Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

When a debate seems intractable, with little agreement as to how one might proceed towards a resolution, it is understandable that philosophers should consider whether something might be amiss with the debate itself. Famously in the last century, philosophers of various stripes explored in various ways the possibility that at least certain philosophical debates are in some manner deficient in sense. Such moves are no longer so much in vogue. For one thing, the particular ways they have been made have (...) themselves undergone much critical scrutiny, so that many philosophers now feel that there is, for example, a Quinean response to Carnap, a Gricean reply to Austin, and a diluting proliferation of Wittgenstein interpretations.2 Be that as it may,3 there do of.. (shrink)

First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general.

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.

Four- Dimensionalism defends the thesis that the material world is composed of temporal as well as spatial parts. This defense includes a novel account of persistence over time, new arguments in favour of the four-dimensional ontology, and responses to the challenges four- dimensionalism faces." "Theodore Sider pays particular attention to the philosophy of time, including a strong series of arguments against presentism, the thesis that only the present is real. Arguments offered in favour of four- dimensionalism include novel arguments based (...) on time travel, the debate between spacetime substantivalists and relationalists, and vagueness. Also included is a comprehensive discussion of the paradoxes of coinciding material objects, and a novel resolution of those paradoxes based on temporal counterpart theory. In conclusion Sider replies to prominent objections to four- dimensionalism, including discussion of the problem of the rotating homogenous disk. (shrink)

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic (...) problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians. (shrink)

Hilary Putnam has been convinced for some time that the present situation in philosophy calls for revitalization and renewal; in this latest book he shows us ...

Now in a new edition, this volume updates Davidson's exceptional Inquiries into Truth and Interpretation (1984), which set out his enormously influential philosophy of language. The original volume remains a central point of reference, and a focus of controversy, with its impact extending into linguistic theory, philosophy of mind, and epistemology. Addressing a central question--what it is for words to mean what they do--and featuring a previously uncollected, additional essay, this work will appeal to a wide audience of philosophers, linguists, (...) and psychologists. (shrink)

This volume which was compiled as a festschrift on the occasion of Russell's ninety-fifth birthday, contains a well-chosen combination of articles dealing with Russell's personality, political life, and influence upon public issues; as well as some rather more technical papers concerning Russell's theory of knowledge, sense-perception, his contribution to logical theory, and ontology. Of the popular and semi-popular pieces in the first half of this volume, many good things might be said; but especially worthwhile are I. F. Stone's moving article, (...) "To Oppose the Stream," an appreciation of Russell's courage and intellectual integrity, and Linus Pauling's bone chilling apocalyptic arithmetic in "Would Civilization Survive a Nuclear War?" It should also be added that Schoenman's introduction to the volume may be an important document in itself. The articles by Dana Scott and Georg Kreisel will be accessible only to readers with considerable training in logic and set theory, but will be of utmost interest to such specialists, as they concern the philosophically significant issues of improper descriptions, existence statements, and the extent to which logic can serve as "foundations" for mathematics. Putnam argues against the traditional logicist view and bases a good deal of his criticism upon the occurrence of impredicative definitions in Principia-type systems. The one sour note in the book is Quine's article on Russell's ontology, which occupies eleven rather insubstantial pages. Surely, this important facet of Russell's work deserves a more detailed exposition and analysis.—H. P. K. (shrink)

In , Peter van Inwagen asked a good question. (Asking the right question is often the hardest part.) He asked: what do you have to do to some objects to get them to compose something---to bring into existence some further thing made up of those objects? Glue them together or what?1 Some said that you don’t have to do anything.2 No matter what you do to the objects, they’ll always compose something further, no matter how they are arranged. Thus we (...) learned of the fusion of the coins in our pockets and the Eiffel tower. (shrink)