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"An infinite number of mathematicians walk into a bar..."
January 9, 2014 10:55 PM   Subscribe

1 + 2 + 3 + 4 + 5 ... = -1/12 -- Numberphile explains a counter-intuitive summation of an infinite series.

There's a more mathematically rigorous proof here.
posted by empath (136 comments total) 49 users marked this as a favorite

 
If you want to understand how this can be, you have to consider that there's no obvious way to sum an infinite series using traditional arithmetic, so a method has to be defined. For a convergent series, you use limits, but the sum of natural numbers doesn't converge using the traditional definition.

So one has to come up with new methods. One way to solve this is using Ramanujan summation. Another method is using the Riemann Zeta function, which has obvious and intuitive solutions for certain inputs, and really non-obvious and counter-intuitive, but internally consistent solutions for other inputs, one of which happens to be 1+2+3+4+5...

The two methods both agree that the answer is -1/12.
posted by empath at 11:02 PM on January 9 [10 favorites]


" If I tell you this you will at once point out to me the lunatic asylum as my goal." --Ramanujan, if the Wikipedia page is to be trusted.
posted by Earthtopus at 11:08 PM on January 9


If you enjoyed that you may also enjoy this previous post.
posted by benito.strauss at 11:23 PM on January 9


Yeah, this builds off of that previous video.
posted by empath at 11:25 PM on January 9


Yeah, Infinity is not the biggest number you can think of plus one. The trickery here is that an alternating series doesn't converge. Limits are useful because they are intuitive if the series converges.

There are also different magnitudes of inifinity. Some are more infinite than others.

Math gets weird after inductive proofs.
posted by lkc at 11:31 PM on January 9


I'm having a very hard time with this.
posted by dazed_one at 11:42 PM on January 9 [9 favorites]


Number of times I shouted GET ON WITH IT as he went on about how ASTOUNDING the thing they were about to tell me was and then told me about things I already knew or could easily figure out: 3.

His smirking presentation makes me distrust his proof more than if he just presented it. Instead, I feel more like Lou Costello just tried to prove to me that 7 X 13 = 28.
posted by JHarris at 11:50 PM on January 9 [5 favorites]


Number of times I shouted GET ON WITH IT as he went on about how ASTOUNDING the thing they were about to tell me was and then told me about things I already knew or could easily figure out: 3.

You kind of have to start with obvious things in order to go on and prove non-obvious things.
posted by empath at 11:58 PM on January 9 [1 favorite]


I loved this, but I’m confused about the right-shift in the first addition step. Why is that okay?
posted by migurski at 11:59 PM on January 9 [8 favorites]


Infinite number of mathematicians walk into a bar...
Bartender says "why the infinite number of long faces?"
Mathematicians say "Barkeep, you got it wrong. That's an infinite number of horses joke!"
Mick Jagger walks in and sings "an infinite number of wild horses couldn't drag me away..."
Keith Richards outlives them all.
The end.
posted by flapjax at midnite at 11:59 PM on January 9 [2 favorites]


Ummm... Burn the witch?
posted by rmxwl at 12:06 AM on January 10 [1 favorite]


Wait so is -1 + 1 -1 + 1 ... also 1/2 (since it's just +1 -1 +1 -1 ... with every other term swapped) or is it -1/2, since it's the negation of +1 -1 +1 -1? I'm asking since we're supposedly subtracting these monsters from each other in his proof.

Basically I feel like none of this makes any sense and he's trying to prove it like it does, but there's lots of other ways that you could go that would make just as much sense but give different answers that I don't find this proof to be at all convincing. I'm much more willing to just accept on blind faith that freaking Ramanujan is right than I am to accept that this bunch of mathematical jiggery-pokery is valid.
posted by aubilenon at 12:10 AM on January 10 [2 favorites]


So on the one hand this proof requires that the sequence 1 + 2 + 3 + ... keeps going, because if you stop somewhere then the result of the equation is just going to be Some Big Number instead of -1/12.

But on the other hand, we know the proof works because it's being used in physics and that field deals with actual existing number sequences, not infinities.

... yeah, I don't get it.
posted by Caconym at 12:12 AM on January 10 [2 favorites]


basically, it's not a "proof"... there are just various summation algorithms that can give unintuitive results that happen to be useful in some cases.
posted by dilaudid at 12:25 AM on January 10 [7 favorites]


I see that empath. It's just, I dunno. This makes me want to interrogate the process more than accept it. The thing that annoys me is the sign shift. If you don't add the non-converging sequence into it, then the sequence continues up and up. That seems like a dodgy step.

Note that I said seems. It is true, I don't have a lot of experience with sequences, and my playing with infinity was largely confined to Calculus. I'm not claiming RARR IM RIGHT THEY WRONG, but more like, this looks more like a parlor trick if you don't have the appropriate maths background. I'm going to have to have a look into this.
posted by JHarris at 12:26 AM on January 10 [1 favorite]


It sounds like dilaudid might be onto it. Could you go into a bit more detail?
posted by JHarris at 12:27 AM on January 10


Basically I feel like none of this makes any sense and he's trying to prove it like it does, but there's lots of other ways that you could go that would make just as much sense but give different answers that I don't find this proof to be at all convincing.

Basically when mathematicians run into something that's undefined in whatever system they're using, they like to figure out a way to define it. The complex numbers were the result of trying to define a solution for the square root of -1, for example. It doesn't really matter if it 'makes sense' in some intuitive way, as long as it doesn't cause contradictions.

So you start with some input (the series of natural numbers) and run it through a particular formula (Ramanujan summation) and call that the 'sum of the series', and while it's not a 'sum' in the way it's traditionally thought of, it is an internally consistent property of the series. You can call it a sum, or a Ramanujan characteristic or whatever else you want to call it.
posted by empath at 12:30 AM on January 10 [5 favorites]


From a straightdope thread:
The key is the Riemann zeta function. If you take any s > 1, then the series Sum(1/n2,n=1..infinity) converges, and the value of that sum is defined as zeta(s). So zeta is a perfectly well-defined function for s > 1. But it turns out that there's a very powerful and somewhat surprising theorem that if you have a function which is defined on some region, there is at most one function which is smooth on the complex plane and which matches up with that function. This complex function is called the analytic continuation of the original function, and since folks like analytic functions, there's a tendancy to think of the analytic continuation as being the "same" as the original function. So, if we take the analytic continuation of the Riemann zeta function, we can evaluate it anywhere we like, not just for s real and greater than 1. Suppose, for instance, that we evaluate zeta(-1): It happens that that gives us 1/12. But if that's really the same zeta, then from the original definition, zeta(-1) is (or should be) also equal to 1 + 2 + 3 + 4 + ....
I want to think about this more but if I start I will never go to bed, soooo --
posted by en forme de poire at 12:44 AM on January 10 [4 favorites]


analytic continuation is (I think?!?) the same thing that gets you from factorial to the gamma function, btw.
posted by en forme de poire at 12:47 AM on January 10 [1 favorite]


so it basically looks like it's one principled way of extrapolating out from other infinite sums of similar form that actually converge. the spooky part to me is that you can get the same result with this stuff that looks deceptively like basic algebra. but knowing that the complex plane is involved in the background makes me feel a little better about seeing that damn minus sign and now I really have to go to bed
posted by en forme de poire at 12:51 AM on January 10 [1 favorite]


en forme de poire: "analytic continuation is (I think?!?) the same thing that gets you from factorial to the gamma function, btw."

And I was literally thinking this morning about how absurd the gamma function is: .5! is the sqrt(pi). How did that pi sneak in!?!
posted by pwnguin at 1:04 AM on January 10 [2 favorites]


it's probably the fucking complex plane again
posted by en forme de poire at 1:05 AM on January 10 [13 favorites]


that thing is the worst
posted by en forme de poire at 1:06 AM on January 10 [16 favorites]


"Good Morning," said Deep Thought at last.
"Er..good morning, O Deep Thought" said Loonquawl nervously, "do you have...er, that is..."
"An Answer for you?" interrupted Deep Thought majestically. "Yes, I have."
The two men shivered with expectancy. Their waiting had not been in vain.
"There really is one?" breathed Phouchg.
"There really is one," confirmed Deep Thought.
"To Everything? To the great Summation of One, plus Two, plus everything?"
"Yes."
Both of the men had been trained for this moment, their lives had been a preparation for it, they had been selected at birth as those who would witness the answer, but even so they found themselves gasping and squirming like excited children.
"And you're ready to give it to us?" urged Loonsuawl.
"I am."
"Now?"
"Now," said Deep Thought.
They both licked their dry lips.
"Though I don't think," added Deep Thought. "that you're going to like it."
"Doesn't matter!" said Phouchg. "We must know it! Now!"
"Now?" inquired Deep Thought.
"Yes! Now..."
"All right," said the computer, and settled into silence again. The two men fidgeted. The tension was unbearable.
"You're really not going to like it," observed Deep Thought.
"Tell us!"
"All right," said Deep Thought. "The Answer to the Great Summation..."
"Yes..!"
"Of One, plus Two, plus Everything..." said Deep Thought.
"Yes...!"
"Is..." said Deep Thought, and paused.
"Yes...!"
"Is..."
"Yes...!!!...?"
"Negative One-Twelfth," said Deep Thought, with infinite majesty and calm.”
posted by cotterpin at 1:19 AM on January 10 [20 favorites]


So, I just got done Calc II last semester, which deals with some of this stuff, and this makes no sense.
posted by runcibleshaw at 1:45 AM on January 10


Riemann zeta functions are indeed endowed with astounding knicker-twisting properties, this is true. Which should come as no surprise, really, as old Bernhard was notoriously fond of making firsthand observations about the topology of the atomic wedgie.
posted by the painkiller at 1:47 AM on January 10


0.5 factorial? What?
posted by Pruitt-Igoe at 1:57 AM on January 10


0.5 factorial? What?

Plot the x! function on a graph. Draw a curve that connects the points. Come up with a function that generates that curve, subject to the restriction that f(x+1) = x*f(x). That's basically the gamma function.
posted by empath at 2:03 AM on January 10 [2 favorites]


Well except it starts going all cray cray below one with asymptotes all over the place, and technically it is shifted over by one such that Gamma(n) = (n-1)! where factorial is defined. empath what did you do now I can't sleep
posted by en forme de poire at 2:28 AM on January 10 [2 favorites]


I consider the "lets show them the weirdest most confusing tricks without any sensible explanation" style to be extremely poor exposition.

Real numbers are defined to be equivalence classes of Cauchy sequences of rational numbers, meaning our usual notion of convergence is built into the equality sign. We departed that world the moment we touched a divergent series, so our equality sign must be based on another notion of convergence. Analytic continuation supplies another useful notion of convergence.

Imho 1 + 2 + 4 + 8 + ... = -1 an easier summation to wrap one's head around because the 2-adic numbers' convergence notion agrees.

p-adic numbers : If e fix a prime p, the any positive integer has a base p expansion of the form a0 + a1 p + a2 p^2 + ... + an p^n where the a0,...,an lie in {0,1,...,p-1}. For any rational x, there is a unique integer n such that x = p^n (a/b) where a and b are integers not divisible by p. We define p-adic absolute value |x|_p = p^{-n} so that larger p divisors mean smaller numbers.

Intuitively 1 + (1 + 2 + 4 + 8 + ..) = 2 + 2 + 4 + 8 + .. = 4 + 4 + 8 + .. = 8 + 8 + .. = .. = 0 makes sense because each successive cancelation is getting smaller by virtue of starting with a larger power of p. So 1 + 2 + 4 + 8 + .. = -1 in the 2-adic integers.

All we did was change the notion of convergence which changes the meaning of equality.
posted by jeffburdges at 2:29 AM on January 10 [5 favorites]


There's a hierarchy of ways of assigning values to infinite sequences. At the simplest, you take the sequence of partial sums and see what it converges to. For example, 1 + 1/2 + 1/4 + 1/8 + 1/16 + … has partial sums S1=1, S2=3/2, S3 = 7/4, S4 = 15/8, S5 = 31/16, etc, and you can prove via induction that Sn = (2n-1)/n = 2 - (1/n) which goes to 2 as n goes to infinity.

More generally, if |x|<1, you can show that 1 - x^(n+1) = (1-x)(1+x+x^2+…+x^n) which suggests that the sum of x^n for n from 0 to infinity is 1/(1-x). Note that if you plug x=1/2 into that new expression you get 1/(1-1/2) = 1/(1/2) = 2. Now this identity only holds for |x|<1, but you can see what happens if you plug in other numbers. For example, if you plug in x = -1, you get 1/(1-x) = 1/2, which corresponds to 1 + (-1) + 1+ (-1) + …, from the previous FPP.

The way I think about it is that the alternating sum of 1s and -1s doesn't really converge, but if I HAD to assign it a value, I'd say it was 1/2.

There are other methods. I made a joke about Cesaro sums in the last FPP, which is when you assign an infinite series a value by looking at the sequence of the means of the partial sums, instead of the sequence of partial sums itself. So for our alternating series, the partial sums are 1, 0, 1, 0, 1, 0, 1 …, but the averages are 1, 1/2, 2/3, 1/3, 3/5, 2/5, … which also converges to 1/2. The nice thing about the hierarchy of these summation methods is once you've found one that gives you a value for a series, all of the "more generous" methods will (usually) assign the sum the same value, like how the zeta function sum and the Ramanujan summation of the integers both give you -1/12.
posted by Elementary Penguin at 2:53 AM on January 10 [5 favorites]


And on preview, the real numbers are Dedekind cuts and I will fight you.
posted by Elementary Penguin at 2:53 AM on January 10 [4 favorites]


empath, I have to hand it to you, this is the most WTF I've seen (and experienced) in a math thread in some time. Good post.
posted by JHarris at 2:54 AM on January 10 [2 favorites]


Imho 1 + 2 + 4 + 8 + ... = -1 an easier summation to wrap one's head around because the 2-adic numbers' convergence notion agrees.

I was actually kinda-sorta thinking about doing a fpp on p-adic numbers.
posted by empath at 2:57 AM on January 10 [3 favorites]


All we did was change the notion of convergence which changes the meaning of equality.

This is why physicists should not be allowed to explain math. Sure, they get all excited because now they can solve physics problems. The moment the prof. went, "let's interpret this as an average, 1/2", you're implicitly employing theories or mappings between spaces.

Also, I don't get his explanation, which begs the question—if he really meant that the reason was to avoid infinity, well why didn't you avoid these infinite structures in the first place?
posted by polymodus at 2:58 AM on January 10 [1 favorite]


And on thinking about it for six minutes, if you plug x=2 into my identity, you get jeffburdges' 1+2+4+8+16+…=1/(1-2) =-1. Math! *jazz hands*
posted by Elementary Penguin at 3:00 AM on January 10 [4 favorites]


The way I think about it is that the alternating sum of 1s and -1s doesn't really converge, but if I HAD to assign it a value, I'd say it was 1/2.

Yes, this is fitting in with what dilaudid noted above. So Numberphile's right, from a certain point of view. He left out the part where the sum becomes a matter of interpretation, which is something about math that's always been there, but in a way that it doesn't matter for most areas of math we have opportunity to commonly think about, and is thus subtly misleading.

It is a useful interpretation in one sphere, which is why Numberphile is careful to mention its applications in physics, and that gives it weight. But it doesn't mean it's the only interpretation available, or even the best one, depending on your problem. Does that sound accurate, anyone?
posted by JHarris at 3:01 AM on January 10


Also, I don't get his explanation, which begs the question

Okay now, I've accepted that an infinite series of positive ascending consecutive integers might add up to -1/12.

But c'mon! Asking me to deal with misusing begs the question is showboating.
posted by JHarris at 3:03 AM on January 10 [2 favorites]


But c'mon! Asking me to deal with misusing begs the question is showboating.

All I mean is there is circular reasoning in his explanation about the need for getting away from infinity.
posted by polymodus at 3:05 AM on January 10 [2 favorites]


Also, I don't get his explanation, which begs the question—if he really meant that the reason was to avoid infinity, well why didn't you avoid these infinite structures in the first place?

Because you can't always avoid infinity in physics. The example they gave was dealing with vibrating strings with an infinite number of harmonics, which gives you exactly this series.
posted by empath at 3:07 AM on January 10


The 1 -1 + 1 + ... =1/2 thing he does is pretty suspect, I'd call the infinite sum undefined.

Basically just because the summation is equal to 1 infinitely often and 0 infinitely often doesn't mean that we get to split the difference and call the "answer" 1/2. I can jump over a fence or crawl under it, but averaging (or even interpolating) the two strategies will probably lead to me breaking my face :)

Also, I'm highly bothered by adding up many positive numbers and getting a negative number. Like, the summation is monotonically increasing with n, so to get a result smaller than any partial sum results in a contradiction.
posted by pzad at 3:12 AM on January 10 [6 favorites]


Because you can't always avoid infinity in physics. The example they gave was dealing with vibrating strings with an infinite number of harmonics, which gives you exactly this series.

That's the same kind of circular reasoning as demonstrated by the Prof. E.g., why are you utilizing a model that contains infinite numbers of harmonics, in the first place? And so on. Basically, the issue is a need for semantic clarity. Like jeffburdges was talking about, with the equivalence classes. Also I'm pretty sure I've seen an article about finite physics or something like that, so although I could be mistaken I think at least one researcher has tried to think about this somewhat.
posted by polymodus at 3:20 AM on January 10


Why are you utilizing a model contains infinite numbers of harmonics?

The world is the way it is. Maybe it contains infinities and maybe it doesn't, but you can't rule it out just because it makes you feel icky.
posted by empath at 3:22 AM on January 10


The world is the way it is. Maybe it contains infinities and maybe it doesn't, but you can't rule it out just because it makes you feel icky.

That's not fair at all. The guy in the video is the one who wants to avoid infinity. Not me. Watch it again and follow his line of reasoning carefully [basically the parts where he explains in words, not the pseudo-proof on paper which is fine in and of itself]. See also jeffburdges re: the notion of convergence, i.e. model theoretic mathematics.
posted by polymodus at 3:24 AM on January 10


The guy in the video is the one who wants to avoid infinity.

Well in physics we never measure infinities, right?

However infinite series do come up in the mathematics all of the time. If an infinite series contributes to some observable in nature, it has to sum up to a finite number that we can measure. That's why you want to get rid of infinities in the final result. That doesn't necessarily mean that you need or want to get rid of infinities in the model.

See also renormalization
posted by empath at 3:29 AM on January 10


However infinite series do come up in the mathematics all of the time.

Then ask yourself why this is the case. When you refer to "the mathematics", that is especially telling.
posted by polymodus at 3:40 AM on January 10


I described the 2-adic convergence explicitly because almost everyone knows about base, thanks to binary's use in computers, but 2-adic numbers agree with an analytic continuation as well, polymodus. Just envision analytic continuation as saying "give me a notion of convergence that makes this series make sense bloody well everywhere". It lets you reason sensibly about that, and related, series at the cost of breaking much other mathematics, which means you must take care when using them.

As an aside pzad, if a series s_n is conditionally convergent, but not absolutely convergent, then, for any real x, some rearrangement of s_n conditionally converges to x.
posted by jeffburdges at 3:41 AM on January 10 [1 favorite]


Then ask yourself why this is the case. When you refer to "the mathematics", that is especially telling.

It's the case because it gives us results that match with experiment. You start with the axioms of quantum mechanics or the standard model or what have you, you work through all the infinities, and you get a result that is a finite number that matches with experiment. If you take out the infinite series, you don't have a model that works.
posted by empath at 3:53 AM on January 10 [1 favorite]


Spock: "Computer, calculate the last digit of pi."
posted by sydnius at 4:32 AM on January 10


If we're quoting Douglas Adams I think this one is more appropiate:

There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened.
posted by DreamerFi at 4:37 AM on January 10 [1 favorite]


It would go to zero, but the infinite series has to pour a little for its homies it lost along the way. Don't get lost watching this, or it will have to pour a little extra for you too.
posted by Nanukthedog at 4:46 AM on January 10


Hmm, this video in the same series explains the 1/2 part better.
posted by I-baLL at 4:51 AM on January 10 [1 favorite]


Good grief. Just look at the second link.
posted by Quilford at 5:27 AM on January 10


I literally just started crying reading this thread, not sure if it's just nice to be around smart people explaining things clearly or if the infinite spaces leaked into my head and started the number madness churning again but either way good work y'all.
posted by Potomac Avenue at 5:29 AM on January 10 [1 favorite]


> Don't get lost watching this, or it will have to pour a little extra for you too.

A lot. Like -1/5.
posted by gilrain at 5:35 AM on January 10


Basically just because the summation is equal to 1 infinitely often and 0 infinitely often doesn't mean that we get to split the difference and call the "answer" 1/2. I can jump over a fence or crawl under it, but averaging (or even interpolating) the two strategies will probably lead to me breaking my face :)

This is the place where I had a problem, too, probably because I spend a lot of time explaining statistics (in a fairly casual way) to students. One of the problems with the mean is that it is not always an actual value -- the average family might have 2.34 children, but you will never meet that .34 child....

So the answer is either 1 or 0, depending on where you stop, but it's never .5, so why can you pretend that it is? Aren't you better off saying that the value oscillates between the two actual values? It just seemed kind of hand-wavy.
posted by GenjiandProust at 6:05 AM on January 10


By the way, I am not claiming that I have found a hole in this -- I assume a bunch of mathematicians and physicists with a lot more mathematical knowledge know what they are talking about. I'm just perplexed.
posted by GenjiandProust at 6:06 AM on January 10


From the video (in a quiet calm voice):

"...this result is critical to getting the 26 dimensions of String Theory..."
posted by sammyo at 6:19 AM on January 10


See this for an explanation.

Basically summation functions should have a few properties.

If used for a normal convergent series, they should get the usual result.
It should be linear, so that you can add series and multiply them by scalars, and you get the same result (for example if S = (1+2+3+4...), then 2S = (2+4+6+8...) = 1+1+2+2+3+3+4+4...
And it should be stable (I'll not quote the explanation, but basically S(1+2+3+4+5...) should equal S1+ S(2+3+4+5...)

I believe that if a summation formula meets all these criteria, then they will all get the same results, when they're able to get results.

These formulas are called 'sums' because when used on a normal converging series, you get the same sum that everyone agrees are sums, but they're also able to give results for divergent or conditionally convergent series, which you can call a sum if you like, or the result of a summation function or whatever else you want to call it.
posted by empath at 6:19 AM on January 10 [1 favorite]


From the video (in a quiet calm voice):

"... it gets a little hairy..."
posted by sammyo at 6:21 AM on January 10


> I consider the "lets show them the weirdest most confusing tricks without any sensible explanation" style to be extremely poor exposition.

I know you're trying to help. And I'm not meaning to pick on you, so what follows is really an attempt to illustrate a particular kind of problem that's not specifically a math problem.

Every one of these bolded terms in your next paragraph are not laypeople's terms; I don't understand what many of them mean and there's not enough context to infer sufficiently:

> Real numbers are defined to be equivalence classes of Cauchy sequences of rational numbers, meaning our usual notion of convergence is built into the equality sign. We departed that world the moment we touched a divergent series, so our equality sign must be based on another notion of convergence. Analytic continuation supplies another useful notion of convergence.

...and so, when a couple paragraphs later, you write the following:

> p-adic numbers : If e fix a prime p, the any positive integer has a base p expansion of the form a0 + a1 p + a2 p^2 + ... + an p^n where the a0,...,an lie in {0,1,...,p-1}.

I can't tell if e is intended to be a constant or a typo for we. Not that, at this point, it matters: You'd already lost me a couple paragraphs earlier. Your explanation is really only useful to somebody who already knows what you know, or is not far from knowing what you know. (In a twist of odd fate, being sufficiently math-illiterate to not know what e is might be an advantage when reading that sentence.)

Your comment could be filled out with links to other places to provide further info, but would have kind of demonstrated the point that it might not be possible to both summarize this particular thing concept and explain it to laypeople. The presenter in the video provided an explanation by skipping the specificity you provide; the result is superficial and avoids a lot of the conceptual glue that allows 1+2+3...=-1/12 to stay together. But the alternative, an accurate and provable explanation, is not more useful to me, because I understand too few of the concepts for your explanation to work.

The problem I have with most technical documentation in my own field is that the author handwaves with the adjective "easy!" instead of writing comprehensible instructions. Things aren't easy just because somebody says they are. An explanation that makes things clear can be hard to write. And might be verbose. And might have to start with first principles, depending on the audience.

So, again, this comment isn't meant to throw your words back in your face. I'm singling your comment out because I can accept on good faith that you are trying to be honest and rigorous, but the result is impenetrable.

On the other hand, I'm in no circumstance where I have to do math on infinities. Using a superficial explanation to connect a valid premise and a valid conclusion is going to be OK, as long as that explanation itself is valid -- meaning it does not have to be unlearned or wholly contradicted if I want to learn the details later. Math can be entertainment, as long as no harm is done by it.

(And if I could tag comments, I would tag this comment, "This is also why all the math entries in Wikipedia, no matter how comprehensive and accurate, are bad.")
posted by ardgedee at 6:26 AM on January 10 [8 favorites]


Good grief. Just look at the second link.

7:25 head explodes
posted by sammyo at 6:31 AM on January 10 [1 favorite]


It does seem like there should be a kernel of insight about why a non-convergent series would be something other than infinity.
posted by sammyo at 6:36 AM on January 10 [1 favorite]


As far as I'm concerned, the trickery is not that s1 = 1/2, but that in s2, they introduce subtraction of infinities. Once you've done that, all bets are off, kind of like division by zero.

So from that aspect, I would say that it's not true that the sum of all positive numbers is -1/12, but I would buy that there are functions defined by summations such that this result is returned.
posted by CheeseDigestsAll at 6:45 AM on January 10 [4 favorites]


Plot the x! function on a graph. Draw a curve that connects the points. Come up with a function that generates that curve

I'm not particularly reactionary about the math here; while the result listed in the FPP annoys my brain, I understand that it's about finding a useful result for a limited set of purposes. Kinda like "horsepower" being useful for comparing power, but not useful for determining how many horses you'd need to outrun a Ferrari.

Similarly, I assume that your description is a useful application of a certain definition of the factorial function. However, that function is commonly defined only for positive integers, so your method would be irrelevant for popular understanding.

An ant has six legs, a bear has four legs, a canary has two legs, does that mean a desk has zero legs and an economy has -2 and a feeling has -4? In other words, just because a set contains all of the valid inputs for a function doesn't mean the function can be validly applied to all of the other members of the set. That's where your method trips me up.
posted by Riki tiki at 6:58 AM on January 10


That's why it's called the gamma function and not the factorial function, though the notation is the same. It's a function that coincides with the factorial at the natural numbers, but nicely extends it to the complex plane.
posted by empath at 7:03 AM on January 10 [2 favorites]


For a concrete use of the Gamma function, see the formulas for the volume and surface areas of n-dimensional spheres.
posted by Elementary Penguin at 7:05 AM on January 10 [1 favorite]


Basically mathematicians like to play with the largest possible set of numbers, so if you tell them that a function isn't well defined for some set of numbers, they'll fiddle with definitions and rules until it is and then see what happens. Maybe it has some relevance for the real world and maybe it doesn't, but in the mean time, as long as it doesn't contradict itself, they'll run with it.
posted by empath at 7:10 AM on January 10


(And if I could tag comments, I would tag this comment, "This is also why all the math entries in Wikipedia, no matter how comprehensive and accurate, are bad.")

Actually, the average math entry on Wikipedia is really quite good, especially once you hit roughly the amount of math knowledge someone has maybe two thirds of the way through an undergraduate degree. Take the entry on real numbers. One will either understand either the first four sentences or basically the whole article (the 'in physics' section is kind of useless to me). But I'm not sure what's so wrong about that in that I'm not sure what non-technical information people would want to know about real numbers that isn't contained in those first four sentences. Even the article on integration contains a decent basic calculus level summary, though it's laid out with conceptual explanation, then more rigorous explanation, then some properties of the integral that a calculus student might want to know. You probably shouldn't be trying to learn math from Wikipedia, but it's generally quite a good reference.
posted by hoyland at 7:11 AM on January 10 [2 favorites]


So if an infinite amount of matter exists, do less than zero kilograms of matter exist?
posted by gubo at 7:27 AM on January 10


I mean I can accept that the answer is "yes because math" — it's just kind of mind-boggling.
posted by gubo at 7:30 AM on January 10


So if an infinite amount of matter exists, do less than zero kilograms of matter exist?

Well, with that much matter, it's too heavy for the floor (of the universe) and falls through.

(I'm no mathematician, but I don't think applying real world metaphors to unreal things like infinities helps in understanding)
posted by device55 at 7:31 AM on January 10 [1 favorite]


Infinity is weird:

Suppose you have three buckets. Bucket A is empty, Bucket B is empty, and Bucket C has infinitely many balls in it, numbered 1, 2, 3, 4, 5, …. Do the following:

1) Take the two balls in C with the smallest numbers on them, and put them both in B.
2) Take the ball in B with the smallest number on it and put it in A.
3) Repeat forever.

If you do each step in half the time of the previous one, you will eventually take all the balls out of C. How many balls will be in A and how many will be in B?

The answer is that B will also be empty and all the balls will be in A, because after the nth repetition of steps 1 and 2, the first n balls will be in A, so by letting n go to infinity, you get that all the balls are in A, even though after the nth step there are n balls in A, n balls in B and infinitely many balls in C.
posted by Elementary Penguin at 7:44 AM on January 10 [3 favorites]


So if an infinite amount of matter exists, do less than zero kilograms of matter exist?

Well, with the Menger sponge, as iterations increase, the surface area heads towards infinity while the volume heads toward zero. Infinity, as they say, is weird.
posted by GenjiandProust at 8:12 AM on January 10 [1 favorite]


If you do each step in half the time of the previous one, you will eventually take all the balls out of C.

How's that, if C has a countably infinite number of balls? How do you "eventually" (as in, at some time point in the future) empty C while only taking a finite number of balls out at a time?
posted by Philosopher Dirtbike at 8:12 AM on January 10


what CheeseDigestsAll said about s2 - you can add and subtract aleph-null all you want but in the end you are going to get is... aleph-null. now, if you want to start multiplying it, or talk about R or Q or C instead of N or Z, that's when scary shit starts to happen. also I even find the claimed result of s1 more than a little handwavy.

if it makes things fit in other fields, I'm definitely interested to read about it. but obviously more than a few mathematicians we are pissed about some handwaving.
posted by dorian at 8:12 AM on January 10


Take the first two out in 1 second, the second two out in 1/2 a second, the third two in 1/4 a second, etc. I mean obviously you can't actually do that, but you also can't actually have a bucket with infinitely many things in it. It's a math thought experiment.
posted by Elementary Penguin at 8:15 AM on January 10 [1 favorite]


Ah, by "step" you meant "iteration of the three steps", not each step 1-3.
posted by Philosopher Dirtbike at 8:17 AM on January 10


Yeah, sorry. I'm inarticulate this year.
posted by Elementary Penguin at 8:27 AM on January 10


Ain't my job to hyperlink all the world's unfamiliar jargon all the time, ardgedee, try google. I haven't proved anything here. I accused the video linked above of poor exposition because they leave the viewer with "woo woo math" rather than "yes because math". And I tersely stated what a "yes because math" explanations requires. I'll address the terms you bolded however :

- I assume listeners took a basic calculus class in high school or university that briefly mentioned convergence, divergence, sequences, and series, making these words vaguely familiar, well unless I know otherwise. At least sequence should be reasonably clear though even without calculus. You've no need to understand the term "Cauchy sequences" here beyond knowing that it's some type of sequence of numbers.

- I'll grant that equivalence classes, and relations, are only taught in university courses, like math, comp. sci., etc., but obviously equivalence shares the same root with equal, which I'm discussing at length and linking it to.

- We're all dancing around not telling you what analytic continuation means, well some math PhDs won't even know that. I attempted a shallow layman's explanation later though.

In more layman's terms, real numbers aren't quite so simple as integers or rational numbers because understanding when two different representations are equal can involve tricky "convergence" issues, ala 1 = 0.9999.. 

I dislike the video because it suggests that 1+2+3... = -1/12, 1+2+4+8+... = -1, etc. are mysteriously true in "situations [you] encounter in real life". That's false. In real life, you'd usually encounter infinity in the context of regular calculous. And 1+2+3... = -1/12 and 1+2+4+8+... = -1 are false, according to the usual equality derived from the convergence in ordinary calculous.

There are however different notions of convergence that give a different equality that makes these true. And that's useful even if it screws up other equalities "because math". Just doing them by symbolic manipulations is misleading though.
posted by jeffburdges at 8:45 AM on January 10


dorian, while I can't justify s1=1/2, I can at least rationalize it by treating it as a probability. If I attempt to sum the alternating sequence some large number of times, then there's p=1/2 I'll get 1 and p=1/2 I'll get 0, so that sort of makes sense. The other result I can neither justify nor rationalize.
posted by CheeseDigestsAll at 8:52 AM on January 10


So you start with some input (the series of natural numbers) and run it through a particular formula (Ramanujan summation) and call that the 'sum of the series', and while it's not a 'sum' in the way it's traditionally thought of, it is an internally consistent property of the series. You can call it a sum, or a Ramanujan characteristic or whatever else you want to call it.

So, in essence this is saying The sum of 1 + 2 + 3 + 4 + 5 ... is -1/12 If we redefine the word "sum" to mean something different than regular addition?

That seems kind of a pointlessly misleading hook, regardless of how worthwhile the actual math going on here is.
posted by straight at 8:53 AM on January 10 [2 favorites]


So, in essence this is saying The sum of 1 + 2 + 3 + 4 + 5 ... is -1/12 If we redefine the word "sum" to mean something different than regular addition?

Sure, but the key idea here is that the thing we redefined "sum" to mean still gives the results we expect for convergent sequences.
posted by singletee at 8:57 AM on January 10


Hmmm.
Let S1 = 1 − 1 + 1 − 1 + · · ·
So -S1 = -1 + 1 - 1 + 1 - · · ·
Since subtraction is the same as adding a negative, -S1 = (-1) + 1 + (-1) + 1 + (-1) + · · ·
But if addition is commutative, I can swap each pair of terms, to get
-S1 = 1 + (-1) + 1 + (-1) + · · ·
But that's just S1 again. So -S1 = S1.
So if S1 is 1/2, that means -1/2 = 1/2. And adding 1/2 to both side, I have now proved that 0 = 1.

Which means there is an error somewhere.
posted by fings at 9:05 AM on January 10


But if addition is commutative

Bzzzzt. We're using "addition", not addition.

(Sorry. This hurts my brain too, and I totally disagree with the "on average it's 1/2" explanation. What's this, statistical guessing?)
posted by RedOrGreen at 9:13 AM on January 10


Everything is undefined, until you define it.
posted by grog at 9:16 AM on January 10


sorry CheeseDigestsAll, the remark about s1 wasn't directed at you but I poorly phrased it such that it appeared to.
posted by dorian at 9:22 AM on January 10


I'd agree that "pointlessly misleading hook" describes this quite accurately, straight, quite poor pedagogy.

We mathematicians view an infinite sum as a sequence of partial sums. So 1+2+3+... is the infinite sequence (s_n) given by s_n = 1+2+..+n. We declare two infinite sequences to be equal, or represent the same number, if they converge to the same value, which requires a notion of convergence. Calculous teaches you the usual convergence notion, which works great for most mathematics.

Analytic continuation is an extremely confusing mask to cover up changing the convergence notion, but actually 2-adic convergence isn't too hard so you could understand why 1+2+4+8+.. = -1 in the 2-adics much more easily and completely.

Also, these infinite sums are commutative under finite permutations whenever they converge, but infinite sums are not necessarily commutative under infinite permutations. As I said, if a series s_n is conditionally convergent, but not absolutely convergent, then, for any real x, some rearrangement of s_n conditionally converges to x.
posted by jeffburdges at 9:23 AM on January 10 [2 favorites]


does anyone have a written proof that's accessible? I'm at work and can't watch the videos but this looks super interesting
posted by Riton at 9:25 AM on January 10


The way I make sense of it is that summing to infinity causes a buffer overflow in the universe and we end up back at -1/12.
posted by mach at 9:53 AM on January 10 [4 favorites]


The way in which assigning these non-intuitive sums works for me is when you can get a intuitive result by manipulating them in some other way.

For example, say you wanted to sum 1 + 2 + 4 + 8 + 16:

1 + 2 + 4 + 8 + 16 = (1 + 2 + 4 + 8 + 16 + 32 ....) - (32 + 64 + 128 + ...).

If you use the formula for summing geometric series
(a0 + a0*r + a0*r^2 + a0*r^3 + ...) = a0/(1-r)

then you get

(1 + 2 + 4 + 8 + 16 + 32 ....) = 1/(1-2) = -1
(32 + 64 + 128 + ...) = 32/(1-2) = -32.

So

1 + 2 + 4 + 8 + 16 = (1 + 2 + 4 + 8 + 16 + 32 ....) - (32 + 64 + 128 + ...).
1 + 2 + 4 + 8 + 16 = -1 - (-32) = 31, which is good and right and proper.

The infinite sums by themselves are very non-intuitive, but if you take the difference you get exactly what you'd expect. I've used this concept when I'm idly trying to calculate investment results. You know, if you invest x dollars every month at y% per year, how much will you have in 40 years?

So it seems to me like the 1+2+3+4... is just a continuation of the same idea. Individually you get answers which seem very non-intuitive, but once you've defined this concept you can do all kinds of useful manipulations, even in finite realms. To those of you who are mathematicians, is this a fair way to conceptualize it?

(At this point it seems like the necessary disclaimer comes in that my formal math education stopped at multi-variable calc and linear algebra).
posted by scottcal at 10:01 AM on January 10


|The way I make sense of it is that summing to infinity causes a buffer overflow in the universe and we end up back at -1/12.

Take a look at a graph of 1/x and try telling me that asymptote isn't wrapping around the back of the Cartesian sphere!
posted by I-Write-Essays at 10:28 AM on January 10 [1 favorite]


Holy shit, I just got what Jeff budges was saying about 2-adics. .....1111111111+ ...000001= 0 and ...1111111 is the 2-adic representation of 1+2+4+8... now that makes more sense to me... BUT the 2-adic representation of -1/12 would be the inverse of ....00001100 or ...111110100 (?) and that's not equal to 1+2+3+4+ is it? Or maybe it is, because you'd constantly be flipping bits around as you added numbers and that's the 'end' result? If so, that just blew my mind more than this video did. Also, does it matter what base the adic number is in?

Wait, it has to matter what base the number is in because the 3-adic representation of -1 is ....2222222 which is 2+6+18+54.... Is that sequence equal to the other sequence in some sense?
posted by empath at 11:32 AM on January 10 [1 favorite]


Oh nm, that's was -12. I'd have to find the multiplicative inverse of that which is too complicated for me to figure out right now.
posted by empath at 11:37 AM on January 10


1+2+3+4+... does not converge in any p-adics. In the 2-adics, the odd terms keep flipping the 1s bit, which means successive partial sums remain distance 1 apart. A similar argument works for other primes. I suspect 1+2+3+4+... hasn't even got any accumulation points in the p-adics, not quite sure.
posted by jeffburdges at 11:46 AM on January 10


How about an analogy?

There are these things called "analytic functions". They are defined on the complex numbers (actually, the complex sphere, which is all the a + bi that we're used to, plus one extra point called "∞").

These have great properties, but can be hard to work with if you just want to know "What happens to the analytic function z ↦ ez near the point 1 + 5i"?. Turns out you can always write an infinite series that is equal to the analytic function in some area around any given point. And by "equal to" I mean the simplest possible sense; the infinite series converges in the way you learned in high school.

However, these infinite series don't converge everywhere, and you get a different-looking series when you look near different points.

Compare this to the spherical surface of the Earth, and the flat maps that we make of it. The surface of the Earth exists; it's a single consistent object. But our flat maps are only good for a certain area. Go away from their center and they are no longer a good representation of reality. Go far enough and they break down completely (e.g. The North Pole, a single point in reality, is an entire line on your Mercator projection. Well, it's also infinitely far up the top of your map - every real Mercator map cuts out a small circle around the pole.)

Now pull up a globe for the next bit. Suppose you are on the 30° East line of longitude, 9990 km north of the Equator. (That puts you 10 km away from the North Pole). Someone tells you "Walk north 20 km". You point yourself north, and go forward 20 km. Where are you? Just doing the math you might say
Answer A) 30° East longitude (I walked north so I stayed on the same longitude),
10010 km north of the Equator (I'm 20 km further from the Equator).
but anyone else would say
Answer B) 150° West longitude, 9990 km north of the Equator.
We can all follow your math, and understand how you got your answer. Heck, if you told us to meet you at the point specified by Answer A we could probably locate that point on the Earth. But it's really not the correct description of your location. Because you passed outside the region where latitudes and longitudes are valid (by passing through the North Pole), the standard math you did to get your location doesn't actually work.

Bringing this back to the infinite series: Take an infinite series, identify it as representing (in some region) an analytic function that has a formula we can write down. Then pick a point outside that region, and plug it into both the series and the formula. The formula gives us a nice answer while the series looks ugly and divergent. We can follow the steps, and, like Answer A above it makes some sense (it's not just some arbitrary number), but having gone outside the region of validity for our representation, it doesn't feel like "the" right answer.

[Caveats:
- It's not a perfect analogy and will break if pushed.
- My description is historically backwards. Infinite series were around for centuries before people discovered analytic functions and how they connected to them.
- Geniuses like Ramanujan could do all sorts of things with infinite series that went above and beyond what I've described and get sensible answers. I've just given how a standard, conservative mathematician of the late 20th century would understand it.]
posted by benito.strauss at 11:56 AM on January 10 [3 favorites]


Ah, my friend, the complex plane. My boyfriend (electrical engineer) and I (physicist) occasionally get into tussles over the usefulness of complex numbers, but I think Jacques Hadamard said it best:
"The shortest path between two truths in the real domain passes through the complex domain."
posted by nat at 12:01 PM on January 10 [2 favorites]


1 + 2 + 3 + 4 + 5 ... = -1/12

One reason everyone sees this as sort of mind boggling is that they are taking three symbols that most people are quite familiar with (+, =, and ... - the ... denoting summation of a series with our normal notions of convergence etc) and redefining all three of them somewhat radically.

I mean if someone said, "There is a function that takes the series 1 + 2 + 3 + 4 + 5 ... and gives the result -1/12, and that function is kinda neat and useful for certain things" everyone would just nod and no one's mind would be blown.

So it really is a bit of a bait and switch when they trot out that formally and say "Mind blown!??!?!?11?" without bothering to explain that they are talking about a different type of convergence than usual and a different type of equivalence (ie, "=") than usual.

Though it is kind of neat and mindblowing that methods like that Ramanujan summation and the Riemann Zeta function can give interesting and useful results for sequences like this.
posted by flug at 1:20 PM on January 10 [2 favorites]


Ah, my friend, the complex plane. My boyfriend (electrical engineer) and I (physicist) occasionally get into tussles over the usefulness of complex numbers ...

That's probably because you can't even agree on the i's and j's.
posted by JackFlash at 1:35 PM on January 10


Genjiandproust: So the answer is either 1 or 0, depending on where you stop, but it's never .5, so why can you pretend that it is? Aren't you better off saying that the value oscillates between the two actual values? It just seemed kind of hand-wavy.

Exactly my sentimonies.
Looking at it from a different angle, you could also argue that:
S1 = 1 - 1 + 1 - 1... = 1 + 1 + 1 + 1 .... - 1 - 1 - 1 - 1 = 0

==> S2 = 0

... and since S-S2 = 4S ==> S = 4S

... so it follows that S = 1, or S = 0, or possibly S = infinity, if we accept that "4 x infinity = infinity", which looks more reasonable than S = -1/12.
posted by sour cream at 2:07 PM on January 10


I'll just take this chance to suggest that silly-ass 'proofs' like (SUM 1-infinity=-1/12) just add to the confusion of many people that don't find math to be 'recreational' .... and, wisely, they throw up their hands and walk away.

I could just as well 'prove' that a fish is a jaguar. Which for most people would, deservedly, put me into the 'clown' camp. Velikovsky got hung for suggesting that Venus was 'young'. What should the reward be for 'clever' people that put others off math? (Ask the millions who suffered their compulsory education by mediocrities.)
posted by Twang at 2:15 PM on January 10 [1 favorite]


I see many instances in this thread of people saying this "doesn't make sense". This brings to mind the following quote; one which I find more astute and important all the time.

"What you think "makes sense" has nothing to do with reality. It just has to do with your life experience. And your life experience may only be a small smidgen of reality. Possibly even a distorted account of reality at that. So what this means is that, beginning in the 20th century as our means of decoding nature became more and more powerful, we started realizing our common sense is no longer a tool to pass judgment on whether or not a scientific theory is correct."
Neil Degrasse Tyson


Now he was referring to scientific knowledge which has an entirely different philosophical basis to mathematics (math is not a science as it is based on proof from postulates, not empirical evidence) but the quote is just as relevant.

If this doesn't "make sense" to you that doesn't make it any less valid. Nor any more valid. It just means you lack the life experience to properly evaluate it. There is an deeply unfortunate tendency for people to take pride in their ignorance in the name of "common sense". They devalue things that they lack the experience to understand in order to absolve themselves of their own failing.
posted by Riemann at 3:42 PM on January 10 [2 favorites]


btw: it would help a lot of people in this thread to remember that "infinity" is not a number. Statements like "4 x infinity" are not equal to anything they are undefined.

Hence the need for constructs like limits which are indeed rigorously defined.
posted by Riemann at 3:44 PM on January 10


here's john baez (mentioned in the second video ;) on "the deep and mysterious links between geometry, number theory and physics" and, from the comments, here's terry tao on The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation :P
Clearly, these formulae do not make sense if one stays within the traditional way to evaluate infinite series, and so it seems that one is forced to use the somewhat unintuitive analytic continuation interpretation of such sums to make these formulae rigorous. But as it stands, the formulae look "wrong" for several reasons. Most obviously, the summands on the left are all positive, but the right-hand sides can be zero or negative. A little more subtly, the identities do not appear to be consistent with each other...

However, it is possible to interpret (4), (5), (6) by purely real-variable methods, without recourse to complex analysis methods such as analytic continuation, thus giving an "elementary" interpretation of these sums that only requires undergraduate calculus; we will later also explain how this interpretation deals with the apparent inconsistencies pointed out above.
more here! like re: freeman dyson on quasicrystals - "The problem of classifying one-dimensional quasi-crystals is horrendously difficult, probably at least as difficult as the problems that Andrew Wiles took seven years to explore. But if we take a Baconian point of view, the history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible."

Take a look at a graph of 1/x and try telling me that asymptote isn't wrapping around the back of the Cartesian sphere!

also btw in the comments baez sez: "The simplest and most fundamental case of 'looping around' is the function 1/x, which goes down to negative infinity and then suddenly reappears up around positive infinity, hinting that maybe these two places are the same place. It's even more evident when you work with complex numbers: there are lots of different directions you could march off to infinity, but it turns out to be best to treat these marches as all leading to the same point, to make the function 1/z continuous. I don't know who first came up with the idea of adding a 'point at infinity' to the complex plane, but the result is the Riemann sphere so maybe it was him. I wonder how much this discovery was influenced by the points at infinity in projective geometry; they turn out to be two aspects of the same thing."
posted by kliuless at 3:45 PM on January 10 [3 favorites]


The extra hilarious thing about this to me is that by the same definition, we seem to have:

1 + 2 + 3 + 4 + 5 + ... = -1/12
Weird! Counterintuitive! The subject of the FPP!

1 + 4 + 9 + 16 + 25 + ... = 0
Well, this is bizarre, but not any more bizarre than the first one.

1 + 1 + 1 + 1 + 1 + ... = -1/2
Again, now that I've accepted that these weird non-convergent infinite sums of whole numbers can give you a rational and even negative number, this isn't that shocking.

1 + 1/2 + 1/3 + 1/4 + 1/5 + ... = ?
So this one probably has a totally finite and tidy solution in this same space, right?
Haha NOPE this one is actually still infinity!

(╯°□°)╯︵ ┻━┻
posted by en forme de poire at 3:45 PM on January 10 [2 favorites]


oh and: Is the Universe Made of Math?
posted by kliuless at 4:04 PM on January 10 [1 favorite]


It's the case because it gives us results that match with experiment. You start with the axioms of quantum mechanics or the standard model or what have you, you work through all the infinities, and you get a result that is a finite number that matches with experiment. If you take out the infinite series, you don't have a model that works.

My second sentence in this thread already addressed this. Not remotely the concern I have been trying to express. The point is that if you are a physicist, you must at some point acknowledge or ponder the broader research issues of:

a) Why is there no strictly finite mathematics for representing theories of the physical world?

and

b) There needs to be more rigorous justification and analysis of "new math", other than "because it works experimentally". The response isn't merely "oh isn't it cool, we can show that ∑n = -1/12", I think a more pertinent response is to ask "What are the salient properties of mathematical theories consistent with this formula"? So I think it's natural that physicists are too busy solving their problems, to entertain this angle of the issue. But if the object is to de-confuse people, this additional level of rigor can help.

On a), possibly I am having a false memory, but I feel sure that the question has been brought up by at least some computer science theorists, if not other physicists, so while I'm not familiar with the details I hope I'm not just making up this issue.

As for b), here's an excerpt from Terence Tao's notes, parts bolded for emphasis:

Remark 1.8.1. One can also perform these completions in a different order, leading to other important number systems such as the positive rationals Q+, the positive reals R+, the Gaussian integers Z[i], the algebraic numbers Q ̄ , or the algebraic integers O.

There is just one slight problem with all this: technically, with these constructions, the natural numbers are not a subset of the integers, the integers are not a subset of the rationals, the rationals are not a subset of the reals, and the reals are not a subset of the complex numbers! For instance, with the above definitions, an integer is an equivalence class… A natural number such as 3 is not then an integer. Instead, there is a canonical embedding of the natural numbers into the integers…

So, rather than having a sequence of inclusions N ⊂ Z ⊂ Q ⊂ R ⊂ C,

what we have here is a sequence of canonical embeddings N ↪ Z ↪ Q ↪ R ↪ C.

In practice, of course, this is not a problem, because we simply identify a natural number with its integer counterpart, and similarly for the rest of the chain of embeddings. At an ontological level, this may seem a bit messy - the number 3, for instance is now simultaneously a natural number, an equivalence class of formal differences of natural numbers, and equivalence class of formal quotients of equivalence classes of formal differences of natural numbers, and so forth; but the beauty of the axiomatic approach to mathematics is that it is almost completely irrelevant exactly how one chooses to model a mathematical object such as 3, so long as all the relevant axioms concerning one’s objects are verified, and so one can ignore such questions as what a number actually is once the foundations of one’s mathematics have been completed.

[Maybe there's a better reference on this question, but this reference is what came to mind.]

My take-home from this passage is two points. First, that "-1/12" is NOT "-1/12". This is explained in the sense above. At what point in the videos is this properly given due consideration? That's the problem. I have no issue with physicists using it, and further suggesting that it seems to have deep connections due to how it shows up in their theories. But it needs to be recognized that the source of confusion for laypeople lies precisely in the lack of metamathematical rigor in their explanations.

Secondly, for non laypeople, Tao's last two criteria are what's missing. There's no verification and there's no complete foundation. Physical evidence is great and intuitive and compelling. But it is not a justification that any scientist should be content with, i.e. it does not substitute for having to show that if you introduce this "averaging" representation you don't break things elsewhere. That amounts to the task of learning and proving properties of the mathematical theory, e.g. formal consistency or whatever have you. Without verifying your new model you can't sensibly be going around telling everyone that this is okay just because contemporary physics seems to require it. It's this follow-through that's missing from the explanations in the video. It's actually a trivial point but one that seems to me isn't being acknowledged much.
posted by polymodus at 4:26 PM on January 10 [1 favorite]


Imaginary numbers.
posted by 4ster at 4:51 PM on January 10


There are an infinite number of numbers between 0 and 1.

There are an infinite number of numbers between 0 and 2.

Therefore, 1 == 2.

Most of these sort of tricks reduce to the above trickery. Doesn't make 'em correct. They're still tricks, just obfuscated better.
posted by effugas at 4:55 PM on January 10 [1 favorite]


And by the way, the whole point is to produce confusion in lay people. That's the idea.
posted by effugas at 4:56 PM on January 10


The set of real numbers between 1 and 2 is homomorphic to the reals and the set of real numbers between 0 and 1, so there are senses in which they're basically equivalent, but that doesn't meant that 1=2.

A lot of the problem is that mathematical notation is heavily overloaded, where the same symbols can mean different things depending on the context.
posted by empath at 5:05 PM on January 10


The set of real numbers between 1 and 2 is homomorphic to the reals and the set of real numbers between 0 and 1, so there are senses in which they're basically equivalent, but that doesn't meant that 1=2.

Homeomorphic?

(They might be homomorphic, but I don't want to think about it.)
posted by hoyland at 5:12 PM on January 10


they are more than homomorphic, they are isomorphic.
posted by Riemann at 5:19 PM on January 10


I knew it was something-morphic :)
posted by empath at 5:30 PM on January 10


empath,

Of course, 1 is not equal to 2. And we've got well-worn mechanisms to show how that's the case, despite the presence of two infinite sets.

The point is that we can put together other constructs, with not quite so well worn mechanisms, and come up with equally ridiculous results. And then math nerds get to confuse the lay people, which is the point. And it's bullshit, because it makes people doubt what little math they know. Double bullshit because the lay people are right and the math nerds, no matter how much they show off, are just as wrong as if they were claiming 1==2.
posted by effugas at 5:44 PM on January 10


Love Numberphile; thanks for sharing.

What I like most about digging into this is that the explanations they provide can lead so swiftly to an infinite series of MORE QUESTIONS.

For instance, I've just learned these two facts:
1 + 2 + 3 + 4 .... = -1/12
1 + 1 + 1 + 1 .... = -1/2

Now, it's also clear that if I take the following:
1 + 1 + 1 + 1 ....
...+ 1 + 1 + 1 + 1...
.........+ 1 + 1 + 1 + 1...
and do that to infinity, I've just made myself:
1 + 2 + 3 + 4 ....

In other words, -1/2 * ∞ = -1/12, and I now have to go find out why THAT result makes sense.
posted by Room 101 at 8:26 PM on January 10


Room 101,

Who says it has to?
posted by effugas at 8:34 PM on January 10


Room 101: I think that's (1 + 1 + 1 + 1 ...)²

Not that that helps anything.
posted by aubilenon at 10:25 PM on January 10


I think that's (1 + 1 + 1 + 1 ...)²

first, outside, inside, fuck it
posted by en forme de poire at 10:38 PM on January 10 [2 favorites]


Ahh, 1+2+3+4+.. and Zeta function regularization have more on why the physicists care. And that's why we're talking about the Zeta function and 1+2+3+4+.. = -1/12 as opposed to some conceptually simpler weirdness 1+2+4+8+..=-1 in the 2-adics.
posted by jeffburdges at 3:11 AM on January 11


And then math nerds get to confuse the lay people, which is the point

I learned a lot about about regularization and divergent series and analysis and so on, which is what I thought was the point.
posted by empath at 3:14 AM on January 11 [1 favorite]


There are an infinite number of numbers between 0 and 1.

There are an infinite number of numbers between 0 and 2.

Therefore, 1 == 2.


Well, no, all that shows is that arithmetic with infinite cardinals is weird. 2 times any infinite cardinal is equal to that same infinite cardinal.
posted by Elementary Penguin at 4:16 AM on January 11


(divides by zero)

/thread
posted by ostranenie at 4:46 AM on January 11


The EFF hopes to counteract the NSA's recruitment efforts at the Joint Mathematics Meetings in Baltimore this week by having their own booth.

If you're attending the JMMs, then drop by to say hi. And consider volunteering you time to discourage young mathematicians form taking jobs at the NSA.
posted by jeffburdges at 6:28 AM on January 14 [1 favorite]


Excellent. Thanks for that news, jeffburdges!
posted by JHarris at 6:33 AM on January 14


Some good explanations of the answer(s) to the FPP question at Quora.
posted by zittrain at 3:09 PM on January 16 [1 favorite]


Ahem.. Such maths
posted by jeffburdges at 2:48 AM on January 17


Academics Against Mass Surveillance
posted by jeffburdges at 10:13 AM on January 17 [1 favorite]


In case anybody's still reading this thread, I spent about 45 minutes on the phone yesterday with Phil Plait, and he has a followup post this morning with a quote from me which I think more or less expresses the mainstream mathematical view on this stuf.
posted by escabeche at 6:55 AM on January 18 [4 favorites]


There's a lot of mea culpa in that post, escabeche, but after reading it I still don't have any better idea of how they got 1..2..3..->infinity equal to -1/12.
posted by JHarris at 7:42 AM on January 18


Nicely done, escabche. The way that people couldn't resist saying "equals" reminds me of how others couldn't resist forwarding the Iron Maiden story.

JHarris, the consistent method used to assign values to series like this is what they identified as analytic continuation. If you read my comment, and skip the middle part talking about maps, you'll get a quick sketch of what's going on.
posted by benito.strauss at 9:18 AM on January 18


I suppose. Kind of. I already had the kind of understanding of it alluded to by your (pretty good actually) maps analogy, but I'm not sure on the specifics.

Don't bother trying to tell me, I can tell I'm not going to "get" this one without diving in and piecing it together, which would probably be more text that would be feasible here. I wish I had the time to look up all this stuff and figure it all out.
posted by JHarris at 9:28 AM on January 18


Yeah, there's a lot of background stuff leading up to analytic continuation, more than I'd write up. If you're ever in Boston, twenty minutes over a beer and I'll bring you up to speed.
posted by benito.strauss at 10:58 AM on January 18 [1 favorite]


I started wondering what the Riemann surface associated to the analytic continuation of the Riemann Zeta function looked like. It's time to ask an analytic number theorist apparently :

"According to the idea of Tate's thesis, the Riemann zeta function ζ(s) should be thought of as a distribution on the idèles GL_1(A_Q) of Q. Integrating this distribution against the complex characters of GL_1(A_Q) gives the values of the zeta function." (see Tate's thesis & Idele group)
posted by jeffburdges at 4:57 PM on January 19


The Means Justify the Ends, or, Mathematicians are Sherlocks and Physicists are Mycrofts
posted by homunculus at 8:16 PM on February 3


Of course, I'm only giving an intuitive feel for analytic continuation here. In truth it really involves analytic functions defined on open subsets of the complex plane, as opposed to the real line, but you get the gist.

HAAAAAAAAAAH! </Pam>
posted by en forme de poire at 8:54 AM on February 4


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