What he means is that we only really have intuition for 1, 2, and 2.5D visuals, but many areas of mathematics don’t map into low dimensions very well, or do but lose essential properties in the process. Building a low dimensional projection of he problem might prime intuition, but it will also introduce fundamental biases as well.
For example, learning geography by flat map projections only. No matter what projection you use there is a trade off, and you end up instilling both the pro and the con of that trade off as intuition.
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:
The comment refers to lebesgue measure (I don't even what), but I'd intuitively and ignorantly assume we count all faces of all n-1 balls (recursively) whereas the Volumes overlap and so the total (in lebesgue ...space?) is less than the sum of it's parts (in euclidean space) - how far off am I? (will delete if too far)
It's related to the geometric problems the parent described because probability distributions roughly describe geometric regions (of high probability density) where observations are likely.
...what? I don’t understand. It’s accurate at “close zoom” because the limit as you scale in to the surface of a sphere is a flat surface. I’m not sure what compression of any sort has to do with this.
For example, learning geography by flat map projections only. No matter what projection you use there is a trade off, and you end up instilling both the pro and the con of that trade off as intuition.