That sounds like a really excellent answer, but as usual, I don't parse things in the same way as the typical LW reader, so I didn't understand most of what you wrote. Specificity and technical language ironically cloud things for me.
Have you read Alejandro Jodorowsky's autobiography? Here's an excerpt (pg 28).
Thanks, that was an awesome read!
I'll attempt a translation. If I'm engaging with the world, then I notice new things about it, or I see things in new ways. For example: once, looking at the sky, I noticed that it was brightest near the horizon and darkest at the zenith. Suddenly I realized the reason: there was more air between me and the horizon than there was between me and the space directly above me. The scene snapped into focus, and I found I could distinctly see the atmosphere as a three-dimensional mass. If I turn my attention inward, on the other hand, I tend to draw connections between pieces of information I already have - suddenly intuiting the behaviour of quantum wave packets, for example, or drawing an analogy between my social networking behaviour and annealing. These two cases are the "dots" and the connections between the dots, respectively.
Information theory generally defines information to be about some event with an uncertain outcome: if you know a coin has been flipped, you will need additional information to determine whether it came up heads or tails. By contrast, if you already know that five coins were flipped and three came up heads, you don't need any additional information to deduce that the number of heads was prime. Anything that you could in principle figure out from the information you've already got isn't treated as new information; In this sense, mathematical truths (connections) are separated from information proper ("dots").
While this separation may be useful for theory, it doesn't capture all aspects of the way we learn and process information. For starters, we're often rather surprised to learn mathematical facts; this is because we need to use physical hardware (our brains) to compute proofs, and we don't know what the outcome of the computation will be. Also, our brains seem to treat things, states, patterns, and pieces of information all in the same way - hence, for example, we can refer to "the economy" as if it were a single thing rather than a complex system of interrelationships; or, going in the opposite direction, we can break down a tree into its component cells and, moreover, recognize each cell as a fantastically complicated system, and so on.
Meanwhile, our ability to draw links between different levels of organization allows us, in particular, to see that certain mathematical patterns are reflected the world around us. Once we've found a model that fits the system, we can make predictions we couldn't make before: the more confident I am, say, that every fifth coin flip will come up as heads, the less information will be conveyed when this does indeed happen. It goes in the other direction too: sometimes we see patterns in nature which point us toward new mathematical understanding. The example I gave was of soliton waves, which you can read about here - even if you have no technical background, I think you'll find the History section enlightening.
For all these reasons, I suspect that a better model of information might loosen the hard distinction that's made between new information and new deductions.
First of all, thank you so much for posting this. I've been contemplating composing a similar post for a while now but haven't because I did not feel like my experience was sufficiently extensive or my understanding was sufficiently deep. I eagerly anticipate future posts.
That said, I'm a bit puzzled by your framing of this domain as "arational." Rationality, at least as LW has been using the word, refers to the art of obtaining true beliefs and making good decisions, not following any particular method. Your attitude and behavior with regard to your "mystical" experiences seems far more rational than both the hasty enthusiasm and reflexive dismissal that are more common. Most of what my brain does might as well be magic to me. The suggestion that ideas spoken to you by glowing spirit animals should be evaluated in much the same way as ideas that arise in less fantastic (though often no less mysterious) ways seems quite plausible and worthy of investigation. You seem to have done a good job at keeping your eye on the ball by focusing on the usefulness of these experiences without accepting poorly thought out explanations of their origins.
It may be the case that we have the normative, mathematical description of what rationality looks like down really well, but that doesn't mean we have a good handle on how best to approximate this using a human brain. My guess is that we've only scratched the surface. Peak or "mystical" experiences, much like AI and meta-ethics, seem to be a domain in which human reasoning fails more reliably than average. Applying the techniques of X-Rationality to this domain with the assumption that all of reality can be understood and integrated into a coherent model seems like a fun and potentially lucrative endeavor.
So now, in traditional LW style, I shall begin my own contribution with a quibble and then share some related thoughts:
Many of them come from spiritual, religious or occult sources, and it can be a little tricky to tease apart the techniques from the metaphysical beliefs (the best case, perhaps, is the Buddhist system, which holds (roughly) that the unenlightened mind can't truly understand reality anyway, so you'd best just shut up and meditate).
As far as I understand it, the Buddhist claim is that the unenlightened mind fails to understand the nature of one particular aspect of reality: it's own experience of the world and relationship to it. One important goal of what is typically called "insight meditation" seems to be to cause people to really grok that the map is not the territory when it comes to the category of "self." What follows is my own, very tentative, model of "enlightenment":
By striving to dissect your momentary experience in greater and greater detail, the process by which certain experiences are labeled "self" and others "not-self" becomes apparent. It also becomes apparent that the creation of this sense of a separate self is at least partially responsible for the rejection of or "flinching away" from certain aspects of your sensory experience and that this is one of the primary causes of suffering (which seems to be something like "mental conflict"). My understanding of "enlightenment" is as the final elimination (rather than just suppression) of this tendency to "shoot the messenger." This possibility is extremely intriguing to me because it seems like it should eliminate not only suffering but what might be the single most important source of "wireheading" behaviors in humans. People who claim to have achieved it say it's about as difficult as getting an M.D. Seems worthy of investigation.
Rather than go on and on here, I think it's about time I organized my experience and research into a top-level post.
These are some interesting points. I meant "arational" in the sense that our actions are arational - rationally motivated, perhaps, but it would be incorrect to say that the action itself is either rational or irrational, hence it's arational. What intrigues me is the fact that these arational phenomena are deeply embedded in the way our minds are structured, and therefore can perhaps inform and augment the process of rationality. Indeed, some of them may be extremal states of the same systems that allow us to be rational in the first place.
I'd definitely like to see this post on Buddhism; you seem to have an excellent grasp on it.
Did your experiences lead you to believe that there are mostly unknown dots to connect to commonly known dots, or just unconventional connections between existing dots?
I have two different answers to your question: one practical, one more theoretical. On a practical level, what I gain from peak experiences depends on where my attention is. If I'm out and about, or doing something materially, then the main advantage I gain is noticing new aspects of a situation, or seeing the same aspects in a different light; I believe this is a result of greater flexibility in choosing the cognitive map I apply to the territory. These, I suppose, would be the "unknown dots": information that was present in the environment, but which my brain never bothered to record. If I'm sitting around thinking, on the other hand, then I tend to find a lot of unconventional connections. Even here, though, there is new information to be gained; I've learned a lot about my own nervous system simply by careful observation of my internal experience.
On a theoretical level, I have some trouble with the distinction between information and deduction. In the strictest sense, mathematical truths contain zero information, since they are automatically true in every possible world (or insert your own X-Rationalist translation of this claim). Yet we are still surprised to learn, for example, that e^(pi.i)+1=0 - or I know I was surprised, at the very least. I think this is a result of the fact that our evaluation of mathematical claims is based on manipulation of tangible stuff: we can check our memory to determine if we've ever seen a proof before, or we can manipulate symbolic expressions, encoded in our wetware, to attempt to prove or disprove it. Even wood pulp and graphite can be leveraged for this purpose, and so the result of this computation comes in as an observation of an uncertain outcome in the world.
Yet the no-information claim takes on new complications when we realize that our brains use the same basic processes to encode relations between facts, as it does to encode facts. This leads to kind of "flat" ontology, in which we can treat relations as facts and draw relations between them, or even between a relation and a fact. We can even draw relations between internal experience (including mathematical knowledge) and external information. Once we have recognized these connections, natural systems can actually point us toward mathematical truths (for example: we might never have learned about solitons if we had not observed them in nature). The hierarchy of "levels of organization", and the division between information and deduction from information, therefore appear to be constructed post hoc, and I'm not sure if I can cleanly distinguish between unknown dots and unknown relations.
You should definitely write at least the first post in this attempted sequence; either it will work or it won't.
Though I will advise that you lead with your strongest and most useful point - you can try writing things in an optimal educational order after that; first you have to hook your readers.
Thanks for the suggestion; I'll definitely keep that in mind as I'm writing.
This post originated in a comment I posted about a strange and unpleasant experience I had when pushing myself too hard mentally. People seemed interested in hearing about it, so I sat down to write. In the process, however, it became something rather different (and a great deal longer) than what I originally intended. The incident referred to in the above comment was a case of manic focus gone wrong; but the truth is, often in my life it's gone incredibly right. I've gotten myself into some pretty strange headspaces, but through discipline and quick thinking I have often been able to turn them to my advantage and put them to good use.
Part 1, then, lays out a sort of cognitive history, focusing on the more extreme states I've been in. Part 2 continues the narrative; this is where I began to learn to ride them out and make them work for me. Part 3 is the incident in question: where I overstepped myself and suffered the consequences.
Some of you, however, may want to skip ahead to part 4 (unless you find my autobiographical writings interesting as a case study). There, I've written a proposal for a series of posts about how to effectively use the full spectrum of somatic and cognitive states to one's advantage. I have vacillated for a long time about this, for reasons that will be discussed below, but I decided that if I was already laying this much on the line, I might as well take it a step further. Read if you will; and if you're interested, please say so.
Does LaTeX support mean using LaTeX to generate images which are "transcluded" (inlined) into the text? This is better than using Unicode's math symbols? Really? Does Math Overflow have LaTeX support?
I don't really care how it renders, I mainly just want to be able to type LaTeX code directly into comments and posts.
Add LaTeX support (I mean inline LaTeX, not this thing).
EDIT: Based on comments below, I think I misused the word "inline". What I meant was simply the ability to type LaTeX directly into comments and posts. How it gets rendered doesn't matter much to me; some legitimate objections have been raised, but I don't feel like hard math gets used enough on the site that this would get out of hand. Restricting its use to posts rather than comments might be a good compromise.
I have had a couple of experiences in which intense study of math and physics led me to some pretty dark psychological places
Why do I feel the irrational urge to beg you to do a post on this? What could possibly go wrong? :-)
Yeah, I could write about this. Look for it tomorrow (Wednesday) or Thursday evening.
Not that I disagree with you in general, but I can think of a few cases in which you may actually want to cultivate blankness toward a given subject. In particular, deep and difficult questions have been known to occasionally drive people mad - it's an occupational hazard for mathematicians in particular, and perhaps also for people in other fields. One might reasonably object that correlation does not imply causation in this case, but I have had a couple of experiences in which intense study of math and physics led me to some pretty dark psychological places, and I had to back off for awhile and think about more mundane matters while my mind reset. It's possible that, for some people, some areas of thought really are inaccessible, insomuch as they could irrevocably damage themselves in trying to get there.
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Can you post references to new mathematical (or philosophical) proofs that you have solved so we can check the assertion that such altered states are beneficial? Have the results been peer reviewed or published (not that them not being peer-reviewed or published makes them any less valid but this gives a baseline of that others have checked your work)?
Also, what was your reasoning for doubting that you exist? How was Descartes proof insufficient?
The nonexistence thing was an error of judgment. In retrospect, it originated in an unconscious assumption I was making that there must be some ground to reality, a kind of "bottom level" of which everything else is epiphenomenal. A materialist might look to quantum fields to fill that role, but when I rejected all my former beliefs, that included my belief in an external reality independent of perception. So all I was left with was thought and sensory experience, and as they were interdependently defined, rather than any one aspect taking ontological primacy, I concluded that they - and hence I - must not exist. There are any number of holes in this argument, but that's how I was thinking at the time.
Unfortunately I'm not yet at the point where I have papers published. For the most part the ideas that come to me in peak states are not specific, easily formalizable facts. In some cases they are directives to do certain things (like the bike trip mentioned in my original post); in other cases, they give a broad direction to my studies. The Tegmark vision is one example: higher category theory seems like it could furnish me with the tools to formally analyze the mathematical universe (or parts thereof) as a topological space; but since my knowledge of category theory is rather patchy, for now I'm simply working on learning some more of the prerequisites (I just finished a course in algebraic topology).
Two cases spring to mind, however, of fairly specific and well-polished ideas that have come from peak experiences. One was a metric on the space of events over a given probability space; it popped into my head as I was waking from a dream during the peak of my mania. If you're interested: for events A and B, we can define d(A,B)=1-P(A|B)P(B|A). You can check that it satisfies the properties of a metric [EDIT: This doesn't actually work, as Sniffnoy pointed out below]; couldn't say for sure whether it's useful for anything, since I got swine flu shortly after that, at which point it got shelved with the rest of the stuff I'd been working on. A more promising example happened quite recently: it was a game theoretic analysis of the relationship between government and citizen which, as I mentioned, might end up as another post here - probably in the discussion area. If you'd be interested to see it, that'd be all the more reason to write it.