Config Space via Mathematical Impressionism. Part 2

This series is intended to provide a very general conceptual frame work for thinking about evolution. In the first part I introduced the following graphical representation of configuration space:

The horizontal axis represents the size of a configuration. The vertical axis is the logarithm of the total number of logically conceivable configurations consistent with a configuration size of value S. For a given size S, each possible configuration is counted by mapping it to a point on the Log Z-S plane. In order to organize this count of points the area under L₀ has been divided up into wedge shaped bands using lines L₁ to L_n. If we take a given size S, then the vertical distance across a band is the Log measure of the number of configurations that have a particular disorder value, where disorder increases from bands 1 to n respectively.

Arranging configurations of a particular value of disorder and size into a 1 dimensional line doesn't do justice to the multidimensional nature of configuration space, a fact I alluded to in the first part of this series. Mapping configurations of a particular disorder and size onto a single vertical line in one of the wedges above will have the effect of forcing a separation on otherwise natural near neighbors in configuration space. In fact this is similar to the effect that occurs when one maps a multidimensional space onto linear computer memory; neighbouring points get separated. 

In spite of the limitations of my graphical representation we can nevertheless use it to help talk about the conditions needed for evolution to occur.

In the first part I defined living structures as configurations with powers of self-perpetuation - a process that includes self-repair and reproduction. Therefore the sort of self-perpetuation I'm thinking of is very proactive in that it is not simply down to atomic bonding stability (as it is for strong crystalline structures), but instead a form of maintenance that depends on a blend of proactive repair and reproduction; in fact in terms of molecular bonds living materials are by and large very fragile.

One of the fairly obvious requirements of evolution as conventionally understood is that of “reducible complexity” (I have talked about this point many times in this blog).  Given axioms 3 and 4 (Seen part 1), conventional evolution requires that living configurations, when mapped to configuration space, give rise a set of points in this space that are close enough to one-another to form a completely connected region; very likely this region would be the multidimensional equivalent of a “spongy structure” made up of extremely thin membrane walls.  This connectedness will mean that the random agitations of evolutionary gradualism can set up a diffusional migration across configuration space without resort to highly improbable saltational leaps. It is this connected structure that defines what “reducible complexity” means. It also explains why so many in the de-facto “Intelligent Design” community are quite sure that living structures are “Irreducibly Complex” rather than “reducibly complex”. A class of structures is irreducibly complex if they form a scattered set in configuration space - that is, they do not form a connected set but are by and large individually isolated.  If self-perpetuating structures are arranged as an irreducibly complex set in configuration space then this means these structures can only be reached by saltational leaps. The de-facto ID community then contend, (with some plausibility), that if this is the case then the only agent we know capable of literally engineering these leaps is intelligence.

To be fair to the ID community, the notion that organic structures form a reducibly complex set is moot on at least three counts

ONE) If a reducibly complex set of self-perpetuating structures exists then it is likely to be highly sensitive to the selected physical regime. I suspect, although I have no proof, that the physical regimes implying reducible complexity is a very small class indeed; I guess that any old selected physical regime won’t do. But even if physical regimes that favour reducible complexity have at least a mathematical existence we are still left with the question of whether our particular physical regime is one of them!

TWO) Axiom 2 tells us that the set of living structures is tiny compared to the set of all possible non-self-perpetuating structures. This fact is an outcome of axiom 1 and the nature of disorder: If living structures occupy the mid regions between high order and high disorder then the logarithmic nature of the vertical axis on the LogZ-S graph will imply that disordered configurations are overwhelmingly more numerous. This raises the question of whether there are simply too few self-perpetuating structures to populate configuration space even with a very thin spongy structure; in fact the spongy structure may be so thin that although mathematically speaking we will have an in-principle reducible complexity, in terms of practical probabilities the structure is so tenuous that it may as well not exist!

THREE) My definition of life in terms of self-repair and reproduction would seem to imply a threshold of sophistication of configuration that is relatively high. Even if this set of structures form a completely connected set in configuration space how did the first structures come about? Their sophistication would seem to demand a size that is too large  to have come about spontaneously (see Axioms 2 and 3). Therefore if evolution is to work our reducibly complex set of structures must be continuously connected to and blend with a set of small stable structures toward the lower size end of our graph where small configuration sizes mean that the probability of spontaneous appearance is relatively high. (An implication of axiom 2). This is the subject of the Origins of Life (OOL) which as far as I’m aware doesn't have any substantive scenarios on the table.

I must express (again) my feeling that solutions to the above questions are not likely to be succinctly analytical, because I suspect that attempts to solve them analytically will hit Wolfram’s computational irreducibility barrier. That is, that the only way of probing these questions is to do a full simulation,  because there may be no other shorter way of computing the result than working, event by event, through the full natural history of the world. But perhaps I'm being too pessimistic!

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The de-facto “ID” community, in my opinion, are not getting the respect and hearing they deserve. After all, the big issues I've outlined above don’t have obvious answers. Nevertheless, as I have expressed many times before, I continue to feel uneasy about the de-facto “ID” community’s ulterior philosophy and underlying motivation.  This uneasiness stems from: a) Their failure to register that even bog-standard evolutionary theory presupposes highly computational complex pre-condition;, that is high information conditions (Which is essentially the lesson from their very own William Dembski. b) That many de-facto IDists still see the subject through the fallacious God did it vs. Naturalism did it dichotomy.  This dichotomy is seductive to both theists and atheists. The polarized and acrimonious state of the debate in North America, where it is cast in the mould of a “Masculine God vs. Mother Nature” paradigm, has probably help keep this dichotomy alive. In this context the natural history question is framed entirely in terms of whether it is guided or unguided - guided by a driving masculine homunculus or left unguided by a scatty mother nature. So, in my next part I will look into the subject of whether evolution, as it is conventionally conceived,  has direction. 

North American Paradigm: Mother Nature or Guiding Homunculus?

Finally I must add this caveat: Although I eschew the North American paradigm that swings so much on the question of whether natural history is "guided or unguided”, this is not to say that the established picture of evolution is correct. As I have said before the game of chess is considerably constrained by its rules, but if you try moving chess pieces about at random even under the constraint of those rules you are unlikely to end up with a sensible game. Physics, as we currently understand it, may not be strong enough constraint to imply a computation that follows the established evolutionary paradigm. In later parts of this series I may probe whether there are ways round the problems outlined above.

End Note: Clearly living structures capable of self-perpetuation in one environment may not survive in another. Therefore the way self-perpetuating structures are arranged in the Log Z-S plane will depend on the environment we are looking at.  Moreover, since the presence of organic structures are  part and parcel with the environment this will introduce non-linear feedback effects. For simplicities sake these non-linear issues have been left out of the above discussion.