language and reference, and the character of ourselves. (shrink)

The subject with which Mark Colyvan's book deals is timely indeed. While discussions of mathematical ontology have been a mainstay in philosophy of mathematics for the last century (at least), for the last thirty years or so this discussion has begun with (and often not left) the Quine/Putnam indispensability argument. Though the argument is widely cited, to my knowledge this is the first book-length project exclusively dedicated to articulating and defending the Quine/Putnam indispensability argument for mathematical platonism. In (...) the first section below, I shall run through a summary of the book (following its chapter layout) with a few critical remarks. In the second section I shall raise a few critical issues organized topically. (shrink)

In this collection of recent and unpublished essays, leading analytic philosopher Scott Soames traces milestones in his field from its beginnings in Britain and Germany in the late nineteenth and early twentieth century, through its subsequent growth in the United States, up to its present as the world's most vigorous philosophical tradition. The central essay chronicles how analytic philosophy developed in the United States out of American pragmatism, the impact of European visitors and immigrants, the midcentury transformation of the Harvard (...) philosophy department, and the rapid spread of the analytic approach that followed. Another essay explains the methodology guiding analytic philosophy, from the logicism of Frege and Russell through Wittgenstein's linguistic turn and Carnap's vision of replacing metaphysics with philosophy of science. Further essays review advances in logic and the philosophy of mathematics that laid the foundation for a rigorous, scientific study of language, meaning, and information. Other essays discuss W.V.O. Quine, David K. Lewis, Saul Kripke, the Frege-Russell analysis of quantification, Russell's attempt to eliminate sets with his "no class theory," and the Quine-Carnap dispute over meaning and ontology. The collection then turns to topics at the frontier of philosophy of language. The final essays, combining philosophy of language and law, advance a sophisticated originalist theory of interpretation and apply it to U.S. constitutional rulings about due process. (shrink)

Much of the current thought concerning mathematical ontology in volves in some way the Quine/Putnam indispensability argument. The indispensability approach needs to be more thoroughly specified, however, before substantive progress can be made in assessing it. To this end I examine in some detail the ways in which pure mathematics is applied to physical theory; such considerations give rise to three specific issues with which the indispensability approach must come to grips.

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)

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)

Quine and Putnam argued for mathematical realism on the basis of the indispensability of mathematics to science. They claimed that the mathematics that is used in physical theories is confirmed along with those theories and that scientific realism entails mathematical realism. I argue here that current theories of confirmation suggest that mathematics does not receive empirical support simply in virtue of being a part of well confirmed scientific theories and that the reasons for adopting a realist (...) view of scientific theories do not support realism about mathematical entities, despite the use of mathematics in formulating scientific theory. (shrink)

In this article, I analyze Quine’s conception of science, which is a radical defence of extensionalism on the grounds that first‐order logic is the most adequate logic for science. I examine some criticisms addressed to it, which show the role of modalities and probabilities in science and argue that Quine’s treatment of probability minimizes the intensional character of scientific language and methods by considering that probability is extensionalizable. But this extensionalizing leads to untenable results in some cases and is (...) not consistent with the fact that Quine himself admits confirmation which includes probability. Quine’s extensionalism does not account for this fact and then seems unrealistic, even if science ought to be extensional in so far as it is descriptive and mathematically expressible. (shrink)

Thomas Kelly, “Quine and Epistemology”: For Quine, as for many canonical philosophers since Descartes, epistemology stands at the very center of philosophy. In this chapter, I discuss some central themes in Quine's epistemology. I attempt to provide some historical context for Quine's views, in order to make clear why they were seen as such radical challenges to then prevailing orthodoxies within analytic philosophy. I also highlight aspects of his views that I take to be particularly relevant to contemporary epistemology.

The essential role of language in rational cognition is analysed. The approach is functional: only the results of the connection between language, reality, and thinking are considered. Scientific language is analysed as an extension and improvement of everyday language. The analysis gives a uniform view of language and rational cognition. The consequences for the nature of ontology, truth, logic, thinking, scientific theories, and mathematics are derived.

Quine was one of the foremost philosophers of the Twentieth century. In this outstanding overview of Quine's philosophy, Peter Hylton shows why Quine is so important and how his philosophical naturalism has been so influential within analytic philosophy. Beginning with an overview of Quine's philosophical background in logic and mathematics and the role of Rudolf Carnap's influence on Quine's thought, he goes on to discuss Quine's famous analytic-synthetic distinction and his arguments concerning the nature of the a priori. He also (...) discusses Quine's philosophy of language and epistemology, his celebrated theory of the indeterminacy of translation and his broader views of ontology and modality. This book is essential reading for anyone interested in Quine, twentieth century philosophy and the philosophy of language. (shrink)