5 ms·

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 i

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.
  - roywiggins 5y ago
    But you can't actually use real numbers in calculations, you have to use approximations or proceed symbolically as far as possible. Nobody has a real computer[0], so the best you can do is pick a really accurate value for Pi and arrange your calculations as best you can to avoid pathological error propagation. [0] https://en.m.wikipedia.org/wiki/Real_computation https://en.m.wikipedia.org/wiki/Real_computation
    - katmannthree 5y ago
      Right, this is essentially why theoretical and applied mathematics are separate branches. Applied techniques like you described make theory incredibly cumbersome (and, importantly, not better). My argument here is not that applied mathematics is inferior (it’s what my degree is in), just that it’s generally not a good idea to carry applied techniques back into theory. You start with doing something the most correct way possible on paper and then convert that into the fastest possible method within your allowable bounds on precision and/or convergence. Operational reordering to keep additions in floats with similar exponents is great but you save that concern until it’s time to crunch numbers. When you’re trying to build an entire theory on how something complex works you’ll have a much better time using the available abstractions to manage complexity without getting bogged down in implementation details. Edit: addressing your point more directly, numerical computation itself must necessarily be done over fixed precision numbers but the tools we use to decide what and how to do that computation come out of theory done over the reals because of those specific properties of the reals. You can make things work over the rationals but the theory is tedious and the results of generally lower utility.
roywiggins 5y ago
It seems quite likely that cutting off Pi at the quintillionth decimal place will not hurt your simulation simply because your knowledge of the initial conditions in the physical universe won't be exact either. If a quintillion digits aren't enough, run your simulation with another quintillion places. Eventually you will match reality to within the accuracy of your ability to measure. Any issues introduced by using a finite approximation to Pi will eventually be swamped by the uncertainty in the initial conditions. If there's no uncertainty in the initial conditions, there will still be some finite approximation to Pi that will give you results as accurate as you can measure...
- katmannthree 5y ago
  Yes, you can generally find ways to manage life with limited precision. Doing so is useful enough that it’s an entire field (applied mathematics). For many cases though it is a better choice to stick with these particular carefully designed constructs (real, imaginary, and complex numbers) because they are easier to deal with and help keep you from getting bogged down in avoidable numerical tangles. Managing error accumulation is a huge deal for some types of simulations and every little bit counts, to the point of carefully ordering your operations to minimize precision loss with the widest numbers you can afford to use.

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.
- roywiggins 5y ago
  But you can't actually use real numbers in calculations, you have to use approximations or proceed symbolically as far as possible. Nobody has a real computer[0], so the best you can do is pick a really accurate value for Pi and arrange your calculations as best you can to avoid pathological error propagation. [0] https://en.m.wikipedia.org/wiki/Real_computation https://en.m.wikipedia.org/wiki/Real_computation
  - katmannthree 5y ago
    Right, this is essentially why theoretical and applied mathematics are separate branches. Applied techniques like you described make theory incredibly cumbersome (and, importantly, not better). My argument here is not that applied mathematics is inferior (it’s what my degree is in), just that it’s generally not a good idea to carry applied techniques back into theory. You start with doing something the most correct way possible on paper and then convert that into the fastest possible method within your allowable bounds on precision and/or convergence. Operational reordering to keep additions in floats with similar exponents is great but you save that concern until it’s time to crunch numbers. When you’re trying to build an entire theory on how something complex works you’ll have a much better time using the available abstractions to manage complexity without getting bogged down in implementation details. Edit: addressing your point more directly, numerical computation itself must necessarily be done over fixed precision numbers but the tools we use to decide what and how to do that computation come out of theory done over the reals because of those specific properties of the reals. You can make things work over the rationals but the theory is tedious and the results of generally lower utility.

roywiggins 5y ago

It seems quite likely that cutting off Pi at the quintillionth decimal place will not hurt your simulation simply because your knowledge of the initial conditions in the physical universe won't be exact either. If a quintillion digits aren't enough, run your simulation with another quintillion places. Eventually you will match reality to within the accuracy of your ability to measure. Any issues introduced by using a finite approximation to Pi will eventually be swamped by the uncertainty in the initial conditions. If there's no uncertainty in the initial conditions, there will still be some finite approximation to Pi that will give you results as accurate as you can measure...

katmannthree 5y ago
Yes, you can generally find ways to manage life with limited precision. Doing so is useful enough that it’s an entire field (applied mathematics). For many cases though it is a better choice to stick with these particular carefully designed constructs (real, imaginary, and complex numbers) because they are easier to deal with and help keep you from getting bogged down in avoidable numerical tangles. Managing error accumulation is a huge deal for some types of simulations and every little bit counts, to the point of carefully ordering your operations to minimize precision loss with the widest numbers you can afford to use.