So a friend of mine got a little riled up about this video:
You know, I write and read poetry, and there’s this thing that happens when I talk about poetry, a thing that I know also happens all the time when people who write and read math talk about math. People say, “I don’t like poetry.” Or sometimes, more charitably, “I don’t understand poetry.” Sometimes — if they like me — they think my interest in poetry is adorable. But they don’t want to talk about it with me. And meanwhile, I’m thinking, what do you mean you don’t like poetry? Poetry is a big thing! It’s like saying you don’t like music! Or food! There’s so much of it, I’m sure we could find something you would like.
Well. That’s poetry. Because you know what my reaction was to my friend’s curiosity about that video up there? It was, “I can’t help you with that.” I might as well have said, “I don’t like math. I don’t understand math.” And in some ways that’s true. But it’s not my best answer.
I’d like to live in a world where my mind was alive. I want to be able to walk up to whatever I don’t understand and sit down and learn from it. I want to sit up late and learn from the people I love about the things I don’t, until I have some of their passion for their subjects reflected back to me, if not actually lighting me on fire myself. My vision of a life well lived includes dinner tables full of people talking, teaching, learning. And the responses, “I don’t like ____” or “I don’t understand ____” or even “You’re so cute when you talk about ____” do not belong at that table. Because the only thing that can come after those responses? Is silence.
Which is maybe why I spent most of the evening and dreamed off into the night reading everything in sight and thinking hard about the sum of a divergent series of natural numbers, trying to make sense of the claims here, and whether they made any sense to me. So here’s what I came up , with my mind that doesn’t understand and doesn’t like math. (And since I don’t understand and don’t like math, you’ll have to put up with any errors I make here as I learn. And notice how I felt the need to put this disclaimer here, even, which tells you something about how people including me feel about looking stupid when they play with numbers.)
It’s a little weird to talk about adding up an infinite series of numbers at all, and getting a result of any kind. After all, you can never be finished adding up to infinity, can you? You can always add one more. So how could you ever get the sum of an infinite series? But apparently people have tried. And in trying, they’ve observed that some infinite series of numbers seem to be *almost* getting to a single definite answer, and others do not.
For example, if I add 1/2 + 1/4 + 1/8 + 1/16+ 1/32 …. and so on up to infinity, I’m going to get closer and closer and closer to 1. I’ll never quite get there, of course — I can always add yet a smaller fraction — but I can certainly see my way to my unreachable destination. That’s apparently called a “convergent series,” because it converges on the answer as it heads toward infinity.
In a divergent series this doesn’t happen. And it doesn’t happen in several different ways that I found kind of interesting. In some divergent series, the number just keeps getting bigger as you add more to it. 1 +2 + 3 + 4 + 5 and so on. Hard to imagine ever finding the *answer* to that infinite series, despite the weird video, because you can always add one more. But other divergent series seems to be equivocal about their “answers,” to have answers they are “considering” but can’t quite ever commit to . Like 1 – 1 + 1 – 1 + 1 – 1 + 1 and so on. Depending on where you stop adding, the answer is always 0 or 1, and if I wanted to know what the “answer” was to the infinite sum of that series, I’d have trouble making up my mind between the two. In the video the answer to that divergent series is the assumption on which everything else rests, and they simply tell you the answer is 1/2. Which makes a certain kind of weird sense. 1/2 is just the average sum over the series. But they don’t actually prove this in the video; they just tell you it is so.
Here’s a more scholarly article that goes into a lot more detail about it, more detail than I can follow, myself. But what I gathered from that is that these divergent series are seductive precisely because you can readily do the simple algebraic manipulations they show in the video, but when you do that, weird things sometimes happen. For example, you can rearrange the terms of that 1-1+1-1+1… series in such as way as to get 0 as the answer, and there are other divergent series where you can rearrange the terms and get any number you want as the answer. No wonder one of the mathematicians whose name is linked with these series described them as an “invention of the devil”!
Still, that paper describes a wish list for how to derive an answer to these infinite series: 1) you should be able to do linear operations on the series without changing the answer, and 2) you’d better not allow re-arranging the terms, since clearly that creates problems, and 3) whatever rules you come up with to sum divergent series, if they work at all, you should get the correct answer when you apply them to convergent series, too. And I guess (at least?) two systems have been invented to meet this “wish list,” one called Cesaro summation, and one called Abel summation. And these systems do indeed yield the answers shown in the video above, which I guess lends credence to the idea that those simple algebraic solutions they do on the video are not *just* hocus pocus. But evaluating those systems other than that really is beyond my understanding without instruction. Still, my cousin said that his opinion of the claim 1 + 2 + 3 + 4 + 5 … = -1/12 was that this was “true in a certain sense,” rather than true the way 1 + 1 = 2 is true, and I guess I agree with him. These solutions may not be hocus pocus of the kind an old boyfriend of mine used to prove 0 = 1 (you had to squint to see at one point he divided by zero), but I think it’s more accurate to say -1/12 is the best answer we can assign to the series, rather than its sum in any ordinary way.
The thing that really riled up my friend, I think, was the idea that these weird “results” have applicability, apparently in string theory and physics. That doesn’t trouble me as much, and not (or at least not only) because I don’t understand the applications. Somewhere I used to have a list of technological innovations that were based on scientific theories that were later shown to be incorrect — but the technologies still worked. Perhaps applying -1/12 to string theory is similar.
Or perhaps this is just one more example of God’s sense of humor, which I gather is shot all through both higher mathematics and theoretical physics anyway…..