394 



SCIENCE 



[N. S. Vol. XXXII. No. S21 



philosophers, in relation to the precise 

 meaning and implication of the operation 

 and the terms. It will, however, be main- 

 tained, probably by the majority of man- 

 kind, 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 lan- 

 guage of Kant, a synthetic or an analytic 

 judgment. It may, I think, be admitted 

 that there is much cogency in this view; 

 and, were mathematics concerned with the 

 elementary operations of arithmetic alone, 

 it could fairly be held that the mathema- 

 tician, like the practical man of the world, 

 might without much risk shut his eyes and 

 ears to the discussions of the philosophers 

 on such points. The exactitude of such a 

 proposition, in a sufficiently definite sense 

 for practical purposes, is empirically veri- 

 fiable by sensuous intuition, whatever 

 meaning the metaphysician may attach to 

 it. But mathematics can not be built up 

 from the operations of elementary arith- 

 metic without the introduction of further 

 conceptual elements. Except in certain 

 very simple cases no process of measure- 

 ment, such as the determination of an area 

 or a volume, can be carried out with exacti- 

 tude by a finite number of applications of 

 the operations of arithmetic. The result to 

 be obtained appears in the form 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 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 never- 



theless a necessary postulation in mathe- 

 matical analysis. The conception of the 

 infinite, in some form, is thus indispensable 

 in mathematics; and this conception re- 

 qi^ires precise characterization by a scheme 

 of exact definitions, prior to all the proc- 

 esses of deduction required in obtaining the 

 detailed results of analysis. The formula- 

 tion of this precise scheme gives an opening 

 to difllerences 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 al- 

 luded above. Under what conditions is a 

 non-finite aggregate of entities a properly 

 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 uni- 

 versal acceptance. Physical intuition of- 

 fers no answer to such a question ; it is one 

 which abstract thought alone can settle. 

 It can not 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 geonietrj^ and in analysis our 

 standard of what constitutes a rigorous 

 demonstration has in the course of the 

 nineteenth century undergone an almost 

 revolutionarjr change. That oldest text- 

 book of science in the world, "Euclid's 

 Elements of Geometry," has been popu- 

 larly 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 num- 

 ber of points, assumptions not included in 



