Why math?

I do math. Pretty much from sun up to sun down. A lot of people can’t relate to that. Why would a person care so much about a thing like that? Math isn’t going to save the world. The math I do — as is the case for a great many mathematicians — has no practical applications whatsoever. From the “outside view,” it looks like a lot of work that just doesn’t need to be done.

I’ve puzzled over why math might be something worth caring about that much — something worth sacrificing one’s life for. Part of the answer is, art. Math is beautiful. I have been bowled over in aesthetic ecstasy by theorems. I have reacted to proofs and definitions in exactly the same way that I react to a brilliant song or a painting. This isn’t a common experience for those not intimately involved with math; but it is reason enough to justify the funding that math departments get. To keep that experience available to humans is an important social service.

Suppose we were to end war, end poverty, end disease. What then? We would have an awful lot of time to kill, and it would be a sorry situation if we had nothing to beautiful with which to fill that time. Some arts seem to suffer these days from the artists having run out of ideas. But math is not out of ideas.

Despite the outside impression that every mathematical question has been answered, in actual fact we have more open problems than we could possibly begin to approach; and the number only increases as we learn more. I do not doubt that math will continue to keep humans fervently preoccupied until the implosion of the universe. And the indications are that it will only grow more beautiful and more fascinating.

But there are many things of which this could be said. There are many entertaining things other than math; there are many beautiful and fascinating things, other than math. Most of them are more accessible and less stressful to do. What does math uniquely offer to humanity?

I think the unique and central contribution of math is not beauty. It the fact that math is the clearest window which humans have onto the truth. Every academic discipline is a window on the truth; but all agree that math is the brightest and clearest. In math alone, truths can be established beyond all doubt, left to withstand the test of eternity without the possibility of revision or improvement. In all other sciences, truth is tentative and can be overturned by new discoveries and new ways of thinking. In math alone, everything is already perfect, and cannot be any different than it is.

Mathematicians do not disagree for long; every disagreement is quickly resolved, in the direction of the person who is actually right. A totally unknown person can, with no budget, make a mathematical discovery and show it to the top mathematicians in the area. If the result is correct, it will be accepted. There is no room for politics and social games; in math alone, the truth shines so bright that it outshines all of that.

The truths of math usually do not matter, in a worldly sense. But this is, in a sense, beside the point. Humans need truth, independent of all other concerns. We do not need truth only so that we can develop new medicines and navigate the oceans. We need truth simply to know truth and be in relation with reality, and know with assurance that we are in such a relation. Math is the only thing which can provide us with that, with the purity and fullness which it does. And that is why I do it.

A Relocation

Nowadays I am doing my writing over at eh-na.com. I am essentially retiring this blog.

By the way, we are looking for people interested in writing on rational mysticism, to contribute to eh-na.com. Let me know if you are interested in this.

Aliens? Aliens!

A few pieces of evidence regarding aliens:

1. UFO sightings have been regularly reported and carefully recorded for decades.
2. Similarly for crop circles.
3. Similarly for abduction experiences.
4. There are ancient artifacts could not have been constructed by the humans of the time. Who made them?[1]
5. The Mexican government has admitted the existence of aliens, and former US government officials have claimed that the US government knows that aliens exist and is not telling anybody.

Of the first four items, each represents a large area of research, with numerous data points and theories. In each case, some of the data points have been debunked; some of them appear very difficult to debunk, with a variety of skeptical explanations definitively ruled out; and the vast majority are ambiguous.

Is there any piece of evidence that “puts the nail in the coffin” regarding aliens? Well, aliens have never landed in the middle of Times Square on a Monday afternoon.

But what does it take to really nail down a proposition? Presumably none of these individual data points are enough by themselves to nail down aliens. But there are millions of them. You can’t say that they’re all hoaxes. Well, you can, but at that point a certain razor needs to step in.

So I ask again, what does it take to really nail down a proposition? What standard of proof do we require? A lot of ideas are widely accepted and credible, on far less evidence than what we here have on offer. Academics accept the DSM, set theory, Martin Heidegger, and so forth, on far less evidence than what we have put forth here.

Infallible proof is talked about more often than it is actually obtained. Compared to the number of ideas that have been floated since the inception of knowledge, the number of ideas that have been truly “nailed down” is miniscule. (It seems like a larger proportion of the total ideas because we keep on teaching those same few ideas over and over to every person who goes through school.) And even those few ideas aren’t nailed down in an absolute sense.

I would put aliens in an epistemic position similar to that of dark matter. We can’t see dark matter, or aliens. But we can deduce, detective-style, that it’s hard to explain how the universe holds together without dark matter, or aliens. Note, not “impossible to explain,” but “hard to explain.” Which in the case of dark matter, is enough to make us believe.

Why do we feel differently about aliens? I think it’s not because the evidence isn’t good enough. It’s not a lack of evidence that makes us reluctant; it’s something else. What?

1. The issue is important. If aliens are actually interacting with humans behind the curtains, that has huge implications for the future of humanity and our place in the universe. The more important something is, the more evidence we demand.
2. Though there is no lack of evidence, all of the evidence is ambiguous. In some ways, a huge mass of ambiguous evidence is less convincing than a small amount of unambiguous evidence. Statistically speaking, a large amount of ambiguous evidence is probably no less weighty than a small amount of unambiguous evidence; but it’s less cognitively accessible.
3. The idea is completely divergent from how we understand the world. Forget quantum physics; forget black holes and dark matter; forget evolving from apes; this is weird. A lot of people feel like aliens are somehow inconsistent with science. They aren’t; they don’t challenge materialism, reductionism, empiricism, or anything else. But there’s still this unshakeable feeling of, this is inconsistent with reality as I understand it. And it is.

[1] The “ancient aliens” argument has been criticized on the grounds that it is fallacious to infer from an unexplained phenomenon (artifacts that humans of the time could not have built) to the explanation of aliens. But it’s not so fallacious. We can safely say that things like statues and artistic landforms are made by sentient beings. If humans didn’t do it, some other sentient beings did. We know of no appropriate sentient beings, besides humans, that exist on Earth. If they’re not from Earth, they’re from somewhere else.

Subjective and Objective Truth

I am better understanding the “dark night of the soul” that mathematics has been for me.

I think that truth needs to involve an element of fantasy. That sounds like a paradox; but I think that our fantasies are true.

If I visualize something so vividly that it feels at least as real to me as physical reality, what grounds do I say that physical reality is real, and the visualization is unreal?

Well, one ground would be that other people can’t see the visualization; but I’ll address that issue later.

Try it. Visualize a red triangle. Visualize it so strongly that it becomes as real as your own hand. Then visualize it so strongly that it’s more real than your hand. It can be done.

People regularly achieve this sort of transcendence of physical reality on the wings of pure faith. Think of the ascetics who deprive and punish their bodies for the sake of a belief. Think of the feminists, who have managed to make widely accepted a truth that has no evidence other than what people’s hearts tell them. Think of the mathematicians, who study a vast paradise of nonexistent and impossible objects. Think of the fiction writers, who live in an imaginary world of their own creation.

Fantasy, the truth that comes from inside us, is subjective truth. The truth that comes from examining the outside world, is objective truth.

Subjective truth is considered by many to be nonexistent. And indeed, I cannot assert the existence of subjective truth in the same way that I can assert that 2+2=4. The reason is that subjective truth is subjectively true; whereas objective truth is objectively true. But subjective truth is not objectively true. Similarly, objective truth is not subjectively true.

Objective truth can only be determined through the methods of empiricism. For subjective truth, the method is this: whatever you wish to be true, is true.

The nature of our experience at this nexus places great emphasis on objective truth. We are, in the ordinary course of things, absorbed in the shuffle of the physical world and its necessities. And this same emphasis on objectivity is reflected in our intellectual climate.

Subjective truth is hard to find, hard to notice, hard to hold onto. But it is better and more important than objective truth.

People have different subjective truths. I can see my own visualizations; you cannot. My passions are not your passions. My ideals are not your ideals.

There is therefore great paradox in trying to share one’s subjective truth. Subjective truth is not objectively true. What is true for me, may not be true for you. For the most part, therefore, we must have our own truth and let others have theirs. The problem of sharing truths is a hard one.

I cannot wish to share my truth with another, if for them it is falsehood. I can only wish to share it with them, if for them it is truth.

Getting to Sleep

Lately I’ve been having to get up at six in the morning, for Zen practice. This is bad because I have insomnia and generally terrible sleep patterns. Here are some things I have found that help. Some of them are tips from my master; thanks, Haju!

1. Do not sleep at odd hours! Not under any circumstances!

2. Don’t do stimulating things in the evening. In my case these include: computer usage; eating sugar; intellectually demanding tasks. I try to reserve these activities for the morning or afternoon.

3. Chamomile tea.

4. Self-hypnosis works. I gather that a lot of people use elaborate techniques with visualizations, recitations, etc. to hypnotize themselves. In my case, for this task, I just invoke the feeling of comfortably dozing off, without the use of that sort of formal machinery.

Four Unreasonable Poems

1.
Hey guys, look at all this truth I brought for you!
Oh, you don’t want any of it?
…Not a word?

2.
I shouldn’t need
to dress up my intuitions in math
to make them acceptable to you.

But I’ll do it,
if it means you’ll finally listen.

3.
First I believed everything I was told.
That’s what you do when you’re a child.

As a teenager I began to know things for myself,
and they told me I was wrong.

I was epistemically torn, and beaten, and battered
and bruised until finally I believed
that whoever has the power
is the one who’s right.

I fell for that ancient ruse,
like a rock off a cliff.

I played the game.
I believed only what I could force others to believe.
Damn could I play.

And I took all of this and began calling it “truth.”

But y’know, truth sort of has a different melody
than the one we’re singin’ in school.

4.
I am sick of conspiring
in my own downfall because it’s easier
than standing up.

Consistency

I can’t help if I contradict myself sometimes. The truth is a very strange place.

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.

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.

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.