Newsgroups: sci.physics
Subject: Re: Axiom of Physics
From: mcirvin@husc8.harvard.edu (Mcirvin)
Date: 26 Aug 92 23:24:50 GMT

frisinv@alleg.edu writes:

>A friend and I have a small disagreement about the existance of an axiom  
>of physics. She claims that all assumptions in physics are made from  
>empirical evidence and therefore there are no axioms in physics. 

I suppose I'd be somewhere in between.  Physics isn't a single
axiomatic system in the sense that Euclidean geometry or Boolean
logic is.  Physicists certainly don't agree on what the correct
overarching fundamental theory of physics is; most will tell you
that they have no idea what it is.  So the only places where
axioms can really live is in the provisional context of theories
that are known to be incomplete.  Instead of "axioms of physics"
we have "axioms (or postulates, or whatever) of theory X."
But even there, the situation is less than immutable.

Specific theories contain postulates on which things are
built, and one can derive the rest of the theory from the postulates
(though it is rare that the derivations are airtight enough that one
can really use words like "proof" to describe them).  But different
physicists often disagree on what is a postulate and what is a
derived consequence; such things are usually more a matter of
aesthetic taste than logical necessity or even universal convention.
You can, to take a simple example, derive classical particle
mechanics from Newton's laws, or you can state everything in terms
of a Lagrangian and extremize the action.  Either set of postulates 
is a logical and pretty place to start.

>I maintain that the conservation of mass/energy is an axiom. 

In both of the cases I mentioned above, conservation of energy
is actually a derived consequence of the postulates.  Usually
it can be derived simply from the time-translation symmetry
of a theory (the symmetry that says that it doesn't matter *when*
a physical process happens) via Noether's theorem.

In fact (this is in the FAQ) in general relativity there isn't
even always a way to define a conserved quantity that you can
call "energy."  So in that case it is questionable whether
this "axiom" is even true.  Likewise, in some mathematical
treatments of quantum-mechanical problems, one can assume that
energy conservation is violated for short periods of time.
  
>There is very strong evidence that they are conserved but (to  
>my limited knowlege) there is no proof that they are conserved in all  
>cases. Thererfore, conservation of mass/energy is an axiom of physics.

Within specific theories, like classical dynamics with special
relativity, one can derive that they are always conserved.  And 
it's only within the context of specific theories that one can speak
of axioms.  I wouldn't call facts gleaned from evidence "axioms,"
I'd call them "experimental results."  If every such thing were
an axiom, physical theories would have no predictive power,
since there would simply be an axiom for every data point.

Euclid's axioms might have been based on things he observed, but
that wasn't what made them axioms; various branches of mathematics
contain axioms that have absolutely nothing to do with observed
phenomena, and as axioms, they're perfectly fine.

So I'd say that there are axioms *in* physics, but not axioms
*of* physics. 

-- 
Matt McIrvin, Cambridge, Massachusetts, USA