Struggles with the Continuum

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: https://johncarlosbaez.wordpress.com/2016/09/08/struggles-wi... https://johncarlosbaez.wordpress.com/2016/09/09/struggles-wi...

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 possibilities into our theories. I think Baez's paper, Struggles with the Continuum, shows a lot of past difficulties we've had that resulted from this: https://arxiv.org/abs/1609.01421

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 blow-ups in an equation indicate a mismatch between the equation and the true physical world, telling physicists to look for better theories, whose equations don't blow up. For the ultraviolet catastrophe, the mystery was solved through the discovery of quantum mechanics. For GR, it is still unsolved, and the solution is expected to come from a theory of quantum gravity that hasn't yet been invented, but is the target of tons of research. Here's a cool expository article about blow-ups in classical mechanics and elsewhere: https://arxiv.org/abs/1609.01421

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

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 mathematical model to its limits. The simulated reality nonsense is the same bad logic in a different guise. [1]: https://arxiv.org/abs/1609.01421

> 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.

> 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 reality? Why does that entail that no other formalism that doesn't embed infinities / continuity is also not possible? I simply see no reason to accept your conclusion. The infinities you see as essential could very well simply be artifacts of our formalisms. In fact, I'd conjecture that our continuous formalisms are at the very heart of some core problems in physics [1], and that at least some of those problems can be resolved by exploring more discrete formalisms. I suppose we'll see. [1] https://arxiv.org/abs/1609.01421

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 I think that any theory that admits point-like objects or infinities will likely lead to unphysical results and paradoxes.

> 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 escape, which is ultimately incompatible with QM, and that conclusion ultimately depends on the infinite density of the singularity. I think the more charitable reading is that we'll find situations where either no sensible calculation can be done, or the sensible calculations we do churn out nonsense. Divergence in the UV catastrophe was an example of that, and Baez covered more here: Struggles with the Continuum, https://arxiv.org/abs/1609.01421

> 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. Uncountable infinities are baked right into the foundations of how we use reason about these systems, so infinities will naturally result. Physics has repeatedly had to tame infinities by elaborate tricks, or by eliminating them entirely [1]. Some people are increasingly looking to discrete formalisms, and I think this is a promising way forward, both for mathematics and physics. [1] Struggles with the Continuum, https://arxiv.org/abs/1609.01421

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.

> 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 received barely any attention because physicists have simply built upon the formalisms they're used to from classical physics. IMO, baking continuity and uncountable infinities is what's responsible for many of the difficulties we've had, and various forms of discretization have resolved these issues in the past [2]. My conjecture is that one of the next big advances in physics will be to take discretization seriously. [1] https://arxiv.org/abs/1902.08997 [2] https://arxiv.org/abs/1609.01421