Thursday, February 28, 2008

Of Gödel and Galaxies

I've been reading the book Gödel, Escher, Bach by Douglas Hofstadter again, and it's really an amazing book. I recommend it very highly. One of the book's centerpiece's is Gödel's Incompleteness Theorem, which, in a nutshell, shows that there exist statements of number theory (such as "3^3=27," or "if you define a sequence a1=1 a2=1 and a3=2a2-a1, then no term is ever equal to 6.") which are true but cannot be proven within the context of number theory. The two examples I just gave are not of this form. One can prove them or prove there negation. The actual number theoretic statement constructed by Gödel is extremely complicated. The method of proof is brilliant, though. Because the laws governing the manipulation of numbers can be encoded as strings of numbers, one can actually connive to have number-theoretic statements have a secondary coded meaning which has something to say about the machinery of proofs. The coup de grace comes when you construct a statement whose coded meaning is "I cannot be proven in the system of number theory." Now let's consider whether the statement is true or false. If it is false, then it can be proven, which is to say it can't be proven. This is a contradiction, a version of the so-called Epimenides paradox: "Epimenides, a Cretan, says that all Cretans are liars." So the statement is not false, hence it is true. Since it is true it cannot be proven. Thus we have elegantly constructed a true statement of number theory which cannot be proven (within the system) precisely because that is what it asserts! Note that it can be proven outside the system, because we just did that. This is an awesome idea. The formal system of number theory can be utilized to show that it cannot adequately capture all true statements. One might say it contains the seeds of its own demise, but it might be better to say that it contains the seeds of its own limitation. Gödel's theorem doesn't destroy or invalidate number theory, it just shows that it is more subtle than any axiomatic treatment.

This whole story reminds me of an article I read in Scientific American which pointed out, that due to cosmic expansion, in 100 billion years all galaxies away from our local group will be traveling away from us at super-light speed, implying we cannot in principle see them or know anything about them. Inhabitants of the local group, now combined into a supergalaxy, will think that they are an island in infinite space. Furthermore, with all galaxies beyond our observable horizon, we won't be able to detect cosmic expansion, thus removing theoretical justification for a Big Bang. Our progeny, unless they have good records, will not have any reason to believe true physical facts about our Universe. In fact, future scientists might propose the existence of other galaxies or a Big Bang, and be shot down for unprovable junk science. So the current laws of Physics contain the seeds of their own limitation, just as with number theory. We can predict, using current Physics, that our progeny will not be able to detect facts about the Universe which we consider objective hard, facts. From here it's a small leap to realize that there are objective facts about our Universe that we will never be able to know because the information has already been lost.

Both these scenarios point to the fact that knowledge contains the seeds of its own limitation. This doesn't invalidate the reasoning process, but rather shows just how powerful and far-reaching reasoning can be. It is so powerful that it can even detect that it fails to completely capture all of reality!

2 comments:

Anonymous said...

I've been reading a wonderful book, The meaning of quantum theory by Jim Baggott.

As I was reading, it occurred to me that the EPR thought experiment, which spotlights spooky action at a distance, is an attempt to prove that QM is incomplete. Previously Einstein had sought to prove QM is inconsistent, but did not prevail. It seems very likely that EPR were stimulated by Godel's work -- but, if so they may have failed to grasp the point that incompleteness is required, and so one might surmise that QM's incompleteness is a point in its favor. Further, analytically based reality must always be found wanting.

beckett said...

I enjoyed this post very much. Well done.