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I was trying to remember the situation where in physics, under certain conditions, a proof that shows infinity can be represented numerically as -1/12 (it was s

I was trying to remember the situation where in physics, under certain conditions, a proof that shows infinity can be represented numerically as -1/12 (it was some obscure fraction yet after seeing it, I think this was the value). I couldn't for the life of me remember the set of conditions and proof that shows this but it was interesting when reading through it a few years ago.

deleted 6y ago
[deleted]
atemerev 6y ago
In physics, it is something related to removing divergences in bosonic string theory, I am not an expert in that. Here, however, is a fantastic article that shows how -1/12 is actually derived: https://medium.com/cantors-paradise/the-ramanujan-summation-1-2-3-1-12-a8cc23dea793 https://medium.com/cantors-paradise/the-ramanujan-summation-... (spoiler: it is really easy and could be done by a high schooler).
- cyphar 6y ago
  That proof (and the more general statement) is not correct because it makes assumptions about infinite summation of divergent series that are not true. There is a link between the sum of natural numbers -- namely, the analytic continuation of the Reimann Zeta function has Z(-1) = -1/12. But the Zeta function is only defined to equal the sum of inverse powers for Re(z) > 1. And under a specific definition of summation (Ramanujan summation) you can say that "the sum is equal to -1/12" but that isn't the same as normal summation. Mathologer did a fairly in-depth video[1] into why this proof is wrong and what the actual link is between the infinite series and -1/12. The upshot is that even if you use more complicated definitions of summation, you cannot define sums of the kind 1+2+3+... to equal a finite number. If you assume the infinite sum of 1+2+3+... converges to a finite number you can easily prove a contradictory statement using the same summation properties assumed by that proof. Namely: S = 1 + 2 + 3 + ... = -1/12 S2 = S-S = 0 = 1 + 2 + 3 + ... - 1 - 2 - 3 - ... = 0 = 1 + 2 + 3 + ... - 1 - 2 - ... = 0 S2 = 1 + 1 + 1 + 1 + ... = 0 S3 = S2-S2 = 0 = (1-1) + (1-1) + ... = 1 - 1 + 1 - 1 + ... = 0 But we "derived" in the original proof that S3 is equal to 1/2, which is a contradiction. (You could derive S3 is 1/2 from S but that's what the article does in reverse.) [1]: https://youtu.be/YuIIjLr6vUA https://youtu.be/YuIIjLr6vUA
OscarCunningham 6y ago
The Casimir Effect in one dimension: https://en.m.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension https://en.m.wikiversity.org/wiki/Quantum_mechanics/Casimir_...

deleted 6y ago

[deleted]

atemerev 6y ago

In physics, it is something related to removing divergences in bosonic string theory, I am not an expert in that. Here, however, is a fantastic article that shows how -1/12 is actually derived: https://medium.com/cantors-paradise/the-ramanujan-summation-1-2-3-1-12-a8cc23dea793 https://medium.com/cantors-paradise/the-ramanujan-summation-... (spoiler: it is really easy and could be done by a high schooler).

cyphar 6y ago
That proof (and the more general statement) is not correct because it makes assumptions about infinite summation of divergent series that are not true. There is a link between the sum of natural numbers -- namely, the analytic continuation of the Reimann Zeta function has Z(-1) = -1/12. But the Zeta function is only defined to equal the sum of inverse powers for Re(z) > 1. And under a specific definition of summation (Ramanujan summation) you can say that "the sum is equal to -1/12" but that isn't the same as normal summation. Mathologer did a fairly in-depth video[1] into why this proof is wrong and what the actual link is between the infinite series and -1/12. The upshot is that even if you use more complicated definitions of summation, you cannot define sums of the kind 1+2+3+... to equal a finite number. If you assume the infinite sum of 1+2+3+... converges to a finite number you can easily prove a contradictory statement using the same summation properties assumed by that proof. Namely: S = 1 + 2 + 3 + ... = -1/12 S2 = S-S = 0 = 1 + 2 + 3 + ... - 1 - 2 - 3 - ... = 0 = 1 + 2 + 3 + ... - 1 - 2 - ... = 0 S2 = 1 + 1 + 1 + 1 + ... = 0 S3 = S2-S2 = 0 = (1-1) + (1-1) + ... = 1 - 1 + 1 - 1 + ... = 0 But we "derived" in the original proof that S3 is equal to 1/2, which is a contradiction. (You could derive S3 is 1/2 from S but that's what the article does in reverse.) [1]: https://youtu.be/YuIIjLr6vUA https://youtu.be/YuIIjLr6vUA

OscarCunningham 6y ago

The Casimir Effect in one dimension: https://en.m.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension https://en.m.wikiversity.org/wiki/Quantum_mechanics/Casimir_...