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.
I have mixed feelings about Christianity losing its grip over the Western world. It’s not hard to see the reasons why it did. Christianity was forced upon the world: disagreeing with the Church could get you killed. And this is obviously doubly bad in the cases where the dictums being forced upon the world are arbitrary.
Using force on people rarely produces good results, and so I suspect that the compulsory Christianity was not terribly good at helping people to become more enlightened. Similar to how people go through twelve years of school and then forget it all, whereas if they had voluntarily sought teaching out of a pure passion for learning that wouldn’t happen.
I think that the illusion of people believing in Christianity probably exceeded the reality. That is, people professed belief in the various dictums of the Church, without believing them in their hearts. No doubt there was genuine belief also. But the more force you apply to people, the more people are going to start faking it.
So in that sense, I’m glad that Christianity has lost its grip over the Western world. But there’s another side to the issue.
I’ve pointed out, in the past, that it isn’t necessary to hold any particular propositional beliefs in order to be enlightened. And it isn’t necessary to be part of an organized religion in order to be enlightened.
It isn’t necessary, but it’s helpful. What it boils down to, I think, is that it’s difficult to value things other people don’t value, think things other people don’t think, and do things other people don’t do. There’s this incredible pressure to fall in line.
So if you’re living in a situation with no organized religion, it’s harder to be enlightened. This isn’t any remarkable fact. It would be hard to be a mathematician if universities had no mathematics departments; it would be hard to laugh in a society that was always serious; it would be hard to keep a heavy metal band going in a society where all music was mellow and relaxed.
This problem shows up for lots of things besides religion. Every marginalized activity becomes harder to do. In our society these marginalized activities include art, music, poetry, and being a woman.
So here’s the issue with having an atheist society. All of a sudden pursuing God is hard. And that’s a problem, because the religious need is at the heart of all human needs.
When we don’t have a broad consensus on what the meaning of life is, it creates a sort of a moral void, an existential ennui, where we’re just randomly floating through life with no direction or purpose. (Nietzsche, incidentally, predicted this, with his declaration that God was dead.) That’s what it’s like to live in an atheist society.
So do I think we should reinstitute compulsory Christianity? No. I think we need to find our way to a new societal religion — whatever the word “religion” might mean in this case. One that turns the clock forward rather than turning it back. One that isn’t a forced consensus, but something resembling a genuine consensus. One that integrates not only what we learned from Christianity, but what we learned from science, feminism, secularism, the New Age movement, existentialism, music, drug culture, modern art, Islam, Buddhism, Hinduism, Taoism, Judaism, occultism, martial arts, science fiction, capitalism, communism, activism, environmentalism, TV and movies, the invention of birth control, the Internet, abstract algebra, the dating scene, and on and on and on. One that is a cornerstone of our lives which resonates beautifully with the whole, rather than being awkwardly tacked on every Sunday morning.
We are one of few societies to have ever existed which did not have a religion. Some would tout this as a great innovation. I suggest that it is actually a form of poverty.