Academic achievements of mathematical logic
Hu Shihua is one of the few representatives who developed mathematical logic in China. In China, it was Hu Shihua who initiated the study of logic beyond the scope of philosophy and linked it with mathematics. He is a pioneer in this field, and also an advocate of China's combination of logic and computer. In addition, he is also very concerned about the philosophical problems of mathematics and has done some influential work in China.
At the end of 1930s, Hu Shihua completed his doctoral thesis "Pseudo-Boolean Algebra and Topological Basis" under the guidance of professors H Scholz and K Kothe. In this thesis, he established the concept and theory of "nonholonomic point" in topological space. In the nonstandard analysis space developed in 1960s, nonholonomic point can be regarded as "nonstandard point". The full text of this article was not published, and the detailed summary was published in Academic Quarterly.
In the 1940s, Hu Shihua began to publish his research results of mathematical logic. 1943 (thirty-two years of the Republic of China) In the third issue of Shirley, Hu Shihua published On Artificial Language. In this article, he introduced the characteristics and functions of artificial language, and introduced symbolic logic into China. 1945 (thirty-four years of the Republic of China), Hu Shihua published "Reproducing a New Arithmetic System and Its Logical Terms" in the fifth issue of the first volume of Academy. In this article, he established a new recursive algorithm system RA.
From the late 1940s to the early 1950s, Hu Shihua's research field was mainly in multivalued logic. 1949 published the article "M-valued Subsystem of (M+N)-valued propositional calculus" in the third issue of Journal of Symbolic Logic, Volume 14, and 1950 published the construction of a valueless propositional calculus in the second to fourth issues of China Science, Volume 1.1950.
In this series of articles, Hu Shihua established some systems of multivalued logic, and considered the application of multivalued logic in other branches of mathematics.
In the article m- valued subsystem of (m-n)- valued propositional calculus system, he constructed a complete M-valued subsystem for any (m-n)- valued propositional calculus system.
In the article "The Construction of a Valuable Propositional Calculus", he constructed a linguistic rule of valueless propositional calculus, and recorded the established system as n, and he also gave some basic grammar theorems of n in the article.
In the article Four-valued Propositional Calculus and Four-color Problems, He Hechen pointed out the relationship between four-valued propositional calculus and four-color problems, thus reducing the four-color problems to four-valued propositional problems.
In the article "Subsystem of Non-valued Propositional Calculus with Finite Value and Functional Completeness", he gave a method to embed any complete finite-valued propositional calculus with functional completeness into a non-valued propositional calculus and become its subsystem. A. Prior's book Formal Logic quoted Hu Shihua's multi-valued logic achievements.
In the late 1950s, there was an important change in Hu Shihua's work, that is, his research on valueless logic turned to recursive function theory. It was during this period that Hu Shihua established a research group on mathematical logic in the Institute of Mathematics of China Academy of Sciences. He transferred two comrades, Tang and Lu Zhongwan, from China Renmin University, absorbed Huang Zuliang and Huang Zuliang, who graduated from the Department of Mathematics at 1955, and set up a mathematical logic group. Since then, the branch of mathematical logic has developed in China Academy of Sciences.
Research on Recursive Function and Recursive Theory
Before talking about his work in recursive function theory, let's introduce his work in classical predicate calculus. In classical predicate calculus, he focused on the study of natural reasoning. In his article Classical Predicate Calculus published in advances in mathematics in 1964, he constructed a predicate calculus system, proved relevant primitive mathematical theorems, and used them to describe formal mathematical systems.
Hu Shihua has made important contributions to the theory of recursive functions. In the second half of 1950s, he himself (sometimes in cooperation with assistants and students) made an in-depth study of recursive function theory. For example, he and Huang Zuliang used the functions sum(x, y, z), prod(x, y, z) and functional < x \ \, lt; X > deeply studies the representation function of arithmetic predicates. He and Yang Dongping studied the function of primitive recursive operators in computable functions in the article on primitive recursion published in Journal of Mathematics. The first article that attracted international attention in this respect was his article "A Recursive Primitive Recursivity" published in the Journal of Mathematics. In this article, he considered all kinds of rich functionality:
H0x=0f(x)=f(0), Hy'x=0f(x)=β(y, Hyx=0f(a(x)), and its general form can be expressed as F0x=0a(x)=a(0) and FY 'x = 0a (x) = β. This question was put forward by Professor Mo Shaokui of Nanjing University in February 1954. In the above article, Hu Shihua proved that the problem has a positive solution, that is, Fyx=0a(x) recurs to β, a, a 1, …, ak.
The research shows that although only the Institute of Mathematics of China Academy of Sciences and Nanjing University were studying recursive function theory at that time, there was a good academic exchange between them, and his relationship with Mo Shaokui had been developing normally, which created favorable conditions for the development of recursive theory in China.
Hu Shihua's works have attracted international attention. R Peter, a famous expert in recursive function theory, continued his work and extended his results.
By 1960, with the development of electronic computers, scientists in several countries in the world noticed that there should be a computable function directly defined in words to facilitate the processing of symbol strings. Almost at the same time as Hu Shihua, Mc-Carthy in the United States and Vuckvic in Yugoslavia all carried out this research work. Hu Shihua published three articles in 1969 with the general title of recursive algorithm theory. The first one is also called recursive algorithm. In this article, he introduced a computable function defined on words. The second article, called Kernel Function, was jointly published by him and Lu Zhongwan, and it is the most wonderful part of the three articles. In this article, they define a kind of kernel function class with simple structure but powerful function in a very concise way. The way this article was handled was praised by peers. The third article is called Paradigm of Recursive Function. In this article, he gives an example that recursive functions on words are represented by kernel functions.
Hu Shihua originally intended to consider the application of recursive algorithm in computer programming language, but due to the political conditions at that time, this research was interrupted, so the work of recursive algorithm was not as complete as McCarthy's, and it was not included in any application of software.
As early as 1990, Hu Shihua considered the application of word computable function in proof theory on the basis of recursive algorithm. China Science published Recursive Structure-Solvability Theory Ⅰ and Formal System of Recursive Structure Theory and Determinism-Solvability Theory Ⅱ. In the first article, he proposed an algebraic structure called recursive structure. In the second article, he established a formal system of recursive structure and gave the necessary and sufficient conditions for judging sentences. Later, Mr. Hu wrote the third article "On Solvability Ⅲ", in which he gave a criterion called condition. He proved that many unsolved problems in number theory can be determined, that is, they can be proved or refuted.
Driven by Hu Shihua, the research of recursive function and recursive theory in Institute of Software of Chinese Academy of Sciences has developed well. Later generations such as, Qi, Jiang Zhigen, Li Ansheng, etc. have all done some useful work.
Contribution to the development of computer science and technology in China
In 1950s, Hu Shihua advocated the combination of mathematical logic and computer. Today, the close relationship between mathematical logic and computer is universally acknowledged, but at that time, Hu Shihua made great efforts to overcome various difficulties to illustrate this point.
Hu Shihua has repeatedly publicized the relationship between mathematical logic and electronic computers in newspapers and periodicals. His representative article is "Basic Characteristics and Scientific Significance of Mathematical Logic" published in Philosophical Research 1957. In this article, Hu Shihua reviewed the history of general electronic computers and pointed out that it was J von Neumann who was inspired by the general Turing machine defined by A.M. and designed the first general electronic computer EDVAC. Turing himself led the design of computer ACE. Therefore, Hu Shihua expounded the close relationship between the feasibility study in mathematical logic and the development of electronic computers. Hu Shihua also participated in the China Computer Development Planning Group and went to the Soviet Union for advice with other members of the group. Hu Shihua has written a monograph on mathematical logic and several papers on the characteristics and significance of mathematical logic and mathematical foundation, such as Basic Mathematical Logic (co-authored with Lu Zhongwan), Basic Characteristics and Scientific Significance of Mathematical Logic, Occurrence, Development and Present Situation of Mathematical Logic, and Mathematical Logic and Mathematical Foundation. From 65438 to 0958, the Institute of Mathematics of Chinese Academy of Sciences started a training course on mathematical logic at the initiative of Hu Shihua. Participants not only learned the knowledge of mathematical logic in class, but also learned a lot of computer knowledge. Many assistants and students of Hu Shihua participated in the concrete work of computer logic design and programming, and many of them turned to the work of computer science theory and technology. Many people in the training class also played a role in the computer department of colleges and universities.
Tang, the first prize winner of the Natural Science Award of the Institute of Software Research, Chinese Academy of Sciences, Zhou, the second prize winner, and Tao, the third prize winner, are all Hu Shihua's assistants or students and have made important contributions to computer theory and technology. Many people in the training class in those days are also important figures in the computer field at present, such as Hao of Northwest University and others.
Hu Shihua also contributed to the philosophy of mathematics. He wrote some influential articles, including introducing the basic characteristics of mathematical logic and its relationship with other disciplines, as well as his views on the basic problems of mathematics. 1, Hu Shihua. On artificial language. Academic quarterly, 1943, 1 (3).
2. Hu Shihua. Copy the new arithmetic system and its logical quantifiers. Academy,1945,5.
Hu Shihua. A four-valued propositional calculus and four-color problems. China Science, 1950, 1 (3): 273-294.
5. The finite-valued complete function subsystem of Hu Shihua's propositional calculus. Zhongke,1955,5:173-191.
Hu Shihua. Primitive recursion of recursive formula. Journal of Mathematics,1956,6 (1): 93-104.
7. Hu Shihua. Basic characteristics and scientific significance of mathematical logic. Philosophical research,1957,6:1-44.
8. Hu Shihua. Some problems about classical deductive logic. Philosophical research,1957,6.
9. Hu Shihua, Lu Zhongwan. Examples of general recursive functions. Journal of Mathematics, 1958, 8: 507-5 19.
13, Hu Shihua. The occurrence, development and present situation of mathematical logic. Science bulletin,1960,6:1-8.
14, Hu Shihua, Huang Zuliang. Addition and multiplication. Advances in mathematics,1963,6.
Yang Dongping Hu Shihua 15. About primitive recursion. Journal of Mathematics, 1964, 14 (4): 607-6 18.
16, Hu Shihua. Classical predicate calculus. Advances in mathematics,1964,7 (4): 349-396.
17, Hu Shihua. Recursive algorithm in multiple recursive algorithms. Proceedings of the National Academic Conference on Mathematical Logic, Beijing: National Defense Industry Press, 1965, 1 1-24.
18, Hu Shihua. The development of cybernetics. Science bulletin, 1965, 10: 862-869.
19, Hu Shihua. Description of algorithmic language. 748 conference paper, Institute of Computing Technology, Chinese Academy of Sciences, 1973, 1-48.
20. Hu Shihua. How to describe a programming language? Computer dynamics,1976,9:1-29.
2 1, Hu Shihua. The influence of computer on mathematics. Encyclopedia knowledge, 198 1 (10).
22. Hu Shihua, Lu Zhongwan. Fundamentals of mathematical logic (I and II). Beijing: Science Press, 19 1.
23. Hu Shihua. Mathematics in the information age. Advances in mathematics, 1988, 17: 1 1-20.
24. Hu Shihua. Mathematical logic. Mathematics volume of encyclopedia of China, Beijing: encyclopedia of China publishing house,1988,588-593.
25. Hu Shihua. Basic mathematics. Mathematics Volume of Encyclopedia of China, Beijing: Encyclopedia of China Publishing House,1988,605-608.
Hu Shihua. Recursive structure solvable theory. Zhongke,1990 (11):1137—1145.
27. Hu Shihua. Recursive structure theory-the formal system of solvable theory and the certainty of language. Zhongke,1990 (12):1235-1242.
(The above reference source)