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.