FOM: Precision and Turing-computability

[wiman lucas raymond]

Hi,

With regards to Lang's quote, my guess is that there is some confusion
about what each side is talking about when they refer to "the numbers of
classical analysis."  Lang was, I believe, talking about numbers which
actually come up in computing things, or in theorems of classical
analysis (pi, e, log(x), etc).  Lang has argued some odd things in the
past, but I doubt he would argue that classical analysis involved only
computable numbers.  Classical analysis of course requires a complete
metric space, and without noncomputable numbers, we cannot have the
metric closure of the rational numbers (the reals).

It's not terrifically clear what D. Richardson thinks the numbers of
classical analysis are, but every all the classical analysists worked
with noncomputable numbers to a significant degree.  Without them,
notions of convergence, continuity, and topology become much more
complex.  

-Lucas Wiman