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I think you posting #281 does get at most of notion of "randomness" in evolutionary theory. I would like to give some illustrations of "randomness" in computation and physics. These are not exhaustive nor deeply explained (if they were, I would just publish them somewhere, like in Erkenntnis, just because it has a nice consonant cluster.)

For this discussion, I'll just "random" to mean processes that satisfy the usual axioms of probability. (Kolmorogov is sufficient, but other interpretations are OK, Fineti for example.) The idea is that probability applies to any system that satisfies these axioms. In one sense, "random" phenomena must (or may) be described by averages.

One example is in the computation of averages or distributions in a game. One has a complete description (example: a die has probablity of 1/6 to show the numbers 1 to 6), and thus one can compute everything. It's sort of randomness through saturation. One assumes a large number of trial games and also assumes that these games will obey the same rules each time.

A second and much more interesting "random" system is given by Brownian motion. Consider a particle (dust, pollen, dust mites, etc.) being bombarded by even smaller particles (molecules) many times per second. Einstein (and others) developed the theory of the motion of such particles. There are some surprises; the velocity of the test particle cannot be defined, but it's position can. A test probe small enough to measure velocities would be subject to Brownian motion of the same size as the test particle and thus would yield no useful information. (Experiments bear this out; by 1900 or so, people knew that velocity could not be defined for Browinian particles.) Even though this system is deterministic in the sense of Laplace, there is no method (even in theory) to measure the exact conditions of the experiment. One must resort to averages. The system can be easily simulated deterministically though.

A third type of randomness would be that implied by quantum mechanics. Single particles act "randomly" and there is no method of resolving such even with simulation. (Exact simulation of quantum systems takes an exponentially large amount of time.) In this case, one must resort to probabilistic descriptions (albeit, not classical probability) to describe such systems even in principle.

The fourth "random" system would just to consider "relative independence" of events. For example, a cosmic ray may be produced on Sirius and strike a germ cell on Earth, causing a mutation. An observer won't see any connection between the local environment of the germ cell and goings on at Sirius. Similarly, a volcano (or a pack of wolves or a piano falling from the 13th floor of a hotel) may wipe out a person (dog, cat, plant) before that person can reproduce and thus kill off the person's genetic contribution. However, nothing in the physics or chemistry of DNA caused the volcano to errupt.

"Random" events (as I'm using the term) are those which may affect the outcome of an observation, but are not themselves (necessarily) implied by the physics of that observation. The lightning example is more like Brownian motion that the other forms. One cannot measure the boundary conditions well enough to exactly predict a lightning bolt, but one can do very well with averages. For example, high points (steeples, trees, golf clubs during a backswing) get struck relatively often.

Thank you oh so very much for your discussion of randomness!

All I can think to add to what you have said is this from Chaitin’s talk on Mathematics in the Third Millennium

Some final words on Stephen Wolfram's fascinating, and unfortunately unpublished, ideas. Wolfram has a very different view of complexity from mine. In my view pi is not at all complex, but to Wolfram it's infinitely complex, because it looks completely random. Wolfram's view is that simple laws, simple combinatorial structures, can produce very complicated unpredictable behavior. pi is a good example. If you didn't know where they come from, its digits would look completely random. In fact, Wolfram says, maybe the universe contains no randomness, maybe everything is actually deterministic, maybe it's only pseudo-randomness! And how could you tell the difference? The illusion of free will is because the future is too hard to predict, but it's not really unpredictable. [To Wolfram's exceedingly bright and sharp mind, the idea of indeterminacy, of randomness, of something irrational, that escapes the power of reason, of simple unifying principles, that happens for no reason—and that he will never be able to understand—is totally abhorrent. The horror of a vacuum of the ancients becomes a modern horror of randomness.

To such a mind, I must appear, because of my belief in randomness, as a muddle-headed mystic!... I'm also reminded of Feynman's fury in a conversation we had near the end of his life when I suggested that there might be wonderful new laws of physics waiting to be discovered. Of course!, I told myself later, how could he bear the thought that he wouldn't live to see it?... Science and magic both share the belief that ordinary reality is not the real reality, that something more fundamental is hidden behind everyday appearances. They share a belief in the fundamental importance of hidden secret knowledge. Physicists are searching for their TOE, theory of everything, and kabbalists search for a secret name of God that is the key that unlocks all understanding. In a way the two are allies, for neither can bear the thought that there is no secret meaning, no final theory, and that things may be arbitrary, random, meaningless, incompressible and incomprehensible. For a dramatization of this idea, see D. Aronofsky's 1998 film pi. See also G. Johnson, Fire in the Mind—Science, Faith, and the Search for Order, and P. Davies, The Mind of God—The Scientific Basis for a Rational World.]

Wolfram also has some fascinating ideas about biology, the origin of life and evolution. One of my big disappointments, the big disappointment in my scientific life, is that I couldn't use my program-size complexity to make a mathematical theory out of Darwin. [I was strongly influenced by von Neumann. For an early report of von Neumann's ideas, see J.G. Kemeny's 1955 article in Scientific American, ``Man viewed as a machine.'' For a statement by von Neumann himself, see ``The general and logical theory of automata'' in volume 4 of J.R. Newman's The World of Mathematics. For a posthumous account assembled by A.W. Burks, see von Neumann's Theory of Self-Reproducing Automata. For samples of contemporary thought on these matters, see P. Davies, The Fifth Miracle—The Search for the Origin of Life, and C. Adami, Introduction to Artificial Life.]

My complexity is conserved, it's impossible to make it increase, which is great if you're doing metamathematical incompleteness results, but hell if you want to get evolution. So I asked Wolfram his thoughts on this matter, and his reply was absolutely fascinating. He has amassed much evidence of the ubiquity of universality. In other words, he's discovered that many, many different kinds of simple combinatorial systems achieve computational universality, and have rich, complicated unpredictable behavior. pi is just one example... So what's so surprising about getting life, about getting clever organisms that exhibit rich, complicated behavior, that need it to survive? That's easy to do!!! And I suspect that Wolfram is right, I just want to get a copy of his 800-page book on the subject and be able to read it and think about it at my leisure. I have held its two volumes in my hands, briefly, once, during a fascinating visit to Wolfram's home...

The point of the foregoing is that randomness may not actually exist in this universe since everything we perceive as random – from lightning bolts to Chaitin’s Omega to Brownian motion – are effects of physical causation.

Danke für diesen ausgezeichneten Überblick Doktor Stochastic.

Of the four examples you gave I think the "Brownian motion" one is the best applicable to what we're dealing with here, since most of the explanation for abiogenesis will be rooted in chemistry, biochemistry, electromagnetism, and geophysics. The example of the die, which I know gets into frequencies, appears much too simplistic for application to a situation such as a "theoretical pre-biotic earth." Quantum mechanics does not appear suitable to chemistry and "relative independence" would seem applicable if we were dealing with "panspermia" as an explanation for the origins of life on earth, but not abiogenesis in its usual form.

I think the central problem we will have is that there are both a large number of known variables of unknown quantities and a large number of unknown variables of unknown quantities which seems to mirror your example of "some surprises" when you described the type of randomness inherent in Brownian motion. But I do want to think about this a bit more.

Your post was extremely helpful. Thanks again.

Just outstanding, Doc -- and so helpful to a non-specialist!

This fourth type of "random" system seems to go straight to the idea of contingency, which is more a philosophical idea than a strictly scientific one. The germ cell here on earth is affected by a discrete event on Sirius -- which probably no observer of the germ cell here on earth could be aware of. And yet the mutation we observe is contingent on the action of that unseen event. From the standpoint of the observer of the germ cell, the mutation may appear to be a "random" happenstance. Yet for a hypothetical observer whose view includes the cosmos in toto, there would be nothing at all "random" about the mutation: That observer would see that it had a cause on Sirius. And because it was "caused," from the point of view of the local observer it becomes fair game for scientific observation, and the natural laws are then found to apply from that point on. But the fact that the cause of the mutation was "hidden from view" means that an important predictive factor remains undisclosed to the scientific method. At least for a while. :^) If I might express it that way.

As you say, nothing in the physics or chemistry of DNA caused the volcano to erupt, or the germ cell to be bombarded by cosmic rays. But we can certainly perceive the effects.... Whereupon we may start speculating on the nature of the cause, and never even come close to actually, correctly identifying it. Which is to say, We usually don't know what we don't know. But we have to carry on any way.

Thanks so much for your excellent post.