Brouwer's intuitionism originated from philosophy, that is, the basic intuition is the feeling that appears in chronological order. When the time process is abstracted, mathematics comes into being.
Brouwer believes that mathematics is the free creation of thinking. It uses a self-evident primitive concept-primitive intuition to construct mathematical objects. Mathematical concepts are embedded in people's minds before language, logic and experience. It is intuition, not experience and logic, that determines the correctness and acceptability of concepts. A system constructed like formal logic can only exist as a means to describe regularity, and can not be used as the basis of mathematics at all.
In his doctoral thesis, Brouwer criticized G. Cantor's set theory and other basic theories of mathematics-there is no doubt that they all rely on formal logic. He insisted that the axiomatic foundation of mathematics must be ruthlessly abandoned no matter how to repair it with Hilbert's compatibility proof. Although Hilbert finite program is retained as the premise, it cannot prove the compatibility of arithmetic. He pointed out that logic belongs to language, and the purpose of logic law is to deduce more statements. However, logic is by no means a reliable tool to reveal the truth. Truth that cannot be obtained by other methods cannot be deduced by logic. Brouwer famously said that logic depends on mathematics, not mathematics. Therefore, Brouwer naturally solved the paradox crisis: logic is not transcendental and inviolable. There is no mathematics based on axioms. So it doesn't matter if the paradox appears. He also pointed out that axiomatic method and formalistic method will certainly avoid contradictions. However, nothing of mathematical value will be obtained by this method. A wrong theory is still wrong, even if it doesn't end in contradiction. Whether Brouwer's most commendable achievement determines the effectiveness of law of excluded middle. He questioned law of excluded middle in On the Unreliable of Logical Principles. He pointed out that law of excluded middle, the cornerstone of indirect proof method, originated from the application of subset reasoning of difference sets in history, but it was later regarded as an independent transcendental principle and applied to infinite sets without foundation.
Since 1923, Brouwer has discussed law of excluded middle's role in mathematics and its reliability in a series of papers, which convinced mathematicians that law of excluded middle must be abandoned in the effective means of proof.
Brouwer reconstructed the mathematical system according to the principle of intuitionism. At first, he made no progress. The reason is that he lacks the concept of structural continuum that meets the requirements. In 2004, he finally had such a concept. This was put forward when he commented on the progress report of set theory of A. Schon Wool and H. Hahn. The following year. He inspected the structural basis of set theory and deeply understood the role of law of excluded middle. In 2008, he published a set theory based on this concept. In 2009, he put forward the constructive theory of measurement. In 2008, he gave the constructor theory. 39860 . 63868688686
Compared with axiomatic set theory, the difficulty that puzzles constructive set theory is that the concept of set cannot be the original concept, but the concept that must be explained and explained. In Brouwer's exposition, he introduced "free choice of strings" to accomplish this task. That is, make a series of choices from a bunch of objects (such as natural numbers) without restriction. All choices are determined by a rule. Moreover, after each choice, the possible choices that followed increased the restrictions. He called the law followed by choice "extension law", and the endless series of free choices allowed were called "elements" of extension law. If the expansion law only allows choices in limited possibilities, it is called "bounded expansion". As a special case, the intuitive continuum can be regarded as given by bounded extension. Brouwer pointed out that the statement that "all elements of an extension have the property p" means, "I have a construction method, which allows me to judge that the selected elements have the property p after a limited number of selections of the string α." According to this explanation and understanding of the essence of this construction method, Brouwer got his theorem, which is called the basic theorem of bounded extension-sector theorem. This theorem states that the integer-valued function f defined on bounded extension S is calculated as follows: For natural number n, if any two free choice strings α and β in S have the same first N choices, then F (α) = F (β). 1924, Brouwer proves that the functions defined everywhere on the unit closed interval are uniformly continuous. In this proof process,
The proof of the basic theorem of intuitionistic mathematics-the sector theorem has never been successfully accepted by people. However, Brouwer has made achievements quite different from the original mathematical knowledge that people are familiar with, such as the indecomposability of intuitive continuum, the uniform continuity of real functions to a certain extent and so on.
Applying the sector theorem, Brouwer fundamentally shook law of excluded middle, especially its non-contrast-┐ ┐ principle (A ┐ ┐ ┐ ┐ ┐ ┐ ┐ ┐ ┐.
After 1920, the attention of logicians was attracted by Brouwer's logic. People have studied its relationship with classical logic. Thanks to the decisive work of K. Gdel, Hilbert's basic program was broken. After World War II, intuitionism became the foundation because of Kleene's pioneering research, the rise of recursive function theory and the wide use of computers.