Force of Mathematical reasoning.—The question arises whence the peculiar force of mathematical, in distinction from other reasoning?—a fact observed by every one, but not easily explained: how happens this, and on what does it depend, this irresistible cogency which compels our assent? Is it owing to the pains taken to define the terms employed, and the strict adherence to those definitions? I think not; for other sciences approximate to mathematics in this, but not to the cogency of its reasoning. The explanation given by Stewart is certainly plausible. He ascribes the peculiar force of demonstrative reasoning to the fact, that the first principles from which it sets out, i. e., its definitions, are purely hypothetical, involving no basis or admixture of facts, and that by simply reasoning strictly upon these assumed hypotheses the conclusions follow irresistibly. The same thing would happen in any other science, could we (as we cannot) construct our definitions to suit ourselves, instead of proceeding upon facts as our data. The same view is ably maintained by other writers.
If this be so, the superior certainty of mathematical, over all other modes of reasoning, if it does not quite vanish, becomes of much less consequence than is generally supposed. Its truths are necessary in no other sense than that certain definitions being assumed, certain suppositions made, then the certain other things follow, which is no more than may be said of any science.
Confirmation of this View.—It may be argued, as a confirmation of this view, that whenever mathematical reasoning comes to be applied to sciences involving facts either as the data, or as objects of investigation, where it is no longer possible to proceed entirely upon hypothesis, as, e. g., when you apply it to mechanics, physics, astronomy, practical geometry, etc., then it ceases to be demonstrative, and becomes merely probable reasoning.
Mathematical reasoning supposed by some to be identical.—It has been much discussed whether all mathematical reasoning is merely identical, asserting, in fact, nothing more than that a=a; that a given thing is equivalent to itself, capable of being resolved at last into merely this. This view has been maintained by Leibnitz, himself one of the greatest mathematicians, and by many others. It was for a long time the prevalent doctrine on the Continent. Condillac applies the same to all reasoning, and Hobbes seems to have had a similar view, i. e., that all reasoning is only so much addition or subtraction. Against this view Stewart contends that even if the propositions themselves might be represented by the formula a=a, it does not follow that the various steps of reasoning leading to the conclusion amount merely to that. A paper written in cipher may be said to be identical with the same paper as interpreted; but the evidence on which the act of deciphering proceeds, amounts to something more than the perception of identity. And further, he denies that the propositions are identical, e. g., even the simple proposition 2×2=4. 2×2 express one set of quantities, and 4 expresses another, and the proposition that asserts their equivalence is not identical; it is not saying that the same quantity is equal to itself, but that two different quantities are equivalent.
II. Probable Reasoning.
Not opposed to Certainty.—It must be borne in mind, as already stated, that the probability now intended is not opposed to certainty. That Cæsar invaded Britain is certain, but the reasoning which goes to establish it, is only probable reasoning, because the thing to be proved is an event in history, contingent therefore, and not capable of demonstration.
Sources of Evidence.—Evidence of this kind of truths is derived from three sources: 1. Testimony; 2. Experience; 3. Analogy.
1. Evidence of Testimony.
In itself probable.—This is, à priori, probable. We are so constituted as to be inclined to believe testimony, and it is only when the incredibility of the witness has been ascertained by sufficient evidence, that we refuse our assent. The child believes whatever is told him. The man, long conversant with human affairs, becomes wary, cautious, suspicious, incredulous. It is remarked by Reid that the evidence of testimony does not depend altogether on the character of the witness. If there be no motive for deception, especially if there be weighty reasons why he should speak truth, or if the narrative be in itself probable and consistent, and tallies with circumstances, it is in such cases to be received even from those not of unimpeachable integrity.
Limits of Belief.—What are the limits of belief in testimony? Suppose the character of witnesses to be good, the narrative self-consistent, the testimony concurrent of various witnesses, explicit, positive, full, no motive for deception; are we to believe in that case whatever may be testified? One thing is certain, we do in fact believe in such cases; we are so constituted. Such is the law of our nature. Nor can it be shown irrational to yield such assent. It has been shown by an eminent mathematician that it is always possible to assign a number of independent witnesses, so great that the falsity of their concurrent testimony shall be mathematically more improbable, and so more incredible, than the truth of their statement, be it what it may.