Book Reviews

Note: For a Romanian translation of this post, see here.

Felix Berezin

Misha Shifman has edited a wonderful book about the mathematician Felix Berezin, which recently appeared with the title Felix Berezin: Life and Death of the Mastermind of Supermathematics. Berezin was a Soviet mathematician largely responsible for many new ideas about “supermathematics”, working out the analog for anticommuting variables of many standard concepts in analysis. Path integrals for fermions crucially use an analog of the standard integral that is now known as the Berezin integral.

Berezin began his mathematical career working with Gelfand on representation theory. While Gelfand thought very highly of him, at some point the two of them had a falling out, which is alluded to without any details in several of the contributions to this book. Since Berezin’s mother was Jewish, his professional life was often difficult due to the anti-semitism that was prevalent in the Soviet mathematical establishment. Between this and being on the outs with Gelfand, he had continual problems with things like getting his papers published, as well as being able to travel or effectively communicate with people in the West.

Tragically, Berezin died at the age of 49, under somewhat unclear circumstances on a trip to Siberia he took with a geological team. The largest segment of the book is a wonderful and touching piece by Elena Karpel, who lived with him for many years (they had a daughter together, Natasha). Karpel describes their life together in detail, as well as the circumstances following his death. It is a moving portrayal of a complex relationship of two highly intelligent and cultured people, with one of them, Berezin, extremely seriously devoted to his work, one cause of stress in his relations with Karpel. Together with contributions from his colleagues, the book gives a fascinating portrayal of the mathematical culture that Berezin was an important part of.

With his interest in quantum mechanics, quantum field theory, path integrals, and anticommuting variables, Berezin helped to transform the field of mathematical physics into something much more modern. His book written during the sixties, The Method of Second Quantization remains one of the classics of quantum field theory. I remember being especially impressed by his paper with Marinov Particle spin dynamics as the Grassmann variant of classical mechanics, which gives an amazing interpretation of the physics of a spin-1/2 particle by invoking anti-commuting variables in a very simple way. The book contains a summary of some of Berezin’s scientific work by Andrei Losev, and this article is available on-line.

The Mathematician’s Brain

Princeton University Press seems to be trying to corner the market on popular books about mathematics, bringing out in quick succession a novel about mathematics (A Certain Ambiguity), a book about The Pythagorean Theorem, and two books trying to explain what it is that mathematicians do: How Mathematicians Think by William Byers, and The Mathematician’s Brain by mathematical physicist David Ruelle. The Ruelle book is the only one of the four that I’ve had a chance to read.

The New York Sun recently published a review of The Mathematician’s Brain by David Berlinski. It’s one of the great mysteries of the popular science book business why anybody publishes the writings of Berlinski. His recent claim to fame is as an affiliate of the Discovery Institute, critic of Darwinism and proponent of Intelligent design, but he has also authored various popular books, including some on mathematics. Some web-sites claim that he has a Ph.D. in mathematics from Princeton, but it appears that the truth of the matter is that he was in the philosophy department there, writing a doctoral thesis on Wittgenstein. His writings on math and science that I’ve seen over the years have always struck me as singularly incoherent and confused.

Berlinski actually doesn’t do that bad a job with the Ruelle review, picking up on one of the things that might interest mathematicians and physicists about the book, the part about Alexandre Grothendieck (I confess to skimming some of the material explaining what mathematicians do, since I spend far too much of my life watching them do it). Ruelle has some interesting stories to tell about Grothendieck and the IHES, where they both worked for many years. The IHES was founded in the late 1950s by Leon Motchane, who had studied mathematics before going into business. Ruelle describes well the IHES during the 1960s, including the various conflicts which existed between Motchane and the IHES members, one of which ended up leading to Grothendieck’s resignation.

Ruelle also has quite a lot to say about the structure of power in mathematics, and how the desire for recognition and honors motivates people. His portrayal of mathematicians is a very well-rounded one, examining not just how they do mathematics, but how they live their lives, noting that:

But one should not forget that, besides beautiful mathematical ideas, there are many more obscure things that crawl in the mind of a mathematician.

Many of the footnotes in the back are well worth reading, such as one that tells us:

As my wife puts it, there are fewer bastards and fewer frauds among mathematicians than in the general population, but maybe also fewer amusing people!

Ruelle also tells a favorite anecdote I’ve heard from several mathematicians. The version I’ve heard is somewhat different than Ruelle’s, and goes:

At the Institute for Advanced Study in Princeton, a visitor once came up to Armand Borel and asked him

“Do you know about algebraic groups?”

Borel answered that, yes, he did. The visitor then went on

“Good. Can I ask a stupid question then?”

to which Borel responded:

“That’s two already.”

La Theorie des Cordes

A colleague brought me back from France a science fiction novel written by the Spanish writer Jose Carlos Somoza. In French the book is called La Theorie des Cordes (String Theory), but the Spanish and English versions have the title Zig Zag. The plot revolves around a discovery about string theory that allows physicists to look back into the past. It begins with some promise, describing the world of theoretical physics as seen from Spain, with references to Witten and other theorists. But it soon degenerates into a long tale revolving around a threatened attractive young female scientist. The string is somehow responsible for forcing her into sexual depravity and the prospect of nearly infinitely long and horrific bloody torture, with time suspended and no end in sight. OK, I guess maybe this does have to do with present-day particle theory, except for the sexual depravity part…

Reviews by Atiyah in the Notices

The October Notices of the AMS contains very interesting reviews by Michael Atiyah of two books about Bourbaki: Bourbaki: A Secret Society of Mathematicians by Maurice Mashaal, and The Artist and the Mathematician by Amir Aczel. Atiyah speaks from personal experience, knowing many of the members of Bourbaki and their work well, and having attended one of the Bourbaki gatherings where they hashed out the text of one of their books. He gives an excellent summary of the Bourbaki story and its place in recent mathematical history, finding the Mashaal book to be both highly readable and reliable on the facts and personalities involved. As for the Aczel book, he’s much more dubious. Aczel tries to claim an important impact of Bourbaki on sociology and structuralism via Claude Levi-Strauss, but Atiyah is not convinced by this, and takes issue with what Aczel has to say about Grothendieck, someone Atiyah knew well. Atiyah’s characterization of Grothendieck goes as follows:

I greatly admired his mathematics, his prodigious energy and drive, and his generosity with ideas, which attracted a horde of disciples. But his main characteristic, both in his mathematics and in social life, was his uncompromising nature. This was, at the same time, the cause both of his success and of his downfall. No one but Grothendieck could have taken on algebraic geometry in the full generality he adopted and seen it through to success. It required courage, even daring, total selfconfidence and immense powers of concentration and hard work. Grothendieck was a phenomenon.

But he had his weaknesses. He could navigate like no one else in the stratosphere, but he was not sure of his ground on earth—examples did not appeal to him and had to be supplied by his colleagues.

He ends with the following critical remarks

Aczel’s total endorsement of Grothendieck leads him to make such fatuous statements as: “Weil was a somewhat jealous person who clearly saw that Grothendieck was a far better mathematician than he was.” Subtle balanced judgement is clearly not Aczel’s forte, and it hardly encourages the reader to take seriously his confident and sweeping assertions in the social sciences.

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27 Responses to Book Reviews

  1. chimbote says:

    I share your appreciation for Berezin’s book and his paper with Marinov. Those were among my first reading assignments when I started my phd.

  2. Jason Starr says:

    You might also mention another quote from Atiyah’s article, which seems to be a pronouncement on EGA (among other things):

    “Where I part company with Aczel is in his assertion that Bourbaki made a fatal mistake in his not taking Grothendieck’s advice and rewriting its foundations in the new language of category theory. Aczel believes that Bourbaki had turned its face away from the future in not following Grothendieck. I doubt whether history will come to this verdict. Grothendieck’s own EGA, as well as the general fate of the over-confident universalists, might suggest otherwise.”

  3. Thomas Love says:

    It is easy to check if someone has a PhD in math. I went to the mathematics geneology website:

    and checked out David Berlinski. He isn’t listed. Neither are you Peter, because your PhD is in Physics. There are two Thomas Loves. How weird is that.

  4. Moeen says:


    Keep in mind that while the Math Genealogy website is a good resource, it is not comprehensive since all of the information on their database is entered voluntarily by individuals. So not only is it likely not comprehensive, but may even be wrong in some cases.

  5. John Baez says:

    The anecdote about Armand Borel only makes sense if you know his specialty was algebraic groups. He’s the author of the well-known book Linear Algebraic Groups.

  6. Arun says:

    The anecdote about Armand Borel only makes sense if you know his specialty was algebraic groups.

    One can deduce that from the anecdote and the fact that it is worth repeating.

  7. Peter Woit says:

    In my experience, the anecdote is invariant under the interchange of Armand Borel and Andre Weil, as well as substitution of various fields of mathematics. In Ruelle’s version it is Weil, and his is the streamlined, one question version: “may I ask a stupid question?” “You just did”


    I also was struck by that comment of Atiyah’s, but I guess it’s not too surprising. I wouldn’t have guessed that he’d be a big fan of EGA.

  8. Chris Oakley says:

    Reminds me of this, concerning Dirac:

    One famous lecture he gave in Canada: at the very end of the lecture during questions one of the members of the audience stood up and said, ‘I don’t understand the equation you have written on the top right-hand of the blackboard’. Dirac was silent. People were waiting. And then one of the chiefs in the front row said, ‘Professor Dirac would you care to respond to the member of the audience’. And he said, ‘Well that wasn’t a question it was a comment’.

  9. Clark says:

    Subject to being deleted for being utterly off topic, some of you may not have noticed that in last summer’s Clay Institute, there were 3 self-contained lectures by Manin from the ground to the frontier. As an aside there were some very funny moments such as when he admitted that he could never remember how the weights are specified in modular forms.

  10. gaspard says:

    About the Aczel book: the only connection I have heard of between mathematicians and Claude Levi-Strauss is that a PhD student of Levi-Strauss had to stop ethnological trips for health reasons and then turned to math. His name is André Avez, who among other things coauthored a book with V.I.Arnold on Ergodic Theory in the 1960s. Perhaps there have been interactions with Bourbaki too, but I don’t know them.

  11. David Berlinski says:

    I have never claimed to have a Ph.D in mathematics from Princeton University. My Ph.D. from Princeton is in philosophy. This is what my resume says; it is how I am described at the DI website; and it is how I am described on the dust jacket of my books. If there is a website that claims otherwise, I revile and denounce it. As long as I am correcting misapprehensions, I might add that I am a critic of intelligent design and not one of its supporters. In this regard, you might consider my essay “Has Darwin met his Match,” in the December 2002 issue of Commentary. It is devoted perceptively to attacking Johnson, Behe and Dembski. I cannot say that my friends at the DI were pleased to see what I wrote, but they were made wiser by reading it. My feelings toward intelligent design remain what they have always been: Warm but skeptical. Nonetheless, I regard the general hysteria about these issues as intellectually disgusting. As for the question why so many editors are interested in publishing what I write, I suspect that this is because so many readers are interested in reading it.

  12. James Milne says:

    I was at IAS at the time of the “Borel” anecdote, and knew all the people involved. I believe my version is the most accurate.

  13. Mike says:

    David Berlinski,

    Thanks for answering the question, I am one of your readers, and find your “humor” interesting. You might have noticed a pattern of, as Dan says, ” baseless ad hominem” !

  14. Peter Woit says:

    David Berlinski,

    One of your most prominent on-line biographies

    starts off

    “Berlinski (Ph.D. in mathematics, Princeton University) is a lecturer and essayist.”


    starts off

    “David Berlinski (born in 1942 in New York City) is a mathematician. He is the author of works on systems analysis, differential topology, theoretical biology, analytic philosophy and the philosophy of mathematics..”

    and yet a third

    claims that you were a “Fellow of the Faculty in Mathematics” here at Columbia. I don’t know what this refers to since presumably it was long before my time (I’ve been here about 20 years).

    You’ve done an excellent job of convincing people you’re a professional mathematician, including PZ Myers who writes:

    “This is a guy who is a competent mathematician with a degree from Princeton”

    I’ll stand corrected about your views on “Intelligent Design”, but direct people to the same post by Myers for a discussion of your views on the theory of evolution:

    And, to anyone who would like an excuse to start arguing about evolution here, don’t even think about it.

  15. Peter Woit says:

    Many thanks to James Milne for the source for the more accurate anecdote, as well as the link to the web page of other ones. Anyone who is not aware of his web-site

    and the fabulous array of high-level expository writings about mathematics there, should definitely take a look.

  16. Peter Woit says:


    Andre Weil did contribute an appendix (entitled “Sur l’etude algebrique de certains types de lois do mariage”) to Levi-Strauss’s book “Les structures elementaires de la parente”

  17. David Berlinski says:

    Thanks for the links. Of the three you cite, the first is incorrect. I shall correct it, The second and third are correct:


    On Systems Analysis: An Essay on the Limitations of Some Mathematical Methods in the Social, Political and Biological Sciences, The MIT Press, 1976.

    The Rise of Differential Topology, LITP, Universite Paris, 7, 1980.

    My essays about theoretical biology, catastrophe theory, model theory (with Daniel Gallin), epistemic logic, and philosophy are scattered in various journals. Just look around: You’ll find them. If not, I’ll be happy to send them to you.

    I am now more than sixty five years old, and a great many things in my life took place before your time. I was a Fellow of the Faculty in the department of mathematics — courtesy entirely of Lipman Bers — in 1973. I did nothing of value and achieved no distinction. I have never suggested otherwise.

    I have made no effort to convince anyone of anything. On the other hand, I have taught mathematics for many many years, both in the United States and in France, and I have written more than eight books about mathematics. I regard myself as a writer — nothing but, but if anyone asks, I know a lot about mathematics.

    I am quite sure that P.Z. Myers disapproves of my views concerning Darwin’s theory of evolution. I have been exhilarated by his criticisms.

  18. Peter Woit says:

    Thanks for the clarifications about your mathematics activities. I suppose I should know about why readers are interested in your books since at one point a copy of “Black Mischief” was on my bookshelves (can’t seem to find it, maybe it didn’t survive one or another move over the years). Good luck with the Darwin denialism business…

  19. Mike says:

    I can’t believe you found that link, Peter, lol.

    And David, very much looking forward to “The Devil’s Delusion” !

  20. Patrick Orlando says:

    I disagree with Prof. Woit’s comments on Dr. David Berlinski’s popular math books. I have read them all and he is an excellent writer able to describe the heart of the mathematical ideas in plane language. His style is very readable and his made up converstions with historical figures are very good. I only have a B.S. in Physics from Columbia so I guess they are on my level.

  21. The man says:

    I take two issues with Atiyah’s comments on Grothendieck. First, his comment “both of his success and his downfall” betrays a nasty petty streak in Atiyah. I doubt Grothendieck, or a lot of other people, see his path after IHES as “downfall”. In fact, Grothendieck, very courageously, resigned and went his own way because of his ethics and morality. Sticking to what you feel is right is not called “downfall”, as least not in the ass-kissing power-hungry circles I’m guessing Atiyah moves in (given how far up the non-academic ladder he’s gone). This phrase pretty much betrays Atiyah’s worldview. Similarly, I’m sure Atiyah would classify Grothendieck’s turning down Crafoord (or Perelman’s decision) as blunder etc, however not the way others would see it.

    Second, his comment that Grothendieck didnt deal with examples is meaningless if it doesnt interfere with Grothendieck’s ability to do good correct maths. So he should have given some examples where Grothendieck was led to publish incorrect math cause of it.

  22. Peter Woit says:

    The man,

    Your attack on Atiyah is really uncalled for. From my personal experience with him and everything else I know about him, he’s an extremely decent and responsible person. He has gotten where he is through his fantastic mathematical achievements (which are at the level of Grothendieck’s), and his encouragement of other people’s. You claim that you are “sure” he would be critical of Grothendieck and Perelman for turning down prizes says more about you than him. I’ve never heard any evidence that he was critical of them for this, and I doubt that that would be his attitude at all. His comments about Grothendieck’s style of mathematics not being grounded in examples, and his uncompromising nature being part of the reason he stopped doing math at the level he was doing it during the 50s and 60s are not at all controversial, but rather widely-held opinions by mathematicians who know the subject and its history. His comments on Grothendieck seemed to me well-balanced, recognizing his amazing achievements while not engaging in hero-worship.

  23. The man says:


    You missed my point. I agree that he was not doing mathematics at his earlier levels post ’70. What I’m disagreeing is whether it should be called his “downfall”. For example, if some mathematician stops doing mathematics to address political issues or protest nuclear armament etc., that’s his priority (which some would call a very nobel one). Calling it a “downfall” is 1. very demeaning to the person, and 2. betrays your own priorities. If Atiyah’s priorities are not with urgent political issues, there is no reason to demean others whose are.

  24. woit says:

    The man,

    I don’t think that Atiyah was specifically calling the decision to stop doing math and do other things Grothendieck’s “downfall” or criticizing his priorities. He was referring explicitly to Grothendieck’s “uncompromising nature”. If you read accounts by friends and allies who were very sympathetic to his political activities, I think you’ll find that many of them feel that the way he went about such activities was unproductive or even counter-productive, exactly because of this “uncompromising nature”.

    I found “Recoltes et Semailles” a fascinating document to read, but at the same time a sad one. Grothendieck’s “uncompromising” personality unfortunately evolved in later years to something that could be described as paranoid and delusional, and I think this is actually what Atiyah had in mind by referring to a “downfall”.

  25. From the second of the AMS Notices articles on Alexander Grothendieck.

    One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right,take 57.”

  26. shameless ass-kisser says:

    The man,

    I think you at least partly misinterpret Atiyah’s use of `downfall’. I understood that Atiyah was speaking about Grothendieck’s mathematics as much as anything else – the next paragraph he speaks explicitly of Grothendieck’s `weaknesses’. I warrant Atiyah feels he is as good a mathematician as Grothendieck, and in a position to point out weaknesses as well as strengths of the man’s approach to mathematics; moreover, there is a tendency (cultlike?) to idealize the math and the man, and Atiyah may be gently (I think he speaks gently) suggesting that neither is healthy. He later refers obliquely to the failure of EGA; this can only be taken as a judgment by a very serious mathematician that EGA did not attain its own goals. Atiyah also calles Grothendieck a `phenomenon’ and refers to his courage and concentration. Nowhere in these paragraphs does he refer to Grothendieck’s lifestyle or politics, except to say that Grothendieck’s failures as a mathematican, and his failures as a man, were both rooted in what Atiyah calls an uncompromising nature. And we should remember that Grothendieck was, at least, a terrible father.

  27. Carl Brannen says:


    I may not like your blog, but I loved your book and couldn’t put it down. I recommended it to my father who bought it quite some time ago, but it sat around on his shelf. He has a PhD in math. His comment in an email today:

    Started on “not even wrong”. I like the perspective
    it gives.

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