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.) 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.

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 has a strong affinity with finitism.

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