Sanford Shieh

The days when Frege was more footnoted than read are now long gone; still, until very recently he has been read rather selectively. No doubt many had an inkling that there’s more to Frege than the sense/reference distinction; but few, one suspects, thought that his philosophy of mathematics was as fertile and intriguing as the present collection demonstrates. Perhaps, as Paul Benacerraf’s essay in this collection suggests, logical positivism should be held partly responsible for the neglect of this aspect of Frege’s thought, since it contributed to the following common impression of Frege’s philosophy of mathematics: Frege’s project was to reduce arithmetic to logic, in order to provide an empiricist account of mathematical knowledge; Russell’s paradox showed that this was not possible, and that the best that can be done is to reduce arithmetic to set theory; so, Frege’s logicism is demonstrably a philosophical dead end.

This book is a study of the philosophy of arithmetic in one of the most significant periods of its history—from Frege to Carnap—prefaced by an account of Kant. Potter aims at a philosophical history, a story told from an explicit interpretative perspective. These theories of arithmetic are seen as attempts to account for its “source of content” and “source of concepts.” Potter never explains these terms; I take the former to be the thing that, when we have knowledge of it or insight into it, provides the ultimate justification for pure and applied arithmetical discourse. That is, Potter’s principal concern is both ontological and epistemological: what determines, and how do we know, the truth of the statements that express our putative arithmetical knowledge? The criterion of success of a theory of arithmetic is how well it explains arithmetic’s simultaneous necessity or apriority and empirical applicability. Potter divides the theories he discusses into four main types, determined by their accounts of the “source of content” and “concepts”: sensibility, thought, language, and the world. At the end, Potter finds that none succeeds in accounting for all of arithmetic: at best some capture the infinity of the natural numbers, but none gives the “source” of arithmetical concepts. But the “similarities … between the scopes of the various approaches” suggest a Gödelian moral for Potter’s tale: the philosophy of arithmetic must treat concepts of “thought structures or … contents,” where “reflection upon the meanings” of “propositions about these mental objects” is needed for their proofs.

Alongside Richard Rorty, Hilary Putnam and Jacques Derrida, Stanley Cavell is arguably one of the best-known philosophers in the world. In this state-of-the-art collection, Alice Crary explores the work of this original and interesting figure who has already been the subject of a number of books, conferences and Phd theses. A philosopher whose work encompasses a broad range of interests, such as Wittgenstein, scepticism in philosophy, the philosophy of art and film, Shakespeare, and philosophy of mind and language, Cavell has also written much about Henry Thoreau and Ralph Waldo Emerson. Including contributions from Hilary Putnam, Cora Diamond, Jim Conant and Stephen Mulhall, this book is a must-have for libraries and students alike.