462 MATHEMATICS 



are used. But more than this. Our A. -objects, our B-objects, and our 

 relation R may be given an interpretation, if we choose, very different 

 from that we had at first intended. 



We may, for instance, regard the A-objects as the straight lines in 

 a plane, the .^-objects as the points in the same plane (either finite 

 or at infinity), and when an ^.-object stands in the relation R to a 

 .B-object, this may be taken to mean that the line passes through the 

 point. Our statement would then become: Any two lines being given, 

 there exists one and only one point through which they both pass. 

 Or we may regard the A-objects as the men in a certain community, 

 the 5-objects as the women, and the relation of an A-object to a 

 B-object as friendship. Then our statement would be: In this com- 

 munity any two men have one, and only one, woman friend in com- 

 mon. 



These examples are, I think, sufficient to show what is meant 

 when I say that we are not concerned in mathematics with the 

 nature of the objects and relations involved in our premises, except 

 in so far as their nature is exhibited in the premises themselves. 

 Accordingly mathematicians of a critical turn of mind, during the 

 last few years, have adopted more and more a purely nominalistic 

 attitude towards the objects and relations involved in mathematical 

 investigation. This is, of course, not the crude mixture of nominalism 

 and empiricism of the philosopher Hobbes, whose claim to mathe- 

 matical fame, it may be said in passing, is that of a circle-squarer. 1 

 The nominalism of the present-day mathematician consists in treating 

 the objects of his investigation and the relations between them as 

 mere symbols. He then states his propositions, in effect, in the fol- 

 lowing form: If there exist any objects in the physical or mental 

 world with relations among themselves which satisfy the conditions 

 which I have laid down for my symbols, then such and such facts 

 will be true concerning them. 



It will be seen that, according to Peirce's view, the mathematician 

 as such is in no wise concerned with the source of his premises or with 

 their harmony or lack of harmony with any part of the external 

 world. He does not even assert that any objects really exist which 

 correspond to his symbols. Mathematics may therefore be truly 

 said to be the most abstract of all sciences, since it does not deal 

 directly with reality. 2 



This, then, is Peirce's definition of mathematics. Its advantages 

 in the direction of unifying our conception of mathematics and of 

 assigning to it a definite place among the other sciences are clear. 



1 Hobbes practically obtains as the ratio of a circumference to its diameter 

 the value vTO. Cf. for instance Molesworth's edition of Hobbes's English Works, 

 vol. vn, p. 431. 



2 Cf. the very interesting remarks along this line of C. S. Peirce in The Monist, 

 vol. vii, pp. 23-24. 



