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 æsthetics, 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 elsewhere discussed at length, in its general 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 Kantian 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 modern 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, however, there are signs that group theory will soon prove to be of importance for the definition of some of the fundamental concepts of that most refractory branch of philosophical inquiry, æsthetics. Dr. Emch, in an important paper in the Monist, called attention, some time since, to the symmetry groups to which certain æsthetically pleasing forms belong, and endeavored to point out the empirical relations between these groups and the æsthetic effects in question. The grounds for such a connection between the groups in question and the observed æsthetic effects, seemed, in the paper of Dr. Emch to be left largely in the dark. But certain papers recently published in the country by Miss Ethel Puffer, bearing upon the psychology of the beautiful (although the author has approached the subject without being in the least consciously influenced, as I understand, by the conceptions of the mathematical group theory), still actually lead, if I correctly grasp the writer's meaning, to the doctrine that the æsthetic object, viewed as a psychological whole, must possess a structure closely, if not precisely, equivalent to the ideal structure of what the mathematician calls a group. I myself have no authority regarding æsthetic concepts, and speak subject to correction. But the unexpected, and in case of Miss Puffer's research, quite unintended, appearance of group theory in recent æsthetic analysis is to me an impressive instance of the use of relatively new mathematical conceptions in philosophical regions which seem, at first sight, very remote from mathematics.
That both the group concept and the concept of the self just suggested are sure to have also a wide application in the ethics of the future, I am myself well convinced. In fact, no branch of philosophy is without close relations to all such studies of fundamental categories.
These are but hints and examples. They suffice, I hope, to show that the workers in this division have deep common interests, and will do well, in future, to study the arts of coöperation, and to regard one another's progress with a watchful and cordial sympathy. In a word: Our common problem is the theory of the categories. That problem can be solved only by the coöperation of the mathematicians and of the philosophers.
THE UNIVERSITY OF PARIS IN THE THIRTEENTH CENTURY
Hand-painted Photogravure from a Painting by Otto Knille. Reproduced
from a Photograph of the Painting by permission of the
Berlin Photograph Co.
This famous painting is now in the University of Berlin. Thomas Aquinas, one of the greatest of the scholastic philosophers, surnamed the "Angelic Doctor," is delivering a learned discourse before King Louis IX. To the right of the King stands Joinville, the French chronicler. The Dominican monk with his hand to his face is Guillaume de Saint Amour, and Vincent de Beauvais, and another Dominican are seated with their backs to the platform desk from which Thomas Aquinas is making his animated address. The picture is thoroughly characteristic of a University disputation at the close of the Middle Ages.