[FOM] R: R: R: constructivism and physics
Antonino Drago
drago at unina.it
Sun Feb 19 18:13:44 EST 2006
> >Antonino Drago commented:
>
> > For your curiosity, there exist also examples of physical notions which
> > can be defined by means of infinitesimal, but not by rigorous calculus.
> > An instance is the notion of reversible process in thermodynamics: a
> > process composed by states of equilibium; i.e here the state is defined
by an
> > exact number which moreover has to express its belonging to a series
> >(process): this notion is exactly that of an infintesimal. No definition
is
> > possble in rigorous calculus, because either you have a process of
limit, i.e a
> > series of distinct approximations, or you have the final number only,
not both.
> Alasdair Urquhart replied:
> I don't see why this is inexpressible in rigorous calculus. For example,
> in Giovanni Gallavotti's "Statistical Mechanics: A short treatise"
> (Springer 1999), the notion of a reversible system is explained on
> pp. 283-284 in terms of a reversible flow, that is to say, there is
> a certain volume preserving smooth map representing time reversal
> defined on the phase space of the system. All of this is perfectly
> rigorous and expressible using the normal concepts of classical
> calculus
(Sorry, I am answering afters some days. I liked to see Gallavotti's book,
by
being I retired and far from libraries, I lost some days unsuccessfully to
obtain it).
When Gallavotti considers "a flow" it is natural that he considers a smooth
function in order to represent it. As a consequence of this definition he is
dealing with a reversible realm (apart the feature of the time reversal)
without any
possibility of distinguishing this situation from an irreversible one, for
instance an abrupt gas diffusion.
By comparing all theories of thermodynamics you can notice that
when the continuum is assumed, then reversibility merely means invariance
to time reversal; but usually reversibility is not even mentioned (for
instance in
Carathéodory's formulation).
Also when you assume that the phenomenon of friction (which is even
microscopic in nature) is a continuous function, you accept to ignore when
and whether this function may be applied to reality. More in general, you
can study any kind of geometry, but this study nothing says about the
question of which kind of geometry has to be applied to your physical
situation.
In more accurate words; the physical notion of reversibility specifies the
conditions under which you can recognise when a process can be represented
by the theory.
Thermodynamics is the very interesting theory where (Ernst Mach remarked it
in Principles of Theory of Heat (1896), Dordrecht, 1986) the continuum is
not given a priori (instead in mechanics, analytical geometry made
"natural", if not trivial, to make use of the continuum). This notion of
reversibility can be interpreted as a pre-requisite under which continuum is
applicable.
What I suggested in my previous posting is that the conditions defining
physical reversibility are fullfilled by infinitesimals (and hence by
non-standard analysis), but not by rigorous mathematics, which cannot
represent a state which at the same time is a measureless point and a
process.
[By the way, Timothy Y. Chow wrote:
Todd Wilson asked for examples, to which
Antonino Drago responded;...
clearly they were *not* examples (or even attempts at providing exmaples) of
results in nonstandard analysis.
I answer: The definition of the basic notion of a theory is not a result of
that theory? When the very basic notions of two theories are proved to be
different, the whole bodies of the results of these two theories are
affected; or not? ]
Now I add that constructive mathematics also is capable to represent this
thermodynamic state, because its point is a mere interval of approximation
belonging to an ever increasing process of approximation; hence this point
includes both the statical representation and the dynamical one. The time
reversal is interpreted as the equality of the left limit and the right
limit in any point, i.e. the punctual continuity; and the absence of
viscosity and friction along a path, as the uniform continuity; which in
thermodynamics is the pre-requisite (likely as in constructive mathematics)
to introduce the first operation of calculus, the integration (definition of
entropy).
Surprisingly enough, only rigorous mathematics, i.e. the more known
foundation
of calculus, is unable to represent a reversible process. Almost all
thermodynamics textbook testify this fact, by defning a reversible process
as a "quasi-statical" process.
It is interesting that the same interpretation of the relationship between
continuum and a theory can be applied to game theory. Both the
lemma of the supporting hyperplane and the lemma of the alternative for
matrices can be interpreted as the requirements for connecting the theory to
the continuum.
Even more interesting is the fact that Hermann Weyl himself ("An elementary
proof of a minimax theorem due to von Neumann", in H.W. Kuhn, A.W. Tucker
(eds.): Contributions to Game Theory, Princeton U.P., Princeton, 1950, vol.
1, 19-25) showed that these two lemmas require at least his kind of
mathematics
(whereas in constructive mathematics they are manifesly undecidable). Hence,
game theory can be considered a clear example of theory choosing its
appropriate kind of mathematics in Weyl mathematics. (more details in my
Finite game theory
according to constructive, Weyl's elementary, and set-theoretical
mathematics, Atti Fond. Ronchi, 57 (2002) 421-436).
Best greetings
Antonino Drago
via Benvenuti 5
Castelmaggiore Calci Pisa 56010
tel. 050 937493
fax 06 233242218
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