Septembek 23, 1910] 



SCIENCE 



395 



the preliminary axioms and postulates are 

 made use of. The fact that these assump- 

 tions usually escape notice is due to their 

 nature and origin. Derived as they are 

 from our spatial intuition, their very self- 

 evidence has allowed them to be ignored, 

 although their truth is not more obvious 

 empirically than that of other assumptions 

 derived from the same source which are 

 included in the axioms and postulates ex- 

 plicitly stated as part of the foundation of 

 Euclid's treatment of the subject. The 

 method of superimposition, employed by 

 Euclid with obvious reluctance, but form- 

 ing an essential part of his treatment of 

 geometry, is, when regarded from his point 

 of view, open to most serious objections as 

 regards its logical coherence. In analysis, 

 as in geometry, the older methods of treat- 

 ment consisted of processes of deduction 

 eked out by the more or less surreptitious 

 introduction, at numerous points in the 

 subject, of assumptions only justifiable by 

 spatial intuition. The result of this devia- 

 tion from the purely deductive method was 

 more disastrous in the case of analysis than 

 in geometry, because it led to much actual 

 error in the theory. For example, it was 

 held until comparatively recently that a 

 continuous function necessarily possesses a 

 differential coefSeient, on the ground that a 

 curve always has a tangent. This we now 

 know to be quite erroneous, when any rea- 

 sonable definition of continuity is em- 

 ployed. The first step in the discovery of 

 this error was made when it occurred to 

 Ampere that the existence of the differen- 

 tial coefficient could only be asserted as a 

 theorem requiring proof; and he himself 

 published an attempt at such proof. The 

 erroneous character of the former belief on 

 this matter was most strikingly exhibited 

 when Weierstrass produced a function 

 which is everywhere eontinuoi;s, but which 

 nowhere possesses a differential coefficient; 



such functions can now be constructed ad 

 libitum. It is not too much to say that no 

 one of the general theorems of analysis is 

 true without the introduction of limitations 

 and conditions which were entirely im- 

 known to the discoverers of those theorems. 

 It has been the task of mathematicians 

 under the lead of such men as Cauehy, 

 Riemann, "Weierstrass and G. Cantor, to 

 carry out the work of reconstruction of 

 mathematical analysis, to render explicit all 

 the limitations of the truth of the general 

 theorems, and to lay down the conditions 

 of validity of the ordinary analytical oper- 

 ations. Physicists and others often main- 

 tain that this modern extreme precision 

 amounts to an unnecessary and pedantic 

 purism, because in all practical applica- 

 tions of mathematics only such functions 

 are of importance as exclude the remoter 

 possibilities contemplated by theorists. 

 Such objections leave the true mathemati- 

 cian unmoved; to him it is an intolerable 

 defect that, in an order of ideas in which 

 absolute exactitude is the guiding ideal, 

 statements should be made, and processes 

 employed, both of which are subject to un- 

 expressed qualifications, as conditions of 

 their truth or validity. The pure mathe- 

 matician has developed a specialized con- 

 science, extremely sensitive as regards sins 

 against logical precision. The physicist, 

 with his conscience hardened in this respect 

 by the rough-and-tumble work of investi- 

 gating the physical world, is apt to regard 

 the more tender organ of the mathemati- 

 cian with that feeling of impatience, not 

 unmingied with contempt, which the man 

 of the world manifests for what he consid- 

 ers to be over-scrupulositj' and unprae- 

 ticality. 



It is true that we can not conceive how 

 such a science as mathematics could have 

 come into existence apart from physical 

 experience. But it is also true that phys- 



