864 



SCIENCE 



[N. S. Vol. XXXVIII. No. 990 



new method is the one who understands 

 best the fundamental ideas on which the 

 methods of his subject are based and the 

 relation of these ideas and methods to cor- 

 responding ones in allied fields of study. 



It is, therefore, important to the stu- 

 dent of every science to analyze the growth 

 of method in his science and to ascertain 

 the fundamental basis on which it has de- 

 veloped. This analysis requires a wider 

 grasp of the subject than the student can 

 possess in the early years of his labor. 

 But he can appreciate, to a large extent, 

 the results of such an analysis and profit 

 by a knowledge of them, if they are pre- 

 sented by some one of a fuller experience 

 than himself. 



It is my purpose this evening to present 

 to you the outcome of such an analysis of 

 the nature of mathematical and of scien- 

 tific demonstration. 



A method which was considered useful 

 and legitimate in one generation has often 

 been discarded in the next. Sometimes it 

 has been replaced by another which was 

 merely more powerful and at least equally 

 convenient. At other times it has been 

 found to be not a legitimate method; and 

 it has been necessary to abandon it be- 

 cause investigators could no longer be sure 

 of results obtained by means of it. This 

 has been true both of mathematics and of 

 experimental science, but less frequently 

 of the former than of the latter. 



For a mathematical method a first requi- 

 site is that the mind shall assert with the 

 strongest emphasis that the method is legiti- 

 mate. We shall say nothing about how 

 this conviction may have arisen: we shall 

 first demand of it only that it shall be a 

 profound and universal conviction of the 

 human mind. 



I shall illustrate what I mean here by 

 an example. Let us take the principle or 

 method of mathematical induction. It is 



convenient to consider a particular case of 

 its use. Suppose that we wish to demon- 

 strate the binomial theorem, 



(a-)-J)» = a'>-)- nan-^l -[-...+ nab'^-'^ + 6", 

 for every positive integer exponent n. 

 Our method of procedure is as follows: 

 We first observe that the theorem is true 

 for n equal to 1. The next step is to 

 prove that if it is true for n equal to fc, 

 where k is any positive integer, it is like- 

 wise true for n equal to ^ + 1 ; and we 

 shall suppose now that this step has been 

 made by the necessary argumentation. 

 Now we know that the theorem is true for 

 11 equal to 1; from the result last men- 

 tioned we conclude further that the theo- 

 rem is true for n equal to 2. Since it is 

 true for n equal to 2 we may apply our 

 previous result again and conclude that is 

 is true for n equal to 3. Likewise we pro- 

 ceed to the case when n is equal to 4 ; and' 

 so on. 



Now, if i)ne analyzes the principle on 

 which this argument is based, the conclu- 

 sion comes home to him with a compelling 

 force ; and he can not fail to have confidence 

 in it. He has verified the theorem per- 

 haps in only a few cases; but he has no 

 fear that a case will ever be found to con- 

 tradict it. 



The first requirement of a mathematical 

 method, as I have said, is that it shall pos- 

 sess just this property of compelling con- 

 fidence in the conclusions reached by its 

 means. The ground of this compelling 

 power in the method the mathematician 

 (as such) does not seek to find; that is a 

 problem for the philosophers. 



But such credentials as those mentioned, 

 however good they may appear to be, are 

 never accepted by the mathematician as 

 entirely satisfactory. He does not, indeed, 

 dispute their legitimacy. But, through 

 much experience, he has found that meth- 

 ods exist concerning which the uninitiated. 



