[anyfoo]

Sorry for asking the same questions as I did above, but this is very interesting to me. When you work with equations in your head, you do not work with visual representations of your equations? What about electronic circuits (if you do that) and, say, the current flowing through them, do you trace that in an actual representation (one out of almost infinitely many possible), or something more abstract? If you remember things from a textbook, do you sometimes remember where on a page you've seen it (e.g. a table, a graph, a picture, or just text) and the general shape of it, or is that impossible as well?

[v64]

Not OP, but also have visual aphantasia. Not speaking for others, but for me personally, my mind is entirely auditory. I hear my thoughts as spoken words, and if I'm thinking about something complex, it resembles a crowd of chatter where I can focus in on certain conversations while tuning out the rest.

I majored in math, and when working with equations, I will literally hear in my head things like "eff of ex equals two ex squared plus ex plus five". If I'm multiplying 36 by 7 in my head, I will hear "seven times six is forty-two, hold the two, carry the four, seven times three plus four is twenty-five, the answer is two fifty two."

If that sounds like a difficult way to mentally calculate, I'll note that I'm not a good mental calculator. :) Abstract algebra and logic are much easier for me to grasp than fields requiring more visual intuition like geometry and topology.

Remembering things from a textbook, I usually just remember the content, although there are also cases too where I'll remember I got it from the textbook with the bicycle on the cover or some detail like that, not because I visually remember the bicycle, but rather because I've textually committed that book in my mind as "the book with the bicycle on the cover". If you asked me what color the bicycle is, I won't remember because I didn't note that in my mental description.

[godshatter]

When I'm working with equations in my head, which I don't do often, I think of them as a list of terms and what's done with them. No visual representation, just a memorized list. For electronic circuits, I would probably remember what connects with what, but not how they are laid out spatially. For textbooks, I do remember sometimes that the specific text I'm think of is found on a left page near the top, but I can't see it. That's just where my eyes will scan when I look for it again. Most often I don't, though, unless I poured over it a lot when learning it.

I doubled majored in math and cs in school, and I found that the 400-level math courses were easier for me than others and I think it's because most people were trying to visualize things that were hard to visualize. For me, it was just another equation to work with.

[whatshisface]

I don't think about the visual representation of equations. I think about the equations, not in any specific representation, but as what they are. I think my mental abilities are roughly average, including my ability to picture things when reading books, but I have noticed that people with extremely good imaginations don't often have a mental slot for "equations," as they actually are, but only for images of written expressions that represent equations.

It's probably all the same in the end. After all, the only paper shortage that our world seems to be prone to is the persistent and reoccurring problem with toilet tissue. It's funny how all the different ways of doing the same thing average out in the end, but I suppose that's evolutionary inevitable - if one way of going about it was better than any other, we'd all be descendant from someone who had those genes.

[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).