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PEIRCE TO STRAWSON circular, but in fact it is not since the notion of equivalence between classes can be defined without making use of the notion of number Two classes are equivalent to each other if they can be mapped onto each other without residue Thus, to take an example of Frege’s, a waiter may know that there are as many knives as there are plates on a table without knowing how many of each there are All he needs to is to observe that there is a knife to the right of every plate and a plate to the left of every knife Thus, we could define four as the class of all classes equivalent to the class of gospel-makers But such a definition would be useless for the logicist’s purpose since the fact that there were four gospel-makers is no part of logic Frege has to find, for each number, not only a class of the right size, but one whose size is guaranteed by logic He does this by beginning with zero as the first of the number series This can be defined in purely logical terms as the class of all classes equivalent to the class of objects that are not identical with themselves: a class that obviously has no members (‘the null class’) We can then go on to define the number one as the class of all classes equivalent to the class whose only member is zero In order to pass from these definitions to definitions of the other natural numbers Frege needs to define the notion of ‘succeeding’ in the sense in which three succeeds two, and four succeeds three, in the number series He defines ‘n immediately succeeds m’ as ‘There exists a concept F, and an object falling under it x, such that the number of Fs is n and the number of Fs not identical with x is m’ With the aid of this definition the other numbers can be defined without using any notions other than logical ones such as identity, class, and class-equivalence Begriffsschrift is a very austere and formal work The Foundations of Arithmetic sets out the logicist programme much more fully, but also much more informally Symbols appear rarely, and Frege takes great pains to relate his work to that of other philosophers According to Kant, our knowledge of both arithmetic and geometry depended on intuition: in the Critique of Pure Reason he had maintained that mathematical truths were synthetic a priori, that is to say that while they were genuinely informative, they were known in advance of all experience.3 John Stuart Mill, as we have seen, maintained that mathematical propositions were empirical generalizations, widely applicable and widely confirmed, but a posteriori nonetheless See vol III, p 103 40
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