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I think I heard that all numbers in the real world are irrational. So that means most of math is not real, except of course the irrationals like pi :)

adastra22 3y ago
That seems like a very weird statement to make. Many numbers in the real world are integral (two objects, one electron, etc.). Thanks to quantum physics, most measurements are integral too.
- beiller 3y ago
  Explained better above. There are more irrational numbers, almost guaranteeing any number come across in nature is irrational. Interesting thought since I think one thing that makes irrational numbers is there is no function for them. So it's kind of a cheeky way to say no math formulas can ever describe the real world since all the numbers are irrational.
  - dagw 3y ago
    There are more irrational numbers, almost guaranteeing any number come across in nature is irrational There is nothing that says that the distribution between rational and irrational numbers that show up in nature is the same as the distribution in our construction of the real numbers.
  - adastra22 3y ago
    You’re missing the point: physics has shown the world to be discretized. Almost every number you go out and measure IS ACTUALLY AN INTEGER. In that sense the real number system doesn’t exist. It’s super useful, yes, but it’s an abstraction away from reality.
    - 2snakes 3y ago
      I thought that "infinities of infinities" ala Cantor exist though. I agree with you that reality being quantized means everything in physical reality can be normalized to integers.
    - eru 3y ago
      > physics has shown the world to be discretized. It's a bit more complicated than that.
      - adastra22 3y ago
        I am a physicist, and I believe what I said is accurate. Although yes I am (1) oversimplifying, and (2) assuming that some form of quantum gravity is correct.
        eru 3y ago
        To expand a bit with some examples: the energy a photon in a standing wave in a specific cavity can have is quantised. But a photon out in space can have any old energy it wants to. (Of course, a given energy level will correspond to a specific wave length etc.) Similar, an electron in a single isolated atom has specific quantised energy levels. But if you look at the electrons in a hunk of copper, they are essentially free to absorb and emit energy in almost arbitrary amounts. An even stronger example is time: as far as I am aware, time is not quantised in any of our accepted theories. There's some reasonable speculation that ultimately everything is quantised at the Planck scale, including time. But that's just a very reasonable hunch, not something that 'physics has shown'. (And you already point out that trying to marry quantum mechanics with general relativity is a hot mess.) I agree that 'quantisation all the way down' is the way to bet. But that's just speculation. (But I strongly disagree with your claim that physics is build on integers. Yes, it might be discretised, but there are plenty of discrete structures that are not integers. Look at a Rubik's cube for a simple example. On top of that: almost any real world measurement is better described by a probability distribution than by single number, be that an integer or otherwise.)
lrc 3y ago
It is better stated that: since there are "so many more" irrational numbers than rational ones, if you were to pick a real number "at random," the probability that it would be rational is zero. The "many more" and "random" ideas are made precise in measure theory (and elsewhere).
- jaza 3y ago
  There is an infinite quantity of both rational and irrational numbers, so isn't it therefore impossible for there to be more of one than of the other? Or is the reasoning that, because there is an infinite quantity of irrational numbers between any two given rational numbers, there are therefore many more irrational numbers than rational numbers? I would have thought that there being an infinite quantity of both, makes it impossible to compare the quantities.
  - defrost 3y ago
    There is a mapping from counting numbers (1, 2, 3, ...) to rationals and back again that shows these quantities are the same; for every element in set A there's an element in set B and vice versa. This is not the case for irrationals... therefore it is concluded that the infinity of irrationals is a larger infinity than the infinity of rationals. See: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://mathworld.wolfram.com/CantorDiagonalMethod.html https://mathworld.wolfram.com/CantorDiagonalMethod.html
    - thaumasiotes 3y ago
      > There is a mapping from counting numbers (1, 2, 3, ...) to rationals and back again that shows these quantities are the same; for every element in set A there's an element in set B and vice versa. That is true, but it's never taught. I don't even know what that mapping is, though I've seen it mentioned once in a popular treatment. What's taught is always the mapping from naturals to rationals that overcounts the rationals, hitting them all an infinite number of times. (Because it's very easy to show a bijection between the naturals and the ordered pairs, but while (2,3) and (4,6) are distinct ordered pairs, they do not represent distinct rationals.) But then all you've shown is that the naturals are at least as numerous as the rationals. To show that the naturals and the rationals have the same cardinality, you either rely on the idea that the naturals are the smallest infinite set, or you appeal to the fact that the naturals are a subset of the rationals.
hliyan 3y ago
I've come to the conclusion that all numbers in the real world are integers and real numbers are a human construct necessary evey time we select a unit that is too large. There is evidence (e.g. quantization) that at the most fundamental level, reality is discrete. Math is layer upon layer of abstract toolsets to operate on integers. In that sense for me, it is very real, but invented.
- russdill 3y ago
  The more interesting question is are the numbers in the universe the subset of real numbers that are computable.
  - adastra22 3y ago
    Why is that interesting?
    - russdill 3y ago
      If they are non computable, it would indicate that the universe cannot be modeled by mathematics.
      - j16sdiz 3y ago
        Modellability or Computability? There are incomputable models.
- mayd 3y ago
  Pi is no more of a human construct than any of the integers. Pi is inherent in Nature as are the integers, but more mysterious. If you accept the existence of the integers then infinity exists and therefore pi exists too; it does not need to be constructed.
  - adastra22 3y ago
    There are no circular objects in fundamental physics. Pi shows up in many physics equations, but that’s entirely due to our choice of units.
jaza 3y ago
But pi is ONE "real" irrational number. The golden ratio is ONE more. That's TWO. QED the rational numbers 1 and 2 are "real"?
- Tao3300 3y ago
  What's "real" about π? It's a tool. Circles are useful, but there aren't any.
Garlef 3y ago
This point of view is conflating two meanings of the world "real". The "real" in "real numbers" has ultimately not much to do with our everydays notion of real. I'd rather treat it as an arbitrary name. You could as well call them "asdfasdf numbers" and they'd remain the same.
mbork_pl 3y ago
Last time I counted my kids, their number was definitely rational.
eru 3y ago
2 is also a real number.

adastra22 3y ago

That seems like a very weird statement to make. Many numbers in the real world are integral (two objects, one electron, etc.). Thanks to quantum physics, most measurements are integral too.

beiller 3y ago
Explained better above. There are more irrational numbers, almost guaranteeing any number come across in nature is irrational. Interesting thought since I think one thing that makes irrational numbers is there is no function for them. So it's kind of a cheeky way to say no math formulas can ever describe the real world since all the numbers are irrational.
- dagw 3y ago
  There are more irrational numbers, almost guaranteeing any number come across in nature is irrational There is nothing that says that the distribution between rational and irrational numbers that show up in nature is the same as the distribution in our construction of the real numbers.
- adastra22 3y ago
  You’re missing the point: physics has shown the world to be discretized. Almost every number you go out and measure IS ACTUALLY AN INTEGER. In that sense the real number system doesn’t exist. It’s super useful, yes, but it’s an abstraction away from reality.
  - 2snakes 3y ago
    I thought that "infinities of infinities" ala Cantor exist though. I agree with you that reality being quantized means everything in physical reality can be normalized to integers.
  - eru 3y ago
    > physics has shown the world to be discretized. It's a bit more complicated than that.
    - adastra22 3y ago
      I am a physicist, and I believe what I said is accurate. Although yes I am (1) oversimplifying, and (2) assuming that some form of quantum gravity is correct.
      - eru 3y ago
        To expand a bit with some examples: the energy a photon in a standing wave in a specific cavity can have is quantised. But a photon out in space can have any old energy it wants to. (Of course, a given energy level will correspond to a specific wave length etc.) Similar, an electron in a single isolated atom has specific quantised energy levels. But if you look at the electrons in a hunk of copper, they are essentially free to absorb and emit energy in almost arbitrary amounts. An even stronger example is time: as far as I am aware, time is not quantised in any of our accepted theories. There's some reasonable speculation that ultimately everything is quantised at the Planck scale, including time. But that's just a very reasonable hunch, not something that 'physics has shown'. (And you already point out that trying to marry quantum mechanics with general relativity is a hot mess.) I agree that 'quantisation all the way down' is the way to bet. But that's just speculation. (But I strongly disagree with your claim that physics is build on integers. Yes, it might be discretised, but there are plenty of discrete structures that are not integers. Look at a Rubik's cube for a simple example. On top of that: almost any real world measurement is better described by a probability distribution than by single number, be that an integer or otherwise.)

lrc 3y ago

It is better stated that: since there are "so many more" irrational numbers than rational ones, if you were to pick a real number "at random," the probability that it would be rational is zero. The "many more" and "random" ideas are made precise in measure theory (and elsewhere).

jaza 3y ago
There is an infinite quantity of both rational and irrational numbers, so isn't it therefore impossible for there to be more of one than of the other? Or is the reasoning that, because there is an infinite quantity of irrational numbers between any two given rational numbers, there are therefore many more irrational numbers than rational numbers? I would have thought that there being an infinite quantity of both, makes it impossible to compare the quantities.
- defrost 3y ago
  There is a mapping from counting numbers (1, 2, 3, ...) to rationals and back again that shows these quantities are the same; for every element in set A there's an element in set B and vice versa. This is not the case for irrationals... therefore it is concluded that the infinity of irrationals is a larger infinity than the infinity of rationals. See: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://mathworld.wolfram.com/CantorDiagonalMethod.html https://mathworld.wolfram.com/CantorDiagonalMethod.html
  - thaumasiotes 3y ago
    > There is a mapping from counting numbers (1, 2, 3, ...) to rationals and back again that shows these quantities are the same; for every element in set A there's an element in set B and vice versa. That is true, but it's never taught. I don't even know what that mapping is, though I've seen it mentioned once in a popular treatment. What's taught is always the mapping from naturals to rationals that overcounts the rationals, hitting them all an infinite number of times. (Because it's very easy to show a bijection between the naturals and the ordered pairs, but while (2,3) and (4,6) are distinct ordered pairs, they do not represent distinct rationals.) But then all you've shown is that the naturals are at least as numerous as the rationals. To show that the naturals and the rationals have the same cardinality, you either rely on the idea that the naturals are the smallest infinite set, or you appeal to the fact that the naturals are a subset of the rationals.

hliyan 3y ago

I've come to the conclusion that all numbers in the real world are integers and real numbers are a human construct necessary evey time we select a unit that is too large. There is evidence (e.g. quantization) that at the most fundamental level, reality is discrete. Math is layer upon layer of abstract toolsets to operate on integers. In that sense for me, it is very real, but invented.

russdill 3y ago
The more interesting question is are the numbers in the universe the subset of real numbers that are computable.
- adastra22 3y ago
  Why is that interesting?
  - russdill 3y ago
    If they are non computable, it would indicate that the universe cannot be modeled by mathematics.
    - j16sdiz 3y ago
      Modellability or Computability? There are incomputable models.
mayd 3y ago
Pi is no more of a human construct than any of the integers. Pi is inherent in Nature as are the integers, but more mysterious. If you accept the existence of the integers then infinity exists and therefore pi exists too; it does not need to be constructed.
- adastra22 3y ago
  There are no circular objects in fundamental physics. Pi shows up in many physics equations, but that’s entirely due to our choice of units.

jaza 3y ago

But pi is ONE "real" irrational number. The golden ratio is ONE more. That's TWO. QED the rational numbers 1 and 2 are "real"?

Tao3300 3y ago
What's "real" about π? It's a tool. Circles are useful, but there aren't any.

Garlef 3y ago

This point of view is conflating two meanings of the world "real". The "real" in "real numbers" has ultimately not much to do with our everydays notion of real. I'd rather treat it as an arbitrary name. You could as well call them "asdfasdf numbers" and they'd remain the same.

mbork_pl 3y ago

Last time I counted my kids, their number was definitely rational.

eru 3y ago

2 is also a real number.