[FOM] R: constructivism and physics

Antonino Drago drago at unina.it
Wed Feb 8 17:31:12 EST 2006

> Steve Awodey
> Can someone please provide some references on connections between
> constructivism in mathematics and mathematical physics? There must
> be some literature on this topic.

I posted 19/1/06 the reference of J. Phil Logic 1993 which started a debate
(about constructive mathematics and quantum mechanics). In this message I
suggested that even inertia principle, in Newton's version, is not
constructive; whereas Lazare Carnot's version is constructive. Hence, there
exist two ways of constructing theoretical physics. My paper "Which kind of
mathematics for quantum mechanics? The relevance of H. Weyl's program of
research", in A. Garola, A. Rossi (eds.): Foundations of Quantum Mechanics.
Historical Analysis and Open Questions, World Scientific, Singapore, 2000,
167-193 gives more references and suggests a solution for quantum mechanics.

>John McCarthy wrote :
>I wouldn't be surprised if chaos theory got itself into non-measurable

Chaos theory and constructive mathematics is dealt, inside a set theoretical
framework, by N.C.A. da Costa and F.A. Doria: "Undecidability and
incompleteness in classical mechanics", Int. J. Theor. Phys., 30 (1991)
1041-1073; "Dynamical systems where proving chaos is equivalent to proving
Fermat's Conjecture", Int. J. Theor. Phys., 32 (1993) 2187-2206.

>Physics has often gone in the opposite direction [of making use of
constructive mathematics].

Surely since its beginnings theoretical physicists (Cavalieri, Torricelli,
Newton) made use of infinitesimals, whereas Galilei, Huygens and Leibniz
resisted to them. Almost two centuries after, rigorous calculus confirmed
almost all the results obtained by infinitesimals; but not all.
However a not negligeable part of physicists gone in the direction of
constructive mathematics. In 1824 S. Carnot founded thermodynamics by
deliberately avoiding calculus. Then Kelvin and Clausius completed the
formulation of this theory by confirming Carnot's elementary mathematics. A
formulation of thermodynamics by means of calculus was done by Carathéodory
not before 1909; however this formulation is useless for physicists, because
its main principle is too abstract in nature.
The further historical case of quanta is too easy to be evoked as a case of
reverse path. Notice thet first Heisenberg formulated quantum mechanics by
means of matrices only, not by means of differential equations.
Similar cases occurred even in the history of mathematics. Lobachtevsky
obtained non-Euclidean geometry by claiming to be following constructive
mathematics (he re-founded calculus as an Algebra of finites, a book edited
in 1835. The case of Galois is maybe the most interesting one. He claimed to
be performing the "calculus of calculus", not calculus; indeed, before him
Lagrange by making use of sophisticated mathematics was unable to solve the
same problem.

These cases have to be stressed as counter-evidence for the sentence :
Stronger mathematics, more results.


Antonino Drago
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