

ON THE NATURE OF MATHEMATICAL 

 KNOWLEDGE. 



"TV MATHEMATICALLY certain and unequivocal" is a phrasa 

 **'*> which is often heard in the sciences and in common life, 

 to express the idea that the seal of truth is more deeply imprinted 

 upon a proposition than is the case with ordinary acts of knowledge. 

 We propose to investigate in this article the extent to which math- 

 ematical knowledge really is more certain and unequivocal than 

 other knowledge. 



The intrinsic character of mathematical research and know- 

 ledge is based essentially on three properties : first, on its conserv- 

 ative attitude towards the old truths and discoveries of mathematics; 

 secondly, on its progressive mode of development, due to the inces- 

 sant acquisition of new knowledge on the basis of the old ; and thirdly, 

 on its self-sufficiency and its consequent absolute independence. 



That mathematics is the most conservative of all the sciences 

 is apparent from the incontestability of its propositions. This last 

 character bestows on mathematics the enviable superiority that no 

 new development can undo the work of previous developments or 

 substitute new in the place of old results. The discoveries that 

 Pythagoras, Archimedes, and Apollonius made are as valid to-day 

 as they were two thousand years ago. This is a trait which no 

 other science possesses. The notions of previous centuries regard- 

 ing the nature of heat have been disproved. Goethe's theory of 

 colors is now antiquated. The theory of the binary combination of 

 salts was supplanted by the theory of substitution, and this, in its 

 turn, has also given way to newer conceptions. Think of the pro- 



