28 ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 



found changes which the conceptions of theoretical medicine, zool- 

 ogy, botany, mineralogy, and geology have undergone. It is the 

 same, too, in the other sciences. In philology, comparative linguis- 

 tics, and history our ideas are quite different from what they formerly 

 were. 



In no other science is it so indispensable a condition that what- 

 ever is asserted must be true, as it is in mathematics. Whenever, 

 therefore, a controversy arises in mathematics, the issue is not 

 whether a thing is true or not, but whether the proof might not be 

 conducted more simply in some other way, or whether the proposi- 

 tion demonstrated is sufficiently important for the advancement of the 

 science as. to deserve especial enunciation and emphasis, or finally, 

 whether the proposition rs not a special case of some other and 

 more general truth which is just as easily discovered. 



Let me recall the controversy which has been waged in this 

 century regarding the eleventh axiom of Euclid, that only one line 

 can be drawn through a point parallel to another straight line. This 

 discussion impugned in no wise the truth of the proposition ; for 

 that things are true in mathematics is so much a matter of course 

 that on this point it is impossible for a controversy to arise. The 

 discussion merely touched the question whether the axiom was 

 capable of demonstration solely by means of the other propositions, 

 or whether it was not a special property, apprehensible only by 

 sense- experience, of that space of three dimensions in which the 

 organic world has been produced and which therefore is of all others 

 alone within the reach of our powers of representation. The truth 

 of the last supposition affects in no respect the correctness of the 

 axiom but simply assigns to it, in an epistemological regard, a dif- 

 ferent status from what it would have if it were demonstrable, as 

 was at one time thought, without the aid of the senses, and solely by 

 the other propositions of mathematics. 



I may recall also a second controversy which arose a few de- 

 cades ago as to whether all continuous functions were differentiate. 

 In the outcome, continuous functions were defined that possessed 

 no differential coefficient, and it was thus learned that certain truths 

 which were enunciated unconditionally by Newton, Leibnitz, and 



