[FOM] Incompleteness and Physics: comment 1
Allen Hazen
allenph at unimelb.edu.au
Mon Oct 13 23:18:46 EDT 2008
This string started with the question of whether Gödel's First
Incompleteness Theorem, given the incorporation of serious mathematics in
serious physical theory, implies the incompleteness of (general, i.e. "TOE")
physical theory, and if so why this doesn't seem to bother people.
At a guess, many people see the implication as obvious and so move on.
It's not QUITE an implication, for (in the abstract) it seems possible that
true physics might be formulable in a way that required only a bit of
mathematics to which the Incompleteness Theorem doesn't apply: a physics on
which the physical universe is finite and discrete might be like this. But
Gödel's theorem implies the incompleteness of a very wide range of candidate
theories.
Note that the relevant sort of completeness for a physical theory is not
that of being a Complete Theory in the logician's sense: a physical theory
could surely count as "complete" in the sense interesting to physicists
without implying every (historically given) matter of particular fact. As a
first approximation, a physical theory would count as complete if it implied
every "natural law": every generalization about the behavior of (perhaps
very precisely specified) physical systems.
Even on that understanding, however, any physical theory, for example, which
admits as physically possible a physical realization of an arbitrary Turing
Machine is bound to be incomplete: otherwise it would imply theorems of the
form "If machine X were started in configuration Y it would (would not)
eventually halt," contradicting the undecidability of the Halting Problem.
Real physical machines, of course, wear out or break down, but it seems
likely that the argument above can be generalized to cover plausible
candidates for (total) physics. If, for example, the wearing out is due to
(perhaps microscopic) random accidents, there will be some finite
probability of a machine that "ought" to halt lasting long enough to reach
its halting state, and this would be enough to ensure the incompleteness of
the physical theory.
Allen Hazen
Philosophy Department
University of Melbourne
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