[btilly]

But I think that if you see a complex equation, you should be able to roughly imagine it in your head. Working only with symbols won't get you there.

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.

[fnbr]

Hmmm. I would disagree with you. It is impossible to visualize the equations as that isn't possible to do in 3 dimensions, but it is possible to develop an intuition. After taking functional analysis, for instance, I began to develop an intuition for how function spaces work, and was able to visualize that to a certain extent (for a given definition of "visualize").

[adamnemecek]

You get it. Few people understand what's happening in every dimension at every point, but getting a feel is doable.

[posterboy]

The whole argument is that its not doable without the right tools and that visualizations are a very good tool for this.

[posterboy]

You might be thinking visually without even recognizing, because to call that kind of imagination visual (as in optical) is misleading. A Hydrogen 2 molecule has 6 degrees of freedom already, give that color and you have 8.

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

[adamnemecek]

I mean this particular equation not really no but that might be because I haven't done much general relativity. But I can kinda visualize other somewhat complicated equations.

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.

[btilly]

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

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.