This book is concerned with `the problem of existence in mathematics'. It develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. It explores the philosophical implications of such an approach through an examination of the writings of Field, Burgess, Maddy, Kitcher, and others.

A powerful challenge to some highly influential theories, this book offers a thorough critical exposition of modal realism, the philosophical doctrine that many possible worlds exist of which our own universe is just one. Chihara challenges this claim and offers a new argument for modality without worlds.

This paper explores some lines of argument in wittgenstein's post-Tractatus writings in order to indicate the relations between wittgenstein's philosophical psychology, On the one hand, And his philosophy of language, His epistemology, And his doctrines about the nature of philosophical analysis on the other. The authors maintain that the later writings of wittgenstein express a coherent doctrine in which an operationalistic analysis of confirmation and language supports a philosophical psychology of a type the authors call "logical behaviorism." they also maintain (...) that there are good grounds for rejecting the philosophical theory implicit in wittgenstein's later works. In particular, They first argue that wittgenstein's position leads to some implausible conclusions concerning the nature of language and psychology; second, They maintain that the arguments wittgenstein provides are inconclusive; and third, They sketch an alternative position which they believe avoids many of the difficulties implicit in wittgenstein's philosophy. (shrink)

Nominalism is the view that abstract mathematical objects like numbers, functions, and sets do not exist. The chapter articulates and defends a variety of nominalism, based on a reading of mathematical statements in terms of possible linguistic constructions. The chapter responds directly to a recent study of nominalism by Gideon Rosen and John Burgess, and develops a reply to the Quine-Putnam indispensability argument for the existence of mathematical objects.

In the final chapter of their book A Subject With No Object, John Burgess and Gideon Rosen raise the question of the value of the nominalistic reconstructions of mathematics that have been put forward in recent years, asking specifically what this body of work is good for. The authors conclude that these reconstructions are all inferior to current versions of mathematics (or science) and make no advances in science. This paper investigates the reasoning that led to such a negative appraisal, (...) and it produces a rebuttal to this reasoning. I am grateful to the following mathematicians who were kind enough to provide me with their thoughts about nonstandard analysis: Martin Davis, Laura Chihara, Ted Chihara, Steve Galovich, Bonnie Gold, and especially Roger Simons, whose comments about an earlier version of this paper were very helpful. Thanks also go to two referees for their useful suggestions and criticisms of an earlier version of this paper. (shrink)

The present paper will argue that, for too long, many nominalists have concentrated their researches on the question of whether one could make sense of applications of mathematics (especially in science) without presupposing the existence of mathematical objects. This was, no doubt, due to the enormous influence of Quine's "Indispensability Argument", which challenged the nominalist to come up with an explanation of how science could be done without referring to, or quantifying over, mathematical objects. I shall admonish nominalists to enlarge (...) the target of their investigations to include the many uses mathematicians make of concepts such as structures and models to advance pure mathematics. I shall illustrate my reasons for admonishing nominalists to strike out in these new directions by using Hartry Field's nominalistic view of mathematics as a model of a philosophy of mathematics that was developed in just the sort of way I argue one should guard against. I shall support my reasons by providing grounds for rejecting both Field's fictionalism and also his deflationist account of mathematical knowledge—doctrines that were formed largely in response to the Indispensability Argument. I shall then give a refutation of Mark Balaguer's argument for his thesis that fictionalism is "the best version of anti-realistic anti-platonism". (shrink)

This paper addresses John Burgess's answer to the ‘Benacerraf Problem’: How could we come justifiably to believe anything implying that there are numbers, given that it does not make sense to ascribe location or causal powers to numbers? Burgess responds that we should look at how mathematicians come to accept: There are prime numbers greater than 1010That, according to Burgess, is how one can come justifiably to believe something implying that there are numbers. This paper investigates what lies behind Burgess's (...) answer and ends up as a rebuttal to Burgess's reasoning. (shrink)

The fundamentals of cardinality theory are laid out within the framework of the Constructibility Theory. Finite cardinality theory is developed along the lines described by Frege in his Foundations of Arithmetic, and applications of theory are discussed.

Since the constructibility quantifiers, used in the mathematical system to be developed, will all assert the constructibility of open sentences, an explanation is given of the kinds of open sentences that will be asserted to be constructible. Each of these open sentences will be assigned to a specific ‘level’, depending on the kind of objects or open sentences that can satisfy it, thus providing the basis for the Simple Type Theoretical characteristic of the system to be developed. The satisfaction relation (...) under discussion will be taken to be a primitive of the system. The relevance of Quine's objections to modality for this system is taken up. (shrink)

Takes up Field's version of Logicism—a position that he calls ‘deflationism’. Unlike traditional Logicists, Field does not analyse mathematical propositions into purely logical ones, but he does analyse mathematical knowledge into logical knowledge. Several objections are raised to deflationism, revolving around Field's contention that mathematics consists mostly of falsehoods. Contends that, although mathematics, literally and platonically construed, is not true, it does convey genuine information.

In this article, I compare Gottlob Frege's and Bernard Bolzano's rationalist conceptions of arithmetic. Each philosopher worked out a complicated system of propositions, all of which were set forth as true. The axioms, or basic truths, make up the foundations of the subject of arithmetic. Each member of the system which is not an axiom is related (objectively) to the axioms at the base. Even though this relation to the base may not yet be scientifically proven, the propositions of the (...) system include all of the truths of the science of arithmetic. I conclude the article by analyzing the respective views of Frege and Bolzano in the light of Gödel's first incompleteness theorem. (shrink)

Compares and contrasts Philip Kitcher's Ideal Agent account of mathematics with the constructibility view of this work. Raises a variety of doubts about the cogency of Kitcher's account and points out several weaknesses in the account.

Briefly sketches a standard form of the development of analysis within the Constructibility Theory. Then develops an axiomatized theory of lengths, in terms of which a system of rational and real numbers is specified. These developments are used to provide the basis for a theory of functions of real and complex variables.

The first of six chapters in which rival views are critically evaluated and compared with the Constructibility view described in earlier chapters. The views considered here are those of Stewart Shapiro and Michael Resnik. A number of difficulties with these two views are detailed and it is explained how the Constructibility Theory is not troubled by the problems that Structuralism was explicitly developed to resolve.

Penelope Maddy has attempted to develop a form of realism in mathematics that is not plagued by the sort of epistemological problems that beset traditional Platonism. Maddy advances the radical doctrine that we can and do causally interact with sets. We can see them, feel them, smell them, and even taste them. This chapter raises a series of objections to Maddy's version of realism.

In "human beings", "studies in the philosophy of wittgenstein" (ed. By p winch), J cook presents a radical solution to the problem of other minds and then suggests that this treatment of the problem is to be found in the writings of wittgenstein. According to cook's interpretation, Wittgenstein's analysis of the problem does not involve in any essential way any special doctrines about criteria, Nor does it commit him to any form of behaviorism. In the course of arguing these theses, (...) Cook raises a number of objections to an earlier paper I wrote with j fodor entitled "operationalism and ordinary language: a critique of wittgenstein", "american philosophical quarterly". In the present paper, I respond to cook's article by arguing that his solution to the problem of other minds is defective, By pointing out that some of his objections are based on misreadings of our paper, And by showing how the bulk of his objections are supported by confused arguments and implausible interpretations of wittgenstein's writings. (shrink)