5 ms·

Bold to assume that reality is discrete.

lupire 4y ago
Even if it's not discrete at the bottom of QFT or something, it is discrete at the level where physics is applied in classical mechanics.
- posterboy 4y ago
  Bold to assume that "it" is reality. Conversely, classical mechanics is discrete at the level where physics is applied. That it is applied to realia goes without saying. But it is not applied to reality as a whole.
  - Tainnor 4y ago
    Well, it's bold to assume anything about reality, because we only ever experience individual aspects of it. We can only try to model what we can observe. But as a counter to the assertion that "AC must be wrong because we can't really experience Banach-Tarski in the real world", it's perfectly valid to comment that "well, we also can't really observe continuous functions R -> R either". Or have you ever seen a least upper bound in real life?
syzarian 4y ago
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
      - 4y ago
andrewla 4y ago
If by "discrete" you mean rejecting the continuum, then I don't think it's as scary as it sounds. The rationals do not form a continuum, but they are "effectively" continuous -- between any two of them there is another. There are no "leaps" between them, and no concept of consecutive numbers. Computable numbers are similarly countable but allow the expression of numbers that can only be approximated by rationals. It's the reals that require "bold" assumptions -- numbers exist which are not possible to compute even approximately?

Even if it's not discrete at the bottom of QFT or something, it is discrete at the level where physics is applied in classical mechanics.

posterboy 4y ago
Bold to assume that "it" is reality. Conversely, classical mechanics is discrete at the level where physics is applied. That it is applied to realia goes without saying. But it is not applied to reality as a whole.
- Tainnor 4y ago
  Well, it's bold to assume anything about reality, because we only ever experience individual aspects of it. We can only try to model what we can observe. But as a counter to the assertion that "AC must be wrong because we can't really experience Banach-Tarski in the real world", it's perfectly valid to comment that "well, we also can't really observe continuous functions R -> R either". Or have you ever seen a least upper bound in real life?

syzarian 4y ago

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
    - 4y ago

andrewla 4y ago

If by "discrete" you mean rejecting the continuum, then I don't think it's as scary as it sounds. The rationals do not form a continuum, but they are "effectively" continuous -- between any two of them there is another. There are no "leaps" between them, and no concept of consecutive numbers. Computable numbers are similarly countable but allow the expression of numbers that can only be approximated by rationals. It's the reals that require "bold" assumptions -- numbers exist which are not possible to compute even approximately?