That being said, nature has given us very strong hints at discretization. For example: https://en.m.wikipedia.org/wiki/Planck_length. Because of uncertainty principle, we can't measure distances less than that. In a similar way, time also appears to be discretized, along with energy and mass.

From the article:

> Importing the atomic philosophy of Democritus into modern physics might be essential for reconciling general relativity (which assumes a continuous reality) with quantum mechanics (which very much does not).

As far as infinite vs finite space: because there is a fixed rate of information propagation, we only need to show that time is finite in extent, and it follows that space is also finite. We know that the visible universe is finite (by looking in the sky) and we can explain why: the big bang happened at a finite time in the past.

We don't know if the universe will have a finite future, but it might be the case. One possible cause of a finite future could be accelerating expansion of the universe due to dark energy. [2]

Of course there are other interpretations of "real space", but this one (causally connected) seems parsimonious to me.

[1] https://news.ycombinator.com/item?id=13466254

[2] https://en.wikipedia.org/wiki/Ultimate_fate_of_the_universe

That does not follow: As our universe is spatially flat (or close to it), it is consistent with the infinite-space Friedmann model. The cosmic horizon that bounds the observable universe would still exist due to the metric expansion of space.

Just to clarify, FLRW is a class of models, which includes ones with finite as well as infinite space. We've been able to determine that our universe is close to spatially flat. However, that does not mean it's infinite: If spatial curvature is positive but small compared to the cosmic horizon (which is due to spatial expansion), we won't be able to tell.

That said, even given finite space, spacetime can of course still be infinite if the universe does not recollapse.

Your criticism is right and that is a bad argument. The real reasons they are hard to reconcile are technical (see for example [1]), and not mentioned in the source I referred to.

The article only says that a discrete theory of spacetime "might be essential" for the reconciliation; the arguments for this are philosophical - it would be beautiful and historically completing. Read the article if you're interested in the arguments, which aren't included in my comments.

> And it's possible to come up with a model where the universe is infinite but the visible universe is finite. In fact, I believe FLRW is one such a model, and is supported with high accuracy by the latest experimental observations.

I agree with you that the universe could be infinite, but that doesn't contradict my previous comment. I explained that I was talking about local (causally connected to us) space-time, only. I admit this is a bit of a trick, but the parent comment used the term "real space" which I interpreted in such a way to favour an argument for finite space.

> So again, your argument doesn't seem that compelling to me.

I've given reasons but I think it might be helpful to restate what I am actually claiming. I admit I don't know that space and time are discrete. The source I gave is a poetic and philosophical argument that it might be. I think it's established fact that energy and matter are, and that time is finite in the past, so the visible universe is finite. I think it's possible that the future is finite too.

I don't know what FLRW is. My best guess is https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–W...

[1] http://www.theory.caltech.edu/people/jhs/strings/str115.html

None of that is established, actually:

Metric expansion of space comes with a cosmic event horizon where redshift goes to infinity. While the energy contained within that region is finite, there's no reason to assume that the universe outside the horizon looks any different than inside the horizon, and it might very well be spatially infinite.

As to time being finite in the past, it is possible to generalize Einstein's equations so that they can be extended past the big bang singularity, and what you get is a mirror universe on the other side. The issue here is that as we approach Planck scale, quantum gravity becomes relevant, and we have no predictive theory of that.

It's also worth pointing out that the original statement is nonsensical because the theory of real closed fields doesn't suffer from incompleteness in the first place, so there's no need to "save" it from paradox the way you have to save Peano arithmetic.

I do not see how that's true for quantum mechanics (take linear superposition as just one example). People may speculate about discrete underpinnings, but at this time, it's merely speculation.

Eg for the qubit, the state space is the Bloch sphere, which is continuous.

Furthermore, quantum computing and classical computing are known to have the same computational power, just different computational complexity, ie. any quantum system can be simulated by a classical system with exponential slowdown, at worst.

There are plenty of papers exploring this territory [1,2,3] if you want further details. There are older references but they're not easily available online.

[1] Computability in Quantum Mechanics, 1995, https://philpapers.org/rec/MYRCIQ

[2] Effectively calculable quantum mechanics, 2015, https://arxiv.org/pdf/1508.03879.pdf

[3] Constructive physics, https://arxiv.org/pdf/0805.2859.pdf

But this is exactly my point. The fact that our observations have finite precision doesn't say anything about the underlying reality. That's the part that you're asserting. If it also turns out that assuming the underlying system is continuous results in much more compact theories than assuming otherwise, why is the obvious conclusion that spacetime is in fact discrete?

I'm not sure what you think I've asserted, but my only actual assertions are that all of modern physics requires only computable abstractions, and thus, continuous abstractions aren't strictly necessary.

I can certainly make the case that a continuous ontology requires more ontological commitments than a discrete ontology, but I haven't said anything along those lines thus far.

> If it also turns out that assuming the underlying system is continuous results in much more compact theories than assuming otherwise, why is the obvious conclusion that spacetime is in fact discrete?

What is the meaning of "compactness"? A proper comparison of theoretical parsimony requires comparing the Kolmogorov complexity. Compressing ontologies positing reals and/or rationals are definitely less parsimonious than ontologies positing only naturals.

As far as I can tell, by this argument there is literally nothing that would convince you that anything but a discrete theory was correct. So I don't think it's a very interesting argument, sorry. But thank you for explaining your viewpoint.