[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.