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I would phrase the crux as follows: Any physics theory is basically a map from one set of measurements (past) to another set of measurements (future). Any phys

I would phrase the crux as follows: Any physics theory is basically a map from one set of measurements (past) to another set of measurements (future). Any physical measurement can only have finite precision, due to fundamental constraints from physics and information theory. The set of finite precision numbers has vanishing measure in the set of all real numbers (since finite precision is a subset of rationals). So are real numbers really necessary? As a physicist, while this question is definitely interesting, it’s not clear what will come out of this line of questioning (aka research).

jwatte 8y ago
I think you're suggesting that "real" numbers are an abstraction that is convenient to model with, but aren't actually fully representable in the universe, because of Planck and Heisenberg and such? I would agree with that view.
- ssivark 8y ago
  It's not even necessarily about quantum stuff. The point is that any practical measurement system (even classical ones) can only deal in finite precision numbers. And since that set is of vanishing measure on the real line, there isn't an obvious notion of continuity/smoothness (my knowledge here is limited). That can lead to some surprises. For the more advanced readers: this result reminds me of how Haag's theorem shows (rigorously) that relativistic interacting QFTs can't exist (assuming operators sit at points indexed by real numbers), while in a Wilsonian sense QFTs are perfectly fine, because there's a UV cutoff, and all measurements have a finite resolution.
contravariant 8y ago
If we're going with the theory that things with a wavelength shorter than the Planck length don't exist then the sampling theorem should allow us to describe the universe with a discrete set of points. General relativity complicates things (as usual) but the sampling theorem can be generalised to non-uniform samples so it should probably still work somehow. In both cases this discrete theory is equivalent to a theory based on real numbers though, so in that sense there's not really any way to say which is better.

jwatte 8y ago

I think you're suggesting that "real" numbers are an abstraction that is convenient to model with, but aren't actually fully representable in the universe, because of Planck and Heisenberg and such? I would agree with that view.

ssivark 8y ago
It's not even necessarily about quantum stuff. The point is that any practical measurement system (even classical ones) can only deal in finite precision numbers. And since that set is of vanishing measure on the real line, there isn't an obvious notion of continuity/smoothness (my knowledge here is limited). That can lead to some surprises. For the more advanced readers: this result reminds me of how Haag's theorem shows (rigorously) that relativistic interacting QFTs can't exist (assuming operators sit at points indexed by real numbers), while in a Wilsonian sense QFTs are perfectly fine, because there's a UV cutoff, and all measurements have a finite resolution.

contravariant 8y ago

If we're going with the theory that things with a wavelength shorter than the Planck length don't exist then the sampling theorem should allow us to describe the universe with a discrete set of points. General relativity complicates things (as usual) but the sampling theorem can be generalised to non-uniform samples so it should probably still work somehow. In both cases this discrete theory is equivalent to a theory based on real numbers though, so in that sense there's not really any way to say which is better.