[Tazerenix]

Mathematicians are well aware of complaints like these about introductions to their subjects, by the way.

It is for a reason that this book introduces the theory of abstract vector spaces and linear transformations, rather than relying on the crutch of intuition from Euclidean space. If you want to become a serious mathematician (and this is a book for such people, not for people looking for a gentle introduction to linear algebra for the purposes of applications) at some point it is necessary to rip the bandaid of unabstracted thinking off and engage seriously with abstraction as a tool.

It is an important and powerful skill to be presented with an abstract definition, only loosely related to concrete structures you have seen before, and work with it. In mathematics this begins with linear algebra, and then with abstract algebra, real analysis and topology, and eventually more advanced subjects like differential geometry.

It's difficult to explain to someone whose exposure to serious mathematics is mostly on the periphery that being exposed forcefully to this kind of thinking is a critical step to be able to make great leaps forward in the future. Brilliant developments of mathematics like, for example, the realisation that "space" is an intrinsic concept and geometry may be done without reference to an ambient Euclidean space begin with learning this kind of abstract thinking. It is easy to take for granted the fruits of this abstraction now, after the hard work has already been put in by others to develop it, and think that the best way to learn it is to return back to the concrete and avoid the abstract.

[bscphil]

The point of starting with physical intuition isn't to give students a crutch to rely on, it's to give them a sense of how to develop mathematical concepts themselves. They need to understand why we introduce the language of vector spaces at all - why these axioms, rather than some other set of equally arbitrary ones.

This is often called "motivation", but motivation shouldn't be given to provide students with a reason to care about the material - rather the point is to give them an understanding of why the material is developed in the way that it is.

To give a basic example, high school students struggle with concepts like the dot and cross products, because while it's easy to define them, and manipulate symbols using them, it's hard to truly understand why we use these concepts and not some other, e.g. the vector product of individual components a_1 * b_1 + a_2 * b_2 ...

While it is a useful skill to be adroit at symbol manipulation, students also need an intuition for deciding which way to talk about an unfamiliar or new concept, and this is an area in which I've found much of mathematics (and physics) education lacking.

[chongli]

Physical intuition isn’t going to help when you’re dealing with infinite-dimensional vector spaces, abstract groups and rings, topological spaces, mathematical logic, or countless other topics you learn in mathematics.

[12345hn6789]

You're just throwing jargon at the wall to dismiss folks who want to learn

[chongli]

Not at all! I fully endorse learning. My point is that physical intuition will only get you so far in mathematics. Eventually you have to make the leap to working abstractly. At some point the band-aid has to come off!

[peanutcrisis]

Well, can you visualize vector spaces beyond 3 dimensions?

[ndriscoll]

You just visualize 2 or 3 and say "n" or "infinite" out loud. A lot of the ideas carry over with some tweaks, even in infinite dimensions. Like spectral theorems mostly say that given some assumption, you have something like SVD.

Now module theory, there's something I don't know how to visualize.

[resource0x]

>... rather than relying on the crutch of intuition from Euclidean space

Euclidean space is not a good crutch, but there are other, much more meaningful, crutches available, like (orthogonal) polynomials, Fourier series etc. Not mentioning any motivations/applications is a pedagogical mistake IMO.

I think we need some platform for creating annotated versions of math books (as a community project) - that could really help.

[Tazerenix]

On that of course I agree, but mathematicians tend to "relegate" such things to exercises. This tends to look pretty bad to enthusiasts reading books because the key examples aren't explored in detail in the main text but actually those exercises become the foundation of learning for people taking a structured course, so its a bit of a disconnect when reading a book pdf. When you study such subjects in structured courses, 80%+ of your engagement with the subject will be in the form of exercises exploring exactly the sorts of things you mentioned.

[fakecontent000]

God I just hate you so much.