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From: Patrick Leahy <jpl.domain.name.hidden>

Date: Tue, 24 May 2005 22:10:19 +0100 (BST)

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 <jpl.domain.name.hidden>
*

*> Sent: 23 May 2005 00:03
*

*> .
*

<SNIP>

*>> 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.

<SNIP>

*>
*

*> 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

Received on Tue May 24 2005 - 17:20:06 PDT

Date: Tue, 24 May 2005 22:10:19 +0100 (BST)

On Tue, 24 May 2005, Alastair Malcolm wrote:

<SNIP>

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.

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.

<SNIP>

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

Received on Tue May 24 2005 - 17:20:06 PDT

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