FOM: Precision and Turing-computability

[Insall]

On 29 May 2002, Daniel Richardson wrote:
``It may be that exact computation is quite possible for the real numbers of
classical analysis (which are only a small subset of computable reals).''

I would like to know which ``computable reals'' are not ``real numbers of
classical analysis''.  To me, classical analysis (generally) proves theorems
about ``the real numbers'', as many quotes from various sources would seem
to indicate.  (Frequently, theorems are formulated as ``Let r be a real
number...'', but usually not ``Let r be a real number of classical
analysis...''.  I was not aware that any of the classical analysts thought
that when they claimed to be working with ``the real numbers'' they were
actually working with ``only a small subset of computable reals''.)

The symbolic calculation to which much of the rest of your post refers is
also possible for arbitrary real numbers.  Consider, for example, the
reduction of the expression ``x-x=0''.  For which real numbers can this fail
to be exactly verified by a symbolic processor, such as that employed by
MathCad, or by Maple, or by Mathematica?  I'd say none, because the
processor has been provided with enough axioms for real number arithmetic to
allow perfect, exact, complete computation for ``x-x'' as an ``expression''.
(The field axioms are more than enough, and comprise a small amount of the
information we know about all of the real numbers, not just ``the computable
ones''.)

As I see it, the problem here is one of vagueness.  Your comment that

``There is not complete agreement as to what these classical numnbers are.
But they certainly include the rational numbers, and are closed under field
operations, exp and log.''

Why is there not complete agreement on what real numbers are ``the real
numbers of classical analysis''?  Well, even the concept of ``classical
analysis'', or even ``classical'', is suspect.  Does the term ``classical''
refer to some specific time period, during which specific mathematicians
lived, or do you mean specific mathematicians working on measure and
integration theory?  Do you mean only those who lived while certain
classical composers produced ``classical music''?  What about the time
period when ``classic rock and roll'' was being played on the radios?  Now,
what do you mean by ``analysis'', as opposed to some other parts of
mathematics?  Personally, I generally think of measure and integration
theory, but if you mean high school calculus, I guess even there the books
refer to ``the real numbers'', rather than some ``small subset of computable
reals''.

Matt Insall
Associate Professor of Mathematics
University of Missouri - Rolla
Department of Mathematics and Statistics
insall at umr.edu
montez at rollanet.org
http:www.umr.edu/~insall