6 ms·

Unfortunately Bourbakism polluted teaching of Mathematics in Europe c.a. 1960s. With visible effects. Except for Eastern Europe were it was opposed by Arnold.

by chewz 6y ago

Unfortunately Bourbakism polluted teaching of Mathematics in Europe c.a. 1960s. With visible effects. Except for Eastern Europe were it was opposed by Arnold. > These discoveries of connections between heterogeneous mathematical objects can be compared with the discovery of the connection between electricity and magnetism in physics or with the discovery of the similarity between the east coast of America and the west coast of Africa in geology. > The emotional significance of such discoveries for teaching is difficult to overestimate. It is they who teach us to search and find such wonderful phenomena of harmony of the Universe. > The de-geometrisation of mathematical education and the divorce from physics sever these ties. For example, not only students but also modern algebro-geometers on the whole do not know about the Jacobi fact mentioned here: an elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system. Arnold - On teaching mathematics - V.I. Arnold, On teaching mathematics – https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html The Arnold – Serre debate - MathOverflow – https://mathoverflow.net/questions/153604/the-arnold-serre-debate https://mathoverflow.net/questions/153604/the-arnold-serre-d... Vladimir Arnold - Wikipedia – https://en.m.wikipedia.org/wiki/Vladimir_Arnold https://en.m.wikipedia.org/wiki/Vladimir_Arnold

wbhart 6y ago
That's interesting. I'd seen Arnold's work on teaching mathematics, and I'm very much aware of the dry formalism in Bourbaki books. However, I was not aware of the Arnold-Serre debate. But the thing that really caught my eye here is your claim that there were visible effects of Bourbakism polluting mathematics. Do you have any examples or specifics of that "visible effect". My question is a genuine one, not a challenge to what you are saying.
- dnpp123 6y ago
  If you speak/read French, try to have a look at elementary/middle school math books from the 60's/70's in France. It was a disaster and forced a lot of kids to imagine they were bad at maths.
  - rq1 6y ago
    I had discussion about it with Cedric Villani once and he told me that was not a disaster because of the books or the material, but the teachers were not good enough. Eg. when teaching set theory, teachers were debating what should be a perfect representation of a set (is it an ellipse or circle... etc.?) Children were completely lost because of this. If you don’t draw a perfect ellipse to represent a set then you had a bad mark and so on and so forth.
    - dnpp123 6y ago
      Sure but what good is an education method if the teachers don't understand it. I don't doubt that if you had a great teacher that would work, but considering how focused French education is on mathematics, this was one of the main reason why the social ladder broke.
      - rq1 6y ago
        I understand, but on the same period, France grew a whole generation of brilliant mathematicians. I don't think that the two are completely unrelated. There is basically two solutions: reduce the quality of the material, or teach the teachers. I tend to prefer the second.
        dnpp123 6y ago
        > I understand, but on the same period, France grew a whole generation of brilliant mathematicians. I don't think that the two are completely unrelated. If you could prove me that there were more brilliant mathematicians educated in France during the 60's/70's compared to other periods, I'd be interested. Even if this is the case (which I'm not sure), at least it's pretty clear it lowered the median (whether this is good or not is another debate ;)
  - wbhart 6y ago
    But how did Bourbaki influence elementary/middle school mathematics? Bourbaki mathematics is basically at the level of graduate school. Except in countries where a graduate degree is required to teach mathematics, I can't imagine how there would even be a connection between the two worlds. Is that the case in France? Again, I'm not disputing what you are saying either, merely trying to understand what happened. Is it still going on in French schools today? Or was it 60's/70's only?
- chewz 6y ago
  The anectodal example in my country is the old (pre-WW2) math problems book that was used then and after war at middle education level (lycee). Later on the problems from the book could be solved only by university students (engineering etc.) and nowadays only by math students. This is anecdotal of course and there are still lots of bright kids eager to learn math today.
- roi 6y ago
  Benoit Mandelbrot often complained about this style of mathematics, and how its dominance in France forced him to go to the US. See for example: https://www.youtube.com/watch?v=BNyuXKOMU5k https://www.youtube.com/watch?v=BNyuXKOMU5k https://www.youtube.com/watch?v=R9k1o2RmqVc https://www.youtube.com/watch?v=R9k1o2RmqVc https://www.youtube.com/watch?v=_jRId9PNd6A https://www.youtube.com/watch?v=_jRId9PNd6A (his uncle, Szolem Mandelbrojt was a founding member of Bourbaki).
amai 6y ago
"By the way, in the 1960s I taught group theory to Moscow schoolchildren. Avoiding all the axiomatics and staying as close as possible to physics, in half a year I got to the Abel theorem on the unsolvability of a general equation of degree five in radicals (having on the way taught the pupils complex numbers, Riemann surfaces, fundamental groups and monodromy groups of algebraic functions). This course was later published by one of the audience, V. Alekseev, as the book The Abel theorem in problems." (Arnold - On teaching mathematics) A english translation of this book is available here https://staff.math.su.se/shapiro/Articles/AbelNew.pdf https://staff.math.su.se/shapiro/Articles/AbelNew.pdf
andi999 6y ago
Somehow from the link it seems Arnold didnt manage to make his point in the Arnold Serra debate, and audience thought he is ranting. i find this quite disappointing, since I think he really has a point.
Syzygies 6y ago
I hiked with Arnold once, in New Hampshire. He at one point described walking far enough out onto a frozen lake to watch the ice crack around him, to see if the pattern matched his intuition. He was very hands-on.
asdf_snar 6y ago
> For example, not only students but also modern algebro-geometers on the whole do not know about the Jacobi fact mentioned here: an elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system. This one resonated with me. At one point I had a problem that had to do with Kahler manifolds, but I knew nothing about them and only had some worked examples using basic ideas from Hamiltonian mechanics that I learned from Arnol'd's book. In hopes of resolving my issues, I spoke to some symplectic geometers and used Hamilton-Jacobi language. They vaguely knew what I was talking about but couldn't carry out any calculations. I once heard an anecdote about Arnol'd that he came to France and lambasted the French mathematicians on a similar basis -- for all their writing, they couldn't carry out "simple" calculations with a clear vision (as once can when one has a physics motivation). It might have been the anecdote was referring to the Serre-Arnold debate -- thanks for that link. Of course, Arnol'd has... high standards, to put it mildly. In his book, "Problems for children from 5 to 15", he writes in the preface, "My long experience has shown that, very frequently, dimwits falling at school behind solve them better than A-grade pupils, since – for their survival at the back of the classroom – they must permanently think more than required “for governing the whole Seville and Granada”, as Figaro used to say about himself, while A-graders cannot catch “what should be multiplied by what” in these problems. I have also noticed that five year old kids solve similar problems better than pupils spoiled by coaching, which in their turn cope with the questions better than university students used to swotting who anyway beat their professors (the worst in solving these simple problems are Nobel and Fields prize winners)." One of the problems is to sum 1/n^2 from 1 to infty (this is for children not older than 15, remember). Not only would I be unable to do this without modern technology (like Fourier analysis), I also find it amazing that Arnold writes: "prove that the sum is pi^2/6, that is, approximately 3/2", as though the approximation were harder (or perhaps more important) than finding the exact value. To me, that small comment really underlines just how hands-on he was. Edit: I remembered a joke. Why did Bourbaki stop writing textbooks? They found out Serge Lang was one person.

by chewz 6y ago

wbhart 6y ago

That's interesting. I'd seen Arnold's work on teaching mathematics, and I'm very much aware of the dry formalism in Bourbaki books. However, I was not aware of the Arnold-Serre debate. But the thing that really caught my eye here is your claim that there were visible effects of Bourbakism polluting mathematics. Do you have any examples or specifics of that "visible effect". My question is a genuine one, not a challenge to what you are saying.

dnpp123 6y ago
If you speak/read French, try to have a look at elementary/middle school math books from the 60's/70's in France. It was a disaster and forced a lot of kids to imagine they were bad at maths.
- rq1 6y ago
  I had discussion about it with Cedric Villani once and he told me that was not a disaster because of the books or the material, but the teachers were not good enough. Eg. when teaching set theory, teachers were debating what should be a perfect representation of a set (is it an ellipse or circle... etc.?) Children were completely lost because of this. If you don’t draw a perfect ellipse to represent a set then you had a bad mark and so on and so forth.
  - dnpp123 6y ago
    Sure but what good is an education method if the teachers don't understand it. I don't doubt that if you had a great teacher that would work, but considering how focused French education is on mathematics, this was one of the main reason why the social ladder broke.
    - rq1 6y ago
      I understand, but on the same period, France grew a whole generation of brilliant mathematicians. I don't think that the two are completely unrelated. There is basically two solutions: reduce the quality of the material, or teach the teachers. I tend to prefer the second.
      - dnpp123 6y ago
        > I understand, but on the same period, France grew a whole generation of brilliant mathematicians. I don't think that the two are completely unrelated. If you could prove me that there were more brilliant mathematicians educated in France during the 60's/70's compared to other periods, I'd be interested. Even if this is the case (which I'm not sure), at least it's pretty clear it lowered the median (whether this is good or not is another debate ;)
- wbhart 6y ago
  But how did Bourbaki influence elementary/middle school mathematics? Bourbaki mathematics is basically at the level of graduate school. Except in countries where a graduate degree is required to teach mathematics, I can't imagine how there would even be a connection between the two worlds. Is that the case in France? Again, I'm not disputing what you are saying either, merely trying to understand what happened. Is it still going on in French schools today? Or was it 60's/70's only?
chewz 6y ago
The anectodal example in my country is the old (pre-WW2) math problems book that was used then and after war at middle education level (lycee). Later on the problems from the book could be solved only by university students (engineering etc.) and nowadays only by math students. This is anecdotal of course and there are still lots of bright kids eager to learn math today.
roi 6y ago
Benoit Mandelbrot often complained about this style of mathematics, and how its dominance in France forced him to go to the US. See for example: https://www.youtube.com/watch?v=BNyuXKOMU5k https://www.youtube.com/watch?v=BNyuXKOMU5k https://www.youtube.com/watch?v=R9k1o2RmqVc https://www.youtube.com/watch?v=R9k1o2RmqVc https://www.youtube.com/watch?v=_jRId9PNd6A https://www.youtube.com/watch?v=_jRId9PNd6A (his uncle, Szolem Mandelbrojt was a founding member of Bourbaki).

amai 6y ago

"By the way, in the 1960s I taught group theory to Moscow schoolchildren. Avoiding all the axiomatics and staying as close as possible to physics, in half a year I got to the Abel theorem on the unsolvability of a general equation of degree five in radicals (having on the way taught the pupils complex numbers, Riemann surfaces, fundamental groups and monodromy groups of algebraic functions). This course was later published by one of the audience, V. Alekseev, as the book The Abel theorem in problems." (Arnold - On teaching mathematics) A english translation of this book is available here https://staff.math.su.se/shapiro/Articles/AbelNew.pdf https://staff.math.su.se/shapiro/Articles/AbelNew.pdf

andi999 6y ago

Somehow from the link it seems Arnold didnt manage to make his point in the Arnold Serra debate, and audience thought he is ranting. i find this quite disappointing, since I think he really has a point.

Syzygies 6y ago

I hiked with Arnold once, in New Hampshire. He at one point described walking far enough out onto a frozen lake to watch the ice crack around him, to see if the pattern matched his intuition. He was very hands-on.

asdf_snar 6y ago

> For example, not only students but also modern algebro-geometers on the whole do not know about the Jacobi fact mentioned here: an elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system. This one resonated with me. At one point I had a problem that had to do with Kahler manifolds, but I knew nothing about them and only had some worked examples using basic ideas from Hamiltonian mechanics that I learned from Arnol'd's book. In hopes of resolving my issues, I spoke to some symplectic geometers and used Hamilton-Jacobi language. They vaguely knew what I was talking about but couldn't carry out any calculations. I once heard an anecdote about Arnol'd that he came to France and lambasted the French mathematicians on a similar basis -- for all their writing, they couldn't carry out "simple" calculations with a clear vision (as once can when one has a physics motivation). It might have been the anecdote was referring to the Serre-Arnold debate -- thanks for that link. Of course, Arnol'd has... high standards, to put it mildly. In his book, "Problems for children from 5 to 15", he writes in the preface, "My long experience has shown that, very frequently, dimwits falling at school behind solve them better than A-grade pupils, since – for their survival at the back of the classroom – they must permanently think more than required “for governing the whole Seville and Granada”, as Figaro used to say about himself, while A-graders cannot catch “what should be multiplied by what” in these problems. I have also noticed that five year old kids solve similar problems better than pupils spoiled by coaching, which in their turn cope with the questions better than university students used to swotting who anyway beat their professors (the worst in solving these simple problems are Nobel and Fields prize winners)." One of the problems is to sum 1/n^2 from 1 to infty (this is for children not older than 15, remember). Not only would I be unable to do this without modern technology (like Fourier analysis), I also find it amazing that Arnold writes: "prove that the sum is pi^2/6, that is, approximately 3/2", as though the approximation were harder (or perhaps more important) than finding the exact value. To me, that small comment really underlines just how hands-on he was. Edit: I remembered a joke. Why did Bourbaki stop writing textbooks? They found out Serge Lang was one person.