Paul P. Budnik Jr. wrote:
> 
>>>In the 1930's, Goedel proved that no consistent nontrivial mathematical
>>>system could be complete. One consequence is that, even in an ideal
>>>deterministic universe where all the laws and initial conditions are
>>>known, it is not possible to always predict the consequences of ones 
>>>choices.
>>
>>I'm unaware of this implication and would appreciate a reference or two
>>if you have some in mind.
>  
> Take a look at:
> http://www.mtnmath.com/whatrh/node48.html
> 

This is your own argument.  I was hoping for something in publication
by an established author.  That's not to say that you are wrong, its
just that I was hoping for a bit of corroboration for what appears
to be an assumption that doesn't appear elsewhere in the literature,
at least not to my knowledge.

> 
>>My understanding of Godel is that, strictly
>>speaking, it demonstrates that mathematical reasoning can not be
>>formalized, i.e., reduced to rules, principles, computations, etc.
>>More generally, it suggests that reasoning itself is not a formal,
>>rule driven, algorithmic process, that its NOT a matter of logic,
>>or certainly not a matter of logic alone.
> 
> 
> This is subtle and can get tricky. Roger Penrose made this argument.
> The argument is false.. A nondeterministic process like biological evolution

Why do you assume that biological evolution is non-deterministic?   All
our models of natural selection suppose the opposite, don't they (e.g.,
Maynard Smith's ESS)?

> can explore any level of mathematical reasoning by exploring an ever
> increasing
> number of paths. (A path is a sequence of species each descendant from the
> previous.) One can construct a algorithmic process for this.
> What one cannot do is construct
> an algorithmic process that decides which branches of the tree of
> evolution are pursuing true mathematics.
> 

I'm having a little trouble with understanding you here.  Why would
anything in evolution pursue "true mathematics"?  I've always thought
of evolution as a blind groping process without any intent or purpose.
I suspect you have something in mind here, but I'm having trouble
following you so far.

> 
>>Surprisingly enough, this
>>is more or less implicit in David Hume's (1739) contention that "all
>>forms of reason are nothing but comparing", suggesting a notion of
>>reasoning that might be considered more ANAlogical than logical in
>>nature.  In terms of the famous syllogism, 'All men are mortal..
>>blah blah blah, the heart of reasoning, or at least of what philosophers
>>might refer to as ampliative inference, lies in THE DISCOVERY of the
>>category "man" (Pierce's abductive reasoning), with the
identification
>>of Socrates as a member of said category with attendant properties
>>a trivial matter more akin to remembering than reasoning in the full
>>blown ampliative inference sense of the term.
> 
> 
> I hope you check out the reference. I think one needs some understanding
> of logic and the foundation of mathematics to think clearly about such
> issues.

I'm aware that Penrose failed to convince many of his peers.
But I also noticed that in the first paragraph at
Gregory Chaitin's web site, he BEGINS with the acknowledgement that
"formalism failed for reasoning", almost as an after thought.  I'm
also aware that just about every mathematician on the planet abandoned
Hilbert's plan of formalizing all of mathematics, and that Godel also
had opinions not unlike those of Penrose.  I'm also aware that
computationalist have rather fiercely resisted this interpretation,
i.e., that reason is itself "unknowable" to use Chaitin's term.
(Chaitin is currently the world's most eminent metamathematician for
those unfamiliar).

> 
> In a sense all human reasoning is analogical. But for analogical reasoning
> to exist there must be some structures that you can compare your observations
> with. 

One of the many nice things about Hume's view of reason is that it lends
itself to an evolutionary view of how reasoning might have evolved from
conditioning, one in which simple conditioning might be defined as:

    'The cognition of obvious similarity and difference' (e.g., this
     A + B sequence is similar to the one I observed previously).

and "reasoning" (ampliative inference) viewed as:

    'The cognition of abstruse similarity and difference' (e.g.,
     electricity is like water flowing in a pipe),

Viewed this way, the "structure" in a simple conditioned cognition
would become represented in the A is connected to B association, and
comparison would eventually give rise to an Aha! experience in which
it was recognized that A + B is a widespread phenomena, a category
of A + B events, if you will.  All this
would transpire at subconscious levels, precisely as it does in
full blown reasoning, which is one of the many frustrations
associated with attempting to understand the damn stuff.   :)

> It is the evolution of those brain structures that are limited by
> Goedel's theorem in any single path approach. Only unlimited diversity can 
 > lead to the unlimited evolution of structures that we can then compare to
 > the external world. It is evolution itself that creates the environment
 > where such complex structures are useful. The deepest and most subtle
 > human reasoning is needed in dealing with other evolved creatures.

I'm also not following you on this Paul.  You seem to think Godel
tells us about the nature of the physical world, where I think it
tells us about the nature of cognition.  Would this be a fair way
to say it?

PR
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