The induction principle (Q691682)

The author gives a survey about the principle of induction, its formulations and its use in the development of mathematics. First he defines the principle of complete (mathematical) induction that he distinguishes from the principle of summative or infinite induction which he defines too. Then he looks at the occurrence of the method of induction by Aristotle, Quintilian and Bacon. The author argues that this type of incomplete induction was mostly used in the empirical sciences and has, however, only the status of a heuristic principle. The emergence of the principle of complete induction forms the contents of the following section. Analysing the use of the principle of induction by Euclid, Maurolico and Fermat, the author points out that they applied in their works a kind of induction that is weaker than the principle of mathematical induction. Furthermore, Maurolico possessed a vision of the mathematical induction, but he did not give it the status of a method of proof. This was done firstly by Pascal about 1654. Complete induction was then systematically used by Jacob Bernoulli and became more and more known during the 18th century. At the end of the 18th century, a special name was coined for that principle, firstly ``Kästner's method'', then ``complete induction'', the latter probably for the first time by Fries in his ``System of logic'' (1819). However, it was through Dedekind only that complete induction became a common term. Finally, the author sketches some attempts to prove the principle of complete induction that resulted in the insight to take the principle as a postulate, its use for characterizing the series of natural numbers through Dedekind and Peano, and its generalizations in modern set theory.