Editing Philosophy of mathematics (section)

===Intuitionism===
{{Main|Mathematical intuitionism}}

In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" ([[Luitzen Egbertus Jan Brouwer|L. E. J. Brouwer]]). From this springboard, intuitionists seek to reconstruct what they consider to be the corrigible portion of mathematics in accordance with Kantian concepts of being, becoming, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the ''a priori'' forms of the volitions that inform the perception of empirical objects.<ref>[[Robert Audi|Audi, Robert]] (1999), ''The Cambridge Dictionary of Philosophy'', Cambridge University Press, Cambridge, UK, 1995. 2nd edition. Page 542.</ref>

A major force behind intuitionism was [[L. E. J. Brouwer]], who rejected the usefulness of formalized logic of any sort for mathematics. His student [[Arend Heyting]] postulated an [[intuitionistic logic]], different from the classical [[Aristotelian logic]]; this logic does not contain the [[law of the excluded middle]] and therefore frowns upon [[Reductio ad absurdum|proofs by contradiction]]. The [[axiom of choice]] is also rejected in most intuitionistic set theories, though in some versions it is accepted.

In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of [[Turing machine]] or [[computable function]] to fill this gap, leading to the claim that only questions regarding the behavior of finite [[algorithm]]s are meaningful and should be investigated in mathematics. This has led to the study of the [[computable number]]s, first introduced by [[Alan Turing]]. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical [[computer science]].

====Constructivism====
{{Main|Constructivism (philosophy of mathematics)}}
Like intuitionism, constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse. In this view, mathematics is an exercise of the human intuition, not a game played with meaningless symbols. Instead, it is about entities that we can create directly through mental activity. In addition, some adherents of these schools reject non-constructive proofs, such as using proof by contradiction when showing the existence of an object or when trying to establish the truth of some proposition. Important work was done by [[Errett Bishop]], who managed to prove versions of the most important theorems in [[real analysis]] as [[constructive analysis]] in his 1967 ''Foundations of Constructive Analysis.''<ref>{{Citation|last=Bishop|first=Errett|author-link=Errett Bishop|year=2012|orig-year=1967|title=Foundations of Constructive Analysis|publisher=Ishi Press|location=New York|edition=Paperback|isbn=978-4-87187-714-5}}</ref>

====Finitism====
{{Main|Finitism}}
[[File:Leopold Kronecker (ca. 1880).jpg|thumb|[[Leopold Kronecker]]]]
[[Finitism]] is an extreme form of [[mathematical constructivism|constructivism]], according to which a mathematical object does not exist unless it can be constructed from [[natural number]]s in a [[finite set|finite]] number of steps. In her book ''Philosophy of Set Theory'', [[Mary Tiles]] characterized those who allow [[countably infinite]] objects as classical finitists, and those who deny even countably infinite objects as strict finitists.

The most famous proponent of finitism was [[Leopold Kronecker]],<ref>From an 1886 lecture at the 'Berliner Naturforscher-Versammlung', according to [[H. M. Weber]]'s memorial article, as quoted and translated in {{cite web
|url=http://www.cs.nyu.edu/pipermail/fom/2000-February/003820.html
|title=FOM: What were Kronecker's f.o.m.?
|access-date=2008-07-19
|author=Gonzalez Cabillon, Julio
|date=2000-02-03
|archive-date=2007-10-09
|archive-url=https://web.archive.org/web/20071009235907/http://cs.nyu.edu/pipermail/fom/2000-February/003820.html
|url-status=live
}}
Gonzalez gives as the sources for the memorial article, the following: Weber, H: "Leopold Kronecker", ''Jahresberichte der Deutschen Mathematiker Vereinigung'', vol ii (1893), pp. 5-31. Cf. page 19. See also ''Mathematische Annalen'' vol. xliii (1893), pp. 1-25.</ref> who said:
{{Blockquote|God created the natural numbers, all else is the work of man.}}

[[Ultrafinitism]] is an even more extreme version of finitism, which rejects not only infinities but finite quantities that cannot feasibly be constructed with available resources. Another variant of finitism is Euclidean arithmetic, a system developed by [[John Penn Mayberry]] in his book ''The Foundations of Mathematics in the Theory of Sets''.<ref name="Mayberry-2001">{{cite book |first=J.P. |last=Mayberry |author-link=John Penn Mayberry |title=The Foundations of Mathematics in the Theory of Sets |year=2001 |publisher=[[Cambridge University Press]]}}</ref> Mayberry's system is Aristotelian in general inspiration and, despite his strong rejection of any role for operationalism or feasibility in the foundations of mathematics, comes to somewhat similar conclusions, such as, for instance, that super-exponentiation is not a legitimate finitary function.