Marx and Mathematics - Annette Vogt

In order to better understand capitalism, Karl Marx taught himself parts of algebra and calculus. Nevertheless, he was not a mathematical genius. The historian of science and mathematician Annette Vogt explains why the editorial history of Marx’s mathematical manuscripts resembles a detective novel, and how he used math to deal with personal crises.

Professor Vogt, is it true that Karl Marx made numerous mathematical errors in Capital?

Annette Vogt: That’s true, there are all kinds of calculation errors. But that’s human. And Marx was also just a human being.

Only a few people know that Marx left behind mathematical manuscripts numbering almost 1000 pages. Why did he engage with mathematics at all?

One reason was that he wanted to predict economic crises; in the case of the first one, he was rather euphoric that capitalism was now collapsing. He then asked himself: are they regular, for example every five or ten years or – as is actually the case – irregular. Marx was friends with the chemist Carl Schorlemmer, who told him that it might be possible with the aid of calculus – more specifically, with differential calculus – to calculate when the next crisis would come. When Marx attended Gymnasium in Germany, differential and integral calculus were not yet part of the curriculum, that was first the case after 1900. So he had no knowledge of it and did what a scientist does…

Pick up a book first?

Exactly. He went to the library and sought out books that he could learn it from. However, as the Dutch-American historian of mathematics Dirk Struik, who was one of the first to write about the manuscripts, accurately put it: for studying capitalism, Marx was in the right country, England; for studying mathematics, he was in the wrong one. He wasn’t familiar with the newest mathematical literature on calculus, because it was all from continental Europe and was not yet available in England. So he studied the textbooks that were available to him. The mathematical manuscripts consisted largely of excerpts that he created on the basis of his readings, and his notes on them. That’s how Marx taught himself differential calculus.

Were there further reasons for his engagement with mathematics?

Yes. A further reason was – and I understand it quite well, as a mathematician – that it helped him through personal crises. We know this from letters to Engels: when one of his children died young, he did arithmetic in order to distract himself. That might sound incredible to people who are afraid of mathematics, but of course this way of keeping busy can help somebody not to grieve all the time.

What other areas of mathematics did Marx devote himself to?

He also did a little bit of algebra. Algebra consists of equations, from the most simple 2+2=4 to abstract equations up to those – think of the Pythagorean theorem – that can be illustrated geometrically.

That simply had to do with the fact that there are equations in economics.

So his interest was largely pragmatic?

There are two interpretations regarding Marx and mathematics. One – the hagiographic one, making him into a pillar saint – is that Marx was such a universal genius, that he was also a mathematical genius. That’s simply wrong. The other one is: he was a scientist, and as such, he appropriated knowledge that he needed via self-study. He also wrote geological excerpt notebooks – but luckily, it never occurs to a geologist to claim that Marx was a great geologist. (laughs)

With regard to the editorial history of the excerpt notebooks, the hagiographical element plays a role, however: those who wanted to publish the mathematical manuscripts were disappointed by their content.

Because they didn’t find in them the genius they were hoping for?

Exactly. However, his notes are nonetheless significant, simply because they show us the areas he was concerned with, and because they help us to understand and reconstruct his thought. However, Marx can be a role model for everyone who is afraid of math: there’s no reason for that, anyone can learn it.

In your entry on the manuscripts in the Historisch-kritisches Wörterbuch des Marxismus, you write: “his notes on the history of ‘infintesimal calculus’, that is, of differential and integral calculus, have a charm of their own.” What did he write?

He studied textbooks – for example those of the French mathematicians Lagrange or Cauchy – and attempted to understand what the crux of differential calculus is. One can actually see this quite nicely when looking at its historical development and asking why which thing was done at what time. For example that it started with physics, because people wanted to calculate the speed of something. Well, that’s exactly what Marx did, he chose a historical approach, and asked: why does Lagrange take this step, why does he examine that function, why didn’t somebody else do that – these notes are simply interesting for historians of mathematics. He did that completely correctly, he understood the core of the matter.

What do you know about the period of time in which he concerned himself with that?

There were three phases in which notes were made, each in the British Museum Library. Using the borrowing slips, it was exactly reconstructed when he read which books there, that’s how we know he wasn’t familiar with the most modern literature. He knew French, that helped him to read Lagrange and Cauchy in the original.

To what extend did his concern with mathematics have an influence on Engels’ work?

While Engels was writing Dialectics of Nature, Marx – we know this from letters – had told him a bit about the history of mathematics. I suspect that Engels for that reason also therefore thought that Marx was a talented mathematician, since Engels didn’t know anything about math and Marx at least knew a little bit. Thanks are due to Engels for the fact that the mathematical manuscripts were preserved after Marx’s death. He considered them important. Marx never intended to publish them; they were working material.

Even today, the manuscripts are – despite Engels’ intention – only partially published. Why?

After the victory of the October Revolution, the Marx-Engels-Institut was founded in Moscow, later the Marx-Engels-Lenin-Institut, and charged with the task of publishing a Marx-Engels-Gesamtausgabe, the MEGA I. The father of this edition was David Borisovich Ryazanov, who later became, along with many other members of the Institut, a victim of Stalin’s persecution. The project of the MEGA I was interrupted. After 1945, the MEGA II began publication, later the project of MEGA III was begun with the participation of the Berlin-Brandenburg Academy of Sciences and Humanities, the International Institute of Social History Amsterdam, and collaborators from Moscow. It is not yet completed, and within the framework of MEGA III, the mathematical manuscripts are also supposed to be published completely.

However, there is a volume with part of the manuscripts: in 1968 a special edition was published, which until today is the basis for all engagement with the manuscripts, including the English and French translations and the – strongly abridged – German edition.

Who was responsible for this edition?

It goes back to work by the mathematician and specialist for logic, Sofia Yanovskaya, and Konstantin Rybnikov, who was a professor of history of mathematics at Lomosonov University in Moscow. However, they “forgot to mention” – in scare quotes – the work of Ernst Kolman, a Czech-Soviet Comintern functionary who lectured and published articles on the mathematical manuscripts at international conferences from 1932 on. In 1968, he distanced himself from Soviet leadership due to the Prague Spring, that’s why he isn’t named in Yanovskaya and Rybnikov’s edition. When I first dealt with this in the 1980s and noticed it, I thought: that’s really unfair.

And it is!

Yes. But here’s the exciting part. I then found out: Kolman himself had deliberately covered up who had been the person commissioned by Riazanov in the 1920s to prepare the mathematical manuscripts for publication in the MEGA I: the mathematician and political author Emil Julius Gumbel. Gumbel was a co-founder of the modern statistics of extreme values, which are used to calculate extreme events, such as the Corona pandemic. Gumble had basically finished editing the manuscripts, at the end of the 1920s he read the galley proofs, but the publication never happened: work on the MEGA fell victim to the repression under Stalin. Gumbel was later driven from Germany by the Nazis; he worked in Paris and Lyon, and later in American exile.

You see, in a certain way it’s tragic: over the decades, almost a hundred years, a few people have already worked on the editing of these mathematical manuscripts, and many sad stories are involved. If I were a writer of crime novels, I’d write a book about it and call it “The Curse of the Manuscripts.”

Annette Vogt has a degree in mathematics and a doctorate in the history of mathematics. From 1994 to 2018, she was a research scholar at the Max-Planck-Institut für Wissenschaftsgeschichte. Since 1997 she has taught at the Humboldt University in Berlin, and since 2014 she has been an honorary professor of the economics faculty of the HU. Among other things, she is co-author of a traveling exhibition on the life and work of Emil J. Gumbels.

Nelli Tügel is an editor at ak.

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