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Intuitionism
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Intuitionism is the belief that truth can be arrived at through direct, unmediated awareness of reality. In ethics, intuitionism argues that moral judgements can be created by an immediate awareness of moral value. In aesthetics, intuitionism is the view that aesthetic value, or "beauty," is something nonnatural.
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Intuitionism takes the validity of a mathematical statement to be equivalent to its having been proved; what other criteria can there be for truth (an intuitionist would argue) if mathematical objects are merely mental constructions? This means that an intuitionist may not believe that a mathematical statement has the same meaning that a classical mathematician would. For example, to say A or B, to an intuitionist, is to claim that either A or B can be proved. In particular, the law of excluded middle, A or not A, is disallowed since one cannot assume that it is always possible to either prove the statement A or its negation.
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Intuitionism's history can perhaps be traced to the nineteenth century. Cantor and his teacher Kronecker — a confirmed finitist — disagreed, (and the details of Cantor's subsequent hospitalization are well known). Frege's effort to reduce all of mathematics to a logical formulation was counted a failure — in the face of a letter from Bertrand Russell received by Frege just as his life's work was about to be published, and which outlined some very troubling paradoxes. For more see Davis (2000) Chapters 3 and 4: Frege: From Breakthrough to Despair and Cantor: Detour through Infinity. See van Heijenoort for the original works and Heijenoort's commentary.
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Intuitionism's history can perhaps be traced to the nineteenth century. Cantor and his teacher Kronecker — a confirmed finitist — disagreed. Frege's effort to reduce all of mathematics to a logical formulation was a scientific breakthough in the department of logic, and it greatly inspired the younger generation, including a youthful Bertrand Russell.
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Heyting's formalization of intuitionistic logic in 1930 was an important step in making Intuitionism accessible to the mathematical community. However, the price of this formalization was the abandonment of L.E.J. Brouwer's philosophical ideas. These ideas have only been expressed rather obscurely in his writings, and most early papers were not available until the early 1980s, long after the dispute between Intuitionism and Formalism had ceased.
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