CONCEPTIONS AND METHODS OF MATHEMATICS 469 



universal and particular propositions respectively. 1 It is only during 

 the last fifty years or so that mathematicians have become conscious 

 of the fundamental importance in their science of existence theorems 

 which until then they had frequently assumed tacitly as they needed 

 them, without always being conscious of what they were doing. 



It is sometimes held by non-mathematicians that if mathematics 

 were really a purely deductive science, it could not have gained 

 anything like the extent which it has without losing itself in trivial- 

 ities and becoming, as Poincare puts it, a vast tautology. 2 This 

 view would doubtless be correct if all primitive propositions were 

 universal propositions. One of the most characteristic features of 

 mathematical reasoning, however, is the use which it makes of aux- 

 iliary elements. I refer to the auxiliary points and lines in proofs 

 by elementary geometry, the quantities formed by combining in 

 various ways the numbers which enter into the theorems to be 

 proved in algebra, etc. Without the use of such auxiliary elements 

 mathematicians would be incapable of advancing a step; and 

 whenever we make use of such an element in a proof, we are in reality 

 using an existence theorem. 3 These existence theorems need not, 

 to be sure, be among the primitive propositions; but if not, they must 

 be deduced from primitive propositions some of which are existence 

 theorems, for it is clear that an existence theorem cannot be deduced 

 from universal propositions alone. 4 Thus it may fairly be said that 

 existence theorems form the vital principle of mathematics, but these 

 in turn, it must be remembered, would be impotent without the 

 material basis of universal propositions to work upon. 



VII. Russell's Definition 



We have so far arrived at the view that exact mathematics is 

 the study by deductive methods of what we have called a mathe- 

 matical system, that is, a class of objects and a class of relations 

 between them. If we elaborate this position in two directions we 

 shall reach the standpoint of Russell. 5 



In the first place Russell makes precise the term deductive method 



1 "All men are mortals" is a standard example of a universal proposition; 

 while as an illustration of a particular proposition is often given: "Some men are 

 Greeks. " That this is really an existence theorem is seen more clearly when we 

 state it in the form: "There exists at least one man who is a Greek." 



2 Cf. La Science et I'Hypothese, p. 10. 



3 Even when in algebra we consider the sum of two numbers a + b, we are using 

 the existence theorem which says that, any two numbers a and b being given, 

 there exists a number c which stands to them in the relation which we indicate in 

 ordinary language by saying that c is the sum of a and b. 



4 The power which resides in the method of mathematical induction, so called, 

 comes from the fact that this method depends on an existence theorem. It is, 

 however, not the only fertile principle in mathematics as Poincare' would have 

 us believe (cf. La Science et I'Hypothese). In fact there are great branches of 

 mathematics, like elementary geometry, in which it takes little or no part. 



5 The Principles of Mathematics, Cambridge, England, 1903. 



