Marchal wrote:
>
> This is linked to something said by Russell Standish in his paper
> and in some posts. Russell Standish pretends that (I quote him):
>
> Each self-consistent mathematical structure is completely described
> by a finite set of symbols, axioms and rule.
>
> This is totaly incorrect, I'm afraid. It is known that you cannot
> find a recursively enumerable (RE) set of axioms to specify the
> (N,+,x) structure (N) of the natural numbers with addition and
> multiplication. Each RE description of N admit plenty of non
> isomorphic models which belongs to any reasonable set of
> mathematical structures.
I don't understand this. Do you have a reference? I thought that
such a system as N with operations of addition and multiplication
are easily definable as a formal system. I must be missing what
you mean by "structure".
> And things are worst with sets,
> categories, etc.
> Of course you can guess this from the fact that the set of RE
> theories are countable, and the set of possible mathematical
> structure is ... much much bigger, to say the least.
>
> People should realize that notion like "definability",
> "provability" etc. are highly relative mathematical concept, by
> which I mean that they are formalism-dependent.
> "computability" is the first and the last (until now) purely
> absolute concept, which doesn't depend on the formalism choosed.
> Godel said that with the computability concept there is some
> kind of miracle. The miracle is the apparent truth of Church's
> thesis.
If this hadn't been a reference to Goedel, I'd have been tempted
to dismiss it. As it is, I'm merely quite skeptical. It seems
to me that computability is just a mathematical formalism like
any other, where you define symbols, axioms, and rules. Whenever
I hear anyone say that mathematical truths are "relative", my
guard goes up. It's one of the things I didn't like about David
Deutsch's book.
IMO, mathematical truths are fundamentally tautologies. They are
equivalent to "If A then A". I am not one to question formal
propositional calculus (the rules for making sylogisms) and
therefore I adhere to the belief that mathematical truths are
absolute, because they are tautologies. I.e. any correct proof
of a theorem T based on axioms A, B, and C could be stated
simply, "If A, B, and C, then T".
So I certainly don't see that computability theory could be in
any way more fundamental.
>
> That is why Tegmark or GSlevy approach, although very interesting,
> is still ill-defined. Tegmark will need some powerfull constructive
> axiom to give a sufficiently precise sense to the "whole set
> of mathematical structures" so that he can make the white
> rabbit disappearing, by isolating the right prior.
His axiom was explicit: there is one of each structure, up to
isomorphism. Making anything useful out of that might be difficult.
>
> Schmidhuber has chosen the "right" (absolute) objective realm:
> the universe of computations, but still doesn't realise that
> with comp, the prior cannot be defined on the computation,
> but only on the relative continuation, and this in a "trans-universe"
> manner (cf the UDA).
> The Universal Dovetailer Argument (UDA as I
> now call it, hoping Chris appreciates)
Much better! =:-)
> shows also that we must
> take into account the continuum.
> It seems that the UDA makes some links between Tegmark and
> Schmidhuber approach, indeed. Because it shows that with comp
> some prior needs non RE mathematical structures.
Which brings me to my question: what is your basis for establishing
the measure? I think I understand Juergen's now, that it's based on
what he calls a "universal prior", that the measure of each program
is related to the chance of guessing it -- and is therefore
directly related to its length.
I don't think I understand Hal's post of several weeks ago where he
discussed that the measure must be related not only to the structure
but also to the ability to locate that structure within the
universe. But this question is motivated by his post. I thought
that I was comfortable with the idea that a Universal Program, or a
Universal Dovetailer, or a Great Programmer, would naturally result
in our witnessing a coherent universe. I was thinking that Hal and
others weren't giving enough consideration to the explanatory power
of the principle of computational indeterminacy.
But I've since changed my mind. I am now completely stumped to the
core.
For one thing, I reject Juergen's principle because it is ad hoc.
I insist that any measure must result from zero information -- that
is the whole appeal of the AUH, after all.
I also reject James Higgo's contention that the WAP can explain it
all. After all, I can think of plenty of scenarios where I survive
but the universe nevertheless acts chaotically. This doesn't
explain the very deep connections I see between my memories and
what I experience now -- such as the fact that my dog responds to
the name "Steve".
Tegmark says "one of each structure up to isomorphism". But now I
keep thinking that that won't do at all, either. As the newcomer
Fritz just recently said:
Considering that every possible state does exist in some world,
it seems safe for me to conclude that there is only one world
corresponding to every state, and the chance of finding
ourselves in any possible universe is just as likely as any
other. The result would be total chaos.
If there's one of each, and each has an equal measure, then how come
I don't find myself embroiled in a chaotic universe?
Same problem, as far as I can see, with any computational theory.
Tegmark recently wrote that (paraphrasing) any Turing machine could
be described as a function on the natural numbers, f(n). It's
state at any time would be n. Then, it's state at the next time
interval would be f(n), then f(f(n)), and so on. This is just a
HLUT (Huge Look-Up Table). Now, if all Turing machines exist in
equal measure, then it seems to me, once again, that we should
expect chaos.
I think you've probably written on your solution to this before,
Bruno, so perhaps I should look back through the archives. Perhaps
you could send me a keyword or two to use.
>
> I have still a question for Russell, what is the meaning of
> giving a "physical existence" to a mathematical structure ? This is
> not at all a clear statement for me.
I don't have a problem with this at all. Tegmark, again, was pretty
clear: ME == PE (Mathematical Existence == Physical Existence).
>
> Bruno
Great posts recently, everyone! I do very much enjoy this group!
--
Chris Maloney
http://www.chrismaloney.com
"Donuts are so sweet and tasty."
-- Homer Simpson
Received on Fri Nov 12 1999 - 19:37:56 PST