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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 from natural numbers in a finite number of steps.
Most constructivists, in contrast, allow a countably infinite number of steps.
In her book Philosophy of Set Theory, Mary Tiles characterized those who allow contably 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."
Although most modern constructivists take a weaker view, they can trace the origins of constructivism back to Kronecker's finitist work.
In 1923, Thoralf Skolem published 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 (eg full Peano Arithmetic) was to be given. (See Hilbert'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 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 Aristotle may be characterized as a Classical Finitist.
Even stronger than finitism is ''ultrafinitism'' (also known as ''ultraintuitionism''), associated primarily with Alexander Esenin-Volpin.

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External links

External links

★ Finitism in Geometry, entry in the Stanford Encyclopedia of Philosophy

★ Wittgenstein's writing about the infinite