402 



SCIENCE 



[Vol. LVI, No. 1450 



physical reality -at all. It is impossible to 

 prove, by mathematical reasoning, any propo- 

 sition whatsoever concerning the physical 

 world, and only a mathematical crank would 

 be likely now to imagine it his function to do 

 so. There is plainly one way only of ascer- 

 taining the facts of experience, and that is bj' 

 observation. It is not the business of a mathe- 

 matician to suggest one view of the universe 

 or another, but merely to supply the physicists 

 with a collection of abstract schemes, which it 

 is for them to select from, and to adopt or dis- 

 card at their pleasure. 



The most obvious example is to be found in 

 the science of geometry. Mathematicians have 

 constructed a very large number of diiferent 

 systems of geometry, Euclidean or non- 

 Euclidean, of one, two, three, or any number of 

 dimensions. All these systems are of complete 

 and equal validity. They embody the results 

 of mathematicians' observations of their reality, 

 a reality far more intense and far more rigid 

 than the dubious and elusive reality of physics. 

 The old-fashioned geometry of Euclid, the en- 

 tertaining seven-point geometry of Veblen, the 

 space-times of Minkowski and Einstein, are all 

 absolutely and eciually real. When a mathe- 

 matician has constructed, or, to be more accu- 

 rate, when he has observed them, his profes- 

 sional interest in the matter ends. It may be 

 the seven-point geometry that fits the facts the 

 best, for anything that mathematicians have to 

 say. There may be three dimensions in this 

 room and five next door. As a professional 

 mathematician, I have no idea; I can only ask 

 some competent physicist to instruct me in 

 the facts. 



The function of a mathematician, then, is 

 simply to observe the facts about his own intri- 

 cate system of reality, that astonishingly beau- 

 tiful complex of logical relations which forms 

 the subject-matter of his science, as if he were 

 an explorer looking at a distant range of 

 mountains, and to record the results of his ob- 

 servations in a series of maps, each of which is 

 a branch of pure mathematics. Many of these 

 maps have been completed, while in others, and 

 these, naturally, are the most interesting, there 

 are vast uncharted regions. Some, it seems, 

 have some relevance to the structure of the 



physical world, while others have no such tan- 

 gible application. Among them there is per- 

 haps none quite so fascinating, with quite the 

 same astonishing contrasts of sharp outline and 

 mysterious shade, as that which constitutes the 

 theory of numbers. 



The number system of arithmetic is, as we 

 know too well, not without its applications to 

 the sensible world. The currency systems of 

 Europe, for example, conform to it approxi- 

 mately; west of the Vistula, two and two make 

 something approaching four. The practical 

 applications of arithmetic, however, are tedious 

 beyond words. One must probe a little deeper 

 into the subject if one wishes to interest the 

 ordinary man, whose taste in such matters is 

 astonishingly correct, and who turns with joy 

 from the routine of common life to anything 

 strange and odd, like the fourth dimension, or 

 imaginary time, or the theory of the represen- 

 tation of integers by sums of squares or cubes. 



It is impossible for me to give you, in the 

 time at my command, any general account of 

 the problems of the theory of numbers, or of 

 the progress that has been made towards their 

 solution even during the last twenty years. I 

 must adopt a much simpler method. I will 

 merely state to you, with a few words of com- 

 ment, three or four isolated questions, selected 

 in a haphazard way. They are seemingly 

 simple questions, and it is not necessary to be 

 anything of a mathematician to understand 

 them; and I have chosen them for no better 

 reason than that I happen to be interested in 

 them myself. There is no one of them to which 



1 know the answer, nor, so far as I know, does 

 any mathematician in the world; and there is 

 no one of them, with one exception which I 

 have included deliberately, the answer to which 

 any one of us would not make almost any 

 sacrifice to know. 



1. When is a number the sum of two cubes, 

 and what is the number of its representations? 

 This is my first question, and first of all I will 

 elucidate it by some examples. The numbers 



2 ^ 1^ -h 1^ and 9 = 2^ -|- 1^ are sums of two 

 cubes, while 3 and 4 are not: it is exceptional 

 for a number to be of this particular form. 

 The number of cubes up to 1,000,000 is 100, 

 and the number of numbers, up to this limit 



