philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. In her book Philosophy of Set Theory, Mary Tiles characterized those who allow countably infinite as Classical Finitists, and those who deny even countably infinite as Strict Finitists.
The most famous proponent of finitism was
Leopold Kronecker, who said::"God created the natural numbers, all else is the work of man." [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.? |accessdate=2008-07-19 |author=Gonzalez Cabillon, Julio |date=2000-02-03Gonzalez gives these sources for the memorial article:
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] Although most modern constructivists take a weaker view, they can trace the origins of constructivism back to Kronecker's finitist work.
Thoralf Skolempublished a paper in which he presented a semi-formal system, what is now known as Primitive recursive arithmetic, which is widely taken to be a suitable background for finitist mathematics. This was adopted by Hilbert and Bernays as the 'contentual', finitist system for metamathematics, in which a proof of the consistency of other mathematical systems (e.g. full Peano Arithmetic) was to be given. (See Hilbert's program.) Reuben Goodsteinis another proponent of finitism. Some of his work involved building up to analysis from finitist foundations. Although he denied it, much of Ludwig Wittgenstein's writing on mathematics has a strong affinity with finitism. If finitists are contrasted with transfinitists (proponents of e.g. Cantor's hierarchy of infinities), then also Aristotlemay be characterized as a Strict Finitist. Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity. (Note that Aristotle's actual infinity means simply an actualization of something neverending in nature, when in contrast the Cantorist actual infinity means the transfinite cardinal and ordinal numbers, that have nothing to do with the things in nature):
"But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences:there will be a beginning and end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. If, then,in view of the above considerations, neither alternative seems possible,an arbiter must be called in;" -Aristotle, Metaphysics, Book 3, Chapter 6.
Even stronger than finitism is "
ultrafinitism" (also known as "ultraintuitionism"), associated primarily with Alexander Esenin-Volpin.
* [http://plato.stanford.edu/entries/geometry-finitism/ Finitism in Geometry] , entry in the
Stanford Encyclopedia of Philosophy
* [http://firstname.lastname@example.org/cantor/wittgensteinquotes.htm Wittgenstein's writing about the infinite]
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