Alastair Malcolm wrote:
> I have found your recent posts to everything-list very interesting, and the
> ideas presented overlap to a degree with my own, but there is one question
> that I have, if I may, which I mention below.
> From: Christopher Maloney <[EMAIL PROTECTED]>
> >In Tegmark's paper,
> >in section 2G, he makes a crucial point that the fewer axioms
> >you use to define your mathematical structure, the larger is
> >the ensemble. This provides a concrete justification for the
> >principle of Occam's Razor. Similarly to the argument given
> >above, we would expect to find ourselves in worlds with fairly
> >few laws of physics, since those admit the most SAS's. You
> >can always add any bizarre behavior to the structure by adding
> >ad hoc axioms, but worlds in which that is the case
> >have a smaller measure than those that do not.
> Could you please explain how Tegmark justifies that fewer axioms give rise
> to larger ensembles? I have read his article and can't see how he has
> convincingly made a case for this. Naively one might think that there are
> more complex mathematical structures than simple ones, so that we ought to
> be in a more complex universe than we are (hence by a kind of reductio ad
> absurdem, Tegmark's scheme could not hold up).
I've reread section two of his paper, and noticed what you are talking
about -- he never does support that claim. In fact, in section 2-G, he
twice refers us to earlier in the paper, but as I said, I've just reread
it, and can't find anything that's refered to. It almost seems as if
section 2-G were originally in some other part of the paper, and he
moved it to its current spot as a last-minute edit. Evidence for that
is that he refers to "ensembles" before ever having defined what he
means by that term.
When you say "there are more complex mathematical structures than simple
ones", that is not the same as talking about the ensemble of structures
that hold a given SAS, and I think that's what Tegmark was refering to.
"Knowledge is good"
-- Emil Faber