The Principles of Mathematics (1903)

§ 494

Arithmetic. Frege gives exactly the same definition of cardinal numbers as I have given, at least if we identify his range with my class[137]. But following his intensional theory of classes, he regards the number as a property of the class-concept, not of the class in extension. If u be a range, the number of u is the range of the concept range similar to u. In the Grundlagen der Arithmetik, other possible theories of number are discussed and dismissed. Numbers cannot be asserted of objects, because the same set of objects may have different numbers assigned to them (Gl. p. 29); for example, one army is so many regiments and such another number of soldiers. This view seems to me to involve too physical a view of objects; I do not consider the army to be the same object as the regiments. A stronger argument for the same view is that 0 will not apply to objects, but only to concepts (p. 59). This argument is, I think, conclusive up to a certain point; but it is satisfied by the view of the symbolic meaning of classes set forth in § 73. Numbers themselves, like other ranges, are things (p. 67). For defining numbers as ranges, Frege gives the same general ground as I have given, namely what I call the principle of abstraction[138]. In the Grundgesetze der Arithmetik, various theorems in the foundations of cardinal Arithmetic are proved with great elaboration, so great that it is often very difficult to discover the difference between successive steps in a demonstration. In view of the contradiction of Chapter X, it is plain that some emendation is required in Frege's principles; but it is hard to believe that it can do more than introduce some general limitation which leaves the details unaffected.(§ 494 ¶ 1)

§ 494 n. 1. See Gl. pp. 79, 85; Gg. p. 57, Df. Z.

§ 494 n. 2. Gl. p. 79; cf. § 111 supra.