Editing Mathematical induction (section)

== History ==
In 370 BC, [[Plato]]'s [[Parmenides (dialogue)|Parmenides]] may have contained traces of an early example of an implicit inductive proof,{{sfn|Acerbi|2000}} however, the earliest implicit proof by mathematical induction was written by [[al-Karaji]] around 1000 AD, who applied it to [[Arithmetic progression|arithmetic sequences]] to prove the [[binomial theorem]] and properties of [[Pascal's triangle]]. Whilst the original work was lost, it was later referenced by [[Al-Samawal al-Maghribi]] in his treatise ''al-Bahir fi'l-jabr (The Brilliant in Algebra)'' in around 1150 AD.{{sfn|Rashed|1994|pp=62–84}}<ref>[https://books.google.com/books?id=HGMXCgAAQBAJ&pg=PA193 Mathematical Knowledge and the Interplay of Practices] "The earliest implicit proof by mathematical induction was given around 1000 in a work by the Persian mathematician Al-Karaji"</ref><ref>{{Cite web |title=The Binomial Theorem |url=https://mathcenter.oxford.emory.edu/site/math108/binomialTheorem/ |access-date=2024-12-02 |website=mathcenter.oxford.emory.edu|quote=That said, he was not the first person to study it.  The Persian mathematician and engineer Al-Karaji, who lived from 935 to 1029 is currently credited with its discovery. (''Interesting tidbit: Al-Karaji also introduced the powerful idea of arguing by mathematical induction.'')}}</ref>

Katz says in his history of mathematics {{quote|text=Another important idea introduced by al-Karaji and continued by al-Samaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus al-Karaji used such an argument to prove the result on the sums of integral cubes already known to [[Aryabhata]] [...] Al-Karaji did not, however, state a general result for arbitrary ''n''. He stated his theorem for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer. [...] Al-Karaji's argument includes in essence the two basic components of a modern argument by induction, namely the [[truth]] of the statement for ''n'' = 1 (1 = 1<sup>3</sup>) and the deriving of the truth for ''n'' = ''k'' from that of ''n'' = ''k'' - 1. Of course, this second component is not explicit since, in some sense, al-Karaji's argument is in reverse; this is, he starts from ''n'' = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument in ''al-Fakhri'' is the earliest extant proof of [[Squared triangular number|the sum formula for integral cubes]].<ref>Katz (1998), p. 255</ref>}}

In India, early implicit proofs by mathematical induction appear in [[Bhāskara II|Bhaskara]]'s "[[Chakravala method|cyclic method]]".<ref name="Induction Bussey">{{harvp|Cajori|1918|p=197|ps=: 'The process of reasoning called "Mathematical Induction" has had several independent origins. It has been traced back to the Swiss Jakob (James) Bernoulli, the Frenchman B. Pascal and P. Fermat, and the Italian F. Maurolycus. [...] By reading a little between the lines one can find traces of mathematical induction still earlier, in the writings of the Hindus and the Greeks, as, for instance, in the "cyclic method" of Bhaskara, and in Euclid's proof that the number of primes is infinite.'}}</ref>

None of these ancient mathematicians, however, explicitly stated the induction hypothesis. Another similar case (contrary to what Vacca has written, as Freudenthal carefully showed){{sfn|Rashed|1994|p=62}} was that of [[Francesco Maurolico]] in his ''Arithmeticorum libri duo'' (1575), who used the technique to prove that the sum of the first {{mvar|n}} [[parity (mathematics)|odd]] [[integer]]s is {{Math|''n''<sup>2</sup>}}.

The earliest [[Rigour#Mathematical proof|rigorous]] use of induction was by [[Gersonides]] (1288–1344).{{sfn|Simonson|2000}}{{sfn|Rabinovitch|1970}} The first explicit formulation of the principle of induction was given by [[Blaise Pascal|Pascal]] in his ''Traité du triangle arithmétique'' (1665). Another Frenchman, [[Pierre de Fermat|Fermat]], made ample use of a related principle: indirect proof by [[infinite descent]].

The induction hypothesis was also employed by the Swiss [[Jakob Bernoulli]], and from then on it became well known. The modern formal treatment of the principle came only in the 19th century, with [[George Boole]],<ref>"It is sometimes required to prove a theorem which shall be true whenever a certain quantity ''n'' which it involves shall be an integer or whole number and the method of proof is usually of the following kind. ''1st''. The theorem is proved to be true {{nowrap|1=when ''n'' = 1}}. ''2ndly''. It is proved that if the theorem is true when ''n'' is a given whole number, it will be true if ''n'' is the next greater integer. Hence the theorem is true universally. … This species of argument may be termed a continued ''[[Polysyllogism|sorites]]''" (Boole c. 1849 ''Elementary Treatise on Logic not mathematical'' pp. 40–41 reprinted in [[Ivor Grattan-Guinness|Grattan-Guinness, Ivor]] and Bornet, Gérard (1997), ''George Boole: Selected Manuscripts on Logic and its Philosophy'', Birkhäuser Verlag, Berlin, {{isbn|3-7643-5456-9}})</ref> [[Augustus De Morgan]], [[Charles Sanders Peirce]],{{sfn|Peirce|1881}}{{sfn|Shields|1997}} [[Giuseppe Peano]], and [[Richard Dedekind]].<ref name="Induction Bussey"/>