This is my first post! I'm really not good at math, and I'm trying to re-learn on Khan Academy. As I'm progressing, this question came up for me.
Learning math concepts that I can visualize in my mind or relate to some observable phenomenon in life are easy. For example, 2+2=4 is easy for me to understand. I can "see" in my mind two marbles in one cup, and then two more marbles being placed in that cup, and then seeing there are four marbles in the cup. I can even do this in real life, and not only imagine them, but literally see them, and I could even feel them with my fingers.
Using concrete, physical materials to teach abstract math concepts is the foundation of several early childhood instructional approaches, especially Montessori's (see here - https://www.montessoriprintshop.com/materialized-abstractions.html). Also, on this blog post - https://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/ - if I understand the author correctly, in the first paragraph he states that in history, numbers were not always used in some forms of math, but rather shapes.
Yet, there always comes a point where I - and many other students I know - can no longer visualize, or give concrete representations of, the math we are learning. This is when math becomes difficult for me to understand. Complex statistical equations or engineering calculations baffle me. Without observable phenomenon, or even metaphors, to relate to the complex, abstract math, I feel lost. The only way forward that I have found is to exercise total faith in the theorems upon which my current math learning is being built. This ever-growing foundation upon which the more abstract math is built has to be a sure one, because I can no longer rely on my imagination or intuition to guide me. Only strong theorems can serve as the next layer of foundation to take me up further and further into the heights of more abstract math, more theorems.
My questions are:
Is it actually possible to visualize, or physically represent, any mathematical concept, no matter how abstract, and I'm just weak-minded?
Are there mathematicians, perhaps certain geniuses, who can visualize in their minds extremely abstract mathematical concepts? Or, at some point, do we all have to simply trust the foundational theorems as we progress, leaving visualization behind?