Editing Philosophy of mathematics (section)

====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.