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Intuitionism is a mathematical philosophy which holds that mathematics is a purely formal creation of the mind. It was originated in the early twentieth century by Dutch mathematician L.E.J. Brouwer. Intuitionism posits that mathematics is an internal, content-empty process whereby consistent mathematical statements can only be conceived of and proven as mental constructions. In this sense, intuitionism contradicts many core principles of classical mathematics, which holds that mathematics is the objective analysis of external existence.
Intuitionism differs from classical philosophies of mathematics, such as formalism and Platonism, in that it does not assume the existence of an external mathematically coherent reality. Additionally, it does not assume that mathematics is a symbolic language which has to follow certain fixed rules. Thus, since symbolic figures commonly used in mathematics are considered pure mediation, they are used only to transmit mathematical ideas from the mind of one mathematician to another, and do not in themselves suggest further mathematical proofs. The only two things assumed by intuitionism are the awareness of time and the existence of a creating mind.
Intuitionism and classical mathematics each posit different explanations of what it means to call a mathematical statement true. In intuitionism, the truth of a statement is not strictly defined by its provability alone, but rather by the ability of a mathematician to intuit the statement and prove it by the further elucidation of other rationally consistent mental constructions.
Intuitionism has serious implications which contradict some key concepts in classical mathematics. Perhaps the most famous of these is the rejection of the law of the excluded middle. In the most basic sense, the law of the excluded middle says that either “A” or “not A” can be true, but both cannot be true at the same time. Intuitionists hold that it is possible to prove both “A” and “not A” as long as mental constructions can be built which prove each consistently. In this sense, proof in intuitionist reasoning is not concerned with proving whether or not “A” exists, but is instead defined by whether both “A” and “not A” can be coherently and consistently constructed as mathematical statements in the mind.
Although intuitionism has never supplanted classical mathematics, it still receives a great deal of attention today. The study of intuitionism has been associated with a wide degree of advancement in the study of mathematics, as it replaces concepts about abstract truth with concepts about the justification of mathematical constructions. It has also been given some treatment in other branches of philosophy for its concern with an idealized and pan-subjective creating mind, which has been compared to Husserl’s phenomenological conception of the “transcendental subject.”