456 



SCIENCE. 



[N. S. Vol. XX. No. 510. 



But now, in comparatively recent times, 

 there has developed a region of inquiry 

 which one may call by the general name of 

 modern logic. To the constitution of this 

 nev/ region of inquiry men have principally 

 contributed who began as mathematicians, 

 but who, in the course of their work, have 

 been led to become more and more philos- 

 ophers. Of late, however, various philos- 

 ophers, who were originally in no sense 

 mathematicians, becoming aware of the 

 importance of the new type of research, are 

 in their turn attempting both to assimilate 

 and to supplement the undertakings which 

 were begun from the mathematical side. 

 As a result, the logical problem of the cate- 

 gories has to-day become almost equally a 

 problem for the logicians of mathematics 

 and for those students of philosophy who 

 take any serious interest in exactness of 

 method in their own branch of work. The 

 result of this actual cooperation of men 

 from both sides is that, as I think, we are 

 to-day, for the first time, in sight of what is 

 still, as I freely admit, a somewhat distant 

 goal, viz., the relatively complete rational 

 analysis and tabulation of the fundamental 

 categories of human thought. That the 

 student of ethics is as much interested in 

 such an investigation as is the metaphysi- 

 cian, that the philosopher of religion needs 

 a well-completed table of categories quite 

 as much as does the pure logician, every 

 competent student of such topics ought to 

 admit. And that the enterprise in ques- 

 tion keenly interests the mathematicians is 

 shown by the prominent part which some 

 of them have taken in the researches in 

 question. Here, then, is the type of recent 

 scientific work whose results most obviously 

 bear upon the tasks of all of us alike. 



A catalogue of the names of the workers 

 in this wide field of modern logic would be 

 out of place here. Yet one must, indeed, 

 indicate what lines of research are espe- 

 cially in question. From the purely 



mathematical side, the investigations of the 

 type to which I now refer may be viewed 

 (somewhat arbitrarily) as beginning ^vith 

 that famous examination into one of the 

 postulates of Euclid's geometry which gave 

 rise to the so-called non-Euclidean geom- 

 etry. The question here originally at 

 issue was one of a comparatively limited 

 scope, viz., the question whether Euclid's 

 parallel-line postulate was a logical con- 

 sequence of the other geometrical prin- 

 ciples. But the investigation rapidly de- 

 velops into a general study of the founda- 

 tions of geometry— a study to which contri- 

 butions are still almost constantly appear- 

 ing. Somewhat independently of this line 

 of inquiry there grew up, during the latter 

 half of the nineteenth century, that reex- 

 anxination of the bases of arithmetic and 

 analysis which is associated with the names 

 of Dedekind, Weierstrass and George Can- 

 tor. At the present time, the labors of a 

 number of other inquirers (amongst whom 

 we may mention the school of Peano and 

 Fieri in Italy, and men such as Poincare 

 and Couturat in Prance, Hilbert in Ger- 

 many, Bertram Russell and Whitehead in 

 England and an energetic group of our 

 American mathematicians— men such as 

 Professor Moore, Professor Halsted, Dr. 

 Huntington, Dr. Veblen and a consider- 

 able number of others) have been added 

 to the earlier researches. The result is 

 Ihat we have recently come for the first 

 time to be able to see, with some complete- 

 ness, what the assumed first principles of 

 pure mathematics actually are. As was 

 to be expected, these principles are capable 

 of more than one formulation, according 

 as they are approached from one side or 

 from another. As was also to be expected, 

 the entire edifice of pure mathematics, so 

 far as it has yet been erected, actually 

 rests upon a very few fundamental con- 

 cepts and postulates, however you may 

 formulate them. What was not observed, 



