From: holmes@garnet.idbsu.edu (Randall Holmes)
Newsgroups: sci.logic
Subject: Re: The Mathematical Universe
Keywords: inconsistent multiplicities
Date: 9 Oct 92 16:24:17 GMT
Organization: Boise State University

In article <1992Oct8.155519.20021@tamsun.tamu.edu> cmenzel@tamsun.tamu.edu (Chris Menzel) writes:
>In article <1992Oct8.050325.22915@guinness.idbsu.edu> holmes@opal.idbsu.edu (Randall Holmes) writes:
>>The reason why I maintain that alternate physical universes exist is
>>that I also maintain that the physical world is a "mathematical
>>object" (say, for the sake of argument, a function with domain R^{4}
>>and values expressing the intensities of various fields at given
>>places and times; I am not committed to a specific description of this
>>kind).  
>
>Zur\"uck zu Pythagoras!  As I recall, Quine suggested a similar idea
>some years ago, but I can't remember where; "Ontological Relativity"
>maybe.  Pythagoreanism notwithstanding, I take it you don't think the
>physical world is *literally* a mathematical object, if only for the
>fact that, as you yourself hint, any number of other mathematical
>objects would have served just as well.

He did.  Indeed many other objects would have served as well (do in
fact serve as well?); our position makes it impossible to discover
which one has in fact served :-)

>
>>There are, of course, many other functions of the same general
>>kind as the one to be identified with the physical universe, some of
>>which satisfy the same "natural laws" as ours.  These latter ones are
>>the "possible worlds" to which I am referring.
>
>Seems to me that, on this story alone, there is no basis for saying
>that one of these alternative functions satisfies the same natural
>laws as ours, since those functions only describe what actually
>happens in fact in those worlds.  To say that certain natural laws
>obtain you have to be able to say that those worlds support certain
>sorts of couterfactuals, i.e., relative to those worlds you have to be
>able to be able to talk about what *would* happen under certain
>conditions, and for this you need at least some notion of "closeness"
>or "similarity" of worlds a la Robert Stalnaker or David Lewis for
>understanding counterfactual discourse.  (E.g., p counterfactually
>implies q iff q is true in the closest world in which p is true.)
>Even this though is rather uncomfortably Humean for us gonzo
realists.

To say that they satisfy certain natural laws is to say that they have
certain mathematical properties -- generally ones which explain a lot
of their structure "economically" (and so make predictions possible,
on our level).  Once I have indicated what the natural laws are, I
know which worlds are "possible" :-) (there are plenty of "impossible"
ones left over which do not satisfy those laws).  There may be
different systems of natural law, even ones which are incompatible
under certain circumstances which do not actually arise in the present
world, which are equally true of the present world and generate quite
different systems of "possible worlds".  How do I have any problem
defining closeness or similarity of worlds or parts of worlds?  It is
a straightforward mathematical exercise to define metrics on the
"space" of "possible worlds"; if I adhere to the kind of structure for
the world indicated above, metrics leap to mind.

Our determination of what natural laws obtain in our world is based on
what is actually the case.  We do not have physical access to any
counterfactual situations; anything we deduce about such situations is
based on prior determination of what laws appear to hold in the world
we are in.  We know about counterfactual worlds through our
understanding of counterfactual implication (application of natural
law to situations we have not encountered), not vice versa.

David Hume was a smart guy.  So was Pythagoras.

>
>>The opinions expressed		|     --Sincerely,
>>above are not the "official"	|     M. Randall Holmes
>>opinions of any person		|     Math. Dept., Boise State Univ.
>>or institution.			|     holmes@opal.idbsu.edu
>
>--Chris Menzel
>  Philosophy Department
>  Texas A&M University

-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu