I found an amazing Quora answer on “What is it like to have an understanding of very advanced mathematics?” in my RSS at both Boingboing.net and Kottke.org:

• You can answer many seemingly difficult questions quickly. But you are not very impressed by what can look like magic, because you know the trick. The trick is that your brain can quickly decide if question is answerable by one of a few powerful general purpose “machines” (e.g., continuity arguments, the correspondences between geometric and algebraic objects, linear algebra, ways to reduce the infinite to the finite through various forms of compactness) combined with specific facts you have learned about your area. The number of fundamental ideas and techniques that people use to solve problems is, perhaps surprisingly, pretty small — see http://www.tricki.org/tricki/map for a partial list, maintained by Timothy Gowers.
• You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It’s not so much that you can imagine the situation perfectly, but you can quickly imagine many other things that are logically connected to it.
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That’s just the first two bullet points.

It’s an amazing read.

Personal anecdote regarding math (well physics, really)

My wife’s father-in-law has a Ph.D. in Physics and is a semi-retired Nuclear Physicist. When I read this Quora answer, it made me think of him.

His sheer genius is pretty impressive, and in one instance, when my wife and her friends were getting their asses kicked by 3rd-year Quantum Mechanics, he offered to help.¹

The conversation, as I understand it, went like this:

My Wife: “Dad, we need help in Quantum Mechanics or the three of us are going to fail.”

Father-in-law: “Okay, give me a weekend to study up.”

And that was it. He retaught himself Quantum Mechanics in a weekend.

Over the next week, he taught my wife and her friends Dan and Bryan quantum.

Unbelievable.

I mean, I was pretty good at math and physics, but I could not relearn something like high school calculus in a weekend, much less teach a person how little I know about it.

Personal anecdote #2 regarding math

I’ve always been good at math and physics, but never brilliant.

One of the points in the article really resonated with me:

• 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.) Instead…

Ah. This is the exact problem I fell into when reading those popular physics books while still in university. I was pondering, over the span of weeks, on what does a blackhole look like in 3D space. Often, it is depicted as a weighted ball sitting on a stretched out blanket (2D plane), but this belies its complex nature–it actually looks like this from any angle you look at. It’s hard to visualize.

There’s a point where my understanding of mathematics and physics couldn’t help me anymore with understanding things like superstrings and general relativity. No simple metaphor or visual mental model would help anymore. Things just couldn’t be intuited because they went against common sense.

However, it did pique my future-wife’s interest in me when I sat down at her table in the student union and asked her and her physics buddies “What does a blackhole look like? No, seriously, what does it look like in 3D space?”

I guess she thought, “Why would a guy studying gym want to know about this?”²

I can’t say it worked out too badly for me in the end.