Victor Katz wrote in a discussion at the Historia mathematica discussion list, "as Barnabas Hughes notes, it is difficult to answer the question of the "first instances of proof by induction" unless one carefully defines what one means by "proof by induction. Noicomachus certainly has an argument which we would find very easy to convert into a modern formal "proof by induction.
See my book, pp. Pascal may well be the first to state the modern principle of mathematical induction explicitly, but even he does not give proofs in the modern style - because he has no notation for a general "n". Thus he generally gives proofs by what I call the "method of generalizable example. In another post Barnabus Hughes suggests yet an earlier "first use" of induction: If the essence of math induction lies in a process that begins at some small value, which process can be continued to larger values which regardless of their size maintain the pattern one wishes to accept, then I would hazard that Nicomachus of Geresa used the essence of math induction where he discussed figurate numbers Arithmetica , Bk.
I think that Nick established the pattern by induction. According to Wikipedia this was Levi ben Gershon a. The work is notable for its early use of proof by mathematical induction, and pioneering work in combinatorics. Gersonides was also the earliest known mathematician to have used the technique of mathematical induction in a systematic and self-conscious fashion.
The word "induction" is used in a different sense in philosophy. One has to distinguish Mathematical induction from "induction" in philosophy.
These are very different things. I am not aware of any ancient Greek use of Mathematical induction. The process of reasoning called "mathematical induction" has had several independent origins. Pascal and P. Fermat, and the Italian F. It seems Fermat might have been the first. Peirce says "mathematical induction" is an improper term for "Fermatian inference" CP 6. In truth, of infinite collections there are but two grades of magnitude, the endless and the innumerable.
Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Who introduced the Principle of Mathematical Induction for the first time?
Ask Question. Asked 6 years, 11 months ago. Active 4 years, 11 months ago. Viewed 9k times. Improve this question. Conifold Add a comment. Active Oldest Votes. New York: Dover. Section 3. Franklin, J. Proof in Mathematics: An Introduction. Sydney: Quakers Hill Press. History Acerbi, F. A Proof by Complete Induction? Archive for History of Exact Sciences 55 : 57— Bussey, W. The American Mathematical Monthly 24 5 : — Cajori, Florian The American Mathematical Monthly 25 5 : — Did They Use It?
Physis XXXI : — Freudenthal, Hans Archives Internationales d'Histiore des Sciences 6 : 17— Katz, Victor J. History of Mathematics: An Introduction. Rabinovitch, Nachum L. Archive for the History of Exact Science 6 : — Rashed, Roshdi Archive for History of Exact Sciences 9 : 1— Ungure, S. While the roots of formal mathematical induction are nested in works from Fermat all the way back, one might say, to Euclid's proof of the infinity of the primes, the work of Maurolycus was unique in the formal use of attaching one term to the next in a general way.
The method of Maurolycus was repeated and extended in the works of Pascal to be a much more clear illustration of the present method but none of them used a particular name for their logical process. Then in his Arithmetica infinitorum in Wallis decided to name the term. His inductive method followed very much the unnamed method of Maurolycus. Bernoulli gives no specific name to his process, but uses his method as an improvement on the "incomplete induction" earlier used.
Then early in the 19th century, George Peacock uses the term "demonstrative induction" in his Treatise on Alebra. Then several years later, Augustus De Morgan proposes the name "successive induction" but then at the end of the article he talks about the method as "mathematical induction.
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