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.
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).
This doesn’t appear to actually be true. :-/ Say we take our probability space to be [0,1], and we take A=[0,2/3], C=[1/3,1], and B=[0,1]. Then d(A,B)=d(B,C)=1/3, so d(A,B)+d(B,C)=2/3, but d(A,C)=3/4>2/3. Any ideas on how to fix?
(Also strictly speaking it would be a pseudometric on the set of positive probability events, with two events being equivalent if they differ by a set of probability 0, but that’s nitpicking.)
Eeeesh. You’re right. In my defense, I think I checked the properties while I was still half-asleep, and I must have fudged the triangle inequality. I fiddled with it a bit, but couldn’t find any obvious way to make it work. Thanks for your correction.
Happens to the best of us. However, it is worth emphasising that you have provided little evidence with your writing that the actual ideas coming from peak experiences are worth much. You have provided a great deal of indication that the motivational aspect of these ideas is useful, though.
However, it is worth emphasising that you have provided little evidence with your writing that the actual ideas coming from peak experiences are worth much. You have provided a great deal of indication that the motivational aspect of these ideas is useful, though.
You may be right. I will have to think about this. A lot of the imperative ideas (“Go do this!”) that I’ve had while manic have had decidedly positive results—notably my bike trip to Georgia and the decision to devote a lot more of my time and mental energy to mathematics, founding the communal house I currently live in, but I’m going to have to try and remember some concrete examples of declarative ideas that have come to me in that state before I continue to make that claim.
I was thinking about how to calculate a metric on a probability space.
One thing that makes sense is Arccos( P(A|B) P(B|A)) . This is the metric you get if you view events as vectors in a Hilbert space and look at the angle between the two vectors, angle, of course, being a metric. It generalizes to the space of random variables in general, which is where I first discovered it. There you get Arccos ( E(XY)^2/ E(X^2) E(Y^2) )
Just on probability events, I think one thing that also makes sense is—Log (P(A|A or B)P(B|A or B)). This should be a metric and should have geodesics in the space of events. The geodesic between A and B passes through (A or B). But I don’t have as clear an argument as to why this works.
So your idea isn’t actually that far from correct, if you look at my angle idea.
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.
This doesn’t appear to actually be true. :-/ Say we take our probability space to be [0,1], and we take A=[0,2/3], C=[1/3,1], and B=[0,1]. Then d(A,B)=d(B,C)=1/3, so d(A,B)+d(B,C)=2/3, but d(A,C)=3/4>2/3. Any ideas on how to fix?
(Also strictly speaking it would be a pseudometric on the set of positive probability events, with two events being equivalent if they differ by a set of probability 0, but that’s nitpicking.)
Eeeesh. You’re right. In my defense, I think I checked the properties while I was still half-asleep, and I must have fudged the triangle inequality. I fiddled with it a bit, but couldn’t find any obvious way to make it work. Thanks for your correction.
Happens to the best of us. However, it is worth emphasising that you have provided little evidence with your writing that the actual ideas coming from peak experiences are worth much. You have provided a great deal of indication that the motivational aspect of these ideas is useful, though.
You may be right. I will have to think about this. A lot of the imperative ideas (“Go do this!”) that I’ve had while manic have had decidedly positive results—notably my bike trip to Georgia and the decision to devote a lot more of my time and mental energy to mathematics, founding the communal house I currently live in, but I’m going to have to try and remember some concrete examples of declarative ideas that have come to me in that state before I continue to make that claim.
BTW, I must say I would love to hear about the founding of this communal house, even if this isn’t necessarily the place for it.
I was thinking about how to calculate a metric on a probability space.
One thing that makes sense is Arccos( P(A|B) P(B|A)) . This is the metric you get if you view events as vectors in a Hilbert space and look at the angle between the two vectors, angle, of course, being a metric. It generalizes to the space of random variables in general, which is where I first discovered it. There you get Arccos ( E(XY)^2/ E(X^2) E(Y^2) )
Just on probability events, I think one thing that also makes sense is—Log (P(A|A or B)P(B|A or B)). This should be a metric and should have geodesics in the space of events. The geodesic between A and B passes through (A or B). But I don’t have as clear an argument as to why this works.
So your idea isn’t actually that far from correct, if you look at my angle idea.