September 16, 1922] 



NA TURE 



381 



The Theory of Numbers. 1 

 By Prof. G. H. Hardy, M.A., F.R.S. 



I FIND myself to-day in the same embarrassing 

 position in which a predecessor of mine at 

 Oxford found himself at Bradford in 1875, the presi- 

 dent of a Section, probably the largest and most 

 heterogeneous in the Association, which is absorbed 

 by a multitude of divergent professional interests, none 

 of which agree with his or mine. 



There are two courses possible in such circumstances. 

 One is to take refuge, as Prof. Henry Smith did then, 

 with visible reluctance, in a series of general pro- 

 positions to which mathematicians, physicists, and 

 astronomers may all be expected to return a polite 

 assent. The importance of science and scientific 

 method, the need for better organisation of scientific 

 education and research, are all topics on which I could 

 no doubt say something without undue strain either 

 on my own honesty or on your credulity. That there 

 is no finer education and discipline than natural 

 science ; that it is, as Dr. Campbell has said, " the 

 noblest of the arts " ; that the crowning achievements 

 of science lie in those directions with which this Section 

 is professionally concerned : all this I could say with 

 complete sincerity, and, if I were the head of a deputa- 

 tion approaching a Government Department, I suppose 

 that I would not shirk even so unprofitable a task. 



It is unfortunate that these essential and edifying 

 truths, important as it is that they should be repeated 

 as loudly as possible from time to time, are, to the 

 man whose interest in life lies in scientific work and 

 not in propaganda, unexciting, and in fact quite intoler- 

 ably dull. I could, if I chose, say all these things, but, 

 even if I wanted to, I should scarcely increase your 

 respect for mathematics and mathematicians by 

 repeating to you what you have said yourselves, or 

 read in the newspapers, a hundred times already. I 

 shall say them all some day ; the time will come when 

 we shall none of us have anything more interesting to 

 say. We need not anticipate our inevitable end. 



I propose therefore to adopt the alternative course 

 suggested by my predecessor, and try to say some- 

 thing to you about the one subject about which I 

 have anything to say. It happens, by a fortunate 

 accident, that the particular subject which I love the 

 most, and which presents most of the problems which 

 occupy my own researches, is by no means over- 

 whelmingly recondite or obscure, and indeed is sharply 

 distinguished from almost every other branch of pure 

 mathematics, in that it makes a direct, popular, and 

 almost irresistible appeal to the heart of the ordinary 

 man. 



There is, however, one preliminary remark which 

 I cannot resist the temptation of making. The present 

 is a particularly happy moment for a pure mathe- 

 matician, since it has been marked by one of the 

 greatest recorded triumphs of pure mathematics. 

 This triumph is the work, as it happens, of a man who 

 probably would not describe himself as a mathe- 

 matician, but who has done more than any mathe- 

 matician co vindicate the dignity of mathematics, and 

 to put that obscure and perplexing construction, 



1 Presidential address delivered to Section A (Mathematics and Physics) 

 of the British Association at Hull on Sept. 8. 



no. 2759, VOL. 1 10] 



commonly described as " physical reality," in its 

 proper place. 



There is probably less difference between the 

 methods of a physicist and a mathematician than is 

 generally supposed. The most striking among them 

 seems to me to be this, that the mathematician is in 

 much more direct contact with reality. This may 

 perhaps seem to you a paradox, since it is the physicist 

 who deals with the subject-matter to which the 

 epithet " real " is commonly applied. But a very 

 little reflection will show that the " reality " of the 

 physicist, whatever it may be (and it is extraordinarily 

 difficult to say), has few or none of the attributes 

 which common-sense instinctively marks as real. A 

 chair may be a collection of whirling atoms, or an idea 

 in the mind of God. It is not my business to suggest 

 that one account of it is obviously more plausible 

 than the other. Whatever the merits of either of 

 them may be, neither draws its inspiration from the 

 suggestions of common-sense. 



Neither the philosophers, nor the physicists them- 

 selves, have ever put forward any very convincing 

 account of what physical reality is, or of how the 

 physicist passes, from the confused mass of fact or 

 sensation with which he starts, to the construction of 

 the objects which he classifies as real. We cannot be 

 said, therefore, to know what the subject-matter of 

 physics is ; but this need not prevent us from under- 

 standing the task which a physicist is trying to perform. 

 That, clearly, is to correlate the incoherent body of 

 facts confronting him with some definite and orderly 

 scheme of abstract relations, the kind of scheme, in 

 short, which he can borrow only from mathematics. 



A mathematician, on the other hand, fortunately 

 for him, is not concerned with this physical reality at 

 all. It is impossible to prove, by mathematical 

 reasoning, any proposition 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 ascertaining the facts 

 of experience, and that is by observation. It is not 

 the business of a mathematician 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 discard 

 at their pleasure. 



The most obvious example is to be found in the 

 science of geometry. Mathematicians have con- 

 structed a very large number of different systems of 

 geometry, Euclidean or non-Euclidean, of one. two, 

 three, or an}' 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 entertaining seven- 

 point geometry of Veblen, the space-times of Minkowski 

 and Einstein, are all absolutely and equally real. 

 When a mathematician has constructed, or, to be more 

 accurate, when he has observed them, his professional 

 interest in the matter ends. It may be the seven-point 



