What was Vladimir Arnold’s opinion of Grothendieck and his work?

Answer by John Baez:

Vladimir Arnold expressed many negative opinions of French mathematics education, and of the influence of Bourbaki.  I'm not aware of him saying anything about Grothendieck.  Here are some things he said:

To the question "what is 2 + 3" a French  primary school pupil replied: "3 + 2, since addition is commutative". He did not know what the  sum was equal to and could not even understand what he was asked about!

Another French pupil (quite rational, in my opinion) defined mathematics as follows: "there is a square, but that still has to be proved".

Judging by my teaching experience in France, the university students' idea of mathematics (even of those taught mathematics at the École Normale  Supérieure – I feel sorry most of all for these obviously intelligent but deformed kids) is as poor as that of this pupil.

For example, these students have never seen a paraboloid and a question on the form of the surface given by the equation [math]xy = z^2[/math] puts the  mathematicians studying at ENS into a stupor. Drawing a curve given by parametric equations (like [math]x = t^3 – 3t, y = t^4 – 2t^2[/math]) on a plane is a totally impossible problem for students (and, probably, even for  most French professors of mathematics).

Beginning with l'Hospital's first textbook on calculus ("calculus for  understanding of curved lines") and roughly until Goursat's textbook, the  ability to solve such problems was considered to be (along with the knowledge of the times table) a necessary part of the craft of every  mathematician.

Mentally challenged zealots of "abstract mathematics" threw all the geometry (through which connection with physics and reality most often takes place in mathematics) out of teaching. Calculus textbooks by Goursat, Hermite, Picard were recently dumped by the student library of the Universities Paris 6 and 7 (Jussieu) as obsolete and, therefore, harmful (they were only rescued by my intervention).

ENS students who have sat through courses on differential and algebraic  geometry (read by respected mathematicians) turned out be acquainted neither with the Riemann surface of an elliptic curve [math]y^2 = x^3 + ax + b[/math] nor, in fact, with the topological classification of surfaces  (not even mentioning  elliptic integrals of first kind and the group  property of an elliptic curve, that  is, the Euler-Abel addition theorem). They were only taught Hodge  structures and Jacobi varieties!

How could this happen in France, which gave the world Lagrange and Laplace, Cauchy and Poincaré, Leray and Thom? It seems to me that a reasonable explanation was given by I.G. Petrovskii,  who taught me in 1966: genuine mathematicians do not gang up, but the weak need gangs in order to survive. They can unite on various grounds (it could be super-abstractness, anti-Semitism or  "applied and industrial" problems), but the essence is always a solution of the social problem – survival in conditions of more literate  surroundings.

You'll note that in this quote he respects Rene Thom, as well he should, though he was probably quite critical of the hype surrounding catastrophe theory.

What was Vladimir Arnold's opinion of Grothendieck and his work?


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