The Effect of Goedel's Theorem on Natural Laws:
         Comments on: "Are Maxwell's Equation's Enough?"                
        _______________________________________________
 
                      Timothy Paul Smith
 
                      Department of Physics
                      University of New Hampshire
                      Durham, New Hampshire 03824
                      tps@fermi.unh.edu

                     (received: September 29, 1994)
 
 
	A few comments on the effect of Goedel's theorem on such 
physical equations as Maxwell's ["Are Maxwell's Equation's Enough?", 
Metaphysical Review,  August 1994].  When discussing the consistency and 
completeness (and related undecidables) of Maxwell's Equations, or any 
set of equations in physics, we are walking a thin line between 
mathematics and nature. To understand the relevance of  Goedel's theorem 
on Maxwell's equations we must first understand what they are and what 
is their realm of application.  I will divided this problem into three 
parts:  nature, physics and mathematics.
 
	Nature can have no paradoxs. This is perhaps my `self-evident 
truth'.  However statements such as `self-evident' have too often lead 
investigators astray and into trouble (it is self-evident that the earth 
is flat,  and the speed of light in infinite),  so I will examine it, 
and try to present some rudimentary evidence.  As a corollary, nature 
can have no paradox because it is consistent and complete.
 
	Nature can have no paradoxs.  A ball in flight can not curve 
both right and left simultaneously.  When there appears to be a paradox 
in physics it is generally caused by our ignorance or our 
approximations. A textbook example of a paradox due to our ignorance is 
the appearance of a particle being in two places at once in the case of 
the quantum mechanics double slit experiment. If we knew nothing of 
wavefunctions it appears to be a paradox.  With the enlightment of 
quantum mechanical wavefunctions the paradox is resolved.  A textbook 
example of a paradox arising from approximation is provided by the so 
called `twins paradox' of relativity.  If we ignored the acceleration of 
the spacecraft we create for ourselves a paradox.  But nature makes no 
approximations.
 
	Nature is complete.  An inconsistent or incomplete set of 
`natural laws' would allow for situations without solutions.  Try to 
imagine a circumstance where a combination of quarks and/or galaxies can 
not evolve in time because there are no rules for their evolution.  
Excluding perhaps the initial "Big Bang" and final "Big Crunch" nature 
always proceeds in time. This does not necessarily imply or require 
determinism. Nature's consistency and completeness even allows for such 
possibility as the `many worlds' interpretation of quantum mechanics.  
By being complete there are no undecidable statements in nature.
 
	The paradox is of great importance in the development of 
Goedel's Theorem,  as was pointed out by O'Connor.  If there was no 
possibility of a paradox, there would be no such thing as Goedel's 
Theorem.  In a very real sense I believe that Goedel's Theorem doesn't 
apply to natural laws.
 
	What is physics (or biology or chemistry or any other physical 
science)?  It is the attempt to map nature into a descriptive system or 
language.  For example we can map the phenomena called electromagnetism 
into the descriptive language of vector calculus.  We call the mapping 
`Maxwell's Equations'. Since we chose as our language vector calculus we 
gain certain advantages such as calculatability,  over a different 
choice of language, such as English. We also acquire all the 
disadvantages of mathematics,  such as Goedel's Theorem.
	
	There really is no escaping Goedel's Theorem.  In mathematical 
systems such as vector calculus we will necessarily have undecidable 
statements.  These are statements whose truth can not show based upon 
the axioms of the systems, the axioms of vector calculus.
 
	Do gauge transformations qualify as undecidables?  I do not 
believe so.  I raise the example of gauges because they demonstrate that 
a set of equations may be useful,  but not absolute necessary.  Without 
gauge transformation one can still calculate the same problems that one 
calculates with the transformations,  it may just be messy.  There are 
equations consistent with  electromagnetism which you can not derive 
from Maxwell's equation,  which are just not necessary to describe the 
natural phenomena.
 
	The question remains is there anything beyond Maxwell's 
equations which are necessary to completely describe nature?   Let us 
return to the problem of mapping electromagnetism into vector calculus.  
If Maxwell's equations are a perfect map of electromagnetism then any 
vector calculus additions which Goedel's theorem tells us exist are not 
necessary for the description of nature.
 
	How could one test for consistent and completeness in Maxwell's 
equations? I suggest that the best we can do is construct a `model' as 
describe by O'Connor.  The best model of Maxwell's equations is a set of 
physical phenomena known as electromagnetism.  Now this many seams like 
cyclic logic.  Rather, it is just a consistence check ( if 
electromagnetism is equivalent to Maxwell's equations,  then Maxwell's 
equations are equivalent to electromagnetism).  It leads us to looking 
for the necessary undecidables of Maxwell's equations in the laboratory.
 
                                ***
 
	This was not the conclusion which I had hoped for, although 
perhaps it is what one should have expected.  Still, there are two 
things to note.  Experimental verification is always limited to the 
range of the experiment.  Outside that range one can never know in an 
absolute sense if Maxwell's equations are applicable.  The other point 
is that experimental evidence with is not consistent with Maxwell's 
equations may mean that they are additional  undecidables, perhaps even 
an infinite number.  It many also mean that the language we have chosen, 
ie. vector calculus,  is not the best choice for describing 
electromagnetism.
 
	In the realm which Maxwell's equations are generally applied 
they have stood the test of time and experiment remarkedly unscathed.  
They withstood, and in fact help foster the relativistic revolution.  In 
quantum mechanics Maxwell's equations have in some since been 
incorporated into Quantum Electrodynamics (QED).
 
 
                                ***
 
	Up to this point I have treated Maxwell's equations as obtained 
from experimental results.  Historically I believe this is in fact where 
Maxwell obtained them.  What may be of more interest to the 
metaphysicist is when you consider Maxwell's equations as a derived 
result,  starting from some metaphysical axioms.  If I start with what?  
Conservation of charge?  Three spatial and one temporal dimension?  The 
universality and invariance of the `laws of nature' (and physics) in  
all references frames? The invariance of the speed of light?
 
	This raises the question does Goedel's theorem apply to 
metaphysical systems? Can a valid metaphysical system support a paradox? 
I will confess that I do not know.
 
 
					Timothy Paul Smith
					Durham, New Hampshire
					September 29, 1994