# 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+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+5…
+ 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
1+2+3+4+5…
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.

• 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.

• In the “proof” pre­sented, the se­ries 1-1+1… is “shown” to equal to 12 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 13 or in­deed AFAICT any ra­tio­nal num­ber be­tween 0 and 1.

So… why is the par­tic­u­lar in­ter­leav­ing that gives 12 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 12 for that se­ries. There’s a pretty good overview on this wiki ar­ti­cle about sum­ming 1-2+3-4.. .

• Good shit.

• Surely as soon as you see the formula

1 + 2 + 3 + … = −1/​12

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.

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.

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:

Phase 1: Wait, 1234x5678 can be solved by mul­ti­ply­ing 4x8, then 4x70, then 4x600, ect., then adding it all up??!! What are the chances of that al­gorithm in par­tic­u­lar work­ing? Of all the pos­si­ble pro­ce­dures, why not liter­ally any­thing else?

Phase 2: Ok, I’ve done some sim­ple ex­am­ples, and it seems to pro­duce the cor­rect re­sult. I guess I’ll just have to grudg­ingly ac­cept this as a brute fact about re­al­ity. It’s an ir­re­ducible law that some an­cient math­e­mat­i­cian stum­bles upon by ac­ci­dent, and then maybe did some com­plex an im­pen­e­tra­ble sor­cery to ver­ify. Maybe some­day I’ll get a PhD in math­e­mat­ics, and maybe then I’ll un­der­stand what’s go­ing on here. Or maybe noone re­ally un­der­stands it, and they just use a brute force solu­tion. They just try ev­ery pos­si­ble al­go­ry­thm, in or­der of in­creas­ing kol­mogrov com­plex­ity un­til one works. Pythago­ras tried `A+B+C=0`, `A+B=C`, etc un­til find­ing that `A^2 + B^2 = C^2`. Progress in math­e­mat­ics is just an au­to­mated, me­chan­i­cal pro­cess, like su­per­com­put­ers do­ing things en­tirely at ran­dom, and then spit­ting out things that work. No one re­ally un­der­stands the pro­cess, but just blindly ap­ply­ing it seems to pro­duce more use­ful math the­o­rems, so they keep blindly turn­ing the crank.

So, upon be­ing told that `A^2 + B^2 = C^2`, or that `1+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:

1. 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.

2. 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.

3. 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.

• From Surely You’re Jok­ing Mr. Feyn­man:

Topol­ogy was not at all ob­vi­ous to the math­e­mat­i­ci­ans. There were all kinds of weird pos­si­bil­ities that were “coun­ter­in­tu­itive.” Then I got an idea. I challenged them: “I bet there isn’t a sin­gle the­o­rem that you can tell me—what the as­sump­tions are and what the the­o­rem is in terms I can un­der­stand—where I can’t tell you right away whether it’s true or false.”

It of­ten went like this: They would ex­plain to me, “You’ve got an or­ange, OK? Now you cut the or­ange into a finite num­ber of pieces, put it back to­gether, and it’s as big as the sun. True or false?”

“No holes.”

“Im­pos­si­ble!

“Ha! Every­body gather around! It’s So-and-so’s the­o­rem of im­mea­surable mea­sure!”

Just when they think they’ve got me, I re­mind them, “But you said an or­ange! You can’t cut the or­ange peel any thin­ner than the atoms.”

“But we have the con­di­tion of con­ti­nu­ity: We can keep on cut­ting!”

“No, you said an or­ange, so I as­sumed that you meant a real or­ange.”

So I always won. If I guessed it right, great. If I guessed it wrong, there was always some­thing I could find in their sim­plifi­ca­tion that they left out.

Ac­tu­ally, there was a cer­tain amount of gen­uine qual­ity to my guesses. I had a scheme, which I still use to­day when some­body is ex­plain­ing some­thing that I’m try­ing to un­der­stand: I keep mak­ing up ex­am­ples. For in­stance, the math­e­mat­i­ci­ans would come in with a ter­rific the­o­rem, and they’re all ex­cited. As they’re tel­ling me the con­di­tions of the the­o­rem, I con­struct some­thing which fits all the con­di­tions. You know, you have a set (one ball)—dis­joint (two halls). Then the balls turn col­ors, grow hairs, or what­ever, in my head as they put more con­di­tions on. Fi­nally they state the the­o­rem, which is some dumb thing about the ball which isn’t true for my hairy green ball thing, so I say, “False!”

If it’s true, they get all ex­cited, and I let them go on for a while. Then I point out my coun­terex­am­ple.

“Oh. We for­got to tell you that it’s Class 2 Haus­dorff ho­mo­mor­phic.”

“Well, then,” I say, “It’s triv­ial! It’s triv­ial!” By that time I know which way it goes, even though I don’t know what Haus­dorff ho­mo­mor­phic means.

I guessed right most of the time be­cause al­though the math­e­mat­i­ci­ans thought their topol­ogy the­o­rems were coun­ter­in­tu­itive, they weren’t re­ally as difficult as they looked. You can get used to the funny prop­er­ties of this ul­tra-fine cut­ting busi­ness and do a pretty good job of guess­ing how it will come out.

• 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:

There is an en­dur­ing de­bate about how far we should de­vi­ate from the rigor­ous aca­demic ap­proach in or­der to en­gage the wider pub­lic. From what I can tell, our video has en­gaged huge num­bers of peo­ple, with and with­out math­e­mat­i­cal back­grounds, and got them de­bat­ing di­ver­gent sums in in­ter­net fo­rums and in the office. That can­not be a bad thing and I’m sure the sim­plic­ity of the pre­sen­ta­tion con­tributed enor­mously to that. In fact, if I may re­turn to the origi­nal ques­tion, “what do we get if we sum the nat­u­ral num­bers?”, I think an­other an­swer might be the fol­low­ing: we get peo­ple talk­ing about Math­e­mat­ics.

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.