(By the way, it is not at all clear to me what Euclid’s intended audience was for *his* “Elements” - did he really intend them as what we would now call a textbook?)

It was only 10 or 20 years later that some people started to use Bourbaki style throughout mathematics teaching, and I agree that it was, on the whole, an unhealthy development: pure theory, and almost a certain pride in shunning examples, pictures, applications, etc.

So the Bourbaki treatise and Bourbakist teaching style are two different things and should not be confused (though I admit that the members of the Bourbaki group have not always been as vocal about the confusion as they should have been).

Many people will agree that Bourbaki will not be the place to start learning Mathematics so it will never be appropriate for many starting to learn math.
The Bourbakians may have thought otherwise since they name their work the Elements like in Euclid’s elements. So maybe as simple as that. Theorem - Proof approach of many Math books and from the General to the Particular.

Many students will quickly loose touch of the importance of the topic without the use of some good examples if they are not able to come up with some non trivial example or even trivial ones.

One question many students always come up is what is this thing used for?
Why is it important? if they do dare to ask it then it means you have not shown them sufficient examples or sufficiently interesting examples of why this particular result is a theorem or something you like to show as some important result.

Naturally not every one might agree with the above statements.

A mathematical material that is written in a way that is accessible to many more people will be probably read by a group of people that might not read it otherwise.
The trick to mathematics is not to discourage people from learning it by making expositions obscure and accessible only to a select minority of specialist but to attract more people to it to show why we like it.

So anything that can make a mathematical exposition more understandable should be use and examples are a good way to make something easier to understand.

Where it gets irritating is it sometimes defines an important concept simultaneously with an exercise. For example: on page 89, problem 1, it defines a conjugate for the first time and asks a problem about it. The conjugate is important enough it really ought to be in the main text.

Here’s how it handles the Orbit-Stabilizer Theorem (page 140):
Definition: Stabilizer of a Point
Definition: Orbit of a Point
Example: Listing the orbit and stabilizers of a specific permutation
Example: Same on permutations of a square
Theorem: Orbit-Stabilizer Theorem
Followed by: a proof
Then: An example using the rotation group of a cube
Finally: An eample using the rotation group of a soccer ball

The book isn’t perfect (I wouldn’t recommend relying on it as a sole reference), but I wouldn’t call it a hotbed of unrigorous blashphemy either.

Do you perhaps have any advice on how much differential geometry I should learn on my own or perhaps by taking a few classes before seeking out somebody to help me learn the rest? How would you go about approaching people? (I am not sure I would have the courage to ask someone like Sir Atiyah or Chern much of anything!)

Oh, and I quite agree with you, by the way, that Sir Michael is a very hard act to follow. Offhand, the only person who seems to me possibly up to the job is Vladimir Arnol’d. And both of them, in different ways, remind me of Chern.

Gosh, I miss that guy.

(A small postscript: I wrote to the People’s Archive to suggest that they visit Stony Brook to interview Milnor, but I’ve never heard back from them.)

Harvard
MIT
NYU
Stanford
Princeton
Stony Brook