[soVeryTired]

I think Bourbaki are partly to blame for this state of affairs. Vladimir Arnold had a lot to say on the subject: he was a big fan of keeping the intuition and motivation for development in the subject clear.

At times he went a bit overboard, but he makes some valid points in this lecture: https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

[goldenkey]

I don't know if I can agree with that. I bought every book Serge Lang produced. He was a member of Bourbaki. His Linear Algebra book is far better at providing intuition and motivations than the rest. Yet it is also rigorous. Lang is #1 for Mathematics books in my world.

[soVeryTired]

Fair - I don't doubt that there are some great educators in the Bourbaki school. But as Arnold mentions in his lecture I do think mathematics lost something when it embraced formalism so fully.

For me, the definition "A group is a set of transformations on an object such that..." is so much more enlightening than "A group is a set G together with a binary operation * such that..."

Only yesterday I was trying to learn about differential forms. Most of the notes I found online introduce the wedge product in a deeply unhelpful way, by listing some axioms that it satisfies and deducing results from the axioms. It takes hours of work to understand why those specific axioms were chosen. For me - and maybe I'm wrong here - that's Bourbaki's formalist approach in a nutshell.

It was only when I found Terry Tao's notes [0] and Dan Piponi's notes [1] that I could actually see the use of differential forms. It's an unfortunate state of affairs for the discipline that in order to learn about X, you have to google "X intuition", since it's not given to you as a matter of course.

[0] https://terrytao.wordpress.com/2007/12/25/pcm-article-differ... [1] https://github.com/dpiponi/forms/raw/master/forms.pdf

[goldenkey]

I totally agree. A lot of times you have to search for intuition rather than it being provided. Rote memorization is useless because mathematical research requires the intuition to give your mind the right direction to go in.

[dboreham]

Got to love a piece that begins "Mathematics is a part of physics" !

[currymj]

i realize the entire speech is a troll, but it successfully made me very angry. i'm not a physicist, and have no interest in physics, yet I do need mathematics.

so it's always been irritating to me how much introductory or "applied" texts bias themselves towards physics and engineering. like how a lot of multivariate calculus/introductory real analysis texts pretty much limit themselves exclusively to three dimensions and teach notation to match.

it even plagues more advanced books; i'm struggling through Villani's text on optimal transport and nearly every example is from physics even though the theory was developed for economics and has extremely broad applications across many disciplines. half of the "motivating examples" i can't even make sense of what they're talking about.

as much as I struggle with the hyper-abstract bourbaki-style stuff, at least I have a hope of understanding it without needing to dip into an irrelevant discipline where i have no background or historical context.

[mbag]

As a former physicist it also made me angry that they used geometry, when teaching us about multi-dimensional algebra. Professor would tell us that vectors are orthogonal to each other, but not telling us what it actually means is, that changing one of them doesn't affect other. I mean, yes, in hindsight, this information was there, however, when you are just starting to learn something, you have a problem filtering, what is really important and what not.

And for example, a real revelation with regard to infinity came, when I read (or heard somewhere, it was long time ago) that infinity can sometimes be few millimeters or even micrometers. Up until then I always imagined some really large number, but at that point I realized, you need to put problem before you need to put problem before you into perspective.