Category Archives: People

Thừa giấy vẽ voi

Có một bài viết hay của Freeman Dyson, “A meeting with Enrico Fermi. How one intuitive physicist rescued a team from fruitless research“, kể lại một sự kiện rất quan trọng trong cuộc đời của ông. Dyson kể lại rằng sau khi giải quyết xong vấn đề tương tác điện từ, ông bắt đầu nghiên cứu tương tác mạnh (lực hạt nhân). Năm 1953, ông cùng một số học trò rất say sưa với một lý thuyết tương tác mạnh, mô tả tương tác giữa proton, neutron và hạt pion gọi là “pseudoscalar meson theory”. Những tính toán của nhóm ông cho ra kết quả gần với kết quả thí nghiệm của Fermi. Ông ta rất tự hào với kết quả của mình và hẹn gặp Fermi để trình bày kết quả. Ông lên xe bus đi từ Ithaca, NY đến Chicago. Câu chuyện tiếp theo như sau (bản dịch của Nguyễn Đình Đăng):

Khi tới văn phòng của Fermi, tôi trao các đồ thị cho Fermi, nhưng ông hầu như không thèm nhìn chúng. Ông mời tôi ngồi, thân thiện hỏi thăm sức khỏe vợ tôi và con trai mới sinh của chúng tôi, hiện nay 50 tuổi. Rồi, với giọng nhẹ nhàng và đều, ông ra phán quyết, “Có hai cách tính toán trong vật lý lý thuyết,” ông nói. “Một cách, là cách tôi thích hơn, đó là có một bức tranh vật lý rõ ràng về quá trình anh tính toán. Cách kia là có một hình thức luận toán học chính xác và nhất quán. Anh chẳng có bất kỳ cách nào cả.” Tôi hơi choáng, nhưng đánh bạo hỏi ông vì sao ông không coi lý thuyết meson giả vô hướng (pseudoscalar meson theory) là một hình thức luận toán học nhất quán. Ông trả lời, “Điện động lực học lượng tử là một lý thuyết tốt vì các lực yếu, và khi hình thức luận còn mơ hồ, chúng ta có một bức tranh vật lý rõ ràng dẫn dắt chúng ta. Với lý thuyết meson giả vô hướng thì không có bức tranh vật lý nào cả, còn các lực thì quá mạnh đến nỗi chẳng có gì hội tụ được. Để đạt được kết quả tính toán, anh đã phải đưa vào một quy trình cắt bớt tùy tiện chẳng dựa trên cơ sở vật lý hay toán học chắc chắn nào cả.”

Tuyệt vọng, tôi hỏi liệu Fermi có ấn tượng với việc các con số tính toán của chúng tôi phù hợp với các con số mà ông đã đo được hay không. Ông trả lời, “Anh đã dùng bao nhiêu tham số tự do trong các tính toán của anh?” Tôi nghĩ một thoáng về quy trình cắt bớt của chúng tôi rồi nói, “Bốn.” Ông nói, “Tôi nhớ ông bạn Johnny von Neumann của tôi từng nói, với bốn tham số tôi có thể mô tả được con voi, còn với năm tham số tôi có thể làm nó ngọ nguậy cái vòi.”

Fermi cuối cùng đã đúng. Phải 20 năm sau đó người ta mới tìm được lý thuyết đúng đắn của tương tác mạnh, “sắc động học lượng tử”, mô tả tương tác giữa các quark. Các hạn như proton, neutron, pion đều làm từ quark. Trong lý thuyết meson giả vô hướng, các hạt này được coi là các hạt cơ bản, nên không thể là lý thuyết đúng. Tính toán của Dyson và học trò hầu như không còn được ai nhớ đến nữa. Không biết gì về quark, chỉ dùng trực giác vật lý, Fermi biết ngay là không thể tin được lý thuyết meson giả vô hướng. Dyson viết rằng nếu không có Fermi, ông ta và học trò chắc chắn sẽ mất nhiều năm vào một hướng nghiên cứu vô ích.

Đọc câu chuyện này tôi cũng nhận ra chính mình: bản thân tôi đã làm nhiều tính toán không có bức tranh vật lý rõ ràng, không có hình thức luận chặt chẽ, và thậm chí cũng không có cả thực nghiệm để so sánh!

Trở lại với vấn đề con voi. Tôi loay hoay mãi không vẽ được con voi mà chỉ dùng 4 tham số, như von Neumann nói. Tìm đọc trên internet tôi thấy đã có người viết là đã vẽ được voi dùng 4 tham số, nhưng tìm hiểu kỹ hơn thì hoá ra họ ăn gian: 4 tham số của họ là số phức, thực chất là 8 tham số thực. Chi tiết có thể đọc ở đây. Tuy nhiên chỉ cần dùng 2 tham số thực có thể vẽ được bức tranh “con trăn nuốt con voi” trong truyện Hoàng tử bé của Saint-Exupery. Bức tranh như sau:

và đây là đồ thị hàm số

y= \exp(-x^2) + a \exp(-(x-b)^2)

với a = 0.8 và b = 1.75.

Tầm quan trọng của việc học ngoại ngữ

Tôi vẫn nhớ một câu thơ của Mayakovsky mà tôi đọc được từ rất lâu khi đang học tiếng Nga, dịch ra đại khái như sau (xin lỗi là dịch không vần):

Kể cả nếu tôi là người da đen cao tuổi
tôi cũng sẽ không chán nản lười biếng
học tiếng Nga chỉ vì đó là tiếng nói của Lênin

Trên thực thế, tôi không biết người nào (tất cả các màu da và tuổi tác) mà học tiếng Nga chỉ vì đó là tiếng nói của Lênin. Tuy vậy gần đây tôi đọc được một câu chuyện làm tôi nhớ tới câu thơ của Mayakovsky. Câu chuyện được nhà toán học Vladimir Voevodsky kể lại:

Năm 1984, Alexander Grothendieck nộp cho CNRS một đề án có tên là “Esquisse d’un Programme”. Ngay sau đó giới toán học bắt đầu chuyền tay nhau các bản sao của đề án này.

Vài tháng sau đó, thầy hướng dẫn khoa học đầu tiên của tôi, ông George Shabat, đưa đề án này cho tôi đọc. Lúc đó tôi là sinh viên năm thứ nhất trường Đại học Tổng hợp Mátxcơva.

Sau khi học một ít tiếng Pháp với mục tiêu duy nhất là để đọc được tài liệu này, tôi bắt đầu triển khai một số ý tưởng mà Grothendieck phác thảo trong đó…”

18 năm sau khi bắt đầu học tiếng Pháp, Voevodsky được huy chương Fields.

Kerson Huang: Chen Ning Yang and I Ching

When I was a postdoctoral fellow in physics at the Institute for Advanced Study in Princeton, I worked with Chen Ning Yang on a problem of statistical mechanics. Every morning we would have heated arguments in his office, but rarely, if ever, did we speak about anything other than physics, so concentrated was our interest.

Earlier, Yang had collaborated with Tsung Dao Lee of Columbia University in an attempt to resolve an outstanding puzzle of the time concerning the so-called “weak interactions.” In a series of now-classic papers, they had made the bold proposal that nature is not left-right symmetric. Specifically, they suggested that left-right symmetry is violated because the neutrino, a spinning subatomic particle important for the weak interactions (which also happens to be indispensable in the nuclear process that causes the sun to shine), always “spins to the left,” like an advancing left-handed screw.

The proposal led to very specific experimental predictions, and Chien Shiung Wu, an experimental physicist at Columbia, set out to test it with a team at the National Bureau of Standards. After six months of hard work, she and her co-workers verified that left-right symmetry was indeed violated. The news sent shock waves through the physics community, and Lee and Yang were awarded the Nobel Prize in Physics the following year.

I remember the morning when Yang learned of the news of the downfall of parity. He was excited about the new outlook on physics the discovery brought. Then he said suddenly, “Let’s ask the I Ching.” We threw the coins in his office and got the hexagram 53 PROGRESS:

Favorable for a maiden’s marriage.
Auspicious omen.

The body of the hexagram emphasizes that progress comes only gradually.

I think Yang was a little disappointed, but the I Ching has proven to be prophetic. By knocking down a sacred cow, Lee and Yang had led physics across a threshold, beyond which an immense vista opened up. A long fuse was lit, which has been sputtering for thirty years, illuminating vast domains in particle physics and leading to furious attempts to probe matter at a deeper level, even to plumb dimensions beyond space-time. But why the neurino should be a “left-handed screw” still remains a deep mystery, and perhaps holds the key to further progress.

Strangely enough, the I Ching had never come up in our conversations until that morning. Yet, by the mere fact that we shared a certain Chinese cultural background, it was taken for granted that we both knew about the I Ching. Neither of us believed that the I Ching could predict the future, in the sense that physics predicts the future in certain systems, but there was the unspoken understanding that to consult it was to solemnize the moment.

From chapter 6, “I Ching and Physics” of the book I Ching by Kerson Huang and Rosemary Huang (Workman Publishing, New York, 1987).

The Hellmann of the Hellmann-Feynman theorem

Portrait of Hans Hellmann by Tatjana Livshits (1999)

The Hellmann-Feynman theorem is well known in quantum mechanics, perhaps even more so in quantum chemistry. Feynman is of course Richard Feynman, who gave a proof of the theorem in his undergraduate thesis. But who is Hellmann?

Hans Hellmann was born in Germany in 1903. He received his PhD in 1929 from University of Stuttgart. He proved the famous theorem in 1933. Shortly after, in 1934 Hellmann escaped from Nazi Germany to the Soviet Union. It was in the USSR where he wrote the first textbook in quantum chemistry, predating Linus Pauling’s book by a few years. He tragically died in 1938 during Stalin’s Great Terror. Below is an excerpt from a biography written by his son Hans Hellmann Jr., which appeared in the new edition of Hellmann’s quantum chemistry textbook Einführung in die Quantenchemie (Springer Spektrum, 2015). Caution: this translation relies heavily on Google Translate. All inaccuracies in translation belong to me.


… On November 1, 1931, at the age of 28, my father got a job as a lecturer in physics (Physik-Dozent) at the University of Veterinary Medicine Hanover, despite the fact that his Habilitation has not been completed yet. His mentor, Professor Fues, assured that it was imminent. In March and July of 1933, two important papers written by my father appeared in the journal Zeitschrift für Physik. In the first paper, he presented a method by which one can make quantitative statements about the energy of polyatomic molecules on the basis of spectroscopic data of their diatomic fragments. The other work highlights the role of the kinetic energy of the electrons in the covalent bond and contained the virial theorem and the theorem known today as the Hellmann-Feynman theorem. Both works would form the basis of my father’s Habilitation thesis.

But my father was denied the Habilitation. Following the appointment of Adolf Hitler as Chancelor on January 30, 1933, several new laws directed against political opponents, and especially against Jews, were passed. The first set of laws, including the “Law for the Restoration of the Professional Civil Service” of April 7, 1933, took aim at Jewish officials. They could not fulfill their new obligation – to prove their “Aryan origins” – and lost their job. A little later, with the Reich Civil Service Act of June 30, 1933, those in the so-called “mixed marriages” were also affected.

My parents faced hard times. With the Habilitation application, my father must declare the “racial origins” of his wife [who was from a Jewish family in Ukraine], but he refused to submit the requested information. My father was never a member of any political party, but he had political beliefs and never concealed his negative attitude towards National Socialism. In contrast, most students at the University of Veterinary Medicine welcomed the “new order” with enthusiasm. They would greet my father at his lectures with disruptive and hostile noise.

My parents had in their library some books that were banned at the time: Heine, Zweig, Fallada, and magazines with articles by progressive authors. Since keeping them was dangerous, they had to destroy them. My mother told me that once, I reported enthusiastically at my kindergarten: “Yesterday my parents burned a lot of red books!” (they were issues of the [leftist] magazine “Die Weltbühne”). The teacher became scared and brought me home. That was the first, but unfortunately not the last, political lesson in my life.

In the autumn of 1933, the Prussian Culture Ministry prohibited the University of Veterinary Medicine from conferring Habilitation on my father. On December 24, 1933, he was informed that his lecturer position was to be terminated effective March 31, 1934, as his Habilitation was no longer expected due to the “non-Aryan origins” of his wife. His doctoral adviser Erich Regener would be dismissed “only” in 1938.

The continuation of scientific collaboration with colleagues at the Technical University of Hanover became impossible. There, my father had had very intensive discussions with Wilhelm Jost, Privatdozent of Physical Chemistry, on the “problem of the nature of the chemical forces.” They still managed to publish, as two papers in the journal Zeitschrift für Elektrochemie (1934/1935), the clear understanding based on quantum mechanics that they worked out, but the second paper had Wilhelm Jost listed as the sole author. Later, these two papers would form the basis of the first chapter of my father’s two textbooks of quantum chemistry.

Like many scientists, particularly those from Göttingen, a center of the development of quantum mechanics, my father had to emigrate with his family. But to where? A fateful decision was made in favor of the Soviet Union. Why? There were probably two deciding factors. First, my father had a certain sympathy for socialist ideas. Second, my mother came from there (she did not lose her citizenship) and still had relatives there. Later I learned from her that my father also had other invitations, including some from America. From about 1930, he had been looking for a job in the Soviet Union. Perhaps he talked about it with his sister back then.

Hans Hellmann and his sister Greta (c. 1930)

Through the mediation of Victor Weisskopf, still in Göttingen at the time, he got two invitations. The first was from the Ukrainian Physico-Technical Institute in Kharkov (now Kharkiv), where several well-known physicists (among others Alexander Weissberg and Lev Shubnikov, a few years later also Lev Landau) were working at that time. The second invitation was from the Physics Institute of the University of Dnepropetrovsk, where Boris Finkelstein was very interested in problems of quantum chemistry. But in both occasions the Soviet authorities refused to issue the necessary entry documents. Then came 1932 when, again with the help of colleagues in Göttingen, he made contact with the Karpov Institute in Moscow, then a leading center of physico-chemical research in the Soviet Union. After a meeting in Berlin with academician Alexander Frumkin, Deputy Director of the Karpov Institute, my father received an official invitation to Moscow and an attractive job offer. In March 1934 we got the necessary entry and exit documents. We bid farewell to our dear relatives and left Germany. Traveling by train from Berlin, we arrived in Moscow on April 31, 1934. My mother’s aunt Maria Minchina picked us up at Belorussian-Baltic Station. Her first words were: “How did you decide to come here? You are crazy!” [“Сумасшедшие, куда вы приехали?”, according to a Russian source].

*
*     *

The Karpov Institute in Moscow was at that time very well financed by the government. The research carried out there were important both for the economy and the military. The directors of the Institute were two academicians, Alexei Bach, a biochemist, and Alexander Frumkin, a physicist. In the “Department of the Structure of Matter” under the leadership of Yakov Syrkin, my father was hired as the “Head of the Theory Group.” He was well received at the Institute and, which was particularly important for him, could now fully devote his knowledge and work to science. At that time, foreign scientists in the Soviet Union had some privileges. Of course, complete loyalty was demanded in return. My father had the freedom to plan his working hours, and he was also able to work at home a lot. In his letters to his mother, who was then living in Hamburg, he described the atmosphere at the Institute and the local working conditions in enthusiastically positive terms. Not a single word about the small two-room apartment or the lack of certain foods. He talked about his work and his contacts with foreign colleagues, for example, at an international conference in Kharkov in 1934.

This photo from the AIP archive was mislabeled. The person to the left of Niels Bohr, under the red arrow, is Hans Hellmann, and to the left of him, Yuri Rumer.

After numerous insults and humiliations in Hanover during the first year of the Nazi regime, he was very happy with his new life in Moscow, and the whole family was also mostly happy. In the summer of 1935 my grandmother came to the Soviet Union to visit us in the for the first and only time. She spent a short vacation in the Crimea with her son during this visit.

Early in 1935, the doctorate, the Russian equivalent of Habilitation, was conferred on my father. Approximately every two months he would publish a scientific paper. At a meeting in Dnepropetrovsk he was elected to the organizing committee of a quantum chemistry conference held in 1936. He was honored several times with monetary awards for his research and results. In June 1936 my father became a Soviet citizen, in November his starting salary of 700 rubles was increased to 1200 rubles, and in December he was invited to give a lecture at the Academy of Sciences. Shortly afterwards, on January 1, 1937, he was appointed a “Full Member of the Karpov Institute” (which corresponds to the title of a professor at a university), and in autumn 1937 became a “Senior Scientist.” In this period of about three and a half years in Moscow, my father had advised and mentored a number of young PhD students and postdocs: V. Kasatochkin, K. Maevsky, M. Mamotenko, S. Pshezhetsky, N. Sokolov, and M. Kovner.

At least since 1933 my father had been planning to write a monograph on his field of quantum chemistry. The above-mentioned join work with Wilhelm Jost in Hanover would form part of this monograph. The first version of the manuscript was finished before emigration, but remained with Jost, who tried in vain to find a publisher in Germany. Based on the manuscript, by now translated into Russian, my father in 1935-1936 gave a lecture course at the Karpov Institute, attended also by young researchers from other institutes in Moscow. Since my father’s Russian was not perfect, his PhD students sometimes had to help him find appropriate terms for notions that were new also in German. The eager Russian listeners offered their criticism and proposed corrections, for which my father expressed gratitude in the preface of “Quantum Chemistry” (Квантовая Химия, Volume 1 in the series “Physics in Monographs” ONTI, Moscow and Leningrad, 546 pages), which appeared in early 1937. More specifically, it includes an acknowledgment to his friend and colleague Yuri Rumer.

Even before completing the Russian version, my father had started revising and tightening the German version. It carried the title “Einführung in die Quantenchemie” (Deuticke, Leipzig and Vienna, 350 pages) and appeared in late 1937. While the Russian version was written for a largely unprepared reader, the shorter German version, with about the same content, placed significantly higher requirements on the reader’s preparation. But while the Russian book was selling well and was soon out of print, the German book had far fewer buyers. The reasons are possibly the contemporary historical circumstances on the one hand and the events which occurred after the publication of the books on the other hand.

*
*     *

In 1937, mass arrests of the “enemies of the people” began in the Soviet Union. Among those arrested were German and Russians, writers and farmers, engineers and artists, officers and soldiers. No one could sleep peacefully any more. The total number of innocent victims in those years amounts to over twenty million. In a letter to his mother from December 1937, my father wrote that “the current international situation has become complicated” and that he did not want to write her more often. In the night from the 9th to the 10th of March 1938, my father was arrested. I was eight and a half years old and I can still remember this event. They woke me up and searched my bed for anti-Soviet writings and evidence of espionage activities.

My father’s doctoral student M. Kovner, who often visited our house, came to Moscow from Voronezh a few days later. He wanted to visit us, but a neighbor warned him against it and told him about my father’s arrest. As the result he had to leave immediately. Later, M. Kovner published two articles about his dear teacher.

After my father’s arrest, my mother tried several times to get information on his fate from the People’s Commissariat for Internal Affairs (NKVD), the predecessor of the KGB, all in vain. She was forced, by the threats from local officials, to end her inquiries. We had to leave Moscow. My mother found a job as a German language teacher at a middle school in a village 120 km west of Moscow (near Volokolamsk). We knew nothing about my father’s fate. Former friends disappeared likewise or avoided us. Only a few people maintained friendly contact with us: the Livshits family (my mother’s relatives) and the translator Nadezhda Volpina.

Several months after the start of the Russian campaign of the German Wehrmacht, on September 9, 1941, when German troops were already on the march to Moscow, my mother was arrested. We found each other again until after the war. She was accused of “anti-Soviet propaganda,” and it was claimed that she, a Jew who had fled Nazi Germany, had been waiting for the German troops to work for them as a translator. After several months in Moscow prison, she was exiled to the Semipalatinsk area in Kazakhstan.

Only after Stalin’s death, and the beginning of the politics of “the Thaw,” my mother was “fully rehabilitated.” She then requested information about her husband. Initially she received a certificate, which later turned out to be wrong, that he died in prison from a disease (peritonitis). Then my mother applied for a certificate of rehabilitation for my father, which she received in 1957. Now my father was “fully rehabilitated.” Sadly, this happened only after his death. Only in 1989, during the “perestroika,” did we get the real death certificate. The documents showed that my father was convicted to “high treason” and “espionage in favor of Germany” in accordance with Article 58 of the Criminal Code and was shot dead on May 29, 1938.

Already in May 1937, shortly after the beginning of mass arrests, Albert Einstein sent a letter to Stalin expressing his great concern for the fate of many well-known scientists. A similar letter from three Nobel laureates Irène Joliot-Curie, Frédéric Joliot-Curie, and Jean Baptiste Perrin was sent to Stalin in June 1937. But their voices were not heard. When Wilhelm Jost noticed in 1938 that the name “Hellmann” no longer appeared among the authors in the journal Acta Physicochimica URSS where my father worked and regularly published until October 1937, he asked his British colleague John Lennard-Jones for help. Lennard-Jones sent a request for reprints to my father’s address at the Karpov Institute, but there was no reply.

Due to the historical and political circumstances and the tragic fate caused by them, for decades my father’s name disappeared almost completely from science. Although his book “Einführung in die Quantenchemie” appeared in the US in 1944 as a war booty, it hardly found the wide dissemination which it deserves. The name “Hellmann” is now remembered mostly through the term “the Hellmann-Feynman theorem”…

“Hôn lý thuyết”

To my wife
tác giả: Ludwig Boltzmann (1844-1906)

Ludwig Boltmann và vợ, Henriette Boltzmann

Abroad, should I not entertain myself with foreign ladies?
Are they not the ones most similar to you in everything there is?
Quite often I kissed your picture on a paper: Oh please forgive me
that I also kissed your picture if I found it made of flesh and blood!
Beside, I am a theorist from bottom to the top
and so, believe me, did I only kiss in theory

(trích từ Reise eines deutschen Professors ins Eldorado của Ludwig Boltzmann, viết năm 1905 sau một chuyến đi Mỹ; bản dịch tiếng Anh của Walter Kutschera: Journey of a German Professor to Eldorado)

From Valery Rubakov’s interview

From Natalia Demina’s interview of Valery Rubakov (November 2014, all translation mistakes are mine):

…There was one rather peculiar point in my life. It was 1981. It seemed to me I’d found an interesting theoretical effect, although I was merely a graduate student then. I wrote up an article about that nontrivial theoretical phenomenon and sent it to Physics Letters. They rejected it. I was very young, and it was my own article, without co-authors. I wrote it all by myself, submitted it myself, and was turned down. What to do?

So I wrote up another article on the same topic, but with a description of possible experimental manifestations of the phenomenon. I sent it to JETP Letters and thought to myself that if JETP Letters wouldn’t take it, then I was an idiot. I thought I’d found something interesting. But if the paper was not approved by a journal, foreign or domestic, then evidently I was no good.

I remember well this moment in my life. With my heart shuddering, I called JETP Letters to learn about the fate of my paper. “Now they’ll tell me they won’t take it. Do I really have to quit physics?” I called the editorial office, turned out they’d accepted my paper! I felt a great relief. The paper later became well known. And in Princeton, Callan published a paper about the same effect. And after, the whole thing got accepted by the physics community. I remember that I wrote a letter to Callan, saying that his paper talked about the same effect as my paper, attached reprints, that everything had been published in the English translation of JETP Letters. To his credit, he immediately started to refer to my paper and recognized my priority. He and everyone else.

I must say that if I were the editor of JETP Letters, then I wouldn’t accept my own paper! It’s because it only outlined the basic theoretical phenomenon – “one can show that.” And I would never trust “one can show that.” I would ask the author to show the proof. The Editor displayed his intuition, publishing the paper in the form it was submitted. No way I would publish it. It was too much against the mainstream.

But the story has a sequel. Time passed, and once I came to Cambridge. And out of the blue, someone whom I only knew by name invited me to dinner at Trinity College. It’s such a solemn and honorable event. I was surprised, but I went. We ate, and as we were leaving he said: “Valery, impose a penance upon me.” I was perplexed, I didn’t understand. I didn’t know the English word then. I said, “What are you talking about?” “I’ve committed a sin. I want you to free me from the sin.” “And what kind of sin?” “Impose a penance upon me first, then I’ll tell you.” “What’s the matter?” “I was your referee at Physics Letters and rejected your paper.” Before that, during dinner, we were discussing if it would be a good idea to write a book on the subject he was working on. He disagreed and told me that he was too lazy. After his confession, I told him: “Write the book, that will be your penance.”

(Natalia Demina’s interview of Valery Rubakov, polit.ru November 29, 2014, accessed December 9, 2014).

Các Mác học toán

Trong lời giới thiệu bản năm 1885 của cuốn “Chống Duhrinh” Ăng-ghen có viết:

Nhưng từ khi Các Mác qua đời, thời giờ của tôi phải ngừng công việc nghiên cứu của tôi lại. Lúc bấy giờ tôi đành tạm bằng lòng với những phác thảo đã đưa ra trong sách này và đợi sau này có dịp thì sẽ tập hợp và công bố những kết quả đã thu nhận được, có thể là cùng một lúc với những bản thảo toán học rất quan trọng do Mác để lại.

Đoạn tôi nhấn mạnh bằng chữ đậm ở trên liên quan đến một điều chắc ít ai biết: Các Mác lúc về già bỏ rất nhiều thời gian học toán, chủ yếu là môn giải tích. Có lẽ Mác cảm thấy cần thêm công cụ toán học để nghiên cứu kinh tế. Các ghi chép của Mác về toán cuối cùng phải đợi gần 50 năm sau khi Ăng-ghen viết những lời trên mới được công bố. Có thể đọc những ghi chép này (đã dịch sang tiếng Anh) ở đây:

http://www.marxists.org/archive/marx/works/1881/mathematical-manuscripts/

Đây là một trang trong bản thảo “On the Differential”

Rõ ràng Các Mác biết tính đạo hàm hàm số x^m.

Đọc qua bản thảo thì có lẽ không có kiến thức gì mới, và có lẽ cũng không có gì mới so với những gì người ta biết thời đó. Nhưng một người đã ở tuổi ngoài 50 mà chăm chỉ học toán như vậy có lẽ cũng là hiếm.