### Saturday, February 12, 2005

## Flotsam and Jetsam

A current sampling of the odds and ends bubbling through my mind:

I'm thinking a major refinement of the Observer Class Hypothesis (i.e. that observers form the largest set of information, precisely because they can and do extract information from any structure, in effect becoming the power set of all information) is the necessity of constraining observers, so that they, say, can only think thoughts of a finite complexity at a given point in time. In other words a particular observer Yi can only think thought Xj if the number of bits I(Xj) it takes to express Xj is less than or equal to Yi's memory Mi: I(Xj) < Mi. Then all of these variables are promoted to functions of time Mi(t) etc. and those Yi that can increase Mi(t) the most then go on to dominate the counting - so you are most likely to be a Yi(t) such that Mi(t) ~ Exp(t), as is generally the case now. But why would the observers be constrained to finite memories? - because they are physically realized computers! - embedded, say in some general spacetime with some physics and composed of the basic mathematical building blocks described by that physics. You then have to specifically manipulate that physics in order to increase M. All of these constraints help to build actual structure as opposed to noise, you only count over the real structure. This is why we are not, for instance, completely abstract turing machines out floating unmoored in idea space and extracting information randomly - all possible rules for extracting any amount of information would thus exist, and so really there is very little information in this set up, so it is thus a very unlikely mode of existence.

I've thought of a nice analogy for this, and it starts with a question: what fraction of the integers are prime? Well, given different arrangements, they converge to any percentage you want, say 50-50: 2,4,3,6,5,8,7,9,11,10,13,12... - indeed there are uncountably many different ways to get any real ratio. This is the long way of saying that we are trying to do something silly - comparing the sizes of 2 countably infinite sets (actually this reminds me of the fun real analysis result that if you have a sequence of positive numbers that converges to zero, but whose sum diverges to infinity, then you can make it so the alternating sign sum (i.e. multiplying the k'th term in the sum by (-1)^k) converges to any real number you want, by arranging the order of the terms). But if we introduce a constraint, namely considering the fraction of integers less than or equal to N that are prime, everything changes! This constraint on the problem immediately produces a huge amount of non-trivial structure - namely, if the Riemann Hypothesis is correct, then the fraction converges to 1/ln(N) in the limit of large N. On a moment's consideration, the similarity between the two cases is startling...

Let's see, in other news I've been emailing David Chandler which is great fun. He pointed me to the beta version of maps.google.com which is a lot of fun because it is so fast - you can find your childhood home quickly, then zoom out several levels, scroll over the state and see where you went to college, zoom out a ways and follow an interstate across the country... Also, the fact that David works at google made me then realize that I don't know how it actually works, but I think I figured it out while riding the bus to campus over the last few days. Say we have a million proper nouns that the engine has cataloged, each of which perhaps occurs on average on a million webpages. OK, no problem, arrange the nouns in alphabetical order over a series of computers, with each noun linked to a rank-ordered list of all the websites that contain it, so when that noun is searched for you go directly to it's location and fetch the links to the top ranked sites. But it's much more complicated than that - you can search for two words at once - say "giraffe" and "lagrangian", which, what do you know, apparently occur together in 137 websites! Obviously you can't keep an alphabetical listing of all words pairs - we'd then be up to a trillion entries, and you still can search for 3 words at a time and so on. Simple brute force comparison of the websites in "giraffe" and "lagrangian"'s lists would also be crazy - that's a trillion comparisons to make! It seems like you need to keep the website lists themselves alphabetically arranged, and then just run the lists together through the processor, essentially like you were going to make a larger alphabetical list out of the two, and whenever a common entry occurs, spit it out into the results. But there is complication after complication - for instance you can also search for "giraffe lagrangian", i.e. those words occuring in direct succession on a website (which doesn't occur, well at least not for a few more hours presumably) - so now you have to keep at which points the word occurs in the webpage so you can compare those as well - you're going to need a lot of data tags. Luckily the problem parallelizes quite well - I wonder how many processors are used on average per query. And then you want to have evolutionary neural nets crawl over all the data and extract general ideas so that they actually understand the information and so forth...

Let's see, I also went out to dinner with Katie to a new upscale mexican restuarant last night. I drank a beer and thoroughly enjoyed the meal, the orange-yellow-blue southwestern art, and the pink noise of the chattering intelligentsia around us. I realized towards the end that my consiousness was almost completely composed of sensory input during the meal - no thoughts or ideas, or at least only those running on autopilot. I then, amusingly, formed a theory that the beer helped to revert me to this ancient form of consciousness, and expounded upon my revelation on the way to the car to Katie, not getting the meta-Magrittean joke in doing so for a few more moments... Still, interestingly, in a way I feel most 'awake' (i.e. the least amount of running on autopilot, as it were), when talking internally to myself, and fully aware that I am doing so. Which only happens now and again, maybe a couple times a week. Note to self: do this at least once a day...

Also, the air today was very clear, with the bare tree branches in sharp contrast with the faint blue cloudless sky. It's almost like being in a hard vacuum, except it was also windy, and I could breathe, among other things.

I'm thinking a major refinement of the Observer Class Hypothesis (i.e. that observers form the largest set of information, precisely because they can and do extract information from any structure, in effect becoming the power set of all information) is the necessity of constraining observers, so that they, say, can only think thoughts of a finite complexity at a given point in time. In other words a particular observer Yi can only think thought Xj if the number of bits I(Xj) it takes to express Xj is less than or equal to Yi's memory Mi: I(Xj) < Mi. Then all of these variables are promoted to functions of time Mi(t) etc. and those Yi that can increase Mi(t) the most then go on to dominate the counting - so you are most likely to be a Yi(t) such that Mi(t) ~ Exp(t), as is generally the case now. But why would the observers be constrained to finite memories? - because they are physically realized computers! - embedded, say in some general spacetime with some physics and composed of the basic mathematical building blocks described by that physics. You then have to specifically manipulate that physics in order to increase M. All of these constraints help to build actual structure as opposed to noise, you only count over the real structure. This is why we are not, for instance, completely abstract turing machines out floating unmoored in idea space and extracting information randomly - all possible rules for extracting any amount of information would thus exist, and so really there is very little information in this set up, so it is thus a very unlikely mode of existence.

I've thought of a nice analogy for this, and it starts with a question: what fraction of the integers are prime? Well, given different arrangements, they converge to any percentage you want, say 50-50: 2,4,3,6,5,8,7,9,11,10,13,12... - indeed there are uncountably many different ways to get any real ratio. This is the long way of saying that we are trying to do something silly - comparing the sizes of 2 countably infinite sets (actually this reminds me of the fun real analysis result that if you have a sequence of positive numbers that converges to zero, but whose sum diverges to infinity, then you can make it so the alternating sign sum (i.e. multiplying the k'th term in the sum by (-1)^k) converges to any real number you want, by arranging the order of the terms). But if we introduce a constraint, namely considering the fraction of integers less than or equal to N that are prime, everything changes! This constraint on the problem immediately produces a huge amount of non-trivial structure - namely, if the Riemann Hypothesis is correct, then the fraction converges to 1/ln(N) in the limit of large N. On a moment's consideration, the similarity between the two cases is startling...

Let's see, in other news I've been emailing David Chandler which is great fun. He pointed me to the beta version of maps.google.com which is a lot of fun because it is so fast - you can find your childhood home quickly, then zoom out several levels, scroll over the state and see where you went to college, zoom out a ways and follow an interstate across the country... Also, the fact that David works at google made me then realize that I don't know how it actually works, but I think I figured it out while riding the bus to campus over the last few days. Say we have a million proper nouns that the engine has cataloged, each of which perhaps occurs on average on a million webpages. OK, no problem, arrange the nouns in alphabetical order over a series of computers, with each noun linked to a rank-ordered list of all the websites that contain it, so when that noun is searched for you go directly to it's location and fetch the links to the top ranked sites. But it's much more complicated than that - you can search for two words at once - say "giraffe" and "lagrangian", which, what do you know, apparently occur together in 137 websites! Obviously you can't keep an alphabetical listing of all words pairs - we'd then be up to a trillion entries, and you still can search for 3 words at a time and so on. Simple brute force comparison of the websites in "giraffe" and "lagrangian"'s lists would also be crazy - that's a trillion comparisons to make! It seems like you need to keep the website lists themselves alphabetically arranged, and then just run the lists together through the processor, essentially like you were going to make a larger alphabetical list out of the two, and whenever a common entry occurs, spit it out into the results. But there is complication after complication - for instance you can also search for "giraffe lagrangian", i.e. those words occuring in direct succession on a website (which doesn't occur, well at least not for a few more hours presumably) - so now you have to keep at which points the word occurs in the webpage so you can compare those as well - you're going to need a lot of data tags. Luckily the problem parallelizes quite well - I wonder how many processors are used on average per query. And then you want to have evolutionary neural nets crawl over all the data and extract general ideas so that they actually understand the information and so forth...

Let's see, I also went out to dinner with Katie to a new upscale mexican restuarant last night. I drank a beer and thoroughly enjoyed the meal, the orange-yellow-blue southwestern art, and the pink noise of the chattering intelligentsia around us. I realized towards the end that my consiousness was almost completely composed of sensory input during the meal - no thoughts or ideas, or at least only those running on autopilot. I then, amusingly, formed a theory that the beer helped to revert me to this ancient form of consciousness, and expounded upon my revelation on the way to the car to Katie, not getting the meta-Magrittean joke in doing so for a few more moments... Still, interestingly, in a way I feel most 'awake' (i.e. the least amount of running on autopilot, as it were), when talking internally to myself, and fully aware that I am doing so. Which only happens now and again, maybe a couple times a week. Note to self: do this at least once a day...

Also, the air today was very clear, with the bare tree branches in sharp contrast with the faint blue cloudless sky. It's almost like being in a hard vacuum, except it was also windy, and I could breathe, among other things.

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I've been noticing the way the barren branches sharply stand out against the sky a lot lately too. Seems like we are observing similiar things. Guess we siblings are wired along some of the same lines.

Perhaps its more general than that. I've read that its hypothesized that sunsets are so striking because it was a dangerous time for our ancestors out on the savannah as all the predators went to work - i.e. watch out! Perhaps being struck by the bare branches against the sky linked with a subconscious response: hey it's f'n cold and there's not much food! Conserve your energy you fool! At least, while certainly beautiful, the sight is also slightly un-nerving in way, at least for me. It's part of what makes it so striking. How about you?

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