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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

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?
        naasking 10y ago
        > 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. 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.

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?
        naasking 10y ago
        > 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. 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.