Intuitionism is a mathematical philosophy that views mathematics as a purely mental creation and rejects the existence of an external, mathematically consistent reality. It differs from classical mathematics in its definition of truth and rejects the law of excluded middle. Intuitionism has advanced the study of mathematics and is associated with an idealized, pan-subjective creative mind.
Intuitionism is a mathematical philosophy which holds that mathematics is a purely formal creation of the mind. It was created in the early 20th century by the Dutch mathematician LEJ Brouwer. Intuitionism postulates that mathematics is an internal process, empty of content, so that coherent mathematical statements can be conceived and proved only as mental constructions. In this sense, intuitionism contradicts many fundamental tenets 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 consistent reality. It also does not assume that mathematics is a symbolic language that must follow certain fixed rules. Hence, since the symbolic figures commonly used in mathematics are considered pure mediation, they are only used to convey mathematical ideas from the mind of one mathematician to another, and do not by themselves suggest further mathematical proofs. The only two things intuitionism assumes are awareness of time and the existence of a creative mind.
Intuitionism and classical mathematics postulate 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 further elucidation of other rationally consistent mental constructions.
Intuitionism has serious implications that contradict some key concepts of classical mathematics. Perhaps the most famous of these is the rejection of the law of excluded middle. In the most basic sense, the law of excluded middle says that “A” or “not A” can be true, but both cannot be true at the same time. Intuitionists argue that it is possible to prove both “A” and “not A” as long as one can construct mental constructs that demonstrate each consistently. In this sense, proof in intuitionistic reasoning is not concerned with proving whether or not ‘A’ exists, but instead is defined by whether both ‘A’ and ‘not A’ can be coherently and consistently constructed as mathematical statements in the mind.
While intuitionism never supplanted classical mathematics, it still receives a great deal of attention today. The study of intuitionism has been associated with a large 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 treated in other branches of philosophy for its concern with an idealized, pan-subjective creative mind, which has been compared to Husserl’s phenomenological conception of the “transcendental subject”.
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