This is a fun one, because of course the kneejerk rejection of the statement "1+2+3+4+5+... = -1/12" is correct

Under everyday, ordinary grade-school arithmetic, the answer can't be "-1/12" -- but the answer can't be any other number either, the operation itself is not possible https://twitter.com/tomgabion/status/1289857027381002241

You cannot do anything an infinite number of times

The answer to "1+2+3+4+5+... = x" is that x can't be anything because you will never actually finish adding up numbers and you will never get an answer

"Infinite" is another way of saying "nonexistent"

You can't say x="infinity" or ∞ because in ordinary arithmetic that's not a number, it's a meaningless word

The *rules of arithmetic* say this -- numbers are *defined* by the fact that if you add 1 or subtract 1 from a number, you get a different number, if "∞" breaks this rule, it's not a number at all and you can't use it

So people's intuitions about this are correct but they don't take it far enough

Caught between the dueling intuitions between "Well the answer to this question can't be any ordinary number" and "It must HAVE an answer though"

All new forms of math are built on that second part

"Okay, I get that this is a stupid-ass question and Pythagoras or whoever would've just said 'shut the fuck up' if I asked him but WHAT IF you COULD add up numbers an infinite number of times"

All summations of infinite series are based on making up new rules and saying "Okay let's pretend you can do this, what happens if you do, what new stuff do you discover if we just fuck around and act like this makes sense"

So like, let's be clear

This classic series:

1+1/2+1/4+1/8+1/16...

Is, from a pure old-school POV, just as bad as the other one

Even though this one looks like it has an answer (it adds up to 2 in the end)

*You can't do things infinite times*

If we imagine the addition as a real-world activity that takes time to do -- a second to write the new sum on a piece of paper, a fraction of a microsecond for a processor to encode it in memory -- then the Sun will go out before you get 2

Ancient Greek philosophers knew about this shit

This is, famously, Zeno's paradox

Getting around this and saying "Well let's just pretend you CAN do it infinite times" is not *answering* Zeno, it's just telling him to shut the fuck up

Specifically, saying Zeno was wrong, and I *can* just wave a magic wand and say "skip to the end of something I just said was by definition endless", is formally inventing the idea of a "limit"

Which has made many people very angry and been widely regarded as a bad move

The fact that being able to say "1+1/2+1/4+1/8...=2" is very useful, and all of calculus is based on it, doesn't actually mean we were *right*

Whether "infinite converging series" are, like, a real thing that exists in the world of atoms is this big heavy question (probably not)

There's a famously sexist quote apocryphally attributed to Shaw that I'll paraphrase as "If you wouldn't sell out for $10, but you will sell out for $10 million, then you are a sellout and you're just haggling over the price"

When you agreed you could just shrug off the paradoxical-ness of doing anything an infinite number of times because pretending like you can is useful, you left the original rules of arithmetic behind

Now we're just arguing over how weird we want to get

This is, in fact, a *whole field of mathematics*, and the summation of infinite series can be done according to any number of different methods, which mathematicians invent at their pleasure, designed according to different criteria

(I'm verging into stuff I only half-remember here, full disclosure)

1+1/2+1/4+1/8+... is one of the easiest series to just up and say has a solution, that it's 2

Because you're only adding, not subtracting, so you can move around all those numbers at will

This is called "absolute convergence"

If the numbers change sign, you have "conditional convergence"

Paradoxically, for people used to doing things a finite number of times, it suddenly matters how the numbers are arranged

1-2+3-4, if I stop at 4, is always going to give the same result (-2) even if it's -2+3+1-4 or -4-2+1+3 (commutative property of addition)

This doesn't work with infinite sums (which doing things an infinite number of times is a filthy lie)

The classic example is the alternating harmonic series, 1-1/2+1/3-1/4+1/5..., which converges on the number ln(2)

If you rearrange the numbers differently, so it's positive+negative+negative and not positive+negative+positive, 1-1/2-1/4+1/3-1/6-1/8+1/5-1/10-1/12... it's ln(2)/2

Without writing out the proof, you see what kind of dirty trick I'm pulling here, right?

For each positive fraction with an odd denominator, I'm "borrowing" an "extra" negative fraction with an even denominator from "the future"

(Dug up a more in-depth discussion here http://larryriddle.agnesscott.org/series/rearrang.pdf)

If this were a finite set of numbers, then I would eventually run out of "future numbers" to borrow from, and this trick to make the sum constantly be lower wouldn't work, and the sums would come out the same

But it's an *infinite* series, so I *never* run out

And so the same set of numbers rearranged converges to 1/2 the original sum

This is why doing things an infinite number of times is, again, a filthy lie

(Riemann proved you can rearrange this series to converge on ANY sum, at all, or to not converge and instead spit out ∞ or -∞

Which is one of those things mathematicians do that really pisses people off)

Anyway

There are series that are absolutely convergent, and series that are conditionally convergent, and series that are divergent

The series 1-1+1-1+1... doesn't converge, it doesn't "get closer" to anything the longer you do it, it just flips from 1 to 0

So does 1-1+1-1+1... equal anything at all?

According to the method people used when they started talking about this ("classical summation"), no, absolutely not

There is no way to rearrange this series so it converges on anything, it's a divergent series

A Norwegian guy named Abel (a deeply weird, extremely smart dude who invented all kinds of new math before he died of TB at the age of 26) was like "Sure you can, it's 1/2"

You probably looked at that and said "Yeah it's 1/2", if you thought about it at all

The thing is in math you are *completely allowed* to say "Okay the rules say you can't do it but it looks like it should be 1/2 so I'm gonna say it is"

You just have to go on to explain to everybody what that *implies* if you decide to do it, which is the hard part

I.e. by the delta-epsilon definition of a limit, which is "It has to get closer to the number every time you do the next thing", 1-1+1-1 by definition is not approaching any limit

You do it once, it's 1, you do it again, it's 0, then 1, then 0

The "distance" between each new sum and the answer 1/2 is exactly the same every time, you *never* get closer

A so-called Abelian summation method involves using the idea of *averaging* instead of the idea of *limits*

Does that make sense? Should that be allowed? The authorities haven't come to a moral conclusion on the matter, but hey it's fun and cool prizes come out when you do

(Philosophically, taking an average of an infinite number of sums is worse than adding numbers together an infinite number of times, because it means you have to "divide by infinity", and that's a big can of worms

Which Abel happily opened)

It goes on from there

Every new bleeding-edge summation method has more scary problems with it -- or, rather, it lacks certain properties that were assumed under classical summation

Abelian summation does not "work" in many ways, but we accept that

It "works" in other ways and that's okay

Cesaro summation and Abelian summation are two methods that give you 1/2 for a "mildly divergent" series like Grandi's series (1-1+1-1)

Abelian summation is slightly more "powerful" and can also handle weirder cases like 1-2+3-4+5, which according to him = 1/4

If that bothers you -- "The number keeps *increasing* in absolute value, how can I go up to 2 then down to -2 then up to 3 then down to -3 and then at the very end of it get 1/4?!" -- well, you're in good company, Abel summation is said to have brought mathematicians to blows

You see at every step we're leaving a little more of "common sense" math as a child would learn in grade school behind, we're redefining the meaning of the symbols "+", "..." and especially "=" a little more to say something else

And that's fine

And that's how we get to the real weirdness, where "1+2+3+4...=-1/12"

This is a sum that does not "work" at all with Abel's summation method, it contains no "oscillation" (the switching from + to -) that was the secret sauce to make the 1-1+1 and the 1-2+3 stuff work

*Any* answer that's actually a number is going to be wildly counterintuitive

But asking "Okay, but what happens if you try it" is how math works

The dude who tried it was Ramanujan, a real weirdo whose life story is tremendously inspiring, and who was kind of an awesome troll

The method Ramanujan used to prove "1+2+3+4...=-1/12" is summarized in this famous YouTube people, which made a lot of people very angry and was widely considered a bad move

It's important to note that this method, stated in the form the guy gives in this video, is wrong

He's doing multiple things you're just not allowed to do

Ramanujan liked that kind of shit

I.e. if you naively just assume everything he's doing -- adding infinite series to each other, multiplying a finite term across an infinite series, etc. -- is allowed in every circumstance, you get contradictions

Trying to build a summation method this way for a diverging series 1+2+3+4... does not work, the way 1+2+3+4... diverges makes it neither "linear" nor "stable"

I.e. if I stick an extra zero in there ("0+1+2+3+4..."), I can use his method to prove 1=0

And 1=0 is generally frowned upon, people don't like it, it's dogs and cats living together

As a great philosopher once said "If SOMETHING is the SAME THING as NOTHING then YOU COULD HAVE ANYTHING

You can't just have anything, you've got to keep the riffraff out"

And yet it's not *arbitrary*

Ramanujan's "dirty trick" isn't just bullshit, I can't actually use it to prove anything I want

If you do the specific thing he's doing the way he's doing it, consistently, every time, you get consistent answers

Whatever the hell he's doing, even if it isn't "adding up numbers" in any recognizable way as we normally do it, it is a *real thing* and it always gives you the answer "-1/12"

Ramanujan famously told Hardy in his letter about this "Please read to the end of the page before sending me to the insane asylum", and Hardy let himself get infected with the brainworms before calling the authorities in time to stop it, and now here we are today

Formalizing the method Ramanujan used here -- explicitly stating what rules actually define the things Ramanujan was doing so you can't just do anything and get 1=0 -- created the method known as "Ramanujan summation"

Which ended up being a special case of the regularization of the Riemann zeta function, a form of analytic continuation (which is a phrase for "doing things you're not allowed to do")

Which the guys who made the above video go into detail about here

I fully admit I do not really understand what they are talking about, and that my brain plasticity at my age is already probably too low to ever actually feel like I have the energy to really get into it

But it's a real thing, and it works

It doesn't map onto the real world in a familiar way very well, at all

But it wasn't meant to

Counting things up infinitely isn't something you can do in the real world AT ALL either, remember?

And yet, bizarrely, this result is meaningful in an applied context

It comes up in quantum physics, where the real physical world gets extremely weird and we need to use weird math to talk about it

It's part of the mathematical description of the Casimir effect, which has been experimentally verified

https://en.wikipedia.org/wiki/Casimir_effect

The extremely broad metaphor the guy gives us for trying to imagine this is -- the actual positive numbers that we add up in 1+2+3+4... can't give us a real sum

The number keeps getting bigger, it leads us only toward ∞, and ∞ isn't a number, so the sum just doesn't exist

Analytic continuation is about saying "Okay, so if you take out everything that you're not allowed to do that absolutely by the rules cannot give you an answer, can you find something about it that *does* give you an answer"

Paradoxical Zen koan shit

Like Death in Discworld saying that the universe by definition is everything that exists and everywhere is inside it so you can't be outside it and standing outside it is a contradiction in terms

But if you could, then from the outside it'd be blue

-1/12 is a "residue", it's a "leftover piece" of 1+2+3+4..., the one bit that can be made to act like a number

And that, in its way, is connected to the paradoxical thing the Casimir effect is, this thing where in a total vacuum where nothing should be exerting any force on anything, two metal plates very close together will attract each other

I.e. -- getting really sloppy and vague from my layman's POV again, in real life in quantum physics a "vacuum" is actually filled with "virtual particles", which kinda almost do exist but then don't

The vacuum energy is adding up all the particles that *could* be interacting with those metal plates, an *infinite number* of ways those plates could be pushing against each other and exerting force via an electromagnetic field

But infinite force can't exist, doesn't mean anything, and what this actually adds up to instead is the "residue", the -1/12, i.e. a very small negative number -- a force that goes the opposite direction and attracts things together, but only at a very small distance

Yes, actual physicists, I don't know what I'm talking about

The point here is just me dreaming of one of the many things in heaven and earth I hadn't previously had in my philosophy

The knowledge that a summation of 1+2+3+4...=-1/12 exists doesn't affect my life -- if I am ever asked to add up all of the natural numbers I will probably stop at 1+2+3=6, honestly -- nor is the proof that they gave the real, final proof

But damn if it isn't a beautiful, powerful demonstration that you can fuck around with seemingly simple things and open a Pandora's Box of wonders

Like the motto of Stephen Fry's show QI, "If you can't be right, be interesting"

At every step along this process someone could have, like Dostoyevsky's coxcomb, blocked the path and waved their arms and said "No you can't do that turn back"

(Hell, the ancient Greeks, who believed you could not in fact do that, were still pretty lax traffic cops and didn't *enforce* not doing that because they didn't have the rigor, which is why Archimedes was allowed to discover a limit-based method for approximating pi)

But it's always more interesting to just ignore him and keep walking anyway

I mean what's the worst that can happen? You try to say something makes sense but everyone just looks at it a while and goes "Nah this is stupid" and moves on?

You gotta at least try