PRESIDENTIAL ADDRESS. 515 



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 psychologist. Whether they arc 

 regarded as possessing apodictic certainty or as purely hypothetical in character, 

 these ideas represent the data or premisses 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 permanence 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 formulation of the principles which form the founda- 

 tions of the science, and the views of mathematicians in this regard have always 

 necessarily been largely affected by the conscious or unconscious attitude of par- 

 ticular 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 mathematicians 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 unceasing 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 mathematicians. 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 extend- 

 ing through the whole detailed treatment of that subject — although critical in its 

 origin, has been constructive in its results. The Mengenlehre, 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 controversy 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 onlologj' ; it turns 

 upon the question of what constitutes a valid definition of a mathematical object. 

 The school known as mathematical ' idealists ' admit, as valid objects of mathe- 

 matical 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 

 regards 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 con- 

 siderable 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 discus- 

 sion, disclosing much difference of opinion, of the proper definitions and principles 

 on which the subject should be based. 



The proposition 7 + 5 = 12, taken as typical of the propositions expressing the 

 results of the elementary operations 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, how- 

 ever, 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, lor 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 Mathematics concerned with the elementary operations of arithmetic 

 alone, it could fairly be held that the mathematician, 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 suffi- 



