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Everything appears to be discrete except possibly space-time. Is there a reason to believe space-time isn’t discrete other than using manifolds makes things ea

Everything appears to be discrete except possibly space-time. Is there a reason to believe space-time isn’t discrete other than using manifolds makes things easier mathematically?

leni536 4y ago
There is no reason to believe that spacetime is anything but an approximation, which breaks down at Planck-length scales. We have events, and causal relationship between events. Events may not have precise spacetime coordinates.
- syzarian 4y ago
  Events are discrete, right? Doesn’t everything then appear to be discete?
  - leni536 4y ago
    Events themselves are discrete, but they might have contiguous properties, like the associated point in spacetime, assuming a the contiguous spacetime is not just an approximation.
lanstin 4y ago
I believe the numbers in QFT are continuous. The electrons and what not are quantized but quantized over complex number spaces. U less one be Unless the fine structure constant or speed of light, or other apparent constants of the theories, are computable some how? Disclaimer: I can only get a few chapters into a QFT book without it all becoming meaningless to me.)
- syzarian 4y ago
  I believe the universe is finite and so there can only be a finite number of realizations of quantum numbers. I have no real intuition in physics or training in physics so I know I could be completely wrong. It seems to me though that our experiences are such that believing the universe is discrete in all things is not completely unreasonable. Though this belief of mine might itself be unreasonable.
- andrewla 4y ago
  The mathematical notion of "discrete" does not require that there be big gaps, or that there be fundamental limits to precision. The rationals are discrete (i.e. they are countable and have measure zero) but they are "continuous" in every way that matters -- between any two rationals there is another rational. Even the computable/constructible/rationally-approximable numbers are discrete, and those sets contain basically all the numbers that are possible to compute (by definition) but they are still "discrete", forming a set of measure zero, The idea of the continuum is the unintuitive one; the idea that there are numbers that cannot be constructed or computed is just kind of silly
  - red_trumpet 4y ago
    In topology the rationals are not discrete[1]: every open interval around zero contains other rational numbers! Also, what do you mean by "rationally-approximable"? I think the construction of the reals using Cauchy sequences is /exactly/ meant to formalise this. [1] https://en.wikipedia.org/wiki/Discrete_space https://en.wikipedia.org/wiki/Discrete_space
    - andrewla 4y ago
      The problem is that Cauchy sequences are too powerful. The intuitionist/constructive version of "rationally approximable" requires a procedure to approximate a number to any requested degree of precision, but cannot guarantee that equality is decideable. My use of discrete was informal and in response to the thread here which equated discarding the notion on the continuum with saying the universe is "discrete". More precise in the topology sense is that the rationals are not compact
  - lanstin 4y ago
    Unintuitive till you move your hand from here to there, though I wouldn’t notice a Planck length digitization, but my point was more the maths for reals or complex over reals is so much simpler than over rationals, and so that is what the physics is based on. I have not seriously looked at constructivist analysis, but I do understand there is promise there, but it is a lot more complicated. Interesting to imagine an integral over all possible paths with only computable paths. Such a smaller set of thing affecting the solution.

Everything appears to be discrete except possibly space-time. Is there a reason to believe space-time isn’t discrete other than using manifolds makes things easier mathematically?

leni536 4y ago

There is no reason to believe that spacetime is anything but an approximation, which breaks down at Planck-length scales. We have events, and causal relationship between events. Events may not have precise spacetime coordinates.

syzarian 4y ago
Events are discrete, right? Doesn’t everything then appear to be discete?
- leni536 4y ago
  Events themselves are discrete, but they might have contiguous properties, like the associated point in spacetime, assuming a the contiguous spacetime is not just an approximation.

lanstin 4y ago

I believe the numbers in QFT are continuous. The electrons and what not are quantized but quantized over complex number spaces. U less one be Unless the fine structure constant or speed of light, or other apparent constants of the theories, are computable some how? Disclaimer: I can only get a few chapters into a QFT book without it all becoming meaningless to me.)

syzarian 4y ago
I believe the universe is finite and so there can only be a finite number of realizations of quantum numbers. I have no real intuition in physics or training in physics so I know I could be completely wrong. It seems to me though that our experiences are such that believing the universe is discrete in all things is not completely unreasonable. Though this belief of mine might itself be unreasonable.
andrewla 4y ago
The mathematical notion of "discrete" does not require that there be big gaps, or that there be fundamental limits to precision. The rationals are discrete (i.e. they are countable and have measure zero) but they are "continuous" in every way that matters -- between any two rationals there is another rational. Even the computable/constructible/rationally-approximable numbers are discrete, and those sets contain basically all the numbers that are possible to compute (by definition) but they are still "discrete", forming a set of measure zero, The idea of the continuum is the unintuitive one; the idea that there are numbers that cannot be constructed or computed is just kind of silly
- red_trumpet 4y ago
  In topology the rationals are not discrete[1]: every open interval around zero contains other rational numbers! Also, what do you mean by "rationally-approximable"? I think the construction of the reals using Cauchy sequences is /exactly/ meant to formalise this. [1] https://en.wikipedia.org/wiki/Discrete_space https://en.wikipedia.org/wiki/Discrete_space
  - andrewla 4y ago
    The problem is that Cauchy sequences are too powerful. The intuitionist/constructive version of "rationally approximable" requires a procedure to approximate a number to any requested degree of precision, but cannot guarantee that equality is decideable. My use of discrete was informal and in response to the thread here which equated discarding the notion on the continuum with saying the universe is "discrete". More precise in the topology sense is that the rationals are not compact
- lanstin 4y ago
  Unintuitive till you move your hand from here to there, though I wouldn’t notice a Planck length digitization, but my point was more the maths for reals or complex over reals is so much simpler than over rationals, and so that is what the physics is based on. I have not seriously looked at constructivist analysis, but I do understand there is promise there, but it is a lot more complicated. Interesting to imagine an integral over all possible paths with only computable paths. Such a smaller set of thing affecting the solution.