The world is your body.

Our starvation is a physical manifestation of our inner hunger. Our infections are manifestations of our inner disease. Our wars are manifestations of our divisions within ourselves.

It is for this reason that people in America are depressed. When you remove the physical manifestations of suffering, the suffering does not go away; its spiritual nature simply becomes more obvious. Then we call it depression.

If you want to end world hunger, learn to feel full and happy. If you want to cure cancer, learn to be in harmony with yourself. If you want to fight pollution, learn to think pure and beautiful thoughts.

You can’t remove other people’s pain. But people can’t learn to remove their own pain if they don’t have an example to follow.

That poor man that you refused to help? That suffering animal that you neglected and let die? You are him. You are it. The world is your body; end your suffering and you end all suffering. Save yourself and you save the world.

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On self-justifying epistemologies

Mathematical logic can basically be characterized as the search for an epistemology. We want to design some formal system which lets us prove all true facts: so then we can decide what to believe just based on what the formal system says. And of course the system needs to prove itself to be correct; because otherwise, how would we know that the system is correct?

That’s the basic, obvious, naive hope of mathematical logic. Basically, we want an epistemology: a formal, mathematically rigorous epistemology that can prove all true facts, including the true fact that it is the correct epistemology.

Now there’s already some magical thinking inherent in this idea. How do we know the epistemology is correct? Why, because the epistemology says so!

So that’s a circular argument; but what’s the alternative? How could we possibly know that our epistemology was correct, other than by using our epistemology?

There’s that problem, and it shows up even before Gödel’s theorem. But then Gödel’s theorem tells us that self-validating epistemologies aren’t even logically possible. It’s not just that if you make a self-validating epistemology, its proof of its own correctness doesn’t mean anything because it’s a circular argument. No, it’s worse: no coherent epistemology even contains a proof of its own coherence.

And we’re not even talking about the epistemology proving itself to be true. No, we’re talking about a much weaker condition: the epistemology proving itself to be consistent, i.e., not contradicting itself. An epistemology can’t even prove itself consistent — at least not if it really is consistent.

Oh, and then things get even worse. Along comes my inconsistency theorem, which says that all of our existing epistemologies do in fact contradict themselves. Damn!

So then we try to come up with an epistemology that doesn’t contradict itself. I came up with one recently. At any rate, my inconsistency theorem doesn’t work in this epistemology; and I can argue pretty forcefully that it is consistent.

Basically, the reason I can say that is this. In this theory, whether or not a statement is true is determined by the outcome of a certain computer algorithm. The computer algorithm always halts, and always outputs exactly one of true or false, no matter what statement you put in. So unless the world magically splits in two and the computer program says both yes and no, despite being programmed to say only one of those, the theory is clearly consistent.

But how do we prove it to be consistent? In math we don’t accept those sorts of vague verbal arguments, especially not for something as important as the internal coherence of the very foundation of our belief system. We want a formal proof, not a verbal argument.

Well, interestingly, Gödel’s theorem still applies to this theory, and it can’t prove itself consistent. But even if it could, what would that mean?

Logic is absolutely filled with consistency proofs. Take any well-known formal system, and you can find half a dozen different consistency proofs for it. But not only are all such consistency proofs circular and therefore meaningless — they’re wrong! Every well-known formal system is inconsistent. We can still prove them consistent in half a dozen different ways.

I hope that demonstrates the utter futility and fatuity of consistency proofs. We can prove any system to be consistent — whether or not it actually is! We can prove any system, consistent or inconsistent, to be consistent. Given that fact, proving that a system is consistent basically forms no argument for it actually being consistent.

So let’s return to the beginning. We want to adhere to an epistemology. But we need to justify that our epistemology is correct. If we do that using our epistemology, then we’re arguing in a circle. If we do that by stepping outside our epistemology, then we’ve just violated our epistemology. (Every epistemology is a dictum of the form, “thou shalt believe all and only the things determined by these rules.”) But then things get worse because we provably can’t construct a self-validating epistemology, most of our epistemologies contradict themselves, and every argument to the effect that any epistemology is coherent is demonstrably meaningless.

Long story short, skepticism wins. We can’t achieve an ironclad belief system, where everything is justified to the point of absolute certainty. It’s just obvious that we can’t do that with anything empirical; but now formal reasoning has taught us that we can’t even do it in the rarefied realm of pure math.

When you penetrate to the very heart of reason, when you gaze upon its hidden starry essence, you find not a perfect and indestructible core of truth, but a bug in the Matrix, a crack in the wall, a strange sort of Zen joke about epistemology. And you go home having learned that you can’t know anything for certain.

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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 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.

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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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