From: holmes@garnet.idbsu.edu (Randall Holmes)
Subject: Re: Belief and science (Einstein)
Organization: Boise State University
Date: Thu, 12 Nov 1992 17:44:00 GMT

In article <emc.721579695@tomalak> emc@doe.carleton.ca (Eli Chiprout) writes:
>In <BxJ4y8.5Bs@mentor.cc.purdue.edu> hrubin@mentor.cc.purdue.edu (Herman Rubin) writes:
>>Putting on my professional hat, when one examines the process of inference
>>carefully, what is being done is using data to update beliefs.  That we do
>>not carry this out explicitly, and in fact cannot, is due the the sheer
>>size of the problem--massively larger than the size of the universe.
>
>Agreed
>
>>God knows His laws of the operation of the universe, but we are commanded
>>to work at getting approximations to them.  It is only in this way that we
>>can have mastery over haaretz.
>
>But you have come to this conclusion using the process of inference that
>is incomplete, as you have mentioned.  So how do you stand above that 
>incompleteness to propose the above as a fact rather than as a theory?
>
>I submit as well, that any being that knows the universe in its totality
>would have to contain more information than the universe: MUCH, MUCH more.
>But then we have just expanded the universe by that statement.

It is mathematically provable that the class of all true sentences in
a reasonable language is impossible to define in that language
(Tarski's Paradox).  In order to avoid this phenomenon, one has to
make the expressive capabilities of one's language _weaker_ (making
the language more expressive does not help!).  In particular, it can
be applied to ideas in the mind of God, as follows:

Let a "concept" in the mind of God be an idea of the form "x has
property P" (where property P may be utterly inconceivable to us -- it
might be infinitely complex).  Certainly God is aware of all of His
concepts.  If a is an object and "x has P" is a concept, we say that a
satisfies "x has P" exactly if a has P.  Examples of concepts: "x
loves God" is satisfies by Amir if Amir loves God.  "x loves x" is
satisfied by Amir if Amir loves Amir (note that the variable x can
appear in more than one place in the definition of a concept).  Now
consider the following concept in the mind of God: "x is a concept in
the mind of God and it is not true that x satisfies x".  Call this
concept C.  Does C satisfy C?  This asks whether the statement which
results from substituting C for x in the definition of C is true: "C
is a concept (true!) and it is not true that C satisfies C".  In other
words, the question of whether C satisfies C is precisely equivalent
to the question of whether C does _not_ satisfy C.  This is a
contradiction.  But it certainly seems reasonable that God is familiar
with concepts in this sense and that God can conceive C.

	The solution to the paradox in its normal form (which does not
mention God!) is as follows: an adequate language will be able to
express the idea of a concept (by talking about sentences in the
language and substituting names in the language for the variable x)
and thus the only ingredient of the argument which is open to
challenge is the idea that we can define truth for sentences of a
given language in that language (which blocks the definition of C).
This solution seems unsatisfactory for the theological form of the
paradox.  God must know the truth or falsehood of any proposition.

	Any suggestions?

>
>>Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
>>Phone: (317)494-6054
>>hrubin@snap.stat.purdue.edu (Internet, bitnet)  
>>{purdue,pur-ee}!snap.stat!hrubin(UUCP)
>--
>Eli Chiprout                 
>Dept. of Electronics, 
>Carleton University, Canada
>emc@doe.carleton.ca                                     


-- 
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