Re: [asa] Godel's theorem [Was: Re: Dawkins, religion, and children]

From: <>
Date: Wed May 09 2007 - 11:00:56 EDT

I do believe in free moral agency (although like the rest of us I have no firm idea how it intersects with physics). But if we want to address materialism, then we must discuss their model, which omits free moral agency. IMO, their model implies through Godel's theorem an irreducible infinity of logic that lies at the origin of nature, and this implies the plausibility of faith in God. Then, having re-introduced God to their model, we can also re-introduce the notion of free moral agency. But this comes after Godel, not before, when we take this line of argument.
-----Original Message-----
Sent: Wed, 9 May 2007 9:33 AM
Subject: RE: [asa] Godel's theorem [Was: Re: Dawkins, religion, and children]

It is true that the mathematical models of the physical aspect of Nature allow
us to make predictions of the future behavior of that aspect of Nature in terms
of initial conditions. However, the whole of Nature---physical, nonphysical,
and supernatural---is governed also by human free will and, more importantly,
God's will. This reality can never be reduced to a mathematical model, no matter
how complex.



From: on behalf of
Sent: Wed 5/9/2007 1:48 AM
Subject: [asa] Godel's theorem [Was: Re: Dawkins, religion, and children]

I changed the subject line since this is way off the original topic.
thanks for the interesting reply.
People have claimed many things about Godel's theorem and so you will find many
mathematicians reacting to them by taking a minimalist view of how the theorem
can be applied. I understand their reaction against many of the claims, but I
don't agree with the minimalist view because I am a theoretical physicist and
have been infected with the desire to describe nature as a mathematical system.
I am a realist and a believer that mathematics and its arithmentic are not
merely constructs of the human mind but integral to reality in that we apprehend
them in physics. Physics as physicists conceive of it is a Turing machine in
the sense that nature performs sequential computations through time in each
locality of space (or nonlocally in view of QM) according to well-defined
"laws", which are reducible to a minimum set. This set of laws (and I am
talking about the real laws that govern nature and not just our attempted
description of them) acts like "axio ms" that define how nat!
 ure computes the state of the universe at the next moment of time as a
deduction from the present moment of time. (I would like to modify that
statement to take out the arrow of time, but that would muddle things.)
You wrote about Godel's theorem:

> In fact, Godel's theorem
> cannot be solved by just adding more and more complexity, even in
> infinity, the problem still arises.
That is correct if we are only discussing countable infinities (cardinality
aleph-null), and that is why I used the adjective "countable". It is why the
description of Godel's theorem that you cited (not repeated here) uses the term
"computationally enumerable," which is a synonym for "countable". It is true
that in a countable infinity, the problem does indeed still arise -- a countably
infinite set of axioms will still be incomplete. But Godel's theorem does not
prove that there is any problem with completeness in a system with an
uncountable infinity of axioms (a higher cardinality), or in a system not
reducible to axioms. A realist view of physics, if it sees arithmetic as being
a part of physics and hence a part of reality, must conclude that nature cannot
be reduced to a countable set of axioms, not even a countably infinite set.
Thus, (in that view of reality) reductionist science must fail at some point,
and Dawkins' belief that everything arises from something !
 simpler cannot be true; whatever lies behind reality must in the final analysis
be infinite to a cardinality greater than aleph-null or must be irreducible to
simple laws. This does not say that anything within nature taken individually
cannot be reducible to a finite number of laws and Turing-type computations in
time (e.g., the formation of a star is described simply by physical laws), or
that the ID arguments are necessarily wrong. It only says that whatever lies
behind reality and gives the arithmetic to nature is irreducible. Also, this
does not prove that whatever lies behind reality is a Mind, but let me say
You wrote:
> Of course completeness can be obtained at the cost of
> consistency, or consistency comes at the cost of completeness, or
> perhaps the mind is not a Turing machine.
This is not really correct. Completeness and consistency are not two
alternatives that allow us to successfully obtain one of them. They are both
unreachable according to Godel's theorem for an axiomatic system containing
arithmetic and defined by a countable set of axioms. (Your quoted source was
rather muddled on this point.) Completeness cannot be obtained at the cost of
consistency. (In an inconsistent system "completeness" becomes even more
unobtainable in the sense that it becomes meaningless.) The answer to
completeness (but not to consistency) is that a complete system, if such can
exist, must not be reducible to axioms; or if it does derive from axioms, then
it must have an uncountably infinite number of axioms (cardinality greater than
Similarly, the way to obtain consistency is not to choose it as an alternative
to completeness, because according to Godel's theorem even an incomplete system
cannot be proven to be consistent by any countable set of deductions. (Note: a
finite number of Turing machines operating for eternity can produce only a
countable set of computations.) Thus, if a consistent system does exist, then
it must not be reducible to axioms, or else if it does derive from axioms then
it must have been used to complete an uncountably infinite number of
computations to discover whether its axioms are consistent. (In an axiomatic
system, the method of doing the computations is defined by the axioms.) So if
physics is real and reducible to axioms, contains arithmetic, and cannot exist
in an inconsistent state, then something must have done an uncountably infinite
number of computations before the physics can exist. We can also apply this to
God or anything else further back behind physics, i!
 n the style for which you praise Dawkins. But how can anything's existence
depend upon an outcome of its existence? I see in this a statement that nature
(or God, or whatever lies furthest back without an infinite regress) is not
reducible and was not simple at its origin. Either physics is not reducible, or
God is not reducible, etc.
This is not a complete argument that whatever lies behind reality must be a
Mind, but doesn't this begin to strike you that believing in an irreducible
Being like God as opposed to reductionist materialism is not foolishness?
I believe the kind of self-existent "complexity" in God's nature implied by this
argument still fits the theologian's definition of divine "simplicity". In its
essence the entity behind nature still has no parts and is indivisible. But it
is complex in that it must exist with knowledge of the outcome of an uncoutably
infinite number of computations as an essential characteristic. This kind of
"complexity" theologians do admit within divine simplicity. So I think Dawkins
and Platinga may be talking past each other on this point of complexity versus
simplicity. God is simple in essence but his knowledge contains everything, and
so in Dawkins' language that would mean He is complex.
We can probably hypothesize a basis for reality that is not a Mind, but which
has computed an uncountably infinite number of deductions to find the consistent
(and uncountable) set of axioms that allows reality to be both consistent (and
complete). Perhaps some analog to QM can perform this miracle by finding zero
probability in the uncountably infinite set of inconsistent descriptions for
itself -- the miracle of quantum computing! But it is strange that such a
system must be defining **itself**, not something else, while doing those
computations, and so it must perform the uncountable number of calculations
before it can exist to perform the calculations. Or maybe the parts of this
Thing that do the calculations with inconsistent laws fail to materialize
precisely because they are inconsistent. How can we ever penetrate this? You
are right that this does not prove that an infinite Mind exists, such as we may
call "God". However, I di d include the three letters "IMO"!
  because I feel that it is more reasonable to call such a basis for reality "a
Mind" rather than "physics" and to believe that it is probably a Being rather
than a thing. (And there is more to this argument than I have stated here.)
> As far as the concept of eternal, I believe the Greeks had Gods which
> were not eternal, some were born, others died.
Precisely, and that is what we find it in almost all polytheistic religions.
The gods emerged from chaos in a way that could (with some stretching) be made
out as consistent with Dawkins' world view. But Dawkins is very clear that he
is not addressing these polytheistic religions, but only the monotheistic faiths
that look back to Abraham: Judaism, Christianity, and Islam. All of these hold
that God is self-existent, "I AM that I AM. Tell them that I AM has sent you."

God bless,

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Received on Wed May 9 11:01:37 2007

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