652 Transactions. — Miscellaneous. 



by which we satisfy ourselves of the truth of identical pro- 

 positions. We think out the truth of such a proposition ; and 

 there could be no better definition of a necessary truth than 

 one which "we see to be true by merely thinking of it." It 

 is obvious, indeed, that we could not have started on our course 

 of mental experiment without being already in possession of 

 all that experience could possibly furnish us with in regard to 

 what straight lines would or would not do in any given cir- 

 cumstances. The fact that we could think out any fresh 

 knowledge about them without refereiice to the world of fact 

 is itself surely evidence sufficient that such knowledge was 

 already implicated in the more obvious knowledge which we 

 had before us about them, and that all that we required 

 to do in regard to it was to unfold it. Kant's great 

 division, therefore, of a priori truths into " analytic " or 

 " implicative," and synthetic or augmentative, seems to be 

 misleading. Analytic or implicative truths may be them- 

 selves augmentative. Indeed, if the truths are there a 

 'priori, though not on the surface, what else can they be 

 but implicated? 



To recapitulate, then, we may lay down the following in 

 regard to necessary truths : (1) That they are always con- 

 cerned with abstractions, never with concrete realities ; 

 (2) that the opposite of them is in- the strictest sense of the 

 word inconceivable, not merely unbelievable ; (3) that this is 

 so because if we think of their opposite we find that the last 

 half of the statement sublates the first ; (4) that they are 

 truths which can be seen to be truths by merely thinking 

 about them ; and (5) that they are in reality truths of 

 sequence only, not of fact. 



The last affirmation brings us face to face with an ap- 

 parently formidable difficulty. How, it may be asked, can it 

 be that if the truths of geometry tire truths of sequence only 

 they can be applied to practical use in the world of fact ? Mr. 

 Mill's theory, of course, does not enable him to escape this 

 difficulty. Indeed, he expressly indorses Stewart's view that 

 the truths of mathematics rest on hypotheses. Any difficulty 

 that there is, however, is not peculiar to mathematical reason- 

 ing, though it comes out in a more obvious light in connection 

 with it than in connection with other sorts of reasoning. All 

 deductive reasoning, it appears, rests on hypotheses, from the 

 simple affirmation that blue is not green to the latest applica- 

 tion of the theory of natural selection in the field of politics or 

 sociology. A fact of sequence, such as the equality of the 

 square on the hypotenuse to the two squares on the other 

 sides of such right-angled triangles as we suppose ourselves to 

 construct, has a real interest and importance for us only when 

 we find that it is, at any rate, approximately true of the right- 



