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.
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!
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.