Newsgroups: sci.physics
Subject: Re: Lowneheim-Skolem theorem (was: Continuos vs. discrete models)
From: burns@latcs1.lat.oz.au (Jonathan Burns)
Date: Mon, 23 Nov 1992 11:39:47 GMT
Organization: Comp Sci, La Trobe Uni, Australia



Applauding Paul Budnik's consistency in trying to find this debate
a better home in sci.logic, I nevertheless think it _is_ theoretical
physics, so here goes.

One could read the history of space and motion as follows:

(1) Pre-Continuum Era, in which a distinction between discrete and
continuous had not been clearly defined. Zeno then deduces a countably
infinite set of positions in a neighbourhood of any point, the
Pythagoreans produce irrationals, and the game is on.

(2) Continuum Era, from Liebnitz and Newton, in which ratios of
quantity measured to measuring unit are supposed to take all values
in a real interval, if they change at all from one value to another.

(3) Functional Era, from Heisenberg, in which measurables are identified
with operators on functions on a continuum.


To put this a bit more crudely, without I hope introducing any fallacy:

(1) Either here or there.

(2) Everywhere between here and there.

(3) A functional on a Hilbert space with a basis of two elements
labelled here and there.


Or, recognizing that this is assuming the Continuum Hypothesis,

(1) Aleph-0

(2) Aleph-1

(3) Aleph-2


I have no sense that an experiment could eliminate the possibility of
a "physics" built from any of these classes. It might yet turn out that
all the kinematics we need to build physics models _can_ be generated
from one countable set of positions. But surely each of the three
is good physics, in the sense of unifying descriptions and allowing
free traffic of inferences between diverse fields of observation.

For all we know, we might one day need such further abstraction in
physical description, that we'll need sets of Aleph-3,4,5,6...

What I seriously want to share, is that the "theory of change" at any
stage in physics is paradigmatic. Newton assumed an entity (Fluxions)
that could paper over Zeno's paradox. This calculus subsumed such a
wealth of description, it was then supposed that he had given an
answer to the paradox - even The Answer. Really it was the point-set
topologists who gave a rigorous answer, including Newton's as a well-
behaved part.

But then, along come Heisenberg and Dirac; they reach into pure
mathematics, and pull out ANOTHER answer! I find it just absolutely
mind-boggling that physics could encompass an entirely novel theory
of change, and that the theory should be hanging around in mathematics
to meet the need.

So, how many answers could there be? In how many ways can the concept
of change be discovered anew in mathematics? What kind of observation
could prompt us to go looking for the next one?


~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Jonathan Burns        | It's a bonanza when Veronica plays piannica
burns@latcs1.lat.oz.au| On my granda-momma's oldio piazzica
Computer Science Dept | With the whistle of the B and O
La Trobe University   | Booting out a obligatti-gattigo!  - LaFemme et Owl, '51
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Newsgroups: sci.physics,sci.astro,talk.politics.space