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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

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.

NoahZuniga 1y ago
> You can't do rigorous calculus (i.e. real analysis) on rationals alone. Yep, but that wasn't my point. 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.
- dullcrisp 1y ago
  You can make this statement for any dense subset of the reals, but we don’t because that would be silly. 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.
  - Garlef 1y ago
    I think it makes much more sense to make this statement for the rational numbers: It's the smallest field inside the real numbers that contains the naturals. So every subset that allows you to do your daily calculations contains the rationals.
    - dullcrisp 1y ago
      They’re a field by construction, and yes, the initial field of characteristic zero, but otherwise don’t arise in any natural way. They’ll be there if you’re studying fields, but exact division by arbitrary integers doesn’t seem to be a very natural property outside the reals. Again, imagine doing any practical computations with rationals and see how far you get before resorting to decimal approximation. 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.
  - tomasson 1y ago
    Here’s my plug for p-adic numbers! So cool
Kranar 1y ago
You don't need the full set of real numbers to do physics, only the computable subset of the real numbers. Using the full reals is mostly done out of simplicity.
- gpm 1y ago
  What do you mean by "the computable subset of the reals" formally? 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?
  - Kranar 1y ago
    A computable real number is a real number for which a Turing Machine exists that can compute it to any arbitrary precision. 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.
    - skybrian 1y ago
      That doesn’t sound right to me. What about the machines that don’t halt? You can’t compute whether or not to skip them directly. > 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-numbers-that-overwhelm-ordinary-math-20250822/ https://www.quantamagazine.org/busy-beaver-hunters-reach-num...
      - Kranar 1y ago
        What specifically doesn't sound right?
BeetleB 1y ago
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).
- omnicognate 1y ago
  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 https://www.dpmms.cam.ac.uk/~wtg10/reals.html
- variadix 1y ago
  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.

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.

NoahZuniga 1y ago

> You can't do rigorous calculus (i.e. real analysis) on rationals alone. Yep, but that wasn't my point. 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.

dullcrisp 1y ago
You can make this statement for any dense subset of the reals, but we don’t because that would be silly. 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.
- Garlef 1y ago
  I think it makes much more sense to make this statement for the rational numbers: It's the smallest field inside the real numbers that contains the naturals. So every subset that allows you to do your daily calculations contains the rationals.
  - dullcrisp 1y ago
    They’re a field by construction, and yes, the initial field of characteristic zero, but otherwise don’t arise in any natural way. They’ll be there if you’re studying fields, but exact division by arbitrary integers doesn’t seem to be a very natural property outside the reals. Again, imagine doing any practical computations with rationals and see how far you get before resorting to decimal approximation. 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.
- tomasson 1y ago
  Here’s my plug for p-adic numbers! So cool

Kranar 1y ago

You don't need the full set of real numbers to do physics, only the computable subset of the real numbers. Using the full reals is mostly done out of simplicity.

gpm 1y ago
What do you mean by "the computable subset of the reals" formally? 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?
- Kranar 1y ago
  A computable real number is a real number for which a Turing Machine exists that can compute it to any arbitrary precision. 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.
  - skybrian 1y ago
    That doesn’t sound right to me. What about the machines that don’t halt? You can’t compute whether or not to skip them directly. > 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-numbers-that-overwhelm-ordinary-math-20250822/ https://www.quantamagazine.org/busy-beaver-hunters-reach-num...
    - Kranar 1y ago
      What specifically doesn't sound right?

BeetleB 1y ago

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).

omnicognate 1y ago
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 https://www.dpmms.cam.ac.uk/~wtg10/reals.html
variadix 1y ago
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.