Septembeb 23, 1910] 



SCIENCE 



393 



pendent 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 per- 

 fect in respect of precision and perma- 

 nence 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 formula- 

 tion of the principles which form the foun- 

 dations 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 re- 

 markable differences of opinion amongst 

 mathematicians which have come into 

 prominence at various times, and have 

 given rise to much controversy as to funda- 

 mentals. 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 founda- 

 tions 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 cen- 

 turj' in the dominant view of the founda- 

 tions of mathematical analysis — a change 

 which has exercised a great influence ex- 

 tending through the whole detailed treat- 

 ment of that subject— although critical in 

 its origin, has been constructive in its re- 

 sults. The ilengenlehre, or theory of ag- 

 gregates, 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 al- 

 ready found important applications in 

 geometry. In connection with the Men- 

 genlehre 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 ontology; 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 mathematical 

 discussion, entities which the rival "em- 

 piricist" school regard as non-existent for 

 mathematical thought, because insuffi- 

 ciently defined. It is clear that the ideal- 

 ist 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 develop- 

 ments of what has come to be known as 

 Cantorism. The difference of view of these 

 rival schools, depending as it does on deep- 

 seated dift'erences of philosophical outlook, 

 is thought by some to be essentially irrec- 

 oncilable. This controversy was due to 

 the fact that certain processes of reasoning, 

 of very considerable plausibility, which 

 had been employed by G. Cantor, the 

 founder of the Mengenlehre, had led to re- 

 sults which contained flat contradictions. 

 The efforts made to remove these contra- 

 dictions, and to trace their source, led to 

 the discussion, 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 



