[anyfoo]

And you are able to manipulate complex equations without any visual tools, just by "thinking about what they are"? For example mentally multiply a term into nominator and denominator of a rational function? That's just inconceivable to me. Where's your "scratch pad" essentially. To me it works pretty much like it works on paper, only that the "paper" is in my head, and the visuality of it all (being able to "focus" on a particular part of the equation etc.) helps in keeping the problem tractable, otherwise even a relatively simple equation quickly becomes overwhelming to manipulate.

(To say nothing about the other meaning of the word "complex", i.e. complex numbers. Getting a good grasp of Fourier or Laplacians without a complex and/or s-plane in my head is fruitless. I admire anyone who just "gets it" without visual aids... real or imagined ones, because that pun was also too good to pass up).

[whatshisface]

You are able to conceive of it, you're just doing it without realizing what you're doing. Someone with a good imagination but no math knowledge at all could picture the same squiggly lines as you can, but without meaning. In your head there exists both the squiggly lines, and what they mean. All you have to do is fill in the last quadrant, which would be holding the meaning without picturing the lines. I would suggest that you might be using the meaning scratch space without using the imagination scratch space every time you think about something that can't be pictured.

You know how some people can't wink? If one eyelid was picturing an equation, and the other was interpreting it by what it meant, well, you see where the analogy is going, people with bad imaginations would be people with an eyepatch, who happen to all be perfect at winking.

[anyfoo]

And yet in some areas at least so much about the language in math itself seems to be centered about a visual understanding.

I mentioned the s-plane earlier: We talk about poles and zeros in the s-plane, because they form poles and zeroes visualized in an actual plane, a 2-dimensional plate with protrusions into the 3rd dimension. The poles pulling the plane upwards into infinity, the zeroes tacking it down to the "floor". In the z-plane, we talk about getting the spectrum of a signal or filter by tracing the unit circle, because you can imagine tracing a literal circle in the plane. We "shift" signals up, down, left, right, we "flip" spectrums, we "cross" the origin.

To know whether a z domain transfer function is stable, I don't read "the complex roots of the denominator have to have a magnitude < 1" past some initial textbook definitions, it's just "the poles have to be inside the unit circle". In the s-plane, we instead talk about the poles being on the left half.

And yet the subject matter, signals and filters, has nothing to do with "visual objects" per se (like geometry would, for example). Even if the signal is a video signal, what we are manipulating here has nothing directly to do with what the video signal shows. And the signal might be an audio signal or just some nondescript digital data to begin with.

It just amazes me that someone can grasp such a complex subject without working with its ubiquitous visualization, so pervasive that its objects were named accordingly. It's true that I can build enough intuition to say things like "okay, if I put a capacitor there I'll have another pole" and not think about the actual plane for that instant, but as soon as it's something more complex that I don't immediately "know", I have to resort to my visualization again. I probably speak out of envy, because this also means I can never fully have the feeling of "grasping" higher dimensional problems for example, especially when familiar properties that are true for n<=4 break down there. I always only feel my understanding is working with a "shadow" of what's actually happening, to use another visual metaphor.

[whatshisface]

Well, if you're looking for someone who can tell you how to do electrical engineering without being able to visualize a circle, you'll have to find someone with less imagination than me. I can do a circle, it's the "using your mind's eye as scratch paper for doing integrals" bit that's beyond my highly average abilities.

As for higher dimensions, yes, when you can barely visualize R3, R4 doesn't seem much harder, but that's hardly a superpower, it's just a pair of equal inabilities. I find it easy to accept the principles of noneuclidian geometry, but that comes with the side effect that if I was told they applied to euclidian space, that'd come "naturally" too.

[anyfoo]

* I meant Laplace Transform, not Laplacian. Different things (and I can't edit anymore).