ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 37 



arithmetical knowledge lies entirely in its applications. But this 

 constitutes no reason why many mathematicians, pursuing their 

 purely deductive bent of mind, should not devote themselves ex- 

 clusively to pure arithmetical developments and leave it to others 

 at the proper time to turn to the material profit of the world the 

 capital which they have garnered. 



Geometry, on the other hand, must have recourse in a much 

 higher degree than arithmetic to the outside world for its first notions 

 and premises. The axioms of geometry are nothing but facts of ex- 

 perience perceived by o*ur senses. The geometry which Bolyai, Lo- 

 bachevski, Gauss, Riemann, and Helmholtz created and which is 

 both independent of the eleventh axiom of Euclid and perfectly 

 free from self-contradictions, has supplied an epistemological dem- 

 onstration that geometry is a science that rests on the observation 

 of nature, and therefore in the correct sense of the word, is a natu- 

 ral science. 



Yet what a difference there is, for instance, between geometry 

 and chemistry! Both derive their constructive materials from sense- 

 perception. But whilst geometry is compelled to draw only its first 

 results from observation and is then in a position to move forward 

 deductively to other results without being under the necessity of 

 making fresh observations, chemistry on the other hand is still 

 compelled to make observations and to have recourse to nature. 



It follows, therefore, that a given act of geometrical knowledge 

 and a given act of chemical knowledge are with respect to the cer- 

 tainty of the truth they contain not qualitatively but only quantita- 

 tively different. In chemistry the probability of error is greater 

 than in geometry, because more numerous and more difficult ob- 

 servations have to be made there than in geometry, where only the 

 very first premises, which no man with sound senses could ever im- 

 pugn, rest on observation. 



The preceding reflexions deprive mathematical knowledge of 

 that degree of certainty and incontestability which is commonly 

 attributed to it when we say a thing is 'mathematically certain. " 

 This certainty is lessened still more as we pass to the semi-math- 

 ematical sciences, where mechanics has the first claim to our at- 



