alt Hacker NewsThe problem with arbitrary real numbers is not precision, it is computability. Most real numbers we actually work with in the sciences are computable - pi, e, 2, 0.(2), and so on - they are all computable by Turing Machines. However, Turing himself showed that not all real numbers are - you can construct real numbers like Chaitin's constant that are not actually computable by a TM. In fact, almost all real numbers are non-computable (the set of computable numbers is countable, so it has the same "size" as the Naturals, while the Reals are not countable - so they are much larger).
However, this doesn't mean that TMs can't operate on such numbers. You can compute the fact that pi + pi = 2 * pi even if you can't enumerate all of pi's decimals.
That's why I added the word "arbitrary" :)
> However, this doesn't mean that TMs can't operate on such numbers.
Well, it kinda does. For example, you can't compute the sum of two arbitrary real values in a finite number of steps.
That being said, it's possible that the universe actually is not "running" on real numbers. This would also mean that our physics built on top of calculus is just an approximation.
This has always fascinated me. You can't really construct a meaningful amount of calculus without having real-valued variables. The intermediate value theorem depends on it, and it's one of the foundational parts.
I tried to construct something like "bounded error rational calculus", by trying to use rational ranges instead of real values and then constraining the range size. I was able to recover _some_ calculus, but I really have not studied it in enough details.