Wednesday, September 08, 2010

The Narrow Cosmic Performance Envelope

The Cosmos must have a very particular performance envelope if evolution is going to get anywhere very fast. (i.e. 0 to Life in a mere 15 billion years)

Brian Charlwood has posted a comment on my Blog post Not a Lot of People Know That. As it’s difficult to work with those narrow comment columns I thought I would put my reply here. Brian’s comments are in italics.

You say //So evolution is not a fluke process as it has to be resourced by probabilistic biases.// so it is either a deterministic system or it is a random system.

I am not happy with this determinism vs. randomness dichotomy: To appreciate this consider the tossing of a coin. The average coin gives a random configuration of heads/tails with a fifty/fifty mix. But imagine some kind of “tossing” system where the mix was skewed in favour of heads. In fact imagine that on average tails only turned up once a year. This system is much closer to a “deterministic” system than it is to the maximally random system of a 50/50 mix. To my mind the lesson here is that the apparent dichotomy of randomness vs. determinism does no justice to what is in fact a continuum.

A deterministic system requires two ingredients:
1/ A state space
2/ An updating rule
For example a pendulum has as a state space all possible positions of the pendulum, and as updating rules the laws of Newton (gravity, F=ma) which tell you how to go from one state to another, for instance the pendulum in the lowest position to the pendulum in the highest position on the left.

Fine, I’m not adverse to that neat way of modeling general deterministic systems as they develop in time, but for myself I’ve scrapped the notion of time. I think of applied mathematics as a set of algorithms for embodying descriptive information about the “timeless” structure of systems. This is partly a result of an acquaintance with relativity which makes the notion of a strict temporal sequencing across the vastness of space problematical. Also, don’t forget that these mathematical systems can also be used to make “predictions” about the past (or post-dictions), a fact which also suggests that mathematical models are “information” bearing descriptive objects rather than being what I can only best refer to here as “deeply causative ontologies”.

A random system is a bit more intricate. It can be built up with
1/ A state space
2/ An updating rule
Huh? Looks the same. Yeah, but I can now add the rule is updating. Contrary to deterministic systems, the updating rule does not tell us what the next state is going to look like given a previous state, it is only telling us how to update the probability of a certain state. Actually, that is only one possible kind of random system, one could also build updating rules which are themselves random. So you have a lot of possibilities, on the level of probabilities, a random system can look like a deterministic system, but it is really only predicting probabilities. It can also be random on the level of probabilities, requiring a kind of meta-probabilitisic description.

If I understand you right then the Schrödinger equation is an example of a system that updates probabilities deterministically. The meta-probabilistic description you talk of is, I think, mathematically equivalent to conditional probabilities. This comes up in random walk where the stepping to the left or right by a given distance are assigned probabilities. But conceivably step sizes could also vary in a probabilistic way, thus superimposing probabilities on probabilities. i.e. conditional probabilities. In the random walk scenario the fascinating upshot of this intricacy is that it has no effect on the general probability distribution as it develops in space. (See the “central limit theorem”)

Anyway, these are technical details, but let's look at what happens when we have a deterministic system and we introduce the slightest bit of randomness. Take again the pendulum. What might happen is that we don't know the initial state with certainty, the result is that you still have a deterministic updating rule, but you can now only predict how the probability of having a certain state will evolve. Now, this is still a deterministic system, the probability only creeps in because we have no knowledge of the initial state.
But suppose the pendulum was driven by a genuine random system. Say that the initial state of the pendulum is chosen by looking at the state of a radio-active atom. If the atom decayed in a certain time-interval, we let the pendulum start on the left, if not on the right. The pendulum as such is still a deterministic system.
But because we have coupled it to a random system, the system as a whole becomes random. This randomness would be irreducible.

This would classify as a one of those systems on the deterministic/random spectrum. The mathematics of classical mechanics would mean that any old behavior is not open to the pendulum system, and therefore it is not maximally random.; the system is constrained by classical mechanics to behave within certain limits. The uncertainty in initial conditions, when combined with mathematical constraint of classical mechanics, would produce a system that behaves randomly only within a limited envelope of randomness; the important point to note is that it is an envelope, that is, an object with limits, albeit fuzzy limits like a cloud. Limits imply order. Thus, we have here a system that is a blend of order and randomness; think back to that coin tossing system where  tails turned up randomly but very infrequently.

So, if you want to say that there is a part of evolution that is random, the consequence is that the whole of it is random and therefore it is all one big undesigned fluke.

No, I don't believe we can yet go this far. Your randomly perturbed pendulum provides a useful metaphor: Relative to the entire space of possibility the pendulum’s behavior is highly organized, its degrees of freedom very limited. Here, once again, the probabilities are concentrated in a relatively narrow envelop of behavior, just as they must be in any working evolutionary system – unless, of course, one invokes some kind of multiverse, which is one (speculative) way of attempting maintain the “It’s just one big fluke” theory. Otherwise, just how we ended up with a universe that has a narrow probability envelope (i.e. an ordered universe) is, needless to say, the big contention that gets people hot under the collar.

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