Archive for June, 2012

Math Tips

Things I have found to be helpful for doing math:

1. Meditate.

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.

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The Academician

The academician does not see the truth.
He does not open his ears to the warbling sound of his inner teacher,
who speaks of the brightest of noons and the darkest of sunsets,
the highest of foothills and the lowest of mountains.

He does not set his gaze on the soft, caressing glow of truth,
but fades out that glorious sunset
with his Kodak monochrome image capturing software,
the tool of those great census bureau workers of the universe, reason.

The academician speaks the truth about all things outside,
but he omits the truth about himself,
and in this omission all of his words are reduced to so much dust,
so many files and records of tiny bits of data.
In his quest to know everything he finally knew nothing,
and in this miserable not-knowing he might finally learn to hear himself.
Let us hope!

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Thesis

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 na├»ve 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.

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Correspondences

Consider the three-concept archetype system sensation/cognition/emotion. Sensation refers to the body and physicality; cognition to the intellect; and emotion to the feelings, in a more mental sense, as distinct from bodily sensations.

Now consider the two-concept archetype system love/wisdom. We can see a relationship between love and feelings, and between wisdom and cognition. With respect to the love/wisdom duality, we could consider sensation a neutralizing force, in that sensation can be wise (concentrated thought consists of visualization), but sensation can also be loving.

Is cognition equal to wisdom? I’m inclined to say no, because this conception leaves out (as I myself so often leave out!) the idea that wisdom can also be intuitive, as opposed to intellectual. We could better model the situation by turning our three-concept system into a four-concept system, sensation/cognition/emotion/intuition. (This becomes the basis for Jung’s system of personality typology.)

With our new system, we can map cognition and intuition onto wisdom, and sensation and emotion onto love. This removes some of the strain from having sensation as neutral between love and wisdom, because sensation feels to me more loving than wise. And this further illustrates the flexible, flowing nature of these correspondences. The ideas are not rigid and fixed like the ideas of science.

Now let’s throw the chakras into the mix. We can easily say that the rays orange and green are loving, whereas the rays yellow and blue are wise. The correspondences play out, in that yellow and blue both correlate nicely with cognition, and orange and green correlate nicely with emotion.

What I’ve been trying to illustrate is that it seems like all of the archetype systems map onto each other. They don’t always overlap perfectly, but they always seem to be mapping the same territory. Each system puts a slightly different angle on things, puts a slightly different emphasis, and probably each system leaves some things out. But it seems like each of them is expressing the same basic set of qualities.

Now let’s make a math analogy. If I have a polynomial function, I can describe it as an equation, as a graph, as a set of roots, etc. In each case I give the same information, but in a slightly different way, which may be good for slightly different things.

A closer analogy. I’ve been searching for an ideal metamathematical theory: a mathematical theory which best expresses the nature of mathematics itself. The main candidates are category theory, model theory/universal algebra, and set theory. Each of these theories looks very similar to all of the others, and you can map them onto each other, though the pieces don’t quite fit perfectly. They seem tantalizingly easy to unify, so that all of a sudden we’d have just one theory instead of three, but you can’t actually do that.

What a perfect analogy to the archetype systems! They’re almost the same, all of them, but not quite. The differences turn out to be just important enough that you can’t blur them out. So you’re left with a whole pocketful of archetype systems, rather than just one. And which lens you pick up and use at a given moment is a matter of expedience.

In math, sometimes what is most convenient is a category theoretic analysis; sometimes a set theoretic analysis; sometimes a model theoretic analysis. Similarly, in spiritual work, sometimes you require a chakral analysis; sometimes a Tarotic analysis; sometimes a Qabalistic analysis. It’s whichever tool most neatly fits the problem at hand.

But all the same, all these systems are describing one Self, and after a while the descriptions should sound about the same. The Self is love and wisdom. The Self is male and female. The Self is thought, feeling, sensation, and intuition. Aren’t I repeating myself? I hope it seems that way!

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Why I do math, II

Earlier tonight I was walking back from a meditation session, and I concluded that math was nothing but a giant power trip, something people use to control other people and the natural world. So in my mind I renounced math, with that little nagging voice saying that there was no way I was done with it.

Later this evening I found myself reading about sheaves, and ejaculating wild screams of ecstasy as if I were a woman in orgasm. What a deep conundrum this is! What other human could possibly empathize with my soul’s dilemma?

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Proof vs. Listening

Most of the time, we don’t have the option of formally proving our ideas. It’s really only in math and science that we can do this. But most intellectual problems in the real world don’t fit that paradigm. Most real-world intellectual problems are decision problems. You have a group of people who are trying to decide something (a workplace, a family, a country, a group of friends, a romantic partnership). They need to share ideas with each other, and eventually they need to agree on something. But they need to do this without the use of formal proof, because almost always, you can’t prove that your idea for what to do is the best one.

The situation is the same in philosophy and religion. We don’t have the luxury of formal proof in these disciplines, but we still need to think about things together and eventually agree about something.

We want to hold philosophical and religious beliefs, but it’s sort of meaningless to do so unless other people agree with us. I’ve tried being a lone believer who has his own perspective on reality that nobody else shares. But I can’t shake this feeling that I’ve regressed into solipsism when I do that. If my beliefs are unproven and held only by me, by what metric are they tested? How do I know that they’re right? How could I possibly distinguish between a private true belief, and a private delusion?

“Proof” can take many forms. There’s formal proof, which is the best kind. But you can also know something just through having experienced it. It is in this sense which I know, for instance, that my happiness is mostly a function of my inner state, rather than a function of what’s going on in my life. I’ve ascertained that experientially, though I can’t prove it to somebody else.

And a further kind of proof, I think, comes from a lot of people adopting a belief. We’ve never had any formal proof that democracy is a good idea; but the fact that a lot of people believe in it constitutes a strong argument for it. Similarly, the very fact that most people believe in God constitutes an argument in favor of God’s existence.

Now, if I have an idea, and I share it with other people, two things can happen. Other people can think the idea is good and adopt it; which boosts my confidence that the idea is right. Or other people can say, “hmm, I don’t think that’s such a good idea;” which lowers my confidence that the idea is right.

People act as a sounding board for each other’s ideas. You put an idea out there, and if it echoes throughout the social matrix, resounding again and again, then it’s a good idea. If it dies soon after it leaves its maker, then it’s not a good idea; so the theory goes.

The problem is that our social matrix doesn’t seem to be a very efficient sounding board. I’ve probably written up hundreds of philosophical ideas, but the number of these ideas that gained traction among my peers is close to zero. I spent years making music, but nobody really listened to my music.

Am I to conclude, from these facts, that I’m a bad philosopher and a bad musician? That seems false. More like, people aren’t listening to me. But I can’t just blame other people: maybe I’m not listening to other people either. We’re not listening to each other.

I think this is what stymies philosophy and metaphysics. To do philosophy, you really need the sounding board effect, because it’s basically all you’ve got. The social sounding board needs to be sensitive and of high quality. People need to be able to give you honest and sympathetic feedback based on genuine listening to your idea.

We don’t have that for philosophy. Even philosophy departments aren’t that. If you tell your idea to a philosophy professor, they’ll probably say it’s wrong. Everybody thinks everybody else is wrong in philosophy departments. That doesn’t get you anywhere; being told you’re always wrong teaches you nothing.

I think science has flourished over philosophy, religion, art, music, etc., in large part because science has the advantage of formal proof. With science, you can have ideas getting widespread traction, without people, y’know, actually having to listen deeply and sympathetically to each other. We don’t know how to listen, so we like science.

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Why I do math

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.

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