14S REASONING. 



CHAPTER 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 induc- 

 tion ; and if a train of reasoning is but bringing many inductions to 

 bear upon the same subject of inquiry, and dravsdng 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 1 Why are mathematics by almost all philosophers, and (by 

 many) even those branches of natural philosophy which, through the 

 medium of mathematics, have been converted into deductive sciences, 

 considered to be independent of the evidence of experience and ob- 

 servation, and characterized as systems of Necessary Truth 1 



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

 ascribed to the truths of mathematics, and even (with some reserva- 

 tions 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 express the properties of, purely 

 imaginary objects. It is acknowledged that the conclusions of ge- 

 ometry are deduced, partly at least, from the so-called Definitions, and 

 that those definitions are assumed to be correct descriptions, 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 fi-om a definition, follows in reality from 

 an implied assumption that there exists a real thing conformable 

 thereto. This assumption, in the case of the definitions of geometry, 

 is false : there exist no real things exactly conformable to the defini- 

 tions. There exist no points withowt magnitude ; no lines ^\dthout 

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

 equal, nor squares with all their angles perfectly right. It will per- 

 haps be said that the assumption does not extend to the actual, but 

 only to the possible, existence of such things. I answer that, accord- 

 ing to any test we have of possibility, they are not even possible. 

 Their existence, so far as we can form any judgment, would seem to 

 be inconsistent with the physical constitution of our planet at least, if 

 not of the universe. To get rid of this difficulty, and at the same 

 time to save the credit of the supposed systems 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 merely, and are 

 part of our minds ; which minds, by working on their own materials, 

 construct an a priori science, the evidence of which is purely mental, 

 and has nothing whatever to do with outward experience. By 

 howsoever high authorities this doctrine may have been sanctioned, 

 it appears to me psychologically incorrect. The points, lines, circles, 

 and squares, which any one has in his mind, are (I apprehend) simply 



