CHAPTEE V. 



OF DEMONSTRATION, AND NECESSARY TRUTHS. 



1. IF, as laid down in the two preceding chapters, the 

 foundation of all sciences, even deductive or demonstrative 

 sciences, is Induction ; if every step in the ratiocinations even 

 of geometry is an act of induction ; and if a train of reasoning 

 is but bringing many inductions to bear upon the same subject 

 of inquiry, and drawing a case within one induction by means 

 of another ; wherein lies the peculiar certainty always ascribed 

 to the sciences which are entirely, or almost entirely, deduc 

 tive ? Why are they called the Exact Sciences ? Why are 

 mathematical certainty, and the evidence of demonstration, 

 common phrases to express the very highest degree of assur 

 ance attainable by reason ? Why are mathematics by almost 

 all philosophers, and (by some) even those branches of natural 

 philosophy which, through the medium of mathematics, have 

 been converted into deductive sciences, considered to be inde 

 pendent of the evidence of experience and observation, and 

 characterized as systems of Necessary Truth ? 



The answer I conceive to be, that this character of neces 

 sity, ascribed to the truths of mathematics, and even (with 

 some reservations to be hereafter made) the peculiar certainty 

 attributed to them, is an illusion ; in order to sustain which, 

 it is necessary to suppose that those truths relate to, and ex 

 press the properties of, purely imaginary objects. It is 

 acknowledged that the conclusions of geometry are deduced, 

 partly at least, from the so-called Definitions, and that those 

 definitions are assumed to be correct representations, as far as 

 they go, of the objects with which geometry is conversant. 

 Now we have pointed out that, from a definition as such, no 

 proposition, unless it be one concerning the meaning of a 

 word, can ever follow; and that what apparently follows 



