I recall from my Real Analysis course that the rational numbers $\mathbb{Q}$ are not suitable for doing calculus, and I believe the reason was that $\mathbb{Q}$ does not possess the least-upper-bound property, which creates problems when defining continuity and computing limits of Riemann sums. (This is discussed in more detail here: Importance of Least Upper Bound Property of $\mathbb{R}$)
So, $\mathbb{R}$ has some desirable properties. On the other hand, I think it also has some undesirable properties. For example:
- $\mathbb{R}$ is uncountably infinite, which means we could never enumerate all real numbers, even given infinite time and space (as demonstrated by Cantor's diagonal argument)
- In contrast, the set of computable numbers is countably infinite. This means that the vast majority of real numbers cannot be described in any algorithmic way. As a consequence, it is impossible to represent the vast majority of real numbers in a computer algebra system, even given an unlimited amount of working memory.
This seems like a philosophical problem for calculus, and I'd be interested to know whether anyone has found a viable alternative to use in place of $\mathbb{R}$.
In particular, if we use the set of computable numbers instead of $\mathbb{R}$, can we define differential and integral calculus sensibly and avoid the limitations of $\mathbb{R}$ mentioned above? (After all, the "important" irrational numbers, including $e$ and $\pi$ and the algebraic irrationals, are all included in the computable numbers, so I doubt we'd miss out on any number we've ever heard of.)
Note: The Wikipedia article on computable numbers (cited above) discusses this somewhat, but the details (as of this writing) are a bit sparse. It mentions "computable analysis", a "constructivist" branch of mathematics that attempts to use computable numbers instead of $\mathbb{R}$. The article says:
The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.
but it does not say precisely what the computable numbers can or cannot be used for, especially regarding calculus.
Also, I realize that this question is entirely moot in practice, since finite computer memory means that we can only use computers to manipulate a finite set of numbers (ditto regarding our finite brains). But it still seems interesting from a theoretical perspective.
You can see at least Constructive Mathematics from Douglas Bridges and google for : Douglas Bridges, Introducing Constructive Mathematics.
You can also read online D.Bridges' FAQ list on Constructive Mathematics.
For calculus, see :
Errett Bishop, Foundations of constructive analysis (1967).
Errett Bishop & Douglas Bridges, Constructive Analysis (1985).