Mathologer
4 HN recommenders · recommended in 5 months, 2020–2023
41:44 · 3.3M views · Jan 13, 2018
Mathologer examines the infamous Numberphile claim that 1+2+3+... equals -1/12, explaining what is wrong with the calculation and how it connects to real math like the zeta function and analytic continuation.
HN points to it as the in-depth rebuttal to the Numberphile video that explains why the 'proof' is wrong yet the result is meaningful.
7 comments from Hacker News
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]: this video
Burkard Polster's (Mathologer) videos on the subject are probably going to be more useful than Numberphile. Numberphile merely presents a trick; Mathologer points out both that it's patent nonsense [0] and that it's useful patent nonsense [1]. [0] this video [1] [LINK]
I know you're asking about Computerphile, but this video explains how one of the Numberphile videos is wrong: this video
I find a response video to it more enlightening. It's often found as the number one related video. this video
It's similar. Two issues with Numberphile videos I can remember: Their video on 1+2+3+... = -1/12 was so bad that another Math channel (Mathologer, which is highly recommendable btw) did a whole 40min video tearing it apart: this video In their video on fluid mechanics, they say that the Navier-Stokes equations describe all fluids on earth, while they only show the incompressible version (without clarifying). The examples shown on screen at that moment also include a Ketchup bottle, and they don't discuss non-Newtonian fluids either (Ketchup is non-Newtonian, that's why you can get a lot of it coming out at once if you shake a bottle too hard). It's a bit nitpicking though, because apart from these missing caveats it's a great video (with a guest lecturer from Tom Rocks Maths, another great math channel).