5 ms·

I'd deny that our objects do not live in a real space. Spacetime is real-valued. The issue isn't the reals, but that "solid object" isn't defined properly, ie.

I'd deny that our objects do not live in a real space. Spacetime is real-valued. The issue isn't the reals, but that "solid object" isn't defined properly, ie., the sets under question don't have well-defined volumes. As soon as you fix that problem, via measure theory, the paradox resolves. You dont need to ditch real numbers.

zarzavat 5y ago
> Spacetime is real-valued There's really no evidence for this, as far as we know the real numbers are a pure mathematical invention and don't have any physicality. Even if you want to say that spacetime is dense (i.e. infinitely divisible), there's an infinite number of fields like that, the real numbers are just a convenient superset. There's no evidence that spacetime is dense either, and many practical ways in which it is not, as an obvious upper bound if you took all the energy in the observable universe to make one photon, it would still have a finite wavelength.
- Pyramus 5y ago
  That is correct and should be highlighted more. In a way the real numbers are a model (or maybe a 'language') to describe physical phenomena. They work exceptionally well at that, but they are not backed by evidence and do come with (theoretical) limitations. This bachelor's thesis is a good starting point [1], search for 'finite precision physics' or 'intuitionistic math/physics'. [1] https://www.math.ru.nl/~landsman/Tein.pdf https://www.math.ru.nl/~landsman/Tein.pdf
  - mjburgess 5y ago
    Well this is a popular idea imported from comptuer science, but there's absolutely no evidence for it -- and plenty against. Eg., QM is only linear in infinitely-dimensional real-spaces, etc. Essentially of a physics uses real spaces indispensably. There is no evidence whatsoever that this is dispensible; other than the fever dreams of discrete mathematicians.
    - jerf 5y ago
      Remember, though: QM is wrong. Relativity also depends on continuous spaces, but it is also wrong. All the theories in physics that depend on continuous space are also wrong. By "wrong", I mean, we know they can't predict everything correctly. QM itself can't derive relativity. Relativity doesn't have QM in it, and break down at extremes like black holes. They're both very, very, very accurate in their domains, but physics knows that neither theory has the domain of "the entire universe". This is not a wild claim by an HN commenter, this is consensus in the physics world, just perhaps not phrased in the way you're used to. It's possible the eventual Grand Unified Theory will still have continuous space at its bottom, but it's also entirely possible it won't. Loop quantum gravity doesn't. And personally I expect some sort of new hybrid between continuous and discrete based on physics history; whenever in the past we've had a similar situation where it couldn't be X for this reason, but it couldn't be the obvious Not-X for some other reason, it has turned out to be something that had a bit of both in them, but wasn't either of them.
      - mjburgess 5y ago
        They're not "wrong" in tests of their real-valuedness though. I'm somewhat confident there is an empirical test of real-valuedness in areas of physics which require infinite-valued spaces. However, either way -- the positions of the other commenters was that *geometry* is somehow a dispensable approximation in physics! This is an extremely radical claim with no evidence whatsoever. Rather some discrete mathematicians simply wish it were the case. It is true that *maybe* (!) spacetime will turn out discrete, and likewise, Hilbert spaces, etc. -- and all continuous and infinite dimensional things will be discretised. This however is a project without a single textbook. There is no such physics. There are no empirical predictions. There are no theories. This is a project within discrete mathematics.
        jerf 5y ago
        "They're not "wrong" in tests of their real-valuedness though." Yes, they are, or more accurate, they're not right enough for you to confidently assert the structure of space time at scales below the Planck scale. You are doing so on the basis of theories known to be broken at that scale. You are not entitled to use the theories that way. Even the Planck scale being the limit is a mathematical number; I'm not sure we have concrete evidence of that size being the limit. I've seen a few proposed experiments that would measure at that resolution (such as certain predictions made by LQG about light traveling very long distances and different wavelengths traveling at very slightly different speeds) but I'm not aware of any that have panned out enough to have a solid result of any kind.
- jankovicsandras 5y ago
  I have very little understanding of physics and math and might be wrong. But if we accept that the Planck length is the smallest possible length and the Planck time is the smallest possible time, then it seems logical that the universe is an integer lattice of these. ("Spacetime is not real-valued") https://en.wikipedia.org/wiki/Planck_length https://en.wikipedia.org/wiki/Planck_length https://en.wikipedia.org/wiki/Planck_units#Planck_time https://en.wikipedia.org/wiki/Planck_units#Planck_time
  - czzr 5y ago
    It’s not known if those are fundamental limits. However the idea that space time is discrete is a reasonable hypothesis to test, we don’t currently have any ways to probe at those resolutions, though.
anthk 5y ago
Sqrt(2) is a real number yet I bet you to properly measure up to the last point the diagonal of a square.

zarzavat 5y ago

> Spacetime is real-valued There's really no evidence for this, as far as we know the real numbers are a pure mathematical invention and don't have any physicality. Even if you want to say that spacetime is dense (i.e. infinitely divisible), there's an infinite number of fields like that, the real numbers are just a convenient superset. There's no evidence that spacetime is dense either, and many practical ways in which it is not, as an obvious upper bound if you took all the energy in the observable universe to make one photon, it would still have a finite wavelength.

Pyramus 5y ago
That is correct and should be highlighted more. In a way the real numbers are a model (or maybe a 'language') to describe physical phenomena. They work exceptionally well at that, but they are not backed by evidence and do come with (theoretical) limitations. This bachelor's thesis is a good starting point [1], search for 'finite precision physics' or 'intuitionistic math/physics'. [1] https://www.math.ru.nl/~landsman/Tein.pdf https://www.math.ru.nl/~landsman/Tein.pdf
- mjburgess 5y ago
  Well this is a popular idea imported from comptuer science, but there's absolutely no evidence for it -- and plenty against. Eg., QM is only linear in infinitely-dimensional real-spaces, etc. Essentially of a physics uses real spaces indispensably. There is no evidence whatsoever that this is dispensible; other than the fever dreams of discrete mathematicians.
  - jerf 5y ago
    Remember, though: QM is wrong. Relativity also depends on continuous spaces, but it is also wrong. All the theories in physics that depend on continuous space are also wrong. By "wrong", I mean, we know they can't predict everything correctly. QM itself can't derive relativity. Relativity doesn't have QM in it, and break down at extremes like black holes. They're both very, very, very accurate in their domains, but physics knows that neither theory has the domain of "the entire universe". This is not a wild claim by an HN commenter, this is consensus in the physics world, just perhaps not phrased in the way you're used to. It's possible the eventual Grand Unified Theory will still have continuous space at its bottom, but it's also entirely possible it won't. Loop quantum gravity doesn't. And personally I expect some sort of new hybrid between continuous and discrete based on physics history; whenever in the past we've had a similar situation where it couldn't be X for this reason, but it couldn't be the obvious Not-X for some other reason, it has turned out to be something that had a bit of both in them, but wasn't either of them.
    - mjburgess 5y ago
      They're not "wrong" in tests of their real-valuedness though. I'm somewhat confident there is an empirical test of real-valuedness in areas of physics which require infinite-valued spaces. However, either way -- the positions of the other commenters was that *geometry* is somehow a dispensable approximation in physics! This is an extremely radical claim with no evidence whatsoever. Rather some discrete mathematicians simply wish it were the case. It is true that *maybe* (!) spacetime will turn out discrete, and likewise, Hilbert spaces, etc. -- and all continuous and infinite dimensional things will be discretised. This however is a project without a single textbook. There is no such physics. There are no empirical predictions. There are no theories. This is a project within discrete mathematics.
      - jerf 5y ago
        "They're not "wrong" in tests of their real-valuedness though." Yes, they are, or more accurate, they're not right enough for you to confidently assert the structure of space time at scales below the Planck scale. You are doing so on the basis of theories known to be broken at that scale. You are not entitled to use the theories that way. Even the Planck scale being the limit is a mathematical number; I'm not sure we have concrete evidence of that size being the limit. I've seen a few proposed experiments that would measure at that resolution (such as certain predictions made by LQG about light traveling very long distances and different wavelengths traveling at very slightly different speeds) but I'm not aware of any that have panned out enough to have a solid result of any kind.
jankovicsandras 5y ago
I have very little understanding of physics and math and might be wrong. But if we accept that the Planck length is the smallest possible length and the Planck time is the smallest possible time, then it seems logical that the universe is an integer lattice of these. ("Spacetime is not real-valued") https://en.wikipedia.org/wiki/Planck_length https://en.wikipedia.org/wiki/Planck_length https://en.wikipedia.org/wiki/Planck_units#Planck_time https://en.wikipedia.org/wiki/Planck_units#Planck_time
- czzr 5y ago
  It’s not known if those are fundamental limits. However the idea that space time is discrete is a reasonable hypothesis to test, we don’t currently have any ways to probe at those resolutions, though.

anthk 5y ago

Sqrt(2) is a real number yet I bet you to properly measure up to the last point the diagonal of a square.