516 TRANSACTIONS OF SECTION A. 



ciently definite sense for practical purposes, is empirically verifiable by sensuous 

 intuition, whatever meaning the metaphysician may attach to it. But Mathematics 

 cannot be built up from the operations of elementary arithmetic without the intro- 

 duction 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 form of a limit, corre- 

 sponding 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 functionality, lie at the very heart of modern analysis. Essentially 

 bound up with this central doctrine of limits is the concept 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 formulation of this precise scheme gives an opening to differences 

 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 properiy 

 defined object of mathematical thought, of such a character 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 yet 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 with the central conception of the 

 'limit,' 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. Th>xt 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 

 assumptions 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 

 introduction, 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 Ampere that the 

 existence of the differential coefficient could only be asserted as a theorem requir- 

 ing 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 everywhere 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 

 analysis is true without the introduction of limitations and conditions which were 



