The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview
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the Bs; it is the existence of that map that constitutes the coincidence of their referents, so that the identification of those referents plays no role in the determination of the truth or falsity of the sentence. Wright has given no reason whatever to suppose that it will play a role in the determination of the truth or falsity of other sentences, and the presumption is that it will not. Until this very large lacuna is filled, Wright's argument for lemmas (1) and (2) carries no weight.


I wish to add a protest against the use of the name ' Hume's Principle', popularized by George Boolos and adopted from him by Richard Heck, William Demopoulos, Matthias Schirn, and others, though sparingly used by Crispin Wright in the present paper, where he prefers his earlier appellation 'N='. There are two grounds for the protest.

First, Frege's ascription of the principle to Hume in Grundlagen, §63, is surely little more than a joke. The sentence he quotes from Hume--'When two numbers are so combined that the one has always an unit answering to every unit of the other, we pronounce them equal'--is very vague. Moreover, it has to do with units, not with objects falling under a concept. Hume evidently regarded numbers as collections of units, a conception Frege utterly repudiated.

Secondly, the term ' Hume's Principle' blurs a vital distinction between two quite distinct principles:

(1) the definition of 'just as many . . . as' (of 'equinumerous'): There are just as many Fs as Gs iff there is a one-one map of the Fs on to the Gs


(2) the equivalence (EN)--the abstraction principle proper: The number of Fs = the number of Gs iff there are just as many Fs as Gs.

Admittedly, in §63 of Grundlagen, Frege himself blurs the distinction between (1) and (2); but that is no good reason for his commentators to follow him in this, since the two principles are essentially distinct, so that either of them could be accepted while the other was rejected. §63 is a purely preliminary paragraph, and, as soon as Frege treats of the matter in any detail, he keeps (1) and (2) distinct. (1) appears as a definition in §72; (2) as a theorem in §73. (Admittedly, Frege does not keep them so clearly apart in Grundgesetze, where he is less concerned with the justification of his definitions.)

When principles (1) and (2) are distinguished, it becomes at once apparent that the sentence Frege quoted from Hume expresses neither of them. It is not, as it stands, about when it is right to say that there are just as many things of one kind as of another, but about equality of numbers, implicitly taken to be objects. From it


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