Stat-meta chat on plastic with Marduk-kur.

For the first entry on statistical metaphysics, here's a conversation I had with Marduk-kur over on plastic:

Hey, Marduk, why don't you try and take my testable theory on? It's approximately infinitely more interesting than these old debates. I'm calling it Statistical Metaphysics right now, although I'm having second thoughts on the about using the word metaphysics — as an atheist physicist I think it has a clear connotation of the mathematical foundations of physics (and thus reality), but the more I read about it on the web, the more I see it associated with all sorts of supernatural flimflam. Anyways, not important. Here's a new rough draft of the paper I'm working on, and I'm also presenting it this Saturday the 7th at Transvision 2004 in Toronto. Here's a quick review of the theory:

First, it is possible to create a complete mathematical model of the universe (perhaps something like string theory...). Then, since we observers are just permutations of atoms within our universe and give it no special status, one argues that all mathematical structures exist — call this collection the ensemble. This then helps to explain the apparent 'fine tuning' of our universe as to allow for life — i.e. it isn't tuned and there probably isn't any life in much of the rest of the ensemble. Still this begs the (non-intuitive and nonobvious) question of why we find ourselves to be observers at all. This question is put in sharper contrast when we ask what it is to be a conscious being. To point, all of one's conscious experiences are the patterns of firing neurons within the brain, i.e. essentially a giant, abstract connected graph (or an abstract structure of objects and their relationships -ASORs- as I like to call them). Indeed everything else in the ensemble can also be placed in this connected graph/ASOR representation. So why is it that we find ourselves to be in the 'evolving-observer' subset of ASORs within the ensemble? Inspired by the success of statistical mechanics in explaining which types of macroscopic structures exist within the universal wave function (i.e. all permutations of atoms exist, but some classes of permutations are extremely more common than others, like those that follow the 2nd law...), I argue simply that observers form the largest subset in the ensemble — there are fundamentally more permutations of observers than any other type of information. For this to be true it requires that observers can grow to be infinitely complex in the limit of infinite time, which in turn requires that there is no absolute finite upper limit to the power of computers that we can build, which itself requires that there is no final theory of physics (instead each physical theory turns out to be only a local approximation of a more complex theory...). If this is true, then observers essentially become the power set of the ensemble, extracting up to an infinite number of unique pieces of information from their environments. This in turn means that the infinite number of permutations of these infinitely complex observers is of a higher cardinality than the number of permutations of all other ASORs in the ensemble, which is why what it is to exist is to be one of these observers evolving in time.

marduk-kur:

Isn't there an implication inherent in this model that the universe will contain an approaching-infinitely complex observer and thus using the same argument that requires us observers to actually be observers we should in fact be this super-observer? It seems that you have to place the Time element of the model in some special category to avoid that conclusion. Am I mistaken?

me:

In the simplest model you have observer histories which are an infinite time series of ever more complex — yet finite — mental snapshots. So there is no one highest finite complexity level where you should expect to find yourself. Instead you should always expect to find yourself growing yet more complex... and it certainly seems like we will be able to continue the 100,000 year old exponential memetic growth for quite a while yet.

Let me come at it from a different angle — and yeah, time is a special overarching relationship. I think time is necessary in the formation of well-defined ASORs. OK, what am I talking about? — consider just the example of the information in pi. The are many different representations of the information in pi: you can have just the circumference of a circle divided by the diameter, there's the infinite decimal expansion: 3.141592654..., there are series expansions: 4(1-1/3+1/5-1/7+1/9-...). So there are finitely complex implicit representations and infinitely complex explicit representations. It would be much nicer to just count among the finite implicit representations — each one's ASOR can then be encoded into a simple finite integer (sort of like Godel's numbers). Working in complete, compact, implicit representations also gets rid of the 'noise' ASORs, i.e. the randomly connected graphs, which have very little representation invariant information (otherwise they would dominate the counting in the ensemble).

So we are coming around to the special status of time... We want to build ASORs which, unlike pi, or the Mandelbrot set or so on, are irreducibly infinitely complex, even in the most compact implicit representation. They will then dominate the counting. You might at first just try constructing the literal power set on top of the ensemble, but as the elements of the power set just randomly connect different ASORs, there is actually very little information in this. Observers evolving in time, however, selectively pick information from the ensemble (usually from their local environments, but not always), thus evolving novel, irreducible structures through iterative trial and error — i.e. evolution via selection. Then take the limit to infinity...

So a global time-like relationship allows for the gradual, continuous, and well-defined creation of infinitely complex ASORs. These infinite ASORs are a collection of instantaneous, finite brain states that always find themselves growing more complex. There can't be any discontinuous jumping up to 'very high' complexity states, and there isn't any final highest level where you could expect to find yourself.

marduk-kur:

I'm not sure why in a universe of all possible mathematical relationships (and thus all possible ASORs) there would not necessarily exist one that is infinite in complexity (or information content or whatever). Why does infinity have this special characteristic in the universe as it does not seem to have it in mathematics?

Is it because your model assumes our anthropomorphic perception of linear time? If we assume all times exist "simultaneously" (I hope you know what I mean) wouldn't we be infinitely likely to be observers who's complexity = infinity?

And ruling out infinite cases, wouldn't each succeeding set iteration dominate all previous ones? That would seem to place any particular observers short of Star Trek's Q in a very unlikely situation, statistically speaking. Is that correct?

me:

Ah yes, thus my 'simplest model' qualifer in my last post! Yeah, in the simplest model where you have the finitely complex ASORs, and the infinite observer-histories that extract information from them, the counting is pretty clear cut — you can get a finite integer encoding for each of the original ASORs, and infinite integer encodings for the observers, and then apply Cantor's diagonal argument, just like with the integers and irrational numbers. But what about, for example, observers whose each 'time-instantaneous' thought was irreducibly infinitely complex? What about higher order 'power sets' acting on the 'aleph-1' observer histories (and so on and so on, eventually leading to structures higher than any cardinality...)?? Well you can make these statements, but do these fantastically complex ASORs actually exist? I think the correct answer is — Maybe! If they don't exist, and we are essentially just talking about turtle shells that are simultaneously both entirely green and entirely not green, then the 'simplest model' applies simply enough. If, however, you can 'reach the point at infinity', and proceed on in some time-like dimension to think different, infinitely complex thoughts, then stat. meta. may still be correct and reconcilable with your current existence as a puny finite creature. Namely, if you think of the infinite complexity thoughts as a natural continuation of the finite ones (say along the lines of Conway's surreal numbers, constructed with Dedekind cuts), then just as before where there is no finite top level of complexity where you would expect to find yourself — then there is also not even any one infinite level of complexity where you would expect to exist — instead again you expect to find yourself always climbing higher — including starting at a finite level of complexity, reaching the 'point at infinity' and keeping on going up to levels of unbounded cardinality.

Assuming stat. meta. is correct, I'm not sure which scenario I think encompasses reality — I lean both ways at times. In the current draft of the paper I retreat a little and say that I'm just going to focus on the simplest model — I say that it is an 'extraordinary enough' of an argument to be made. But you're quite right — if these infinitely complex ASORs actually do exist, and furthermore if we can not 'reach the point at infinity' in any universe, then since we are currently finitely complex, stat. meta. must be false.

marduk-kur:

Ok good I understood enough to make sense.

Fun!

Thanks for the theory. Let me know how it holds up to review. It's an interesting and probably very productive line of reasoning even if it turns out to have flaws.

Somewhat related: What's your take on the notion that complexity seems to specifically emerge from non-linear dynamical systems composed of ever-simpler basic components? (sort of related to Wolfram's theories, and shout out here to Crick as well) It seems that this specific type of emergence is a fundamental component of almost everything we're able to observe in the universe.

Conversations:

OK, here's one I had on plastic with Marduk-kur. I'll post it in reverse order so it makes sense to read. Wait, what I just wrote is going to be at the bottom. Oh well.

Travis:

Hey, I'm back. Eating dinner and so on.

I'm really happy you're intrigued by the theory. I've talked to a good number of people now, and so far no one has come up a obvious fatal flaw in the logic — which means it actually has a non-zero chance of being correct, which is astounding. We'll see what the transhumanists have to say in Toronto (which I plan on blogging at my site) — I bet Nick Bostrom and Anders Sandberg will have some interesting comments. And after I polish the paper a good bit more, I'm planning on sending it to Woodin and Chaitin and Tegmark and Hofstadter and so on, and let them take a crack at it. If it still stands up, then we'll have to see if that good old exponential technological and scientific growth keeps up, and if Statistical Metaphysics really is correct then we are all in for a HELL of a ride... We'll see.

Hmmm, I'm a bit burned out from too much coffee, so I'll keep this shortish... But yeah, in my more traditional research I work on colliding black holes, which is almost as nonlinear a system as you can get — just working with a model of a nonlinear scalar system brings out all sorts of interesting behavior — I almost feel like a biologist at times grappling with the myriad little interacting phenomenon that pop up. In fact, stepping back, that's almost a definition for nonlinear systems — the elements interact with eachother. Hmmm, I'm probably overgeneralizing there, but with a linear wave equation, you can have two different waves that run into eachother, will pass right through eachother, and separate and go on their ways, their shapes unchanged by the encounter. The same doesn't hold for a nonlinear system! Hmmm, so OK, the subunits (waves, atoms, cellular automata cells...) interact — in what ways? What's the effective rule set? And then I bet simple combinatorics of the rules kicks in, and from the huge number of different scenarios thus made possible some will be 'interesting' (maybe set up a reproducing system?) and higher level heuristic rules can then emerge and so on... I should really think a lot more about this. In fact, yeah, you're right, it segways very smoothly into stat meta! I'll reference you if I work it in somehow. What are you're thoughts?

Intro

Hello, this blog will focus on discussions of my new theory of Statistical Metaphysics.