NATURE 



[September i, 1910 



what constitutes a valid definition of a mathematical 

 object.. The school, known as mathematical "idealists" 

 admit, as valid objects of mathematical discussion, entities 

 which the rival " empiricist " school regard as non- 

 existent for mathematical thought, because insufficiently 

 defined. It is clear that the idealist may build whole 

 superstructures on a. foundation which the empiricist re- 

 gards as made of .sand, and this . is what has actually 

 happened in some of the recent developments of what has 

 come to be known as Cantorism. The difference of view 

 of these rival schools, depending as it does on deep-seated 

 differences of philosophical outlook, is thought by some 

 to be essentially irreconcilable. This controversy was due 

 to the fact that certain processes of reasoning, of very 

 considerable plausibility, which had been employed by 

 G. Cantor, the founder of the Mengenlehre, had led to 

 results which contained, flat contradictions. The efforts 

 made to remove these contradictions, and to trace their 

 source, led to the discussion, disclosing much difference of 

 opinion, of the proper definitions and principles on which 

 the subject should be based. 



The proposition 7-1-5 = 12, taken as typical of the pro- 

 positions expressing the results of the elementary opera- 

 tions of arithmetic, has since the time of Kant given rise 

 to very voluminous discussion amongst philosophers in 

 relation to the precise meaning and implication of the 

 operation and the terms. It will, however, be maintained, 

 probably by the majority of mankind, that the theorem 

 retains its validity as stating a practically certain and 

 useful fact, whatever view philosophers may choose to 

 take of its precise nature — as, for example, whether it 

 represents, in the language of Kant, a synthetic or an 

 analytic judgment. It may, I think, be ' admitted that 

 there is much cogency in this view ; and, were Mathe- 

 matics concerned with the elementary operations of arith- 

 metic alone, it could fairly be held that the mathematician, 

 like the practical man of the world, might without much 

 risk shut his eyes and ear^ to the discussions of the philo- 

 sophers on such points. The exactitude of such a proposi- 

 tion, in a sufficiently definite sense for practical purposes, 

 is empirically verifiable by sensuous intuition, whatever 

 meaning the metaphysician .may attach to it. But Mathe- 

 matics cannot be built up from the operations of 

 elementary arithmetic without the introduction of further 

 conceptual elements. Except in certain very simple cases, 

 no process of measurement, such as the determination of 

 an area or a volume, can be carried out with exactitude 

 by a finite number of applications of the operations of 

 arithmetic. The result to be obtained appears in the for.n 

 of a limit, corresponding to an interminable sequence of 

 arithmetical operations. The notion of " limit," in the 

 definite form given to it by Cauchy and his followers, 

 together with the closely related theory of the arithmetic 

 continuum, and the notions of continuity and functionalilv, 

 lie at the very heart of modern analysis. Essentially 

 bound up with this central doctrine of limits is the con- 

 cept of a non-finite set of entities, a concept which is not 

 directly derivable from sensuous intuition, but which is, 

 nevertheless, a necessary postulation in mathematical 

 analysis. The conception of the infinite, in some form, 

 is thus indispensable in Mathematics; and this conception 

 requires precise characterisation by a scheme of exact 

 definitions, prior to all the processes of deduction required 

 in obtaining the detailed results of analysis. The formu- 

 lation of this precise scheme gives an opening to differ- 

 ences of philosophical opinion which has led to a variety 

 of views as to the proper character of those definitions 

 which involve the concept of the infinite. Here is the 

 point of divergence of opinion among mathematicians to 

 which I have alluded above. Under what conditions is 

 a non-finite aggregate of entities a properly defined object 

 of mathematical thought, of such a chjiracter that no 

 contradictions will arise in the theories based' upon it? 

 That is the question to which varying answers have been 

 offered by different mathematical thinkers. ' No one 

 answer of a completely general character has as vet met 

 with universal acceptance. Physical intuition • offers no 

 answer to such a question ; it is one which abstract 

 thought alone can settle. It cannot be altogether avoided, 

 because, without the notion of the infinite, at least in 

 connection w-ith the central conception of the " limit," 



NO. 2 13 I, VOL. 84] 



mathematical analysis as a coherent body of thought falls 

 to the ground. 



Both in geometry and in analysis our standard of what 

 constitutes a rigorous demonstration has in the course of 

 the nineteenth century undergone an almost revolutionary 

 change. That oldest text-book of science in the world", 



Euclid's Elements of Geometry," has been popularly 

 held for centuries to be. the very model of deductive logical 

 demonstration. Criticism has, however, largely invalidated 

 this view. It appears that, at a large number of points, 

 assumptions not included in the preliminary axioms and 

 postulates are made use of. , The fact that these assump- 

 tions usually escape notice is due to their nature and 

 origin. Derived as they are from our spatial intuition, 

 their very self-evidence has allowed them to be ignored, 

 although their truth, is not more obvious empirically than 

 that of other assumptions derived from the same source 

 which are included in the axioms and postulates explicitly 

 stated as part of the foundation of Euclid's treatment of 

 the subject.-. The method of superimposition, employed by 

 Euclid with obvious reluctance, but forming an essential 

 part of his treatment of geometry, is, when regarded from 

 his point of view, open to most serious objections as 

 regards its logical coherence. In analysis, as in geometry, 

 the older methods of treatment consisted of processes of 

 deduction eked out by the more or less surreptitious intro- 

 duction, at numerous points in the subject, of assumptions 

 only justifiable by spatial intuition. The result of this 

 deviation from the purely deductive method was more 

 disastrous in the case of analysis than in geometry, because 

 it led to much actual error in the theory. For example, 

 it was held until comparatively recently that a. continuous 

 function necessarily possesses a differential coefficient, on 

 the ground that a curve always has a tangent. This we 

 now know to be quite erroneous, when any reasonable 

 definition of continuity is employed. The first step in the 

 discovery of this error was made when it occurred to 

 .■Xnip^re that the existence of the differential coefficient 

 could only be asserted as a theorem requiring proof, and 

 he himself published an attempt at such proof. The 

 erroneous character of the former belief on this matter 

 was most strikingly exhibited when Weierstrass produced 

 a function which is everywMiere continuous, but which 

 nowhere possesses a differential coefficient ; such functions 

 can now be constructed ad libitum. It is not too much 

 to say that no one of the general theorems of analvsis is 

 true without the introduction of limitations and conditions 

 which were entirely unknown to the discoverers of those 

 theorems. It has been the task of mathematicians under 

 the lead of such men as Cauchy, Riemann, Weierstrass, 

 and G. Cantor, to carry out the w-ork of reconstruction of 

 mathematical analysis, to render explicit all the limita- 

 tions of the truth of the general theorems, and to lay 

 down the conditions of validity of the ordinary analytical 

 operations. Physicists and others often maintain that this 

 modern e.xtreme precision amounts to an unnecessary and 

 pedantic purism, because in all practical applications of 

 Mathematics only * such functions are of importance as 

 exclude the remoter possibilities contemplated by theorists. 

 Such objections leave the true mathematician unmoved ; 

 to him it is an intolerable defect that, in an order of ideas 

 in which absolute exactitude is the guiding ideal, state- 

 ments should be made and processes employed, both of 

 which are subject to une.xpressed qualifications, as con- 

 ditions of their truth or validity. The pure mathematician 

 has developed a specialised conscience, extremely sensitive 

 as regards sins against logical precision. The physicist, 

 with his conscience hardened in this lespect by the rough- 

 and-tumble work of investigating the physical world, is 

 apt to regard the more tender organ of the mathe^iatician 

 with tliat feeling of impatience, not unmincled with con- 

 tempt, which the man of the world manifests for what 

 he considers to be over-scrupulosity and unpracticality. 



It is true that \ve cannot conceive how such a ' science 

 as Mathematics could have come into existence apart from 

 physical experience. But it is also true that physical 

 precepts, as given directly in unanalysed experience, are 

 wholly unfitted to form the basis of an exact ■ science. 

 Moreover, physical intuition fails altogether to afford anv 

 trustworthy guidance in connection with the concept of 

 the infinite, which, as we have seen, is in some form 



