It can be seen that the school of mathematics and the school of philosophy before the Middle Ages almost coincided. You can learn a lot by studying the history of western philosophy. Mathematics itself originates from natural philosophy. When mathematical science is gradually separated from philosophy, the mathematical foundation still has a strong philosophical atmosphere. Every school has a long story, and every mathematician has many exciting works and legends. We can learn many stories about mathematicians by reading M. Klein's four-volume "Ancient and Modern Mathematical Thoughts" and E. T. Bell's "Mathematical Elite".
Until modern times, we can have a general understanding of mathematics since the 20th century by referring to books such as Biography of Contemporary Mathematicians-Winners of Fields Prize and Contemporary Mathematicians: wolf prize in mathematics Winners and Their Achievements and Views.
Moscow School and Gottingen School are my two favorite schools. These two places once gathered a large number of famous mathematicians, with a long tradition of mathematical history and profound mathematical culture.
About the Gottingen School:
Gottingen School has long been the leading school in the development of mathematical science in the world. This school insists on the unity of mathematics, and its thought reflects the essence of mathematics and promotes its development.
Gauss initiated the initial era of Gottingen's school of mathematics, and he raised modern mathematics to a new level. Riemann, Dirichlet and jacoby inherited the work of Gauss and made contributions in the fields of algebra, geometry, number theory and analysis. Klein and Hilbert brought the German school of mathematics into its heyday, and the University of G? ttingen became an international mathematical research and education center.
Gottingen School is the cradle and holy land of mathematicians in the world, but Hitler's coming to power gave it a fatal blow. A large number of Jewish scientists were forced to flee to the United States, and the Gottingen School of Mathematics disintegrated. 1
About Moscow School:
In the past century, hundreds of world-class mathematicians have emerged in Soviet Russia, including Jin Lu, Alek Sandrov, Kolmogorov, Gelfand, Shafarevich, Aluluo, Novikov, Lyapunov, Fichkin Goelz, Kovalev Kaya and so on. Are all famous mathematicians. Most of these outstanding mathematicians graduated from Moscow University. Except for the University of G? ttingen at the end of 19 and the beginning of the 20th century, the number and quality of outstanding mathematicians emerged in Moscow University are very high. In the 20th century, no university dared to compare with it. Even the famous Princeton University has never trained so many outstanding mathematicians, and Moscow University is well-deserved as the world's top mathematics school.
In Moscow school, I like Arnold best. All his books are easy to understand, and he has written profound mathematical theories in simple mathematical language, and quoted many examples from life to connect mathematical theories. He is a mathematician with a deep understanding of mathematics. I like reading his works, such as Ordinary Differential Equations, Dynamical Systems and Mathematical Methods of Classical Mechanics.
It's a pity that there has never been such a famous math school in China, let alone a school. The development of mathematics in China needs the investment and struggle of more young people.
The following three schools, there were many first-class mathematicians, logicians and philosophers in the world at that time. They have made great contributions to the improvement of mathematics foundation, and we pay high tribute to them here.
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1 "Note" Just list some mathematicians and physicists who have taken refuge in the United States from Germany (including Austria and Hungary), and you can see the transfer of talents. Einstein (1879 ~ 1955, a great physicist); J. franck J. (1882 ~1964.1925 won the nobel prize in physics); Von Neumann (1903 ~ 1957, one of the outstanding mathematicians); Kurante (1888 ~ 1972, head of the Institute of Mathematics in Gottingen); Godel (1906 ~ 1976, mathematical logician); Nott (1882 ~ 1935, one of the founders of abstract algebra); Feller (W. Feller, 1906 ~ 1970, one of the founders of stochastic process theory); Atin (1896 ~ 1962, one of the founders of abstract algebra); Friedrich (K. Friedrichs, 190 1 ~ 1983, applied mathematician); Wyle (1885 ~ 1955, one of the outstanding mathematicians); Dean (1878 ~ 1952, Hilbert's third problem solver); In addition, there are Paulia, Seg, Hellinger, ewald, Nordheim, Debye, Wigner and so on.
A brief introduction to the three schools of mathematics in the middle and early 20th century.
/kloc-After Cantor established the set theory at the end of 0/9, the concept of set became the most basic and widely used concept. It was once thought that all basic mathematical theories could be unified by the concept of set. 1900, at the international congress of mathematicians held in Paris, Poincare once confidently said, "Now we can say that we have reached complete rigor." However, less than three years after this statement was put forward, the British mathematician Russell put forward a set paradox in a letter to the German mathematician Frege in 1902, which shook the mathematical foundation. In Frege's words, "Suddenly one of its cornerstones collapsed."
Russell's paradox of set:
Sets can be divided into two categories: the first category is characterized by: a set itself is an element in a set, such as the "set composed of all sets" that people often said at that time; The characteristic of the second set is that the set itself is not an element of the set, such as a set of points on a straight line. Obviously, a set must be one and only one of these two kinds of sets. So, what kind of set is R? A popular statement of Russell's paradox is Barber's paradox:
There is a barber in a certain city. His advertisement reads: "My haircut skills are superb and the whole city is famous. I will shave all the people in this city who don't shave themselves. I will only shave these people. I would like to extend a warm welcome to everyone! " When people come to him to shave, they naturally don't shave themselves. One day, however, the barber saw in the mirror that his beard had grown. He instinctively grabbed the razor. Do you think he can shave himself? If he doesn't shave himself, then he belongs to the "person who doesn't shave himself" and he has to shave himself. What if he shaved himself? He belongs to the "person who shaves himself" and should not shave himself.
The fundamental problem of why there is contradiction in set theory involves the credibility of mathematical logic reasoning and the truth of mathematical propositions, which belongs to the category of mathematical philosophy.
During the 30 years from 1900 to 1930, many mathematicians participated in a discussion on the basis of mathematical philosophy, and gradually formed different schools of mathematics, mainly logicism, formalism and intuitionism.
First, logicism.
1. The historical origin of logicism
The formation of logicism can be traced back to Leibniz's time. He imagined logic as a universal science, including some principles that form the basis of all other sciences. This logic precedes all scientific viewpoints, that is, the sprout of logicism. But he failed to finish the work. /kloc-in the 9th century, Dai Dejin, Frege, piano and others inherited Lai Xianmin's works, developed gradually and made great achievements.
2. The basic idea of logicism
The main representative of logicism is Russell, a famous British mathematician, philosopher and logician. In 19 13, he and Whitehead completed the classic representative work of logicism-mathematical principles. The author tries to explain to people in these three volumes of mathematical masterpieces that all mathematics can be strictly deduced from a logical axiom system, that is to say, mathematical concepts can be obtained from clearly defined logical concepts; Mathematical theorems are derived from logical propositions through pure logical deduction. Therefore, all mathematics can be derived from basic logical concepts and rules. In this way, mathematics can be regarded as an extension or branch of logic. So Russell said, "Logic is the youth of mathematics, and mathematics is the prime of logic." "Mathematics is logic."
In Introduction to Mathematical Philosophy, Russell further elaborated his proposition: "To achieve more and more abstraction and logical simplicity through analysis, it is necessary to study whether we can find more general ideological principles. Based on these ideas and principles, what is the starting point now can be defined and deduced. " So what is the principle of thought? Russell went on to say, "We should start from a certain logical premise that has been generally recognized, and then deduce it to achieve those obvious results that belong to mathematics." That is, turning mathematics into logic, which is his basic point of view.
In Principles of Mathematics, Russell and Whitehead strictly deduced the mathematics at that time through pure logic, axiom of choice and infinite axioms of set theory, and achieved success. Therefore, Russell declared: "The work of pure mathematics based on logic has been done in detail by Whitehead and me in Principles of Mathematics." However, this is not the case. When Russell deduced mathematics from a logical system, he used the infinite axioms of axiom of choice and set theory, and both of them were indispensable, otherwise it could not be completed. Without infinite axioms, natural number systems cannot be constructed, not to mention all mathematics. So Russell did not classify mathematics as logic, but as set theory.
To deduce all mathematics from logic, we must develop set theory, which is contradictory and incompatible. But logical systems are not allowed to have contradictions, so paradoxes must be eliminated. But the work done by Russell and Whitehead later did not solve this problem well, and then encountered many difficulties.
Mathematicians generally do not accept the view that "mathematics is logic"; It is also unacceptable that "all mathematical thinking is logical thinking". But, in spite of this. The book Principles of Mathematics, co-authored by Russell and Whitehead, had a great influence on the development of science and technology in the 20th century. It stated the logical system, definition and theorem established by the author in the strictest formal symbolic language at that time, thus marking the success of symbolic logic method. This also shows the significance of the research on the logical basis of mathematics, which further shows the scientific significance of modern logic.
The book Principles of Mathematics became a masterpiece. Although the idea of logicism cannot be realized and the mathematical view of logicism cannot be widely accepted by basic mathematicians, the methodological significance of this book cannot be ignored. They successfully brought classical mathematics into a unified axiom system, so that Cantor's set theory, general arithmetic and most mathematics can be deduced from several logical concepts and axioms and the infinite axioms of set theory. This has developed logical reasoning to an unprecedented height, making people see that many mathematical contents can be derived on the basis of mathematical logic calculus, forming the logical system of axiomatic system of set theory. This is a great event in the history of logic, which plays a decisive role in the later development of mathematical logic and is an important starting point of modern axiomatic methods.
Second, formalism.
It is generally believed that the founder of formalism is Hilbert, and Hilbert's mathematical view and foundation are called "formalism". Both Russell and Brouwer call Hilbert a representative figure of formalism, but they refer to Hilbert's formal method of laying a mathematical foundation, not necessarily to some of his thoughts. Hilbert himself does not pretend to be a formalist, nor does his student Bernards think Hilbert is a formalist.
1. The formation of formalism
The theoretical system of formalism was formed in the atmosphere of "rebuilding the mathematical foundation" in the study of mathematics and mathematical philosophy after the appearance of non-Euclidean geometry.
When non-Euclidean geometry is recognized by people, that is, when the two kinds of geometry that arrive at the contradiction theorem prove that they are not contradictory, people will ask: Where is the truth of mathematics? Imagine, according to a geometric theory, if you cross a point outside a straight line, you can only make a straight line that does not intersect with the original straight line; Another geometry says that crossing a point outside a straight line can make at least two straight lines not intersect with the original straight line; Another geometry says: you can't make any straight line at a point outside the straight line that doesn't intersect with the original straight line. Didn't these three geometries fight each other? At least two of them are considered to be wrong. Why are all three geometries true?
Hilbert, a famous German mathematician, advocates defending classical mathematics and classical mathematical methods and developing them. He believes that classical mathematics, including the new direction of mathematics developed due to the emergence of set theory, is the most precious spiritual wealth of mankind; In order to avoid the paradox in mathematics, we try to prove the non-contradiction of mathematics absolutely, so that mathematics can be based on strict axioms. Mathematical axioms and logical reasoning are as important as telescopes in the hands of astronomers and cannot be discarded. In order to achieve this goal, Hilbert put forward the famous Hilbert Plan in 1922.
2. The basic idea of formalism
The main idea of the Hilbert plan is: to lay a foundation of mathematics, it is necessary to prove the coordination of this mathematics strictly and mathematically (that is, it is neither contradictory nor consistent); The mathematical content of Hilbert plan is the proof theory in mathematical logic.
The two volumes of Basic Mathematics co-authored by Hilbert and bernays are the representative works of Hilbert's project.
Hilbert plans to formalize all mathematics into a formal system, and then prove the compatibility of all formal systems by an elementary method, that is, there is no contradiction, thus deducing the non-contradiction of all mathematics.
He distinguished three mathematical theories: 1. Intuitive informal mathematical theory; 2. Formalize the first mathematical theory to form a formal system, transforming the basic concepts in intuitive mathematical theory into initial symbols in the formal system, transforming propositions into symbolic formulas, transforming deductive rules into deformation relations among symbolic formulas, and transforming proofs into finite sequences of symbolic formulas; 3. It describes and studies the second mathematical theory, which is called meta-mathematics, proof theory or meta-theory. Meta-mathematics is a new mathematics with formal system as the research object, including the description, definition and properties of formal system.
Formalism is the most important turning point in the history of mathematics development, which marks the establishment of meta-mathematics. Since then, the development of mathematics has entered a new stage of studying formal system.
Here we should make it clear that formalism, like logicism, starts from the axiomatic system. The difference is that logicians no longer hold the original viewpoint when catching up with the axiomatic system, but require the axiomatic system to have content and try their best to find out where the truth of logical laws is embodied. Formalists don't. They think that the basic concept of the axiom system of mathematics or logic is meaningless, and its axiom is only a line of symbols. It doesn't matter whether they are true or not. As long as it can be proved that the axiomatic system is compatible and not contradictory, it will be recognized and represent some truth. Even the logical axiom system thinks that it has no content, and its truth cannot be guaranteed by content, so only "compatibility" or "no contradiction" is left as truth.
Hilbert originally conceived that the proof of mathematical compatibility could be limited to limited construction methods. However, research shows that this range should be expanded. Godel's incompleteness theorem says, "In any compatible mathematical formalization theory, as long as it is strong enough to define the concept of natural numbers, a proposition that can neither be proved nor proved can be constructed in the system." Any compatible formal system cannot be used to prove its compatibility. This theorem completely shattered Hilbert's formalistic ideal. However, Hilbert's basic mathematical thought has developed meta-mathematics, promoted formal psychology and promoted the development of mathematics. Metamathematics (proof theory) has developed into one of the four branches of mathematical logic.
The representatives of formalism are American mathematicians Robinson and Cohen. They believe that mathematics should be regarded as a pure symbolic game on paper, and the only requirement for this form is that it will not lead to contradictions.
However, this formalism is obviously different from Hilbert's proposition.
Third, intuitionism
1. The historical roots of intuitionism
Intuitionism can be traced back to Aristotle, who was the first philosopher in history who opposed the infinity of reality and only admitted the infinity of potential. The philosophy of intuitionism is that natural numbers directly derived from Kant and Brouwer are all derived from "primitive intuition", that is, Kant said that "natural numbers are derived from time intuition".
/kloc-Kroneck in the 10th and 9th centuries emphasized feasibility, saying that many theorems at that time were only symbolic games and had no practical significance. He believes: "God created natural numbers, and everything else is man-made. Integers are intuitive and clear, so they are acceptable, and others are suspicious. " It means that only natural numbers are real, and the rest are just some artificial symbols. He also advocated building the whole mathematics on the basis of natural numbers.
At the beginning of the 20th century, Poincare also thought that natural numbers were the most basic intuitive and potentially infinite concept. Other semi-intuitionists or French empiricists, such as Borriel, Leberg and Rukin, also emphasize the concept of feasibility.
They openly denied axiom of choice, thinking that the axiom of choice-based assembly was simply not feasible and could not recognize its existence. They put forward the concept of feasibility, and without feasibility, it will not be recognized. They are all pioneers of intuitionism. All these provided a direct premise for Brouwer's intuitionism. Brouwer collected the achievements of his pioneers and systematically provided intuitionism.
2. Intuition in mathematics.
The founder and representative of intuitionism is the Dutch mathematician Brouwer. From 1907, Dr. Brouwer's thesis "Mathematical Basis", intuitionists gradually and systematically expounded their own mathematical views and the idea of rebuilding mathematical basis.
His view of mathematics includes the following aspects:
(1) His views on mathematical objects.
He put forward a famous slogan: "Being is being constructed." He believes that people's understanding of mathematics does not depend on logic and language experience, but on "primitive intuition" (that is, an ability that everyone has). Pure mathematics is "the mathematical structure of the mind itself" and "the structure of reflexivity", and it is "starting from natural numbers" rather than set theory. This mathematical structure has nothing to do with the nature of this structure, whether it is independent of human knowledge or not, and the philosophical views people hold. No matter what the structure should be, mathematical judgment should be an eternal truth.
Therefore, Brouwer does not admit that there is an objective, closed and complete truly infinite system.
Real infinity theorists believe that "the whole of natural numbers" refers to natural number set {0, 1, 2, 3, ...}, which is a complete set that does exist and can and should be the object of mathematical research.
Potential infinity denies real infinity, thinking that infinity is only potential, not a closed entity that has been completed, but infinite in terms of its development. In their view, nature is 0, 1, 2, 3, ... only in the process of continuous construction and generation, not a complete and closed entity.
So the concept of "all natural numbers" is meaningless.
(2) Views on mathematical logic.
Brouwer's view of mathematical object directly led to his view of mathematical logic. It is believed that "logic is not an absolutely reliable tool to discover truth", and law of excluded middle can't be used in real mathematical proof, because law of excluded middle and other classical logic laws are abstracted from finite sets, so they can't be used in infinite sets indefinitely. Similarly, reduction to absurdity cannot be used.
Intuitionism has a great influence on the development of mathematics in the 20th century. After 1930s, many mathematicians began to attach importance to intuitionism because of Godel's work. Mathematicians try to establish real number theory, mathematical analysis and even all mathematics by construction method, and have achieved many important results.
Structural mathematics has become an important mathematical subject group in mathematical science and is closely related to computer science. 1967, the American mathematician Bishop completed and published the book Structural Analysis, which opened the structuralist period of the intuitionistic school.
History has proved that the three schools have their own advantages and disadvantages, but they all make up for many shortcomings on the basis of mathematics and provide more accurate symbols and languages for the rigor of mathematics. Let's end this article with a sentence by G.H. Hardy: "Beauty is the first test: there is no permanent place for ugly mathematics in the world."