As an ex-pure maths researcher, I agree with everything, except the humble bit, which doesn't quite fit in that phrasing.
I would add: you become a Platoist emotionally, because mathematical research really, really feels like discovery, not creation.
As someone who knows a bit of advanced mathematics, I was impressed by how closely this chimed with my experience.
Hmm, I recognize myself a lot in this, and other people tend to say I'm good at math, but I don't consider myself very knowledgeable about math at all... Anything I should take into consideration before I conclude I'm just suffering from some variation of impostor syndrome?
I think impostor syndrome is a good bet for you, at least by comparison with me, since I only see about 3 of these propositions in myself.
In spite of getting A's up through Calculus II in high school, I stopped taking math after that (except for a couple of applied math subjects like number theory and statistics) because I had reached the point where math problems were starting to (literally, not figuratively) give me a headache when I tried to "hold them in my mind".
I am curious if other people on Lesswrong ever experienced the "this literally hurts my head" barrier at any point in math, and if so when.
"this literally hurts my head"? Hit that many, many, many times (most recent example: today). Got past most of them, though; I think I only failed a couple of times on any math I really focused on. Sometimes took weeks, though - not pleasant weeks.
The head bashing ones are those where you can mentally plot a line from premises to the conclusions you want, modulo a few holes to fill in, and where the holes keep on getting bigger and bigger, but always look fixable.
More like making furniture, realising there's little pieces to add, making the little pieces, realising there's more little pieces to add, making them... and this does happen over and over, and since you can do this in your head, you never get to rest.
Anyway, that's sometimes my experience :-)
Trying to memorize a phone number gives me a headache, but studying mathematics doesn't. I don't think this is a native ability (not entirely), but something you pick up with experience.
The analogy between learning math and "holding something in your mind" might be what Anon_User was trying to criticize with this:
Your intuitive thinking about a problem is productive and usefully structured, wasting little time on being aimlessly puzzled. For example, when answering a question about a high-dimensional space (e.g., whether a certain kind of rotation of a five-dimensional object has a "fixed point" which does not move during the rotation), you do not spend much time straining to visualize those things that do not have obvious analogues in two and three dimensions. (Violating this principle is a huge source of frustration for beginning maths students who don't know that they shouldn't be straining to visualize things for which they don't seem to have the visualizing machinery.)
Yes... not just with math, but with a wide range of problems... but only during the couple of months after my stroke when I was recovering from brain damage.
Does intuition play an important role in the field of mathematics? The essay seems to suggest that mathematicians use their intuition a great deal. Terence Tao seems to agree that it is important:
...“fuzzier” or “intuitive” thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as “non-rigorous”. All too often, one ends up discarding one’s initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one’s mathematical education.
The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems;
Intuition is vital. Theorems can take paragraphs and proofs can go for pages; without intuition, the combinatorics would annihilate you. Interestingly, I'm starting to develop new intuitions (in logic, rather than my old field, differential geometry) which means I might soonbe able to do some work in the field.
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