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.
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The belief that there are cannot enough rules to explain how things work. Finitism rejects the belief that there can be an ideal language, for example, in which the meaning of statements are entirely determined by a set of self-consistent implicit rules. It also rejects the notion that a research program could conceiveably determine a sufficiently elaborate set of rules that completely explains how things work. Both Wittgenstein and Garfinkel were finitists. PMTH has a series of articles on finitism in Wittgenstein and Garfinkel.
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