Monday, August 29, 2005

3 Questions - Part 2

In the last post I asked three fundamental questions:

Since I do happen to exist, I was able to think about things for a while, and came up with two plausible assumptions about reality: first, that our universe actually is a mathematical structure, and second, that all mathematical structures exist. These assumptions lead to a statistical meta-prediction: if they are correct then we will forever be finding new phenomenon that require us to expand our previous theories, i.e. we will find physics to be bottomless. And they answer questions A and B: there is no distinction between possible and realized structures, and so they all exist within the ensemble. Note, however, that this does not mean anything goes! For instance {4 is Prime} and {f(x)=x^2 is the solution to df/dx=f} won't be elements of the ensemble. Indeed, most random sentences that one could form will correspond to no structure within the ensemble. There is natural structure within the ensemble. Clearly observers form one class of information within the ensemble. Thus question C becomes: why does existence entail being a member of the observer class?

The Observer Class Hypothesis (OCH) proposes a statistical answer to question C: observers form by far the largest class of information in the mathematical ensemble, and this explains why what it is to exist is to be an observer. Observers form the largest class precisely because they observe: they can selectively extract information from any other type of object in the ensemble. They can observe things in their physical environment like trees and moons and planets, and then with the scientific method figure out things like photosynthesis and the inverse square law. Or they can directly explore abstract objects in the ensemble like busy beaver programs, or the Banach-Tarski Paradox. Thus the observer class is something like the power set of the ensemble (quick note on the terminology - I suspect that the observer class is not a formal set, but rather a proper class, like the Universe in set theory: V = {x:x=x})

If a set X contains N elements: X = {X1,X2,...,XN}, then the power set P(X) - the set of all subsets of X - will contain 2N elements. Thus P(X) contains vastly more elements than X - in fact, if we let N be countably infinite, then there will be an uncountable number of elements in P(X). It is then tempting to argue that the ensemble is like a countable number of pieces of information, and that there are an uncountable number of observers extracting information from them. If this was the entire contents of the ensemble, then it would make the counting problem very easy, however this is quite likely not the case. In general it seems that there are two likely scenarios: either the ensemble is a countable collection of finitely complex structures, or, if it is meaningful to talk about infinitely complex individual structures (like uncomputable reals), then there is no upper bound to the cardinality of objects in the ensemble.

It is a good bit easier to consider the first case: a countable collection of basic structures, and a countable number of observers extracting information from them. But then how can the observers form the largest class? As I mused in a previous post, I think the key is to introduce a constraint. For instance, there are a countable number of both prime and composite integers, but if we introduce the constraint of only considering integers less than N, then the density of primes falls off as ~ 1/ln(N) as N goes to infinity (if the Riemann Hypothesis is correct!) Likewise, we only consider observers that can process ideas that contain less than N bits at time t. That is, if idea Xi takes I(Xi) bits to express (in some representation), then observer Yj can only consider Xi if I(Xi) < Mj(t), where Mj(t) is Yj's memory at time t. And why is observer Yj so constrained? Because he/she/it is a concretely realized computer within some physical background (the physical background being some mathematical structure in the ensemble).

I posit that if observers can increase their memory capacities Mj(t) boundlessly as time progresses, by redesigning their brain architectures and utilizing ever deeper physical structures (which will be available if the ensemble exists), then they will form the largest class in the ensemble. This is because observers will be able to learn any idea in time, and thus all forms of information are subsets of the observer class. Time evolution is key, since, in contrast to static structures, it allows for each observer to absorb any combination of structures. One might try and construct the power set of observers, in an attempt to break the OCH by forming an even more vast set, but in fact this is just another collection of structures that observers extract information from (indeed, thinking about other people is one of our primary mental activities). Amusingly, any collection of structures that one could try and construct such that they would be more vast than the observer class is still going to be described by an observer, and is thus self defeating. Still, this is a bit vague, and it would be good to place the OCH on a more formal grounding. The technique of forcing from set theory looks intriguing, and might be modified to this end.

The testability of the OCH comes through the requirement that there is no finite upper bound to the power of computers that we will eventually be able to build. Thus scientific and technological progress must never halt, but rather continue to press onward, perhaps even exponentially. The primary next step is the development of human level artificial intelligence, which will then rapidly lead to the evolution of vastly more intelligent computers. These very smart creatures could conceivably run into an absolute design wall at some point, perhaps imposed by physical limitations, and not be able to grow any more intelligent, which would falsify the OCH for them. However, if the ensemble does exist, I posit that these scenarios will be vastly outnumbered by those where there is no upper bound to the power of computers we will build and no limit to the complexity of ideas we can think and sensory information we can absorb. The longer that this proves to be true, then the greater confidence we will have that the Observer Class Hypothesis is correct. We shall see!

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