460 MATHEMATICS 



true; for, as I have said, other modes of reasoning which are now 

 universally recognized as faulty have appealed in just this way to 

 the greatest minds of the past. Such confidence as we feel must, 

 I think, come from the fact that those modes of reasoning which 

 we trust have withstood the test of use in an immense number of 

 cases and in very many fields. This is the severest test to which any 

 theory can be put, and if it does not break down under it we may 

 feel the greatest confidence that, at least in cognate fields, it will 

 prove serviceable. But we can never be sure. The accepted modes 

 of exact reasoning may any day lead to a contradiction which would 

 show that what we regard as universally applicable principles are 

 in reality applicable only under certain restrictions. 1 



To show that the danger which I here point out is not a purely 

 fanciful one, it is sufficient to refer to a very recent example. Inde- 

 pendently of one another, Frege and Russell have built up the theory 

 of arithmetic from its logical foundations. Each starts with a definite 

 list of apparently self-evident logical principles, and builds up a 

 seemingly flawless theory. Then Russell discovers that his logical 

 principles when applied to a very general kind of logical cZass lead 

 to an absurdity; and both Frege and Russell have to admit that 

 something is wrong with the foundations which looked so secure. 

 Now there is no doubt that these logical foundations will be somehow 

 recast to meet this difficulty, and that they will then be stronger 

 than ever before. 2 But who shall say that the same thing will not 

 happen again? 



It is commonly considered that mathematics owes its certainty 

 to its reliance on the immutable principles of formal logic. This, 

 as we have seen, is only half the truth imperfectly expressed. The 

 other half would be that the principles of formal logic owe such 

 degree of permanence as they have largely to the fact that they 

 have been tempered by long and varied use by mathematicians. 

 "A vicious circle!" you will perhaps say. I should rather describe 

 it as an example of the process known to mathematicians as the 

 method of successive approximations. Let us hope that in this 

 case it is really a convergent process, as it has every appearance of 

 being. 



But to return to Peirce's definition. From what are these neces- 



1 If the view which I here maintain is correct, it follows that if the term " abso- 

 lute logical rigor" has a meaning, and if we should some time arrive at this abso- 

 lute standard, the only indication we should ever have of the fact would be that 

 for a long period, several thousand years let us say, the logical principles in ques- 

 tion had stood the test of use. But this state of affairs might equally well mean 

 that during that time the human mind had degenerated, at least with regard to 

 some of its functions. Consider, for instance, the twenty centuries following Euclid 

 when, without doubt, the high tide of exact thinking attained during Euclid's gen- 

 eration had receded. 



2 Cf. Poincar<'s view in La Science et VHypothese, p. 179, according to which 

 a theory never renders a greater service to science than when it breaks down. 



