[FOM] Incompleteness and Physics: comment 2

Allen Hazen allenph at unimelb.edu.au
Fri Oct 17 02:33:49 EDT 2008

As a step toward sorting out the different kinds of completeness question
relevant to physics and mathematics, Harvey Friedman (excerpts from his 9
October post reproduced below) suggested looking at a particular class of
models of a physical theory.  Distinguish between the mathematical and the
physical parts of the ontology (in Quine's sense: objects over which the
variables range) of the theory: numbers and functions, etc, on the one side,
particles and spatial regions, etc, on the other.  Now restrict attention to
models of the theory in which the mathematical part is standard: the REAL
real numbers, etc.  This gets rid of incompletenesses arising from Gödel's
Incompleteness Theorem; differences between "mathematically standard" models
will reveal specifically physical incompletenesses.

A similar proposal was made by Montague(1) to try to analyze the notion of a
DETERMINISTIC theory: a theory is deterministic if, for any two
mathematically standard models of the theory, isomorphism of the
representations in the two models of the state of the physical world at one
moment implies that they will have isomorphic physical pictures for every
other moment of time.

(Montague remarks that it is uncertain whether Newtonian particle dynamics
is deterministic, and says the answer depends on open questions about the
many-body problem.  This is discussed further in John Earman's (2), pp.
30-37, where it is claimed that results of Mather and McGehee (3) and Gerver
(4) suggest that Newtonian mechanics is NOT deterministic.)

Surely the possibility of a complete physical THEORY should not depend on
whether the laws of nature are deterministic!  Suppose, for example, that in
an otherwise well-behaved universe, one particular sort of particle
exhibited irreducibly stochastic behavior.  A theory stating exactly what
the probability of such a particle's doing X in a given time interval would,
it6 seems to me, be as complete as a physicist could hope for.  (A
physicist, that is, willing to countenance the idea that God really does
play dice with the world!)   But two mathematically standard models of such
a theory, even if they have isomorphic "maps" of the location and state of
all particles at one time, could diverge at later times: one representing a
particle of the given kind as doing X and one as not.

Leaving a problem: how should one characterize the physical completeness of
non-deterministic theories?

(1) Richard Montague, "Deterministic theories," in Montague (ed. R.H.
Thomason) "Formal Philosophy," Yale U.P. 1974
(2) John Earman, "A Primer on Determinism," Reidel 1986
(3) J.N. Mather and R. McGehee, "Solutions of the collinear four-body
problem which become unbounded in a finite time," in J. Moser "Dynamical
Systems: Theory and Applications," Springer-Verlag 1975
(4) J.L. Gerver, "A possible model for a singularity without collisions in
the five-body problem," Journal of Differential Equations, vol. 52 (1984),
pp. 76-90

Allen Hazen
University of Melbourne

Comment inspired by:
On 12/10/08 8:59 AM, "Harvey Friedman" <friedman at math.ohio-state.edu> wrote:

> To focus matters, let's assume that we are talking about a theory
> where the mathematics involved is put in purely set theoretic terms.
> So we are looking at first order theories T in many sorted predicate
> calculus with a sort for sets, with epsilon between sets, and equality
> between sets.
> We have the following concepts.
> COMPLETENESS_1. Any two model of T in which the sort for sets is
> interpreted as "all sets, with ordinary membership and equality"
> satisfy the same sentences.
> COMPLETENESS_2. Any two models of T in which the sort for sets is
> interpreted as "all sets, with ordinary membership and equality" are
> isomorphic.
> Here is a semi formal conjecture:
> CONJECTURE. For any reasonably system that purports to formalize some
> physical theory incorporating mathematics, if it is Complete_1 then it
> is Complete_2. Furthermore, if it is not Complete_1, then there is a
> simple sentence that violates Complete_1.
> In particular, this Conjecture suggests that if we look at formalized
> physical theories, and we factor out the math incompleteness, then we
> get completeness issues that are of a totally different character than
> those in mathematics.

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