FOM: Infinite sequences in the physical world

[Fred Richman]

JoeShipman at aol.com wrote:
 
> Richman:
> 
> >> In particular, if our best theories of physics lead us to conclude
> >> that a certain reproducible experimental procedure will generate a
> >> sequence that is mathematically definable but not computable, then
> >> CT-phys will be false,
> 
> >I'm not sure what this means. "Reproducible" seems to refer to an
> >actual procedure that can be carried out. Although we are constantly
> >going back and forth between our mathematical models and the real
> >world, isn't it some sort of categorical mistake to talk about a
> >procedure of this sort generating an infinite sequence of positive
> >integers?
 
> Not at all.  The sequence doesn't have to be infinite to be
> metamathematically significant.

Maybe there is some metamathematics going on in CT-physics that I've
missed. It seems to me that we are dealing with mathematical models
and with the real world. Perhaps the phrase "mathematically definable"
is supposed to take us into the realm of metamathematics. Does the
sequence have to be infinite to be mathematically significant, or
physically significant? What is the point of the qualifier
"metamathematically"?

> The idea is that the reproducible
> experimental procedure, will, in theory, generate integers until you
> run out of raw materials or die of old age or the sun turns into a red
> giant or whatever, which will at any given time represent a finite
> initial segment of a mathematically definable nonrecursive sequence f.  

Of course we may only get a couple of dozen of those integers by the
time the sun turns into a red giant. Suppose, for example, that the
sequence is given by the digits in the decimal expansion of some
dimensionless physical constant. What kind of experiment can we run to
distinguish whether that infinite sequence of digits is recursive or
not? In fact, we may be able to compute the first hundred values of f
even if f is not recursive.

> Even if you never have the completed infinite sequence, at some point
> you have the value x of f(N) for some N large enough that ZFC does not
> settle the truth value of the mathematical assertion "f(N)=x".
> (Because if contrariwise ZFC did settle all such questions then f
> would be recursive after all.)

"At some point"? Only if the universe is very accommodating. In any
event, we will never know when we have reached that point. 

I think I understand this argument, but the question remains as to
whether it even makes (experimental) sense to say that the decimal
expansion of a physical constant is nonrecursive, or even irrational.
The logical positivists may have gotten a lot of things wrong, but
their insight that some statements should be rejected because they
have no meaning still has a lot of appeal.

--Fred