514 TRANSACTIONS OF SECTION A. 



the term; this would include algebraic form, geometrical form, functional rela- 

 tionship, the relations of order in any ordered set of entities such as numbers, 

 and the analysis of the peculiarities of form of groups of operations. A 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 em- 

 phasised 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 procedure in general. A definition which appears to identify all Mathe- 

 matics with the Mengenlehre, or Theory of Aggregates, has been given by E. 

 Papperitz : ' The subject-matter of Pure Mathematics consists of the relations 

 that can be established between any objects of thought when we 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 mathe- 

 maticians under the influence of the ordinary traditions of the science. Mr. 

 Russell writes: 1 'Pure Mathematics is the class of all propositions of the 

 form "p implies q," where p 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 following : Implication, the relation of a term to a class of which 

 it is a member the notion of such 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 which is not a constituent 

 of the propositions which it considers — namely, the notion of truth.' 



The belief is very general amongst instructed persons that the truths of 

 Mathematics have absolute certainty, 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 necessarily 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 disconcerting and would 

 cause some degree of astonishment. At the risk of this, 1 must here mention 

 two facts which are of considerable importance as regards an estimation of the 

 precise character of mathematical knowledge. In the first place, it is a fact 

 that frequently, and at various times, differences of opinion have existed among 

 mathematicians, giving rice to controversies as to the validity of whole lines of 

 reasoning and affecting the results of such reasoning; a considerable amount of 

 difference of opinion of this character exists among mathematicians 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 department of knowledge can be absolutely isolated, and that metaphysical 

 and psychological implications are a necessary 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 



1 Principles of Mathematics, p. 1.- 



