4 ms·

I would say it like so: pi has many transcendental neighbors, which are in no way related to pi. But e and pi, despite being numerically distant, are in one sen

I would say it like so: pi has many transcendental neighbors, which are in no way related to pi. But e and pi, despite being numerically distant, are in one sense "close" - related - via Euler's formula. So there's a "local closeness" - the arithmetic difference of pi and its neighbor - but also a "relationship closeness" where pi and e are intimately linked. Perhaps quantum states obey the relationship closeness, not the arithmetical one. Classical physics obeys "arithmetical closeness" meaning that small differences in dynamical properties are close in state-space. But actually this is an emergent property: highly sensitive quantum experiments reveal that small differences in dynamical properties can be highly separated in state-space. Rationals are all pretty easily related, but pi has nothing in common with a random close transcendental neighbor. More: https://en.wikipedia.org/wiki/Implicate_and_explicate_order https://en.wikipedia.org/wiki/Implicate_and_explicate_order

knzhou 7y ago
Yes, but in order to actually explain the results of existing experiments superdeterministically, this "implicate" order must be vast. It has to link the results of spin measurements, coin flips, radioactive decays halfway across the world, and the twinkling of the light from a long-dead star, all of which are sources of apparent randomness used together in experiments. All of this has to work in concert. If that really is the right way to think about the world, then absolutely nothing of science can survive. There is no such thing as a controlled experiment.
- ganzuul 7y ago
  Naively, superconductivity could serve as a partially closed system for experiments in this theory.
- jcranberry 7y ago
  nonsense, that's saying that no kind of science based on probabilities could survive because it's difficult to run a deterministic experiment. if that was the case then things like climate science, atmospheric science and meteorology would be unable to exist in the first place.
  - knzhou 7y ago
    Once again, the "superdeterminism" promoted here is a very very very very very different thing from either ordinary determinism or ordinary probability. You're like the 50th person to reply to me in exactly this way.
thaumasiotes 7y ago
> But e and pi, despite being numerically distant, are in one sense "close" - related - via Euler's formula. So there's a "local closeness" - the arithmetic difference of pi and its neighbor - but also a "relationship closeness" where pi and e are intimately linked. I don't understand how Euler's formula establishes a special relationship. You can compute a function of any two numbers. Where f(a,b) = a^{ib}, then f(e,pi) = -1, but why is that more meaningful than the fact that pi + e = 5.85987... ? In Euler's formula in particular, the value e cannot be changed -- a complex number is necessarily represented as r * e^{iθ}. (Step one in processing e.g. 3 * 7^{i√2} is to normalize it to 3 * e^{i√2 ln 7}.) Pi is the angle between the vector (1, 0) and the vector (-1, 0). You could change that, and you'd have an equation saying that, if you rotate a unit vector by whatever the new angle is, the result is a complex number of unit magnitude and the angle you specified. For example, e^{2i} = cos(2) + i sin(2) = -0.416 + 0.909i. But I think the claim that this shows a deep, fundamentally special relationship between e and 2 is a hard sell. The cleanness of e^{i * pi} = -1 isn't telling you that pi is related to e -- it's telling you that pi is related to trigonometry.
- pfortuny 7y ago
  So: e is the basis giving rise to a group homomorphism between (R,+) and the unit circumference. That is: e is inherently trigonometric. Not irrelevant.
  - thaumasiotes 7y ago
    e is related to the complex numbers through trigonometry. And pi is related to trigonometry more directly than that. But e and pi are not related to each other; "being related" isn't transitive like that. Consider correlations. A correlation is necessarily "transitive" if the correlation values are very, very close to ±1, and not otherwise. If a correlates with b at 0.7, and b correlates with c at 0.68, the correlation between a and c might be 0.69, but only by coincidence.
    - jcranberry 7y ago
      correlations are not a good analogy, homomorphisms often preserve desirable properties or relationships.

knzhou 7y ago

Yes, but in order to actually explain the results of existing experiments superdeterministically, this "implicate" order must be vast. It has to link the results of spin measurements, coin flips, radioactive decays halfway across the world, and the twinkling of the light from a long-dead star, all of which are sources of apparent randomness used together in experiments. All of this has to work in concert. If that really is the right way to think about the world, then absolutely nothing of science can survive. There is no such thing as a controlled experiment.

ganzuul 7y ago
Naively, superconductivity could serve as a partially closed system for experiments in this theory.
jcranberry 7y ago
nonsense, that's saying that no kind of science based on probabilities could survive because it's difficult to run a deterministic experiment. if that was the case then things like climate science, atmospheric science and meteorology would be unable to exist in the first place.
- knzhou 7y ago
  Once again, the "superdeterminism" promoted here is a very very very very very different thing from either ordinary determinism or ordinary probability. You're like the 50th person to reply to me in exactly this way.

thaumasiotes 7y ago

> But e and pi, despite being numerically distant, are in one sense "close" - related - via Euler's formula. So there's a "local closeness" - the arithmetic difference of pi and its neighbor - but also a "relationship closeness" where pi and e are intimately linked. I don't understand how Euler's formula establishes a special relationship. You can compute a function of any two numbers. Where f(a,b) = a^{ib}, then f(e,pi) = -1, but why is that more meaningful than the fact that pi + e = 5.85987... ? In Euler's formula in particular, the value e cannot be changed -- a complex number is necessarily represented as r * e^{iθ}. (Step one in processing e.g. 3 * 7^{i√2} is to normalize it to 3 * e^{i√2 ln 7}.) Pi is the angle between the vector (1, 0) and the vector (-1, 0). You could change that, and you'd have an equation saying that, if you rotate a unit vector by whatever the new angle is, the result is a complex number of unit magnitude and the angle you specified. For example, e^{2i} = cos(2) + i sin(2) = -0.416 + 0.909i. But I think the claim that this shows a deep, fundamentally special relationship between e and 2 is a hard sell. The cleanness of e^{i * pi} = -1 isn't telling you that pi is related to e -- it's telling you that pi is related to trigonometry.

pfortuny 7y ago
So: e is the basis giving rise to a group homomorphism between (R,+) and the unit circumference. That is: e is inherently trigonometric. Not irrelevant.
- thaumasiotes 7y ago
  e is related to the complex numbers through trigonometry. And pi is related to trigonometry more directly than that. But e and pi are not related to each other; "being related" isn't transitive like that. Consider correlations. A correlation is necessarily "transitive" if the correlation values are very, very close to ±1, and not otherwise. If a correlates with b at 0.7, and b correlates with c at 0.68, the correlation between a and c might be 0.69, but only by coincidence.
  - jcranberry 7y ago
    correlations are not a good analogy, homomorphisms often preserve desirable properties or relationships.