DISCOVERY 



65 



at the solution of Fermat's hist theorem. There is 

 probably no other branch of scientific work where such 

 a remark would apply. The fact that a great deal of 

 recent mathematical research has been concentrated 

 in this direction is an item of additional interest. 

 Progress is being made in two main classes of problems 

 — in the theory of partitions and in the theory of the 

 distribution of primes. In both of these is seen the 

 same fundamental characteristic — -the intense difficulty 

 of the analysis combined with the remarkable simplicity 

 of the results aimed at. It is comparatively simple to 

 enunciate the problem : In how many ways is it 

 possible to express a number « as the sum of other 

 numbers less than n ? — ^but its solution requires the 

 discussion of an elliptic modular function. It is 

 suggested, then, not that the methods here are simple, 

 but that the results achieved and contemplated are in 

 many cases within the comprehension of those of 

 moderate attainments. 



The positive integers, however, form only a very 

 small class of the numbers that exist. The theprj' of 

 rational and irrational numbers brings problems to our 

 notice of a totally different character. It is here that 

 the mathematician has entered the realm of the 

 philosopher, and has seized territory from which it will 

 be difficult to dislodge him. The problems of the 

 infinite and the infinitesimal have been inadequately 

 represented since Zeno lirst propounded them in his 

 famous paradoxes. The great Eleatic, unconscious of 

 the false premises latent in his mind, obtained results 

 inconsistent with experience. A strict investigation 

 into the nature of the mathematical infinite, effectively 

 conpleted by the mathematician, is necessary for the 

 solution of the paradoxes. The fallacy is now seen to 

 consist in applying the properties of finite aggregates 

 to the infinite. That the whole is greater than its 

 part, or that a number can be obtained by the process 

 of counting, seem self-evident statements, but they 

 produce disastrous results when applied to the infinite. 

 The accurate theory of the irrational and of the infinite, 

 first propounded by Dedekind and Cantor about fifty 

 years ago, gives rise to many interesting speculations. 

 What, for example, is the correspondence between a 

 set of points and the continuum of real number ? Are 

 all infinite numbers equal ? Is there a greatest number ? 

 It seems natural to suppose that there must be, namely 

 the number that includes everything, but Cantor proves 

 that it does not exist. 



From such speculations the mind naturally- passes 

 to the consideration of the general philosophy of 

 Mathematical thought : its purpose and ideais. Its 

 value in a technical education will be revealed by its 

 usefulness, but its ideals alone measure its value in a 

 liberal education. The Professor of Mathematics at 

 Oxford said quite recently that the life of a mathema- 



tician was possibly one which no perfectly reasonable 

 man would elect to live, and that " it is something to 

 be able to say that at any rate we do no harm." * 

 There is a chance of such a remark being taken more 

 seriously than was intended, for there is a tendency for 

 the mathematician to concede all that is artistic to the 

 clasdcal and literary scholar. He is apt to think that 

 its manifold u^es in the practical life are sufficient to 

 compensate what many regard as its dsficiency in 

 culture. But there mu^t bs value in a study that tends 

 to regulate the mind in its outlook on life— one freed 

 from those natural prejudices and desires that prevent 

 a clear, uncoloured view of human conduct. The 

 scientific method, as the Hon. Bsrtrand Russell has 

 persuasively shown,^ is free from the disturbing influence 

 of ethical considerations, and might be oftener used 

 in other branches of thought. There have been instances 

 even in the history of science itself where progress has 

 been hindered by the introduction of ethical and 

 religious beliefs. To believe that the earth went round 

 the sun was to show the greatest irreverence, this being 

 inconsistent with the theory, so comforting to human 

 egoism, that the earth was the centre of the universe. 

 The classicist and literary scholar, we are often told, 

 lives in the past, and when contemplating the things of 

 the present, will exclaim in the words of the Ecclesiast, 

 " There is nothing new under the sun." Talk to him 

 of some event in modern history, and he will give j'ou 

 a quotation from the austere historian of Greece, 

 describing an exact parallel in the Peloponnesian War. 

 However that may be, the mathematician is not so 

 encompassed by the presence of a completed system of 

 knowledge, and consequently his endeavours are not 

 obvious repetitions of what has already been done. It 

 may be that the Chinese once knew of Taylor's theorem, 

 or that the chimpanzee was once acquainted with the 

 rule of three, but we certainly do not know these 

 statements to be true ; and we think them highly 

 improbable. There is always, therefore, the hope of 

 discovery of something new, and comfort in thinking 

 of possible achievement. 



In the modern striving after practical results, there 

 is an aspect of Mathematics that is apt to be ignored. 

 The high value with which we estimate a technical 

 education should not detract from the ;esthetic claim 

 of Pure Mathematics based on the beauty of its pure 

 reasoning. It is not a beauty of colour and shade ; 

 it is one of order and austerity. There is nothing 

 incongruous in regarding as a work of art a theorem 

 wherein is displayed a simple exposition of the necessary 

 and sufficient. 



This suggests another characteristic often absent 

 elsewhere — the characteristic of permanence ; and it 



' Some Famous Problems of the Theory of Numbers, 1920. 

 - Scientific Method in Philosophy. 



