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

From: PvM <pvm.pandas@gmail.com>
Date: Wed May 09 2007 - 12:23:51 EDT

Thanks for the in depth response. I still see the problem with your
argument as follows: Godel does not say that the system cannot be
complete, it states that it cannot be proven to be complete within the
theory in question.

As far as completeness and consistent, the two may indeed may be
sacrificed, one for the other. In most cases we abandon completeness
for consistency, but what if we were to abandon consistency for
completeness? You say that this is useless, and yet this may very well
describe the mind

As the following quote says it better than I could ever

<quote>There are two very common but fallacious conclusions people
make from this, and an immense number of uncommon but equally
fallacious errors I shan't bother with. The first is that Gödel's
theorem imposes some some of profound limitation on knowledge,
science, mathematics. Now, as to science, this ignores in the first
place that Gödel's theorem applies to deduction from axioms, a useful
and important sort of reasoning, but one so far from being our only
source of knowledge it's not even funny. It's not even a very common
mode of reasoning in the sciences, though there are axiomatic
formulations of some parts of physics. Even within this comparatively
small circle, we have at most established that there are some
propositions about numbers which we can't prove formally. As Hintikka
says, "Gödel's incompleteness result does not touch directly on the
most important sense of completeness and incompleteness, namely,
descriptive completeness and incompleteness," the sense in which an
axiom systems describes a given field. In particular, the result
"casts absolutely no shadow on the notion of truth. All that it says
is that the whole set of arithmetical truths cannot be listed, one by
one, by a Turing machine." Equivalently, there is no algorithm which
can decide the truth of all arithmetical propositions. And that is
all.</quote>

http://cscs.umich.edu/~crshalizi/notebooks/godels-theorem.html

In other words Godel's incompleteness theorem refers to
1. Axiomatic formulations
2. Incompleteness of proving all truths using the axioms within the system

While concepts like Godel may seem tempting to draw further
conclusions, we have to be careful about the limitations of Godel's
theorems.

On 5/8/07, philtill@aol.com <philtill@aol.com> wrote:
>
> I changed the subject line since this is way off the original topic.
>
> Pim,
>
> 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 "axioms" that define how nature 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 more...
>
> 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 aleph-null).
>
> 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, in 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 did 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,
> Phil
>
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Received on Wed May 9 12:24:10 2007

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