Updated with a third footnote clarifying my use of the term “diverge,” thanks to suggestion by Evelyn Lamb, who has also written an excellent discussion of the problem with the video. At the end of this post I list all the critiques I’ve found so far.
I feel like one of those grizzled action heroes who, having given it all up, is dragged reluctantly out of retirement for one more big mission. Over the past month or so (honestly, I forget how long I was working on things), I wrote a series of blog posts on the “weirdness” of infinity in mathematical set theory. Hopefully, there were two things that I got across in those posts: (1) infinity can be very weird, but (2) it can be comprehended, and even reasonable, once one understands the assumptions and limitations built into the mathematics.
Having retired from writing those posts, the other day I came across the following video:
So, using a seemingly simple series of mathematical manipulations, they “prove” the following astounding result: the infinite sum of increasing positive integers equals a finite, fractional, negative number. In short:
This video was picked up by Phil Plait at Bad Astronomy, who called* it “simply the most astonishing math that you’ll ever see.” It has already spread far and wide across the internet, including making it to the popular site Boing Boing.
But is it true? The video makes it seem so simple, and uncontroversial, almost obvious. But there are some big mathematical assumptions hidden in their argument that, in my opinion, make it very misleading. To put it another way: in a restricted, specialized mathematical sense, one can assign the value -1/12 to the increasing positive sum. But in the usual sense of addition that most human beings would intuitively use, the result is nonsensical.
To me, this is an important distinction: a depressingly large portion of the population automatically assumes that mathematics is some nonintuitive, bizarre wizardry that only the super-intelligent can possibly fathom. Showing such a crazy result without qualification only reinforces that view, and in my opinion does a disservice to mathematics.
I’ve actually discussed this result years ago on this blog, talking about the Riemann zeta function and how -1/12 isn’t really equal to the infinite sum given. But even that discussion is probably a little too abstract, especially since I don’t discuss in any detail how the result -1/12 could be physically accurate. As it has been noted (and I’ve noted myself), the -1/12 result can be used with surprising accuracy in physics problems. But even there, things are much more subtle than they appear.
So let’s take a closer look** at the “proof” that an infinite increasing sum can equal -1/12. We will explain why the answer is not so simple as the video makes it appear, and why it is also not quite so simple to say that physics justifies the answer. We have a lot of ground to cover, so let’s go!