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 12, 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, so 2P = 1, so P = 12
Now con­sider Q = 1-2+3-4+5…
And write out Q+Q this way:
+ 1 −2+3-4…
= 1-1+1-1+1 = 12 = 2Q, so Q = 14
Now con­sider S = 1+2+3+4+5...
Write S-4S as
4 −8 …
=1-2+3-4+5… = Q=1/​4
So S-4S=-3S = 14, 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.