[FOM] R: Could spacetime be discrete?

[Antonino Drago]

By referring to the last message by J.F. Costa (18-1-06),

To my knowledge the debate about the uncomputable numbers in theoretical
physics was inconclusive. In my opinion it is too local, and hence too
dependent from too variables, even implicit philosophical variables.

The relationship between constructive mathematics and differential equations
was discussed in details inside a clear background (Richards-Pour-El's
backgroung is contestable) by J. Cleave: "The primitive recursive analysis
of ordinary differential equations and the complexity of their solutions" J.
Comp. Syst. Sci., 3 (1969) 447-455 and O. Aberth: "Computable Analysis and
Differential Equations", in J. Myhill, A. Kino, R.E. Vesley (eds.):
Intuitionism and Proof Theory, North-Holland, Amsterdam, 1979, 4-52 and also
in: Computable Analysis, Mc-Graw-Hill, New York, 1980, ch. 12.

More than ten years ago G. Hellmann: "Constructive mathematics and quantum
mechanics. Unbounded operators and spectral theorem", J. Phil. Logic, 22
(1993) 221-248, p. 221; "Gleason's theorem is not constructively provable",
J. Phil. Logic, 22 (1993) 193-201 questioned the adequacy of constructive
mathematics to theoretical physics; he suggested that constructive
mathematics is unable to prove the decisive Gleason theorem in quantum
mechanics. Subsequent papers by constructivists in the same Journal showed
that at least a version of Gleason theorem which is suitable for quantum
mechanics belongs to the potentialities of constructive mathematics. In this
occasion the thesis of the indispensability of classical mathematics to
theoretical physics was defeated.

Rather, it is clear that some physicists made appeal to non-operative and
non-constructive notions inside decisive topics of theoretical physics. For
instance, Newton's version of inertia principle ("Any body either in rest or
in uniform motion perseveres in this state unless a force changes it")
requires an infinite accuracy in measuring the rest or the uniform motion,
possibly along an infinite path (this criticism was advanced in the best way
by N.R. Hanson: "Newton's first Law. A Philosopher's door in Natural
Philosophy", in R.G. Colodny (ed.): Beyond the edge of certainty,
Prentice-Hall, 1965, 6-28). This requirement makes no problem in classical
mathematics, but it is rejected by constructive mathematics.
Instead in 1803 Lazare Carnot was able to suggest an operative (R. Dugas:
Histoire de la Mécanique, Griffon, Neuchatel, 1956)  and hence constructive
version of the inertia principle: "Once a body is at rest, it does not
change its state; once it is in motion by itself it does not change its
velocity and direction." In this case all measurements are approximate in
nature.

Of course this version begins a kind of mechanics which is different from
Newton's one. Hence, the kind of mathematics suitable to theoretical physics
depends from the principles - whether operative or not - of the particular
formulation of a physical theory you consider.

Best regards

Antonino Drago
via Benvenuti 5
Castelmaggiore Calci Pisa 56010
tel. 050 937493
fax 06 233242218


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