Le 24-mai-05, à 00:17, Patrick Leahy a écrit :
>
> On Mon, 23 May 2005, Bruno Marchal wrote:
>
> <SNIP>
>
>> Concerning the white rabbits, I don't see how Tegmark could even
>> address the problem given that it is a measure problem with respect
>> to the many computational histories. I don't even remember if Tegmark
>> is aware of any measure relating the 1-person and 3-person points of
>> view.
>
> Not sure why you say *computational* wrt Tegmark's theory. Nor do I
> understand exactly what you mean by a measure relating 1-person &
> 3-person.
This is not easy to sum up, and is related to my PhD thesis, which is
summarized in english in the following papers:
http://iridia.ulb.ac.be/~marchal/publications/CC&Q.pdf
http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.pdf
or in links to this list. You can find them in my webpage (URL below).
> Tegmark is certainly aware of the need for a measure to allow
> statements about the probability of finding oneself (1-person pov,
> OK?) in a universe with certain properties. This is listed in
> astro-ph/0302131 as a "horrendous" problem to which he tentatively
> offers what looks suspiciously like Schmidhuber's (or whoever's)
> Universal Prior as a solution.
Could be promising heuristic, but is deeply wrong. I mean I am myself
very suspicious that Universal Prior can be used as an explanation per
se.
>
> (Of course, this means he tacitly accepts the restriction to
> computable functions).
You cannot really be tacit about this. If only because you can gives
them basic role in more than one way. Tegmark is unclear, at least..
>
> So I don't agree that the problem can't be addressed by Tegmark,
> although it hasn't been. Unless by "addressed" you mean "solved", in
> which case I agree!
To adress the problem you need to be ontologically clear.
>
> Let's suppose with Wei Dai that a measure can be applied to Tegmark's
> everything. It certainly can to the set of UTM programs as per
> Schmidhuber and related proposals.
Most such proposals are done by people not aware of the 1-3
distinction. In the approach I have developed that difference is
crucial.
> Obviously it is possible to assign a measure which solves the White
> Rabbit problem, such as the UP. But to me this procedure is very
> suspicious.
I agree. You can serach my discussion with Schmidhuber on this list.
(search on the name "marchal", not "bruno marchal": it is an old
discussion we did have some years ago).
> We can get whatever answer we like by picking the right measure.
I mainly agree.
> While the UP and similar are presented by their proponents as
> "natural", my strong suspicion is that if we lived in a universe that
> was obviously algorithmically very complex, we would see papers
> arguing for "natural" measures that reward algorithmic complexity. In
> fact the White Rabbit argument is basically an assertion that such
> measures *are* natural. Why one measure rather than another? By the
> logic of Tegmark's original thesis, we should consider the set of all
> possible measures over everything. But then we need a measure on the
> measures, and so ad infinitum.
I mainly agree.
>
> One self-consistent approach is Lewis', i.e. to abandon all talk of
> measure, all anthropic predictions, and just to speak of possibilities
> rather than probabilities. This suited Lewis fine, but greatly
> undermines the attractiveness of the everything thesis for physicists.
With comp the measure *is* on the *possibilities*, themselves captured
by the "maximal consistent extensions" in the sense of the logicians.
I have not the time to give detail, but in july or augustus, I can give
you all the details in case you are interested ....
>
> <SNIP>
>> more or less recently in the scientific american. I'm sure Tegmark's
>> approach, which a priori does not presuppose the comp hyp, would
>> benefit from category theory: this one put structure on the possible
>> sets of mathematical structures. Lawvere rediscovered the
>> Grothendieck toposes by trying (without success) to get the category
>> of all categories. Toposes (or Topoi) are categories formalizing
>> first person universes of mathematical structures. There is a
>> North-holland book on "Topoi" by Goldblatt which is an excellent
>> introduction to toposes for ... logicians (mhhh ...).
>>
>> Hope that helps,
>>
>> Bruno
>
> Not really. I know category theory is a potential route into this, but
> I havn't seen any definitive statements and from what I've read on
> this list I don't expect to any time soon. I'm certainly not going to
> learn category theory myself!
At least you don't need them for reading my work. I have suppressed all
need to it because it is a difficult theory for those who have not a
sufficiently "algebraic mind". In the long run I believe they will be
inescapable though. If only to learn knot theory, which I have reason
to believe as being very fundamental for extracting geometry from the
UTM introspection (as comp forces us to believe unless my thesis is
wrong somewhere ...).
>
> You overlooked a couple of direct queries to you in my posting:
>
> * You still havn't explained why you say his system is "too big
> (inconsistent)". Especially the inconsistent bit. I'm sure a level of
> explanation is possible which doesn't take the whole of category
> theory for granted. Also, if you have a proof that his system is
> inconsistent, you should publish it.
I was alluding to the fact that no mathematician has ever succeed to
define the "whole of math" in a consistent way, and since, logics shows
you just cannot do it. The idea is that if you can define the whole of
math, you transform it into a mathematical object living with other
mathematical objects around. Direct argument can be done by the
diagonalization technic. See perhaps the book by Grim "the incomplete
universe" which is very clear on that point:
Grim, P. (1991). The Incomplete Universe. The MIT Press, Cambridge,
USA.
>
> * Is it correct to say that category theory cannot define "the whole"
> because it is outside the heirarchy of the cardinals?
Not really. This is just a technical problem which can be overcomed in
some nice or less nice way. But category theory is hardly the whole of
math!
>
> And another mathematical query for you or anyone on the list:
>
> I've overlooked until now the fact that mathematical physics restricts
> itself to (almost-everywhere) differentiable functions of the
> continuum. What is the cardinality of the set of such functions? I
> rather suspect that they are denumerable, hence exactly representable
> by UTM programs.
> Perhaps this is what Russell Standish meant.
Beside the error you corrected yourself, I'm afraid that the
mathematical nature of physical objects, if that exists, depends on to
many other questions. Mathematically, with some notion of computability
on the reals (which I prefer not to use, because unlike computability
on the natural numbers, there is no "church's thesis" bearing on it,
and no standard on which everyone agree) it is possible to argue on the
existence of non turing emulable solution of some "resonnable"
differential equation.
But we don't need this. I have argue (let us say) that if WE are turing
emulable, then reality, whatever it is, cannot be turing emulable.
>
> I must insist though, that there exist mathematical objects in
> platonia which require c bits to describe (and some which require
> more), and hence can't be represented either by a UTM program or by
> the output of a UTM.
> Hence Tegmark's original everything is bigger than Schmidhuber's. But
> these structures are so arbitrary it is hard to imagine SAS in them,
> so maybe it makes no anthropic difference.
That's why I insist so much that Schmidhuber's does not take the comp
hyp seriously enough. Comp is necessarily much closer to Tegmark's
ontology, but then Tegmark miss comp. Both seems not to be aware that
the mind-body problem is not yet solved. And both overlooks the
consequence of incompleteness phenomenon (although Schmidhuber on that
point is probably (just) a little bit closer to my work).
Bruno
http://iridia.ulb.ac.be/~marchal/
Received on Tue May 24 2005 - 07:37:52 PDT