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You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (

You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?

IAmBroom 1y ago
> You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). Citation needed. Especially since there are well-established math proofs of irrational numbers.
- NoahZuniga 1y ago
  The argument is essentially that you can only measure things to finite precision. And for any measurement you've made at this finite precision, there exist both infinitely rational and irrational numbers. So it's impossible to rule out that the actual value you measured is one of those infinitely many rational numbers.
  - oskaralund 1y ago
    This argument feels like it's assuming the conclusion. If in principle it is only possible to measure quantities to finite precision, then it follows logically that we couldn't tell the difference between a rational and real universe. The question is, is the premise true here?
    - BalinKing 1y ago
      AFAIK it would take an infinite amount of time to measure something to infinite precision, at least by the usual ways we’d think to do so…. I suppose one could assume a universe where that somehow isn’t the case, but (to my knowledge) that’s firmly in science-fiction territory.
      - oskaralund 1y ago
        I don't think time and measurement precision are necessarily related in that way. You can measure weight with increased precision by using a more precise scale, without increasing the time it takes to do the measurement.
        AndrewDucker 1y ago
        There are limits to precision there too. The amount of available matter to build something out of and the size you can build down to before quantum effects interfere.
        oskaralund 1y ago
        The example was only to illustrate that measurement precision is independent of the time it takes to perform the measurement.
        AIPedant 1y ago
        The real point is that it takes infinite energy to get infinite precision. Let me add that we have no clue how to do a measurement that doesn't involve a photon somewhere, which means that it's pure science fiction to think of infinite precision for anything small enough to be disturbed by a low-energy photon.
        oskaralund 1y ago
        I'm not making the case that it is possible to make measurements with infinite precision. I'm making the case that the argument "It is not possible to make measurements with infinite precision, therefore we cannot tell if we live in a rational or a real world." is begging the question. The conclusion follows logically from the premise. Unless the argument is just "we can't currently distinguish between a rational and a real world", but that seems trivial.
  - BobaFloutist 1y ago
    If I'm carrying a single apple, I can measure the number of apples I'm carrying to infinite precision. I'm carrying 1.000... apples.
    - griffzhowl 1y ago
      You're implicitly assuming your conclusion by calling it a "single" apple, which means exactly one. "Apple" is an imprecise concept, but they're often sufficiently similar that we can neglect the differences between them and count them as if they're identical objects, but this is a simplification we impose for practical purposes. Even for elementary particles, we can't be sure that all electrons, say, are exactly alike. They appear to be, and so we have no reason yet to treat them differently, but because of the imprecision of our measurements it could be that they have minutely different masses or charges. I'm not saying that's plausible, only that we don't know with certainty
- BeetleB 1y ago
  > Especially since there are well-established math proofs of irrational numbers. The logic is circular, simply because mathematicians are the ones who invented irrationals. Of course they have proofs on them. They also have proofs on lots of things that don't exist in this universe. And as I pointed out elsewhere, many analysis textbooks define a real number to be "a (converging) sequence of rationals". The notion of convergence is defined before reals even enter into the picture, and a real number is merely the identifier for a given converging sequence of rationals.
omnicognate 1y ago
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.
andrewla 1y ago
I would argue that even the rational numbers are unphysical in the same way that the integers are! The idea that a quantity like 1/3 is meaningfully different than 333/1000 or 3333333/10000000 is not really that interesting on its own; only in the course of a physical process (a computation) would these quantities be interestingly different, and then only in the sense of the degree of approximation that is required for the computation. The real numbers in the intuitionalist sense are the ground truth here in my opinion; the Cantorian real numbers are busted, and the rationals are too abstract.
shonenknifefan1 1y ago
I think this is right. Any measurement will have finite precision, so while we might be able to discover some maximum precision that the universe uses eventually, we won't ever be able to prove that the universe has infinite precision representations from finite precision measurements.
- andrewla 1y ago
  Only so long as we use the rationals as an approximation. If we expect them to be exact then they are as bad as the integers. The continuum is the reality that we have to hold to. Not the continuum in the Cantor sense, but in the intuitionalist or constructivist sense, which is continuously varying numbers that can be approximated as necessary.
alt187 1y ago
The standard construction for
threatofrain 1y ago
Another popular pedagogical pathway is to construct the reals via convergent sequences of rational numbers, i.e. Cauchy sequences.
tomrod 1y ago
Because the square root of 2 exists.
- tremon 1y ago
  How does it exist though? Does it exist because we have a notation for it, or because we know its definition? Does the number 2 itself even exist? What does it mean to say that the number 2 exists? Calculo, ergo sum?
  - tomrod 1y ago
    The number 2 simply exists independent of human intervention.
    - chopin 1y ago
      I am not convinced. There are no two equal things in nature. Numbering things, say apples, is a completely human abstraction over two different things.
      - tomrod 1y ago
        How many units of protons does Helium have?
  - jacquesm 1y ago
    > Calculo, ergo sum? Pretty much: https://en.wikipedia.org/wiki/Church_encoding https://en.wikipedia.org/wiki/Church_encoding
vessenes 1y ago
To a mathematician saying god created the integers is the same thing as saying god created the rationals: there’s a bijection between the rationals and integers. I’m not convinced that we could have our current universe without irrationals - wouldn’t things like electromagnetism and gravity work differently if forced to be quantized between rationals? Saying ‘meh it would be close enough’ might be correct but wouldn’t be enough to convince me a priori.
- tomasson 1y ago
  Yeah this is an understatement. Modern technology and the world economy require irrational numbers

IAmBroom 1y ago

> You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). Citation needed. Especially since there are well-established math proofs of irrational numbers.

NoahZuniga 1y ago
The argument is essentially that you can only measure things to finite precision. And for any measurement you've made at this finite precision, there exist both infinitely rational and irrational numbers. So it's impossible to rule out that the actual value you measured is one of those infinitely many rational numbers.
- oskaralund 1y ago
  This argument feels like it's assuming the conclusion. If in principle it is only possible to measure quantities to finite precision, then it follows logically that we couldn't tell the difference between a rational and real universe. The question is, is the premise true here?
  - BalinKing 1y ago
    AFAIK it would take an infinite amount of time to measure something to infinite precision, at least by the usual ways we’d think to do so…. I suppose one could assume a universe where that somehow isn’t the case, but (to my knowledge) that’s firmly in science-fiction territory.
    - oskaralund 1y ago
      I don't think time and measurement precision are necessarily related in that way. You can measure weight with increased precision by using a more precise scale, without increasing the time it takes to do the measurement.
      - AndrewDucker 1y ago
        There are limits to precision there too. The amount of available matter to build something out of and the size you can build down to before quantum effects interfere.
        oskaralund 1y ago
        The example was only to illustrate that measurement precision is independent of the time it takes to perform the measurement.
      - AIPedant 1y ago
        The real point is that it takes infinite energy to get infinite precision. Let me add that we have no clue how to do a measurement that doesn't involve a photon somewhere, which means that it's pure science fiction to think of infinite precision for anything small enough to be disturbed by a low-energy photon.
        oskaralund 1y ago
        I'm not making the case that it is possible to make measurements with infinite precision. I'm making the case that the argument "It is not possible to make measurements with infinite precision, therefore we cannot tell if we live in a rational or a real world." is begging the question. The conclusion follows logically from the premise. Unless the argument is just "we can't currently distinguish between a rational and a real world", but that seems trivial.
- BobaFloutist 1y ago
  If I'm carrying a single apple, I can measure the number of apples I'm carrying to infinite precision. I'm carrying 1.000... apples.
  - griffzhowl 1y ago
    You're implicitly assuming your conclusion by calling it a "single" apple, which means exactly one. "Apple" is an imprecise concept, but they're often sufficiently similar that we can neglect the differences between them and count them as if they're identical objects, but this is a simplification we impose for practical purposes. Even for elementary particles, we can't be sure that all electrons, say, are exactly alike. They appear to be, and so we have no reason yet to treat them differently, but because of the imprecision of our measurements it could be that they have minutely different masses or charges. I'm not saying that's plausible, only that we don't know with certainty
BeetleB 1y ago
> Especially since there are well-established math proofs of irrational numbers. The logic is circular, simply because mathematicians are the ones who invented irrationals. Of course they have proofs on them. They also have proofs on lots of things that don't exist in this universe. And as I pointed out elsewhere, many analysis textbooks define a real number to be "a (converging) sequence of rationals". The notion of convergence is defined before reals even enter into the picture, and a real number is merely the identifier for a given converging sequence of rationals.

omnicognate 1y ago

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.

andrewla 1y ago

I would argue that even the rational numbers are unphysical in the same way that the integers are! The idea that a quantity like 1/3 is meaningfully different than 333/1000 or 3333333/10000000 is not really that interesting on its own; only in the course of a physical process (a computation) would these quantities be interestingly different, and then only in the sense of the degree of approximation that is required for the computation. The real numbers in the intuitionalist sense are the ground truth here in my opinion; the Cantorian real numbers are busted, and the rationals are too abstract.

shonenknifefan1 1y ago

I think this is right. Any measurement will have finite precision, so while we might be able to discover some maximum precision that the universe uses eventually, we won't ever be able to prove that the universe has infinite precision representations from finite precision measurements.

andrewla 1y ago
Only so long as we use the rationals as an approximation. If we expect them to be exact then they are as bad as the integers. The continuum is the reality that we have to hold to. Not the continuum in the Cantor sense, but in the intuitionalist or constructivist sense, which is continuously varying numbers that can be approximated as necessary.

alt187 1y ago

The standard construction for

threatofrain 1y ago

Another popular pedagogical pathway is to construct the reals via convergent sequences of rational numbers, i.e. Cauchy sequences.

tomrod 1y ago

Because the square root of 2 exists.

tremon 1y ago
How does it exist though? Does it exist because we have a notation for it, or because we know its definition? Does the number 2 itself even exist? What does it mean to say that the number 2 exists? Calculo, ergo sum?
- tomrod 1y ago
  The number 2 simply exists independent of human intervention.
  - chopin 1y ago
    I am not convinced. There are no two equal things in nature. Numbering things, say apples, is a completely human abstraction over two different things.
    - tomrod 1y ago
      How many units of protons does Helium have?
- jacquesm 1y ago
  > Calculo, ergo sum? Pretty much: https://en.wikipedia.org/wiki/Church_encoding https://en.wikipedia.org/wiki/Church_encoding

vessenes 1y ago

To a mathematician saying god created the integers is the same thing as saying god created the rationals: there’s a bijection between the rationals and integers. I’m not convinced that we could have our current universe without irrationals - wouldn’t things like electromagnetism and gravity work differently if forced to be quantized between rationals? Saying ‘meh it would be close enough’ might be correct but wouldn’t be enough to convince me a priori.

tomasson 1y ago
Yeah this is an understatement. Modern technology and the world economy require irrational numbers