SCIENCE 



[N. S. Vol. XLV. No. 1168 



nitz's second differential was meaningless. 

 But these and many other criticisms never 

 hindered the development of the new cal- 

 culus, but served rather to aid in clearing 

 off certain excrescences which had nothing 

 to do with its essential characteristics and 

 in helping it to that central place of im- 

 portance which it holds to-day. 



The criticism last mentioned is one which 

 is made so often that it is profitable to 

 dwell longer upon it. So often the mathe- 

 matician hears: "What is the use of what 

 you are doing?" He knows a thousand 

 answers to this question; and one of the 

 most effective is that which history has 

 given to the criticism of the illustrious 

 Huygens. The recently developed subject 

 of integral equations has sometimes been 

 confronted with the inquiry: "Why develop 

 this theory? Will not differential equa- 

 tions serve the purpose? But the mathe- 

 matician goes calmly ahead with the devel- 

 opment of those things which interest him 

 just as he did formerly; and in the new 

 case he anticipates with confidence the 

 same triumphant justification in the event 

 which has uniformly crowned his labors in 

 the past. 



Sometimes the criticisms directed against 

 mathematics have grown out of a miscon- 

 ception of the natural limitations to which 

 it is subject. Pages of formulas can not 

 get a safe result from loose data. No 

 amount of computation will remove from 

 a result the errors already existent in the 

 underlying observations. I have several 

 times been confronted with the statement 

 that mathematics made a great mistake in 

 this or that particular case in predicting 

 what was not found on proper examination 

 to be true. But after all there was nothing 

 wrong with the mathematics. It was 

 merely that the supposed laws of phenom- 

 ena on which the investigation was based 

 were not exact. It was they and not the 

 mathematics which were on trial. 



It is true, as one of my friends said to 

 me recently, that no machine can be con- 

 structed and completely theorized on 

 mathematical principles alone. When this 

 is given as a statement of fact I have 

 nothing to say in reply. But if this natural 

 limitation of the subject is spoken of (as I 

 have sometimes heard) as an expression of 

 the failure of mathematics, then an error 

 is made which ought to be corrected. 

 Mathematics does not claim to do the whole 

 thing in the development of science. It 

 simply has its role to perform; and there 

 is a devoted body of workers throughout 

 the world striving to see that it perform 

 this function with eminent success. How 

 it does and is to continue doing this will 

 be apparent from the following discussion 

 of the relation of mathematics to experi- 

 mental verification. 



By their very nature the conclusions of 

 pure mathematics are not subject to ex- 

 perimental examination. One would not 

 say that they are above or beyond experi- 

 ence, but that they are outside of it. Pure 

 mathematics deals with certain creations of 

 the human spirit and with these alone. So 

 far as it is concerned, no import attaches 

 to the inquiry after the impulse which re- 

 sulted in the creation of these things. The 

 mathematician qua mathematician is not 

 interested in this matter, however much it 

 may fascinate him as a philosopher; and 

 he develops his science usually with per- 

 fect indifference to such considerations, 

 rearing it from a small group of postu- 

 lates or perhaps even from the general log- 

 ical premises of all reasoning. In any 

 event, he drags out into the limelight all 

 hypotheses and keeps them vividly before 

 him during the progress of his investiga- 

 tions. 



In applied mathematics the state of 

 things is very different. Here the whole 

 treatment is bristling with implicit assump- 

 tions, some of them being carried con- 



