THEORY OF.
Of the various designations of this mode of argument, "mathematical induction' is undoubt edly the most appropriate, for, though one may not be able to agree with Poincare (see Bibliog raphy below) that the mode in question is characteristic of mathematics, it is peculiar to that science, being indeed, as he has called it, 'mathematical reasoning par excellence.' The nature of mathematical induction as it is ordinarily understood may be made clear by an example. Perhaps the simplest application of the method is found in the proof of the theorem: (a) 1 + 2+ 3 + +nin(n +1) where is denotes any positive integer whatever. Suppose it ascertained by observation or other wise that (1) 1 +2=i2(2+1), (2) Facts (1) and (2)justify the suspicion that maybe a fact. The proof by mathematical induction that (a) is indeed true runs as fol lows: It is assumed that (a) is true for some definite but unspecified integer n. Then by adding n + 1 to each member of the assumed equation, n having the same meaning as in the assumption, one fords ($) .... +41+ n + 1 "'–i(n+ 1) (vs + 2).
So it is seen that, if (a) be true for some in teger n, it is true also for the next greater integer n +1. But by (2), (a) is true when n is 3; it is, therefore, true for 3 + 1, or 4; there fore, for 4 + 1, or 5. The argument is then usually closed by saying "and so on, hence (a) is true for any integer whatever,' or by an equivalent speech. The reader will recall that the binomial theorem, the Newtonian expansion of (a+ b)n, where n is any positive integer, is justified in essentially the foregoing manner. Numerous other examples of propositions sim ilarly established may be found in the better recent textbooks of algebra.
The nature and the role of the foregoing et cetera, 'and so on,* demand consideration. Without it, the argument as stated seems obvi ously incomplete. But how is the et cetera to be logically justified? By reference to some axiom or principle of thought? If so, what? Or can the phrase be in some way dispensed with without damage to the argument? Before attempting to answer them it may be well to show the inevitableness of the questions by a further analysis. Suppose it established, in regard to some property p (where, for ex ample, p might signify the validity of the bi nomial theorem for some integral exponent) : (1) that p belongs to the integer 1, that is, referring again to the mentioned example, the theorem is valid for the exponent I ; (2) that, if p belong to an integer n, it belongs to n + 1. Propositions (1) and (2) furnish the
means of generating, one after another, a se quence of syllogisms by which one proves first that p belongs to 2, then to 3, then to 4, and so on. Note that in order to ascertain by this analytic (syllogistic) method whether p belongs to a specified Integer m, it is necessary to de termine in advance the same question for each of the integers 2, 3, , m-1, in the order as written, a process requiring a number of syllogisms which is greater the greater the number m. Accordingly this method, of suc cessive deductions, is not available for deter mining whether p belongs to each in the (in finite) totality of integers. Equally powerless to that end is experience (including observe tion), for this can take account of the individ uals of a finite assemblage of objects at most. Either analysis or experience may avail if a sequence be finite, but if it be infinite both must fail. Not less vain is it to invoke finally the aid of induction as the term is understood and employed in the physical sciences, for this latter, resting upon a purely assumed order in the external universe, is confessedly inductio imperfecta, and, being such, can yield approxi mate certainty only.
Nevertheless, despite the inadequacy of the means mentioned, as soon as hypotheses (1) and (2) are admitted and the indicated se quence of deductions is berm, °the judgment imposes itself upon us with irresistible evi dence') that p is a property of all the integers. Why? That is, how justify the so on"? It appears to be clear that the answer must be the adduction or invocation of an additional presupposition of formal thought, a presup position whose formulation shall mark a con scious extension of the domain of logic by affirming as axiomatic that apodictic certainty can and does transcend every limited sequence of deductions or observations. Such presup position, which may be called the axiom of in finity, is stated by Poincare, in answer to the foregoing question, as follows: It is the affirmation of the power of the mind which knows it is capable of conceiving the indefinite repetition of a same act as soon as this act is found to be once possible?' The act or opera tion, which cannot indeed be indefinitely re peated, but which by the axiom can be con ceived as so repeated, is, in the present case, the construction of the syllogisms of the se quence above mentioned.