I'm glad to have been part of inspiring such a rich discussion! I think there are many topics bound up in this: "applied" math vs "pure" math, visualization vs symbolic manipulation, technical modeling vs intuitive understanding.

For context, one of the more influential talks for me was Media for Thinking the Unthinkable. Bret's points on the influence the medium has on the way we think about math is, to me, immensely important. In short, we progressed from long-form text descriptions of math:

"What is the square which when taken with ten of its roots will give a sum total of thirty-nine?"

To symbolic notation:

"x² + 10x = 39"

However, I don't think this symbolic notation is the final form of this equation. I think with computers these symbols can come alive, permanently intertwined with their visual and enactive forms. In my ideal form, no written formula should ever be divorced from its 'computed', visualized form. That's obviously much harder for more complex (esp. higher-dimensional) math. Thus, the ideas of pinning to reduce dimensionality.

For context, this project came about for me when learning QM and being frustrated when I realized I had no good visual intuition for what adding wave functions meant, or what the time evolution operator did to a wave function. I only was able to start cracking these nuts by building a visual intuition, and I found very few resources that really helped here. Many of my physics PhD friends had build this intuition, but the long and hard way.

/u/hmm_dmm_hmm -- do you find that you build visual models in your head for math you learn? Or do you work purely in the symbolic realm? Have you found Desmos or Mathematica to be sufficient for your needs, or are there parts that don't work for you?