September i, 1910] 



NATURE 



287 



transient, in view of the power which the science has 

 alwavs shown of constantly extending its borders in un- 

 foreseen directions. Such definitions, many of w'hich have 

 been advanced, are apt to err by excess or defect, and 

 often contain distinct traces of the personal predilections 

 of those who formulate them. There was a time when it 

 would have been a tolerably sufficient description of Pure 

 Mathematics to say that its subject-matter consisted of 

 magnitude and geometrical form. Such a description of 

 it would be wholly inadequate at the present day. Some 

 of the most important branches of modern Mathematics, 

 such as the theory of groups, and Universal Algebra, are 

 concerned, in their abstract forms, neither with magni- 

 tude nor with number, nor with geometrical form. That 

 great modern development, Projective Geometry, has been 

 so formulated as to be independent of all metric con- 

 siderations. Indeed, the tendency of mathematicians under 

 the influence of the movement known as the Arithmetisa- 

 tion of .Analysis, a movement which has become a 

 dominant one in the last few decades, is to banish 

 altogether the notion of measurable quantity as a concep- 

 tion necessary to Pure Mathematics, Number, in the ex- 

 tended meaning it has attained, taking its place. 

 Measurement is regarded as one of the applications, but 

 as no part of the basis, of mathematical analysis. Perhaps 

 the least inadequate description of the general scope of 

 modern Pure .Mathematics — I will not call it a definition 

 — would be to say that it deals with /onti, in a very 

 general sense of the term ; this would include algebraic 

 form, geometrical form, functional relationship, the rela- 

 tions of order in any ordered set of entities such as 

 numbers, and the analysis of the peculiarities of form of 

 groups of operations. .\ strong tendency is manifested in 

 many of the recent definitions to break down the line of 

 demarcation which was formerly supposed to separate 

 Mathematics from formal logic ; the rise and development 

 of symbolic logic has no doubt emphasised this tendency. 

 Thus Mathematics has been described by the eminent 

 American mathematician and logician B. Pierce as " the 

 Science which draws necessary conclusions," a pretty 

 complete identification of Mathematics with logical pro- 

 cedure in general. .-^ definition which appears to identify 

 all .Mathematics with the Mengenlehre, or Theory of 

 Aggn|;ates, has been given by E. Papperitz : " The 

 subject-matter of Pure Mathematics consists of the rela- 

 tions that can be established between any objects or 

 thought when w'e regard those objects as contained in an 

 ordered manifold ; the law of order of this manifold must 

 be subject to our choice." The form of definition which 

 illustrates most strikingly the tendencies of the modern 

 school of logistic is one given by Mr. Bertrand Russell. 

 I reproduce it here, in order to show how wide is the 

 chasm between the modes of expression of adherents of 

 this school and those of mathematicians under the in- 

 fluence of the ordinary traditions of the science. Mr. 

 Russell whites : ' " Pure Mathematics is the class of all 

 propositions of the form ' p implies i;,' where /> and q are 

 propositions containing one or more variables, the same 

 in the two propositions, and neither p nor q contains 

 any constants except logical constants. .And logical 

 constants are all notions definable in terms of the follow- 

 ing : Implication, the relation of a term to a class of 

 which it is a member, the notion of sur/i that, the notion 

 of relation, and such further notions as may be involved 

 in the general notion of propositions of the above form. 

 In addition to these. Mathematics uses a notion w-hich is 

 not a constituent of the prooositions which it considers — 

 namely, the notion of truth." 



The belief is verv general amongst instructed persons 

 i that the truths of Mathematics have absolute certainty, 

 I or at least that there appertains to them the highest 

 , degree of certainty of which the human mind is capable. 

 It is thought that a valid mathematical theorem is neces- 

 sarily of such a character as to compel belief in any mind 

 ; capable of following the steps of the demonstration. .Any 

 considerations tending to weaken this belief would be 

 i disconcerting, and would caiise some degree of astonish- 

 ment. At the risk of this. \ must here mention two facts 

 which are of considerable importance as regards an 

 estimation of the precise character of mathematical know- 

 1 " Principles of Ma'hemat'cs," p. i. 



NO. 2 13 I, VOL. 84] 



ledge. In the first place, it is a fact that frequently, and 

 at various times, differences of opinion have existed among 

 mathematicians, giving rise to controversies as to the 

 validity of whole lines of reasoning, and affecting the 

 results of such reasoning ; a considerable amount of differ- 

 ence of opinion of this character exists among mathe- 

 maticians at the present time. In the second place, the 

 accepted standard of rigour, that is, the standard of what 

 is deemed necessary to constitute a valid demonstration, 

 has undergone change in the course of time. Much of 

 the reasoning which was formerly regarded as satisfactory 

 and irrefutable is now regarded as insufficient to establish 

 the results which it was employed to demonstrate. It has 

 even been shown that results which were once supposed 

 to have been fully established by demonstrations are, in 

 point of fact, affected with error. • I propose here to 

 explain in general terms how these phenomena are 

 possible. 



In every subject of study, if one probes deep enough, 

 there are found to be points in which that subject comes 

 in contact with general philosophy, and where differences 

 of philosophical view will have a greater or less influence 

 on the attitude of the mind towards the principles of the 

 particular subject. This is not surprising when we reflect 

 that there is but one universe of thought, that no depart- 

 ment of knowledge can be absolutely isolated, and that 

 metaphysical and psychological implications are a neces- 

 sary element in all the activities of the mind. A particular 

 department, such as Mathematics, is compelled to set up 

 a more or less artificial frontier, which marks it off from 

 general philosophy. This frontier consists of a set of 

 regulative ideas in the form of indefinables and axioms, 

 partly ontological assumptions, and partly postulations of 

 a logical character. To go behind these, to attempt to 

 analyse their nature and origin, and to justify their 

 validity, is to go outside the special department and to 

 touch on the domains of the metaphysician and the psycho- 

 logist. Whether they are regarded as possessing apodictic 

 certainty or as purely hypothetical in character, these ideas 

 represent the data or premises of the science, and the 

 whole of its edifice is dependent upon them. They serve 

 as the foundation on which all is built, as well as the 

 frontier on the side of philosophy and psychology. A set 

 of data ideally perfect in respect of precision and perman- 

 ence is unattainable — or at least has not yet been attained : 

 and the adjustment of frontiers is one of the most frequent 

 causes of strife. -As a matter of fact, variations of opinion 

 have at various times arisen within the ranks of the 

 mathematicians as to the nature, scope, and proper formu- 

 lation of the principles which form the foundations of the 

 science, and the views of mathematicians in this regard 

 have always necessarily been largely affected by the con- 

 scious or unconscious attitude of particular minds towards 

 questions of general philosophy. It is in this region, I 

 think, that the source is to be found of those remarkable 

 differences of opinion amongst m.athematicians which have 

 come into prominence at various times, and have given 

 rise to much controversy as to fundamentals. Since the 

 time of Newton and Leibnitz there has been almost un- 

 ceasing discussion as to the proper foundations for the 

 so-called infinitesimal calculus. More recently, questions 

 relating to the foundations of geometry and rational 

 mechanics have much occupied the attention of mathe- 

 maticians. The very great change which has taken place 

 during the last half-century in the dominant view of the 

 foundations of mathematical analysis — a change which has 

 exercised a great influence extending through the whole 

 detailed treatment of that subject — although critical in its 

 origin, has been constructive in its results. The Mengen- 

 lehre, or theory of aggregates, had its origin in the critical 

 study of the foundations of analysis, but has already 

 become a great constructive scheme, is indispensable as 

 a method in the investigations of analysis, provides the 

 language requisite for the statement in precise form of 

 analytical theorems of a general character, and, moreover, 

 has already found important applications in geometry. In 

 connection' with the Mengenlehre, there has arisen a con- 

 troversy amongst mathematicians which is at the present 

 time far from having reached a decisive issue. The exact 

 point at issue is one which may be described as a matter 

 of mathematical ontology ; it turns upon the question of 



