Tuesday, March 07, 2006
There Are Limits
The March issue of Scientific American (not yet available online) has a very interesting article by
I want to focus on one particular aspect of Chaitin's work, because I think it extremely informative concerning some modern theories
Why do I find this so important? - a single "scientific" issue -- Global Warming. No theory of global warming can possibly meet the limitations of simplicity that Leibniz sets forth. A while back, I caught a little comment heat over my distinction between mathematical, statistical, and descriptive models in science. Global warming is a classic statistical model as I have described them -- it is not reductionist, it simply massages the data and is as Chaitin puts it a "law that describes some data is an extremely complicated one, then the data are actually lawless." Statistical analysis teases pattern out of chaos and orders the apparently disordered. That's the problem with statistical models, they do not simplify or reduce, they simply order, and are therefore as complex as the data itself.
The mathematics here is pretty complex and much as I find it interesting, I'm betting my average reader does not. So, to simplify matters. There exists mathematical proof that there are limitations to human reason. This fact says we cannot know everything by reason, somethings must be matters of belief. This also implies that so-called "scientific proof" must pass certain sniff tests to gain vaildity.
I am more than willing to conclude that much science, particualrly science about which public policy swirls, does not meet that sniff test.
I urge you to get a copy of SciAm and read the piece for yourself. Then think about it.
Related Tags: reason, mathematics, Godel, Omega, limits of reason, staistical modeling, global warming, theory, law
GREGORY CHAITIN is a researcher at the IBM Thomas J. Watson Research Center. He is also honorary professor at the University of Buenos Aires and visiting professor at the University of Auckland. He is co-founder, with Andrei N. Kolmogorov, of the field of algorithmic information theory. His nine books include the nontechnical works Conversations with a Mathematician (2002) and Meta Math! (2005). When he is not thinking about the foundations of mathematics, he enjoys hiking and snowshoeing in the mountains.Chaitin's article is an expansion, if you will, of Godel's famous proof, racasting it in that "alogorithmic information theory" he helped found. Chaitin summarizes Godel's proof this way
It fascinated me because Kurt Gödel used mathematics to show that mathematics itself has limitations. Gödel refuted the position of David Hilbert, who about a century ago declared that there was a theory of everything for math, a fi nite set of principles from which one could mindlessly deduce all mathematical truths by tediously following the rules of symbolic logic. But Gödel demonstrated that mathematics contains true statements that cannot be proved that way.Chaitin does not shy from painting the consequences of his work in broad brushstrokes
In ancient Greece, if you wanted to convince your fellow citizens to vote with you on some issue, you had to reason with them?which I guess is how the Greeks came up with the idea that in mathematics you have to prove things rather than just discover them experimentally. In contrast, previous cultures in Mesopotamia and Egypt apparently relied on experiment. Using reason has certainly been an extremely fruitful approach, leading to modern mathematics and mathematical physics and all that goes with them, including the technology for building that highly logical and mathematical machine, the computer. So am I saying that this approach that science and mathematics has been following for more than two millennia crashes and burns? Yes, in a sense I am.Now, the ramifications of that are nothing short of astounding - mathematical proof of the fact that reason fails, that ultimately science and math cannot prove everything. But that has been known in some sense since Godel's work which was published before WWII. Scientists and mathematicians have long since dismissed Godel on a philosophical level, because they dare not give up their precious naturalistic assertions. This is a long discussed issue in science, math and science philosophy.
I want to focus on one particular aspect of Chaitin's work, because I think it extremely informative concerning some modern theories
My story begins in 1686 with Gottfried W. Leibniz's philosophical essay Discours de métaphysique (Discourse on Metaphysics), in which he discusses how one can distinguish between facts that can be described by some law and those that are lawless, irregular facts. Leibniz's very simple and profound idea appears in section VI of the Discours, in which he essentially states that a theory has to be simpler than the data it explains, otherwise it does not explain anything. The concept of a law becomes vacuous if arbitrarily high mathematical complexity is permitted, because then one can always construct a law no matter how random and patternless the data really are. Conversely, if the only law that describes some data is an extremely complicated one, then the data are actually lawless.Chaitin's own work on algorithmic information theory quantifies this assertion of Leibniz and this is where he bases his expanded proof of Godel.
Why do I find this so important? - a single "scientific" issue -- Global Warming. No theory of global warming can possibly meet the limitations of simplicity that Leibniz sets forth. A while back, I caught a little comment heat over my distinction between mathematical, statistical, and descriptive models in science. Global warming is a classic statistical model as I have described them -- it is not reductionist, it simply massages the data and is as Chaitin puts it a "law that describes some data is an extremely complicated one, then the data are actually lawless." Statistical analysis teases pattern out of chaos and orders the apparently disordered. That's the problem with statistical models, they do not simplify or reduce, they simply order, and are therefore as complex as the data itself.
The mathematics here is pretty complex and much as I find it interesting, I'm betting my average reader does not. So, to simplify matters. There exists mathematical proof that there are limitations to human reason. This fact says we cannot know everything by reason, somethings must be matters of belief. This also implies that so-called "scientific proof" must pass certain sniff tests to gain vaildity.
I am more than willing to conclude that much science, particualrly science about which public policy swirls, does not meet that sniff test.
I urge you to get a copy of SciAm and read the piece for yourself. Then think about it.
Related Tags: reason, mathematics, Godel, Omega, limits of reason, staistical modeling, global warming, theory, law
