December 19, 1913] 



SCIENCE 



865 



mind asserts emphatically that they are 

 valid, whereas he knows eases in which they 

 lead to inconsistent results. 



Therefore these credentials are treated 

 by the mathematician as affording him 

 only a means of making a first choice of 

 methods to be examined. They are still to 

 be subjected to tests in the laboratory of 

 the mind. 



Tou may ask: To what sort of test may 

 one conceivably subject a method which 

 the mind approves with as much confidence 

 as it does that of mathematical induction, 

 for instance? There seems to be just one 

 such test available. Does it always lead to 

 consistent results? I do not say true re- 

 sults; for there is no one to determine 

 whether the results are true. If several 

 methods are involved at once, it is to be 

 demanded of them also that the results ob- 

 tained by means of any of them shall be 

 consistent with those obtained from others. 



Effectively, what the mathematician does, 

 then, is to select a number of methods in 

 the intuitional way which I have indicated 

 and then to subject them to the most exact- 

 ing requirements in the way of consistency 

 of results obtained by their use — results 

 exact in their nature and deduced from 

 exact data and covering a wide range of 

 thought. 



The only methods which he retains after 

 these extended tests are those which have 

 never been known to lead to a contradic- 

 tion at any time in the history of human 

 thought. One other analysis must finally 

 be made before they can be admitted into 

 the privileged circle of mathematical 

 methods. It must be ascertained of a 

 given method whether it is perfectly pre- 

 cise in its nature in the sense that no two 

 persons of intelligence have a different 

 opinion as to what the method is. There 

 is no disagreement, for instance, among 



thinkers concerning the definition of mathe- 

 matical iaduction. 



Once the mathematician has selected 

 some methods which he is willing to em- 

 ploy, he uses them in argument in the 

 coldest and most formal way. In making 

 discoveries intuition plays a most impor- 

 tant role and is a precious guide which he 

 can not dispense with. But when he states 

 his proofs he does it in terms which are 

 entirely free from intuition. Further, he 

 is careful to make sure that he has used no 

 methods except those which have already 

 successfully passed his most searching 

 scrutiny. Through sore experience he has 

 learned that safety lies in no other direc- 

 tion. 



But this is not aU. Every new use of his 

 methods gives rise to the possibility at least 

 that a contradiction has crept in through 

 some argument which has never before led 

 into such error; and this possibility must 

 be examined — certainly in all cases where 

 the research opens up a new field of thought, 

 if not also in the more common investiga- 

 tions. 



It is due to this extreme carefulness on 

 the part of the mathematician that we have 

 so strong a feeling of certainty in his con- 

 clusions. But if we analyze this feeling 

 with care we shall find, unexpectedly per- 

 haps to most of us, that it is due after all 

 to our experience with the methods em- 

 ployed, since under the most severe tests 

 they have never led us into contradiction. 

 (They are the only methods which possess 

 this latter property.) 



If you will recall what I said about the 

 way in which the mathematician has 

 selected his tools of investigation, you wiU 

 see why he can never be absolutely sure 

 that he has employed a proper procedure in 

 argument. At no stage in the development 

 of his method was there an absolute crite- 

 rion according to which a method was to be 



