160 NORMATIVE SCIENCE 



of the categories 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, 

 namely, 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 meta- 

 physician, 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 question keenly interests the mathematicians is shown 

 by the prominent part which some of them have taken in the re- 

 searches 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 especially in question. From the 

 purely mathematical side, the investigations of the type to which I 

 now refer may be viewed (somewhat arbitrarily) as beginning with 

 that famous examination into one of the postulates of Euclid's 

 geometry which gave rise to the so-called non-Euclidean geometry. 

 The question here originally at issue was one of a comparatively 

 limited scope, namely, the question whether Euclid's parallel-line 

 postulate was a logical consequence of the other geometrical prin- 

 ciples. But the investigation rapidly develops into a general study 

 of the foundations of geometry a study to which contributions 

 are still almost constantly appearing. Somewhat independently 

 of this line of inquiry there grew up, during the latter half of the 

 nineteenth century, that reexamination of the bases of arithmetic 

 and analysis which is associated with the names of Dedekind, Weier- 

 strass, and George Cantor. At the present time, the labors of a num- 

 ber of other inquirers (amongst whom we may mention the school 

 of Peano and Fieri in Italy, and men such as Poincare* and Couturat 

 in France, Hilbert in Germany, Bertrand Russell and Whitehead in 

 England, and an energetic group of our American mathematicians 

 men such as Professor Moore, Professor Halsted, Dr. Hunting- 

 ton, Dr. Veblen, and a considerable number of others) have been 

 added to the earlier researches. The result is that we have recently 

 come for the first time to be able to see, with some completeness, 

 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 



