- Finitism
In the

philosophy of mathematics ,**finitism**is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed fromnatural number s 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:

] Although most modern constructivists take a weaker view, they can trace the origins of constructivism back to Kronecker's finitist work.*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*In 1923,

Thoralf Skolem published a paper in which he presented a semi-formal system, what is now known asPrimitive 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. fullPeano Arithmetic ) was to be given. (SeeHilbert's program .)Reuben Goodstein is another proponent of finitism. Some of his work involved building up to analysis from finitist foundations. Although he denied it, much ofLudwig 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 alsoAristotle may 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 withAlexander Esenin-Volpin .**References****See also***

temporal finitism

*Infinitism **External links*** [

*http://plato.stanford.edu/entries/geometry-finitism/ Finitism in Geometry*] , entry in theStanford Encyclopedia of Philosophy

* [*http://uk.geocities.com/frege@btinternet.com/cantor/wittgensteinquotes.htm Wittgenstein's writing about the infinite*]

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### Look at other dictionaries:

**finitism**— FINITÍSM s.n. Concepţie metafizică care susţine că universul este mărginit. – Din fr. finitisme. Trimis de tavilis, 24.05.2002. Sursa: DEX 98 finitísm s. n. Trimis de siveco, 10.08.2004. Sursa: Dicţionar ortografic FINITÍSM s.n. Concepţie… … Dicționar Român**finitism**— noun 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 … Wiktionary**finitism**— In the philosophy of mathematics, the view that the only legitimate numbers are finite: ‘God created the integers, the rest is the work of man’ (L. Kronecker). In mathematics the restriction to finitary methods was advocated by David Hilbert. It… … Philosophy dictionary**finitism**— [ fʌɪnʌɪtɪz(ə)m] noun Philosophy & Mathematics rejection of the belief that anything can be infinite. Derivatives finitist noun & adjective … English new terms dictionary**finitism**— fi·nit·ism … English syllables**finitism**— Финитизм … Вестминстерский словарь теологических терминов**finitism**— n. belief in the finiteness of the world, God, etc. Derivatives: finitist n … Useful english dictionary**Temporal finitism**— is the idea that time is finite.The philosophy of Aristotle, expressed in such works as his Physics , held that although space was finite, with only void existing beyond the outermost sphere of the heavens, time was infinite. This caused problems … Wikipedia**theistic fïnitism**— Теистический финитизм … Вестминстерский словарь теологических терминов**Philosophy of mathematics**— The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of … Wikipedia