October 7, 1904.] 



SCIENCE. 



461 



The formal identity of the geometrical re- 

 lation called 'between' with a purely log- 

 ical relation which one can define as ex- 

 isting or as not existing amongst the mem- 

 bers of a given triad of logical classes, or of 

 logical statements, is shown by Kempe in a 

 fashion that I can not here attempt to ex- 

 pound. But Kempe 's result thus enables 

 one, as I believe, to simplify the theory of 

 relations far beyond the point which Rus- 

 sell, in his brilliant book has reached. For 

 Kempe 's triadic relation in question can 

 be stated, in what he calls its obverse form, 

 in perfectly symmetrical terms. And he 

 proves very exactly that the resulting log- 

 ical relation is precisely identical, in all its 

 properties, with the fundamental ordinal 

 relation of geometry. 



Thus the order-systems of geometry and 

 analysis appear simply as special cases of 

 the more general order-system of pure 

 logic. The whole, both of analysis and of 

 geometry, can be regarded as a description 

 of certain selected groups of entities, which 

 are chosen, according to special rules, from 

 a single ideal world. This general and 

 inclusive ideal world consists simply of 

 all the oijects which can stand to one 

 another in those symmetrical relations 

 wherein the pure logician finds various 

 statements, or various decisions inevitably 

 standing, 'Let me,' says in substance 

 Kempe, 'choose from the logician's ideal 

 world of classes or decisions, what entities 

 I will; and I will show you a collection of 

 objects that are in their relational struc- 

 ture, precisely identical Avith the points of 

 a geometer's space of n dimensions.' In 

 other words, all of the geometer's figures 

 and relations can be precisely pictured by 

 the relational structure of a selected system 

 of classes or of statements, whose relations 

 are wholly and explicitly logical relations, 

 such as opposition, and whose relations may 

 all be regarded, accordingly, as reducible 



to a single type of purely symmetrical re- 

 lation. 



Thus, for all exact science, and not 

 merely for the logician's special realm, the 

 contrast between symmetrical and unsym- 

 metrieal relations proves to be, after all, 

 superficial and derived. The purely log- 

 ical categories, such as opposition, and such 

 as hold within the calculus of statements, 

 are, apparently, the basal categories of all 

 the exact science that has yet been devel- 

 oped. Series and levels are relational 

 structures that, sharply as they are con- 

 trasted, can be derived from a single root. 



I have restated Kempe 's generalization 

 in my own way. I think it the most 

 promising step towards new light as to the 

 categories that we have made for some gen- 

 erations. 



In the field of modern logic, I say, then, 

 work is doing which is rapidly tending 

 towards the unification of the tasks of our 

 entire division. For this problem of the 

 categories, in all its abstractness, is still a 

 common problem for all of us. Do you 

 ask, however, what such researches can do 

 to furnish more special aid to the workers 

 in metaphysics, in the philosophy of re- 

 ligion, in ethics, or in esthetics, beyond 

 merely helping towards the formulation of 

 a table of categories— then I reply that we 

 are already not without evidence that such 

 general researches, abstract though they 

 may seem, are bearing fruits which have 

 much more than a merely special interest. 

 Apart from its most general problems, that 

 analysis of mathematical concepts to which 

 I have referred has in any case revealed 

 numerous unexpected connections between 

 departments of thought which had seemed 

 to be very widely sundered. One instance 

 of such a connection I myself have else- 

 where discussed at length, in its general 

 metaphysical bearings. I refer to the log- 

 ical identity which Dedekind first pointed 

 out between the mathematical concept of 



