As in why bother using real numbers in physics? Mostly because you need them to make the maths rigorous. You can't do rigorous calculus (i.e. real analysis) on rationals alone.
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).
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
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.
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.”?
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.
Is sqrt(2) computable?
Is BB(777) computable?
Is [the integer that happens to be equal to BB(777), not that I can prove it, written out in normal decimal notation] computable?
So yes sqrt(2) is computable.
Every BB(n) is computable since every every natutal number can be computed. It's the BB function itself that is not computable in general, not the specific output of that function for a given input.
> A busy beaver hunter who goes by Racheline has shown that the question of whether Antihydra halts is closely related to a famous unsolved problem in mathematics called the Collatz conjecture. Since then, the team has discovered many other six-rule machines with similar characteristics. Slaying the Antihydra and its brethren will require conceptual breakthroughs in pure mathematics.
https://www.quantamagazine.org/busy-beaver-hunters-reach-num...
But given the answer, I suppose you could write a program that just returns it. This seems to hinge on the definition of “computable.” It’s an integer, so that fits the definition of a computable number.
My mistake.
Then HH the function itself is not computable, but the numbers 0 and 1, which are the only two outputs of HH are computable.
Integers themselves are always computable, even if they are the output of functions that are themselves uncomputable.
So as you noticed, it only makes sense to talk about whether a function is computable, we can't meaningfully talk of computable numbers.
So a specific BB(n) is just a number and is computable.
My point was that it is possible that all values in our universe are rational, and it wouldn't be possible for us to tell the difference between this and a universe that has irrational numbers. This fact feels pretty cursed, so I wanted to point it out.
I think the conceit is supposed to be that analysis—and therefore the reals—is the “language of nature” more so than that we can actually find the reals using scientific instruments.
To illustrate the point, using the rationals is just one way of constructing the reals. Try arguing that numbers with a finite decimal representation are the divine language of nature, for example.
Plus, maybe a hot take, but really I think there’s nothing natural about the rationals. Try using them for anything practical. If we used more base-60 instead of base-10 we could probably forget about them entirely.
So every subset that allows you to do your daily calculations contains the rationals.
I think teachers lie to children and say that decimals are just another way of representing rationals, rather than the approximation of real numbers that they are (and introduce somewhat silly things like repeating decimals to do it), which makes rationals feel central and natural. That’s certainly how it was for me until I started wondering why no programming languages come with rational number packages.