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Pi is the ratio of a mathematical circle to its diameter, but there are no physical circles which have that ratio as exactly Pi, and even if one existed, you'd

Pi is the ratio of a mathematical circle to its diameter, but there are no physical circles which have that ratio as exactly Pi, and even if one existed, you'd never be able to distinguish it from one that was merely equal to Pi to the accuracy you are capable of measuring, because you'd need to measure with infinite precision, which you can't.

kadoban 5y ago
Orbits over long stretches of time seem likely to need arbitrarily high amounts of accuracy in pi. Sure you can just pick a rational number close enough for the accuracy you need, but why should the definition of pi need to change based on what you're measuring?
- roywiggins 5y ago
  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.
katmannthree 5y ago
That’s not isomorphic to the discussion at hand though, which refers to modeling systems. Being able to physically construct an object within epsilon does not imply that you won’t run into precision issues when modeling said object at the same degree of precision. In other words, cutting your beams to +/- 1/2” may work for each individual beam in a building but that does not imply that your building as a whole can tolerate an average beam length being +.499” above nominal.
- thereisnospork 5y ago
  >In other words, cutting your beams to +/- 1/2” may work for each individual beam in a building but that does not imply that your building as a whole can tolerate an average beam length being +.499” above nominal. The stronger version of the argument is that the length of a steel beam cannot be more precise(-ish) than the radius of an iron atom, so only 10-12 decimal places (in meters) are required to fully describe a steel beam's length. Likewise an actual circle's area isn't a function of Pi, but is rather a 'really large number' regular polyhedron. Which could then be approximated by a fairly pedestrian number of decimal points of pi to atomic precision. That said e.g. orbits are rather smooth, and could probably be considered to be fairly exact w.r.t. an arbitrary reference.
  - katmannthree 5y ago
    You’d want to generalize that to a beam whose length is a significant portion of the width of the universe (at which point you should also consider relativistic effects, so there’s more math you’ll need to define over the rationals), but even so that’s not addressing the issue at hand which is that you still have to propagate your uncertainty through each calculation. Depending on the function(s) and time steps your uncertainty can quickly blow past the threshold of utility (e.g. in the case of orbits, your uncertainty could end up being numerically greater than your ability to deal with it [meaning your orbital prediction for N bodies after T time has passed is so imprecise that your craft is not capable of intercepting at all potential states]). This is the power of the real numbers, being able to bypass the accumulation of error in some cases.

kadoban 5y ago

Orbits over long stretches of time seem likely to need arbitrarily high amounts of accuracy in pi. Sure you can just pick a rational number close enough for the accuracy you need, but why should the definition of pi need to change based on what you're measuring?

roywiggins 5y ago
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.

katmannthree 5y ago

That’s not isomorphic to the discussion at hand though, which refers to modeling systems. Being able to physically construct an object within epsilon does not imply that you won’t run into precision issues when modeling said object at the same degree of precision. In other words, cutting your beams to +/- 1/2” may work for each individual beam in a building but that does not imply that your building as a whole can tolerate an average beam length being +.499” above nominal.

thereisnospork 5y ago
>In other words, cutting your beams to +/- 1/2” may work for each individual beam in a building but that does not imply that your building as a whole can tolerate an average beam length being +.499” above nominal. The stronger version of the argument is that the length of a steel beam cannot be more precise(-ish) than the radius of an iron atom, so only 10-12 decimal places (in meters) are required to fully describe a steel beam's length. Likewise an actual circle's area isn't a function of Pi, but is rather a 'really large number' regular polyhedron. Which could then be approximated by a fairly pedestrian number of decimal points of pi to atomic precision. That said e.g. orbits are rather smooth, and could probably be considered to be fairly exact w.r.t. an arbitrary reference.
- katmannthree 5y ago
  You’d want to generalize that to a beam whose length is a significant portion of the width of the universe (at which point you should also consider relativistic effects, so there’s more math you’ll need to define over the rationals), but even so that’s not addressing the issue at hand which is that you still have to propagate your uncertainty through each calculation. Depending on the function(s) and time steps your uncertainty can quickly blow past the threshold of utility (e.g. in the case of orbits, your uncertainty could end up being numerically greater than your ability to deal with it [meaning your orbital prediction for N bodies after T time has passed is so imprecise that your craft is not capable of intercepting at all potential states]). This is the power of the real numbers, being able to bypass the accumulation of error in some cases.