Sanford Shieh

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.

Three clusters of philosophically significant issues arise from Frege’s discussions of definitions. First, Frege criticizes the definitions of mathematicians of his day, especially those of Weierstrass and Hilbert. Second, central to Frege’s philosophical discussion and technical execution of logicism is the so‐called Hume’s Principle, considered in The Foundations of Arithmetic . Some varieties of neo‐Fregean logicism are based on taking this principle as a contextual definition of the operator ‘the number of …’, and criticisms of such neo‐Fregean programs sometimes appeal to Frege’s objections to contextual definitions in later writings. Finally, a critical question about the definitions on which Frege’s proofs of the laws of arithmetic depend is whether the logical structures of the definientia reflect our pre‐Fregean understanding of arithmetical terms. It seems that unless they do, it is unclear how Frege’s proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes the definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre‐definitional understanding. One or more of these topics may be studied in a survey course in the philosophy of mathematics or a course on Frege’s philosophy. The latter two topics are obviously central in a seminar in the philosophy of mathematics in general or more specialized seminars on logicism, or on mathematical definitions and concept formation.

In this paper I advance an account of the necessity of logic in Wittgenstein’s Tractatus. I reject both the “metaphysical” reading of Peter Hacker, who takes Tractarian logical necessity to consist in the mode of truth of tautologies, and the “resolute” account of Cora Diamond, who argues that all Tractarian talk of necessity is to be thrown away. I urge an alternative conception based on remarks 3.342 and 6.124. Necessity consists in what is not arbitrary, and contingency in what is up to our arbitrary choice, in the symbols we use, in how we picture or model the world. Necessity is not a mode of truth of propositions, but lies in the requirements of their intelligibility. I argue that this conception is implicit in certain “resolute” readings and in some of their critics. Both sides of the dispute are committed to certain logical features of language or thought, patterns of symbolizing constitutive of intelligibility that are not up to us to institute or alter. This conception of non-arbitrary patterns of symbolizing, I argue, is what logical syntax in the Tractatus consists in. I also argue that the well-known Tractarian view of propositions as truth-functions of elementary propositions can be understood in terms of patterns of norms governing our making sense with the affirmation and denial of propositions.