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Quite often, when several of us talk about 'descriptions' and
'specifications' in relation to measure (or relative measure) of worlds, we are
also implicitly or explicitly referring to a corresponding underlying ontology
(so the world would not 'really' be made of 'concrete chunks of stuff') - and it
is *this* underlying ontology that determines the relative frequency or measure
of worlds.
Even if it is just a matter of comparing cardinalities, I can't see why
that of all possible invisible combinations would be less than that of the total
possible number of combinations (even assuming the unit of comparison is a s-t
whole, which I don't in my paper). Lewis says something not too dissimilar here
in 'On the Plurality of Worlds':
"We might ask how the inductively deceptive worlds compare in abundance to
the undeceptive worlds. If this is meant as a comparison of cardinalities, it
seems clear that the numbers will be equal. For the deceptive and
undeceptive worlds alike, it is easy to set a lower bound of beth-two, the
number of distributions of a two-valued magnitude over a continuum of spacetime
points; and hard to make a firm case for any higher cardinality." [p118]
(An upcoming holiday will probably prevent any further contribution to this
discussion unfortunately)
Alastair
----- Original Message -----
Sent: 14 June 2005 17:26
Subject: Re: possible solution to modal
realism's problem of induction
Hi everyone (in this world and all
relevantly similar ones :-),
I like the solution to the Induction /
Dragon / Exploding Cow problem that I see in work by Malcolm, Standish,
Tegmark, and Schmidhuber. So I forwarded references to Alexander Pruss, whose
dissertation raises the Induction Objection to modal realism. The full context
is on my blog at http://blog.360.yahoo.com/knowinghumans?p=8.
I'm interested in how the folks on this list would respond to Pruss's most
recent comment, below. Can anyone recommend a primer on probability
in transfinite contexts like ours?
--------------------------------
Remember that I am working in David
Lewis's framework. Each world is
a physical object: a bunch of matter,
connected together
spatiotemporally. So I do not need to work with
specifications, but
with concrete chunks of stuff. There is nothing
further illuminating
to be said in a lewisian context, really, about what
makes two
concrete chunks of stuff the same chunk, is there?
That
said, I am making an assumption that there is only one copy of
each world.
I suppose one could recover the "measure" the authors
you cite have if you
suppose that there is a copy of each world for
every
arrangement-description of it. But I do not see why one would
suppose
that.
In the Lewisian setting, it is intuitively plausible that the
probability that I exist in w1 should equal the probability that I
exist in w2, as long as w1 and w2 contain intelligent observers in
equal numbers. The "measures" from the authors you cite do not
satisfy
this criterion IF there is one world for a class of
equivalent
descriptions, as is going to be the case under the
assumptions I am
making.
Most observers are going to be in worlds with a much higher
cardinality of stuff than our world contains. Our world probably
only
has a finite number of particles. The cardinality of worlds
just like ours
until tomorrow but where \aleph_8 neutrons appear in
San Francisco
down-town, causing everything in the universe to
collapse is much greater
than the cardinality of regular worlds. In
fact, I think what I am saying
here will apply even on information-
theoretic measures. (The one or two
papers you linked to that I
looked at made the assumption that there was a
fixed maximum
cardinality of things. But why assume
that?)
---------------------------
For one thing, Pruss seems mistaken to
assume that a possible world consists necessarily of matter in a connected
spacetime. (I think he inherits this mistake from Lewis, who uses
spatiotemporal connectedness rather than causal connectedness to define
worlds, because Lewis wants to explain/define causality instead of making
it a primitive.) It seems better to define a possible world as a causal
closure than as a spatiotemporal closure.
But the main problem perhaps is that Pruss
misses (or disagrees with?) the point that in the information-theoretic
paradigm for specifying possible worlds, the number of worlds with
unobserved/unobservable irregularities will vastly outnumber the ones with the
observed irregularities like his example, even if those irregular worlds
vastly outnumber the lucky few worlds that are like ours and have no
irregularities whatsoever, even unobserved/unobservable ones.
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