On Mon, May 23, 2005 at 12:03:55AM +0100, Patrick Leahy wrote:
...
> A very similar argument ("rubbish universes") was put forward long ago
> against David Lewis's modal realism, and is discussed in his "On the
> plurality of worlds". As I understand it, Lewis's defence was that there
> is no "measure" in his concept of "possible worlds", so it is not
> meaningful to make statements about which kinds of universe are "more
> likely" (given that there is an infinity of both lawful and law-like
> worlds). This is not a defense which Tegmark can make, since he does
> require a measure (to give his thesis some anthropic content).
>
> It seems to me that discussion on this list back in 1999 more or less
> concluded that this was a fatal objection to Tegmark's version of the
> thesis, although not to some alternatives based exclusively on UTM
> programs (e.g. Russell Standish's Occam's Razor paper).
>
> Is this a fair summary, or is anyone here prepared to defend Tegmark's
> thesis?
>
> Paddy Leahy
>
I think most of us concluded that Tegmark's thesis is somewhat
ambiguous. One "interpretation" of it that both myself and Bruno tend
to make is that it is the set of finite axiomatic systems (finite sets
of axioms, and recusively enumerated theorems). Thus, for example, the
system where the continuum hypothesis is true is a distinct
mathematical system from one where it is false.
Such a collection can be shown to be a subset of the set of
descriptions (what I call the Schmidhuber ensemble in my paper), and
has some fairly natural measures associated with it. As such, the
arguments I make in "Why Occam's razor paper" apply just as much to
Tegmark's ensemble as Schmidhuber's.
Conversely, if you wish to stand on the phrase "all of mathematics
exists" then you will have trouble defining exactly what that means,
let alone defining a measure.
Cheers
--
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Received on Sun May 22 2005 - 20:38:20 PDT