You're wrong. To take a real example, visualize G_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }. I dare you. That's Einstein's field equations for General Relativity. I don't visualize a system of 2nd order partial differential equations involving many dimensions at each point in a 4-dimensional coordinate system. Do you?

Furthermore people differ on how visual they are. If you're someone who has to visualize things to understand them, you're hardly alone. Lots of people are like that. But then again there are people like me whose thinking is almost entirely non-visual. If I'm dealing with something abstract, not only do I not think visually, but a purely visual explanation doesn't really help me much.

There was a story on HN about activity in the "visual" system of blind people. Here is a similar story google spit out: https://news.ycombinator.com/item?id=14720225

Commutative diagrams are a development that has applications in algebra, too. I'm currently reading Physics, Topology, Logic and Computation: A Rosetta Stone (J. C. Baez, M. Stay) https://news.ycombinator.com/item?id=12317525

Sure but then you are giving an advantage to the more visual people no? Can't I just flip your argument?

Yes, symbols are useful but you should have an understand of what they are actually doing.

Sure, you can flip the argument. You'll be wrong, but you just did it.

People naturally lean towards understanding using different methods. Each mode of thinking has strengths and weaknesses, and so do people. Some things are better understood visually. Other things not. Some things have the best way of understanding varying by person.

So yes, on some things you'll have an advantage over me because you're visual. On some things I'll have an advantage over you because I probably understand complex abstract relations more naturally than you do. And your insistence that everyone is best off trying to think like you do is completely misguided.

Anyways I've explained my position enough. If you don't wish to get it, you won't. I'll let other people take over the task.

This breaks down in many cases. What if your equation involves complex numbers for each component? What if it's an eight dimensional space or perhaps a multi-hundred dimensional space for recommender system?

At some point, you have to be comfortable with more abstract representations.

As for multidimensional spaces, you might not be able to visualize all the dimensions at once but by doing them three dimensions at a time, you can go Pretty far.

You should check that the abstract representations still make sense intuitively.