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)

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)

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)

This book consists of an introduction by the editor, eleven of Plantinga’s previously published pieces, and an index. The previously published works are presented in the following chronological order: “De Re et De Dicto” (1969); “World and Essence” (1970); “Transworld Identity or Worldbound Individuals?” (1973); Chapter VIII of The Nature of Necessity (1974); “Actualism and Possible Worlds” (1976); “The Boethian Compromise” (1978); “De Essentia” (1979); “On Existentialism” (1983); “Reply to John L. Pollock” (1985); “Two Concepts of Modality: Modal Realism and (...) Modal Reductionism” (1987); and “Why Propositions Cannot Be Concrete” (1993). (shrink)