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 from natural numbers in a finite number of steps.

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(Most constructivists, in contrast, allow a countably infinite number of steps.)

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The most famous proponent of finitism was Leopold Kronecker, who said:

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:"God created the natural numbers, all else is the work of man."

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

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Reuben Goodstein is another exponent 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 have a strong affinity with finitism.

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Even stronger than finitism is ultrafinitism (also known as ultraintuitionism), associated primarily with Alexander Esenin-Volpin.

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Philosophy of mathematics: Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: "why is mathematics useful in describing nature?", "in which sense(s), if any, do mathematical entities such as numbers exist?" and "why and how are mathematical statements true?". Various approaches t...

Constructivism: In education, constructivism is a learning theory which holds that knowledge is not transmitted unchanged from teacher to student, but instead that learning is an active process of recreating knowledge. Constructivists teach techniques that place emphasis on the role of learning activities in a goo...

Natural number: Natural number can mean either a positive integer (, , , , ...) or a non-negative integer (, , , , , ...)....