We don't need reals to make the math rigorous. Only to make the math a lot more tractable.
I've solved multiple continuous value problems by discretizing, applying combinatorics to the techniques, and then taking the limit of the result - you of course get the same result if you had simply used regular integration/differentiation, and it's a lot easier to use calculus than combinatorics.
But the point is the "rational", discretized approach will get you arbitrarily close to the answer.
It's why many analysis textbooks define a (given) real number as "a sequence of converging rational numbers" (before even defining what a limit is).
It's more about derivation of theorems than calculations.
Computation can only use rationals, and of course can get arbitrarily close to an answer because they are dense in the reals.
However, the entire edifice of analysis rests on the completeness axiom of the reals. The extreme value theorem, for example, is equivalent to the completeness axiom; the useful properties of continuous functions break down without it; the fundamental theorem of calculus doesn't work without it; Etc. So if the maths used in your physics (the structure of the theory, not just the calculations you perform with it) relies on these things at all, you're relying on the reals for confidence that the maths is sound.
Now you could argue that we don't need mathematical rigour for physics, that real analysis is a preoccupation of mathematicians, while physicists should be fine with informal calculus. I'm not going to argue that point. I'm just pointing out what the real numbers bring to the table.
Here's Tim Gowers on the subject: https://www.dpmms.cam.ac.uk/~wtg10/reals.html
The uncomputable real numbers always seemed strange to me. I can understand a convergent sequence of rationals, or the idea of a program that outputs a number to arbitrary precision, but something that cannot be computed at all is a very bizarre object. I think NJ Wildberger has some interesting ideas in this area, although I’m not sure I agree with his finititist interpretation in all circumstances. Specifically I don’t think comparisons to the number of atoms in the universe or information theoretic limits on storage based on the volume of the observable universe are interesting considerations here.
To me at least, if you can write down a finite procedure that can produce a number to arbitrary precision, I think it is fair to say the number at that limit exists.
This made me think of a possible numerical library where rather than storing numbers as arbitrary precision rationals, you could store them as the combination of inputs and functions that generate that number, and compute values to arbitrary precision.
> I've solved multiple continuous value problems by discretizing, applying combinatorics to the techniques, and then taking the limit of the result
But taking the limit of a sequence of rationals isn’t guaranteed to remain in the rationals (classic example: https://en.wikipedia.org/wiki/Basel_problem. Each partial sum is rational, but the limit of the partial sums is not)
So, how does that statement rebut “You can't do rigorous calculus (i.e. real analysis) on rationals alone.”?
> But taking the limit of a sequence of rationals isn’t guaranteed to remain in the rationals
I'm not saying it does. What I'm saying is that you can make a correspondence with the reals by using only rationals.
You can define convergence without invoking the reals (Cauchy convergence). If you take any such sequence, you give that sequence a name. That name is the equivalent of a real number. You can then define addition, multiplication - any operation on the reals - with respect to those sequences (again, invoking only rational numbers).
So far, we have two distinct entities: The rationals, and the converging sequences.
Then, if you want, you can show that if you take the rationals and those entities we're calling "converging sequences" together, you can make operations involving the two (e.g. adding a rational to that converging sequence) and eventually build up what we know to be the number line.