Re: use of real numbers in mathematics and physics

arivero_at_posta.unizar.es
Date: 12/02/04 

Date: Thu, 2 Dec 2004 12:18:22 +0000 (UTC)

Alfred Einstead wrote:
> baez@galaxy.ucr.edu (John Baez) wrote:
> > >That's why the error bars are intrinsic to measurement results, even to
> > >single readings. Deleting them and claiming exact measurement results is
> > >just laziness, acceptable when the resolution of an instrument is known.
> > If we fully accepted this principle, we'd have to admit the error bars
> > should have error bars themselves... and so on ad infinitum... which
> > gets a bit tiresome. Everyone gets lazy sooner or later. :-)
>
> The root of the matter, addressing the subject header itself, is that
> a real number conveys an infinite amount of information.

Er...

So, I understand that this thread is about the use of non-algebraic
numbers in mathematics and physics. Is it? Because as far as I know,
only a few of these are used (pi, e...). Any other, as sqrr(2) or the
coefficients appearing when composing angular momenta, are always
algebraic, ie, they are solutions of some equation with integer
coefficients. That is not infinite information.

By the way, I supposse this is the right method to focus physicist
approach to numbers. On one side we have measurements, and we can
debate if they are rational numbers or... hmm... "statistic numbers?".
On the next step we have equations having these numbers as
coefficients, and the numbers they appearing can be real or complex, as
they are to be interpreted as the roots of these equations. Finally we
have some deep equations which express an interactions, as for instance
the infinite process of approaching a circle with a polygon, or the
composition of infinitesimal group actions to get a finite move. In
such cases non algebraic numbers are doomed to appear, expressing our
intention of an indefinitely repeated process.

In any case, the question of the error bar seems interesting, because
it is not straightforward how it translates across an equation when
some criticality enters play.

Alejandro

If, on the
> other hand, we're to take limitations seriously such as the Bekenstein
> Bound, a finite region of space-time will only possess a finite
> capacity for information. This puts the real number in the
> spot light as an abstraction that goes further than what nature
> mandates.
>
> But this and all related issues are already handled and superseded in
> large part by the notion of a mixed state. First, the only place
> where real numbers make their full impact (i.e., where both the
> p's and q's potentially have unlimited precision together) is
> classical physics. The pure states then become singular distributions
> over phase space and a gulf of infinite information separates such
> a state from the classical mixed states (such as the grand canonical
> distribution) which may be comprise of it. So, pure states are
> out, except as idealizations and everything is mixed. Therefore,
> the last of precision is already built into the framework -- for
> the case of Classical Physics -- at the outset.
>
> In quantum physics 1/2 of the problem is already gone, since the
> p's and q's in nature no longer contain an infinite amount of
> information when taken together. The pure state then, too, have
> an inherent fuzziness associated with them.
>
> The other 1/2 of the problem still remains: the p's or the q's
> taken by themselves can be made arbitrarily precise -- which seems
> to indicate that either, alone, contains an infinity of information.
> The problem shows up, then, in the question of what q looks like
> after a "wave function collapse" and what a subsequent measurement
> of, say, p yields after q is measured.
>
> The root of the problem is then addressed by noting that the p's
> and q's are singular in the standard interpretation and one is
> forced to either (a) disallow variables with a continuous spectrum
> or (b) bring in a more generalized notion of operator -- the POVM.
>
> Lurking underneath this should be some kind of theorem to the effect
> that there is some kind of equivalence between (a) and (b) so that
> either possibility can be focused on without loss of generality.
>
> Taking the Bekenstein bound seriously seems to point, already, to
> (a). Conversely, allowing either (a) or (b) seems then to provide
> a way framework which will allow for the Bekenstein bound or something
> similar.
>
> Ultimatly, the root of the problem is the continuum: the subalgebra
> of p's or of q's each by themselves is still classical (i.e. a
> commutative algebra). But the framework for going beyond the
> continuum is already there and has already been 1/2 realized by
> the preliminary step of limiting p-q's together. The last step
> of limiting p's or q's by themselves (or calling forth the
> Bekenstein bound) is what remains.

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