brad@looking.UUCP (Brad Templeton) (01/17/85)
I can't believe I'm looking at these postings from Computer Scientists. No computer, now matter how powerful, can predict its OWN future, let alone that of the universe. Or do I have to prove to you that no computer can predict whether an arbitrary algorithm will halt in a limitless universe? So the universe is definitely not predictable from within. Whether it is FIXED or not is another matter, of course. -- Brad Templeton, Looking Glass Software Ltd. - Waterloo, Ontario 519/884-7473
djsalomon@watdaisy.UUCP (Daniel J. Salomon) (01/18/85)
> No computer, now matter how powerful, can predict its OWN future, let > alone that of the universe. Or do I have to prove to you that no > computer can predict whether an arbitrary algorithm will halt in a > limitless universe? > > So the universe is definitely not predictable from within. Whether it > is FIXED or not is another matter, of course. > -- > Brad Templeton, Looking Glass Software Ltd. - Waterloo, Ontario 519/884-7473 The halting problem states that no Turing machine can be built that decides whether some other arbitrary Turing machine will halt. But there is no rule against predicting what a Turing machine will do for the next little while. The halting problem is more of a comment on the inadequacy of our logic systems along the lines of Russell's Paradox, than it is a statement about the universe. Turing machines also have the property that they have an infinite tape. This is the practical problem that one encounters when trying to build a halt detector. But it has not been shown that the universe is infinite. In fact I believe that Einstein proved that the universe is finite.
jlg@lanl.ARPA (01/22/85)
> The halting problem states that no Turing machine can be built > that decides whether some other arbitrary Turing machine will halt. > But there is no rule against predicting what a Turing machine will do > for the next little while. The halting problem is more of a comment on > the inadequacy of our logic systems along the lines of Russell's > Paradox, than it is a statement about the universe. > > Turing machines also have the property that they have an infinite > tape. This is the practical problem that one encounters when trying to > build a halt detector. But it has not been shown that the universe is > infinite. In fact I believe that Einstein proved that the universe is > finite. The halting problem is less like Russell's Paradox than it is like Godel's Theroem. It doesn't require the infinite tape of the turing machine, but only that some state variable have unbounded range (I learned a proof of it with only one such variable in an otherwise finite machine with finite input). However, a corollary of the halting problem does imply that no machine can predict its own behaviour, much less the behaviour of anything more complex than itself (even without the unbounded variable). Einstein not only DIDN'T prove the universe finite, he didn't believe it. For a long time Einstein supported the steady-state theory (more because he was a relativist and didn't want to admit that there were special reference times than because he was attached to an infinite duration universe). He even introduced a so-called cosmological constant into his equations for general relativity in order to prevent it from predicting that the universe would expand or contract. When the evidence came that the universe was expanding, Einstein retracted the cosmological constant - calling it the worst error he had ever made (to introduce an idea merely to support his own preconceived beliefs). Still, the universe could be finite or infinite since general relativity doesn't exclude either possibility.
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (01/29/85)
> Einstein not only DIDN'T prove the universe finite, he didn't believe it. > For a long time Einstein supported the steady-state theory (more because he > was a relativist and didn't want to admit that there were special reference > times than because he was attached to an infinite duration universe). He > even introduced a so-called cosmological constant into his equations for > general relativity in order to prevent it from predicting that the universe > would expand or contract. When the evidence came that the universe was > expanding, Einstein retracted the cosmological constant - calling it the > worst error he had ever made (to introduce an idea merely to support his > own preconceived beliefs). I don't think this abbreviated history is correct. The "steady-state" theory postdates Einstein's cosmological constant by several decades. Einstein preferred a static isotropic cosmology and thought he could only get one if there were a cosmological term. When Friedmann was able to obtain these properties from the original (non-cosmological-term) equations, Einstein retracted the cosmological term. Evidence of the expansion of the universe had nothing to do with this, and indeed is contrary to the development that caused Einstein to repudiate the cosmological constant. Interestingly, the cosmological constant reappears in a pure-affine version of the theory. Its presence or absence cannot be tested on the medium scale, although it would have an impact on the very large or the very small. Because with a cosmological constant the universe becomes "self-gauging" (Eddington's terminology), such a theory has some attractive aspects. Einstein's regret at warping a theory to fit preconceived notions should not prevent investigation of such theories when they arise naturally.
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (01/30/85)
Thanks to Ethan for a good summary of the history of the cosmological term. > Actually, within standard GR there appears no theoretical reason to discard L > except those of simplicity. It may very well be demanded in any case from > particle physics (although the only 'natural' values are experimentally absurd). Indeed, the deSitter solution is a highly symmetric one and apparently the "natural" background space-time when a cosmological term is present. There is some relation (not fully known to me) with the "deSitter group", which I recall was of some interest several years ago in elementary particle theory. I think the point is sufficiently important that I will repeat it and elaborate a bit: When one produces a unified field theory along the lines of the classical development of General Relativity but with various symmetry constraints removed, if only ONE field is taken as fundamental it has to be the "affine" connection and not the metric tensor. (There are actually several similar connections but they're all related.) Schr"odinger investigated such theories in the late 1940s and found that the "pure affine" theory produced from a particular variational principle could be formulated to look just like Einstein's equation but WITH a "cosmological term" automatically appearing. In my Master's thesis I investigated alternative integrands for the variation and determined that they all produced either identical results or a symmetry-restricted subset (or, in many cases the integral variation vanished identically). The conclusion is that any generalization of general relativity along "classical" lines based on the affinity as the sole fundamental structure field will yield field equations identical in structure to those of the Einstein-Straus theory (under a particular choice of gauge) and to those of the Einstein-Kaufman reformulation (except for an additional term in the Bianchi identity), if the cosmological constant is set equal to zero. The interesting point is that in such a theory the cosmological constant arises in such a way that it CANNOT be precisely zero; indeed, its numerical value may be set to any positive real value by appropriate choice of units. This shows that the constant is an intrinsic measurement of SOMETHING in the local space-time structure, and since it is a constant therefore of the whole cosmology. (It is like an inverse square of the "size of the universe".) One of the nice features of the pure affine field theory is that the scaling argument of C. P. Johnson, Jr. against the E-S-K theories does not apply to the pure affine field laws.
jlg@lanl.ARPA (01/31/85)
After using several hours of Cray time on this problem I have some bad news for some of you on the net. We have predicted the entire state of the universe for the next five years. Two submitters to the net will get diverced, one will meet a tall dark stranger, exactly four will have trouble with a lock during the next week, and in a surprisingly rare event one of you will be struck down by a meteor! There are other predictions, I don't think we should panic too many people at this time. I would like to thank the little green men from Alpha Centauri for their assistance in setting up the initial conditions for this model. J. Giles ------------------------------------------------------------------------------ Consistency is the Hobgoblin of thick soup!