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

From: PvM <pvm.pandas@gmail.com>
Date: Sun May 13 2007 - 14:56:35 EDT

On 5/13/07, philtill@aol.com <philtill@aol.com> wrote:
>
> Well, Pim, I am going to bow out of the conversation at this point because
> it is taking way too much time. I still contend that you are not hearing
> me. I'll close with one small final defense:
>
> I believe the main disconnect is that you do not believe that there is a
> profound mathematical basis to nature and you want to allow that nature may
> exist in a form contrary to the conclusions of mathematics as long as it
> merely mimicks mathematics at a gross level.

No no no. Not contrary to the conclusions of mathematics, on the
contrary... What I am saying is that whether or not something can be
proven to be the truth using mathematical axioms is irrelevant to the
truth of the proposition.

> If you follow the arguments of
> theorists working in cosmology, you will see that they do not hesitate to
> accept that nature is profoundly mathematical at its essence. Philosophers
> can break down what this means and why it works, but the standard practice
> is to push mathematics as far as it will go and trust that nature conforms.
> It has been eminently successful and is the basis for all physics. To
> abandon that in defense of reductionism would be to abandon science in
> defense of reductionism, which is nonsense.

And thus this is why I am not taking this strawman route.

> Whether you or I can prove that triangles can add up to 180 degrees (or any
> other such element of reality) is indeed irrelevant to the existence of
> reality as you and I have repeatedly stated. But hypothetically **if**
> nature had to begin with a small set of propositions that alone defined it,
> then it would **not** be entirely defined since it would be unable to decide
> from those propositions what the triangles add up to (or many other such
> questions). But that is nonsense, since obviously the triangles do add up
> to something because nature is entirely defined. Hence, reductio ad
> absurdum, nature does not begin with a small set of propositions that
> entirely define it. Nature is not reducible.

You are confusing the fact that using a set of axioms not all truths
within the system can be proven with 'nature not being reducible'.
Whether or not we can prove that triangles add up to something or not
has no relevance to the simple fact that the sum indeed is 180
degrees. There is a difference between provable truth and realizable
conditions.
Nature does not have to 'decide that triangle add up to a particular
value', they just do.

> Quantum mechanics is also a well-defined mathematical system that contains
> arithmetic and so as a part of reality (a complete and consistent system of
> nature) cannot be reducible to a small set of propositions.

QM just like physics contains arithmetic to allow us to capture the
essence, but as I have pointed out, a particle is not involved in
solving the QM wave equation at any position and time, a particle is
not adding differential values to its speed and position to derive its
path.

> Hier stehe ich.
>
> God bless,
> Phil
>
>
> -----Original Message-----
> From: pvm.pandas@gmail.com
> To: philtill@aol.com
> Cc: asa@calvin.edu
> Sent: Sun, 13 May 2007 1:55 PM
> Subject: Re: [asa] Godel's theorem [Was: Re: Dawkins, religion, and
> children]
>
>
> I believe that you are wrong. Godel says nothing about discovering
> these truths, just about being able to prove that they are truths
> using the axioms that form a particular system. I see this as a subtle
> but very relevant difference.
> Your example about the triangle is a good one. Does the triangle care
> or even cease to exist because a particular set of axioms cannot
> derive that the sum of its angles equals 180 degrees?
> So let's accept that physics were a Turing machine, an axiomatic
> algorithm. All this means is that we cannot prove any and all truths
> within the system of axioms. Of course, anything subjected to physics
> does not care that it cannot be proven, it can still function without
> this knowledge.
>
> You state
>
> <quote> That is, is there any finite quantity of information that is
> sufficient to allow nature itself to decide all of its own truths
> according to the physical algorithms that make up nature? </quote>
>
> I am not sure that this is what Godel argues nor do I really accept
> that nature is built around algorithms. Does an accelerating stone
> calculate it's next position and/or speed based on an algorithm or
> does it just fall in a manner consistent with an algorithm which takes
> an object of mass, calculates the accelleration due to its interaction
> with another mass and derives based on initial position and initial
> speed, it's trajectory? And in spite of all this, due to the chaotic
> nature, nature itself does not even know where the object is going to
> end up, not to mention the unpredictable nature of QM.
> And yet, science still works, physics still works, often as a good
> enough approximation or as an ensemble of many particles.
>
> You then state
>
> <quote>So in summary, I am not really concerned about nature being
> unable to discover its own truths as if those truths did not exist. I
> am instead concerned that we are unable to discover a countable set of
> axioms (or fundamental peices of information) from which nature could
> have computed all its truths if it had been required to do so. I am
> concerned we are unable to discover them not because of our epistemic
> problem, but because ontologically nature is not reducible to that
> small set of information.</quote>
>
> I find this mostly a strawman argument. Why should we expect let alone
> require nature to be able to compute all its truths? So what if we
> cannot discover all truths, that does not mean that they are not
> truths, just that they cannot be proven within a particular system of
> axioms.
>
> <quote>And as for self-existence, we are left with the question
> whether the infinite information in nature and the infinite internal
> connectedness required to ensure the information's consistency is
> better described as material or as Mind? Does nature in fact
> internally compute all the relationships to guarantee its own
> consistency, or is nature adjunct to a Mind that does the computing?
> </quote>
>
> I find your question very much begging the question. Infinite
> information in nature? Infinite internal connectedness required to
> ensure information's consistency?
>
> <quote>The truth (if this argument is correct) is that the algorithms
> of physics that cause evolution to occur (related to the information
> we see in bacterial flagella) and the algorithms of Big Bang cosmology
> (related to the information we see in spacetime curvature and angle
> summations in triangles), along with everything else we see operating
> in nature, is not truly reducible to a small set of information or a
> small set of propositions or axioms. </quote>
>
> On the contrary it IS reducible to such a set (lets not add the
> confusing concept of information lest we get lost in discussing what
> information concept and measures we are considering here) is that not
> what Godel's compleness theorem is all about? What Godel's
> incompleteness theorem states is that there are some truths which
> cannot be proven within the system of axioms. My response is simple:
> So what? My triangles still add up to 180 degrees and even if they may
> not know this, they seem to care little.
>
> On 5/11/07, philtill@aol.com <philtill@aol.com> wrote:
> >
> > Pim,
> >
> > I am sorry. I was in a hurry when I posted that last one and had already
> > regretted sending it before seeing your reply. Thank you for the gracious
> > response.
> >
> > Let me try to explain this idea again because we are still talking past
> each
> > other. While I have spent much time first working through Godel's theorem
> > and then thinking about what it means, on and off over several years, I
> have
> > not tried to **communicate** these ideas very often, and so I am still in
> > the process of learning how.
> >
> > I am not concerned at all about you or me discovering certain
> propositions.
> > (That is what most people think about when they think of Godel's
> theorem.)
> > Instead, I am concerned about NATURE discovering those propositions. If
> > Godel's theorem is correct (and it is), then ANY algorithm defined by
> axioms
> > (if they are sufficient to define arithmetic) is limited in its ability
> to
> > discover all the truths that must exist within its own system; and ANY
> > algorithm is unable to determine the consistency of the axioms that
> define
> > itself. So if nature is reducible to axioms, then the algorithms of
> nature
> > themselves are unable to determine all that nature must be and whether or
> > not its own physical information is self-consistent. ...Unless it was
> > endowed with a complete set of physics irreducible to axioms, that is.
> >
> > > <quote>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>
> >
> > This "no algorithm" includes physics, if physics is an algorithm.
> >
> > A concrete example is the question whether the angles inside a triangle
> add
> > up to pi radians, or less than pi radians, or greater than pi radians. In
> > classical geometry, unless you resolve the question by bruth force --
> that
> > is by adding another axiom to define the answer -- then geometry is
> unable
> > to determine the answer. And yet the same geometry demands that there
> must
> > be an answer. Even before we add the additional axiom we still find
> within
> > the other axioms a method to construct a triangle and to measure its
> angles
> > and to add them up. And so the smaller set of axioms demands that there
> > must be an answer to the question of the sum of the angles. And
> furthermore
> > the three possible answers are mutually exclusive -- the truth must be
> only
> > one of them. So the system defined by the smaller set of axioms may
> include
> > only one of them as truths, and MUST include on of them as truths, but
> > cannot decide which of them is the truth. We must simply decide the
> > question ourselves by further defining the system, by adding another
> axiom.
> > This is what it means when Godel tells us that any axiomatic system (if
> it
> > contains arithmetic) is unable to decide the truthfulness of all the
> truths
> > that must exist in the system.
> >
> > Likewise, if **nature** began by having written upon its fabric only the
> > same set of geometric axioms, then it would not know the answer to this
> > question, either. God would need to add more information to nature by
> > simply defining it to be one of these three possible answers. That is,
> > nature needs more defining than was provided by the other geometric
> axioms
> > in order to exist. (And according to Godel, it needs more defining that
> is
> > provided by ANY countable number of axioms in order to exist.)
> >
> > Now in fact we can break down the axioms of a system in any number of
> > different ways, and so in the physics of this universe we see that this
> > question of angles in a triangle is actually answered by a different set
> of
> > truths: it is defined by the value of the energy-density of space which
> > determines the curvature of spacetime according to the algorithm of
> general
> > relativity. But what in nature decides the energy-density of spacetime?
> > Did something earlier in the process of the Big Bang simply define what
> it
> > would be in this universe by brute force, as by adding another axiom or
> > another arbitrary piece of information to the universe, or was there
> > something even further back causally that made that decision for nature?
> If
> > it was the latter, then we have only moved the question further back one
> > step in the regress. We know that ultimately, excluding infinite
> regresses,
> > if we pursue the course of reductionism in physics we will eventually get
> to
> > a set of axioms (or a set of information) that supposedly decides
> everything
> > in nature including the energy-density of spacetime and hence the
> question
> > of the angles in a triangle.
> >
> > Or will we? Will we discover a finite quantity or even a countably
> infinite
> > set of information that is sufficient to decide the answer to every truth
> > that must exist within the system of nature? That is, is there any finite
> > quantity of information that is sufficient to allow nature itself to
> decide
> > all of its own truths according to the physical algorithms that make up
> > nature? Godel says, NO. Nature's algorithms, no more than any other
> > algorithm, cannot compute the full set of truths that must exist in its
> own
> > system if it is required to do so from an enumerable set of axioms or a
> > countable set of information. Therefore, the information in nature must
> not
> > be reducible to a countable set of equations or parameters or any other
> form
> > of information ala reductionist science. Nature must contain not only an
> > infinite set of information in its physics, but it must contain an
> infinite
> > amount to a cardinality greather than aleph-null if it is to have all the
> > truths that its own self demands that it must have.
> >
> > (BTW, this is not a make-believe example: we have actually considered
> > putting into space a constellation of spacecraft that will measure the
> > angles in a triangle in order to determine the curvature of spacetime.)
> >
> > So the real answer to this poser, if we believe that nature is complete
> and
> > consistent (and has real correspondence to logic ala mathematics
> including
> > arithmetic), is that nature cannot be described by any finite number of
> > parameters and algorithms (laws).
> >
> > So in summary, I am not really concerned about nature being unable to
> > discover its own truths as if those truths did not exist. I am instead
> > concerned that we are unable to discover a countable set of axioms (or
> > fundamental peices of information) from which nature could have computed
> all
> > its truths if it had been required to do so. I am concerned we are unable
> > to discover them not because of our epistemic problem, but because
> > ontologically nature is not reducible to that small set of information.
> >
> > If nature were reducible according to the idea of reductionist science,
> then
> > nature would have an ontological problem, being unable to decide upon all
> > its own truths of how it should BE. But since the truths really do exist
> in
> > nature, then it is actually reductionist science that has an epistemic
> > problem; not the epistemic problem associated with Godel's incompleteness
> > theorem because that is going in the opposite direction from axioms to
> > truths; science has an epistemic problem going from truths to axioms,
> since
> > the information in nature is not reducible that way. My conclusion is
> that
> > I don't think reductionist physics can ultimately prevail. We can play a
> > shell game and move the information around to see how one part of nature
> > relates to another part, and therefore we can gain great insight into how
> it
> > all behaves, but ultimately we cannot reduce the mystery of the totality
> of
> > its existence.
> >
> > Then there is the question of consistency. Again, I am not concerned
> about
> > our ability to compute nature's consistency. I am concerned about
> nature's
> > ability to compute its own consistency, if it had to begin with a axioms
> > without a God to pre-decide whether those axioms are consistent.
> >
> > Godel himself believed in God. Godel even developed his own version of an
> > ontological argument for God's existence (unrelated to his two famous
> > theorems). So Godel believed in the existence of Truth outside of
> material
> > nature. Unlike some physicists, Godel didn't believe that his theorems
> > applied to nature, perhaps because he did not believe that nature was
> > required to determine its own consistency, having God to perform that
> task.
> > However, if you do not believe in God or in the existence of any Truth or
> > Logic or Math floating in a vacuum, but believe in only that which arises
> > within the fabric of the material world, then that fabric has nothing
> other
> > than itself to determine its own consistency and completenes. According
> to
> > Godel, then, that fabric must contain an uncountably infinite amount of
> > truth information and must have performed the equivalent of an
> uncountably
> > infinite number of computations (internal connectedness within the set of
> > its information) in order to determine if it is a consistent universe. I
> am
> > happy believing that an infinite Mind is well described in this manner,
> and
> > I am not so sure that materialism is best served by such a
> non-reductionist
> > view.
> >
> > In any case -- and this at last is getting back to the original point --
> in
> > any case, this argument militates against Dawkins' imperative that
> > everything must have evolved from something simpler, and that therefore
> it
> > is unlikely that a complex God exists at the beginning of all things. The
> > truth (if this argument is correct) is that the algorithms of physics
> that
> > cause evolution to occur (related to the information we see in bacterial
> > flagella) and the algorithms of Big Bang cosmology (related to the
> > information we see in spacetime curvature and angle summations in
> > triangles), along with everything else we see operating in nature, is not
> > truly reducible to a small set of information or a small set of
> propositions
> > or axioms. Instead, reductionism is a shell game. It is a useful game
> > because it allows us to see the internal relationships in nature and to
> > understand where the arbitrary informational choices in nature are NOT.
> But
> > ultimately it does not reduce nature to any finite amount of information
> or
> > to any finite number of algorithmic computations.
> >
> > And as for self-existence, we are left with the question whether the
> > infinite information in nature and the infinite internal connectedness
> > required to ensure the information's consistency is better described as
> > material or as Mind? Does nature in fact internally compute all the
> > relationships to guarantee its own consistency, or is nature adjunct to a
> > Mind that does the computing? Either way, I do not find it improbable (as
> > Dawkins claims) to believe that a Mind, which we call God, may exist at
> the
> > beginning. Dawkins' ultimate 747 argument is not compelling to my
> thinking
> > because nature, no less than God, cannot be simple at its origin, if in
> fact
> > God does not exist. So either way, if He does exist or if He does not,
> > Dawkins' argument is still wrong.
> >
> > If you notice, I have consciously tried to do what you say Dawkins' has
> done
> > with ID. I have tried to put forth a judo maneuver against his own
> argument
> > (based on what I have long believed about Godel's theorems).
> >
> > Sorry for the very long post. I hope that explains the argument a little
> > better. And again, please forgive the impatience in my last, hurried
> post.
> >
> > God bless,
> > Phil
> >
> > Footnote: I have used the term "simple" above not in the theological
> sense
> > of indivisible divine essence, but in Dawkins' sense of information
> content.
> > God's mind knowing everything is not "simple" in this sense.
> >
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Received on Sun May 13 14:57:02 2007

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