CONCEPTIONS AND METHODS OF MATHEMATICS 473 



thrown a zero in my denominator this time." This method, if a pro- 

 ceeding often unconscious can be called a method, has been of great 

 service in the rapid development of many branches of mathematics, 

 though it may well be doubted whether in a subject as highly devel- 

 oped as is ordinary algebra it has not now survived its usefulness. 1 

 While no one of these methods can in any way compare with 

 that of rigorous deductive reasoning as a method upon which to 

 base mathematical results, it would be merely shutting one's eyes 

 to the facts to deny them their place in the life of the mathematical 

 world, not merely of the past but of to-day. There is now. and there 

 always will be room in the world for good mathematicians of every 

 grade of logical precision. It is almost equally important that the 

 small band whose chief interest lies in accuracy and rigor should 

 not make the mistake of despising the broader though less accurate 

 work of the great mass of their colleagues; as that the latter should 

 not attempt to shake themselves wholly free from the restraint the 

 former would put upon them. The union of these two tendencies 

 in the same individuals, as it was found, for instance, in Gauss and 

 Cauchy, seems the only sure way of avoiding complete estrangement 

 between mathematicians of these two types. 



1 Cf. the very suggestive remarks by Study, Jahresbericht d. Deutschen Math- 

 ematiker-Vereinigung, vol. xi (1902), p. 100, footnote, in which it is pointed out 

 how rigor, in cases of this sort, may not merely serve to increase the correctness of 

 the result, but actually to suggest new fields for mathematical investigation. 



