John Burgess

While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics on the one hand and computerized formal proofs on the other hand. Along the way, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed.

Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obscured previous discussions of these projects, and presents clear, concise accounts of a dozen strategies for nominalistic interpretation of mathematics, thus equipping the reader to evaluate each and to compare different ones. The authors also offer critical discussion, rare in the literature, of the aims and claims of nominalistic interpretation, suggesting that it is significant in a very different way from that usually assumed.

John Burgess is the author of a rich and creative body of work which seeks to defend classical logic and mathematics through counter-criticism of their nominalist, intuitionist, relevantist, and other critics. This selection of his essays, which spans twenty-five years, addresses key topics including nominalism, neo-logicism, intuitionism, modal logic, analyticity, and translation. An introduction sets the essays in context and offers a retrospective appraisal of their aims. The volume will be of interest to a wide range of readers across philosophy of mathematics, logic, and philosophy of language.

Philosophical Analysis in the Twentieth Century by Scott Soames reminds me of nothing so much as Lectures on Literature by Vladimir Nabokov. Both are works that arose immediately out of the needs of undergraduate teaching, yet each manages to say much of significance to knowledgeable professionals. Each indirectly provides an outline of the history of its field, through a presentation of selected major works, taken in chronological order and including items that are generally recognized as marking decisive turning points. Yet neither Soames’s work nor Nabokov’s is a history in any conventional sense, both being immediately disqualified from that category by the general absence of coverage of minor and middling works and writers. The emphasis is pedagogical rather than historiographical: the emphasis is on introducing the student to the field through very close examination of the limited number of key texts selected for inclusion. The author’s distinctive personality is also apparent in both works. Each writer has a favorite theme he repeatedly sounds: for Soames, the danger of conflating the analytic, the a priori, and the necessary; for Nabokov, the philistinism of expecting an uplifting “message” from works of literary art. Each also includes some quirky, individual selections: The Right and the Good, The Strange Case of Dr. Jekyll and Mr. Hyde. Few others would have taken R. L. Stevenson to be up there with Dickens, Flaubert, and Proust, or W. D. Ross with Russell, Wittgenstein, and Quine. Each also sets aside for separate treatment elsewhere a major body of work one might have expected to be covered. Nabokov reserves Russian literature for a companion volume, while Soames gives only slight coverage to what he describes as “work in logic, the foundations of logic, and the application of logical techniques to the study of language” — a category that in practice turns out to include the bulk of the relevant material (by such writers as Frege, Carnap, and Tarski) that published originally in German without simultaneous English translation..