950 



SCIENCE 



[N. S. Vol. XXX. No. 7S3 



allow, the more salient among the facts by 

 which the thesis is supported. 



It is no part of my pui'pose to treat the 

 matter historically. As, however, the thesis 

 in question is the goal and culmination of 

 two originally independent but closely re- 

 lated and finally convergent movements of 

 modern thought, I can not refrain from 

 saying a brief preliminary word regarding 

 each of them. They may be characteristic- 

 ally designated as the critico-mathematical 

 movement and the logistical movement. 



The distinctively critical spirit is not a 

 new manifestation in mathematics. The 

 age of Euclid was a critical age. And just 

 now, thanks to the superb edition of the 

 "Elements" by Dr. Heath with its wonder- 

 ful richness of bibliographic citation, quo- 

 tation and critical commentary, one is en- 

 abled to understand better than ever before 

 how very fine and penetrating in funda- 

 mental questions of geometry and of logic 

 was the thought of the age that produced 

 the Alexandrine classic— the age, I say, for 

 the "Elements" is to be attributed not less 

 to the age of Euclid than to Euclid the man. 

 But it is not of antiquity that I wish to 

 speak. I refer to the critical movement in 

 modern mathematics— to the demand for 

 precision of concept, to the process of log- 

 ical rigorization, to the sense and the 

 craving for perfection of intellectual and 

 scientific form, in a word, to that spirit of 

 creative criticism which, following close 

 upon the great Eulerian and pre-Bulerian 

 period of discovery, manifesting itself al- 

 ready in the works of Gauss and Lagrange, 

 finding powerful agencies in the analytic 

 genius of Cauchy and Bolzano, in the geo- 

 metric genius of Lobachevski and Bolyai, 

 waxed in intensity throughout the lapsing 

 decades of the nineteenth century, at length 

 pervading the entire realm of mathematics 

 like a refining and purifying fire. The 

 result of this critical movement, thus orig- 



inating in mathematics and conducted 

 by mathematicians, was, not indeed the 

 grounding of mathematics itself, regarded 

 as a unitary science, but the grounding 

 rather, upon distinct bases of postulated 

 mathematical notions and propositions, of 

 various great hranclies of the science; in 

 witness whereof— to cite bvit one example 

 —behold the theory of the real variable as 

 founded by Weierstrass upon the familiar 

 theory of the cardinal numbers assumed as 

 certain, primordial and fundamental. 



Such bases, however, were destined to 

 appear, in the light of modern researches 

 in another field or in what seemed at all 

 events another, namely, the field of logic, 

 not as constituting the foundation either 

 of mathematics or of any of its branches 

 but as genuine components of the super- 

 structure. For it has ever been the faith 

 of the logician that there are a few ideas in 

 terms of which all definable ideas admit of 

 immediate or mediate definition and a few 

 propositions upon which as a basis or from 

 which as a body of premises all demon- 

 strable propositions admit of proof or de- 

 duction; and it has ever been the chief of 

 the logician's problems to discover such a 

 system of primitive concepts and proposi- 

 tions. It is in nothing less than a closely 

 approximate solution of that hoary problem 

 that modern investigations in logic have 

 culminated. As every one knows, the con- 

 ception of logic as an autonomous science 

 is nothing new. Among the very greatest 

 contributions of antiquity to human knowl- 

 edge is the "Logic" of Aristotle. As a 

 scientific achievement it is comparable to 

 the "Elements" of Euclid— comparable to 

 it also in another respect, namely, that it 

 was not significantly improved upon for 

 nearly two thousand years. Though always 

 indispensable as an instrument of thought, 

 yet logic, regarded as a science, remained 

 stationarj' for so long a time, showing no 



