You’re missing the fact that mathematicians develop an intuition for mathematical concepts, and that this is different from the intuition they have when they begin studying. When teaching real analysis, I have to focus on all kinds of counterexamples to things that seem intuitively true, or else people will be stuck on blind alleys when they’re actually trying to prove things. (For example, most of the mathematical community was shocked at the time when Weierstrass produced a function that is differentiable nowhere, and now the exposure to such functions is an essential part of an analysis curriculum at the high undergraduate level.)
So in particular, it is still a bad idea to use untrained intuition on very abstract problems. Instead, we have to spend years training our mathematical intuitions as we would train an extra sense.
You’re missing the fact that mathematicians develop an intuition for mathematical concepts...
I am aware that humans are not born with an innate intuition about Hilbert spaces. What I was talking about is that intuition is that which allows us humans to recognize structural similarity and transfer that understanding across levels of abstraction to reach a level of generality that allows us to shed light on big chunks of unexplored territory at once.
I (unsupportedly) suspect that something like “recognizing structural similarity” — or rather, a mechanism for finding the structures something has, after which structural similarity is just a graph algorithm — is the foundation for human abstract thought.
You’re missing the fact that mathematicians develop an intuition for mathematical concepts, and that this is different from the intuition they have when they begin studying. When teaching real analysis, I have to focus on all kinds of counterexamples to things that seem intuitively true, or else people will be stuck on blind alleys when they’re actually trying to prove things. (For example, most of the mathematical community was shocked at the time when Weierstrass produced a function that is differentiable nowhere, and now the exposure to such functions is an essential part of an analysis curriculum at the high undergraduate level.)
So in particular, it is still a bad idea to use untrained intuition on very abstract problems. Instead, we have to spend years training our mathematical intuitions as we would train an extra sense.
I am aware that humans are not born with an innate intuition about Hilbert spaces. What I was talking about is that intuition is that which allows us humans to recognize structural similarity and transfer that understanding across levels of abstraction to reach a level of generality that allows us to shed light on big chunks of unexplored territory at once.
I (unsupportedly) suspect that something like “recognizing structural similarity” — or rather, a mechanism for finding the structures something has, after which structural similarity is just a graph algorithm — is the foundation for human abstract thought.