Re: A Questionnaire for Bill Taylor
Le 14-mai-05, à 16:04, Lee Corbin a écrit : If they are furthermore enough rich in complexity to have "abstract inhabitants", it is reasonable or plausible (at least) that for those inhabitants their abstract universe will look as it is real. This rests on the surprising conclusion that the inhabitants actually compute later states from earlier ones. Of course, it can always be made to *look* as though that is what "happened". Eternal Truth #6: seek and ye shall find. It is precisely this latter "surprising conclusion" that is resisted by so many (including me). Just as it would seem that anyone should resist making conclusions about the relationships between the books in Borges' Library of Babel. Borges' Library is different from the UD in this very respect. The UD does not simply generate all programs, but it executes all program. In the platonic sense that it does relate the computational states to each other. Its existence is far from from obvious. It seems easy for a mathematician to refute its existence by simple cantor-like diagonalisation. It takes the genius of Kleene or Post to unravel the paradoxes, to show that the set of programmable functions is close for the diagonal procedure, to show this entails the absoluteness of the computability relation, and the relativeness of any provability relation. It is a whole new mathematical world which has been discovered. And this will make sense if, furthermore again, their relative abstract computational continuations have the right measure. And theoretical computer science can justify the existence of such relative measure. We may take the books in the Borges library---admittedly in his scenario none of which is infinite---and begin willy-nilly assigning a greater measure to some than to others. It is extremely tempting to assign greater measure to short ones. But in the infinite-string version of Borges' library, Russell Standish (for one) begins by assigning equal measure to each bit string. I do not assign measure to short strings. Nor to any strings. I just use the Godel-Lob Logic to define the particular case of "measure one" in the language of a (Lobian) Universal Machine. Lobian is equivalent with having enough provability ability. I use a result from Goldblatt which translates Quantum logic in modal logic. I compute the inverse of Goldblatt transformation on the lobian machine description of that measure one, and I compare with quantum logic. Looks simple but those transformation are not easy to handle and lead to intractable problems and open question. Compare to QM or even Newton, I agree, the comp-physics is still too young (to say the least), but this only with respect to pure 3-person prediction, let us say the quanta. But thanks to incompleteness, which in the modal setting is captured by the gap between two amazing modal logics G and G*, I can say that with respect, to the existence of 1 and 3 person, an to the explanation of why quanta and qualia, and why they are different, the comp-physics is already in advance. Modal logic is just a tool for adding possible nuance to classical logic. My experience is that modal logic is more easy to understand than to understand the difference between atheist and agnostic. Indeed, I am used to explain the difference between atheist and agnostic by using the belief modality. I'm sure you know the difference between atheist and agnostic, but let me explain it with an explicit modality. Let d be the proposition according to which God exists. Let B be the belief modality as applied to someone. And let - represent negation. Then, by definition I would say: -a religious believer is someone who believes in the existence of God, i.e. about who 'Bd' is true. -an atheist is someone who believes God does not exist, i.e. about who 'B-d' is true. -an agnostic is someone who does not believe in the existence of God, and he does not believe in the non existence of God. Both '-Bd' and '-B-d" are true for him (or her, it, ...). People tends to confuse, and natural language does not help, propositions like -Bd and B-d. Most acts of putting the mind-body under the rug, or using Godel theorem against comp, can be seen at some level of description of the argumentations as error of that kind. That's why the discovery of G and G* by Solovay is a formidable event for simplifying the life of those who want to study the counter-intuitive lesson of the universal machine which introspect itself: what machine can prove and cannot prove but can correctly guess about their consistent extensions, and their geometry. The modal logic G is the study of the Beweisbar Gödel probability predicate Bew(x) which is true if there is a number coding a proof of x. You should'nt reduce the whole of computer science in the Library of Babel. Machine have dynamics, even if they are on the type discrete, digital, and, when seen in Platonia, looks like static abstractions. Here the movi
RE: A Questionnaire for Bill Taylor
Bruno writes
> Le 12-mai-05, à 19:14, Peter D Jones a écrit :
>
> > I don't see why. Surely what is being asserted is that there is a set
> > of physically real universes, and it is a subset of logically
> > possible universes ("Platonia") -- but logically possible universes
> > are not real in any sense, they are just an abstraction.
Well, that is the whole (as yet unproven) claim of those like
Balfour, Schmidhuber, and Marchal, etc. Namely, that they are
real (see Bruno's nice "sum up" below).
> But logically possible universes are certainly real in one sense: as
> being logically possible. Or as being logically consistent.
Yes indeed. If I have a string that contains (looks entirely arbitrary)
...071CB7150F1B0571C391BF0194C713100F15070C33149054012F0C59202039D10091...
of one's and zeros (in hex notation) and somewhere else an almost
exactly similar string (both either finite, or infinite), then
indeed there *is* an objective relationship between them if they
are sufficiently similar. They are, in fact, related precisely
to the degree that there is such a mapping from one to the other.
Now the more complex the mapping, that is, the more esoteric the
TM that is needed (the greater the Kolmogorov complexity of the
program that relates them), the less that they are *objectively*
related.
And since by hypothesis, all strings "exist" to platonic mathematicians,
all strings exist in Platonia, then any and every relationship that you
care to name exists there also. But one has to keep in mind a sort of
futility that attends such observations; a futility described first by
Borges.
> If they are furthermore enough rich in complexity to have "abstract
> inhabitants", it is reasonable or plausible (at least) that for those
> inhabitants their abstract universe will look as it is real.
This rests on the surprising conclusion that the inhabitants actually
compute later states from earlier ones. Of course, it can always be
made to *look* as though that is what "happened". Eternal Truth #6:
seek and ye shall find.
It is precisely this latter "surprising conclusion" that is resisted
by so many (including me). Just as it would seem that anyone should
resist making conclusions about the relationships between the books
in Borges' Library of Babel.
> And this will make sense if, furthermore again, their relative abstract
> computational continuations have the right measure. And theoretical
> computer science can justify the existence of such relative measure.
We may take the books in the Borges library---admittedly in his
scenario none of which is infinite---and begin willy-nilly assigning
a greater measure to some than to others. It is extremely tempting
to assign greater measure to short ones. But in the infinite-string
version of Borges' library, Russell Standish (for one) begins by
assigning equal measure to each bit string.
> And, finally, if such mathematical measure leads to the verified
> empirical measure, then, frankly, it seems to me that materialism in
> physics begins to look like ... late vitalism in 19th century biology.
A big "if"! Indeed, if a single new prediction comes from consideration
of Platonia or of Borges' Library (finite or infinite-string version),
then there will be quite a cause for celebrating.
> To sum up: "real" is just (abstract) consistency as seen from inside.
Great summary. Are all these books in libraries whispering to
each other or not? They indeed even quote each other verbatim,
and many can be seen to be holding conversations with each other
at arbitrarily high degrees of abstraction and wisdom. But will
anyone a hundred years from now think there is anything substantial
in this perception, or not?
Lee
Re: A Questionnaire for Bill Taylor
Le 12-mai-05, à 19:14, Peter D Jones a écrit :
I don't see why. Surely what is beng asserted is that there is a set
of physically real universes, and it is a subset of logically
possible universes ("Platonia") -- but logically possible universes
are not real in any sense, they are just an abstrction.
But logically possible universes are certainly real in one sense: as
being logically possible. Or as being logically consistent.
If they are furthermore enough rich in complexity to have "abstract
inhabitant", it is reasonable or plausible (at least) that for those
inhabitants their abstract universe will look as it is real. And this
will make sense if, furthermore again, their relative abstract
computational continuations have the right measure.
And theoretical computer science can justify the existence of such
relative measure.
And, finally, if such mathematical measure leads to the verified
empirical measure, then, frankly, it seems to me that materialism in
physics begins to look like ... late vitalism in 19th century biology.
(And then my thesis shows that the mathematical measure extract from
computer science looks sufficiently like the quantum measure to
considerate that the case for a scientific materialism is at least
premature.
To sum up: "real" is just (abstract) consistency as seen from inside.
Bruno
http://iridia.ulb.ac.be/~marchal/
