Any mathematical object is considered to be a product of a construction of a mind, and therefore, the existence of an object is equivalent to the possibility of its construction. This contrasts with the classical approach, which states that the existence of an entity can be proved by refuting its non-existence. For the intuitionist, this is invalid; the refutation of the non-existence does not mean that it is possible to find a constructive proof of existence. As such, intuitionism is a variety of mathematical constructivism; but it is not the only kind.
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. (See also intuitionistic logic.)
Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: "why is mathematics useful in describing nature?", "in which sense(s), if any, do mathematical entities such as numbers exist?" and "why and how are mathematical statements true?".
This is a prime concern of the philosophy of mathematics.
Three schools, intuitionism, logicism and formalism, emerged around the start of the 20th century in response to the increasingly widespread realisation that mathematics (as it stood), and analysis in particular, did not live up to the standards of certainty and rigour with which it was over-credited.
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans.
Any mathematical object is considered to be a product of a construction of a mind, and therefore, the existence of an object is equivalent to the possibility of its construction.
Intuitionism also rejects the abstraction of actual infinity; i.e., it does not consider as given objects infinite entities such as the set of all natural numbers or an arbitrary sequence of rational numbers.
Feeds |
Contact