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

From: PvM <pvm.pandas@gmail.com>
Date: Fri May 11 2007 - 12:29:42 EDT

As to 2. you claim that materialists claim that nature self exists,
but their definition of self existence is very different from your
definition. Your use of a mathematical argument fails as I have
pointed out because it says nothing about self existence.
Mathematics serve to capture physics in a format suitable for our
consumption, however nature can exist quite well without the axioms of
mathematics.

It seems to me that your use of Godel is incorrect when it comes to
self existence. As far as axioms are concerned, the fact that Godel
only applies to axioms has little relevance on reducibility.

Let me repeat an earlier quote I provided

<quote>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</quote>

In other words, even if we were to limit ourself to an axiomatic
system, the fact that within this system there are propositions we
cannot prove to be true, is only relevant to our ability to prove
something to be true, not the truth of the proposition itself. In
fact, using another system with different axioms, the truth of such
propositions can be proven as well.

You said: Well, Godel tells us about the mathematics; he tells us that
it cannot be reduced to any countable set of axioms if it contains
arithmetic.

It seems to me that your interpretation of Godel is incorrect. Have
you heard of Godel's completeness theorem? What Godel says is that not
every proposition can be proven to be true within such a system.

<quote wikipedia>t states, in its most familiar form, that in
first-order predicate calculus every logically valid formula is
provable.

The word "provable" above means that there is a formal deduction of
the formula. Such a deduction is a finite list of steps in which each
step either invokes an axiom or is obtained from previous steps by a
basic inference rule. Given such a deduction, the correctness of each
of its steps can be checked algorithmically (by a computer, for
example, or by hand).</quote>

Of course, mathematics is just a limited form of how we try to
understand the world around us. At most Godel has shown that using
mathematics alone we may be unable to capture all truths. Many of
those truths may have little relevance to physics anyway.

On 5/10/07, philtill@aol.com <philtill@aol.com> wrote:
>
> Pim,
>
> Answering your two statements, taking the more interesting (2nd) one first:
>
> 2. I'm not assuming that nature must be able to self-exist, because
> personally I believe that it is God who self-exists, not nature. But it is
> the materialists who believe that nature self-exists, and so to examine
> their belief I am trying to see if nature is the kind of thing that really
> can self-exist. To address this I am introducing an argument from
> mathematics. Mathematics does not exist by itself like an equation floating
> in a vacuum. It is not an entity in its own right: it must be written onto
> something. It must be written explicitly in a Mind that is the source of
> all logic and mathematics, or it must be written implicitly on the fabric of
> some physics that displays the patterns of mathematics. Theists believe it
> is both; materialists believe it is only the latter. If it is only the
> latter, and if we believe that the physics emerged from a simpler state,
> then we can rightly discuss that si mpler state in its relationship to its
> own implicit mathematics to see how far back we can reduce it. If we can
> reduce physics all the way back to an origin where it is supremely simple,
> then the mathematics written implicitly onto that physics at its origin must
> also have been supremely simple. Well, Godel tells us about the
> mathematics; he tells us that it cannot be reduced to any countable set of
> axioms if it contains arithmetic. Therefore, remembering that mathematics
> cannot float in a vacuum, if the mathematics have always and only been
> written on physics then the physics cannot be reducible to a state that
> supports no more than a countable set of axioms. It must have been complex
> enough to have the irreducible mathematics written on it. This has
> implications for its ability to self-exist. That was the thrust of the
> argument. I was not claiming that nature self-exists, but trying to argue
> against that belief.
>
>
> 1. I didn't assume nature must be axiomatic. I repeatedly said that one
> alternative is that nature cannot be reduced to axioms. I am happy with
> that alternative because it supports my belief that nature or whatever lies
> behind it cannot be simple in the reductionist sense.
>
> God bless,
> Phil
>
> -----Original Message-----
> From: pvm.pandas@gmail.com
> To: philtill@aol.com
> Cc: asa@calvin.edu
> Sent: Thu, 10 May 2007 12:41 PM
> Subject: Re: [asa] Godel's theorem [Was: Re: Dawkins, religion, and
> children]
>
>
> Your argument is based on several assumptions which I believe have
> made it fail to support your conclusion
>
> First of all consistency versus completeness. What if nature has given
> up consistency (Quantum Mechanics)
>
> But that is but a minor detail, the main problems with your argument
> are as follows
>
> 1. You assume that nature must be axiomatic
> 2. You assume that nature must be able to self-determine or self
> exist. I fail to see why this is true or even relevant to the
> argument. Within a particular system of axioms, one cannot prove that
> all is true. But why should nature care about mathematical proofs of
> truth? And if it were to matter, it can find another system which can
> be used to explain or determine the truth.
>
> As for mathematicians critiquing your argument. I have seen various
> people make very similar assertions about Godel's incompleteness
> theorem and have found similarly various rebuttals to such.
> Not being a mathematician myself, I found it pretty trivial to find
> the weaknesses in your claims which is the need for nature to be able
> to self determine. However I propose that Godel's theorem says nothing
> about such.
>
> On 5/9/07, philtill@aol.com <philtill@aol.com> wrote:
> >
> > Hi Pim,
> >
> > you said,
> >
> > > 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?
> >
> > I'm sorry to have to disagree but this is not correct. You are mixing
> > Godel's first and second theorems. Please research it and you'll find
> out.
> >
> > Also, I am not questioning that consistent and complete systems may
> exist,
> > or that the arithmetic inherent in nature is indeed such a system. I am
> > also not questioning the ability of scientists to develop laws of
> physics,
> > or mankind's ability to do logic and discover truth.
> >
> > What I am questioning is **nature's** ability to self-determine its own
> > truth, and therefore to self-exist. If we conceive of nature as being
> > ultimately simple, being nothing but one (or a few) mindless phenomena
> that
> > can be described as axiomatic laws executing repeatedly to bring forth
> all
> > the appearance of complexity in physics, then it may not strain us as
> hard
> > to imagine the system self-existing. But since a few simple axiomatic
> laws
> > are incapable of describing a complete and consistent arithmetical
> physics,
> > then we must concede that nature in its essence is not simple. Indeed,
> the
> > axioms at the essence of nature must be uncountably infinite, or rather
> > nature must not be reducible to axioms at all. But then it would be truly
> > amazing to think that such a complete and consistent nature self-exists
> > rather than evolving slowly from a simpler state.
> >
> > While mathematicians have criticized the various misapplications of
> Godel's
> > theorem, I have never run across anybody criticizing this particular take
> on
> > it.
>
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Received on Fri May 11 12:30:36 2007

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