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The Standard of Equality of Numbers

by George Boolos

A contribution to the case against Logicism.

The following notes are not intended as a general review of the paper. It is an analysis of the arguments presented in the paper which may bear upon either the claim that Mathematics is reducible to logic or that Mathematics is Analytic.

There are, I believe, two main themes in this paper:

First, the refutation of attempts to prove the existence of infinite collections.
Second, an exposition of the merits of Frege's derivation of arithmetic from a principle dubbed "Hume's Thesis".

The importance of the first theme lies in its relevance to the claims of logicism.

I don't really understand why Boolos attaches as much importance to the role of "Hume's" thesis as he does.

The gross structure of the paper is as follows:

A discussion of Dedekind's proof that "there are infinite systems"
A refutation of the reducibility of arithmetic to second order logic.
Hume's Principle.

created 1994/10/19 modified 1998/01/18