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Every simulation picks some rational approximation to Pi, because they have to. Either they will run out of time or space or collapse into a black hole before n

Every simulation picks some rational approximation to Pi, because they have to. Either they will run out of time or space or collapse into a black hole before needing more than a finite number of decimal places, so for all plausible purposes we can make do with the first googleplex digits (or whatever) of Pi. I guess my argument is, since you can always just pick a rational approximation to Pi, you cannot prove empirically that we live in a universe where more than a finite number of digits of Pi matter. That is, the mathematical irrationality doesn't really matter, physically speaking, since no experiment could ever prove that every digit in Pi actually contributes to the result. If the universe does have ways to do this, to mix an entire irrational number into a physical outcome, that means hypercomputation is probably possible, since Turing machines definitely can't. https://en.m.wikipedia.org/wiki/Real_computation https://en.m.wikipedia.org/wiki/Real_computation

zozbot234 5y ago
> Every simulation picks some rational approximation to Pi, because they have to. Your simulation might need to ask for an increasingly tighter bound on the real value of Pi. You can totally do this with no more than the usual rational numbers, but it's not equivalent to "just picking some rational approximation" and running with it, because what accuracy/precision you pick is outcome-dependent and it's always possible to request more.
- roywiggins 5y ago
  My point is that there is some finite number of digits your simulation will ever access, whether they're precalculated ahead of time or on done on the fly. If you do it on the fly you can always go back and rerun the computation with the number hardcoded. There must be some digit after which no computation will ever access, because it will require more negentropy than the entire universe has to even calculate. The digits after that don't matter to the universe.
  - kadoban 5y ago
    That seems true, but unsatisfying. If you only consider a finite time, you can only use a finite number of digits is the idea? Or can infinite time also work, because laws of thermodynamics? Either way though, then doesn't your model of the universe just need an extra parameter, the number of digits to care about? Seems like everything else being equal, the fewer unmotivated parameters in your model, the better. Especially because this would rely on internal details of what happens in the universe, seems unlikely to be true unless this is a simulation.

zozbot234 5y ago

> Every simulation picks some rational approximation to Pi, because they have to. Your simulation might need to ask for an increasingly tighter bound on the real value of Pi. You can totally do this with no more than the usual rational numbers, but it's not equivalent to "just picking some rational approximation" and running with it, because what accuracy/precision you pick is outcome-dependent and it's always possible to request more.

roywiggins 5y ago
My point is that there is some finite number of digits your simulation will ever access, whether they're precalculated ahead of time or on done on the fly. If you do it on the fly you can always go back and rerun the computation with the number hardcoded. There must be some digit after which no computation will ever access, because it will require more negentropy than the entire universe has to even calculate. The digits after that don't matter to the universe.
- kadoban 5y ago
  That seems true, but unsatisfying. If you only consider a finite time, you can only use a finite number of digits is the idea? Or can infinite time also work, because laws of thermodynamics? Either way though, then doesn't your model of the universe just need an extra parameter, the number of digits to care about? Seems like everything else being equal, the fewer unmotivated parameters in your model, the better. Especially because this would rely on internal details of what happens in the universe, seems unlikely to be true unless this is a simulation.