Vladimir Arnold is known, among other things, for offering a scathing critique of Bourbaki:

The Arnold – Serre debate

Recently I've been reading some Nietzsche, and he chides some Germans in the wake of the Franco-Prussian War for triumphantly proclaiming that the English and French might have civilization, but we have culture.

Inspired by reading this, I'm wondering if there are any critiques by 20th century German mathematicians of Bourbaki out there?


This question has an open bounty worth +50 reputation from user126532 ending in 2 days.

The question is widely applicable to a large audience. A detailed canonical answer is required to address all the concerns.

  • 27
    $\begingroup$ How are Nietzsche's complicated feelings/opinions about the Europe of his time relevant to Bourbaki, or to 20th century German mathematicians? $\endgroup$ – Yemon Choi Aug 10 at 19:50
  • 5
    $\begingroup$ Carl Ludwig Siegel was known to be a vehement adversary of Bourbaki, though I‘m not aware of written accounts (except Grauert‘s funny note at degruyter.com/downloadpdf/j/dmvm.1995.3.issue-1/dmvm-1995-0122/…) and there certainly does not exist an elaborated philosophical criticism from his part. $\endgroup$ – ThiKu Aug 10 at 20:25
  • 6
    $\begingroup$ Armand Borel reminisced: "I was rather put off by the very dry style [of Bourbaki], without any concession to the reader, the apparent striving for the utmost generality, the inflexible system of internal references and the total absence of outside ones. For many, this style of exposition represented an alarming tendency in mathematics, towards generality for its own sake, away from specific problems. Among those critics was H. Weyl, whose opinion I knew indirectly through his old friend and former colleague M. Plancherel." $\endgroup$ – Carlo Beenakker Aug 10 at 20:49
  • 4
    $\begingroup$ Here is GoogleTranslate on the relevant part of ThiKu’s link: He [Siegel] preferred every constructive proof to others, perhaps much simpler arguments. He also considered all developments in mathematics destructive if they had not been forced by a constructive continuation. The worst enemy was Bourbaki. When I introduced myself in Göttingen in 1958, I expressed (as a precaution) some criticism of Bourbaki. Years later, Siegel once walked through the Göttinger Wald with Frau Reidemeister. After a long silence he said quite unexpectedly: "And Grauert is a Bourbakist after all.” $\endgroup$ – Matt F. Aug 10 at 22:16
  • 4
    $\begingroup$ Vladimir Arnnold was an extremely sharp mathematician but his war on abstractness was grossly irrational. $\endgroup$ – Wlod AA Aug 13 at 6:17

This seems to be too long for a comment.

Siegel's emotional letter to Mordell concerning Lang's book "Diophantine geometry" compares Bourbaki style mathematicians (Siegel does not say "Bourbaki") with pig in a garden and even with national socialists:

"Thank you for the copy of your review of Lang's book. When I first saw this book, about a year ago, I was disgusted with the way in which my own contributions to the subject had been disfigured and made unintelligible. My feeling is very well expressed when you mention Rip van Winkle!

The whole style of the author contradicts the sense for simplicity and honesty which we admire in the works of the masters in number theory — Lagrange, Gauss, or on a smaller scale, Hardy, Landau. Just now Lang has published another book on algebraic numbers which, in my opinion, is still worse than the former one. I see a pig broken into a beautiful garden and rooting up all flowers and trees.

Unfortunately there are many "fellow-travelers" who have already disgraced a large part of algebra and function theory; however, until now, number theory had not been touched. These people remind me of the impudent behaviour of the national socialists who sang: "Wir werden weiter marschieren, bis alles in Scherben zerfällt!"$^\ast$

I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction — as I call it: theory of the empty set — cannot be blocked up. Let us hope that your review may be helpful..."

The comments of Serge Lang were published in AMS Notices (42:3, 1995), see here from p.339

$^\ast)$ "We will continue to march, until everything shatters", from this popular Nazi song.

  • 1
    $\begingroup$ Translates as: until everything shatters. Wild (2006, p. 377): “The lines of the refrain are still cause for the occasional debate, since the version adapted by the Hitler Youth (...) “wir werden weiter marschieren, bis alles in Scherben fällt (...)” is a variant of the original refrain “wir werden weiter marschieren, wenn alles in Scherben fällt (...)” $\endgroup$ – Francois Ziegler Aug 13 at 11:23
  • $\begingroup$ @FrancoisZiegler thank you, fixed. $\endgroup$ – Fedor Petrov Aug 13 at 14:26
  • 1
    $\begingroup$ Siegel published similar thoughts himself in e.g. (1969, p. 304) (still doesn’t explicitly target Bourbaki). $\endgroup$ – Francois Ziegler Aug 13 at 22:33

It seems to me that German-(speaking) mathematics at the advent of Bourbaki was primed to be, by and large, comfortable with trends towards axiomatization, abstraction, and structure theory in the footsteps of Hilbert and with the successes of abstract algebra in the Goettingen school around Emmy Noether and van der Waerden. See Alten et al "4000 Jahre Algebra" Springer, 2003 (in German).

There was some engagement around the question of reforming pre-university eduation in the late 60s and 70s. In his 1965 book "Mathematik als Bildungsgrundlage" (Mathematics as foundation of education"), Meschkowski dedicates a chapter titled "Bourbaki in der Schule?" (Bourbaki in schools?) to the question of ideas of Bourbaki (here primarily the advocacy of Dieudonne) making it into highschool education. While there is some critique (particularly the idea of replacing Euclidean geometry with vector spaces), it is nuanced and overall quite positive.

Carl Ludwig Siegel has been mentioned. As an addition to what has been said, I'd warn against reading his rejection of abstraction in algebra as a reaction against French mathematics. Siegel's 1959 letter to Weil give more context:

It is entirely clear to me what circumstances have led to the inexorable decline of mathematics from a very high level, within about 100 years, to its present nadir. The evil began with the ideas of Riemann, Dedekind and Cantor, through which the well-grounded spirit of Euler, Lagrange and Gauss was slowly eroded. Next the textbooks in the style of Hasse, Schreier, and van der Waerden, had further a detrimental effect upon the next generation of scholars. And finally the works of Bourbaki here provided the last fatal shove.

Notice how the mentioned drivers of Siegel's perceived "decline", namely Hasse, Schreier and van der Waerden as well as Riemann, Dedekind and Cantor are all German or German-speaking. (See Grauert's "Wie Gauß die alte Göttinger Mathematik schuf" in "Proceedings of the 2nd Gauss Symposium, 1993". Relevant quote is also cited in these more accessible German and English translated sources.) Grauert incidentally, citing this very letter to Weil, advocates for a nuanced view of Siegel in the letter to the editor linked by ThiKu. I encourage reading the letter (which isn't that long) in full.


Another scathing critic of Bourbaki can be found in the article "The Ignorance of Bourbaki" by the Logician A.R.D. Mathias in THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, pp.4-13. So it is not a German-French argument, but nevertheless a harsh criticism of Bourbaki's stance on Logic and Meta-Mathematics.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.