I would reiterate that simply because there are problems with naive visualization, we shouldn't discredit visual thinking. There are several key elements to ef

I would reiterate that simply because there are problems with naive visualization, we shouldn't discredit visual thinking. There are several key elements to effective visual thinking. The primary importance is to keep it grounded in proofs and theorems, so you know exactly what are your limitations. Often you can use a geometric argument on top of a few theorems and you get a very strong result intuitively, and then use this intuition with a tiny amount of algebra to prove it (which might take you forever to arrive from a purely algebraic perspective). Another key is that there are several ways of visualizing things. You can almost always transform a problem into an equivalent one that is easy to visualize (just need a little bit of care with the transformation, etc). --- For example, you can show functions form a vector space, visualizing some interesting algebraic properties about them, even if it constitutes an infinite-dimensional space. You can show several operators (such as d/dx) are linear, you can give it a norm, internal product, etc. This trick lets you use visual tools (and linear algebra tools) with arbitrary functions. You can visualize projection of a function into a subspace, or into some non-canonical basis -- yielding useful applications -- such as Fourier analysis. Fourier analysis itself is a fertile ground for visual thinking. You'll be finding trivial arguments for seemingly difficult decisions such as "Does this linear system have a bounded output for any bounded input?". There isn't one right way of thinking about anything. --- On the other hand, it can't be stressed enough the importance of keeping track of formal assumptions, axioms, definitions, theorems to construct valid, correct proofs. That way you minimize the risk of fooling yourself, and can safely use your intuition. This 3B1B video exemplifies many of those elements: https://www.youtube.com/watch?v=zwAD6dRSVyI&t=633s https://www.youtube.com/watch?v=zwAD6dRSVyI&t=633s