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.

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)

How can the infinite, a subject so remote from our finite experience, be an everyday tool for the working mathematician? Blending history, philosophy, mathematics, and logic, Shaughan Lavine answers this question with exceptional clarity. Making use of the mathematical work of Jan Mycielski, he demonstrates that knowledge of the infinite is possible, even according to strict standards that require some intuitive basis for knowledge.

An engaging account of the origins of the modern mathematical theory of the infinite, his book is also a spirited defense against the attacks and misconceptions ...

'Mass terms', words like water, rice and traffic, have proved very difficult to accommodate in any theory of meaning since, unlike count nouns such as house or dog, they cannot be viewed as part of a logical set and differ in their grammatical properties. In this study, motivated by the need to design a computer program for understanding natural language utterances incorporating mass terms, Harry Bunt provides a thorough analysis of the problem and offers an original and detailed solution. An (...) extension of classical set theory, Ensemble Theory, is defined, and this provides the conceptual basis of a framework for the analysis of natural language meaning which Dr Bunt calls Two-level model-theoretic semantics. The validity of the framework is convincingly demonstrated by the formal analysis of a fragment of English including sentences with quantified and modified mass terms. Separate chapters of the book are devoted to an axiomatic definition of Ensemble Theory and a detailed discussion of its status as a mathematical formalism. (shrink)

This paper investigates the ontological presuppositions of quantifier logic. It is seen that the actual infinite, although present in the usual completeness proofs, is not needed for a proper semantic foundation. Additionally, quantifier logic can be given an adequate formulation in which neither the notion of individual nor that of a predicate appears.

We define a first-order theory FIN which has a recursive axiomatization and has the following two properties. Each finite part of FIN has finite models. FIN is strong enough to develop that part of mathematics which is used or has potential applications in natural science. This work can also be regarded as a consistency proof of this hitherto informal part of mathematics. In FIN one can count every set; this permits one to prove some new probabilistic theorems.

Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now some of my fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, and there (...) are quite a few highly technical journals in logic, such as The Journal of Sym-. (shrink)

Recovers some of the value in the Wittgensteinian period of philosophy, using certain logical systems: Prior's theory of operators and Hilbert's epsilon calculus. This work applies, discursively, the previous largely technical results published in Prolegomena to Formal Logic (Aldershot, Gower 1989) and Intensional Logic (Aldershot, Ashgate 1994) to resolve matters of current interest in philosophy, logic and linguistics - notably attacking a variety of realisms found in comtemporary cognitive science and the philosophy of mathematics.

This book will appeal to mathematicians and philosophers interested in the foundations of mathematics.

NOWADAYS, even schoolchildren babble about "null sets" and "singletons" and "one-one correspondences," as if they knew what they were talking about. But if they understand even less than their teachers, which seems likely, they must be using the technical jargon with only an illusion of understanding. Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should (...) be child's play. (shrink)