You wrote about the correspondence between physical measurements and the
real numbers
> So far as I know, this arithmetical correspondence has not been
> questioned, but only assumed.
However, it has been questioned:
See
THE EMPEROR'S NEW MIND by Roger Penrose.
On page 86, there is a section on
'Reality' of real numbers
"However, the relationship between the abstractly defined 'real' numbers
and physical quantities is not as clear-cut as one might imagine."
And Roger Penrose goes on to give reasons for doubting the correspondence.
On the next page, 87, he concludes.
"Why is there so much confidence in these numbers for the accurate
description of physics, when our initial experience of the relevance of
such numbers lies in a comparatively limited range? This confidence
--perhaps misplaced-- must rest (although this fact is not often recognized)
on the logical elegance, consistency, and mathematical power of the real
number system, together with a belief in the profound mathematical
harmony of Nature."
Best wishes,
Sam Kutler