Thanks for the correction. I'm most definitely a layman when it comes to infinite series. A couple of things gave me confidence in this video: the result is presented in the string theory text and there's another video demonstrating the same result using Riemann Zeta functions (so it must be legit :)).
I sympathize with your frustration at the lack of rigor but isn't this kind of like taking pot shots at a middle school physics textbook for not covering Lagrangian mechanics?
No. While it's true that ζ(-1) = -1/12 and that the ζ function plays an important role in physics, your reasoning is fallacious.
If X implies Y and we know that Y is true, that does not mean X is true. So just because the video reached a "correct" conclusion does not mean that the means by which they reached that that conclusion are sensible or even consistent.
If I threw a dart at a dartboard labeled "What is 1 + 2 + 3 + ...?" and it landed on the section marked "-1/12", would you believe my answer? Would the fact that it happened to land on "-1/12" and also agreed with the ζ function lend credibility to my dart-throwing method of proof?
Indeed, if I encapsulated the methods used in that video, I could use those methods to have 1 + 2 + 3 + ... turn out to be any number I choose. This is the problem with specious reasoning — one can use it to reach any conclusion.
No. The problem is that for a very large range of applications 1+2+3+4+...=infinity. That’s the standard definition and the result is quite intuitive, even for a layman (infinity = verrry biiig).
For other applications, it’s sensible to define 1+2+3+4+...=-1/12(R) with an (R) to denote that you are not using the standard definition, but the Ramanujan definition. (You can drop the (R) one you are sure all the public has enough technical background.) The problem is that the Ramanujan definition doesn’t have many of the intituive properties of the standard definition. For example, in this article, eq. (8) and eq.(9) say that 0+2+3+4+... != 2+3+4+5+...
This is not similar to not discussing Lagrangian mechanics in a secondary school book. It’s more similar to mix Newtonian mechanics with the properties of the Higgs boson, and mix the density of water and the fact that electron really don’t have mass, and even say that the R and L electrons are different particle in spite the gravity force cancels the centrifugal force of the Moon (in a no Newtonian reference frame). It’s confusing, and mixing the theories can produce a paradox and be unintelligible.
If you mix them correctly and use just a little of the properties of theory inside the other, you can produce a convincing almost intelligible explanation that produce a paradox. The important point is to hide the technical problems in seemingly obvious properties, like in magic. The standard examples are mixing results of special relativity and Newtonian mechanics, or Quantum mechanics and Newtonian mechanics.
For the layman, I prefer an explanation that start saying that 1+2+3+4+...=infinity, then explain that there are other definitions, then a Ramanujan photograph, then some magic and handwaving to show 1+2+3+4+...= -1/12 (R), then enumerate some applications of this new definition, then show that 1+2+3+4+... != 0+1+2+3+4+... , so you must be very careful with this new summation.
It’s impossible to explain all the technical details to a layman, but it’s important to explain that they are hidden there, and why sometime there is necessary to make definitions that are not intuitive.
I sympathize with your frustration at the lack of rigor but isn't this kind of like taking pot shots at a middle school physics textbook for not covering Lagrangian mechanics?