CONCEPTIONS AND METHODS OF MATHEMATICS 471 



and explained how the feast would be served if only the dishes were 

 filled. 1 I fully agree with this opinion, and can only plead in excuse 

 that my subject was the fundamental conceptions and methods of 

 mathematics, not the infinite variety of detail and application 

 which give our science its real vitality. In fact I should like to 

 subscribe most heartily to the view that in mathematics, as else- 

 where, the discussion of such fundamental matters derives its interest 

 mainly from the importance of the theory of which they are the 

 so-called foundations. 2 I like to look at mathematics almost more 

 as an art than as a science; for the activity of the mathematician, 

 constantly creating as he is, guided though not controlled by the 

 external world of the senses, bears a resemblance, not fanciful I 

 believe but real, to the activity of an artist, of a painter let us say. 

 Rigorous deductive reasoning on the part of the mathematician 

 may be likened here to technical skill in drawing on the part of the 

 painter. Just as no one can become a good painter without a certain 

 amount of this skill, so no one can become a mathematician without 

 the power to reason accurately up to a certain point. Yet these 

 qualities, fundamental though they are, do not make a painter or 

 a mathematician worthy of the name, nor indeed are they the most 

 important factors in the case. Other qualities of a far more subtle 

 sort, chief among which in both cases is imagination, go to the 

 making of the good artist or good mathematician. I must content 

 myself merely by recalling to you this somewhat vague and difficult 

 though interesting field of speculation which arises when we attempt 

 to attach value to mathematical work, a field which is familiar 

 enough to us all in the analogous case of artistic or literary criticism. 

 We are in the habit of speaking of logical rigor and the considera- 

 tion of axioms and postulates as the foundations on which the superb 

 structure of modern mathematics rests; and it is often a matter of 

 wonder how such a great edifice can rest securely on such a small 

 foundation. Moreover, these foundations have not always seemed so 

 secure as they do at present. During the first half of the nineteenth 

 century certain mathematicians of a critical turn of mind Cauchy, 

 Abel, Weierstrass, to mention the greatest of them perceived to 

 their dismay that these foundations were not sound, and some of the 

 best efforts of their lives were devoted to strengthening and improv- 

 ing them. And yet I doubt whether the great results of mathematics 



1 Notice that just as the empty dishes could be filled by a great variety of 

 viands, so the empty symbols of mathematics can be given meanings of the most 

 varied sorts. 



2 Cf. the following remark by Study, Jahrcsbericht der deutschcn Mathematiker- 

 Vereinigung, vol. xi (1902), p. 313: 



" So wertvoll auch Untersuchungen liber die systematische Stellung der math- 

 ematischen Grundbegriffe sind . . . wertvoller ist doch noch der materielle Inhalt 

 der einzelnen Disciplinen, um dessentwillen allein ja derartige Untersuchungen 

 liberhaupt Zweck haben. ..." 



