In the philosophy of mathematics, intuitionism can be said to have caused a revolution in modern academic thought. The relationship between mathematics and philosophy is a problem that people talk about. The following are the relevant materials of my thesis on mathematics and philosophy. Welcome to read!

Abstract: In the philosophy of mathematics, intuitionism caused a revolution in modern academic thought. Although intuitionism can be traced back to Kant and even Plato. However, it is modern, and it was famous as an independent trend of mathematical philosophy in the first 20 years of the 20th century. It is a storm in logic philosophy and a powerful challenge to classical mathematics. Intuitionism emphasizes "structure" and starts from "mind". Intuitionism regards the whole theory of natural numbers as the basis of the whole mathematics. Intuitionism rejects law of excluded middle and reduction to absurdity, and advocates the infinite reality and the infinite potential. With the appearance and development of computers, intuitionism has played an active role in digital construction. At the same time, intuitionism has a noticeable influence on the innovative education of mathematical philosophy.

Keywords: Brouwer, traditional logic of intuitionism in mathematical philosophy

First, "existence must be constructed"-the emergence of intuitionism

The word intuition is an intuitive thinking that directly grasps the essence of things without sufficient logical reasoning. Different from the intuitionism of H. Bergson, B. Croce and E. Husserl, the "intuition" we study here refers not to the direct grasp of objective things by the subject, but to an instinctive psychological activity of thinking. Here, intuition advocated by intuitionism is not the "intuitive feeling" of dialectical materialism, but its original intention is "transcendental psychological structure", which forms the requirement of equality between "existence" and "constructability" of mathematical objects. Intuitionistic philosophy is an idealistic philosophy against rationalism. Constructivism in mathematical research is a viewpoint based on mathematics, which advocates that natural numbers and some regular methods, especially mathematical induction, are reliable starting points, and all other mathematical objects and theories should be constructed from natural numbers. [2] "Being must be constructed" is the most famous slogan of intuitionists. Therefore, intuitionism is a structural logic. Intuitionists believe that concepts and methods in mathematics must be constructive, and unstructured proofs are unacceptable to intuitionists. In the field of mathematics, the * * * paradox problem can not be solved by making some regional modifications and restrictions on existing mathematics, but must rely on some credible standards to comprehensively examine and transform existing mathematics. Intuitionism holds that logic depends on mathematics, not mathematics depends on logic. Mathematics is based on intuition. At the same time, intuitionism believes that concepts such as philosophy, logic and even counting are much more complicated than mathematics and cannot be used as the basis of mathematics. The foundation of mathematics needs a simpler and more direct concept, that is intuition, which is a basic function of the mind. [3] An intuitionist mathematician, Ahrend Heiting, pointed out in his paper "The Intuitionistic Basis of Mathematics": "It may be in line with the positive attitude of intuitionists to deal with the structure of mathematics immediately. The most important cornerstone of this construction is the concept of unit, which is the construction principle on which integer sequences depend. Integers must be treated as units, and the difference between them is only their position in this sequence. " [4]6 1

Intuitionists believe that the foundation of mathematics lies in mathematical intuition. In their view, the theory based on mathematical intuition can make "concepts and reasoning very clearly presented to us", that is, "it is so direct to ideas and the results are so clear that it does not need any casting foundation". Ding Hei: An Introduction to 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 the classical method says that the existence of an entity can be proved by denying its non-existence. For intuitionists, this is not correct; 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. The basic philosophical position of intuitionism is that mathematics is an internal creative activity of human mind, an activity of the subject itself, rather than an external description. Mathematical concepts are the result of independent intellectual activities, and intellectual activities are the study of ideological structures governed by self-evident laws. [5]

Second, subvert the traditional logic, counterattack * * *-the characteristics of intuitionism

Intuitionism does not recognize the infinity of reality and rejects the abstraction of the infinity of reality. In other words, it does not regard all infinite entities such as natural numbers or infinite sequences of arbitrary rational numbers as given objects. The idea of real infinity in mathematics is to regard the infinite whole itself as a ready-made unit and a constructed thing, in other words, to regard the infinite object as a self-realization process or an infinite whole. There is a dispute between potential infinity and real infinity in mathematics, just as there is a dispute between materialism and idealism in philosophy. There will be a long and continuous debate. The idea of potential infinity in mathematics refers to the interpretation of infinity as something that is always extending, changing, growing and constantly producing. An example of an image point is that there are infinitely many points forming a straight line, which will go on forever without end. It is always under construction and will never be completed. It is potential, not reality. According to the standard intuitionism of full names and conditional quantifiers, one proof is such a potential infinite structure, which may be reasonable. Philosophical Basis of Damit's Intuitionistic Logic [4] 142 According to this view, all natural numbers can form a * * *, because all natural numbers can be regarded as a complete infinite whole. Obviously, intuitionism supports the view of potential infinity, that is, treating infinity as an infinitely extended sequence.

Intuitionism opposes law of excluded middle, which means that intuitionists may have a different understanding of the meaning of mathematical propositions from classical mathematicians. Law of excluded middle and the law of identity and contradiction are also called the three basic laws of formal logic. Traditional logic first regards law of excluded middle as the law of things, which means that everything has or does not have certain attributes at the same time, and there is no other possibility. Law of excluded middle is also a law of thinking, that is, whether a proposition is established or not, there is no other possibility. For example, to say A or B is to claim that A or B can prove it. However, for law of excluded middle, A or A is not allowed, because it cannot be assumed that people can always prove proposition A or its negative proposition.

Intuitionism is mainly against * * *. For centuries, the belief in the impeccable accuracy of mathematical laws has been the main object of mathematical philosophy research. Intuitionism says that accuracy exists in people's minds, while those who believe that accuracy exists on paper. [4]90

Intuitionism is illogical and holistic. Mathematical intuition is a cognitive method defined as the opposite of logic, so illogicality is the most important feature of mathematical intuition. It can be said that other characteristics of mathematical intuition are determined by its illogicality, which is the same view of many philosophers and scientists. [6] Intuitionism believes that mathematics is a creative activity of the mind, with rich brains and poor logic. Therefore, we must not use poor logical rules to comprehensively and accurately plan rich spiritual activities. Arendt, another representative of intuitionism? Hetin Allen and Hetin said, "Logic belongs to applied mathematics". Regarding the integrity of intuitionism, a Japanese mathematician has the following incisive explanation: when a person has been engaged in research for a long time and has become a fully mature research? A href ='' target =' _ blank'> Cough pen? He has formed a relatively stable knowledge system in his mind. Through his own efforts, this knowledge system has been integrated into a special and definite form. Of course, my comprehensive work itself is a very valuable experience. [7]

Poincare wrote in the article Intuition and Logic in Mathematics:

Philosophers tell us that pure logic can never make us get anything; It can't create anything new, and no science can be created out of thin air. In a sense, these philosophers are right; To form arithmetic, just like geometry or any science, you need something else besides pure logic. To call this thing, we have to use the word intuition. However, after this same Oracle Bone Inscriptions, how many different meanings are hidden? Compare these four axioms: 1 equals the third and the last two quantities are equal; 2 If the logarithm 1 is true, it is assumed to be true for n, and if we prove that it is true for N+ 1, it is true for all integers; 3 is in a straight line, point C is between A and B, and point D is between A and C, so point D will be between A and B; Only one straight line passing through the fixed point is parallel to the known straight line. These four axioms all belong to intuition, but the first one clarifies a law of formal logic; The second is the true transcendental comprehensive judgment, which is the basis of strict mathematical induction; The third resort to imagination: the fourth is false definition. Intuition does not have to be based on feeling and understanding; I soon felt weak. [8]

It is worth noting that intuitionism is not mysticism. The "non-interpretability" of intuition does not mean the "mystery" of intuition. Although intuition is "unexplained", it has a definite essence. We believe that intuition is a leap in the process of cognition, so it is not empirical knowledge, but an interruption to the original ideological line. Intuition is "unexplained" because it cannot be associated with conclusion and reasoning in the usual way of thinking. [9]

Third, from Kant to Mitter, the main figures of intuitionism and their thoughts.

Immanuel Kant, Immanuel Kant, 1724- 1804 In a sense, intuitionism was initiated by the philosopher Kant. From 1755 to 1770, Kant taught physics and mathematics at the University of Konigsberg. He thinks that all our feelings come from a preset external world. Although these feelings can't provide any knowledge, the interaction between perceived objects produces knowledge. The mind sorts out these feelings and gets an intuition about space and time. Kant said that perceptual intuition has two pure forms, which are the principles of innate knowledge, and these two pure forms are space and time. Space is a pure form of external intuition, while time is a pure form of internal intuition. They do not come from the experience of foreign neighbors, but are inevitable and innate ideas. Space and time do not exist objectively, but are the creation of the mind. The mind understands experience, and experience awakens the mind. Although Kant's thought has the shadow of intuitionism, it still does not put forward intuitionism intuitively. Intuitionism is modern as far as the basic methods of mathematics are concerned. [ 10]

Henry Poincare is often translated as Poincare, Henry Poincare, 1854- 19 12, a pioneer of mathematical intuitionism in the contemporary context. The evaluation of later generations is a bridge between mathematical philosophy and contemporary mathematical intuitionism. Logicalism's understanding of mathematical basis is illusory. It makes mathematics lose its foundation. However, the foundation of mathematics exists and is our intuition. It gives mathematical meaning, thus giving mathematical objects. Poincare pointed out the bridge between human spirit and mathematical existence, which is our mathematical intuition. [1 1] poincare believes that natural numbers are the most basic intuition, and mathematical induction is a thinking method that includes intuition and cannot be simply reduced to logic. He advocated the concept defined by limited words and the constructability of mathematical objects. He also understood and emphasized mathematical intuition in another sense, and regarded it as a tool for selection and invention. Poincare thinks we have a lot of intuition. But the most important things can be summarized into two categories: one is "pure intuition", that is, what he usually calls "pure number intuition", "pure logical form intuition" and "mathematical order intuition", which is mainly the intuition of analysts; The second is "perceptual intuition", that is, imagination, which is mainly the "shape" intuition of geometricians. For these two kinds of intuition, he thinks both are necessary and each plays a different role. He believes that these two kinds of intuition "seem to play two different instincts in our hearts", just like "two searchlights to guide strangers to and from two worlds". [ 12]

Brouwer L.E.J Brouwer, 188 1- 1966, the real founder of intuitionism is Brouwer. Brouwer's intuitive position in mathematics comes from his philosophy. 1907, in his doctoral thesis "Fundamentals of Mathematics", he put forward the intuitive view that the foundation of mathematics is transcendental initial intuition. Mathematics is a human activity that originated and produced in the mind, and does not exist outside the mind, so it is independent of the real world. Brouwer believes that mathematical thinking is a process of intellectual construction, which constructs its own world, independent of experience and only limited by basic mathematical intuition. [10] Basic Mathematics published by Brouwer shows that intuitionism emphasizes intuition, which does not mean denying the logic and rigor of mathematics, but only highlighting the position of intuition, inspiration and creativity in mathematics. Intuitionists believe that mathematics is not only the most rigorous science, but also the most creative science. Brouwer thinks that the foundation of mathematics is the transcendental initial intuition. He and his students said that their so-called intuition is a clear understanding of the human heart structure itself. [13] Brouwer revised Kant's transcendental view of time and space and abandoned the transcendental view of "external intuitive pure form" to adapt to the development of non-Euclidean geometry; On the basis of the transcendental time concept of "pure form of internal intuition", he established the basic intuition of mathematics. [14] Brouwer also put forward the duality of the "two-one principle". He thinks this is the basic intuition of mathematics. That is, if n holds, then N+ 1 holds. This process can be repeated indefinitely, creating all finite ordinal numbers, because one of the elements of the "21 principle" can be regarded as a new "21 principle". Brouwer believes that in this basic intuition of mathematics, connectivity and separation, continuity and discreteness are unified, which directly leads to the intuition of linear continuum, that is, the intuition of "between". Brouwer's Intuitionism and * * * [4] 93

Arend Heyting, 1898- 1980, Brouwer's student. Inherited Brouwer's thought about mathematical intuitionism. He believes that intuitionism is based on some arbitrary assumptions. Its theme is constructive mathematical thought. This makes it break away from classical mathematics. The difference between * * * and intuitionism is that intuitionism is independent of formalization, and it can only follow the mathematical structure. Logic is not the foothold of intuitionism, the mathematical structure is very direct in the mind, and the conclusion should be clear without any foundation. Hai Ting argues that when describing intuitionistic mathematics, it should be understood in daily life. For example, when I looked at the tree over there, I was sure I saw the tree, but in fact the light waves reached my eyes, so I spent quite a lot of training to construct the tree. This view is natural. When two people talk, I instill ideas into you, which actually creates the vibration of the air. This is the structure of theory. Arendt Heiting's Debate [4]77-88

Michael Damit also translated Michael Damit, 1925-20 1 1, a representative of the contemporary school of mathematical intuitionism. Damit thinks that mathematics is transcendental first, and then analytical. Damit once defended intuitionism from the perspectives of linguistics and meaning theory. Intuitionism's explanation of the meaning of mathematical statements avoids the deficiency of the meaning theory with the concept of truth as the core concept. It combines the speaker's understanding of mathematical statements with the speaker's ability to actually use mathematical statements, so it has great advantages. Starting from intuitionism's explanation of the meaning of mathematical statements, Damit put forward the idea of new meaning theory with confirmation as the core concept. [15]202 Damit pointed out: "For intuitive logic, law of excluded middle's double negation is an effective semantic principle, just like binary logic thinks that law of excluded middle itself is effective: it is inconsistent to assert that any statement is neither true nor false." [4] 132

Fourthly, the significance and rationality of intuitionism.

Intuitionism holds a negative attitude towards some principles and unstructured conclusions in law of excluded middle's and double negative law principles in classical logic, and does not recognize the real infinite objects and methods in mathematics. The history of mathematics also shows that mathematical knowledge and theory are inseparable from the eternal dependence on the outside world, and mathematical errors are not overcome by limiting mathematics, such as excluding unstructured mathematics and traditional logic. The accumulation of mathematical truth and the abandonment of fallacies are obtained through the continuous growth of mathematical knowledge and the continuous improvement of theory. In a word, the life of mathematics lies in the endless creative process. Fortunately, intuitionism has become the dominant idea in mathematics from some innate weakness in its ideological system. But we should also see the important value of his structural thought. [16]123-124 It can be said that the intuitionistic school is subjective and absurd in essence, and the absolute affirmation of intuitionistic mathematics based on intuitive constructibility can't really solve the problem. Intuitionism reveals that classical logic has only relative truth, which is of great significance in concrete mathematical work.

First of all, the intuitionism school in mathematical philosophy highly recognizes the role of intuition and individual creative thinking in scientific practice, which promotes the establishment and development of modern recursive function theory, which undoubtedly plays a very positive role in the progress of mathematics. Secondly, the structured and feasible research methods advocated by intuitionists promote the development of artificial intelligence and computer science. This method of actively exploring feasibility has important practical significance in computer mathematics and computer science. Thirdly, the thinking method of intuitive mathematical philosophy has valuable reference value in the theoretical research and practice of quality education. In mathematics education, the function of logic is obvious, and its characteristic is to deduce and calculate from known knowledge according to logical rules, and gradually reach the understanding of the research object. Intuition can be recognized by leaps and bounds. Although it can get the correct answer in one step, it cannot explain the steps clearly. Although intuitionism rejects traditional logic, it is closely related to logic and plays an important role in cultivating good concepts of mathematical logic. In addition, intuitionism is helpful to cultivate the thinking habit of bold guessing in mathematics education. This spirit of innovation and exploration is conducive to the progress and development of mathematics.

References:

Fu Min. Intuitionistic mathematical philosophy and its enlightenment to mathematical quality education [J]. Journal of Northwest Normal University, Natural Science Edition, 1996 1.

[2] Zhuge Yintong. A Powerful Challenge to Traditional LogicComment on Classical Logic and Intuitionistic Logic [J]. Philosophical trends, 19904.

[3] Ke Huaqing. Two stages of intuitive mathematical philosophy [J]. Academic Research, 20052.

[4] Paul Benacerraf beauty, Hillary? Pute South America. Philosophy of mathematics [M]. Beijing Commercial Press, 2003.

Qin huangan. Mathematical philosophy and mathematical culture [M]. Xi 'an: Shaanxi Normal University Press, 1999.

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