The Secret Math Society Known as Nicolas Bourbaki

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.

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.

A english translation of this book is available here

https://staff.math.su.se/shapiro/Articles/AbelNew.pdf

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.

Books that claim to appeal more to intuition or rely on visual arguments certainly help a lot with motivating ideas and establishing context, but at a certain point we need to be clear about what exactly we are talking about.

When learning a new topic in math, I personally prefer to start with a more formal, terse text. For me, the texts that focus too much on examples and motivation tend to be chattier -- I have to read an entire paragraph to understand what it is they are getting at, making more challenging the process of chunking the information into digestible bits that I can hold in mind as I shower or go for a walk (and generally less fruitful). Compare to say working through Rudin or Kolmogorov, where I can read a distilled sentence where each word is carefully chosen that I can easily recall and munch on. Part of it is that I have ADHD -- texts that are more formal and less chatty make it easier for me to focus.

That said, I do think Courant had a point with

> Mathematics presented as a closed, linearly ordered, system of truths without reference to origin and purpose has its charm and satisfies a philosophical need. But the attitude of introverted science is unsuitable for students who seek intellectual independence rather than indoctrination; disregard for applications and intuition leads to isolation and atrophy of mathematics. It seems extremely important that students and instructors should be protected from smug purism.

Basically, math doesn't exist in a vacuum, and I think part of the reason there are so many people with low emotional intelligence in math is this belief that it can. But that's another story

For bourbaki, I had trouble figuring out what to read first, there are so many separate books. I'll try to figure it out