472 MATHEMATICS 



seemed less certain to any of them because of the weakness they 

 perceived in the foundations on which these results are built up. 

 The fact is that what we call mathematical rigor is merely one of 

 the foundation stones of the science; an important and essential 

 one surely, yet not the only thing upon which we can rely. A science 

 which has developed along such broad lines as mathematics, with 

 such numerous relations of its parts both to one another and to other 

 sciences, could not long contain serious error without detection. 

 This explains how, again and again, it has come about, that the 

 most important mathematical developments have taken place by 

 methods which cannot be wholly justified by our present canons of 

 mathematical rigor, the logical "foundation" having been supplied 

 only long after the superstructure had been raised. A discussion 

 and analysis of the non-deductive methods which the creative 

 mathematician really uses would be both interesting and instructive. 

 Here I must content myself with the enumeration of a few of them. 



First and foremost there is the use of intuition, whether geometrical, 

 mechanical, or physical. The great service which this method has 

 rendered and is still rendering to mathematics both pure and applied 

 is so well known that a mere mention is sufficient. 



Then there is the method of experiment; not merely the physical 

 experiments of the laboratory or the geometrical experiments I 

 had occasion to speak of a few minutes ago, but also arithmetical 

 experiments, numerous examples of which are found in the theory 

 of numbers and in analysis. The mathematicians of the past fre- 

 quently used this method in their printed works. That this is now 

 seldom done must not be taken to indicate that the method itself is 

 not used as much as ever. 



Closely allied to this method of experiment is the method of 

 analogy, which assumes that something true of a considerable num- 

 ber of cases will probably be true in analogous cases. This is, of 

 course, nothing but the ordinary method of induction. But in mathe- 

 matics induction may be employed not merely in connection with 

 the experimental method, but also to extend results won by deduct- 

 ive methods to other analogous cases. This use of induction has 

 often been unconscious and sometimes overbold, as, for instance, 

 when the operations of ordinary algebra were extended without 

 scruple to infinite series. 



Finally there is what may perhaps be called the method of optim- 

 ism, w r hich leads us either willfully or instinctively to shut our eyes 

 to the possibility of evil. Thus the optimist who treats a problem in 

 algebra or analytic geometry will say, if he stops to reflect on what 

 he is doing: "1 know that I have no right to divide by zero; but 

 there are so many other values which the expression by which I am 

 dividing might have that I will assume that the Evil One has not 



