| Russian Mathematical Surveys |
|
|
|
|
|
Mathematical Life Mathematical culture of the society: its value and development V. V. Kozlov, I. A. Taimanov
Bibliographic databases:
    The mathematical culture of a society is an important prerequisite for its evolution. First of all, it is necessary for the development of sciences and technology. Promoting this culture is traditionally based on popular literature, which serves three purposes:
1.As regards the first aim, there are two prominent texts that we must mention. $\bullet$ V. A. Steklov’s book Mathematics and its value for the mankind [1], which was published (in Russian) by the State Publishing House of the Russian Federation in Berlin in 1923; $\bullet$ A. N. Kolmogorov’s article “Mathematics” [2] in volume 26 of the second edition of the Great Soviet Encyclopaedia. Steklov’s book is exceptional in many respects. At the end of it, on p. 137, the author put the completion data “Petersburg, 27 July 1920” (given that from 1914 to 1924, including in the period when the book was at work, the name of the city was changed to Petrograd). The book was published by the state publishing house abroad, in Berlin, in 1923, when the young republic encountered great economic problems and was short not just of foreign exchange reserves, but of mere hard currency. And even at that time it was decided that a — certainly costly — edition of a book on the importance of mathematics was expedient! Steklov devoted most of his book to the formation of mathematical ideas. He also discussed the influence that mathematics had had on the development of rational philosophy, mentioning in this connection Bacon, Spinoza, and Descartes; individual chapters were devoted to Leibniz, Hume, Kant, while no mention of Marx and Engels can be found in the text, in spite of the growing trend of that time. We would like to point out Steklov’s comments on the importance of physical observations for the development of mathematics ([1], p. 135): “… with the extension of the range of natural phenomena observed and the improvement of methods of observation, approximations provided by the Euclidean geometry can turn out to be insufficient, and then we perhaps will have to improve slightly Euclid’s geometric model or turn to Lobachevskii’s system. Perhaps the most precise physical measurements of recent years indicate the possibility of such an outcome.” Moreover, Steklov also mentioned the development “observed in physical science and prompted by the statement of the relativity principle, the creation of the theory of quanta, new theories of Bohr, Sommerfeld, and other authors…, which only became possible because of the utmost precision of observations attained during the last 20 or 30 years” ([1], p. 123). Steklov’s phrase that improvements in the precision of observations (due to the progress in equipment for experiments, that is, in engineering) lead to the discovery of physical phenomena or the refinements of their explanations, which in turn encourages the development of mathematics, were written in 1920, when relativity theory was not yet generally accepted and quantum theory was at an early stage of its development. These parts of theoretical physics made the most significant impact on the development of mathematics in the 20th century. Kolmogorov’s article, written originally for the first edition of the Great Soviet Encyclopedia in 1938, was published in a revised form in the second edition of 1954 and in the third edition of 1974. This article is different from Steklov’s book in its character: it was written for an encyclopedic compendium and contains a large historical review, making accent on the development of mathematics in Russia and the USSR. A separate subsection is devoted to the role of set theory and mathematical logic in the substantiation of mathematics. The article discusses the subject of mathematics with encyclopedic concision, starting with the phrase “Mathematics (from Greek $\mu\alpha\phi\epsilon\mu\alpha\tau\iota\kappa\alpha$, derived from $\mu\acute{\alpha}\phi\epsilon\mu\alpha$ – knowledge, science) – is a science of quantity relations and space forms of the actual world”, after which the author cites appropriately F. Engels’s Anti-Dühring: “Pure mathematics deals with the space forms and quantity relations of the real world — that is, with material which is very real indeed. The fact that this material appears in an extremely abstract form can only superficially conceal its origin from the external world. But in order to make it possible to investigate these forms and relations in their pure state, it is necessary to separate them entirely from their content, to put the content aside as irrelevant”. Kolmogorov’s article and, in particular, its first part “The definition of the subject of mathematics, its connections with other sciences and technology”, is even now worth of deep reading. On the other hand, in what relates to applications to other sciences and, first of all, the role of calculating machines (the term almost forgotten nowadays), it reflects its time, both concerning the current state of mathematics and the forecasts of its development. 2.The tradition of publishing popular books and journals, aiming first of all to attract young people to science, goes back to decades before the revolution. The journal Vestink Opytnoi Fiziki i Elementarnoi Matematiki (Bulletin of Experimental Physics and Elementary Mathematics) was published from 1887 to 1917, with 24 issues every year. In total, 674 issues were published. Note that The American Mathematical Monthly, the best known popular mathematical journal of our time, began publication only in 1894. From 1904 to 1925 the publishing house Mathesis in Odessa was active in book publishing. The digitized versions of these publications (Vestnik and Mathesis books) are now available at https://en.etudes.ru/. Because of financial and organizational difficulties of the years after the revolution, these publications were stopped and took other forms. For example, V. F. Kagan, who managed both projects during their final years, was in the 1930s the head of the mathematical division of the first edition of the Great Soviet Encyclopedia. From 1934 to 1938, 13 issues of collections of papers Matematicheskoe Prosveshchenie (Mathematical Education) were published; their publication resumed in 1957, when six annual issues were published, and then it resumed again in 1997 and still goes on. The two famous book series Library for Extra-Curriculum Mathematics and Popular Lectures in Mathematics started in 1950. Both the belated start and then the closure of these series in the 1990s were linked, of course, with the economic hardships of the postwar period and the end of the century. Since 1970 the journal Kvant has been published. In recent years popular books for high-school and university students were published on a large scale, first of all, through the effort of the Moscow Center for Continuous Mathematical Education https://mccme.ru/en/. We must separately mention A. Ya. Khinchin’s book Three pearls of number theory [3], which is outstanding in many respects. It was written on a request from a soldier, who had completed one year of university education and, wounded in a battle, wrote a letter from the hospital to his professor, with a request to send him ‘a few mathematical pearls’ to the frontline. Khinchin’s book begins with a “Letter to the front (in lieu of a preface)” and contains an elementary presentation of the proofs of three famous theorems in number theory. Writing the book took Khinchin some time, and the letter to the front was dated 24 March 1945. Published immediately after the war, Khinchin’s book remains a masterpiece of popular mathematics and an example of a civic stand of a researcher. In the mid-1930s mathematical olympiads were organized in the Soviet Union; they started as annual competitions of high-school students, in Leningrad in 1934 and in Moscow in 1935. With time, this trend extended to the whole country, and since 1967 national mathematical olympiads have been organized. The importance and fruitfulness of these events were also understood in other countries: in 1950 the first Mathematical Contest was organized in the USA, and International Mathematical Olympiads are being held since 1959. 3.In 1947 the book What is mathematics? by R. Courant and H. Robbins was translated into Russian. The best characterization of this book is due to Kolmogorov. In fact, the results of Soviet mathematicians were not quite adequately appraised in that book and — as mentioned in the foreword to the third Russian edition ([4], p. 11) — “special arguments were used to save the printed copies of the book from destruction”. Pages with editorial foreword were glued in each of the 15 000 copies. This foreword, written by Kolmogorov, contained, in particular, the following appraisal of the book: “There is a significant gap between school mathematics and the areas of contemporary mathematics which are currently most active and important for natural sciences and technology. The most prominent component of this gap is the lack of elements of calculus in school mathematics, which are indispensable for understanding the central ideas of physics and many areas of engineering… . The Russian translation of the book by R. Courant and H. Robbins can fill these gaps to a certain extent…” . The book by Courant and Robbins contained, in particular, a popular explanation of the main ideas of non-Euclidean geometry and topology, but question relating to mathematical analysis were central to it. In the early 1960s Ya. B. Zeldovich’s book Higher mathematics for beginners and its applications to physics [5] was published, which addressed specifically physicists and engineers. In it, apart from the basics of analysis and differential equations, explained on the physical level of rigour, the reader could find examples of their use in applied problems. Zeldovich’s book was criticised by many mathematicians for lack of mathematical rigour, because mathematics students mainly learn the culture of rigourous proofs and arguments from courses of mathematical analysis. L. S. Pontryagin, who shared this critical attitude, published four pamphlets in the 1970–1980s under the common title Learning higher mathematics [6]. Zeldovich’s and Pontryagin’s brilliant books complement each other and can be advised to a reader wishing to get insight in the basics and applications of mathematical analysis. Note that also these days, the development of mathematical modelling in natural sciences is seriously inhibited by the poor education in mathematical analysis and differential equations which many researchers not specializing in mathematics actually receive. Although mathematical statistics is taught, for example, to medical students, in the current circumstances and, first of all, given the development of neural networks, at least a slight knowledge of some of its more profound chapters is necessary. 4.In the late 1940s a number of conferences and sessions of specialized academies were organized to discuss the methodological aspects of the development of science from the standpoint of Marxism, the dominating doctrine of that time. Similar events in mathematics were not so noticeable, and the hundreds of pages of notices of methodological seminars and academic councils have never been discussed in public. The first of such events was a conference on the methodology of mathematics held in Leningrad in 1948. The agenda of its first working day, 31 May 1948, included two items: 1. Report of A. D. Aleksandrov “What is mathematics”; 2. Discussion. On November 22, 1948, A. D. Aleksandrov gave a lecture “On formalism in mathematics” on a general meeting of the staff of the Steklov Mathematical Institute of the Academy of Sciences. As a result of these events, an idea arose to wrote a book explaining to the wider public what mathematics actually is and how important it is. It must be mentioned that mathematicians themselves took charge of this development and did not let in Marxist philosophers, whose activities often had adverse effects on science at that time. In 1953 the first version of the book Mathematics: its content, methods, and value [7] was published as a manuscript, ‘for discussion’. It consisted of 16 chapters:1 Introduction (A. D. Aleksandrov); Analysis (M. A. Lavrentiev and S. M. Nikol’skii); Analytic geometry (B. N. Delone); Algebra (the theory of algebraic equations) (B. N. Delone); Ordinary differential equations (I. G. Petrovskii); Partial differential equations (S. L. Sobolev); Variational calculus (V. I. Krylov); Functions of a complex variable (M. V. Keldysh); Approximation of functions (S. M. Nikol’skii); Prime numbers (K. K. Mardzhanishvili); Curves and surfaces (A. D. Aleksandrov); Functions of a real variable (S. B. Stechkin); Linear algebra (D. K. Faddeev); Abstract spaces (A. D. Aleksandrov); Functional analysis (I. M. Gelfand); Groups and other algebraic systems (A. I. Malcev). We have listed the chapters in the order of numbering but dropped their ordinal numbers for brevity. So the reader cannot see from this list that two of the planned chapters, namely, X (“Computing”) and XIII (“Probability theory”), were left out as “they were not presented in time by their authors”. In the resulting three-volume set [8] the numbering of chapters was different, but the two missing chapters were added (where, in addition, the chapter on computing was split into two), as well as one new chapter. Thus, four chapters in total were added to the original 16:2 Approximate methods and computing (V. I. Krylov); Electronic calculating machines (S. A. Lebedev); Probability theory (A. N. Kolmogorov); Topology (P. S. Alexandroff). It becomes clear from some documents published only in the 21st century why the article on probability theory was considered ‘unprepared’. These documents include a positive review of Kolmogorov’s paper by the corresponding member of the Academy A. Ya. Khinchin, a letter from Aleksandrov to Kolmogorov, containing the words “as I am in charge of the final stage of the preparation of the monograph Mathematics, its content, methods, and value, … I would ask you to restructure your paper …”, and Kolmogorov’s response and can be found in [9], pp. 433–435. Even before one reads the text itself, it is clear from the contents and the list of authors that this was a unique publication. While we presented a three-part classification of popular books at the beginning of this paper, we cannot definitely ascribe this monograph to one of them. Schoolchildren, students, engineers, and even experts in mathematics can read articles in it, and each of them can find there something new and interesting. Of course, texts on computing are outdated, but the mathematical foundations of the design of calculating machines, which were developed in 1940–1950s, are well described there and thoroughly presented. 5.Over the last 70 years great changes have taken place in mathematics, and some results obtained have determined its development for many years to come. Computerization, digitalization, and numerical modelling of natural processes are among the main factors of the development of modern civilization. Some methods attributed previously to ‘pure’ science or lacking spectacular applications are now widely used. We believe that it is time to continue the mission of the monograph [8]. The reader can regard our paper as an invitation, for the first step, to a discussion of the new fields of mathematics that could supplement the contents of [8], which we described above, and of the material that could be added to the existing chapters from the standpoint of our current knowledge. This would also help the authors of popular mathematics literature to extend the list of their topics and areas. There can also be other forms of presentation, for instance, lectures uploaded to the Internet. Nonetheless, all of this must be founded on new texts, new articles: in the beginning must be the Word. 
Citation in format AMSBIB
\Bibitem{KozTai25} |
|
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|