Re: Skolem's Paradox and why is math the way it is?

From: J.E. (troubled6man_at_yahoo.com)
Date: 09/22/04 

Date: 22 Sep 2004 08:16:37 -0700

> >Electrons are infinitely small as far as we can tell. I don't care
> >about large lumpy projectiles, but the electromagnetic FIELD of an
> >electron IS pretty darn amooth, is it smooth enough to meet your
> >criteria?
>
> Is it? Because when I conduct an experiment I see phenomena that
> suggest quantization. Then there's the whole issue of virtual
> particles, which are in principle unobservable but whose existence
> alters calculated quantities in a way that has been confirmed by
> experiment. From the outside that looks like a significant failure of
> smoothness.

This ALL depends on your model. I prefer a a wavefunction over a
configuration space of ALL particles that "could" exists and all their
possible configurations over a single minkowski vector space.
Anything in my model is observable in principle, but obvioulsy you
can't do multiple contrary experiments in the smae place at the same
time. But you could do any you like, and you'd get the results I
predict. Lumpiness breaks symmetry. If you look at a PORTION of the
wavefuntion, then you can break symmetry, and that might be USEFUL in
a calculation when the other parts don't affect what you are studying.
 
> >The models you describe have ONE function, to EASE the proving of
> >MATHEMATICAL results. Mathematician bring in an uninterpreted "i"
> >(projective definitions instead of contructive definitions) so that
> >they can prove theorems that hold for ANYTHING that meets their
> >definitions, but then it becomes VERY HARD to later make the
> >equations be again about real physical things once you step into
> >such a mathematical trap.
>
> Trap? The whole point of Mathematics is to abstract similar things and
> then prove results once that apply in a lot of different contexts. If
> you're measuring three fields, each 2' by 3', do you prove that 3x3=6
> three times, once for each field? Strangely enough, physicists do the
> same thing that you complain about in Mathematics, and a good thing,
> too; there wouldn't be any progress without it. Do you rederive, e.g.,
> the heat equations, every time that you apply them to a different
> physical system, or to you take the known solution and apply it?

The POINT of physics is to take a large set of observations and
COMPRESS their discription into a smaller set. This would be FOLLY if
the universe were not PREDICTABLE. But HOW we compress data is PART
of our THEORY (MODEL) of the universe, it's PHYSICS and not MATH.
Just because some math guy took a consisten collection of theorem and
COMPRESSED it into a smaller collection of consistent statements that
he then called AXIOMS, does NOT imply that that compression is a GOOD
compression for PHYSICS, it doesn't imply that it is NOT either, it
just doesn't imply ANYTHING about physics.

> >If we forced people to use interpreted results the whole time, then
> >they could trace their calculations to meanings.
>
> What is the interpretation of the phase of the wave function?

There IS no phase of the wave function (that's where electrodynamics
comes from) anymore than there IS a fixed point in space (that's
partly where gravitation comes from). There are RELATIVE positions in
space and RELATIVE phases of the wavefuntion. RELATIVE phases are
POTENTIALS (in the physics sense, not the logic sense), didn't you SEE
the time-evolution equation? It SAYS that the change in the phase for
a "particle" is EQUAL to the potentials acting on the "particles".
The ABSOLUTE phase is meaningless, only the relative phase.

> >Who CARES if complex numbers "solve" polynomial equations? If the
> >polynomial is 5th degree or higher then it isn't going to be solved
> >anyway,
>
> What gives you that idea?

We have to solve them numerically anyway. That's all I was trying to
say.

> >Heisenberg SHOULD have used the OLD mathematics of CLIFFORD
> >algebras,
>
> He couldn't; those are finite dimensional and the commutation
> relations are only possible in an infinite dimensional space.

Who cares about commutations? You want a wave function, you want it
to wiggle in time, you want the way it wiggles in time to be
consistent with observations. You need an infinite space because you
have an infinite number of potential particles that could be "created"
and so the CONFIGURATION space needs to be infinite dimensional. You
don't need an infinite dimensional ALGEBRA to describe how a wave
(function from an infinite configuration space to a finite dimensional
algebra) wiggles. And how it wiggles is ALL one needs for physics.

> >And using fields (in the physics sense) of clifford algebra gives
> >DIFFERNT results (like no big bang) than using manifolds,
>
> Fields over what? A flat manifold is still a manifold.

I acknowledge that one COULD make a theory on a FLAT manifold as
easily as *I* do on a vector space. The definitions of manifolds that
I've seen are a set that contains charts and that contains the flat
sections. The definitions of vactor spaces that I've seen are a set
that contains a set of vectors and that contains a a multiplication
and addition on those vectors. Vector spaces and manifold seem like
different SETS to me, but I won't say that there is NOT a set that
could be consistently interpreted as both a manifold and a vector
space, but I wouldn't be afraid to say that I doubt that they would be
interesting.

It DOES appears that your arguements on mathematics rest entirely on
what physical models I make, at least this justifies my opinion that
the basis for mathematics and the models of physics weren't
independant.

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