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> Real space doesn't appear to be infinite Citation needed. People keep saying things like this ("real space isn't infinite" or "real spacetime must be discre

> Real space doesn't appear to be infinite Citation needed. People keep saying things like this ("real space isn't infinite" or "real spacetime must be discrete") but never show their work.

naasking 10y ago
Google "Bekenstein Bound".
- Jweb_Guru 10y ago
  That doesn't prove anything about the continuity (or lack thereof) of spacetime, just indicates that energy is discretized. Also has no bearing on the original statement about infinity. 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.
  - naasking 10y ago
    The Bekenstein bound, as a starting point, addresses the more general point that all of known physics needs only computable abstractions. Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen, which does address your specific question about whether space is truly continuous.
    - cygx 10y ago
      Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen 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.
      - naasking 10y ago
        We simulate quantum mechanics on classical computers all the time. Superposition requires exponentially more resources in some cases, but it doesn't somehow make QM non-discrete.
        cygx 10y ago
        The point is that according to ordinary QM, any of the states you can construct via superposition are physical. Eg for the qubit, the state space is the Bloch sphere, which is continuous.
        naasking 10y ago
        Which is a property of our formalism, not reality. Any state we'd care to actually observe necessarily has finite precision, so the internal mechanics of our formalism to model this process utilizing infinities is an artifact only of our formalism, not of reality. 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 https://philpapers.org/rec/MYRCIQ [2] Effectively calculable quantum mechanics, 2015, https://arxiv.org/pdf/1508.03879.pdf https://arxiv.org/pdf/1508.03879.pdf [3] Constructive physics, https://arxiv.org/pdf/0805.2859.pdf https://arxiv.org/pdf/0805.2859.pdf
        Jweb_Guru 10y ago
        > Which is a property of our formalism, not reality. Any state we'd care to actually observe necessarily has finite precision, so the internal mechanics of our formalism to model this process utilizing infinities is an artifact only of our formalism, not of reality. 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?
smilliken 10y ago
Nobody has proof, and it's probably unknowable. It could be that we're a discrete system embedded in a continuous system. Or it could be we're in a seemingly continuous system which is actually embedded in a high precision discrete system. That being said, nature has given us very strong hints at discretization. For example: https://en.m.wikipedia.org/wiki/Planck_length 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.
- Jweb_Guru 10y ago
  The fact that we don't know what's going on within the Planck length doesn't demonstrate that spacetime is discretized; I don't even think it strongly indicates it, unless you're already predisposed to think that way. Anyway, that seems to have no bearing on whether the universe is infinite.
warrenpj 10y ago
There was a post here two weeks ago discussing this exact problem. [1] It starts by explaining Democritus' reasoning that matter must be made of discrete atoms. The main point of the article is that a similar logic applies to the future theory of quantum gravity; space and time too must be discrete. 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 https://news.ycombinator.com/item?id=13466254 [2] https://en.wikipedia.org/wiki/Ultimate_fate_of_the_universe https://en.wikipedia.org/wiki/Ultimate_fate_of_the_universe
- Jweb_Guru 10y ago
  To the best of my knowledge, mainstream quantum mechanics assumes continuous spacetime, so the argument that the primary reason it can't be reconciled with general relativity is continuity seems pretty specious to me. 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. So again, your argument doesn't seem that compelling to me.
  - cygx 10y ago
    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 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.

> Real space doesn't appear to be infinite Citation needed. People keep saying things like this ("real space isn't infinite" or "real spacetime must be discrete") but never show their work.

naasking 10y ago

Google "Bekenstein Bound".

Jweb_Guru 10y ago
That doesn't prove anything about the continuity (or lack thereof) of spacetime, just indicates that energy is discretized. Also has no bearing on the original statement about infinity. 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.
- naasking 10y ago
  The Bekenstein bound, as a starting point, addresses the more general point that all of known physics needs only computable abstractions. Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen, which does address your specific question about whether space is truly continuous.
  - cygx 10y ago
    Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen 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.
    - naasking 10y ago
      We simulate quantum mechanics on classical computers all the time. Superposition requires exponentially more resources in some cases, but it doesn't somehow make QM non-discrete.
      - cygx 10y ago
        The point is that according to ordinary QM, any of the states you can construct via superposition are physical. Eg for the qubit, the state space is the Bloch sphere, which is continuous.
        naasking 10y ago
        Which is a property of our formalism, not reality. Any state we'd care to actually observe necessarily has finite precision, so the internal mechanics of our formalism to model this process utilizing infinities is an artifact only of our formalism, not of reality. 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 https://philpapers.org/rec/MYRCIQ [2] Effectively calculable quantum mechanics, 2015, https://arxiv.org/pdf/1508.03879.pdf https://arxiv.org/pdf/1508.03879.pdf [3] Constructive physics, https://arxiv.org/pdf/0805.2859.pdf https://arxiv.org/pdf/0805.2859.pdf
        Jweb_Guru 10y ago
        > Which is a property of our formalism, not reality. Any state we'd care to actually observe necessarily has finite precision, so the internal mechanics of our formalism to model this process utilizing infinities is an artifact only of our formalism, not of reality. 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?

smilliken 10y ago

Nobody has proof, and it's probably unknowable. It could be that we're a discrete system embedded in a continuous system. Or it could be we're in a seemingly continuous system which is actually embedded in a high precision discrete system. That being said, nature has given us very strong hints at discretization. For example: https://en.m.wikipedia.org/wiki/Planck_length 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.

Jweb_Guru 10y ago
The fact that we don't know what's going on within the Planck length doesn't demonstrate that spacetime is discretized; I don't even think it strongly indicates it, unless you're already predisposed to think that way. Anyway, that seems to have no bearing on whether the universe is infinite.

warrenpj 10y ago

There was a post here two weeks ago discussing this exact problem. [1] It starts by explaining Democritus' reasoning that matter must be made of discrete atoms. The main point of the article is that a similar logic applies to the future theory of quantum gravity; space and time too must be discrete. 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 https://news.ycombinator.com/item?id=13466254 [2] https://en.wikipedia.org/wiki/Ultimate_fate_of_the_universe https://en.wikipedia.org/wiki/Ultimate_fate_of_the_universe

Jweb_Guru 10y ago
To the best of my knowledge, mainstream quantum mechanics assumes continuous spacetime, so the argument that the primary reason it can't be reconciled with general relativity is continuity seems pretty specious to me. 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. So again, your argument doesn't seem that compelling to me.
- cygx 10y ago
  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 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.