[Falkens_Maze]

Completely agree. It's what Prof. Susskind states in one of his lectures on QM explaining quantum fields. Eventually pictures and mental abstractions break down, and it's better to not visualize.

At some point, one's limited biological brain will be unable to fully perceive or visualize some mathematical object. That's where intuition comes in.

[jules]

You can still reason by analogy with something you can visualise. You can't visualise 3+1 = 4 spacetime dimensions, but you can pretend that space has 2 dimensions so that you can visualise it as 2+1 dimensions. Blind algebra rarely works. I think that many have an optimistic view of blind algebra because proofs are often stated as algebra without geometric intuition, even when the author of the theorem almost certainly used geometric intuition to come up with it. This can give a false impression. I've seen people struggle tremendously with simple proofs in Hilbert spaces, when the corresponding proof in 2 dimensions is easy by drawing a picture and translating that into algebra that works just as well in the more general setting. For example, let S be a closed subspace of Hilbert space H and x in H. Prove |x-y| is minimised over y in S iff x-y is orthogonal to S.

[AstralStorm]

Analogies break down as easily as they are helpful. On average, they help mathematical intuition less than expected. (Check Bose-Einstein thought experiments and the mathematics they spawned for an example.)

QM went into blind math side way past any analogies. Yet it is currently matching observations which is as good as it gets with true.

[jules]

QM did not go into blind math mode. The way physicists usually do QM is by glossing over a lot of the subtleties of Hilbert spaces and act as if the results from finite dimensional linear algebra work. Concepts physicists have been using for a long time, such as the Dirac delta function and path integrals, only got formalised much later. Many revolutions in QM, such as Feynman diagrams, are very much based on intuition. You can transfer all of that back into formal math, as Dyson did, but Dyson could not have done so without Feynman having invented them first.

Especially in mathematics it is fashionable to write proofs in a style that completely erases the reasoning that lead to the proof. I think that's a pity. So much unwritten knowledge gets lost that way when the originators die. Differential forms, for example, have a beautiful geometric interpretation, but the way they're taught nowadays obscures that completely and makes it seem like they're a formal algebraic tool only. In fact, we're now several generations later, so even some of the instructors may be unaware of that, because their own instructors failed to transfer the original geometric intuition!