There is a new paper on Irreducible Complexity by renowned mathematician Gregory Chaitin: The Halting Probability Omega: Irreducible Complexity in Pure Mathematics Milan Journal of Mathematics, Vol. 75, 2007.

ÃŽÂ© is an extreme case of total lawlessness; in effect, it shows that God plays dice in pure mathematics.

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 Ã¢â‚¬â€ the very ones the solid-staters study Ã¢â‚¬â€ 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.

For the reader’s benefit here is a bit of a tutorial on the idea of IC in physics:

In science’s great chain of being, the particle physicists place themselves with the angels, looking down from the heavenly spheres on the chemists, biologists, geologists, meteorologists Ã¢â‚¬â€ those who are applying, not discovering, nature’s most fundamental laws. Everything, after all, is made from subatomic particles. Once you have a concise theory explaining how they work, the rest should just be filigree.

Even the kindred discipline of solid-state physics, which is concerned with the mass behavior of particles Ã¢â‚¬â€ what metals, crystals, semiconductors, whole lumps of matter do Ã¢â‚¬â€ is often considered a lesser pursuit. “Squalid state physics,” Murray Gell-Mann, discoverer of the quark, dubbed it. Others dismiss it as “dirt physics.”

Recently there have been rumblings from the muck. In a clash of scientific cultures, some prominent squalid-staters have been challenging the particle purists as arbiters of ultimate truth.

“The stakes here are very high,” said Dr. Robert B. Laughlin, a Stanford University theorist who shared a Nobel Prize in 1998 for discoveries in solid-state physics. “At issue is a deep epistemological matter having to do with what physics is.”

Many complex systems Ã¢â‚¬â€ the very ones the solid-staters study Ã¢â‚¬â€ appear to be irreducible. Made of many interlocking parts, they display a kind of synergy, obeying “higher organizing principles” that cannot be further simplified no matter how hard you try.Carrying the idea even further, some solid-state physicists are trying to show that the laws of relativity, long considered part of the very bedrock of the physical world, are not platonic truths that have existed since time began.

…

particle physicists have reason to be wary. The squalid-staters are challenging them in a debate over how the universe is made and how science should be done.….

The particle physicists’ ultimate goal is “grand unification” Ã¢â‚¬â€ recovering the primordial symmetry in the form of a single law Ã¢â‚¬â€ a few concise equations, it is often said, that could be silk-screened onto a T- shirt.This approach, in which the most complex phenomena are boiled down to a unique underlying theory, is called reductionism.

The problem, the solid-staters say, is that many forms of matter Ã¢â‚¬â€ ranging from the exotic like superconductors and superfluids to the mundane like crystals and metals Ã¢â‚¬â€ cannot be described in terms of fundamental particle interactions. When systems become very complex, completely new and independent laws emerge. “More is different,” as the Nobel laureate Philip W. Anderson put it in a landmark paper in 1972. To the solid-staters, it would take something the size of a circus tent to hold all the equations capturing the unruliness of the physical world.

Like Aristotle, they lean toward the notion that it is the equations that flow from nature instead of the other way around. Mathematics is just a tool for making sense of it all.

“For at least some fundamental things in nature, the theory of everything is irrelevant,” declared Dr. Laughlin and Dr. Pines in the Jan. 4, 2000 issue of The Proceedings of the National Academy of Sciences. “The central task of theoretical physics in our time is no longer to write down the ultimate equations but rather to catalog and understand emergent behavior in its many guises, including potentially life itself.”

There may not be a theory of everything, they say, just a lot of theories of things. This is exactly the kind of squalor the particle physicists abhor.

Dr. Grigori E. Volovik, a solid- state physicist at the Helsinki University of Technology in Finland, champions an idea he calls “anti- grand unification.” In a review article last year (xxx.lanl.gov/abs /gr-qc/0104046), he ventured that the universe may have begun not in a state of pristine symmetry but in one of lawlessness. The laws of relativity and perhaps quantum mechanics itself would have emerged only later on.

The notion of emergent laws is not radical in itself. A flask of gas consists of trillions of molecules randomly colliding with one another. From this disorder, qualities like temperature and pressure emerge, along with laws relating one to the other.

So take that idea a level deeper. Physicists now believe that the vacuum of space is, paradoxically, not vacuous at all. It seethes with energy, in the form of “virtual particles” constantly flitting in and out of existence. So perhaps, Dr. Volovik suggests, even laws now considered fundamental emerged from this constant subatomic buzz.

Solid-state physics offers clues to how something like this might occur. The atomic vibrations that ripple through matter are, like all quantum phenomena, carried by particles Ã¢â‚¬â€ called, in this case, phonons.

Just as photons carry light and gravitons carry gravity, phonons carry the subatomic equivalent of sound. Like bubbles in a carbonated beverage, phonons Ã¢â‚¬â€ physicists call them “quasi particles” Ã¢â‚¬â€ appear only when the medium is disturbed.

In the world of solid-state physics, quasi particles abound. In some substances, like the semiconductors used to make computer chips, the displacement of an electron leaves behind a “hole” that behaves like a positively charged particle. An electron and a hole can sometimes stick together to form a chargeless quasi particle called an exciton. Other such ephemera include magnons and polarons.

Evanescent though they are, quasi particles act every bit like elementary particles, obeying the laws of quantum mechanics. This has led some mavericks to wonder whether there is really any difference at all. Maybe elementary particles are just quasi particles Ã¢â‚¬â€ an effervescence in the vacuum.

Particularly intriguing is a phenomenon, occurring at extremely low temperatures, called the fractional quantum Hall effect. In certain substances, quasi particles appear that act curiously like electrons but with one-third the normal charge. (Dr. Laughlin won his Nobel Prize for a theory explaining this.)

Quarks, the basic building blocks of matter, also carry a one-third charge, a coincidence that has fueled speculation that emergence may be somehow fundamental to the very existence of the physical world.

A stumbling block to carrying this idea further has been that the quantum Hall effect seems to work only in two-dimensions Ã¢â‚¬â€ on the surface of a substance. But in a paper published in the Oct. 26 issue of Science, Dr. Zhang and his student Jiangping Hu showed how to extend the phenomenon. In their scheme, the physical world would be a three-dimensional “surface” of a four-dimensional “quantum liquid” Ã¢â‚¬â€ an underlying sea of particles that can be thought of as the vacuum.

Analyzing the ripples that would appear in such a medium, the two scientists were surprised to find that they mathematically resembled electromagnetic and gravitational waves. But there are problems with the model. At this point, the hypothetical photons and gravitons that emerge from the equations do not interact with other particles, as they do in the real world.

“The coupling is zero, so apples are weightless, as is everything else,” said Dr. Joseph Polchinski, a string theorist at the University of California at Santa Barbara, who recently discussed the model with Dr. Zhang.

And there is what the theory’s inventors concede is an “embarrassment of riches” Ã¢â‚¬â€ the equations predict hordes of exotic particles that do not exist.

“The hope is that some modification of the theory, not yet specified in detail, will remove the extra fields and turn on the coupling,” Dr. Polchinski said. “Whether this can be done is at this point a guess. Overall my attitude now is interest with a high degree of skepticism.”

If the theory can be made to work, it may point to a new way of unifying quantum mechanics and relativity. But Dr. Zhang is careful not to oversell what he considers a work in progress.

“Our work only made a tiny step toward this direction,” Dr. Zhang said, “but it seems to indicate that the goal may not be impossible to reach.” At the very least, he said, his work may inspire more collaboration between particle physicists and solid-staters.

Ultimately, though, the two sides know that they are talking across a divide. Taken to its extreme, emergence suggests that all the fundamental laws, even quantum mechanics, may be secondary Ã¢â‚¬â€ that at the base of reality is random noise.

Dr. Polchinski said he found that idea discouraging.

“To me, the history of science seems to be a steady progression toward simpler and more unified laws, and I expect to see this continue and to contribute to it. Things may take many surprising twists and turns,” he said, “but we reductionists are still quite happily and busily reducing.”

Back to the paper on Irreducible Complexity in Mathematics by Chaitin:

The Halting Probability Omega: Irreducible Complexity in Pure Mathematics Milan Journal of Mathematics, Vol. 75, 2007

On the surface this may seem totally decoupled from Irreducible Complexity (IC) in the ID literature. In fact, Hubert Yockey furiously criticized Behe’s notion of IC and contrasted it with Turing notion of IC.

But let us not be to hasty to say the two notions of IC can have nothing to do with each other. Let me suggest, IC in math will permeate physics and therefore biology! There has been a suggestion that emergent IC phenomena in physics are tied to Godel’s incompleteness, thus we ought to expect irreducible systems in physics! And as I pointed out if biology exploits such irreducible phenomena, we have strong theoretical evidence there are not Darwinian pathways to creation of such systems. More nails in the coffin Darwin’s theories.

See this paper:

This paper presents a discussion of the possible influence of incomputability and the incompleteness of the mathematics as a source of apparent emergence in complex systems. The suggestion is made that the analysis of complex systems as a specific instance of a complex process may be subject to inaccessible Ã¢â‚¬Å“emergenceÃ¢â‚¬Â. We discuss models of computation associated with transcending the limits of

traditional Turing systems, and suggest that inquiry into complex systems in the light of the potential limitations of incomputability and incompleteness may be worthwhile.We suggest that what we intuitively define as (strongly) emergent systems may include processes which are not computable in a classical sense. We ask how incomputable processes would appear to an observer and, via a thought experiment, show that they would display features normally defined as Ã¢â‚¬ËœemergentÃ¢â‚¬â„¢.

If this conjecture is correct, then two important corollaries follow: first, some emergent phenomena can neither be studied nor modelled via classical computer simulations and second, there may be classes of emergent phenomena which cannot be detected via standard physical measurements unless the process of measurement exhibits super-Turing properties in its own right. Borrowing from recent literature in

computer science we then show that tools which enable us to break the classical computational barrier are already available and suggest some directions for a novel approach to the problem.