SCIENCE 



[N.B. Vol. XXXTIII. No. 990 



retained. He proceeded entirely by exclu- 

 sion. First, all conceivable methods which 

 did not come up to a certain standard were 

 put aside. Those that remained were sub- 

 jected to further tests, one after another, 

 and some of them were found to be unsatis- 

 factory. Those left over were finally re- 

 tained because they had the negative recom- 

 mendation of never having been caught in 

 an act of deception. 



What shall we say then of the certainty 

 of mathematical doctrine at the present 

 day? To answer this question, let us ob- 

 serve that, in all preceding generations, 

 methods in mathematics have been used 

 with confidence which, in the experience of 

 a later day, were found to be not legiti- 

 mate; they have been discarded, sometimes 

 after generations of confident use. It is not 

 likely that men have heretofore always 

 made mistakes of this kind and that we 

 have suddenly come upon an age in which 

 mathematical methods are certain in the 

 absolute sense. 



We are then forced to the conclusion, 

 however unwelcome it may be, that the cer- 

 tainty of mathematics is after all not abso- 

 lute, but is relative. To be sure, it is the 

 most profound certainty which the mind 

 has been able to achieve in any of its proc- 

 esses; but it is not absolute. The mathe- 

 matician starts from exact data ; he reasons 

 by methods which have never been known 

 to lead to error; and his conclusions are 

 necessary in the sense, and only in the 

 sense, that no one now living can point to a 

 flaw in the processes by which he has 

 derived them. 



When we find ourselves forced to this 

 result, our first feeling is probably one of 

 disappointment. But a deeper analysis of 

 the matter will bring us to a different atti- 

 tude. It gives us a new sense of the prob- 

 lem which lies before us in the development 

 of mathematical thought. We have not 



merely to seek new results; but we have 

 also the larger problem of method to inspire 

 our activity and to lead us perhaps to 

 fundamental achievement. 



It is conceivable that methods may be 

 devised by means of which we shall attain 

 to well-nigh perfect certainty. Let us sup- 

 pose that we have found a method of argu- 

 ment, or a principle A, which has this 

 property, namely: In whatever way we 

 start from a principle not in accord with 

 it we shall be led into results which are 

 themselves mutually contradictory. Now 

 suppose that principle A is itself not a 

 legitimate one. Then there is a legitimate 

 principle B not in accord with it. From 

 this new principle we can get mutually 

 contradictory i-esults. That is, principle B 

 is both legitimate and not legitimate. This 

 being a contradiction in itself, we conclude 

 that the hypothesis from which it is deduced 

 is false. Therefore principle A is legiti- 

 mate. I say that it is conceivable that such 

 principles A will some day be discovered; 

 but they have not yet been found. 



In an earlier day, and of course without 

 the aid of such principles as I have just 

 mentioned, men apparently had come to a 

 feeling of absolute certainty about the accu- 

 racy of mathematical conclusions. Those 

 fundamental methods' of argumentation, of 

 which I spoke in the outset, they conceived 

 to belong to a class of innate or inherent 

 ideas which had been put in the mind of 

 man by the Creator. The initial hypoth- 

 eses and basic notions of a mathematical 

 discipline they thought of as belonging to 

 the same category. If these innate ideas 

 did not have all the elements of absolute 

 certainty, there could be only one conclu- 

 sion : the Creator had deliberately deceived 

 man. Since they considered this to be abso- 

 lutely impossible, they had complete con- 

 fidence in the certainty of mathematical 

 results. 



