[time 472] Re: [time 471] Noncomputability = not by TM

Stephen P. King (stephenk1@home.com)
Fri, 23 Jul 1999 12:12:15 -0400

Dear Matti,

        I am working on my responce to [time 469], my computer lockedup and I
have had to start over...

Matti Pitkanen wrote:
> Part of Roger Penrose's hypothesis regarding the form
> of a final theory is that its dynamics should be
> noncomputable. Recall that this simply means that no
> Turing machine could reproduce this dynamics in the
> output of a calculation. Mathematics already offers
> examples of noncomputable sequences whose first few
> elements we know, but only because we have hit upon
> methods that suffice to identify those particular
> elements.
> [MP] I read Shadows of the Mind and Penrose's comments
> on it on some homepage. He left open whether
> quantum theory os genuinely non-deterministic or
> whether noncomputality gives rise to effective nondeterminism.
> I think the basic problem is that
> quantum nondeterminism is *not* random! On the
> contrary: in quantum jump only states belonging to discrete
> set of states are possible. It is extremely difficult to
> see how this kind of non-randomness could result from
> deterministic but non-calculable dynamics.
> In addition one should understand the reduction
> probabilities. And the prize for all these feats
> would be mystery: we we experience of having free will
> despite that free will is not actual!
> ***********
> When looking for noncomputability in physics, Penrose
> suggests that quantum gravity with topology change
> might be noncomputable, since four-manifolds are
> not classifiable, and four-manifolds would interpolate
> between the spacelike hypersurfaces at either end
> of a sum over histories. This would give us
> noncomputable amplitudes, and so noncomputable
> transition probabilities.
> [MP]
> I have seen this kind of argument somewhere.
> One could however consider quite well the possibility
> that amplitude is calculable after all. Calculation
> need not be done by numerical computing by taking
> actually the sum over all histories.
> On basis of frustrating personal experiences I do not
> believe on the summation over histories:
> a purely formal generalization of
> Feynmanns sum over histories approach is in question. There
> is not guarantee that resulting amplitudes are unitary.
> I spend more than seven years trying to make some sense
> about summation over spacetime surface idea until I realized
> that configuration space geometry is the only possibility
> to achieve manifestly unitary theory.
> **************
> This might be suitable for a noncomputable stochastic
> theory, but I wonder if we could go further and have
> a deterministic noncomputable theory. In this regard
> I find Chaitin's number interesting. Chaitin's number
> is the halting probability for a Turing machine,
> given certain weightings on initial conditions.
> Not only is Chaitin's number noncomputable, it is a
> random real, which means that it is statistically
> indistinguishable from a random series.
> Could the apparent randomness of quantum behavior,
> rather than resulting from real (albeit structured)
> randomness, be the result of a pseudorandom,
> deterministic noncomputable dynamic?
> Best,
> MP

        Umm, I think that Peter's idea that the appearence of randomness is due
to "secondary observer interactions" is the best quess so far! In other
words, yes!, I agree with you! ;-) But "non-computable", to me, means
"not computable by a Turing Machine" It is "computable" by an
Interaction Machine, as Peter defines it!



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