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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

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.
- andrewla 4y ago
  > Unintuitive till you move your hand from here to there Continuity does not require the notion of a continuum. There need not be a "minimum length" even in a non-continuum approach. The rational numbers are perfectly fine for this because Zeno's paradox still applies -- between any two points is a halfway point, no matter what scale you look at. If you need more power you have the computable reals, which include all real numbers that it is possible to compute. The leap from the computable reals to the continuum is a huge, vast leap -- in measure theoretic terms you are going from a measure zero set to a non-zero-measure set. But all you get from that leap is access to a bunch of numbers that it is not possible to construct anyway, where I mean "not possible" in the strict, formal, definitional sense -- it is literally impossible to produce an example of one. Only the Axiom of Choice and similar nonsense even let you reason about these, the "zombies" of the OP.

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.

andrewla 4y ago
> Unintuitive till you move your hand from here to there Continuity does not require the notion of a continuum. There need not be a "minimum length" even in a non-continuum approach. The rational numbers are perfectly fine for this because Zeno's paradox still applies -- between any two points is a halfway point, no matter what scale you look at. If you need more power you have the computable reals, which include all real numbers that it is possible to compute. The leap from the computable reals to the continuum is a huge, vast leap -- in measure theoretic terms you are going from a measure zero set to a non-zero-measure set. But all you get from that leap is access to a bunch of numbers that it is not possible to construct anyway, where I mean "not possible" in the strict, formal, definitional sense -- it is literally impossible to produce an example of one. Only the Axiom of Choice and similar nonsense even let you reason about these, the "zombies" of the OP.