Comments on `The Effect of Goedel's Theorem on Natural Laws:
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                 Comments on, "Are Maxwell's Equation's Enough?"'
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                                  George Lyons
 
                             GLyons@worldnet.att.net
 
                          (received: September 2, 1997)
 
	Regarding your two articles on Goedel's Theorem in 1994 [1,2], I am not 
a physicist, but having investigated Goedel in reference to philosophical 
questions, I find the relationship of Goedel to physics has an additional 
dimension described in some literature which these gentlemen might find 
useful. John Casti in his Beyond Belief (Casti,J. ed. Beyond Belief: 
Randomness, Prediction, and Explanation in Science. Boston: CRC Press, and 
Casti, J. 1994. Complexification. NY: Harper; Casti reports mathematics of 
Chaitin, G. 1987. Algorithmic Information Theory. Cambridge: Univ. Press) 
reports an equivalence between logical incompleteness and strange attractors, 
which your authors did not mention.  They looked more at it in terms of 
consistency proofs, of which Goedel wrote some papers, I think about General 
Relativity (from his association with Einstein at Princeton).
 
	In terms of strange attractors, systems like Maxwell's equations may 
not be solvable in practice through finite procedures.  When physical theories 
are expressed in real numbers, the measurements can incorporate infinite 
amounts of computational information, as transcendental numbers, and cannot be 
represented by a finite symbolic system.  Such a system is required for there 
to be a proof or decision of a proposition, which is not quite the same as a 
determinate result physically (there is a question, however, whether a 
physical state incorporating infinite information is really determinate). The 
abstract equations are represented, but not actual measurements.  The 
predictions of the equations, in a chaotic regime, may not remain accurate, 
even approximately, when represented only approximately; in a stable regime, 
approximations work. Chaotic predictions then become equivalent to undecidable 
propositions about the future. The problem is familiar in fluid dynamics 
computations. Undecidability is thus not a matter of paradoxes per se, but of 
computability, of whether the universe is finite and whether measurements are 
ultimately finite rationals instead of infinitely precise reals.  The issue 
goes back to Zeno's paradoxes which are still found interesting to 
mathematicians.  This situation makes the role of Goedel in physics more 
immediate than the prospect of incomplete systems of equations, as there is a 
deeper problem imbedded within any equations contemplated.
 
	There is a metaphysical question whether reality can really 
incorporate infinities hypothesized in mathematics.  Whatever paradoxes exist 
are just those associated with infinity and its indefinite qualities, the 
strange mappings of which it is capable, like a part being equivalent to the 
whole and so on.
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[1] Smith, Timothy Paul, "The Effect of Goedel's Theorem on Natural Laws:
	Comments on, `Are Maxwell's Equation's Enough?'", 
	Metaphysical Review, Oct. 1994 (vol 1, no. 4) , p 2.

[2] O'Connor, Michael, "Are Maxwell's Equation's Enough?",
	Metaphysical Review, Aug. 1994 (vol 1, no. 2) , p 1.

Additional References:
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Casti, J.L. and Karlqvist, A., eds. 1991. Beyond Belief: Randomness, 
	Prediction, and Explanation in Science. Boston: CRC Press.

Chaitin, G. 1987. Algorithmic Information Theory. Cambridge: Univ. Press.

Davis, M. 1958. Computability and Unsolvability. New York: McGraw Hill.

-------. 1965.  The Undecidable. Basic papers on Undecidable
	Propositions, Unsolvable Problems, and Computable Functions.

-------. 1976. Hilbert's Tenth Problem. Diophantine Equations, 
	Positive Aspects of A Negative Solution. In Proceedings of 
	Symposia In Pure Mathematics. Hilbert's Problems. Vol. XXVIII.
	American Mathematical Society.

Rucker, R. 1983. Infinity and the Mind.  The Science and Philosophy of 
	the Infinite. New York: Bantam.

Smullyan, Raymond. 1992. Godel's Incompleteness Theorems. New York:
	Oxford U. Press.

Stewart Ian. 1996. From Here to Infinity.  A Guide to Today's 
	Mathematics. New York: Oxford U. Press.

Wang, Hao. 1974. From Mathematics to Philosophy. New York: Humanities Press.

-------. 1987. Reflections on Kurt Godel. Cambridge, MA: MIT Press.