238 METAPHYSICS 



itself based on the very principles at stake.) (2) The existence of 

 fundamental problems of this kind which remained almost or wholly 

 unsuspected until revealed in our own time by the creation of a science 

 of symbolic logic should console us if ever we are tempted to suspect 

 that metaphysics is at any rate a science in which all the main con- 

 structive work has already been accomplished by the great thinkers 

 of the past. To me it appears, on the contrary, that the recent enor- 

 mous developments in the purely formal sciences of logic and mathe- 

 matics, with the host of fundamental problems they open up, give 

 promise of an approaching era of fresh speculative construction 

 which bids fair to be no less rich in results than any of the great 

 "golden" periods in the past history of our science. Indeed, but 

 that I would avoid the slightest suspicion of a desire to advertise 

 personal friends, I fancy I might even venture to name some of those 

 to whom we may reasonably look for the work to be done. 



Of the relation of metaphysics to pure mathematics it would be 

 impertinent for any but a trained mathematician to say very much. 

 I must therefore be content to point out that the same difficulty 

 in drawing boundary lines meets us here as in the case of logic. Not 

 so long ago this difficulty might have been ignored, as it still is by too 

 many writers on the philosophy of science. Until recently mathematics 

 would have been thought to be adequately defined as the science of 

 numerical and quantitative relations, and adequatel} 7 distinguished 

 from metaphysics by the non-quantitative and non-numerical char- 

 acter of the latter, though it would probably have been admitted that 

 the problem of the definition of quantity and number themselves is 

 a metaphysical one. But in the present state of our knowledge such 

 an account seems doubly unsatisfactory. On the one hand, we have 

 to recognize the existence of branches of mathematics, such as the 

 so-called descriptive geometry, which are neither quantitative nor 

 numerical, and, on the other, quantity as distinct from number appears 

 to play no part in mathematical science, while number itself, thanks 

 to the labors of such men as Cantor and Dedekind, seems, as I have 

 said before, to be known now to be only a special type of order in 

 a series. Thus there appears to be ground for regarding serial order 

 as the fundamental category of mathematics, and we are thrown back 

 once more upon the difficult task of deciding how many ultimately 

 irreducible types of order there may be before we can undertake any 

 precise discrimination between mathematical and metaphysical 

 science. However we may regard the problem, it is at least certain 

 that the recent researches of mathematicians into the meaning of 

 such concepts as continuity and infinity have, besides opening up new 

 metaphysical problems, done much to transfigure the familiar ones, 

 as all readers of Professor Royce must be aware. For instance I 

 imagine all of us here present, even the youngest, were brought up on 



