Similar to @sdfrost61, I think your reply is excellent. I read all the post on the linked to site with lots of people saying “ho ho, isn’t phil plait a moron” (or more nasty things) and lots of “This is so obvious, it’s clearly just [insert name of mathematician that came up with theorem] functions” without ever explaining themselves. I tend to assume that when people don’t explain things clearly, they don’t understand it properly.
There is an interesting point made in the video. It was a point that I, as an mathematically oriented engineer, couldn’t understand the difficulty behind. Clearly there was more to it than meets the eye, but I needed someone to explain it properly. I thank you for presenting one aspect of the explanation so well.
Boring answer: mathematics is a game where you pick a set of initial assumptions and play with them to see what falls out. Occasionally you get stuck, encountering some question that has either no answer or more than one “correct” answers depending on what route you take (when analysing arithmetic this is unavoidable). At this point you have to decide whether to start over with new assumptions, or modify your existing ones and plough ahead (which risks invalidating some of your earlier results, by enabling new contradictions to be found), or just declaring certain questions to unanswerable dead ends.
ALL RIGHT-THINKING PLATONIST MATHEMATICIANS, SHUN THE FORMALIST!!!1!!!
Jokes aside, it’s results like this that make me a formalist, too. (Under the influence of certain recreational pharmaceuticals, I can even be an intuitionist.)
But don’t tell anyone, or else they’ll lynch me. (A mathematical lynch mob is slow, but pretty thorough.)
I think the worst thing about the “hur, hur, you dummy” and “well that’s just…” answers is that when you miss the connection to the Riemann zeta function, you miss this unexpected (to me anyway) connection between quantum mechanics and number theory. (Properties of the Riemann zeta function are related to the distribution of prime numbers.)
The point is that you have a way of assigning finite values to infinite sums - the method in the video (average the tail of a large sum) will agree with the ‘standard’ answer of just doing the sum whenever the series is convergent. E.g if you sum n=1 … Infinity 1/n^2 with that method you’ll get the right answer. But then you can also get finite answers for sums that don’t converge, which might be useful.
It’s like asking if 3!=3x2x1=6 and 4!=4x3x2x1=24, what is that the factorial of 1/2?
You can show 1/2! = 1/sqrt(4.pi) using the gamma function. E.g. Gamma(n)=(n-1)! for every integer n>1 and whilst it doesn’t make sense to write 1/2!=(1/2)(-1/2)(-3/2)(-5/2)(-7/2)… using the normal definition of factorials, you can still evaluate the gamma function for fractional arguments and claim it continues the factorial function to non-integer values.
Maybe the confusion is in using ‘equals’ to mean equals rather than ‘assign a value in some consistent way’
I’m not a mathematician but played one for long enough to complete my PhD coursework. I read some of Hardy’s Divergent Series, the classic work on this subject, and played with these series a bit in my free time. My position is that there is something “real” going on here, that if you need a finite answer to 1+2+3+… the only reasonable one is -1/12. Programmers will be unsurprised that using these techniques the only reasonable solution to 1+2+4+8+16+… is -1.
To me, the video generally shows a typical physicist’s use of mathematics and the word “proof” meaning, it isn’t really proved at all but fits certain intuitions and happens to give the correct answer, despite not being generalizable to any other situation. If getting the result just happened to require being lucky that multiple hidden infinities cancel each other out, well, that’s physics for you.
