Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel (...) version of mathematical realism. She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

It is one of the tasks of the philosophy of mathematics to explain in a Kantian way, how and to what extent, mathematics as we know it, is possible. It should be clear, however, that, as things stand at present, a philosophical theory of mathematics can in this respect, it seems, be little more than a declaration of faith or the lack of it.

The conflict between Platonic realism and Constructivism marks a watershed in philosophy of mathematics. Among other things, the controversy over the Axiom of Choice is typical of the conflict. Platonists accept the Axiom of Choice, which allows a set consisting of the members resulting from infinitely many arbitrary choices, while Constructivists reject the Axiom of Choice and confine themselves to sets consisting of effectively specifiable members. Indeed there are seemingly unpleasant consequences of the Axiom of Choice. The non-constructive (...) nature of the Axiom of Choice leads to the existence of non-Lebesgue measurable sets, which in turn yields the Banach-Tarski Paradox. But the Banach-Tarski Paradox is so called in the sense that it is a counter-intuitive theorem. To corroborate my view that mathematical truths are of non-constructive nature, I shall draw upon Gödel’s Incompleteness Theorems. This also shows the limitations inherent in formal methods. Indeed the Löwenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to Platonists. In this light, Quine/Putnam’s arguments come to take on a clear meaning. According to the model-theoretic arguments, the Axiom of Choice depends for its truth-value upon the model in which it is placed. In my view, however, this is another limitation inherent in formal methods, not a defect for Platonists. To see this, we shall examine how mathematical models have been developed in the actual practice of mathematics. I argue that most mathematicians accept the Axiom of Choice because the existence of non-Lebesgue measurable sets and the Well-Ordering of reals open the possibility of more fruitful mathematics. Finally, after responding to Benacerraf’s challenge to Platonism, I conclude that in mathematics, as distinct from natural sciences, there is a close connection between essence and existence. Actual mathematical theories are the parts of the maximally logically consistent theory that describes mathematical reality. (shrink)

The paper is concerned with the way in which “ontology” and “realism” are to be interpreted and applied so as to give us a deeper philosophical understanding of mathematical theories and practice. Rather than argue for or against some particular realistic position, I shall be concerned with possible coherent positions, their strengths and weaknesses. I shall also discuss related but different aspects of these problems. The terms in the title are the common thread that connects the various sections.

Maddy has written a concise and lucid book which covers not only her own ideas on set-theoretic realism and the "reality" of mathematics, but also provides a useful summary of the old and continuing debates surrounding mathematical ontology. This feature makes the book ideal as a classroom text. A basic familiarity with elementary set theory and number theory is obviously helpful, particularly in her technical discussions towards the end of the book, but the mathematics should not present (...) a barrier to most advanced students of philosophy. (shrink)

In an attempt to justify research efforts in various branches of science, scholars have tried to capture the essence of the relevant subject-matters in a definition, or at least have declared these subject-matters to exist. Otherwise the study of topics in the branches would be of questionable value, to say the least. For example, when dealing with numbers, their ontological status somehow has to be declared.

In an attempt to justify research efforts in various branches of science, scholars have tried to capture the essence of the relevant subject-matters in a definition, or at least have declared these subject-matters to exist. Otherwise the study of topics in the branches would be of questionable value, to say the least. For example, when dealing with numbers, their ontological status somehow has to be declared.

I show that Wittgenstein's critique of G.H. Hardy's mathematical realism naturally extends to Paul Benacerraf's influential paper, ‘Mathematical Truth’. Wittgenstein accuses Hardy of hastily analogizing mathematical and empirical propositions, thus leading to a picture of mathematical reality that is somehow akin to empirical reality despite the many puzzles this creates. Since Benacerraf relies on that very same analogy to raise problems about mathematical ‘truth’ and the alleged ‘reality’ to which it corresponds, his major argument falls prey to the same (...) critique. The problematic pictures of mathematical reality suggested by Hardy and Benacerraf can be avoided, according to Wittgenstein, by disrupting the analogy that gives rise to them. I show why Tarskian updates to our conception of ‘truth’ discussed by Benacerraf do not answer Wittgenstein's concerns. That is, because they merely presuppose what Wittgenstein puts into question, namely, the essential uniformity of ‘truth’ and ‘proposition’ in ordinary discourse. (shrink)

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 (...) class='Hi'>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)

This article attempts to motivate a new approach to anti-realism (or nominalism) in the philosophy of mathematics. I will explore the strongest challenges to anti-realism, based on sympathetic interpretations of our intuitions that appear to support realism. I will argue that the current anti-realistic philosophies have not yet met these challenges, and that is why they cannot convince realists. Then, I will introduce a research project for a new, truly naturalistic, and completely scientific approach to philosophy (...) of mathematics. It belongs to anti-realism, but can meet those challenges and can perhaps convince some realists, at least those who are also naturalists. (shrink)

The aim of this paper is to identify some of the motivations that can be found for taking a realist position concerning mathematical entities and to examine these motivations in the light of a case study in contemporary mathematics. The motivations that are found are as follows: (some) mathematicians are realists, mathematical statements are true, and finally, mathematical statements have a special certainty. These claims are compared with a result in algebraic topology stating that a certain sequence, the so-called (...) Mayer-Vietoris sequence, has different properties when placed in different categories. The conclusion is that the before mentioned motivations should be modified and it is suggested that they could also be explained by a position claiming that mathematical entities are introduced by mathematicians. (shrink)

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining (...) abstract mathematical entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. (shrink)

In this book Robert Tragesser sets out to determine the conditions under which a realist ontology of mathematics and logic might be justified, taking as his starting point Husserl's treatment of these metaphysical problems. He does not aim primarily at an exposition of Husserl's phenomenology, although many of the central claims of phenomenology are clarified here. Rather he exploits its ideas and methods to show how they can contribute to answering Michael Dummet's question 'Realism or Anti-Realism?'. In (...) doing so he makes a challenging and provocative contribution to the debate. (shrink)

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining (...) abstract mathematical entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. (shrink)

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

In her recent book, Realism in mathematics, Penelope Maddy attempts to reconcile a naturalistic epistemology with realism about set theory. The key to this reconciliation is an analogy between mathematics and the physical sciences based on the claim that we perceive the objects of set theory. In this paper I try to show that neither this claim nor the analogy can be sustained. But even if the claim that we perceive some sets is granted, I argue (...) that Maddy's account fails to explain the key issue faced by an epistemology for mathematics, namely the step from knowledge of the finite to knowledge of the infinite. (shrink)

In his influential book, The Nature of Morality, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles.” What is the epistemological relevance of this contrast, if genuine? This chapter argues that ethicists and philosophers of mathematics have misunderstood it. They have confused what the chapter calls the justificatory challenge for realism about an (...) area, D—the challenge to justify our D-beliefs—with the reliability challenge for D-realism—the challenge to explain the reliability of our D-beliefs. Harman’s contrast is relevant to the first, but not, evidently, to the second. One upshot of the discussion is that genealogical debunking arguments are fallacious. Another is that indispensability considerations cannot answer the Benacerraf–Field challenge for mathematical realism. (shrink)

Tragesser intends to show that Husserl in his phenomenological investigation of the foundations of logic and mathematics undercuts the basis on which the problem of realism and antirealism in epistemology and the philosophy of logic is traditionally conceived. Husserl does this, Tragesser contends, by attempting "to purge logical thinking of [the] assumption [of the law of the excluded middle] while at the same time avoiding the pitfalls of psychologism". Central to this investigation is Husserl's disclosure of the noema, (...) the intentional correlate of an act of consciousness, as the thought-component of acts of thinking. Thus, phenomenology, on Tragesser's account, is the descriptive theory of "the domain of all possible thoughts and thus also all possible entities which can be thought". (shrink)

Wittgenstein is generally supposed to have abandoned in the 1930's a realistic conception of the meaning of mathematical propositions, founded on the idea of tmth-conditions which could in certain cases transcend any possibility of verification, for a realistic one, where the idea of truth-conditions is replaced by that of conditions of justification of assertability. It is argued that for Wittgenstein mathematical propositions, which are, as he says, "grammatical" propositions, have a meaning and a role which differ to a much greater (...) degree from those of ordinary propositions than either platonistic realism or intuitionistic anti-realism would admit, and that is the tendency to assimilate the mathematical proposition to an ordinary descriptive proposition which confers on it an appearance of meaning independent of the possibility of proving it, and not, as Dummett would say, that it is a decision concerning the kind of meaning it has which gives it the status of a proposition describing a determinate objective reality. (shrink)

Wittgenstein is generally supposed to have abandoned in the 1930's a realistic conception of the meaning of mathematical propositions, founded on the idea of tmth-conditions which could in certain cases transcend any possibility of verification, for a realistic one, where the idea of truth-conditions is replaced by that of conditions of justification of assertability. It is argued that for Wittgenstein mathematical propositions, which are, as he says, "grammatical" propositions, have a meaning and a role which differ to a much greater (...) degree from those of ordinary propositions than either platonistic realism or intuitionistic anti-realism would admit, and that is the tendency to assimilate the mathematical proposition to an ordinary descriptive proposition which confers on it an appearance of meaning independent of the possibility of proving it, and not, as Dummett would say, that it is a decision concerning the kind of meaning it has which gives it the status of a proposition describing a determinate objective reality. (shrink)

When we examine the history of astronomy up to the end of the seventeenth century by considering the relation between mathematical astronomy and natural philosophy, it has been argued that there were two groups of philosophers and astronomers: instrumentalists and realists. However, this classification is deficient when we consider attitudes toward the explanatory power of mathematics in determining astronomical theories. I offer the solution of dividing realists into two subcategories—mathematical realists and physical realists. Mathematical realists include those who thought (...) mathematics could provide the real motions of celestial bodies. The physical realists, who include Ibn Rushd (Averroës), believed mathematics could not provide us with the real structure of the heavens. To avoid contradictions between the physical and mathematical models, the physical realists held that a mathematical model of the heavens should be designed by considering physics. I argue, in this article, that Francis Bacon can be considered a physical realist. (shrink)

This book contains ten papers that were presented at the symposium about the realism debate, held at the Center for Logic, Philosophy of Science and Philosophy of Language of the Institute of Philosophy at the Katholieke Universiteit Leuven on 10 and 11 March 1995. The first group of papers are directly concerned with the realism/anti-realism debate in the general philosophy of science. This group includes the articles by Ernan McMullin, Diderik Batens/Joke Meheus, Igor Douven and Herman de (...) Regt. The papers of the second group concentrate on specific problems arising from the realism/anti-realism debate. Theo Kuipers' contribution discusses the problem of truth-approximation. Roger Vergauwen's article pertains to the issue of realism in the philosophy of mind and semantics. Jaap van Brakel's article focusses on the relation between everyday concepts and scientific concepts, and on the theory-dependence of observation. Paul Cortois investigates the relation between the question of realism and Kuhn's concept of incommensurability between scientific theories. The final group contains two papers on the realism/anti-realism debate in the special sciences. James Cushing discusses the problem of underdetermination in quantum mechanics and Jean Paul van bendegem addresses the question of the possiblity of an empiricist philosophy of mathematics. (shrink)

In his 1951 Gibbs Lecture Gödel formulates the central implication of the incompleteness theorems as a disjunction: either the human mind infinitely surpasses the powers of any finite machine or there exist absolutely unsolvable diophantine problems (of a certain type). In his later writings in particular Gödel favors the view that the human mind does infinitely surpass the powers of any finite machine and there are no absolutely unsolvable diophantine problems. I consider how one might defend such a view in (...) light of Gödel's remark that one can turn to ideas in Husserlian transcendental phenomenology to show that the human mind ‘contains an element totally different from a finite combinatorial mechanism’. (shrink)

Edited book on the prospects of non-Platonist realism in the philosophy of mathematics. Physicalism holds that mathematics studies properties realised or realisable in the physical world. This collection of papers has its origin in a conference held at the University of Toronto in June of 1988. The theme of the conference was Physicalism in Mathematics: Recent Work in the Philosophy of Mathematics. At the conference, papers were read by Geoffrey Hellman (Minnesota), Yvon Gauthier (Montreal), Michael (...) Hallett (McGill), Hartry Field (USC), Bob Hale (Lancaster & St Andrew's), Alasdair Urquhart (Toronto) and Penelope Maddy (Irvine). This volume supplements updated versions of six of those papers with contributions by Jim Brown (Toronto), John Bigelow (La Trobe), John Burgess (Princeton), Chandler Davis (Toronto), David Papineau (Cambridge), Michael Resnik (North Carolina at Chapel Hill), Peter Simons (Salzburg) and Crispin Wright (St Andrews & Michigan). Together they provide a vivid, expansive snapshot of the exciting work which is currently being carried out in philosophy of mathematics. (shrink)

Metaphysical anti-realism is a large and heterogeneous group of views that do not share a common thesis but only share a certain family resemblance. Views as different as mathematical nominalism—the view that numbers do not exist—, ontological relativism—the view that what exists depends on a perspective—, and modal conventionalism—-the view that modal facts are conventional—all are versions of metaphysical anti-realism. As the latter two examples suggest, relativist ideas play a starring role in many versions of metaphysical anti-realism. (...) But what does it mean for the existence of something to “depend on” a perspective, or for a modal fact to “depend on” a convention? We can distinguish between various dependence relations, giving rise to an array of drastically different forms of metaphysical anti-realism. This article offers a guided tour. I develop a systematic distinction between various forms of metaphysical anti-realism with a focus on the role of relativist ideas in this landscape. (shrink)

Hilary Putnam suggests that the essence of the realist conception of mathematics is that the statements of mathematics are objective so that the true ones are objectively true. An argument for mathematical realism, thus conceived, is implicit in Putnam's writing. The first premise is that within currently accepted science there are objective truths. Next is the premise that some of these statements logically imply statements of pure mathematics. The conclusion drawn is that some statements of pure (...) mathematics are objectively true. A key principle assumed is that if one statement logically implies a second, then if the first is objectively true so is the second. A question about this principle is raised and answered. The problem with the argument is with the second premise. (shrink)

Just as the accuracy of scientific theories is best tested in extreme physical conditions, it is advisable to verify the accuracy of a recognized conception of language on its extreme parts. Mathematical statements meet this role, thanks to the notion of truth and proof. Michael Dummett’s anti-realism is an enterprise that has attempted on this basis to question the notion of the functioning of language-based primarily on the principle of bivalence, the truth-condition theory of meaning, and the notion that (...) the speaker must be able to demonstrate his knowledge of meaning publicly. In common language practice, one can observe assertions that we can neither verify nor refute in principle. On these so-called undecidable statements, Dummett tried to show that if we apply the traditional description to them, we inevitably reach paradoxical conclusions. Mathematical statements referring to an infinite number may be examples of these assertions. In the submitted paper, I will present Dummett’s position resulting primarily in a manifestation and acquisition argument, according to which it should not be possible to understand undecidable statements at all. In conclusion, however, I will show that his intention – despite many valuable comments – fails, i.e. that there is a way to avoid both arguments while preserving the realistic description of the language in general. Key words: anti-realism, mathematical statements, meaning, Michael Dummett, truth, truth-condition theory of meaning. (shrink)