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 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 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 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 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 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 from 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 not strictly true: there exist no real things exactly conformable to the definitions. 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, existence of such things. I answer that, according 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 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 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
