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induction
A method of reasoning by which one infers a generalization from a series of instances. Say there is a hypothesis H that contains the variable n, which is a whole number. To prove by induction that H is true for every value of n is a two-step process: (1) prove that H is true for n = 1; (2) prove that H being true for n = k implies that H is true for n = k + 1. This is sufficient because (1) and (2) together imply that H is true for n = 2, which, from (2), then implies H is true for n = 3, which implies H is true for n = 4, and so on. H is called an inductive hypothesis. Some philosophers don't accept this kind of proof, because it may take infinitely many steps to prove something; however, most mathematicians are happy to use it.
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• MATHEMATICS
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