Someone is wrong on the Internet. In particular, today my friend Paul sent me a link to this guy, who credulously buys someone’s argument that 1 + 2 + 3 + 4 + 5 + 6 + … equals a small negative number. This is completely false, but it’s false for reasons that trip up a lot of people, so I think it’s worth spending some time on.
This is the same genre of argument by which you can “prove” that 1 = 2. So here’s the first step in arguing against it: think to yourself, “If I find this nonsensical, then it’s probably nonsense.” That’s really an okay way to feel. But people are scared of math, so they often think, “Well, mathematics says a lot of crazy things, so what do I know?” They’re likely to blame mathematicians for being unrealistic and for endorsing absurd conclusions just because their axioms made them say so.
The next step is to ask why mathematicians *don’t* just follow their axioms off a cliff. 1 is not equal to 2, and mathematicians know it. But who knows, maybe some abstruse chain of reasoning would lead a mathematician somewhere absurd. The reason that doesn’t happen is that *mathematics eventually has to collide with the real world*. Eventually physicists are going to use mathematics. Eventually engineers are going to build buildings; if they prove that a steel beam can handle 2 tons of weight, it damn well better not actually be 1 ton of weight. Mathematics is used in all sorts of real contexts. Logic cannot be used to lead us to unreasonable conclusions.
Now, mathematics is nice, because it consists of axioms and logic. You start with some axioms, and you follow some logic, and you get a conclusion. If the conclusion is absurd, then it must be because either the axioms were wrong or the logic was wrong. So you only have a small number of places to check for mistakes. (As opposed to your gut, which is less subject to verification.)
But infinity is weird, right? Surely infinities can do weird things. That’s absolutely true, which is why a couple hundred years of mathematicians and philosophers, starting with Isaac Newton and Bishop Berkeley, worked very hard to create a set of tools that allow us to talk about infinity in a sensible way that makes it hard for us to trip ourselves up. This is what calculus is, and why calculus is one of the monuments of Western civilization. It’s not just a very useful collection of tools used in everything from humdrum contexts like building buildings to literally heavenly pursuits like astronomy, though it is that. It’s also a philosophical marvel that makes the infinite comprehensible to mere finite humans. It is a way of keeping our language precise and avoid getting in hopeless muddles, even when we’re talking about incomprehensible vastness.
The basic trick that the essayist and the video creator are (mis)using, and the trick that lands them in such a muddle, is the following. We start with this:
x = 1 – 1 + 1 – 1 + …
and we add another copy like so:
2x = (1 – 1 + 1 – 1 + …) + (1 – 1 + 1 – 1 + …)
Then we write them on separate lines and shift things, like so:
2x = (1 - 1 + 1 - 1 + …) + (1 - 1 + 1 - 1 + …) = (1 - 1 + 1 - 1 + …) + (1 - 1 + 1 - 1 + …)
Nothing too complicated, right? We just shifted everything down a line and over by a couple of spaces. Great. Now, goes the argument, we see that every +1 on one line is paired with a -1 on the next line, or vice versa. From this they conclude that
2x = 1 + (-1 + 1) + (-1 + 1) + … = 1 + 0 + 0 + …
And that equals 1. So then 2x = 1, which means x = 1/2.
Your intuition should tell you that this is absurd. The sum up to the first term is 1. The sum up to the second term is 0. The sum up to the third term is 1. And on we go, back and forth, forever. The sum never settles down at a single value. Your intuition should tell you this, and your intuition is correct.
Another way to respond to this essayist’s nonsense is to use his argument against him. Take the same chain of reasoning as before: we put the definitions of x and 2x on separate lines, except this time we shift everything ahead *two* positions rather than just one. Like so:
2x = (1 - 1 + 1 - 1 + …) + (1 - 1 + 1 - 1 + …) = (1 - 1 + 1 - 1 + …) + (1 - 1 + 1 - 1 + …)
Again, nothing suspicious about this, right? Only this time, the same chain of reasoning — that we pair the row above with the row below — leads us to conclude that
2x = 1 - 1 + (1 + 1) + (-1 + -1) + (1 + 1) + (-1 + -1) + … = 0 + 2 + -2 + 2 + -2 + …
which lands us back where we started. If just shifting things around by an arbitrary amount leads to wildly varying results, then your intuition should tell you that something is probably wrong with the “shifting” method.
Basically everything in that essay and that video reduces to this “shifting” trick. By repeated application of the method they end up concluding that 1+2+3+4+5+… equals a negative number. It doesn’t, which is obvious. Your intuition doesn’t fail you here.
The actual answer is that talking about the sum of this series makes no sense, because it has no sum. If a sum is going to eventually settle down to something nice and finite, the terms have to get smaller. Here the terms aren’t getting smaller; they’re just oscillating. Likewise, the terms in 1+2+3+4+5+… aren’t getting smaller; they’re increasing. So that sum doesn’t converge either, and for a different reason: it’s blowing up, and will grow without bound.
The mathematical answer is that if a sum “diverges” like this one does, then you can’t arbitrarily rearrange terms in it and expect the sum to keep working out. Your intuition should tell you that the problem with 1+2+3+4+5+… isn’t the sort of problem that can be solved by just shifting things around; the problem with that sum is that *you’re adding things that keep getting larger*. No amount of shifting things is going to make that sum up to something nice.
Indeed, the 1-1+1-1+… example is one that they give you in calculus textbooks to show you that we can’t treat infinite sums the way we treat finite ones. The example shows that you need to be much more careful with infinities. It shows you that the logic and axioms you thought were sensible for finite quantities don’t quite work out for infinite ones.
Your intuition does, then, need help sometimes. In particular, it regularly fails when it’s faced with infinities. But there are times when your intuition leads you the right way, and mathematics can help you confirm it.
There are other examples that are facially similar but differ in crucial ways from this 1-1+1-1+… nonsense. There’s a mathematical proof, for instance, that .99999…=1. That happens to be true. The basic intuition there is that if I can bring two numbers as close together as I want, then those two numbers are indeed equal. If I am standing a foot away from you, and tell you that I’m going to halve the distance between us, then halve it again, then continue halving it forever, then — assuming we both live forever — I will eventually be standing 0.00000… inches away from you.
This can be proven rigorously. It’s important to note, though, that it can be proved entirely with finite numbers. I never need to use an “actual infinity” to prove to you that this works. All I need to say is that, essentially, I have a recipe for coming close to you. The recipe is “at every step, close half the distance between me and you.” Then you challenge me: “I bet you can’t get within 1/4 of a foot of me.” I reply, “My recipe will get me there in two steps: after one step I’m 6 inches away, and after two steps I’m 3 inches away.” So you say, “Fine, but I bet you can’t within an inch of me,” to which I reply, “My recipe will get me there in four steps: after 1 step I’m 6 inches away, after 2 steps I’m 3 inches away, after 3 steps I’m 1.5 inches away, and after 4 steps I’m 3/4 of an inch away. At that point I’m within an inch of you.”
You see what’s happening. I never actually say anything about how “after an infinite number of steps, I’m 0.000… inches away from you.” Instead I just show that I have a recipe that will get me as close as you could wish, in a finite number of steps. That is what we call a “limit” in calculus. The labor that went into making that word intellectually coherent is one of our species’s greatest accomplishments.
So please: use your intuition here. And if you question whether your intuition is the proper guide, learn a little bit of math. The mathematics of infinities is both spectacularly beautiful and really fun. Maybe in subsequent posts I’ll give some examples of how fun it is.
__P.S.__ (same day): This is an excellent response to the #slatepitch quackery, also via my friend Paul.
Yeah, Numberphile is usually better than this.
That said, the -1/12 holds up but only if you use a particular method to assign a value to the divergent series. They just happen to be using one particular method that is commonly used in physics. They allude to it but don’t bother explaining it.
I saw a link recently to a math Stack Exchange discussing this in detail but I can’t find it at the moment.
Right, there are ways of combining the terms in divergent series so that the transformed series doesn’t diverge. See “Cesaro summation”, for instance. But this ain’t that.
I believe it’s Zeta function regularization:
And I think they mention on Zeta function in passing.