I’ll stick to it. It’s easier to perform experiments than it is to give mathematical proofs. If experiments can give strong evidence for anything (I hope they can!), then this data can give strong evidence that pi is normal: http://www.piworld.de/pi-statistics/pist_sdico.htm
Maybe past ten-to-the-one-trillion digits, the statistics of pi are radically different. Maybe past ten-to-the-one-trillion meters, the laws of physics are radically different.
Maybe past ten-to-the-one-trillion digits, the statistics of pi are radically different. Maybe past ten-to-the-one-trillion meters, the laws of physics are radically different.
I was just thinking about the latter case, actually. If g equalled G (m1 ^ (1 + (10 ^ −30)) (m2 ^ (1 + (10 ^ −30))) / (r ^ 2), would we know about it?
Well, the force of gravity isn’t exactly what you get from Newton’s laws anyways (although most of the easily detectable differences like that in the orbit of Mercury are better thought of as due to relativity’s effect on time than a change in g). I’m not actually sure how gravitational force could be non-additive with respect to mass. One would have the problem of then deciding what constitutes a single object. A macroscopic object isn’t a single object in any sense useful to physics. Would for example this calculate the gravity of Earth as a large collection of particles or as all of them together?
But the basic point, that there could be weird small errors in our understanding of the laws of physics is always an issue. To use a slightly more plausible example, if say the force of gravity on baryons is slightly stronger than that on leptons (slightly different values of G) we’d be unlikely to notice. I don’t think we’d notice even if it were in the 2nd or 3rd decimal of G (partially because G is such a very hard constant to measure.)
Just a pedantic note: pi has not been proven normal. Maybe one fifth of the digits are 1s.
I’ll stick to it. It’s easier to perform experiments than it is to give mathematical proofs. If experiments can give strong evidence for anything (I hope they can!), then this data can give strong evidence that pi is normal: http://www.piworld.de/pi-statistics/pist_sdico.htm
Maybe past ten-to-the-one-trillion digits, the statistics of pi are radically different. Maybe past ten-to-the-one-trillion meters, the laws of physics are radically different.
The later case seems more likely to me.
I was just thinking about the latter case, actually. If g equalled G (m1 ^ (1 + (10 ^ −30)) (m2 ^ (1 + (10 ^ −30))) / (r ^ 2), would we know about it?
Well, the force of gravity isn’t exactly what you get from Newton’s laws anyways (although most of the easily detectable differences like that in the orbit of Mercury are better thought of as due to relativity’s effect on time than a change in g). I’m not actually sure how gravitational force could be non-additive with respect to mass. One would have the problem of then deciding what constitutes a single object. A macroscopic object isn’t a single object in any sense useful to physics. Would for example this calculate the gravity of Earth as a large collection of particles or as all of them together?
But the basic point, that there could be weird small errors in our understanding of the laws of physics is always an issue. To use a slightly more plausible example, if say the force of gravity on baryons is slightly stronger than that on leptons (slightly different values of G) we’d be unlikely to notice. I don’t think we’d notice even if it were in the 2nd or 3rd decimal of G (partially because G is such a very hard constant to measure.)