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[LINK] What is it like to have an understanding of very advanced mathematics?
This, apparently:
You can answer many seemingly difficult questions quickly.
You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry).
You are comfortable with feeling like you have no deep understanding of the problem you are studying.
Your intuitive thinking about a problem is productive and usefully structured, wasting little time on being aimlessly puzzled.
When trying to understand a new thing, you automatically focus on very simple examples that are easy to think about, and then you leverage intuition about the examples into more impressive insights.
...the biggest misconception that non-mathematicians have about how mathematicians think is that there is some mysterious mental faculty that is used to crack a problem all at once.
You go up in abstraction, "higher and higher". The main object of study yesterday becomes just an example or a tiny part of what you are considering today.
The particularly "abstract" or "technical" parts of many other subjects seem quite accessible because they boil down to maths you already know. You generally feel confident about your ability to learn most quantitative ideas and techniques.
You move easily between multiple seemingly very different ways of representing a problem.
Spoiled by the power of your best tools, you tend to shy away from messy calculations or long, case-by-case arguments unless they are absolutely unavoidable.
You develop a strong aesthetic preference for powerful and general ideas that connect hundreds of difficult questions, as opposed to resolutions of particular puzzles.
Understanding something abstract or proving that something is true becomes a task a lot like building something.
In listening to a seminar or while reading a paper, you don't get stuck as much as you used to in youth because you are good at modularizing a conceptual space and taking certain calculations or arguments you don't understand as "black boxes" and considering their implications anyway.
You are good at generating your own questions and your own clues in thinking about some new kind of abstraction.
You are easily annoyed by imprecision in talking about the quantitative or logical.
On the other hand, you are very comfortable with intentional imprecision or "hand waving" in areas you know, because you know how to fill in the details.
You are humble about your knowledge because you are aware of how weak maths is, and you are comfortable with the fact that you can say nothing intelligent about most problems.
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Comments (19)
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.
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.
And if anyone has gotten past that barrier.
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:
"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.
When did math first get head-hurtingly difficult, and what was it about the subject matter that made it so?
My inability to solve the problem, and my inability to give up on it.
In what way was the problem more complex? I always picture mental difficulty in terms of objects-to-juggle. Was there a mental juggling threshold?
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.
Do you mean that it's a little like putting together a piece of IKEA furniture, thinking you're done according to the directions, and then noticing there are still pieces left to add, but you don't know where they go?
And this happens over and over?
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 :-)
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:
What is intuition?
Damn you, XiXiDu! I'd already squirreled away the first sentence of the second paragraph for January's rationality quotes thread. Ah well.
It still might be a good idea to post it there. Afaik, duplication in quotes threads is discouraged, but not between the quotes threads and the rest of the site.
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.
As someone who knows a bit of advanced mathematics, I was impressed by how closely this chimed with my experience.