Septembeb 23, 1910] 



SCIENCE 



397 



Dr. Whitehead, and which has distin- 

 guished adherents both in Europe and in 

 America. The ideal of this school is a pre- 

 sentation of the whole of mathematics as a 

 deductive scheme in which are employed 

 a certain limited number of indefinables 

 and unprovable axioms, by means of a pro- 

 cedure in which all possibility of the illicit 

 intrusion of extraneous elements into the 

 deduction is excluded by the employment 

 of a symbolism in which each symbol ex- 

 presses a certain logical relation. This 

 school receives its inspiration from a pe- 

 culiar 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 im- 

 plications in the movements of mathemat- 

 ical 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 mathematics in all 

 the wealth of their present develop- 

 ment, 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 lo- 

 gistics. It is quite impo.ssible 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 pres- 

 ent and future developments of mathe- 

 matics. 



I am very far from wishing to minimize 

 the high philosophic interest of the attempt 

 made by the Peano-Russell school to exhibit 

 mathematics as a scheme of deductive logic, 

 I have myself emphasized above the neces- 

 sity and importance of fitting the results of 

 mathematical research in their final form 

 into a framework of deduction, for the pur- 

 pose of ensuring the complete precision and 

 the verification of the various mathemat- 

 ical theories. At the same time it must 

 be recognized that the purely deductive- 

 method is wholly inadequate as an instru- 

 ment of research. Whatever view may be 

 held as regards the place of psychological 

 implications in a completed body of mathe- 

 matical doctrine, in research the psycholog- 

 ical factor is of paramount importance. 

 The slightest acquaintance with the history 

 of mathematics establishes the fact that dis- 

 coveries 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 mathe- 

 matical theory. With these alone the 

 mathematician would be unable to move a 

 step. In face of an unlimited number of 

 possible combinations a principle of selec- 

 tion of such as are of interest, a purposive 

 element, and a perceptive faculty are essen- 

 tial 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 establi.shed ; 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 



