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In previous papers
I reviewed past contributions untill to 1970 to the debate about the relationship
of theoretical physics (=TP) with classical mathematics (=CM) which notoriously
includes actual infinity (e.g. Zermelo's axiom). In this communication
I will consider the contributions occurred after 1970 for suggesting a
new kinds of mathematics more suitable for TP. Several strategies have
been followed: 1) to deny a real role to mathematics in TP; 2) to enquire
about a suitable formalisation of quantum logic for suggesting new mathematical
tools ; 3) to suggest non-standard-analysis ; 4) to interpret the whole
TP by means of set theory of Suppes' axiomatics; 5) to develop chaos theory;
6) to suggest Weyl's elementary mathematics (=WEM); 7) to inquire in CM
by means of reverse mathematics which mathematical axioms PT needs; 8)
to suggest constructive mathematics (=CoM); 9) to bound mathematics to
discrete fields; 10) to simulate experimental physics and TP by means
of computable objects, e.g. cellular automata; 11) to bound TP by means
of some restriction of physical nature like finitism, or operativism,
or essential inaccuracy. Then, I will consider the well-qualified mathematical
strategies only, i.e. the no.s 3, 6, 7, 8 and 9. Furtherly, I will chose
strategies no.s 6, 7 and 8 in order to consider the lesser powerful kind
of mathematics which covers PT's needs.
CoM was by some authors - myself included - applied CoM to classical physics.
All they evodentiated undecidable results; in particular Pour-El and Richards
proved that some singular solutions of wave equations have not counterparts
in CoM. I interpreted such undecidabilities as pertaining not to the mathematics
of the physical theory at issue, but to which particular formulation of
the theory is investigated. Hence, a successful introduction of CoM in
a physical theory amounts to solve the problem of discovering among the
past formulations of it a constructive one, or at most to invent a new
formulation according to CoM. For ex., I proved that whereas Newton's
formulation of mechanics includes undecidable principles, L. Carnot' formulation
(1783) is free of undecidabilities. In thermodynamics, the old, "phenomenological"
formulation is constructive, while Carathéodory's one does not.
However, is commonly held that modern TP - in particular quantum mechanics
(=QM) -, represents an incomparable level of sophistication in the foundations
of TP; hence, the problem is referred by most people to this theory. In
the last years, an inconclusive debate took place among Bridges, Hellmann
and da Costa. Since, in my opinion, the undecidable problems pertain to
the particular formulation of QM one analyses, in order to obtain a positive
result one has to search among the past formulations one of them which
is in agreement with CoM. A modern version of Heisenberg's formulation
was discovered, yet under the hypothesis that continuous spectra are constructively
representable. Actually, constructive algebra is not well defined at present
time.
One more suggestion is to investigate a QM formulation based upon symmetry.
Actually in 1928 Weyl first introduced group theory in QM in a mathematical
framework which may be recognised as WEM. It is a relevant fact that reverse
mathematics characterises WEM at the lower levels of his hiearchy of kinds
of ever powerful mathematics. Nevertheless in 1988 Feferman guessed that
WEM is able to cover all mathematical need of TP. I instead proved that
WEM covers a part only of TP. I discovered that WEM fits Cavalieri's and
Torricelli's mathematics of indivisibles so that they have to be credited
as the first inventors of calculus, yet in WEM version. Moreover, WEM
supported the first statement of inertia principle in Cavalieri (1632
and then Torricelli in 1644, the same year of Descartes' statement). That
proves that since their beginnings the whole mathematics and the whole
TP severed in several foundations. Under the light of reverse mathematics,
an analysis follows on Weyl's book founding new mathematical foundations
of QM in order to decide which Weyl's results pertain to respectively
CoM, WEM, rigorous and non-standard analysis.
Drago A.: "Which kind of mathematics for quantum mechanics?
A survey and a program of research", Proceedings Conference SILFS,
Urbino, 1999, in press
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