One of the important challenges in the philosophy of mathematics is to account for the semantics of sentences that express mathematical propositions while simultaneously explaining our access to their contents. This is Benacerraf’s Dilemma. In this dissertation, I argue that cognitive science furnishes new tools by means of which we can make progress on this problem. The foundation of the solution, I argue, must be an ontologically realist, albeit non-platonist, conception of mathematical reality. The semantic portion of the problem can (...) be addressed by accepting a Chomskyan conception of natural languages and a matching internalist, mentalist and nativist view of semantics. A helpful perspective on the epistemic aspect of the puzzle can be gained by translating Kurt G ̈odel’s neo-Kantian conception of the nature of mathematics and its objects into modern, cognitive terms. (shrink)

What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the the current debate on (...) Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem. (shrink)

In this paper, I consider an argument for the claim that any satisfactory epistemology of mathematics will violate core tenets of naturalism, i.e. that mathematics cannot be naturalized. I find little reason for optimism that the argument can be effectively answered.

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

During the past few decades, a radical shift has occurred in how philosophers conceive of the relation between science and philosophy. A great number of analytic philosophers have adopted what is commonly called a ‘naturalistic’ approach, arguing that their inquiries ought to be in some sense continuous with science. Where early analytic philosophers often relied on a sharp distinction between science and philosophy—the former an empirical discipline concerned with fact, the latter an a priori discipline concerned with meaning—philosophers today largely (...) follow Willard Van Orman Quine (1908-2000) in his seminal rejection of this distinction. -/- This book offers a comprehensive study of Quine’s naturalism. Building on Quine’s published corpus as well as thousands of unpublished letters, notes, lectures, papers, proposals, and annotations from the Quine archives, this book aims to reconstruct both the nature (chapters 2-4) and the development (chapter 5-7) of his naturalism. As such, this book aims to contribute to the rapidly developing historiography of analytic philosophy, and to provide a better, historically informed, understanding of what is philosophically at stake in the contemporary naturalistic turn. (shrink)

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

Quine is routinely perceived as having changed his mind about the scope of the Duhem-Quine thesis, shifting from what has been called an 'extreme holism' to a more moderate view. Where the Quine of 'Two Dogmas of Empiricism' argues that “the unit of empirical significance is the whole of science” (1951, 42), the later Quine seems to back away from this “needlessly strong statement of holism” (1991, 393). In this paper, I show that the received view is incorrect. I distinguish (...) three ways in which Quine's early holism can be said to be wide-scoped and show that he has never changed his mind about any one of these aspects of his early view. Instead, I argue that Quine's apparent change of mind can be explained away as a mere shift of emphasis. (shrink)

There is little in Quine’s philosophy that is more significant and more puzzling than his commitment to naturalism. On the one hand, naturalism seems to play an unparalleled role in explaining the development and unorthodox nature of Quine’s views. On the other hand, however, naturalism is deeply elusive. Not only is there disagreement amongst commentators about how to understand the nature and development of naturalism, but also Quine’s own characterisations of naturalism are often thinly sketched and leave us with few (...) clues as to how we should understand naturalism and its origins. In light of these reasons, it is a significant challenge to arrive at an accurate picture of what naturalism amounts to and how it fits into Quine’s philosophical development. The purpose of this thesis is to provide an analysis of Quine’s commitment to naturalism, and to highlight the ways in which this analysis illuminates the surrounding landscape of his philosophical views. Here, I endeavour to show that careful analysis into what Quine’s commitment to naturalism amounts to establishes that this commitment is not as longstanding as is often thought. More specifically, I argue that Quine is not committed to naturalism until shortly after ‘Two Dogmas of Empiricism’, and this reveals that Quine’s philosophical work cannot be felicitously understood as revolving around a stable commitment to naturalism. Having established this, I then explain the ways in which this analysis informs a thorough interpretation of several vital aspects of his thought, such as the development of his attitude towards analyticity, and his sustained commitment to empiricism. I believe that these findings motivate and lay the foundations of a cautious and appropriately nuanced understanding of Quine. (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 revisit my 1993 paper on Putnam and mathematical realism focusing on the indispensability argument and how it has fared over the years. This argument starts from the claim that mathematics is an indispensable part of science and draws the conclusion, from holistic considerations about confirmation, that the ontology of science includes abstract objects as well as the physical entities science deals with.

S. Shapiro, The Oxford handbook of philosophy of mathematics and logic Oxford and New York: Oxford University Press, 2005. xv + 833 pp. £52.00. ISBN 0-19-514...