£18 TRANSACTIONS OF SECTION A. 



case, is of great interest not only from the purely mathematical point of view, but 

 also in relation to the general theory of knowledge, in that, owing to the results of 

 this study, some change is necessitated in the views which have been held by 

 philosophers as to what is known as Kant's space-problem. 



The school of thought which has most emphasised the purely logical aspect of 

 Mathematics is that which is represented in this country by Mr. Bertrand Russell 

 and Dr. Whitehead, and which has distinguished adherents both in Europe and 

 in America. The ideal of this school is a presentation of the whole of Mathe- 

 matics as a deductive scheme in which are employed a certain limited number of 

 indefinables and unprovable axioms, by means of a procedure in which all possi- 

 bility of the illicit intrusion of extraneous elements into the deduction is 

 excluded by the employment of a symbolism in which each symbol expresses a 

 certain logical relation. This school receives its inspiration from a peculiar 

 form of philosophic realism which, in its revolt from idealism, produces in the 

 adherents of the school a strong tendency to ignore altogether the psychological 

 implications in the movements of mathematical thought. This is carried so far 

 that in their writings no explicit recognition is made of any psychological factors 

 in the selection of the indefinables and in the formulation of the axioms upon 

 which the whole structure of Mathematics is to be based. The actually worked- 

 out part of their scheme has as yet reached only the mere fringe of modern 

 Mathematics as a great detailed body of doctrine ; but to any objection to the 

 method on the ground of the prolixity of the treatment which would be necessary 

 to carry it out far enough to enable it to embrace the various branches of Mathe- 

 matics in all the wealth of their present development, it would probably be replied 

 that the main point of interest is to establish in principle the possibility only of 

 subsuming Pure Mathematics under a scheme of logistic. It is quite impossible 

 for me here to attempt to discuss, even in outline, the tenets of this school, or 

 even to deal with the interesting question of the possibility of setting up a final 

 system of indefinables and axioms which shall suffice for all present and future 

 developments of Mathematics. 



I am very far from wishing to minimise the high philosophic interest of the 

 attempt made by the Peano-Russell school to exhibit Mathematics as a scheme of 

 deductive logic. 1 have myself emphasised above the necessity and importance of 

 fitting the results of mathematical research in their final form into a framework 

 of deduction, for the purpose of ensuring the complete precision and the verifica- 

 tion of the various mathematical theories. At the same time it must be recog- 

 nised that the purely deductive method is wholly inadequate as an instrument of 

 research. Whatever view may be held as regards the place of psychological 

 implications in a completed body of mathematical doctrine, in research the 

 psychological factor is of paramount importance. The slightest acquaintance with 

 the histdry of Mathematics establishes the fact that discoveiies have seldom, or 

 never, been made by purely deductive processes. The results are thrown into a 

 purely deductive form after, and often long after, their discovery. In many cases 

 the purely deductive form, in the full sense, is quite modern. The possession 

 of a body of indefinables, axioms, or postulates, and symbols denoting logical 

 relation, would, taken by itself, be wholly insufficient for the development of a 

 mathematical theory. With these alone the mathematician would be unable to 

 move a step. In face of an unlimited number of possible combinations a prin- 

 ciple of selection of such as are of interest, a purposive element, and a perceptive 

 faculty are essential for the development of anything new. In the process of 

 discovery the chains in a sequence of logical deduction do not at first arise in their 

 final order in the mind of the mathematical discoverer. He divines the results 

 before they are established; he has an intuitive grasp of the general line of a 

 demonstration long before he has filled in the details. A developed theory, or 

 even a demonstration of a single theorem, is no more identical with a mere com- 

 plex of syllogisms than a melody is identical with the mere sum of the musical 

 notes employed in its composition. In each case the whole is something more than 

 merely the sum of its parts ; it has a unity of its own, and that unity must be, 

 in some measure at least, discerned by its creator before the parts fall completely 

 into their places. Logic is, so to speak, the grammar of Mathematics ; but a 

 knowledge of the rules of grammar and the letters of the alphabet would not be 

 sufficient equipment to enable a man to write a book. There is much room for 



