I have done differential topology and geometry for a time in University. There is a lot of visualisation going on in every lecture every day, even if it is not

I have done differential topology and geometry for a time in University. There is a lot of visualisation going on in every lecture every day, even if it is not computer visualization. Everything in difftop is drawn at some point, just so you can get an understanding of what's happening. In fact, I would say that topology and subfields are formalisations of these beautiful and silly visual ideas. One problem, however is that visualizations are by necessity simplifications, even in pure 2d cases like this one. That's not a sphere, that's a tessellation of one. Everything on a screen is differential and "smooth" (or actually, everything is discrete and thus not differentiable), but some things are symbols for "not differentiable here". Even these break down in pathological cases, singularities and such. This doesn't make the visualisation any less useful, but IMHO it can in most cases not be "the" reasoning, only a guide to reasoning. Besides,as the article points out, there are many non-constructive proofs that don't have any visuals. Have a look at how vsauce explains Banach-Tarsky [1] or any of 3blue1brown's excellent videos to see the edges of what's possible with visualisations. [1] https://youtube.com/watch?v=s86-Z-CbaHA https://youtube.com/watch?v=s86-Z-CbaHA [2] https://youtube.com/watch?v=zjMuIxRvygQ https://youtube.com/watch?v=zjMuIxRvygQ