🚀 go-pugleaf

In article <1992Dec10.192451.9924@lmpsbbs.comm.mot.com>, bhv@areaplg2.corp.mot.com (Bronis Vidugiris) writes:
|> [...]
|> P2:  Mathematically, if some measurable phenomenon C affects the
|> probabilities of both A and B, one can mathematically state that
|> 
|> P(A) = intergal of Pa|c() * delta P(C)
|> P(B) = intergal of pb|c() * delta P(C)
|> 
|> where Pa|c is the 'conditional' probability of A given C
|> (I belive this is known as Baynes thereom)

There are several problems with this.  One is that the formal mathematical
properties of probability are not necessarily applicable to sub-QM physics.
The second is that the definition of integration is not clear when the
underlying functions are not continuous - or, at least, the number of
discontinuities are infinite (and what proof have you that they are
continuous?).  There are many other assumptions buried in the math.  
The idea that conventional mathematics is necessarily applicable to 
these physical domains is pure speculation.  Even the definition of 
*multiplication* is a human invention which we have no reason - beyond
what can be experimentally verified - to believe is applicable.

The mathematical model of QM gives purely descriptive predictions of
measurable event distributions.  Any properties of the *MODEL* which
are not verifiable properties of the real world are just speculation.
I have no qualms with speculating.  I just oppose stating these
speculations as fact.

|> [...]
|> The second is a well accepted axiom of probability theory.

The parallel postulate was a well accepted axiom of geometry too.
It's not universally applicable to reality.  It's still an important
tool in those contexts in which the formal system of Euclidean 
space is a close match to physical reality.  Maybe some of the 
properties of probability theory are not universally applicable.
What verifiable evidence have you that they apply to QM?

This is all a very subtle excursion into the philosophy of science.
The models, and particularly the mathematics, of theories are inventions
of *people*, not artifacts of the natural world.  These models can be
used and accepted only so far as they are directly verifiable.  Those
properties of the models which are not verifiable are speculative at best.  
They cannot be treated as fact.

The idea that formal systems *are* the reality they model is a recurrent
one in the philosophy of science.  Pythagoras wanted to base the entire
cosmos on integers and ratios of integers (and hid the very existence
of irrational numbers because they proved his cosmology was wrong).

The fact is, speculation about the mathematical properties of models 
*does* often lead to new verifiable discoveries.  This encourages the 
comforting notion that physics can be approached as a theorem-proving 
exercise.  But, it's not always true.  The invented formal systems 
(which is what mathematics and models are) may be incomplete.  They 
may be incapable of representing, much less proving all the true 
properties of the  real world.  They may even be contradictory with 
the real world.  It may be that we'll have to invent new kinds of
mathematics - or that the universe may not conform to *any* formal
system.

(This latter is yet a further excursion into the philosophy of science.
The idea that reality conforms to the dictates of some formal system
is what we mean by "rational universe".  The idea of science is that,
though we may be ignorant of the relationships - and perhaps even 
some of the fundamental axioms - of reality, we can - at least in
principle - discover what these are.  In short, our target is for
our models of reality to match the rules of reality *exactly*.  We
don't even know if this is really possible.  We certainly don't 
know if *any* of our present models are complete or universally
applicable.  In fact, we don't know if any of these models are 
applicable beyond the limits of our present experimental technology.)

-- 
J. Giles