On Sat, Oct 15, 2011 at 06:53:59PM +0200, Bruno Marchal wrote:
> On 15 Oct 2011, at 02:50, Russell Standish wrote:
> >On Fri, Oct 14, 2011 at 05:01:26PM +0200, Bruno Marchal wrote:
> >>
> >>On 13 Oct 2011, at 23:50, Russell Standish wrote:
> >>>I don't see why Bayes' theorem assumes a physical universe.
> >>
> >>
> >>Bayes' theorem does not assume a physical universe. But some use of
> >>bayes theorem to justify the laws of physics, presuppose that a
> >>physical universe is an object (may be mathematical, like in
> >>Tegmark) among other objects.
> >
> >Then why couldn't the physical universe be a trace (aka history)
> >of UD*?
> Because the UDA show it to be a sum of infinitely many computations.
> Even 2^(aleph_0) due to the dovetailing of the real (and complex
> ...) inputs of the program generated and executed by the UD. This
> cannot be generated by any programs. It can only be lived or
> inferred by the internal observers experimenting their golabl (on
> UD*) first person indeterminacies.

Fair point. Let me rephrase: Why couldn't the physical universe be a
set of computations, all giving rise to the same experienced history.

> >
> >>
> >>
> >>>All it
> >>>assumes is a prior probability distribution. Something like the
> >>>universal prior of Solomonoff-Levin, or the distribution of observer
> >>>moments within UD*.
> >>
> >>I don't think such a distribution makes sense. What makes sense is a
> >>computational state, and a distribution of (competing) universal
> >>machines relating that state with other states through the
> >>computations that they emulate.
> >>
> >
> >Whenever an observer interprets multiple different input strings (ie
> >observations) as the same thing, the S-L distribution makes
> >sense. Particularly so if the mapping process is a computation.
> I am not sure I understand this.

The S-L distribution is defined as the sum over all programs that halt and
produce a given output (x say) of 2^{- length of program expressed as
a bitstring}.

We can replace the Turing machine with any function that takes
bitstrings, and maps them to a countable set of meanings (which can be
identified with N, obviously), provided the map is prefix free (ie if we
read n bits, and decide the meaning is x, we cannot change our mind
after reading n+m bits).

> >
> >The UDA indicates we must be supervenient on all programs passing
> >through our current observer moment.
> It makes sense with OM = 3-OM = relative computational state. But
> this is not Bostrom's OM a priori (provably with comp).

It seems we've been around the world on this one. There is only one OM
concept, which is defined by the information content of the observer
at a point in time.

But there may be multiple programs instantiating a given observer, so
there will in general be multiple machine states corresponding to a given

> >
> >I know there are only a countable number of programs. Does this entail
> >only a countable number of histories too? Or a continuum of histories?
> >I did think the latter (and you seemed to agree), but I am partially
> >influenced by the continuum of histories available in the "no
> >information" ensemble (aka "Nothing").
> It is a priori a continuum, due to the dovetailing on the infinite
> real input on programs by the UD.

IIUC, the programs dovetailed by the UD do not take inputs. You
expanded a bit on this in your response to Brent, but I don't follow, sorry.

> >
> >Could it be that there are only a countable number of histories after
> >all, given there are only a countable number of programs. That would
> >be one big difference right there.
> We do agree on this. The difference is that the comp statistics is a
> statistics on non-random things, even if those things include
> computations (non random) with random inputs.

Are you agreeing there may only be a countable number of histories
after all? Or something different :).

I'm not sure what you mean by random inputs. Surely, if random inputs
were applicable, then the histories will be random things.

> >>Well, because UDA shows that the laws of physics are logico-
> >>arithmetical, and that they take the form of internal
> >>(epistemological) relative statistics on computation.
> >
> >I actually don't get that conclusion from your work, so it might be
> >worth elaborating more.
> This already happens in the UDA step 7. We don't need the
> immateriality or the 'arithmeticality'. 

Sorry - I think I minsinterpreted what you said previously...

> >The Theatetus definition leading to the AUDA has the feel of something
> >"put in by hand", rather than being a logical consequence of the
> >UDA. Nothing wrong with that, of course, but we should be honest with
> >it, if it is the case.
> I agree I am not always clear on that. That is why I try to
> distinguish comp (used in UDA), and comp+theaetetus, used in AUDA.
> But the theaetetus ca be shown to be the unique definition meeting
> the requirement of computer science, provability logic, and the
> usual definition of knowledge (Kp -> p, Kp -> KKp,
> K(p->q)->(Kp->Kq)). It can be motivated, as it is by Socrates in the
> Theaetetus of Plato, by the dream argument, which is basically step
> 6 of UDA.
> How would you define knowledge axiomatically, accepting that you
> want it to apply to an entity (a machine) whose beliefs are
> rational, in the sense of obeying classical logic on the finite
> things?

As I said - I don't have an answer to that. I'm not an
epistemologist. All I can say is that your axioms seem to treat
knowledge as rather like how we might know a mathematical fact, rather than
how we know a fact of chemistry, or a fact of human life. So it seems
unsatisfactory to me.

It could be that knowledge resists axiomatisation. It could be that a
different set of axioms is more appropriate - eg incorporating ideas
from evolutionary theory. WRT the latter, I would say that life, and
also evolution has also resisted axiomatisation, in spite of a number
of attempts.


Prof Russell Standish                  Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics      hpco...@hpcoders.com.au
University of New South Wales          http://www.hpcoders.com.au

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