[FOM] Incompleteness and Physics

[Timothy Y. Chow]

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.

Tim