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Comment by red_trumpet | Hacker News Reader

Bold to assume that reality is discrete.

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?

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?

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.

Events are discrete, right? Doesn’t everything then appear to be discete?

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.

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.)

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.

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

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

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

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.

> 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.

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.

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.

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?

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