We need formalism, and we need motivation, examples, and pictures. One isn't better than the other. In particular, formalism isn't about hazing (though I think some people abuse it this way).

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

>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

Soviet books (MIR Publishers) are notoriously terse and to the point. This is what we used in college, and it has made it difficult for me to read a different style, because the content tries my patience.

For example, I couldn't consume Andrew Ng's Coursera ML course, but I appreciated the CS229 recorded at Stanford because he dove directly into the maths part. I didn't want to see slides, I needed to see problem statement, pause the video, work through gradient descent, play the video, and check I got it right. Side note: doing this [pause, working on it, play] has its advantages, it helped avoid a confusion the instructor had in the course with notation, for example, which you had to be sorted out in the later part of the course.

As you said, it is a spectrum of content with different styles for different needs and people, but even for the same person, one might need a style at a certain point and the other at another point. I remember in my third year I used some MIT OCW resources because I felt my brain was shot and I needed to be spoon fed on a topic I was so far behind on.

This is really a good question. On the one hand the work of bourbaki was important, the foundation of math was shaky (I heard there were proof of some theorems and the negative of the same theorem). But then it somehow seems to destroy the spirit of mathematics. Arnold wrote about it: https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html#:....

To add: I dont know what the solution to this is. So lets say there are two types of approaching math. Approach one is formal/rigid (caring about Axiom of Choice, Set theoretical foundations ..:-)) the other one still (partially) rigorous but more classical in spirit. A big question for me is, how is actually research done? Is it more the first or more the second approach? So what are possible solutions? I dont know. If you are trained in the first approach, you usually loose the ability to gloss over the gaps in the second approach. If you are trained in the second you cannot (or very difficult) come around the first one.

And then there is physics which uses the sloppy (and often not correct) version of the second approach. I once asked Serge Lang about this (phrasing:'what about physicist getting the right result with wrong mathematics' (thinking about renormalization)). Lang replied:'this is God's way of calculation'.

Oh wow. So it looks like that the reason why I suddenly got 'bad' at math towards the end of high school might be due to the switch to this hyper-formal way of teaching mathematics, instead of using a more geometric/physic way ?

Could be. Although high school (at least where I am) only teaches a tiny drop of mathematics (so the other style is also difficult.). But generally the is an adaptation problem/challenge in the first two semesters of math students where usually only solving a lot of problem and getting them corrected helps.

But dont worry, these things can be overcome (do not fight them at your current stage if you are still a student). Just dont believe the 'hours needed' in the description of the modules (they are politically decided). Generously work around the clock for the first few years.....

edit: more seriously: although working hard is important: go to the office hours of your TA and/or prof and discuss the work you are doing. Usually no student does that and you will learn a lot.