168 EEASONING. 



CHAPTER V. 



OF DEMONSTRATION, AND NECESSARY TRUTHS. 



§ 1. If, as laid down in the two pi'eceding 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, deductive? 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 assurance attainable by reason? Why are mathematics by al- 

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

 losophy which, through the medium of mathematics, have been converted 

 into deductive sciences, considered to be independent of the evidence of 

 experience and observation, and characterized as systems of Necessary 

 Truth? 



The answer I conceive to be, that this character of necessity, ascribed to 

 the truths of mathematics, and (even with some reservations to be here- 

 after made) the peculiar certainty attributed to them, is an illusion ; in or- 

 der to sustain which, it is necessary to suppose that those trutlis relate to, 

 and express the properties of, purely imaginary objects. It is acknowl- 

 edged that the conclusions of geometry are deduced, partly at least, from 

 the so-called Definitions, and that those definitions are assumed to be cor- 

 rect 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 on:; concerning the meaning of a word, can ever 

 follow ; and that what apparently follows from a definition, follows in real- 

 ity from an implied assumption that there exists a real thing conformable 

 thereto. This assumption, in the case of the definitions of geometry, is not 

 strictly true: there exist no real things exactly conformable to the defini- 

 tions. There exist no points without magnitude; no lines without breadth, 

 nor perfectly straight; no circles with all their radii exactly equal, nor 

 squares with all their angles perfectly right. It will perhaps be said that 

 the assumption does not extend to the actual, but only to the possible, exist- 

 ence of such things. I answer that, according to any test we have of possi- 

 bility, they are not even possible. Their existence, so far as we can form 

 any judgment, would seem to be inconsistent with the physical constitu- 

 tion of our planet at least, if not of the universe. To get rid of this diffi- 

 culty, and at the same time to save the credit of the supposed system of 

 necessary truth, it is customary to say that the points, lines, circles, and 

 squares which are the subject of geometry, exist in our conceptions mere- 

 ly, and are part of our minds ; which minds, by working on their own ma- 

 terials, construct an a priori science, the evidence of which is purely men- 

 tal, and has nothing whatever to do with outward experience. By how- 

 soever high authorities this doctrine may have been sanctioned, it appears 



