CHAPTER XIII 

 ORDER, NUMBER, QUANTITY 



The concepts which will be subjected to analysis in this 

 chapter are those which are presumed to lie at the basis of 

 mathematics. Whether they are properly mathematical 

 concepts or not, however, need be of no concern. The point 

 of view seems to be gaining ground in recent years that 

 mathematics has no specific subject matter, but is consti- 

 tuted merely by its form. The famous definition of Benjamin 

 Peirce, "mathematics is the science which draws necessary 

 conclusions," 1 and the views of Russell, Wittgenstein, 

 Hilbert, and others are all in accord with this claim. For- 

 tunately, the dispute need not be settled here. All that is 

 required is the recognition that the concepts of order, number, 

 and quantity are implicitly present in a large part of scien- 

 tific investigation. They are thus basic in the sense that a 

 recognition of their natures is important to an understanding 

 of the sciences. 



The order in which the concepts should be discussed is 

 more or less arbitrary. If one's concern were with the estab- 

 lishment of a postulate scheme in terms of which the con- 

 cepts could be defined, the decision as to order would be 

 important. The more complex of the terms should be de- 

 fined by means of the less complex, and the order of discus- 

 sion would therefore coincide with the order of logical 

 derivation. But since the present concern is also with the 

 problem of empirical foundation, the issue is not so important. 

 As has already been shown, the concepts which are empirically 

 most obviously given are not necessarily those which are log- 

 ically most basic. Hence it is of pedagogical convenience to 

 examine the concepts roughly in the order of psychological ac- 

 cession rather than in the order of logical deducibility. 



1 American Journal of Mathematics, Vol. iv, 1881, p. 97. 



253 



