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.

Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. Chihara builds upon his previous work, in (...) which he presented a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings.A Structural Account of Mathematics will be required reading for anyone working in this field. (shrink)

Charles Chihara gives a thorough critical exposition of modal realism, the philosophical doctrine that there exist many possible worlds of which the actual world--the universe in which we live--is just one. The striking success of possible-worlds semantics in modal logic has made this ontological doctrine attractive. Modal realists maintain that philosophers must accept the existence of possible worlds if they wish to have the benefit of using possible-worlds semantics to assess modal arguments and explain modal principles. Chihara (...) challenges this claim, and argues instead for modality without worlds; he offers a new account of the role of interpretations or structures of the formal languages of logic. (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 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.

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)

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)

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)

Sketches the basic idea for the approach taken. A mathematical system is to be developed in which the existential theorems of traditional mathematics are to be replaced by constructibility theorems: where, in traditional mathematics, it is asserted that such and such exists, it will be asserted in this system that something or other can be constructed. Thus, constructibility quantifiers are introduced in this chapter as logical constants of formal systems. The logic of the constructibility quantifier is explained in each case (...) via possible worlds semantics, which is used as a didactic or heuristic device. (shrink)

Concerns an attempted refutation of nominalism put forward by John Burgess in the form of a dilemma argument. Argues that Burgess's argument is based upon a false dilemma.

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.

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.

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.

Focuses on Hartry Field's Instrumentalism. The ‘Conservation Theorems’, upon which Field bases so much of his form of Instrumentalism, are examined in detail, as is Field's attempt to ‘nominalize’ physics. Doubts are raised about the adequacy of Field's views of mathematics and physics, and a detailed comparison with the Constructibility Theory is presented.

The promised mathematical system—the Constructibility Theory—is presented as an axiomatized deductive theory formalized in a many‐sorted first‐order logical language. The axioms of the theory are specified and a justification for each of the axioms is given. Objections to the theory are considered.