Re: Turing vs math

From: Christopher Maloney <>
Date: Mon, 08 Nov 1999 21:14:32 -0500

I agree with Bruno in this thread, but I'm not sure I completely
understand him, so I'm putting this out for comment.

The question arises when we accept as a starting point what Bruno
calls "comp", that the universe arises as a result of a computation.
Then, we want to know, is it possible for the physical laws of space
and time to be based on a continuum, or must everything be
quantised? Is the space we are embedded within really a fine grid,
with each cell on the order of 10e-23 cm, or some such small number?
Taking the question further, we can ask whether we should expect
the probabilities of, say, a given observation to be a real number,
or rational. If real, how is it possible that a TM can produce a
real number (or an infinite number of real numbers, corresponding
to the infinite number of observations)?

Is this the right question?

Then I agree with Bruno that due to computational indeterminacy, or
what he calls the PE-omega (which name I hate, BTW, Bruno), that the
answer is certainly yes, real numbers, continuous space and time.

The point is that whenever we make measurements, we are in a
1-perspective. I liked his description a few posts back of an oracle
that we can ask for any digit of a real number. Whenever we ask for
a digit, we get one, no matter how far down it is in significance.
It doesn't matter that the UD will never produce _all_ the digits
of any given real. It's enough that from a first person perspective,
that any given digit will eventually be computed.

As far as probabilities for observations, these don't have to
actually be computed by the UTM at all. They are simply manifest
to us, in our 1-perspective, because of the measure and because of
computational indeterminism.

Marchal wrote:
> Hal Finney wrote:
> >Marchal <> writes:
> >> Step n owns 2^(n-1) initial segments.
> >>
> >> Now, could you give me a bitstring which is not generated
> >> by this countably infinite process ?
> >
> >Are you familiar with Cantor's diagonalization argument which proves that
> >the real numbers are greater in cardinality than the integers? It
> >directly disproves your statement.
> >
> >The real number 2/3, = .10101010... is never output by your procedure.
> >If you disagree, tell me at what step it is output.
> Gosh! Please Hal, I have explicitely explained how and why my
> procedure cannot be disproved by Cantor diagonalisation, nor does
> my procedure contradicts the uncountability of the reals.
> I have also explicitely said that I don't ask for ma procedure to
> output the reals at some step. (but that is the case for any
> procedure giving the complete expansion of a real!).
> What I say is that the UD generates the entire decimal or binary
> expansion of all the real: 2/3, = .10101010... is generated at
> the first, second, third, ... steps.
> Note that any procedure outputing the expansion of a real does that.
> The fact is that my procedure generates also non-computable reals.
> There is no contradiction with diagonalisation, because at each step
> the procedure generates a set of decimal expansion and I don't
> provide (and I cannot provide!) a way to pick a particular
> expansion as the one being the expansion of a non computable real.
> Exemple:
> Suppose for simplicity that 0.00010110011110110... is the binary
> expansion of an uncomputable real (like Chaitin's number for exemple).
> The my procedure generates, 0,at the first step 00, at the second
> (among 01,10,11), 000 at the third step (among the 8 others),
> and so one.
> What I am saying is incredibly trivial, even if it looks a little
> paradoxical.
> You know, I can write a procedure which gives me with
> certainty the answer for any well defined mathematical question.
> It looks crazy? Take the question ``is Goldbach conjecture
> true or false"?
> Here is my procedure: generate the set {true, false}. I know that
> the answer is in the set. Of course I don't know which one it is
> but I have never been pretending knowing that.
> In the same way, it is easy to realise that during his infinite
> running the UD generates the binary expansion of each reals
> including the non-computable one. (Exercice: show that any attempt
> to build a LIST of all the reals using my procedure will fail. Hint:
> diagonalisation!).
> NOW, the important and perhaps less trivial point is the following one:
> 1) With comp we survive self-duplication (at a unknown level!, but here
> I will not insist although it is fundamental in other part of the proof).
> 2) From the first person point of views the delays of virtual
> reconstitution in the UD does not change the way of quantifying the
> indeterminism.
> SO we must take the generation of non computable reals into account
> in the search for a measure on the computational histories.
> Bruno
> PS 1) I send this message friday. For obscur reasons it has not been
> send as I discover in the archive. Does someone knows if something
> happens friday with the mailing list ?

Chris Maloney
"Donuts are so sweet and tasty."
-- Homer Simpson
Received on Mon Nov 08 1999 - 18:24:47 PST

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