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. 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, who said:
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, which is now known as primitive recursive arithmetic, that is widely taken to be a suitable background for finitist mathematics. This was adopted by David Hilbert and Paul 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 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. Georg Cantor's hierarchy of infinities), then also Aristotle 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 never-ending 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):
Even stronger than finitism is ultrafinitism (also known as ultraintuitionism), associated primarily with Alexander Esenin-Volpin.
Intuitionistic logic · Constructive analysis · Heyting arithmetic · Intuitionistic type theory · Constructive set theory ·
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