Re: White Rabbit vs. Tegmark

From: Patrick Leahy <jpl.domain.name.hidden>
Date: Mon, 23 May 2005 23:17:04 +0100 (BST)

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. 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.

(Of course, this means he tacitly accepts the restriction to computable
functions).

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!

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. 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. We can get whatever answer we like by picking the
right measure. 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.

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.

<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!

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.

* Is it correct to say that category theory cannot define "the whole"
because it is outside the heirarchy of the cardinals?

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.

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.

Paddy Leahy
Received on Mon May 23 2005 - 18:25:33 PDT