THE SCIENCES OF THE IDEAL 167 



calculus of statements, are, apparently, the basal categories of all 

 the exact science that has yet been developed. Series and levels are 

 relational structures that, sharply as they are contrasted, 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 generations. 



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 religion, in ethics, or in aesthetics, 

 beyond merely helping towards the formulation of a table of cate- 

 gories - - then I reply that we are already not without evidence that 

 such general researches, abstract though they may seem, are bear- 

 ing fruits which have much more than a merely special interest. 

 Apart from its most general problems, that analysis of mathemat- 

 ical 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 elsewhere discussed at length, in its gen- 

 eral metaphysical bearings. I refer to the logical identity which 

 Dedekind first pointed out between the mathematical concept of 

 the ordinal number of series and the philosophical concept of the 

 formal structure of an ideally completed self. I have maintained 

 that this formal identity throws light upon problems which have as 

 genuine an interest for the student of the philosophy of religion as 

 for the logician of arithmetic. In the same connection it may be 

 remarked that, as Couturat and Russell, amongst other writers, 

 have very clearly and beautifully shown, the argument of the Kant- 

 ian mathematical antinomies needs to be explicitly and totally 

 revised in the light of Cantor's modern theory of infinite collections. 

 To pass at once to another, and a very different instance : The mod- 

 ern mathematical conceptions of what is called group theory have 

 already received very wide and significant applications, and promise 

 to bring into unity regions of research which, until recently, appeared 

 to have little or nothing to do with one another. Quite lately, how- 

 ever, there are signs that group theory will soon prove to be of im- 

 portance for the definition of some of the fundamental concepts of 

 that most refractory branch of philosophical inquiry, sesthetics. Dr. 

 Emch, in an important paper in the Monist, called attention, some 

 time since, to the symmetry groups to which certain aesthetically 

 pleasing forms belong, and endeavored to point out the empirical 

 relations between these groups and the aesthetic effects in question . 



