Intuitionism is an ideological and theoretical trend that emphasizes intuition or its role in cognition. It is believed that intuition is a more basic and reliable way to understand the world than abstract reason. This theory or trend of thought usually has strong anti-rationalism, anti-positivism and anti-materialism. Many philosophers in history attached importance to intuition, but it was not until the early 20th century that intuition really became a theory or trend of thought. Typical intuitionists are H. Bergson, B. Croce, E. Husserl and others.

Mathematical intuitionism

In mathematical philosophy and logic, intuitionism, or neo-intuitionism (corresponding to pre-intuitionism), is a mathematical research method with human constructive thinking activities. It can also be translated into intuitionism.

Any mathematical object is regarded as the product of thinking structure, so the existence of an object is equivalent to the possibility of its construction. This is different from the classical method, because according to the classical method, the existence of an entity can be proved by denying its non-existence. For intuitionists, this is incorrect: denying non-existence does not mean that it is possible to find structural proof of existence. Because of this, intuitionism is a mathematical structuralism; But this is not the only one.

Intuitionism equates the correctness of mathematical proposition with that it can be proved; If the mathematical object is a pure mental structure, what other laws can be used to test the authenticity (as intuitionists will argue)? This means that intuitionists may have different understanding of the meaning of a mathematical proposition from classical mathematicians. For example, to say A or B is to declare that A or B can be "proved" instead of saying that either of them is "true". It is worth mentioning that law of excluded middle, which only allows A or non-A, is not allowed in intuitive logic; Because it cannot be assumed that people can always prove proposition A or its negative proposition.

Intuitionism also refuses to admit the abstract concept of infinite reality; In other words, it does not regard infinity as an entity, such as the set of all natural numbers or the sequence of arbitrary rational numbers. This requires that the foundation of set theory and calculus be reconstructed into constructivism set theory and constructivism analysis respectively.

In the philosophy of mathematics, structuralism (constructivism) holds that in order to prove the existence of a mathematical object, it is necessary to construct it. Assuming that an object does not exist, a contradiction is deduced from this assumption, which is not enough to prove the existence of the object for structuralists. See structural proof.

Structuralism is often confused with intuitionism. In fact, intuitionism is only a kind of structuralism. Intuitionism emphasizes that the foundation of mathematics is based on the personal intuition of mathematicians, so mathematics is regarded as a subjective activity in essence. Structuralism does not emphasize this point, which is consistent with the objective view of mathematics.

Intuitionism is a school of mathematics, and one of its representatives is the Dutch mathematician Brouwer (I _ E.J

), whose fundamental view is about the constructibility of mathematical concepts and methods, holds that the theoretical basis of mathematics is not set theory, but natural number theory. A famous slogan of intuitionism is "Being must be constructible". From the basic viewpoint of intuitionism, it is directly decided that one of the main purposes of this school in mathematics work is to completely adopt potential infinity and reject real infinity on infinite problems.

Brouwer (Brouwer，I _。 J.j.

), Heyt-ting (a.), and others are all famous intuitive mathematicians.

Intuitionism has not become the mainstream of mathematics. Intuitionism only recognizes natural numbers as the basis of mathematics, but does not recognize the concept of set. Brouwer skillfully constructed the concept of continuum satisfying the structural requirements by using the concept of "expansion", thus laying the theoretical foundation for intuitive calculus.