Re: Godel's Incompleteness Theorems

[Tim Gwinn <***>]

Jamie,

I do not understand your view of Godel's incompleteness theorems. Godel's
theorems say that the idea that one can turn every mathematical system into
a strictly syntactic, axiomatic system, while retaining most or all of the
richness of the original system, is false. In effect, they imply that
semantic qualities and impredicativities are not rare occurrences in
mathematical system, which can be expunged with sufficient effort. Instead,
semantic qualities and impredicativities are the generic occurrences, and
the fully syntactic, algorithmic occurrences comprise essentially a
degenerate subset of those occurrences.

Rather than imposing "limitations" or "strictures", Godel's theorems
effectively demonstrate that the universe of mathematics is far broader and
richer than only axioms plus algorithms. The genericity of nonformalizable
systems in mathematics suggests that supposing models of physical systems in
science must be restricted to only predicative, syntactic models is
unjustified. In effect, it is far more likely that predicative, syntactic
models of the physical world will be the degenerate cases.

So I do not see how you consider Godel's theorems as "rejecting" anything,
other than rejecting simplicity (in Rosen's sense) as a universal property
of mathematics (and therefore, of mathematical models). I also do not
understand when you say Rosen "had at least identified the incompleteness
theorems as hobbled regional mathematics." This seems rather backwards. The
incompleteness theorems are what demonstrate that it is formalization which
will hobble most mathematical systems.

Further, I disagree with your comments about Rashevsky: "Rashevsky walked
there, but capitulated to the enormity of the "unknown", finding it
comfortable to accept 'unknowability' as if it were synonymous with
unprovable/unverifiable." On what grounds or evidence do you say this?? As
evidenced by the quote from Rashevsky, he was both quite aware and quite
willing to extend physics in ways that exceeded algorithmic means, such as
relational biology, as was Rosen.

Where you say:
> What I don't carry well at the moment is the realization that RR
> never looked to find an access route past Godel and thusly to
> solidify and identify an accessibility between the world of
> limited models and the extended world of the natural.
I disagree. It seems to me that this is precisely what Rosen did. When
confronted with Godel's theorems, one has two choices: either limit one's
universe of discourse (e.g. mathematical system) to one that avoids the
undecidable propositions and impredicativities by confining oneself to
predicative, syntactic, formalizable systems, or to enlarge one's universe
of discourse so as to embrace undecidables and impredicativities as entirely
valid entities in that larger universe. Both of these are conceptual
choices. The "access route" is to accept the latter conceptual view. This is
what "Life Itself" methodically seeks to argue in great detail, and what
"Essays on Life Itself" seeks to demonstrate by using a wide variety of
examples from many areas of science and mathematics.

I also don't know what you mean by "Godel spaces" or "Godel axioms". Those
do not seem to be normal mathematical terms.

Regards,
Tim