Archive for June, 2012
Things I have found to be helpful for doing math:
2. Don’t guess and check. This can take subtle forms. E.g., “I have no real idea how to prove this theorem, but I think this method might work, even though I haven’t really thought it through; but I’ll try it, since I don’t have any other ideas.”
3. Start by understanding the question. If you really understand the question, usually the answer will be obvious. So you could just spend 95% of your thinking time trying to understand the question, and then the time you spend looking for an answer approaches zero, because it will just come to you.
4. If you have a hard question, try rephrasing it. Find a theorem that’s equivalent to the theorem you’re trying to prove, or find a structure that’s equivalent to the structure you’re studying. Hell, invent a new field of math if you need to (viz. Evariste Galois).
5. Math is about showing that things are the same when they obviously aren’t. Deep similarities behind obvious distinctness. 3 * 7 is the same as 21, a circle is homeomorphic to a square, addition and multiplication both form abelian groups. Usually when I solve a hard problem, 90% of the solution consists of ekeing out some deep similarity that seemingly has little to do with the hard problem, and then the remaining 10% is an easy solution to the hard problem that employs the deep similarity.
My eventual graduate thesis will be on the inconsistency of mathematics, and its implications. Topics will include:
- A proof of the inconsistency of all theories of mathematics in first-order logic.
- The equivalence of truth and provability, via Tarski’s truth schema.
- The equivalence of all first-order theories of mathematics, via the truth schema.
- The consequences of naïve set theory:
- Properties of the set of all sets, including the combinatoric indescribability of its cardinality.
- Infinitely deep sets, and a proof of the continuum hypothesis.
- The existence of various large cardinals.
- The existence of indefinable sets.
- Whatever I figure out about the “singularity point” of mathematics: the location of the border between consistency and inconsistency in the hierarchy of increasingly strong theories.
- My philosophy of mathematics, including mathematical nondualism: the view that every statement is ultimately true and false. As well as the view that formal proof does not solely dictate what propositions we are to accept.
Lately I’ve been spending inordinate amounts of time doing math. From sun-up to sun-down it sometimes seems, learning and understanding and deriving. Why am I compelled to do this? I ask myself that a lot. It takes a more explicit form: why am I doing that instead of meditating?
I think the answer is that math is easy and rewarding. Not as rewarding as meditation, but quite a bit easier. Math easy? Esoteric graduate-level math, easy? Compared to meditation, yes. Meditation makes abstract algebra and topology seem like child’s play.
With the qualifier, of course, “for me.” I don’t think there’s some absolute scale of task-difficulty, where meditation is six levels higher than abstract algebra. I’m sure there are lots of people more enlightened than me, but less intellectually adroit, for whom abstract algebra would be quite a lot harder than meditation.
But still, for me abstract algebra is child’s play compared to meditation. And I don’t think this is just a fact about me. I feel comfortable saying that, in some general sense, meditation is immensely harder than math. It’s not just that I happen to find math really easy. Sure, I’m better at math than most people. But I’m also better at meditation than most people. It’s gotta balance out somewhat.
What makes math easier than meditation? No doubt a complex question, but I think the biggest factor is this: meditation is lonelier.
Let’s think about math. When I do math, nobody will tell me I’m wrong. (Except when I occasionally am. But that doesn’t upset me; they’re just bringing me back to the truth.) There are people I know who will actually talk about it with me, and we can actually understand each other, and actually agree with each other. I can ask them questions, and they’re delighted to teach me. I can teach them things, and they’ll learn something. And they’re impressed as hell with me because I’m so damn good at math.
I can even make a god damn career out of talking to people about math! I can get paid money to do this thing I love! There’s a shortage of me! I’m in huge demand!
Compare to meditation. It’s “weird” to be a mystic. It’s not socially accepted. I can’t share my experiences with anybody. Nobody will recognize my achievements. Nobody will appreciate my work. Nobody will even *know* what I achieved. I can have the most earth shattering mystical experience ever, and nobody will ever see that. Nobody will ever pat me on the back. And you can bet your ass that nobody will pay me to do it! Being a mystic is a lonely, lonely, lonely experience.
And I think that’s the reason I spent three hours doing math today, instead of spending three hours meditating. And this aching empty void is still sitting inside me because I didn’t spend enough time with God today.