August 19, 1897] 



NATURE 



375 



further the plan : in particular, to urge upon the Government 

 the establishment of such a Laboratory, and, if possible, to 

 obtain from them the funds which are a preliminary necessity 

 for that purpose. It was a deputation from this joint committee 

 which, headed by Lord Lister, waited upon the Prime Minister 

 on February i6 last. His reply to the deputation was mani- 

 festly sympathetic with the request ; there is consequently 

 reasonable ground for supposing that the Government will take 

 the matter into their favourable consideration. 



After having said, by way of preface, thus much upon the 

 chief event of the past year arising partly from our direct 

 action, I wish to turn to the main line of my address, and to 

 ask, for a brief time, your attention and your consideration for 

 the subject of pure mathematics. If, remembering the brilliant 

 address made at the Montreal meeting, you regret riiat Lord 

 Kelvin is not again now occupying this position : or if, re- 

 membering the interest aroused by Prof. J. J. Thomson's 

 address last year, you regret that the fascinating tale then 

 ojiened is not being resumed by some one with imagination 

 enough and knowledge enough to continue it : I can, not un- 

 selfishly, share your regret. 



It appears, however, from the practice of the Council and 

 the General Committee, to be their policy that mathematicians 

 belonging to the extreme right (if the phrase may be used) shall 

 from time to time be nominated to the presidency of the Section. 

 It is, I think, the case that this Section has always had assigned 

 to it the subjects of Mathematics and Physics. In their de- 

 velopment, pure mathematics has continued to be associated 

 with applied mathematics, and applied mathematics with 

 physics. So far as I know, there is no substantial reason why 

 any change should be made, and so far as I have been able to 

 observe, there is a strong consensus of opinion that no change 

 by way of separation need be tried. Wide a$ is the range of 

 our discussions, distracting as is the occasional variety in the 

 matter of the papers we receive, the complexity of our Section, 

 if in any respect a disadvantage, does not appreciably discount 

 the advantages it otherwise secures. Si:)ecialisation in all our 

 subjects has become almost a necessity for progress ; but exces- 

 sive obedience need not be paid to that necessity. On the one 

 hand, there will be danger of imperfect appreciation if a subject 

 is so completely restricted to a few specialists that it is ignored 

 by all but them ; and, on the other hand, there will be danger 

 of unsound growth if subject and thinkers alike become isolated, 

 and cease to take an active interest in the methods, the pro- 

 cesses, and the results other than those which directly concern 

 them. Accordingly, I think that our group of sciences, which 

 form a continuous range, are better united than divided. 



Aristotle declared that it is unbecoming to praise the gods. 

 Observing his canon, I shall say nothing as to the wisdom and 

 the justice of our Executive in sometimes selecting a pure 

 mathematician to preside over this Section. I shall only appeal 

 to your indulgence in accepting the opportunity they have thus 

 given me of speaking more specially about my own subject. 



I make this appeal the more earnestly, for two particular 

 reasons. One of these is based upon the conflicting views, 

 popularly held and sometimes summarily expressed, about the 

 subject and those who are addicted to it. It is true that the 

 day has gone by, when it is necessary to give serious considera- 

 tion to attacks upon mathematical studies, and particularly upon 

 analysis, such as were made by the metaphysician Hamilton : 

 attacks no longer thought worthy of any answer. Feelings of 

 hostility, if ever they were widely held, have given way toother 

 feelings, which in the mildest form suggest toleration and acqui- 

 escence, and in the most extreme form suggest solemn respect 

 and distant wonder. By common consent, we are allowed with- 

 out reproach to pursue our aims ; though those aims sometimes 

 attract but little sympathy. It is not so long since, during one 

 of the meetings of the Association, one of the leading English 

 newspapers briefly described a sitting of this Section in the 

 words, " Saturday morning was devoted to pure mathematics, 

 and so there was nothing of any general interest": still, 

 such toleration is better than undisguised and ill-informed 

 hostility. But the attitude of respect, I might almost say of 

 reverence, is even more trying : we mathematicians are sup- 

 posed to be of a different mould, to live far up the heights above 

 the driving gales ot controversy, breathing a rarer intellectual 

 atmosphere, serene in impenetrable calm. It is difficult for us 

 to maintain the gravity of demeanour proper to such superior 

 persons ; and perhaps it is best to confess at once that we are of 

 the earth, earthy, that we have our differences of opinion and of 



NO. 1 45 1, VOL. 56] 



judgment, and that we can even commit the Machiavelian crime 

 of making blunders. 



The other of my reasons for claiming your indulgence is of a 

 graver character, and consists in the difficulty of framing general 

 explanations about the subject. The fact is that mathematics 

 do not lend themselves readily to general exposition. Clift'ord, 

 it is true, could lecture and enchant his audience : and yet even 

 his lectures ranged about the threshold of the temple of mathe- 

 matical knowledge and made no attempt to reveal the shrines in 

 the sanctuary. The explanation of this initial difficulty is, how- 

 ever, at hand. Our vocabulary is highly technical, perhaps as 

 technical as is that of moral philosophers : and yet even the 

 technicality of a vocabulary can be circumvented by prolixity 

 of statement. But the ideas and the subject-matter in any 

 branch of our study, when even only moderately developed, are 

 so abstract as to demand an almost intolerable prolixity of state- 

 ment if an attempt is made to popularise them. Moreover, of 

 the many results obtained, there are few that appeal to an unpro- 

 fessional sympathy. Adams could discover a new planet by 

 subjecting observations made of the known planets to the 

 most profound calculations ; and the world, not over curious 

 about the process, could appreciate the significant result. But 

 such instances are rare ; for the most part, our particular results 

 must remain somewhat intangible, somewhat incomprehensible, 

 to those who dwell resolutely and completely outside the range 

 of mathematical knowledge. 



What then am I to do ? It would be pleasant to me, though 

 it might not prove satisfying to you, to discourse of the present 

 state of one branch or of several branches of mathematics, and 

 particularly to indicate what seem to be lines of possible and 

 probable growth in the future. Instead of pursuing this course, 

 I shall keep my remarks of a general character as far as possible, 

 and shall attempt, not merely to describe briefly some of the 

 relations of pure mathematics to other branches of science, but 

 also to make a bold claim that the unrestricted cultivation of 

 pure mathematics is desirable in itself and for its own sake. 

 Some — I should like to believe many — who are here will con- 

 cede this claim to the fullest extent and without reservation ; 

 but I doubt whether this is so in general. And yet the claim is 

 one which needs to be made before an English-speaking audience. 

 For it is a curious fact that, although the United Kingdom has 

 possessed some of the very greatest of pure mathematicians in 

 the second half of this century, the subject has there received 

 but a scant share of attention as compared with that which it 

 has found in France, in Germany, in Italy, in Sweden and 

 Norway, or in the United States. I am not oblivious of the 

 magnificent contributions to other parts of our science made 

 alike by British leaders and British followers ; their fame is 

 known to the world. But apathy rather than attention has been 

 the characteristic feature of our attitude towards pure mathe- 

 matics ; and it seems to me a misfortune, alike for the intellectual 

 activity of the nation and for the progress of the subject, that 

 English thought has had relatively so small an influence upon 

 its vast modern developments. 



Now it is not enough for my purpose to be told that the 

 British Association includes all science in its scope, and con- 

 sequently includes pure mathematics. A statement thus made 

 might be framed in a spirit of mere sufferance ; what I wish to 

 secure is a recognition of the subject as one which, being full of 

 life and overflowing with a power of growth, is worthy of the 

 most absorbing devotion. The most cursory examination of the 

 opinions of scientific men leads at once to the conclusion, that 

 there are two views of the subject, both accurate so far as they 

 go, both inadequate whether alone or combined,'which to some ex- 

 tent explain if they do not justify what may be called the English 

 attitude in the past. Let me deal with these in succession. 



One of these estimates has been framed by what is called the 

 practical man ; he regards the subject as a machine which is to 

 provide him with tables, as far as tables can be calculated ; and 

 with the simplest formulae and the most direct rules, whenever 

 tables cannot be calculated. Results, not methods, are his 

 want ; it is sufficient for him that an authoritative statement as 

 to a result shall be made ; all else is ignored. And for what is 

 beyond, in the shape of work that does nothing to meet his 

 special wants, or of the processes that have led to the results 

 he uses, he cares little or nothing. In fact, he would^ regard 

 mathematics as a collection of formulce and an aggregate of 

 processes to grind out numerical results ; whatever else there is 

 in it, may be vain and is useless. In his view, it is to be the 

 drudge of the practical sciences. 



