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God created the rational numbers. The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Ou

God created the rational numbers. The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Our equations for the universe require a complete set, but that would be confusing the map with the territory. There is no physical evidence for uncountable infinities, those are purely in the imagination of man.

adrian_b 1y ago
The physical evidence is quite irrelevant in this case, and there also is no evidence that uncountable infinities do not exist. This is a problem of modeling optimization. The models based on uncountable "real" numbers are logically consistent and simple to use, so they are adequate for predicting what happens in natural or artificial systems. All attempts to avoid the uncountable infinities produce models that are both more complicated and also incomplete, as they do not cover all the applications of traditional infinitesimal calculus, topology and geometry. Unless someone will succeed to present a theory that avoids uncountable infinities while being as simple as the classic theory and being applicable to all the former uses, I see such attempts as interesting, but totally impractical.
- zarzavat 1y ago
  The real numbers require infinite storage and infinite computation. There are both distinctly unphysical concepts. The real numbers are a useful mathematical trick that make it possible to prove results in calculus. What you surrender in return for being able to prove statements is to give up the ability to compute expressions. This may be a worthwhile trade-off for physicists but for the universe (which does many computations and zero proofs) it's quite a burden.
andrewla 1y ago
Why are rationals special? They represent an exactness in a similarly unphysical way as the integers. The rationals are infinitely precise. 1/3 is not the same as 0.33333 or 0.33333333 or 0.3. The real numbers exist and are approximable, either by rationals or by decimal expansion. The idea of approximability and computability are the critical things, not the specific representation.
- AIPedant 1y ago
  I am confused why you think the exactness of integers and rationals is unphysical. "This egg carton has 12 eggs" is a (boring) physical statement. "You can make 1/3rd of a carton of eggs without cutting an egg" also seems perfectly physical to me. Your problem with zero-point-three-repeating is a quirk of decimal representation, not a mystical property of 1/3. Egg cartons might sound contrived but the reals don't necessarily make sense without reference to rulers, scales, etc. And in fact the defining completeness / Dedekind cut conditions for the reals are necessary for doing calculus but any physical interpretation is both pretty abstract and probably false in reality.
  - tomrod 1y ago
    I take a unit square. It's diagonal is a real number but not rational.
    - spyrja 1y ago
      OK, but surely only because the exact value of 1 exists in the first place.
      - housecarpenter 1y ago
        My first thought on reading your comment was to disagree and say no, we can have the exact value of 1, because we can choose our system of units and so we can make the square a unit square by fiat. A better way to dispute the unit square diagonal argument for the existence of sqrt(2) would be to argue that squares themselves are unphysical, since all measurements are imprecise and so we can't be sure that any two physical lengths or angles are exactly the same. But actually, this argument can also be applied to 1 and other discrete quantities. Sure, if I choose the length of some specific ruler as my unit length, then I can be sure that ruler has length 1. But if I look at any other object in the world, I can never say that other object has length exactly 1, due to the imprecision of measurements. Which makes this concept of "length exactly 1" rather limited in usefulness---in that sense, it would be fair to say the exact value of 1 doesn't exist. Overall I think 1, and the other integers, and even rational numbers via the argument of AIPendant about egg cartons, are straightforwardly physically real as measurements of discrete quantities, but for measurements of continuous quantities I think the argument about the unit square diagonal works to show that rational numbers are no more and no less physically real than sqrt(2).
nwallin 1y ago
> The universe [...] doesn't require infinite precision, Doesn't it though? What happens when three bodies in a gravitationally bound system orbit each other? Our computers can't precisely compute their interaction because our computers have limited precision and discrete timesteps. Even when we discard such complicated things as relativity, what with its Lorentz factors and whatnot. Nature can perfectly compute their interactions because it has smooth time and infinite precision.
- skulk 1y ago
  > Nature can perfectly compute their interactions because it has smooth time and infinite precision That doesn't follow. Nature can perfectly compute them because they are nature. Nowhere is it required to have infinite precision, spatial or temporal.
baxtr 1y ago
A circle seems quite ordinary at first glance, yet its area is pretty irrational.
- zarzavat 1y ago
  The area of a circle is a computable number so it can be put into one-to-one correspondence with the rationals. It's much more like a rational number than a real number, insofar as it doesn't require infinities to represent it. The set of real numbers is almost all extraneous junk that the universe definitely doesn't care about but is very important to mathematicians.

adrian_b 1y ago

The physical evidence is quite irrelevant in this case, and there also is no evidence that uncountable infinities do not exist. This is a problem of modeling optimization. The models based on uncountable "real" numbers are logically consistent and simple to use, so they are adequate for predicting what happens in natural or artificial systems. All attempts to avoid the uncountable infinities produce models that are both more complicated and also incomplete, as they do not cover all the applications of traditional infinitesimal calculus, topology and geometry. Unless someone will succeed to present a theory that avoids uncountable infinities while being as simple as the classic theory and being applicable to all the former uses, I see such attempts as interesting, but totally impractical.

zarzavat 1y ago
The real numbers require infinite storage and infinite computation. There are both distinctly unphysical concepts. The real numbers are a useful mathematical trick that make it possible to prove results in calculus. What you surrender in return for being able to prove statements is to give up the ability to compute expressions. This may be a worthwhile trade-off for physicists but for the universe (which does many computations and zero proofs) it's quite a burden.

andrewla 1y ago

Why are rationals special? They represent an exactness in a similarly unphysical way as the integers. The rationals are infinitely precise. 1/3 is not the same as 0.33333 or 0.33333333 or 0.3. The real numbers exist and are approximable, either by rationals or by decimal expansion. The idea of approximability and computability are the critical things, not the specific representation.

AIPedant 1y ago
I am confused why you think the exactness of integers and rationals is unphysical. "This egg carton has 12 eggs" is a (boring) physical statement. "You can make 1/3rd of a carton of eggs without cutting an egg" also seems perfectly physical to me. Your problem with zero-point-three-repeating is a quirk of decimal representation, not a mystical property of 1/3. Egg cartons might sound contrived but the reals don't necessarily make sense without reference to rulers, scales, etc. And in fact the defining completeness / Dedekind cut conditions for the reals are necessary for doing calculus but any physical interpretation is both pretty abstract and probably false in reality.
- tomrod 1y ago
  I take a unit square. It's diagonal is a real number but not rational.
  - spyrja 1y ago
    OK, but surely only because the exact value of 1 exists in the first place.
    - housecarpenter 1y ago
      My first thought on reading your comment was to disagree and say no, we can have the exact value of 1, because we can choose our system of units and so we can make the square a unit square by fiat. A better way to dispute the unit square diagonal argument for the existence of sqrt(2) would be to argue that squares themselves are unphysical, since all measurements are imprecise and so we can't be sure that any two physical lengths or angles are exactly the same. But actually, this argument can also be applied to 1 and other discrete quantities. Sure, if I choose the length of some specific ruler as my unit length, then I can be sure that ruler has length 1. But if I look at any other object in the world, I can never say that other object has length exactly 1, due to the imprecision of measurements. Which makes this concept of "length exactly 1" rather limited in usefulness---in that sense, it would be fair to say the exact value of 1 doesn't exist. Overall I think 1, and the other integers, and even rational numbers via the argument of AIPendant about egg cartons, are straightforwardly physically real as measurements of discrete quantities, but for measurements of continuous quantities I think the argument about the unit square diagonal works to show that rational numbers are no more and no less physically real than sqrt(2).

nwallin 1y ago

> The universe [...] doesn't require infinite precision, Doesn't it though? What happens when three bodies in a gravitationally bound system orbit each other? Our computers can't precisely compute their interaction because our computers have limited precision and discrete timesteps. Even when we discard such complicated things as relativity, what with its Lorentz factors and whatnot. Nature can perfectly compute their interactions because it has smooth time and infinite precision.

skulk 1y ago
> Nature can perfectly compute their interactions because it has smooth time and infinite precision That doesn't follow. Nature can perfectly compute them because they are nature. Nowhere is it required to have infinite precision, spatial or temporal.

baxtr 1y ago

A circle seems quite ordinary at first glance, yet its area is pretty irrational.

zarzavat 1y ago
The area of a circle is a computable number so it can be put into one-to-one correspondence with the rationals. It's much more like a rational number than a real number, insofar as it doesn't require infinities to represent it. The set of real numbers is almost all extraneous junk that the universe definitely doesn't care about but is very important to mathematicians.