It’s been a while since I’ve replied to anything over at Uncommon Descent. But this entry from Salvador Cordova really caught my eye.

It is based on this paper, by mathematician Gregory Chaitin, The paper’s title: “The Halting Probability Omega: Irreducible Complexity in Pure Mathematics.”

Goodness! There’s irreducible complexity again. Let’s check in with Salvador first:

On the surface Chaitin’s notion of Irreducible Complexity (IC) in math may seem totally irrelevant to Irreducible Complexity (IC) in ID literature. But let me argue that notion of IC in math relates to IC in physics which may point to some IC in biology…

First, of consider this article archived at Access Research Network (ARN) by George Johnson in the NY Times on IC in physics:

Challenging Particle Physics as Path to Truth

Many complex systems — the very ones the solid-staters study — appear to be irreducible.

The concept of “irreducible complexity” has been used by Alan Turing, Michael Behe, and perhaps now by physicists. Behe’s sense of irreducible is not too far from the sense of irreducible in the context of this physics.

If biological systems take advantage of irreducible phenomena in physics (for example, what if we discover the brain uses irreducible physical phenomena ) we will have a strong proof by contradiction that there are no Darwinian pathways for biolgoical systems to incorporate that phenomena.The possibility of IC in physics may be tied to IC in math and this may have relevance to IC in biology. (Emphasis Added)

Let us ignore the arrogance and abusrdity of mentioning Michael Behe in the same sentence with Alan Turing. Instead let’s try to describe all the ways in which that paragraph above is meaningless.

There’s no comparison between Behe’s notion of irreducible complexity and anything Johnson was talking about in his article. In fact, they are almost diamterically opposed, as I will now show.

According to Behe, irreducible complexity is a property that might be possessed by any multi-part system that can be said to have a clear function. He is fond of illustrating his idea with a mousetrap, you might recall. As applied to biology, an IC system is any biochemical system composed of several well-matched parts, such that the removal of any part destroys the function of the system. This property was said to be important because, according to Behe, any biochemical system possessing this property could not have evolved gradually under the aegis of natural selection.

Of course, Behe’s claim is simply wrong as a matter of logic. Leaving aside the fact that biology has many illuminating things to say about the evolution of complex systems, the simple fact is that as a matter of logic Behe’s notion of irreducible complexity is simply irrelevant to any determination of whether a system could have evolved. There are a variety of scenarios through which a system meeting Behe’s definition could have evolved through gradual stages, and that is enough to refute his central argument.

Johnson, by contrast, is contrasting the approaches of particle physicists with those of solid-state physicists. The idea is that the particle physicists like to break down complex systems into their component parts, and then try to understand the whole by formulating laws governing the interactions of the parts. Solid-staters, according to Johnson, regard certain complex systems as “irreducible” that is, they display properties that can not be understood via simple laws governing the interactions of fundamental particles. This, of course, is related to the idea of emergent properties.

So where Behe is saying that something valuable is learned by taking a complex biochemical system, atomizing it into its component parts, and considering the effects of knocking out each part in turn, Johsnon is saying this is precisely the sort of thing you must not do.

Moving on, what could that bold-face remark possibly mean? Salvador bases his notions of irreduciblity in physics on the usage by Johnson in the article linked to above. But Johnson does not apply the word “irreducible” to phenomena. He is talking about complex systems that can not be studied properly by breaking them into their component parts. In other words, it is systems, not phenomena, that are irreducible. So what could it mean for the brain to take advantage of them? And even if we manage to impart some meaning to that phrase, why would it imply that such systems could not have evolved? I suspect Salvador has no more idea than I do.

So what was Chaitin talking about? Well, that gets a bit complicated. Basically, he is using ideas from algorithmic information theory to try to elucidate the nature of incompleteness in mathematics. Godel’s famous incompleteness theorem showed that any finite set of axioms strong enough to include elementary number theory must contain statements that are true but unprovable.

A “complete” mathematical theory would be one in which any statement that is true within the theory could be proved from the axioms of that theory. Since Godel effectively showed this to be impossible, his result is referred to as the incompleteness theorem.

In his paper, Chaitin observes that, as important as Godel’s theorem is, it does not really tell us how serious a problem incompleteness is. In other words, Godel showed that there must be certain propositions that are true but unprovable. But to do this he had to conjure up a pretty bizarre, self-referential kind of statement. Not exactly the usual, humdrum kind of statements with which mathematicians generally concern themselves. The way mathematicians undertook their work was ultimately little affected by Godel’s discovery. It was possible for professional mathematicians to pretty much ignore what Godel did.

Chaitin proceeds to make an argument that mathematics, in a technical sense described in the article, can be said to be infinitely complex. Since any finite axiom system can only encompass a finite portion of this complexity, incompleteness is something that permeates the entire mathematical enterprise. This is said to point to a fundamental inadequacy in the axiomatic method.

The “irreducible” part refers to certain bit strings that can not be realized as the output of a computer program less complex than the string itself. According to Chaitin, most bit strings, that is, most mathematical facts, are of this sort. This sheds some new light both on the results of Godel, and on the later results on computability by Turing.

There is a lot to mull over in Chaitin’s paper; I don’t feel that I fully understand all of his points. I also find much to disagree with in his remarks about the proper conduct of mathematics. But for now the relevant thing is that Chaitin’s notion of irreduciblility has nothing to do with Behe’s worthless notion of the same name, and has only a passing connection at best with the sort of irreducibility to which Johnson refers in his article.

Salvador is simply playing word games.