They definitely don't visualize 24 dimensional spheres. When I did my PhD in pure math I found that gradually I just became comfortable working without any visual or spatial intuition and instead relying on the (algebraic and topological) machinery that had been put in place before me. Terence Tao had a nice essay [1] talking about how the final stage in becoming a professional mathematician is developing the intuition to know what is likely true or not in these very abstract spaces.

I also never used a computer for anything other than latex.

[1] https://terrytao.wordpress.com/career-advice/theres-more-to-...