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arXiv:1609.01421math-ph

Struggles with the Continuum

John C. Baez

Abstract

Our assumption that spacetime is a continuum leads to many challenges in mathematical physics. Singularities, divergent integrals and the like threaten many of our favorite theories, from Newtonian gravity to classical electrodynamics, quantum electrodynamics and the Standard Model. In general relativity, singularities are intimately connected to some of the theory's most dramatic successful predictions. We survey these problems and the large amount of work that has gone into dealing with them.

arXiv comments
Journal reference
DOI
Subjects
math-phMathematical Physics gr-qcGeneral Relativity and Quantum Cosmology hep-thHigh Energy Physics, Theory physics.hist-phHistory and Philosophy of Physics

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Linked by 7 identifiable commenters in 18 comments across 17 threads, from Sep 2016 to Nov 2025. Found in 4 HN submissions from Sep 2016 to Nov 2023. Open submission discussions.

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Discussed in comments17 threads

  1. A new bridge links the math of infinity to computer science 1 comment

    • Math is a tool for structured reasoning. We often want to reason about things that physically exist. Why should we use a tool that assumes the existence of something that literally cannot physically exist? That introduces the danger that of admitting all sorts of unphysical…

      naaskingView on HNParent context

  2. James Webb Space Telescope finds evidence for alternate theory of gravity 1 comment

    • > Can you give some examples? See Baez's paper, Struggles with the Continuum, https://arxiv.org/abs/1609.01421 The UV catastrophe is probably the most well known.

      naaskingView on HNParent context

  3. Large Language Models Are Neurosymbolic Reasoners 1 comment

    • > Here, it is absolutely necessary for QM to work that the unmeasured state is real-valued You're just doubling down on the premise that a theory founded on a formalism based on unbounded numbers requires unbounded numbers to work. Sure, but why is that necessarily reflective of…

      naaskingView on HNParent context

  4. Do black holes have singularities? 1 comment

    • That was the old thinking. Turns out it leads to divergences which was one of the difficulties that quantum theory overcame. Baez has a good overview of these kinds of problems: Struggles with the Continuum, https://arxiv.org/abs/1609.01421 Lots of issues spring from unphysical…

      naaskingView on HNParent context

  5. A physicist who bets that gravity can’t be quantized 2 comments

    • > But when they tried to quantize gravity, they ran into unnatural infinities that had to be sidestepped with clumsy mathematical tricks. Maybe they run into unnatural infinities because all of our formalisms in physics are still fundamentally continuous rather than discrete.…

      naaskingView on HNParent context

    • > Oh no I get that, but two comments above inciampati was suggesting that we will end up in a situation where some variable will have to have an infinite value The black hole information paradox from that article presumably fits. The conclusion from GR is that no information can…

      naaskingView on HNParent context

  6. Towards quantum computers that are robust to errors 1 comment

    • Nobody knows of any certain reasons to not scale, but there's plenty of speculation. Someone else posted about Gil Kalai in this subthread, so he's one high profile person that has documented his skepticism on quantum computing and explained his reasons. For my part, I don't…

      naaskingView on HNParent context

  7. Our reality may be a sum of all possible realities 1 comment

    • Yes, there are historical issues with that calculation, and field theories in general. I've become very suspicious of continuous formalisms period: https://arxiv.org/abs/1609.01421 I think many problems in physics will be solved by moving to more discrete formalisms.

      naaskingView on HNParent context

  8. Norton's Dome 1 comment

    • > As far as we know, time and space are continuous. Our models are continuous. The map is not the territory. > Various aspects of quantum field theories would not work in a discretized universe. If you're referring to symmetries, that's not true [1]. Discretized theories have…

      naaskingView on HNParent context

  9. How Organisms Come to Know the World: Fundamental Limits on AGI 1 comment

    • First, all of physics is computable to arbitrary precision. Continuous quantities in physics are being systematically removed because they almost inevitably introduce absurdities [1]. Measures lose all meaning below Planck length anyway. Second, the point of enumerating all…

      naaskingView on HNParent context

  10. The Axiom of Infinity (2018) 1 comment

    • Nice list! I'll add one more I enjoyed: Struggles with the Continuum, by John Baez, https://arxiv.org/abs/1609.01421

      naaskingView on HNParent context

  11. Computer proof ‘blows up’ centuries-old fluid equations 1 comment

    • There are all kinds of blow-ups in Newtonian mechanics and in other equations of physics. The singularity at the center of a black hole in general relativity is a famous example. The ultraviolet catastrophe in classical thermodynamics was another. The presumption is that…

      throwaway81523View on HNParent context

  12. How to fit any dataset with a single parameter 1 comment

    • As an aside, this idea that reality is "geometrical" leads to all kinds of fascinating paradoxes[1]. [1] https://arxiv.org/abs/1609.01421

      macrolocalView on HNParent context

  13. A proof P = NP was accidentally published in TOCT 1 comment

    • In an accelerating expanding universe like the one we appear to occupy (and the one modelled by our current "concordance" standard model of comology), the comoving horizon is at constant comoving distance. Cast into more Earth-centric coordinates, it is at a finite spatial…

      raattgiftView on HNParent context

  14. Supertasks 1 comment
  15. How real are real numbers? (2004) 1 comment

    • For the parallel historical development of 'the continuum' in physics, I recommend this readable survey: Paper: https://arxiv.org/abs/1609.01421 https://math.ucr.edu/home/baez/continuum.pdf Blog summary with discussion…

      mikhailfrancoView on HNParent context

  16. What If Time Really Exists? (2008) 1 comment

    • Try this physics paper[1] explaining how pretty much every model we have of physics is broken in fundamental ways. Things you take for granted because of what you've read in popular media (eg., heat death of the universe) are simply speculation based on stretching an incomplete…

      j2kunView on HNParent context

  17. Paranoid Android: The Divine Madness of Philip K Dick 1 comment

    • dS space is really simple; you only need a cosmological constant to power a uniform metric expansion, and no matter to retard it. The far future dS space approximation has only localized matter, namely mostly isolated gas molecules and (swarms of) black holes, each occupying…

      raattgiftView on HNParent context

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