INDUCTION, MathematicaL Despite the age-long tyranny exercised by the Aris totelian logic —a tyranny having, at least in the domain of science, scarcely a match except in the case of Euclid's elements — the forms of thought, those diagrammatic representations of the orderliness of the reasoning processes, sus tain to-day perhaps even greater interest than ever before (see SYMBOLIC Looic). The mathe matician's interest in these forms is twofold, attaching to them both as norms for testing the validity of arguments and as constituting exceedingly subtle matter for mathematical in vestigation.
Of all argument forms, there is one which, viewed as the figure of the way in which the mind pins certainty that a specified property belonging but not immediately by definition to each element of a denumerable (see ASSEM BLAGE THEORY) assemblage of elements does so belong, enjoys the distinction of being at once perhaps the most fascinating, and, in its mathe matical bearings, doubtless the most important, single form in modern logic. This form is
that variously known as reasoning by recur rence, induction by connection (De Morgan), mathematical induction, complete induction, and Fermatian induction — so called by C. S. Peirce, according to whom this mode of proof was first employed by Fermat. Whether or not such priority is thus properly ascribed, it is certain that the argument form in question is unknown to the Aristotelian system, for this system allows apodictic certainty in case of deduction only, while it is the distinguishing mark of mathematical induction that it yields such certainty by the reverse process, a move ment from the particular to the general, from the finite to the infinite. See ASSEMBLAGES,