Am I inhibiting my progress by essentially training myself to rely so heavily on 2D and 3D illustrations, at my current level?

Question

My math education/experience is at a low enough level that I can construct 2D or 3D figures which illustrate the vast majority of equations I encounter. Such visualizations tremendously help me, whenever I work on associated problems (especially for checking the reasonableness of attempted solutions).

However, I wish to attain much greater math proficiency in the future. Thus, do you think I'm inhibiting my progress by essentially training myself to rely so heavily on 2D and 3D illustrations, at my current level? 2D and 3D visualizations will surely be irrelevant to the more advanced math I seriously want to study and use, someday.

Thank you.

Answer 1

No, you're not inhibiting. But education is about exploring other learning styles, not just exploiting your pet styles. (This is contrary to the conventional wisdom that says teachers should figure out each student's personal learning style and then cater to it.) My opinion is that you should use your 2D and 3D figures as needed. The only real danger is that you slowly build up an unconscious bias against any math that doesn't fit into your figures. Remain flexible.

Answer 2

It's pretty much impossible to directly visualize something in more than $3$ dimensions. Nevertheless, visualizations can be incredibly useful even in infinite dimensions. Without being too literal about it, you can let one dimension conceptually stand in for infinitely many.