Finitism

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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 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."

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 (e.g. full Peano Arithmetic) was to be given. (See Hilbert's program.) An important thing to understand is that Hilbert's finitism is constrained solely on the length of mathematical proofs. Hilbert did not demand finitism of models, but instead he embraced the very source of transfinitism: "No one shall expel us from the Paradise that Cantor has created for us". It is not hard to understand that an infinitely long proof is impossible: a proof that never ends, is not a proof. Finitists deny the infinity of models too. According to Löwenheim-Skolem theorem LwS, all talk about innumerable infinite models can be substituted by the talk about numerably infinite models. Therefore, LwS is at least somewhat finitist in nature.

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. 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.

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