sci.physics.research

In article <11ab6$47c5a5f0$d88ac3c2$3834@>,
hendrik@ says...
> [Mod. note: Please keep replies on topic, . relevant to physics. -ik ]
> On Wed, 20 Feb 2008 09:37:35 +0000, Doug Sweetser wrote:
> > Hello Salviati:
> >
> >> The reals are uncountable.
> >
> > That had deep implications for our description of Nature. What those
> > implications are can be confusing.
>
> But our measurements are rationals. The rationals are countable. This
> too has deep implications for our description of Nature. The reals are
> formed by some sort of completion of the rationals (Cauchy sequences or
> Dedekind cuts; I don't care). The idea that such completion is possible
> assumes we can have arbitrarily small intervals of rational numbers. If
> we're talking about Nature, this would involve arbitrarily precise
> measurements, which we know do not exist.
>
> The closest we get to real numbers is limits of averages of ever-
> increasing numbers of measurements. But we can't make infinite numbers
> of measurements ...

I don't see that our measurements are necessarily rationals. If we
observe that something has rotated exactly once, we can also say it has
rotated through 2*pi radians, an irrational number. If we measure a
value by determining its square, the quantity itself will be a square
root, which is typically irrational.

What is correct, I think, is the second point - we cannot make infinite
numbers of measurements. That means that our description of any
physical quantity - whether it uses rationals, irrationals, or any other
sort of number - must be finite. 2*pi is written as an infinitely long
non-recurring decimal, but it can be expressed quite briefly in various
ways.

Does that mean that the quantities *themselves* (as distinct from our
descriptions) must be finitely describable? That brings us back on
topic... one possible answer, it seems to me, is that classical objects,
. objects entangled with an effectively infinite environment, may
have a state that cannot be finitely described.

- Gerry Quinn