Inductive Reasoning: Mathematical Induction and Induction in Mathematics

property of the natural number system that is responsible for the correctness of math induction. It is still possible, of course, that math induction picks up quasi-inductive support from its fruitfulness in proving further theorems, but in this respect it doesn’t differ from any other math axiom. Although we think that math induction doesn’t threaten the distinction between deductive and inductive reasoning, there is a related issue about generalization in math that might. Math proofs often proceed by selecting an “arbitrary instance” from a domain, showing that some property is true of this instance, and then generalizing to all the domain’s members. For this universal generalization to work, the instance in question must be an abstraction or stand-in (an “arbitrary name” or variable) for all relevant individuals, and there is no real concern that such a strategy is not properly deductive. However, there’s psychological evidence that students don’t always recognize the difference between such an abstraction and an arbitrarily selected exemplar. Sometimes, in fact, students use exemplars in their proofs (and evaluate positively proofs that contain exemplars) that don’t even look arbitrary but are simply convenient, perhaps because the exemplars lend themselves to concrete arguments that are easy to understand. In these cases, students are using an inductive strategy, since the exemplar can at most increase their confidence in the to-be-proved proposition. It’s no news, of course, that people make mistakes in math. And it’s also no news that ordinary induction has a role to play in math, especially in the context of discovery. The question here is whether these inductive intrusions provide evidence that the deduction/induction split is psychologically untenable. Does the use of arbitrary and not-so-arbitrary instances show that people have a single type of reasoning mechanism that delivers conclusions that are quantitatively stronger or weaker, but not qualitatively inductive versus deductive? We’ve considered one way in which this might be the case. Perhaps people look for counterexamples, continuing their search through increasingly arbitrary (i.e., haphazard or atypical) cases until they’ve found such a counterexample or have run out of steam. The longer the search, the more secure the conclusion. We’ve seen, however, that this procedure doesn’t Mathematical Induction and Induction in Mathematics / 25 extend to deductively valid proofs; no matter how obscure the instance, it will still have an infinite number of properties that prohibit you from generalizing from it. 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