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

From: <philtill@aol.com>
Date: Sat May 12 2007 - 00:34:37 EDT

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 Sat May 12 00:35:08 2007

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