Wittgenstein Objects and Russell Particulars

Wittgenstein Objects and Russell Particulars

Introduction

What is a number actually? — A question of great mathematical significance yet only vaguely understood in philosophy, dealt with by people of different background and persuasion, and even by now without a final answer. A well-known approach is the Frege-Russell definition of number. It defines the "cardinality" of a concept C as the class of all concepts that have a one-one correlation (also called a "one-one correspondence," or a "bijection") with C, which is, intuitively speaking, the class of all concepts that have the same number of objects falling under them as C has. The collection of numbers is then defined to be the collection of such classes a collection that can be explicitly characterized by inductively defining (what I will call) "standard" concepts of arbitrary cardinality.

Discussion

Several philosophers have, with varying degrees of success, impugned this definition. Its most famous critic, the French philosopher Henri Poincare, blamed the principle of mathematical induction for a circulus vitiosus in the definition (see his (Poincare 1952), while other critics simply observed that if for instance the number two were to be the class of all classes of two elements, it would change every time a pair of twins were born. Throughout his philosophical carrier, Wittgenstein, too, has been concerned with the notion of number. He first explained it by means of the "general form of a natural number" (TLP 6.03), a view he later gave up in favor of an account in terms of family resemblance (Diamond, 1975). I will not deal in this paper with these positive contributions, but rather discuss two explicit formulations of objections against the Frege-Russell definition. The first one appears in Friedrich Waismann's notes of a meeting of the Vienna Circle on January 4, 1931 (Waismann 1967), while the second and considerably different -- objection is taken from Wittgenstein's 1939 lectures at Cambridge (Diamond 1975). While Wittgenstein's positive contributions to the problem of number have received attention by a number of commentators (see, e.g., (Waismann, 1959), and the entry "number" in (Glock 1996), to my knowledge there has been no discussion of his negative criticism among Wittgenstein scholars.

Perhaps this is due to the questionable status of the textual sources: no written statements by Wittgenstein himself appear to be available. Another reason, however, may be that it is extremely difficult to make sense of the ...