[FOM] Incompleteness and Physics
Timothy Y. Chow
tchow at alum.mit.edu
Tue Oct 28 20:54:01 EDT 2008
On Thu, 23 Oct 2008, Joe Shipman wrote:
> It's really very simple. Suppose a physically measurable quantity is,
> according to a physical theory (which one believes in for physical
> rather than mathematical reasons), a real number X that is DEFINABLE in
> ZFC but not recursive. Then some mathematical sentence of the form "the
> nth bit of X is 1" or "the nth bit of x is 0" is independent of ZFC;
> however physical experiments might nonetheless give us KNOWLEDGE of the
> nth bit of X, and therefore of the truth of a ZFC-independent statement.
While this scenario is conceivable in theory, I have a hard time imagining
a concrete instantiation that would work as you describe.
It's true that we might have a physical theory that predicts that
such-and-such a physical constant is equal to (say) Chaitin's Omega.
However, theories in physics, even the most strongly confirmed ones, are
never believed "absolutely." They are always by nature tentative, subject
to future refutation by experiment. If we were to measure the constant in
question, and started getting alleged digits of Omega that we had no other
way of obtaining, why would we believe that we were "acquiring new
mathematical knowledge" as opposed to entering a regime where we were
unable to use our theory to make empirical predictions any more? In other
words, instead of believing that we were getting new digits of Omega, we
could just as well believe that the new digits were *not* digits of Omega
and that they were empirical refuting the theory. How could we tell which
was the case? We couldn't.
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