I’m one of these professional mathematicians, and I’ll say that this article completely fails to demonstrate it’s central thesis that there is a valid intuitive argument for concluding that 1 + 2 + 3 + … = −1/12 makes sense. What’s worse, it only pretends to do so by what’s essentially a swindle.
In my understanding, it’s relatively easy to reason that a given divergent series “should” take an arbitrary finite value by the kind of arguments employed here, so what is being done is taking a foregone conclusion and providing some false intuition for why it should be true.
On a less serious note, speaking to the real reason why 1 + 2 + 3 + … = −1/12, that’s actually what physicists will tell you, and we all know one should be careful around those.
In bosonic string theory, the attempt is to compute the possible energy levels of a string, in particular, the lowest energy level. Speaking informally, each harmonic of the string can be viewed as a collection of ‘D’ − 2 independent quantum harmonic oscillators, one for each transverse direction, where D is the dimension of spacetime. If the fundamental oscillation frequency is ω, then the energy in an oscillator contributing to the n-th harmonic is nħω/2. So using the divergent series, the sum over all harmonics is −ħω(D − 2)/24. Ultimately it is this fact, combined with the Goddard–Thorn theorem, which leads to bosonic string theory failing to be consistent in dimensions other than 26.
I’m one of these professional mathematicians, and I’ll say that this article completely fails to demonstrate it’s central thesis that there is a valid intuitive argument for concluding that 1 + 2 + 3 + … = −1/12 makes sense. What’s worse, it only pretends to do so by what’s essentially a swindle. In my understanding, it’s relatively easy to reason that a given divergent series “should” take an arbitrary finite value by the kind of arguments employed here, so what is being done is taking a foregone conclusion and providing some false intuition for why it should be true.
On a less serious note, speaking to the real reason why 1 + 2 + 3 + … = −1/12, that’s actually what physicists will tell you, and we all know one should be careful around those.
is the = −12 a typo or a joke?
Typo, thanks for pointing it out. Also, see here for the physics reference: https://en.m.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_⋯
Haha what the fuck
It’s astonishing, but yes, that is the reason why that form of string theory takes place in a 26 dimensional space-time.
wait till you see E8*E8 heterotic string theory,
https://en.wikipedia.org/wiki/Heterotic_string_theory
This might be a better description: link
I did warn about the prereqs.