Research background
In the beginning, Marx discussed elementary mathematics in correspondence with Engels and others. For example, in a letter he wrote in 1864 about numerical calculations: “It can be seen that calculations that are not too large, such as in household expenses and business, never use numbers but only use stones and other similar marks. On the abacus, set several parallel lines on this kind of abacus. The same few stones or other prominent marks indicate a few in the first row, a few dozen in the second row, and a few hundred in the third row. The fourth line means a few thousand, and so on. This kind of abacus was used almost throughout the Middle Ages, and is still used by the Chinese today. As for larger mathematical calculations, the ancient Romans had multiplication before there was such a need. A watch or a Pythagorean watch, of course, this kind of watch is still very inconvenient and cumbersome. Because part of this watch is made up of special symbols, and part of it is made up of Greek letters (later Roman letters)....... The old method caused insurmountable obstacles when doing large calculations. This can be seen in the tricks of the outstanding mathematician Archimedes."[2](P650)
Marx's notes on mathematics are closely related to his materials on political economy. In an economics notebook in 1846, the last few pages are all algebraic operations; in many later notebooks, there are also mathematical formulas and graphics, as well as whole pages of arithmetic; in writing "Politics" He drew some geometric figures in the notebook of the preparation materials of the "Critique of Economics" and recorded the formulas of the fractional index and logarithm.
On January 11, 1858, Marx said in a letter to Engels: “In formulating the principles of political economy, calculation errors greatly hindered me. Quickly review it. I have always been very poor in arithmetic, but indirectly using algebraic methods, I will soon calculate the correct one." [4] (P247) Marx once believed that he could apply certain formulas of advanced mathematics to economics. I am very pleased with the research. When Marx wrote to Engels on the study of wages on January 8, 1868, he said: "Wages are described for the first time as an unreasonable manifestation of a relationship hidden behind it. This is through wages. Two forms, hourly wages and piece rate wages, have been precisely explained (such formulas can often be found in advanced mathematics, which is very helpful to me)."[5](P12)
It seems , Marx’s interest in mathematics is related to his hope to apply mathematics to economic research. In a letter to Engels on May 31, 1873, talking about the study of economic crises, he said: "In order to analyze the crisis, I wanted to calculate these as irregular curves more than once, and I wanted to use mathematical formulas. From this, we can draw the main law of the crisis (and now I still think it is possible if there are enough tested materials)." [6] (P87) In "Das Kapital" we can also see the use of mathematics According to Lafarge’s recollection, Marx once emphasized that a science can only be truly developed when it can successfully use mathematics. [7] (P8) I understand that the use of mathematics mentioned by Marx here is not only the use of mathematical calculation methods, but also the use of mathematical thinking methods and demonstration methods.
Learning and thinking
After the 1860s, Marx successively read a large number of books on calculus, including J.L.B.ucharlat and Sind (J. Hind), Lakua (S.F. Lacr.ix), Hall (G. Hall) and others respectively compiled calculus textbooks, as well as Newton-related mathematics original works, etc., wrote detailed reading notes. Marx compared these textbooks and started his own independent thinking on some problems in differential calculus. Around 1881, Marx wrote research drafts on the historical development of differential calculus, on the concept of introductory functions, on differential, and on Taylor's theorem. He has written drafts on these issues many times, for example, on Taylor's theorem. Eight drafts were left.
Marx regards differential calculus as a new discovery and new thing in science, and examines how it was produced, what difficulties it encountered after it was produced, and what tortuous development it experienced. Marx had a vivid and philosophical description of calculus: "People themselves believe in the mystery of the newly discovered algorithm. This algorithm is correct through mathematical methods that must be incorrect (especially in geometric applications). The result is amazing. People have mystified themselves in this way and valued this new discovery even higher, making a group of old orthodox mathematicians even more irritated, and aroused hostile clamor, even in the mathematics world. It is inevitable to open up a path for new things."[8](P88)
Marx took Newton (1642-1727) and Leibniz (1646-1716) From the creation of differential calculus to the development of Lagrange (J.L. Lagrange 1736-1813), the development process of about 100 years is divided into three stages, namely: "mysterious differential calculus" and "rational differential calculus" ", "Pure Algebraic Differential Calculus". In the period of Newton and Leibniz, the freshman calculus quickly achieved amazing success in application, but from the perspective of old traditional mathematics, this new algorithm, such as the differential process, is through incorrect mathematical methods. Get the correct result. In the process of deriving the same formula, Δx and dx are both finite quantities, but disappear to zero, showing a logical contradiction; why can there be a definite value, etc., can not give a reasonable theoretical value explain. People think that differential calculus is mysterious. Newton and Leibniz, as well as their successors, hoped to find a logical explanation for differential calculus, and they made great efforts to this end. The "rational differential calculus" represented by D'Alembert (J.L.R.D'Alembert, 1717-1783) and the "pure algebraic differential calculus" represented by Lagrange are both such efforts. The results of a certain stage. Marx pointed out: "Here, as elsewhere, it is important to tear off the veil of mystery to science." [8] (P139)
Marx tried to use dialectics to analyze the difficulties of differential calculus. He believes that "understanding all the difficulties in differential operations" is "just like understanding the negation itself of negation." It is necessary to understand "negation" as a link of development, and to treat quantitative changes from the unity of quantity and quality. In the process of differentiation, in the negation of quantity, such as the disappearance of quantity, it can be seen that there is still a specific qualitative relationship, that is, the qualitative relationship restricted by the functional relationship of y to x. Therefore, when the increment Δx becomes zero and Δy also becomes zero, it can have a specific value, that is, the derivative function. Marx said that to grasp the true meaning, "the only difficulty is the dialectical view of determining a ratio between the gradually disappearing quantities." [9] (P16)
Take the differential process of a polynomial function as an example. I compared a variety of textbooks with reference to the above point of view and selected a specific derivation step to illustrate the rationality of the differential process of this function, thus showing that the mystery of differential calculus can be got rid of . Such content, although it seems to be very simple now, is not enough to explain the differentiation process of general functions. But this is also a historic effort by Marx to tear off the mystery of differential calculus.
Marx once persuaded Engels to study calculus. He said in a letter to Engels on July 6, 1863: "I will study calculus when I have time. By the way, I have many books on this subject. If you are willing to study, I am going to send you one. I think this is almost indispensable for your military research. Moreover, this department of mathematics (only in terms of technology), for example, is much easier than advanced algebra. Except for ordinary algebra and trigonometry, it is not What knowledge is required, but a general understanding of conic section is necessary."[2](P357)
Marx’s interest in and research in advanced mathematics influenced and inspired Engels. After 1865, they In the communication, it is more about calculus. In an attachment to a letter to Engels, Marx said: "All differential calculus is to find the tangent to any point on any curve. I just want to use this example to illustrate the essence of the problem." Marx uses a parabola. The example of the tangent line at a certain point m on y[2]=ax is carefully drawn and explained to Engels in detail. [3] (P168-169)
In 1881, Marx copied a manuscript of "On the Concept of Derivatives" and a manuscript of "On Differentials" clearly and sent them to Engels. Engels carefully read these manuscripts and wrote a very long reply to Marx on August 18, 1881 discussing derivative functions. The letter said: "This matter aroused my great interest, so that I not only thought about it. All day, and dreaming about it: Last night I dreamed that I gave my collar buckle to a young man for differentiation, and he slipped away with the collar buckle." [10] (P21-23)< /p>
Under the influence of Marx, Engels became more and more interested in calculus. In his philosophical works such as Anti-Duhring and Dialectics of Nature, he not only talked about micro Integral, an incisive analysis of the difference between advanced mathematics and elementary mathematics, and there is also a high praise for calculus: "In all theoretical achievements, there may not be anything like the invention of calculus in the second half of the seventeenth century. That is regarded as the highest victory of the human spirit. If somewhere we see the pure and unique merits of the human spirit, it is here."[11](P611)
Dialectics h2>
Both Marx and Engels clearly believe that mathematics is an important foundation for the establishment of dialectical materialist philosophy. Engels pointed out: "To establish a dialectical and materialistic view of nature at the same time, knowledge of mathematics and natural science is required." [12] (Preface to the third edition)
In the old philosophy, Haig Er is more on mathematics. Engels once pointed out: "Hegel's knowledge of mathematics is extremely rich, and even none of his students has the ability to sort out and publish a large number of mathematics manuscripts left over by him. As far as I know, he knows enough about mathematics and philosophy to do this job. The only person in the world is Marx." [3] (P471) Marx was busy with his own research and revolutionary activities, and did not undertake this work. However, he linked the development of differential calculus with the development of German idealist philosophy in his mathematical manuscripts, and made an interesting comparison. When he discussed the relationship between Newton and Leibniz and their successors, he said: "Just like this, Fichte inherits Kant, Schelling inherits Fichte, Hegel inherits Schelling, regardless of Fichte , Schelling, and Hegel have never studied Kant’s general basis, that is, idealism itself; otherwise they would not be able to further develop Kant’s idealism."[8] (P88)
Marx studied mathematics As a source of rich materialist dialectics. Through his years of research in mathematics, he has a deep understanding that he has found the most logical and simplest dialectical movement in advanced mathematics. This description can be seen in Marx's mathematical manuscripts.
Overall evaluation
Marx once planned to write some of his research results on mathematics into formal papers, but he repeatedly rewritten the draft many times, but he did not have time to finish it. Before his death, he instructed his youngest daughter Elena: "I want her to handle all his manuscripts together with Engels, and care about publishing those things that should be published, especially the second volume (press: refers to the second volume of "Capital") and some Mathematics works."[13] (P42) After the death of Marx, Engels also hoped to publish his research results in dialectics of nature together with the mathematical manuscripts left by Marx. [11] (Preface to the third edition) But because he was tasked with organizing and publishing Marx's most important works-Volumes 2 and 3 of Capital, the above wish was not realized.
Several drafts of Marx’s essays on differential calculus and some readings were translated into Russian in 1933 and met with readers, that is, it was first published in Soviet theoretical journals in commemoration of the fiftieth anniversary of Marx’s death. "Under the Banner of Marxism", subsequently included in the essay "Marxism and Natural Science". In 1968, a relatively complete German-Russian version of Marx's mathematics manuscript was published in the former Soviet Union [14], in which a more detailed description of the manuscripts of various periods was written. In addition, Marx’s mathematical manuscripts have been published successively in German, Japanese, Italian, etc., with different contents and layouts. It has aroused the attention and interest of scholars in the international academic circle. For example, Japan’s Tamaki Mihiko and Imano Takeo have long written articles describing the content of Marx’s mathematical manuscripts. At the International Conference on the History of Mathematics held in West Germany in 1977, American scholar H.C. Kennedy gave an academic report entitled "Marx and the Basics of Calculus". The famous American historian of mathematics D.J. Struik wrote an article in the "Mathematics Review" magazine in 1978 and introduced this report. In the past few years, there were also graduate students in the history of science in the United States studying the spread and influence of Marx's mathematical manuscripts.
In China, Xu Mofu has published articles on Marx’s mathematical manuscripts as early as 1949 (Note: Xu Mofu’s articles on Marx’s mathematical manuscripts were published in "Northeast Daily" (1949). May 5), "Natural Science" (Volume 1, 1951), "Bulletin of Mathematics" (1958, No. 12), "New Science" (1955, No. 2) and other newspapers.), some scholars later Translate part of the content from Japanese or Russian text. In January 1973, Peking University established the Marx Mathematics Manuscript Compilation Team, which translated it based on the German-Russian version published by the Soviet Union in 1968. In order to translate accurately and directly from the original German text into Chinese, Peking University purchased all the photocopies of the original mathematics manuscripts from the Netherlands through diplomatic channels in 1974, and translated most of the calculus and some of the notes on elementary mathematics. It was written into Chinese and arranged into a book, which was officially published by People's Publishing House in 1975. (Note: In January 1973, Comrade Wang Huide, then head of the Compilation Bureau of Marx, Engels and Lenin, handed over a copy of "Marx's Mathematics Manuscript" (German-Russian version in 1968, given to him by a Swiss journalist) Sun Xiaoli suggested that Peking University should organize the translation. Peking University readily accepted this suggestion and immediately established the Peking University Marx Mathematics Manuscript Compilation Team. Deng Donggao and Sun Xiaoli were specifically responsible for mobilizing teachers from the Department of Mathematics, Department of Western Languages, Department of Russian, and Department of Philosophy. In the translation work, there are Jiang Zehan, Yao Baocong, Leng Shengming, Ding Tongren and others in German, and Wu Wenda, Huang Dun, Guo Zhongheng, Bao Liangjun, Yan Pin and others in Russian. In March 1974, he translated Marx's calculus After proofreading most of the essays by Yu Guangyuan, Hu Shihua, Lu Ruqian, and Yang Yanjun of the Translation Bureau, a special issue: Marx's Mathematics Manuscript (Trial Version) was printed by the Journal of Peking University in May 1974. Marx was purchased in the winter of 1974. After the photos of the original mathematics manuscripts, two professors, Jiang Zehan and Yao Baocong, who are well versed in German, carefully identified the handwriting of Marx's manuscript. Supplement. Finally, Professor Zhang Herui and Professor Jiang Shuomin of Beijing Normal University were asked to make detailed revisions of all the translated manuscripts from German before the People's Publishing House published Marx's Mathematics Manuscript in July 1975.)
Extreme views
After Marx’s "Manuscripts of Mathematics" was compiled and published in China in 1975, two extreme views appeared:
One is too much in mathematics Uplifting Marx, saying that Marx laid the theoretical foundation for calculus, and regarded the important achievements of many outstanding mathematicians in the 19th century as metaphysics. Only Marx’s expositions are dialectical, and Marx’s Mathematics Manuscript should be used in teaching. "Replace the calculus textbook. This approach is obviously extremely wrong. It runs counter to Marx's original intention and does not conform to the actual development of mathematics. It can only have a harmful effect on higher mathematics teaching.
Another extreme view is that Marx doesn't understand mathematics at all, at least not advanced mathematics. The "Mathematics Manuscript" written in the 19th century has no academic value and is not worthy of translation and publication. This completely negative attitude also lacks historical analysis and is not in line with reality.
Since these two views have continued to the present to varying degrees, I feel that it is appropriate to put Marx’s "Manuscripts of Mathematics" under the historical conditions at the time, and to seek truth from facts based on its specific content. The evaluation is necessary and meaningful.
Valuable documents
By reading Marx’s mathematics manuscripts, as well as Marx’s works and correspondence on mathematics, we can connect with the translation, introduction and publication of Marx’s mathematics manuscripts in our country for decades And the influence, I wrote this article to talk about my understanding and views on Marx's mathematical manuscripts. I will teach it to friends who are interested in it, and it will also serve as a commemoration of the 120th anniversary of the death of Marx.
Reading Marx’s mathematics manuscripts, I feel that Marx has delved into mathematics. Indeed, as Engels said: "Marx is proficient in mathematics." [12] Of course, the so-called "proficient" cannot Marx is required to be acquainted with all of mathematics at that time, just as it is impossible for an expert who can be called "proficient" in mathematics to know all the contents of current mathematics. In fact, as Engels said: "For natural science, we can only do sporadic, intermittent, fragmentary research", and "natural science itself is also in such a huge process of change, even those People who have all their free time to work on it are also difficult to track without losing" [12]. Before his death, Marx had not had time to track important achievements in mathematical analysis in the 19th century, and he had not read some important works like Cosi's "Analysis Course" (the first edition of 1821) that had been published at that time. Because Marx did not yet understand that calculus passed through Bolzano (B. B. lzan, 1781-1848), Cosi (ALCauchy, 1789-1857), Weierstrass (KWT Weierstrass, 1815-1897), etc. After the mathematician's efforts, he has obtained the gradually "perfect" form, so it is impossible for him to use the limit theory to clarify the essence of calculus as people understand later.
Marx is not a full-time mathematician, nor has he made major contributions to mathematics itself. The reason why his mathematics manuscripts are valued is first of all because he is a great thinker in human history, and he is a great thinker in mathematics. Decades of persistent hard work in this field have been done day by day. This deed is rare in the history of human culture and is incomparable to any thinker in history. The mathematics manuscript we read now is the historical footprint of his hard work in his own unique way. This footprint can be preserved and is known to the world. It is precious, and it is worth studying and recollecting, and getting useful from it. Enlighten.
Secondly, in Marx's mathematics manuscripts, there are indeed ideas and insights that are still shining today. For example, after investigating the specific historical development process of differential calculus, Marx once made this thesis: "The real thing between the new and the old is thus the simplest connection, always after the new thing itself acquires a perfect form. It was discovered.” [8] (P144) This is a philosophical generalization of the relationship between the new and the old, and it is also a philosophical generalization of the laws of human knowledge, which is very inspiring to the progress of people’s knowledge.
Thirdly, in Marxist theories, great emphasis is placed on people, especially their all-round development. Marx has a profound discussion on the importance of free time or leisure time, that is, non-working time. He regards free time as wealth and leisure as an important part of human life. So, how does Marx spend his free time? According to Marx’s son-in-law Lafarge, “In addition to reading poetry and novels, Marx also has a unique way of spiritual recuperation, which is his favorite mathematics. Algebra even gave him With spiritual comfort; in some of the most painful periods of his turbulent life, he always used this to masturbate."[7](P8)
Marx once said to Engels: I—of course I can’t always write—I just do differential calculus. I don’t have the patience to read other things. Any other reading always drives me back to the desk.” [3] (P124) Marx’s knowledge of mathematics Special hobbies enable him to immerse himself in mathematics under any circumstances. When Marx’s wife, Jenny, was seriously ill with liver cancer, he wrote to Engels, “It’s almost impossible for me to write an article now. The only thing I can use to keep the mind calm is necessary, It's mathematics." [2] (P113) His draft of his research on differential calculus was written during those painful days when Yanni was critically ill in 1881.
In Marx's mathematics manuscripts, we can see a lot of humorous language and vivid and interesting metaphors. It is conceivable that mathematics used to be Marx’s leisure kingdom seeking joy and comfort. Marx spent many days happily here. Thousands of pages of mathematics manuscripts are the truth of Marx’s unique spiritual recuperation method. record.
To sum up, I believe that Marx's mathematical manuscript is a valuable historical document with special value.
References
[1] Selected Works of Marx and Engels: Volume 3 [M]. Beijing: People's Publishing House, 1971.
[2] Marx and Engels Complete Works: Volume 30[M]. Beijing: People's Publishing House, 1975.
[3] The Complete Works of Marx and Engels: Volume 31[M]. Beijing: People's Publishing House, 1972.
[4] The Complete Works of Marx and Engels: Volume 29[M]. Beijing: People’s Publishing House, 1972.
[5] The Complete Works of Marx and Engels: Volume 32[M]. Beijing: People’s Publishing Society, 1971.
[6] The Complete Works of Marx and Engels: Volume 33 [M]. Beijing: People's Publishing House, 1973.
[7][法] Lafarge. Recalling Marx[M]. Beijing: People’s Publishing House, 1954.
[8] Marx. Mathematics Manuscripts[M]. Beijing: People’s Publishing House, 1975.
[9] Marx's Mathematics Manuscript[J]. Special Issue of Journal of Peking University, 1974.
[10] The Complete Works of Marx and Engels: Volume 35[M]. Beijing: People's Publishing House, 1971.
[11] The Complete Works of Marx and Engels: Volume 20 [M]. Beijing: People's Publishing House, 1971.
[12] Engels. Anti-Duhring Theory [M]. Beijing: People's Publishing House, 1971.
[13] The Complete Works of Marx and Engels: Volume 36 [M]. Beijing: People's Publishing House, 1975.
[14]К. МАРКС МАТЕМАТИЧЕСИЕ РУКОПИСИ, ИЗДАТЕЛЪСТВО "НАУКА" ГЛАВНАЯ РЕДАКЦИЯ ФИЗИКО-МАТЕМАТИЧЕСКОЙ ЛИТЕРАТУРЫ, МОСКВА, 1968p>div>