1923, when von Neumann was a university student in Zurich, he published a paper exceeding the ordinal number. The first sentence of the article bluntly stated that "the purpose of this article is to make Cantor's ordinal number concept concrete and precise". His definition of ordinal number has been widely adopted.
It is von Neumann's wish to explore axiomatization vigorously. From about l925 to l929, most of his articles tried to carry out this axiomatic spirit, even in theoretical physics research. At that time, his expression of set theory was particularly informal. At the beginning of his doctoral thesis on axiomatic system of set theory in 1925, he said, "The purpose of this thesis is to give an axiomatic exposition of set theory logically and irreproachable."
Interestingly, von Neumann foresaw the limitations of any form of axiomatic system in his paper, which vaguely reminded people of the incompleteness theorem proved by Godel later. Professor frankl, a famous logician and one of the founders of axiomatic set theory, once commented on this article: "I can't insist that I have understood everything, but I can safely say that this is an outstanding work, and I can see a giant through him".
1928, von Neumann published the article "Axiomatization of Set Theory", which is an axiomatic treatment of the above set theory. The system is very concise. It uses the first type object and the second type object to represent the set and the properties of the set in naive set theory. It takes a little more than one page to write the axioms of the system, which is enough to establish all the contents of naive set theory, thus establishing the whole modern mathematics.
Von Neumann's system may give the first foundation of set theory, and the finite axiom used has a logical structure as simple as elementary geometry. Starting from axioms, Von Neumann's ability to skillfully use algebraic methods to deduce many important concepts in set theory is simply amazing, which has prepared conditions for his interest in computers and "mechanization" proof in the future.
In the late 1920s, von Neumann participated in Hilbert's meta-mathematics project and published several papers to prove that some arithmetic axioms were not contradictory. 1927 The article "On Hilbert's Proof" has attracted the most attention, and its theme is to discuss how to get rid of contradictions in mathematics. The article emphasizes that the question put forward and developed by Hilbert and others is very complicated and has not been answered satisfactorily at that time. It is pointed out that Ackerman's proof of eliminating contradictions is impossible in classical analysis. Therefore, von Neumann gave a strict finiteness proof of the subsystem. This seems to be not far from the final answer Hilbert is looking for. Just then, 1930 Godel proved the incompleteness theorem. Theorem assertion: In an incongruous formal system containing elementary arithmetic (or set theory), the incongruity of the system is unprovable in the system. At this point, von Neumann can only stop this research.
Von Neumann also got a special result about set theory itself. His interest in mathematical basics and set theory continued until the end of his life. During the period of 1930 ~ 1940, von Neumann's achievements in pure mathematics were more concentrated, his creation was more mature and his reputation was higher. Later, in a question-and-answer table for the National Academy of Sciences, von Neumann chose the mathematical basis of quantum theory, operator ring theory and ergodic theorem of states as his most important mathematical work.
1927 von Neumann has been engaged in the research work in the field of quantum mechanics. He co-published a paper "Fundamentals of Quantum Mechanics" with Silvito and Nordam. This paper is based on Hilbert's lecture on the new development of quantum mechanics in the winter of 1926. Nordem helped prepare the lecture, and von Neumann devoted himself to the mathematical formalization of the subject. The purpose of this paper is to replace the exact function relation in classical mechanics with probability relation. Hilbert's metamathematics and axiomatic scheme have been put into use in this dynamic field, and the isomorphic relationship between theoretical physics and corresponding mathematical system has been obtained. We can't overestimate the historical importance and influence of this article. In his article, Von Neumann also discussed the operational outline of observable operators in physics and the properties of self-adjoint operator. There is no doubt that these contents constitute the prelude to the book Mathematical Basis of Quantum Mechanics.
1932 The world-famous springer Publishing House published his Mathematical Basis of Quantum Mechanics, which is one of von Neumann's main works. The first edition is in German, the French edition is published in 1943, the Spanish edition is published in 1949, and the English translation is published in 1955. It is still a classic in this field. Of course, he has also done a lot of important work in quantum statistics, quantum thermodynamics, gravitational field and so on.
Objectively speaking, in the history of the development of quantum mechanics, von Neumann made at least two important contributions: Dirac's mathematical treatment of quantum theory was not strict enough in a sense, and von Neumann developed Hilbert operator theory through the study of unbounded operators, which made up for this deficiency; In addition, von Neumann clearly pointed out that the statistical characteristics of quantum theory are not caused by the unknown state of the observer engaged in measurement. With the help of Hilbert space operator theory, he proved that all the assumptions of quantum theory, including the correlation of general physical quantities, must lead to this result.
For von Neumann's contribution, Wegner, the winner of the Nobel Prize in Physics, once commented: "His contribution to quantum mechanics ensures his special position in the field of contemporary physics."
In von Neumann's works, operator spectrum theory and operator ring theory in Hilbert space occupy an important position, and the articles in this field account for about one-third of his published papers. They include a very detailed analysis of the properties of linear operators and an algebraic study of operator rings in infinite dimensional space.
Operator ring theory began in the second half of 1930. Von Neumann was very familiar with the noncommutative algebra of Nott and Adin, and soon applied it to the algebra of bounded linear operators on Hilbert space, which was later called von Neumann operator algebra.
During the period of 1936 ~ 1940, von Neumann published six papers on noncommutative operator rings, which can be described as the analysis masterpieces of the 20th century, and its influence continues to this day. Von Neumann once said in "Mathematical Basis of Quantum Mechanics" that the ideas first put forward by Hilbert can provide an appropriate foundation for the quantum theory of physics without introducing new mathematical ideas into these physical theories. His research achievements in operator rings have achieved this goal. Von Neumann's interest in this subject runs through his whole career.
An amazing growth point of operator ring theory is continuous geometry named by von Neumann. The dimensions of general geometry are integers 1, 2, 3, etc. As von Neumann saw in his works, it was actually rotation group who decided the dimensional structure of a space. So the dimension can no longer be an integer. Finally, the geometry of continuous series space is proposed.
1932, von Neumann published a paper on ergodic theory, which solved the proof of ergodic theorem and expressed it with operator theory. This is the first accurate mathematical result obtained in the whole research field of ergodic hypothesis of statistical mechanics. Von Neumann's achievements may once again be attributed to his mastery of mathematical analysis methods influenced by set theory and the methods he created in the study of Hilbert operators. It is one of the most influential achievements in the field of mathematical analysis in the 20th century, and it also marks that a field of mathematical physics has begun to approach the general research of accurate modern analysis.
In addition, von Neumann has also made many achievements in mathematical fields such as real variable function theory, measure theory, topology, continuous group and lattice theory. In the famous speech 1900, Hilbert raised 23 questions for mathematical research in the 20th century, and von Neumann also contributed to solving Hilbert's fifth question. 1940 is a turning point in von Neumann's scientific career. Before that, he was a pure mathematician who was familiar with physics. From then on, he became a superb applied mathematician who firmly grasped pure mathematics. He began to pay attention to the most important tool for applying mathematics to physics at that time-partial differential equations. At the same time, he constantly innovated and applied non-classical mathematics to two new fields: game theory and electronic computer.
On the one hand, this change of von Neumann comes from his long-term love for mathematics and physics problems; On the other hand, it came from the needs of the society at that time. After the outbreak of World War II, von Neumann was called to participate in many military scientific research plans and engineering projects. From 1940 to 1957, he was the scientific consultant of the experimental ballistic research laboratory in Aberdeen, Maryland. Washington Naval Ordnance Bureau1941~1955; 1943 ~ 1955 served as the consultant of Los Alamos Laboratory; 1950 ~ 1955, member of the Army Special Weapons Design Committee; 195 1 ~ 1957. Member of the Washington Scientific Advisory Board of the US Air Force; 1953 ~ 1957, member of the atomic energy technical advisory group; 1954 ~ 1957, Chairman of Missile Advisory Committee.
Von Neumann studies continuum mechanics. He has been interested in turbulence for a long time. In l937, he paid attention to the discussion on the possibility of statistical processing of Naville-Stokes equation, and 1949 wrote the latest theory of turbulence for the Naval Research Department.
Von Neumann studied the problem of excessive waves. Most of his work in this field comes directly from the needs of national defense. His contribution to the collision shock wave interaction is remarkable. One of the results is that the Capman-Ruger hypothesis is strictly proved at first, which is related to the combustion caused by shock wave. The systematic study of shock wave reflection theory began with his Progress Report on Shock Wave Theory.
Von Neumann studies meteorology. For quite some time, he has been attracted by the extremely difficult problems raised by the hydrodynamic equations of the earth's atmospheric movement. With the appearance of electronic computer, it is possible to study and analyze this problem numerically. The first high-scale calculation by von Neumann involved a two-dimensional model, which was related to the geostrophic approximation. He believes that people can finally understand, calculate and control climate change.
Von Neumann also put forward the proposal of detonating nuclear fuel by fusion and supported the development of hydrogen bombs. 1947, the army issued a commendation order praising him as a physicist, engineer, weapon designer and patriot. Von Neumann not only played his talents in weapons research, but also in social research. 1928, von Neumann proved the basic principles of game theory, thus announcing the formal birth of game theory. The game theory he created is undoubtedly his most enviable outstanding achievement in the field of applied mathematics. The current game theory mainly refers to the study of social phenomena with specific mathematical methods. Its basic idea is to pay attention to the similarity of bargaining, negotiation, team formation and profit distribution among competitors in indoor games such as chess and cards when analyzing the interests of multiple subjects.
Some ideas of game theory existed in the early 1920s, and the real creation has to start with von Neumann 1928' s paper on social game theory. In this article, he proved the minimax theorem for dealing with a basic two-person game problem. If either party considers the maximum possible loss for each possible strategy and chooses the one with the maximum loss as the "optimal" strategy, then from the statistical point of view, he can guarantee that the scheme is optimal. The work in this area has been basically improved. In the same paper, von Neumann also clearly put forward the general countermeasures among N players.
Game theory is also used in economics. Mathematical research methods in economic theory can be roughly divided into pure theory with qualitative research as the goal and econometrics with empirical and statistical research as the goal. The former is called mathematical economics, which was formally established after 1940s. No matter in thought or method, it is obviously influenced by game theory.
In the past, mathematical economics, which imitated classical mathematical physics skills, mainly used calculus and differential equations, and regarded economic problems as classical mechanical problems. Obviously, the trade fair attended by dozens of businessmen is handled by classical mathematical analysis, and its complexity far exceeds that of planets moving in the solar system. The effect of this method is often unpredictable. Von Neumann resolutely gave up this simple mechanical analogy and replaced it with a novel game theory and a new mathematics-convex thought.
From 65438 to 0944, Game Theory and Economic Behavior, co-authored by von Neumann and Morgenstein, is a basic work in this field. The two-person game is extended to the n-person game structure, and the game theory system is applied to the economic field, thus laying the foundation and theoretical system of this discipline. The paper includes the explanation of pure mathematical form of game theory and the detailed explanation of practical application. This paper and the discussion of some basic problems of economic theory have triggered various studies on economic behavior and some sociological problems. Today, this is a widely used and increasingly rich mathematical discipline. Some scientists enthusiastically praised it as "one of the greatest scientific contributions in the first half of the 20th century". The last subject that contributed to von Neumann's popularity was electronic computer and automation theory.
As early as Los Alamos, von Neumann clearly saw that even if the study of some theoretical physics is only to get qualitative results, it is not enough to rely solely on analytical research, but also to be supplemented by numerical calculation. The time required for manual calculation or using a desktop computer is unbearable, so von Neumann began to make great efforts to study electronic computers and calculation methods.
From 1944 to 1945, von Neumann formed the basic method of transforming a set of mathematical processes into computer instruction language. At that time, electronic computers (such as ENIAC) lacked flexibility and versatility. Von Neumann's idea of fixed and universal circuit system in machines, the concepts of "flow diagram" and "code" have made great contributions to overcoming the above shortcomings. Although this arrangement is obvious to mathematical logicians.
The development of computer engineering should also be largely attributed to von Neumann. The logic schema, storage, speed, the choice of basic instructions and the design of interaction between circuits in modern computers are deeply influenced by von Neumann's thought. He not only participated in the development of electronic tube component ENIAC computer, but also personally supervised the construction of computer in Princeton Institute of Advanced Studies. Not long ago, Von Neumann and Moore's team worked together to write a brand-new general-purpose electronic computer program EDVAC with a stored program, and the report of 10L page caused a sensation in the mathematics field. According to this report, the Princeton Institute for Advanced Studies, which has always been good at theoretical research, approved von Neumann to make computers.
The electronic computer, which is 10 million times faster than manual calculation, not only greatly promotes the progress of numerical analysis, but also stimulates the emergence of new methods in the basic aspects of mathematical analysis itself. Among them, the vigorous development of Monte Carlo method for dealing with deterministic mathematical problems with random numbers formulated by von Neumann and others is a prominent example.
/kloc-the precise mathematical expression of mathematical physics principles in the 0/9th century seems to be very lacking in modern physics. The complex structure in the study of elementary particles is dazzling, and the hope of finding a comprehensive mathematical theory soon is still very slim. On the whole, not to mention the analytical difficulties encountered in dealing with some partial differential equations, there is little hope of obtaining accurate solutions. All these forces people to look for new mathematical models that can be processed by electronic computers. Von Neumann contributed many ingenious methods to this: most of them were included in various experimental reports. From solving numerical approximate solutions of partial differential equations to reporting long-term weather values, and finally controlling climate.
In the last few years of von Neumann's life, his thoughts were still very active. He integrated the results of logic research and his early work on computers, and extended his vision to the general automata theory. With his unique courage, he overcame the most complicated problem: how to design a reliable automaton with unreliable components and build an automaton that he can replicate. From this, he realized some similarities between computer and human brain mechanism, which was reflected in Hillemann's speech; It was not until after his death that someone published a pamphlet under the name of Computer and Human Brain. Although this is an unfinished work, some quantitative results obtained by his accurate analysis and comparison of human brain and computer system still have important academic value.