"Quantum theorists often speak of the world as being pointillist
[i.e., discrete] at the smallest scales. Yet a closer
look at the laws of nature suggests that the physical world is
actually continuous—more analog than digital" [i.e.,
more continuous than discrete]
Hayes, Brian (2009),
"The Higher Arithmetic",
American Scientist 97 (September-October): 364–368.
"How to count to a zillion without falling off the end of
the number line". Discusses the computer representation of the
continuous real numbers using a discrete and finite subset of the natural numbers.
"From Fish to Infinity" is the first of 15 blog posts on the
nature of both discrete and continuous math. It appeared in the
New York Times. The complete set of posts—in
reverse chronological order, but you should read them
chronologically!!—can be found
here.
On Beyond Z:
Baez, John C.; & Huerta, John
(2011),
"The Strangest Numbers in String Theory",
Scientific American
304(5) (May): 60–65.
On "octonians", "a forgotten number system invented
in the 19th century [that] may provide the simplest explanation for why
our universe could have 10 dimensions. Also discusses "quaternions":
Both quaternions and octonians are generalizations of C, the
complex (or "imaginary") numbers, which are themselves a generalization
of R, the real numbers.
On the relationship between "computable" (in the intuitionistic sense of
"constructive") and "real" numbers, see this article by a famous
logician and former UB professor:
Myhill, John (1972),
"What Is a Real Number?",
American Mathematical Monthly
79(7) (August-September): 748–754.