358 THE POPULAR SCIENCE MONTHLY. 



of necessary truth, merely by the unfolding of the inherited intellectual 

 forms and faculties. 



" To state the case more specifically : Before a necessary truth can 

 be known as such, two conditions must be fulfilled. There must be a 

 mental structure capable of grasping the terms of the proposition and 

 the relation alleged between them; and there must be such definite 

 deliberate mental representation of these teims as makes possible a 

 clear consciousness of this relation. Non-fulfillment of either con- 

 dition may cause non-recognition of the necessity of the truth ; and 

 may even lead to acceptance of its contrary as true. Let us take 

 cases. 



" The savage who cannot count the fingers on one hand, can frame 

 no definite thought answ^ering to the statement that 7 and 5 make 

 12; still less can he frame the consciousness that no other total is 

 possible. 



"The boy adding np figures inattentively, says to himself that 7 

 and 5 make 11 ; and may repeatedly bring out a wrong result by re- 

 peatedly making this error. 



"Neither the non-recognition of the truth that T and 5 make 12, 

 which in the savage results from undeveloped mental structure, nor 

 the assertion, due to the boy's careless mental action, that they make 

 11, leads us to doubt the necessity of the relation between these two 

 separately-existing numbers, and the sum they make when existing 

 together. Nor does failure from either cause to apprehend the neces- 

 sity of this relation make us hesitate to say that, when its terms are 

 distinctly represented in thought, its necessity will be seen ; and that, 

 apart from any multiplied experiences, this necessity becomes cogni- 

 zable w^hen structures and functions are so far developed that groups 

 of 7 and 5 and 12 can be intellectually grasped. 



" Manifestly, then, there is a recognition of necessary truths, as 

 such, which accompanies mental evolution. Along with acquirement 

 of more complex faculty and more vivid imagination, there comes a 

 power of perceiving to be necessary truths what were before not 

 recognized as truths at all. And there are ascending gradations in 

 these recognitions. Thus a boy who has intelligence enough to see 

 that things which are equal to the same thing are equal to one another, 

 may be unable to see that ratios which are severally equal to certain 

 other ratios, that are unequal to each other, are themselves unequal ; 

 though to a more developed mind this last axiom is no less obviously 

 necessary than the first. 



" All this, which holds of logical and mathematical truths, holds, 

 with change of terms, of physical truths. There are necessary truths 

 in Physics, for the apprehension of which, also, a developed and dis- 

 ciplined intelligence is required ; and, before such intelligence arises, 

 not only may there be failure to apprehend the necessity of them, but 

 there may be vague beliefs in their contraries. Up to comparatively 



