EAI OnInductionabout induction in formal mathematical systems: given a structure all of whose objects can be built from a given set of constructors from simpler objects, properties that are preserved by all given constructors hold for all objects in the structure. Give examples with natural integers, lists, etc.
about induction in informal systems: we cannot ever know for sure that all the possibilities have been covered, that the actual structure at hand can be inductively defined by any given set of constructors. as with deduction, we must make hypotheses, only these are now "higher-order" hypotheses. Deductive reasoning with induction principles is not the same as inductive reasoning.
Happily, we have a criterion as to what hypotheses to make: the "simplest" one, as popularized by Occam's Razor, and as formalized by Kolmogorov's Complexity: given an existing knowledge base of facts, and a new fact that arises, choose whichever explanation of that fact minimizes the resources necessary to store the augmented knowledge base. [This minimization condition directly accounts as an evolutionary principle]
Actually, we should refine that into not only facts, but their explicit interexplanations, too; for Kolmogorov Complexity not being computable, we cannot have the scientific method rely upon using it constructively; however, we may rely on dynamically simplifying whole-world explanations, which is a computable process for which Kolmogorov Complexity gives a lower bounding limit.