On Tue, 24 May 2005, Alastair Malcolm wrote:

Perhaps I can throw in a few thoughts here, partly in the hope I may learn
something from possible replies (or lack thereof!).

----- Original Message -----
From: Patrick Leahy <[EMAIL PROTECTED]>
Sent: 23 May 2005 00:03


This is not a defense which Tegmark can make, since he does
require a measure (to give his thesis some anthropic content).

I don't understand this last sentence - why couldn't he use the 'Lewisian
defence' if he wanted - it is the Anthropic Principle (or just logic) that
necessitates SAS's (in a many worlds context): our existence in a world that
is suitable for us is independent of the uncountability or otherwise of the
sets of suitable and unsuitable worlds, it seems to me. (Granted he does use
the 'm' word in talking about level 4 (and other level) universes, but I am
asking why he needs it to provide 'anthropic content'.)

You have to ask what motivates a physicist like Tegmark to propose this concept. OK, there are deep metaphysical reasons which favour it, but the they arn't going to get your paper published in a physics journal. The main motive is the Anthropic Principle explanation for alleged fine tuning of the fundamental parameters. As Brandon Carter remarks in the original AP paper, this implies the existence of an ensemble. Meaning that fine tuning only ceases to be a surprise if there are lots of universes, at least some of which are congenial/cognizable. But this bare statement is not enough to do physics with. But suppose you can estimate the fraction of cognizable worlds with, say the cosmological constant Lambda less than its current value. If Lambda is an arbitrary real variable, there are continuously many such worlds, so you need a measure to do this. This allows a real test of the hypothesis: if Lambda is very much lower than it has to be anthropically, there is probably some non-anthropic reason for its low value.

(Actually Lambda does seem to be unnecessarily low, but only by one or two orders of magnitude).

The point is, without a measure there is no way to make such predictions and the AP loses its precarious claim to be scientific.

There are hints that it may be worth exploring fundamentally different
approaches to the White Rabbit problem when we consider that for Cantor the
set of all integers is the same 'size' as that of all the evens (not too
good on its own for deciding whether a randomly selected integer is likely
to come out odd or even); similarly for comparing the set of all reals
between 0 and 1000, and between 0 and 1. The standard response to this is
that one *cannot* select a real (or integer) in such circumstances - but in
the case of many worlds we *do* have a selection (the one we are in now), so
maybe there is more to be said than that of applying the Cantor approach to
real worlds, and also on random selection.

This is very reminiscent of Lewis' argument. Have you read his book? IIRC he claims that you can't actually put a measure (he probably said: you can't define probabilities) on a countably infinite set, precisely because of Cantor's pairing arguments. Which seems plausible to me.

Lewis also distinguishes between inductive failure and rubbish universes as two different objections to his model. I notice that in your articles both you and Russell Standish more or less run these together.


A final musing on finite formal systems: I have always
considered formal systems to be a provisional 'best guess' (or *maybe* 2nd
best after the informational approach) for exploring the plenitude - but it
occurs to me that non-finitary formal systems (which could inter alia
encompass the reals) may match (say SAS-relevant) finite formal systems in
simplicity terms, if the (infinite-length) axioms themselves could be
algorithmically generated. This would lead to a kind of 'meta-formal-system'
approach. Just a passing thought...

I think this is the kind of trouble you get into with the "mathematical structure" = formal system approach. If you just take the structure as mathematical objects, you are in much better shape. For instance, although there are aleph-null theorems in integer arithmetic, and a higher order of unprovable statements, you can just generate the integers with a program a few bits long. And the integers are the complete set of objects in the field of integer arithmetic. Similarly for the real numbers: if you just want to generate them all, draw a line (or postulate the complete set of infinite-length bitstrings). No need to worry about whether individual ones are computable or not.

Paddy Leahy

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