> 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
The Arnold – Serre debate - MathOverflow – https://mathoverflow.net/questions/153604/the-arnold-serre-d...
Vladimir Arnold - Wikipedia – https://en.m.wikipedia.org/wiki/Vladimir_Arnold
A english translation of this book is available here
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.
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.
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.
There is basically two solutions: reduce the quality of the material, or teach the teachers. I tend to prefer the second.
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 ;)
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?
> Is it still going on in French schools today? Or was it 60's/70's only?
Unless you are in a reaally elitist high school in Paris with a really old teacher, this should'nt happen nowadays.
The first thing I learned in the 1st year of schools were basic concepts of set theory. We were drawing circles, ellipses and Venn diagrams (even though we didn't call them like that) filled with images of apples, plums and cherries. Teachers explained to us what an intersection, union and set differences are and we were supposed to draw items into one set, but not in another set, etc.
I recall these exercises as funny and playful. They were similar to IQ tests in the sense that the exercise is logical, slightly entertaining, but highly abstract and loosely related to the world you know.
And I think this was the main issue. The 2nd topic we learned was simple arithmetics as in standard educational systems. However, at that time, I didn't see any relationship with the concepts of set theory.
Was the system any good? Hard to say. AFAIK, it was dropped after a few years. Eventually, I obtained a PhD in computer science, so at least, the system wasn't a complete disaster for me. :-)
Perhaps it made sense in ye olde days when many students would not go on to junior high.
- There was an American school about a lifetime ago that tried that strategy, and the principal claimed it worked fine.
- A general impression that before mass education it wasn't uncommon for people who got schooling to start years older than our start, with no impression that they learned arithmetic any worse. E.g. http://www.scientiasocialis.lt/pec/files/pdf/vol57/90-101.Pi... "In general the pupils began their arithmetical instruction at 10–11 and this education prosecuted for two years."
- Piaget's picture of stages of development (in my vague understanding) suggests that arithmetic beyond a very concrete level would be developmentally unnatural for younger kids, and more natural later. Apparently Montessori schools do better on this score?
- Unschoolers sometimes reach adulthood with less understanding of math than state-schoolers, but if the average is worse, I haven't heard of it. Anecdotally they're fine.
- Hate and ignorance of math is very widespread (I've read similar claims about average French people with their substantially different school system)
- This jibes with my general experience, having gone to school, etc.
- There's not much reason to expect a claim this far from mainstream to have been carefully studied. Maybe it has been and refuted -- I just don't very much respect the status quo and so I expect there are improvements that would 'easy' except for the obstacle that it's very hard to meaningfully change the system. And this strikes me as a plausible (though unambitious) one.
In my country I have been still learning from old schoobooks while classes below had new schoolbooks - thats why I remember that well.
Recently I have been helping 11 year old with classes (remote learning now). And my impression is that this top-down approach is still present but the schoolbook was full of practical life examples (money issues, understanding newspaper articles with pecentages and percent points etc.)
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.
I don't really know how to phrase all that, but discarding the CS/axiomatic side of maths while praising the intuitive physics-inspired one is not the right approach imo.
I also think it’s going a little far to say “remove the beauty” - but it is true that if you don’t already have an appreciation for the specific subject, that beauty is awfully hard to find.
For some reason I often find papers and thesis I read in French way more interesting, well thought and presented than most things I read in English. I have a few hypotheses on why I get this impression:
1) the publish or perish culture originated from the Anglo-Saxon world. Until recently the research written in the national language was kinda shielded from it. In Japan for instance, where the higher education is modeled upon the US one, master students are expected to publish at least a conference paper. In France students generally don’t publish anything.
2) the PhD thesis are written differently. Where I’m studying it’s basically slapping three papers together with an introduction and a conclusion. This sometimes leads to awkward thesis and shallow work. Thesis (and HDR) written in Europe are more like a very well structured monograph.
3) the way to use the language is different as well. In French, any intellectual written work will use long and complex sentences (sometimes to a fault) that are cramming few ideas and their relationships. In English the style is to write short sentences, with at most one idea each. I sometimes feel I have to dumb down my writing and splitting sentences while in my native language a Proust-like sentence would be more appropriate.
The much simpler explanation is that there are way more English as a foreign language PhD students than there are French as a foreign language PhD students.
I went to a top CS research school and there were several students from China that had such a high language barrier that their papers had to be professionally rewritten by a service the university offered. Their research was Amazing but motivating the problem, describing the methodology, etc all in English just devastated the signal.
So yeah, you’re going to see good quality French writing because French is no longer the lingua Franca of science so a relatively tiny minority of non-French researchers are going to use it.
https://johncarlosbaez.wordpress.com/2020/04/13/bigness-part...
I have learned from both Bourbaki and Stewart and to each his own. I enjoyed Bourbaki’s Topology a lot, and their Real Analysis is quite good.