COSMIC PHILOSOPHY 



that of the known. But no other standard is 

 possible for us." ^ 



The same general considerations will apply to 

 the truths of mathematics, which some Kantians 

 regard as the necessary truths par excellence^ — 

 habitually speaking of them as if they were 

 in some way truer than physical and chemical 

 truths. Bearing in mind what was said a mo- 

 ment ago, it will be sufficient to observe that in 

 mathematics we utter propositions with respect 

 to certain particular relations alone, without 

 regard to other conditions, and hence there is 

 absolutely no room for contingency. Let me 

 conclude this portion of the subject by a cita- 

 tion from Mr. Lewes : " When we say that 

 twice two is four, or that the internal angles of 

 a triangle are equal to two right angles, we ab- 

 stract the relations of Number and Form from 

 all other conditions whatever, and our proposi- 

 tions are true, whether the objects counted and 

 measured be hot or cold, large or small, heavy 

 or light, red or blue. Inasmuch as the truths 

 express the abstract relations only, no change 

 in the other conditions can affect these relations ; 

 and truths must always remain undisturbed un- 

 til a change take place in their terms. Alter the 

 number two^ or the figure triangle^ by an infini- 

 tesimal degree, and the truth is thereby altered. 

 When we say that bodies expand by heat, the 



^ Lewes, History of Philosophy y 2d edition, p. 668. 



84 



