Infinite Summations: A Rationality Litmus Test
You may have seen that Numberphile video that circulated the social media world a few years ago. It showed the ‘astounding’ mathematical result:
1+2+3+4+5+… = −1/12
(quote: “the answer to this sum is, remarkably, minus a twelfth”)
Then they tell you that this result is used in many areas of physics, and show you a page of a string theory textbook (oooo) that states it as a theorem.
The video caused quite an uproar at the time, since it was many people’s first introduction to the rather outrageous idea and they had all sorts of very reasonable objections.
Here’s the ‘proof’ from the video:
First, consider P = 1 − 1 + 1 − 1 + 1…
Clearly the value of P oscillates between 1 and 0 depending on how many terms you get. Numberphile decides that it equals 1⁄2, because that’s halfway in the middle.
Alternatively, consider P+P with the terms interleaved, and check out this quirky arithmetic:
1-1+1-1…
+ 1-1+1…
= 1 + (-1+1) + (1-1) … = 1, so 2P = 1, so P = 1⁄2
Now consider Q = 1-2+3-4+5…
And write out Q+Q this way:
1-2+3-4+5…
+ 1 −2+3-4…
= 1-1+1-1+1 = 1⁄2 = 2Q, so Q = 1⁄4
Now consider S = 1+2+3+4+5...
Write S-4S as
1+2+3+4+5…
− 4 −8 …
=1-2+3-4+5… = Q=1/4
So S-4S=-3S = 1⁄4, so S=-1/12
How do you feel about that? Probably amused but otherwise not very good, regardless of your level of math proficiency. But in another way it’s really convincing—I mean, string theorists use it, by god. And, to quote the video, “these kinds of sums appear all over physics”.
So the question is this: when you see a video or hear a proof like this, do you ‘believe them’? Even if it’s not your field, and not in your area of expertise, do you believe someone who tells you “even though you thought mathematics worked this way, it actually doesn’t; it’s still totally mystical and insane results are lurking just around the corner if you know where to look”? What if they tell you string theorists use it, and it appears all over physics?
I imagine this is as a sort of rationality litmus test. See how you react to the video or the proof (or remember how you reacted when you initially heard this argument). Is it the ‘rational response’? How do you weigh your own intuitions vs a convincing argument from authority plus math that seems to somehow work, if you turn your head a bit?
If you don’t believe them, what does that feel like? How confident are you?
(spoilers below)
It’s totally true that, as an everyday rationalist (or even as a scientist or mathematician or theorist), there will always be computational conclusions that are out of your reach to verify. You pretty much have to believe theoretical physicists who tell you “the Standard Model of particle physics accurately models reality and predicts basically everything we see at the subatomic scale with unerring accuracy”; you’re likely in no position to argue.
But—and this is the point—it’s highly unlikely that all of your tools are lies, even if ‘experts’ say so, and you ought to require extraordinary evidence to be convinced that they are. It’s not enough that someone out there can contrive a plausible-sounding argument that you don’t know how to refute, if your tools are logically sound and their claims don’t fit into that logic.
(On the other hand, if you believe something because you heard it was a good idea from one expert, and then another expert tells you a different idea, take your pick; there’s no way to tell. It’s the personal experience that makes this example lead to sanity-questioning, and that’s where the problem lies.)
In my (non-expert but well-informed) view, the correct response to this argument is to say “no, I don’t believe you”, and hold your ground. Because the claim made in the video is so absurd that, even if you believe the video is correct and made by experts and the string theory textbook actually says that, you should consider a wide range of other explanations as to “how it could have come to be that people are claiming this” before accepting that addition might work in such an unlikely way.
Not because you know about how infinite sums work better than a physicist or mathematician does, but because you know how mundane addition works just as well as they do, and if a conclusion this shattering to your model comes around—even to a layperson’s model of how addition works, that adding positive numbers to positive numbers results in bigger numbers --, then either “everything is broken” or “I’m going insane” or (and this is by far the theory that Occam’s Razor should prefer) “they and I are somehow talking about different things”.
That is, the unreasonable mathematical result is because the mathematician or physicist is talking about one “sense” of addition, but it’s not the same one that you’re using when you do everyday sums or when you apply your intuitions about intuition to everyday life. This is by far the simplest explanation: addition works just how you thought it does, even in your inexpertise; you and the mathematician are just talking past each other somehow, and you don’t have to know what way that is to be pretty sure that it’s happening. Anyway, there’s no reason expert mathematicians can’t be amateur communicators, and even that is a much more palatable result than what they’re claiming.
(As it happens, my view is that any trained mathematician who claims that 1+2+3+4+5… = −1/12 without qualification is so incredibly confused or poor at communicating or actually just misanthropic that they ought to be, er, sent to a re-education camp.)
So, is this what you came up with? Did your rationality win out in the face of fallacious authority?
(Also, do you agree that I’ve represented the ‘rational approach’ to this situation correctly? Give me feedback!)
Postscript: the explanation of the proof
There’s no shortage of explanations of this online, and a mountain of them emerged after this video became popular. I’ll write out a simple version anyway for the curious.
It turns out that there is a sense in which those summations are valid, but it’s not the sense you’re using when you perform ordinary addition. It’s also true that the summations emerge in physics. It is also true that the validity of these summations is in spite of the rules of “you can’t add, subtract, or otherwise deal with infinities, and yes all these sums diverge” that you learn in introductory calculus; it turns out that those rules are also elementary and there are ways around them but you have to be very rigorous to get them right.
An elementary explanation of what happened in the proof is that, in all three infinite sum cases, it is possible to interpret the infinite sum as a more accurate form (but STILL not precise enough to use for regular arithmetic, because infinities are very much not valid, still, we’re serious):
S(infinity) = 1+2+3+4+5… ≈ −1/12 + O(infinity)
Where S(n) is a function giving the n’th partial sum of the series, and S(infinity) is an analytic continuation (basically, theoretical extension) of the function to infinity. (The O(infinity) part means “something on the order of infinity”)
Point is, that O(infinity) bit hangs around, but doesn’t really disrupt math on the finite part, which is why algebraic manipulations still seem to work. (Another cute fact: the curve that fits the partial sum function also non-coincidentally takes the value −1/12 at n=0.)
And it’s true that this series always associates with the finite part −1/12; even though there are some manipulations that can get you to other values, there’s a list of ‘valid’ manipulations that constrains it. (Well, there are other kinds of summations that I don’t remember that might get different results. But this value is not accidentally associated with this summation.)
And the fact that the series emerges in physics is complicated but amounts to the fact that, in the particular way we’ve glued math onto physical reality, we’ve constructed a framework that also doesn’t care about the infinity term (it’s rejected as “nonphysical”!), and so we get the right answer despite dubious math. But physicists are fine with that, because it seems to be working and they don’t know a better way to do it yet.
I have a different approach when I see a suspicious-looking claim from an authoritative-seeming source. The first thing I do is rephrase the claim so that instead of a yes/no question it’s an open-ended question. (In this case: “What is the sum of 1+2+3+4+...”) Then I go to Google (or PubMed or wherever) and try to it, without using any references or other clues from the original source. If the first answer I find matches, this is evidence that the original source is trustworthy; if it doesn’t match, and the claim was presented as though it were uncontroversial, then I give the original source a big credibility hit.
The idea is to escape the original source’s framing, because if it isn’t trustworthy, then any thinking you do on its terms will also be suspect. I find this works much better than trying to engage with suspicious claims on their author’s terms.
That’s a good approach for things where there’s a ‘real answer’ out there somewhere. I think it’s often the case that there’s no good answer. There might be a group of people saying they found a solution, and since there no other solutions they think you should fully buy into theirs and accept whatever nonsensities come packaged with it (for instance, consider how you’d approach the 1+2+3+4+5..=-1/12 proof if you were doing math before calculus existed). I think it’s very important to reject seemingly good answers on their own merits even if there isn’t a better answer around. indeed, this is one of the processes that can lead to finding a better answer.
I’ll try to come back and engage more substantively with the material later when I’m not actually supposed to be working, but for now just wanted to say bravo—this is exactly the kind of thing I was hoping to see when you mentioned making math posts. I’d take posts like this every day if I could get ’em.
Thanks! Validation really, really helps with making more. I hope to, though I’m not sure I can churn them out that quickly since I have to wait for an idea to come along.
Surely as soon as you see the formula
you know that you are dealing with some notion of addition that has been extended from the usual rules of addition. So I don’t think it’s meaningful to just ask whether or not 1 + 2 + 3 + … = −1/12. The only sensible question to ask is “Is there some sensible extension of the rules of arithmetic such that 1 + 2 + 3 + … = −1/12 under these new rules?”, and I think the video is enough evidence to see that the answer to that question is “Yes” even though the video itself doesn’t make clear how or why arithmetic is being extended.
Edit: TL;DR: Mathematics is largely Ra worship, perhaps worse than even the more abstract social sciences. This means that That Magic Click never happens for most people. It’s a prime example of “most people do not expect to understand things”, to the point where even math teachers don’t expect to understand math, and they pass that on to their students in a vicious cycle.
Only if you know that it’s possible to have multiple rules of addition. That’s an unknown unknown for almost everyone on the planet. Most people aren’t even familiar with the concept of unknown unknowns, and so are hopelessly far away from this in idea space. For them, they are more likely to just reject logic and math entirely as obviously wrong.
That requires being aware of the fact that addition can be constructed in multiple ways, which is very much NOT something you learn in school. They basically just present you with a series of weird looking “facts”, and give a handwaving explanation. I suspect the vast majority of people, maybe even a narrow majority of LessWrongers, wouldn’t even know that disagreeing with mathematics is something you’re allowed to do. (“It’s math, it’s totally unambiguous, you can’t just disagree about the results.”) I suspect that’s why this post has as many upvotes as it does, even if most of us are dimply aware of such things.
Let me try and explain where I’m coming from with this. I don’t know about the rest of you, but I always went through the exact same procedure after learning each new layer of mathematics. It goes something like this:
So, upon being told that
A^2 + B^2 = C^2, or that1+2+3+4+5+… = -1/12, my initial reaction is the usual disbelief, but with the expectation that after an hour or two of toying with numbers and banging my head against the wall trying to make sense of it, I’ll invariably just give up and accept it as just one more impenetrable brute fact. After all, I’ve tried to punch holes in things like this ten thousand times before and never had any success. So, the odds of making any sense of it this time can’t be more than 0.01% at most, especially with something so far above my head.How can someone even do math without understanding what math is? Well, I can only offer my own anecdata:
I was always good at math through highschool, but I suspect I spent twice as much time as everyone else doing the homework. (When I did it. I didn’t bother if I could get A’s despite getting 0′s on my homework.) Most of this time was spent trying to decipher how what we were doing could possibly work, or solving the problems in alternate ways that made more sense to me.
When I hit Calculus in college, I promptly failed out because I didn’t have enough time to do the homework or complete the tests my way. (I rarely just memorized formulas, but instead beet my head against the wall toying with them until I more or less knew the algorithm to follow, even if I didn’t understand it. I didn’t know about spaced repetition yet, so I was unable to memorize enough of the formulas to pass the tests, and didn’t have time to derive them.)
I concluded that I was just bad at math, especially since I could never follow anything being written on the board, because I would get stuck trying to make sense of the first couple lines of any proof. I considered my mathematical curiosity a useless compulsion, and assumed my brain just didn’t work in a way that let me understand math. In retrospect, I don’t think anyone else in any of the classes actually understood either, but were just blindly following the algorithms they had memorized.
Personally, I have acquired 3 clues that math isn’t just a series of random brute facts:
Philosophy of Mathematics has a divide between Mathematical Platonism and Empiricism. I was really confused to hear a calculus professor make an offhand empiricist remark, because I wasn’t aware that there was an alternative to Platonism. I had always just assumed that math was a series of platonic ideal forms, suspended in the void, and then physics was just built up from these brute facts. The idea of math as a social construct designed to fit and understand reality was bizarre. It wasn’t until I read Eliezer’s The Simple Truth and How to Convince Me That 2+2=3 that it really clicked.
I stumbles upon A Mathematician’s Lament, and gained a bunch of specific insight into how new mathematical ideas are created. It’s difficult to sum up in just a few words, but Lockhart argues that how we teach mathematics would be like teaching music by having kids memorize and follow a vastly complex set of musical rules and notations, and never let them touch an instrument or hear a note until graduate school. After all, without the proper training, they might do it wrong. He argues that mathematics should be a fundamentally creative process. It is just a bunch of rules made up by curious people wondering what would happen to things if they applied those rules. Previously, whenever I saw a new proof, I’d spend hours trying to figure out why they had chosen those particular axioms, and how they knew to apply them like that. I could never understand, and figured it was way beyond my grade. Lockhart provides a simple explanation, which has since saved me many hours of handwringing: They were just playing around, and noticed something weird or cool or interesting or potentially useful. They then played around with things, experimenting with different options to see what would happen, and then eventually worked their way toward a proof. Their original thought process was nothing like the mysterious series of steps we memorize from the textbook to pass the test. It was exactly the sorts of things I was doing when I was toying with numbers and formulas, trying to make sense of them.
I recently taught myself some lambda calculus. (“Calculus” here doesn’t mean integration and differentiation, but only the simplest forms of operations. In fact, the basics are so simple that someone made a children’s game called Alligator Eggs out of the rules of lambda calc.) It’s basically just a simple set of rules, that you can string together and use to build up some interesting properties, including AND, OR, IF, IFF operators, integers, and addition/subtraction.
Let me tie it all back together. Apparently there are multiple ways of building up to operators like this, and lambda calc is just 1 of several possibilities. (And, I would have been mystified as to why the rules of lambda calc were chose if it weren’t for reading The Mathematician’s Lament first.) Under the mathematical empiricist view, by extension, it’s not just how we build up to such operators that’s arbitrary. It’s ALL OF MATHEMATICS that’s arbitrary. We just focus on useful operators instead of useless ones that don’t fit reality. Or not, if we find other things interesting. No one expected non-Euclidian geometry to be useful, but as it turns out spacetime can warp, so it drifted into the domain of applied mathematics. But it started as someone toying around just for lolz.
Yeah I definitely agree with all of this. It’s just that the original post was phrasing it as “Someone has claimed that 1+2+3+...=-1/12, do you believe them or not?” and it struck me that it doesn’t mean anything to believe it or not unless you first understand what it would even mean for 1+2+3+… to equal −1/12. In order to understand this you first have to be aware that the notion of addition can be extended. If you aren’t aware of this (as you point out most people aren’t) the original post is even less useful; it’s asking a question that you can’t possibly answer.
In the “proof” presented, the series 1-1+1… is “shown” to equal to 1⁄2 by a particular choice of interleaving of the values in the series. But with other methods of interleaving, the sum can be made to “equal” 0, 1 1⁄3 or indeed AFAICT any rational number between 0 and 1.
So… why is the particular interleaving that gives 1⁄2 as the answer “correct”?
Interleaving isn’t really the right way of getting consistent results for summations. Formal methods like Cesaro Summation are the better way of doing things, and give the result 1⁄2 for that series. There’s a pretty good overview on this wiki article about summing 1-2+3-4.. .
Good shit.
From Surely You’re Joking Mr. Feynman:
When this sum hit the mainstream interwebz a while back, we had some discussion about it in the physics department where I work. The consensus was that it was misrepresented as a spooky non-intuitive fact about adding numbers, when really it’s closer to a particular notation for assigning a finite value to a diverging sum that happens to be useful in physics*. Some of us were annoyed, because it feels like it’s reinforcing this idea that math is impossibly opaque, a notion that we have to deal with on a regular basis when trying to teach physics to undergraduates.
Also, FWIW, I don’t recall seeing this presented in my QFT class, but then again, I only took one semester.
*I think you’re actually characterizing a it a little differently and a little more precisely, as a way of actually evaluating the sum, while subtracting off a term of order infinity, in a way that allows for certain kinds of manipulations that happen to be useful in physics.
Yeah, that’s the exact same conclusion I’m pushing here. That and “you should feel equipped to come to this conclusion even if you’re not an expert.” I know.. several people, and have seen more online (including in this comment section) who seem okay with “yeah, it’s negative one twelfth, isn’t that crazy?” and I think that’s really not ok.
My friend who’s in a physics grad program promised me that it does eventually show up in QFT, and apparently also in nonlinear dynamics. Good enough for me, for now.
I think that in the interests of being fair to the creators of the video, you should link to http://www.nottingham.ac.uk/~ppzap4/response.html, the explanation written by (at least one of) the creators of the video, which addresses some of the complaints.
In particular, let me quote the final paragraph:
In light of this paragraph, I think a cynical answer to the litmus test is this. Faced with such a ridiculous claim, it’s wrong to engage with it only on the subject level, where your options are “Yes, I will accept this mathematical fact, even though I don’t understand it” or “No, I will not accept this fact, because it flies in the face of everything I know.” Instead, you have to at least consider the goals of the person making the claim. Why are they saying something that seems obviously false? What reaction are they hoping to get?
What a wonderful post, like that classic days of LW. I hope to see more posts like this again in the future.
Personally, I encountered this in the wild. My brother asked me “do you know what the series 1 + 2 + 3 + 4 + 5 + and so on sums up to?” “Well,” I said, “That sums up to infinity.”
“No, it’s −1/12!”, he exclaims. I exclaimed that this was bullshit—there are only positive numbers in the series, there are only additions in the series, and since adding positive numbers together produces a positive number, the “negative a twelfth” result is just plain wrong.
We had a bit of an argument after that, after which he said “yes, you’re right, but when you sum them up ALL AT ONCE then you get negative a twelfth”. And I accepted that, because, well, summing them up all at once is a different operation than repeated addition, so then you might get some different result. I didn’t study complicated maths, I don’t know what happens when you wrap things with complicated functions, and I’m willing to concede that there exists some fancy way of wrapping repeated addition in a mathematical construct so that you can get a negative result.
So this post comes along, and yes. You take a fancy mathematical function and apply it over the repeated addition, just as expected.
Infinities are an interesting case for rationality. On one side, they are totally made up: we do not have any example of an infinite quantity in our experience. On the other side, they seem to have a quality of coherence and persistence that is independent from our mind, in contrast with other kinds of fiction. They are complex and unintuitive, and it’s a problem that shines a light on the fact that even amongst the experts there are different degrees of expertise.
A set theorist migh reply that due to the Transfinite Recursion Theorem, that summation really sums to omega, not any finite value, and that the only way to create a coherent infinite function is to specify a limit step: but someone who is not versed in mathematics will believe what a nice and lively professor seen on Youtube will say. It is unfortunate that seen from ‘below’, all experts seem alike, and there are few ways to discern someone who is treading out of their area of expertise.
I actually know a bit about summing series, so I recognize the proof as completely bogus but the actual sum as probably correct, for a certain sense of “sum”. You can make a divergent series add up to anything at all by grouping and rearranging terms. On the other hand, there actually are techniques for finding the sum of a convergent series that sometimes don’t give nonsensical answers when you try to use them to find the “sum” of a divergent series, and in this sense the sum of 1 + 2 + 3 + etc. actually can be said to equal −1/12.
I know about Cesaro and Abel summation and vaguely understand analytic continuation and regularization techniques for deriving results from divergent series. And.. I strongly disagree with that last sentence. As, well, explained with this post, I think statements like “1+2+3+...=-1/12” are criminally deceptive.
Valid statements that eliminate the confusion are things like “1+2+3...=-1/12+O(infinity)”, or “analytic_continuation(1+2+3+)=-1/12“, or “1#2#3=-1/12”, where # is a different operation that implies “addition with analytic continuation”, or “1+2+3 # −1/12”, where # is like = but implies analytic continuation. Or, for other series, “1-2+3-4… #1/4” where # means “equality with Abel summation”.
The massive abuse of notation in “1+2+3..=-1/12” combined with mathematicians telling the public “oh yeah isn’t that crazy but it’s totally true” basically amounts to gaslighting everyone about what arithmetic does and should be strongly discouraged.
I don’t like infinity at all. But if one permits it, this is quite an understandable conclusion.
I have a (bit ironic) view on it:
https://protokol2020.wordpress.com/2016/08/18/why-is-1248-1/
Infinite sums/sequences are a particular area of me. I would love to know about how these sums appear in string theory—what’s the best introduction/way into this? You said these sums appear all over physics. Where do they appear?
Well, Numberphile says they appear all over physics. That’s not actually true. They appear in like two places in physics, both deep inside QFT, mentioned here.
QFT uses a concept called renormalization to drop infinities all over the place, but it’s quite sketchy and will probably not appear in whatever final form physics takes when humanity figures it all out. It’s advanced stuff and not, imo, worth trying to understand as a layperson (unless you already know quantum mechanics in which case knock yourself out).
All I ever covered in university was taking the Scrodinger equation and then quantum physics did whatever that equation said.