MATHEMATICS AND METAPHYSICIANS 75 



Pure mathematics consists entirely of assertions to the 

 effect that, if such and such a proposition is true of any- 

 thing, then such and such another proposition is true of 

 that thing. It is essential not to discuss whether the first 

 proposition is really true, and not to mention what the 

 anything is, of which it is supposed to be true. Both 

 these points would belong to applied mathematics. We 

 start, in pure mathematics, from certain rules of infer- 

 ence, by which we can infer that if one proposition is 

 true, then so is some other proposition. These rules of 

 inference constitute the major part of the principles of 

 formal logic. We then take any hypothesis that seems 

 amusing, and deduce its consequences, //our hypothesis 

 is about awy/M^^and not about some oneormore particular 

 ' things, then our deductions constitute mathematics. Thus 

 mathematics may be defined as the subject in which we 

 never know what we are talking about, nor whether what 

 we are saying is true. People who have been puzzled by the 

 beginnings of mathematics will, I hope, find comfort in 

 this definition, and will probably agree that it is accurate. 



As one of the chief triumphs of modern mathematics 

 consists in having discovered what mathematics really 

 is, a few more words on this subject may not be amiss. 

 It is common to start any branch of mathematics ^foi 

 instance. Geometry ^with a certain number of primitive 

 ideas, supposed incapable of definition, and a certain 

 number of primitive propositions or axioms, supposed 

 incapable of proof. Now the fact is that, though there 

 are indefinables and indemonstrables in every branch of 

 applied mathematics, there are none in pure mathematics 

 except such as belong to general logic. I^ogic, broadly 

 speaking, is distinguished by the fact that its propositions 

 can be put into a form in which they apply to anything 

 whatever. All pure mathematics Arithmetic, Analysis, 



