And indeed, he was right. Phil Plait the Bad Astronomer, of all people, got taken in by a bit of mathematical stupidity, which he credulously swallowed and chose to stupidly expand on.

We'll consider three infinite series:

S1 = 1 - 1 + 1 - 1 + 1 - 1 + ...
S2 = 1 - 2 + 3 - 4 + 5 - 6 + ...
S3 = 1 + 2 + 3 + 4 + 5 + 6 + ...


S1 is something called Grandi's series. According to the video, taken to infinity, Grandi's series alternates between 0 and 1. So to get a value for the full series, you can just take the average - so we'll say that S1 = 1/2. (Note, I'm not explaining the errors here - just repeating their argument.)

Now, consider S2. We're going to add S2 to itself. When we write it, we'll do a bit of offset:

1 - 2 + 3 - 4 + 5 - 6 + ...
1 - 2 + 3 - 4 + 5 + ...
==============================
1 - 1 + 1 - 1 + 1 - 1 + ...


So 2S2 = S1; therefore S2 = S1=2 = 1/4.

Now, let's look at what happens if we take the S3, and subtract S2 from it:

   1 + 2 + 3 + 4 + 5 + 6 + ...
- [1 - 2 + 3 - 4 + 5 - 6 + ...]
================================
0 + 4 + 0 + 8 + 0 + 12 + ... == 4(1 + 2 + 3 + ...)


So, S3 - S2 = 4S3, and therefore 3S3 = -S2, and S3=-1/12.

So what's wrong here?

To begin with, S1 does not equal 1/2. S1 is a non-converging series. It doesn't converge to 1/2; it doesn't converge to anything. This isn't up for debate: it doesn't converge!

In the 19th century, a mathematician named Ernesto Cesaro came up with a way of assigning a value to this series. The assigned value is called the Cesaro summation or Cesaro sum of the series. The sum is defined as follows:

Let . In this series, . is called the kth partial sum of A.

The series is Cesaro summable if the average of its partial sums converges towards a value .

So - if you take the first 2 values of , and average them; and then the first three and average them, and the first 4 and average them, and so on - and that series converges towards a specific value, then the series is Cesaro summable.

Look at Grandi's series. It produces the partial sum averages of 1, 1/2, 2/3, 2/4, 3/5, 3/6, 4/7, 4/8, 5/9, 5/10, ... That series clearly converges towards 1/2. So Grandi's series is Cesaro summable, and its Cesaro sum value is 1/2.

The important thing to note here is that we are not saying that the Cesaro sum is equal to the series. We're saying that there's a way of assigning a measure to the series.

And there is the first huge, gaping, glaring problem with the video. They assert that the Cesaro sum of a series is equal to the series, which isn't true.

From there, they go on to start playing with the infinite series in sloppy algebraic ways, and using the Cesaro summation value in their infinite series algebra. This is, similarly, not a valid thing to do.

Just pull out that definition of the Cesaro summation from before, and look at the series of natural numbers. The partial sums for the natural numbers are 1, 3, 6, 10, 15, 21, ... Their averages are 1, 4/2, 10/3, 20/4, 35/5, 56/6, = 1, 2, 3 1/3, 5, 7, 9 1/3, ... That's not a converging series, which means that the series of natural numbers does not have a Cesaro sum.

What does that mean? It means that if we substitute the Cesaro sum for a series using equality, we get inconsistent results: we get one line of reasoning in which a the series of natural numbers has a Cesaro sum; a second line of reasoning in which the series of natural numbers does not have a Cesaro sum. If we assert that the Cesaro sum of a series is equal to the series, we've destroyed the consistency of our mathematical system.

Inconsistency is death in mathematics: any time you allow inconsistencies in a mathematical system, you get garbage: any statement becomes mathematically provable. Using the equality of an infinite series with its Cesaro sum, I can prove that 0=1, that the square root of 2 is a natural number, or that the moon is made of green cheese.

What makes this worse is that it's obvious. There is no mechanism in real numbers by which addition of positive numbers can roll over into negative. It doesn't matter that infinity is involved: you can't following a monotonically increasing trend, and wind up with something smaller than your starting point.

Someone as allegedly intelligent and educated as Phil Plait should know that.

• Having been trained in mathematics, I find that almost nobody but mathematicians[*] really understands more than the most trivial of mathematical concepts. Confused math-fandom is even more annoying than confused science-fandom...

[*] "almost nobody" = except a set of measure zero.

• sugarfrosted says:

But since the set of all people are finite, all sets of people are of measure zero.

• justinformal says:

Depends on the measure

• I found the discussion generated by Phil's post very enlightening. But can I ask a favor? Please stop calling people who struggle with mathematical concepts "stupid" or "allegedly intelligent" or any of the other unnecessary slams that you sprinkled throughout this otherwise very informative post. It's counter-productive and in the long run, if your goal is to bring people into the math-loving and math-understanding fold, you're shooting yourself in the foot by doing so. It just makes people not want to try to engage at all, ever, with anything math related. And that hurts science, and society, in the long run.We learn through failure after all: by getting things wrong, sometimes even publicly, and learning when our errors are pointed out to us. If it's done in the right way, it makes us want to try harder and get it right in the future. That's what you're shooting for. I hope. 🙂

• To be fair to Marc, if Plait would have shown this to a single mathematician, he would have been educated quickly that this is a cute stunt, not a significant result. Plait's readers deserve better - enthusiasm for amateur mathematics should be fed with presentations of valid results - amateurs deserve better than being "led down the garden path".

• It's not even about asking a mathematician. This is literally someone with a doctorate in astronomy who credulously accepts something that should set off the bullshit detector of a freshman in second-semester calculus.

No, Phil isn't stupid or unintelligent. He just doesn't really give a damn about mathematics beyond the point where it serves his particular purpose. He learned enough to pass the required tests and never looked back, just like most other scientists and engineers.

• Ian Kaplan says:

Ow. First those who make math mistakes are slammed and then scientists and engineers (those of a non-math color) are slammed. No wonder people hate math.

I just finished a Masters in Applied Math. I will not miss doing proofs. I suppose that this throws me into the outer darkness (I'm a software engineer, so perhaps I get doubly cursed).

• Will White says:

The way to reduce mistakes is to apply a cost to them. Public humiliation of someone with a doctorate in astronomy falls in that category.

To frequently these days smart people are taken in by the "Malcolm Gladwell" effect. A desire to explain something outside of their field of interest with a simple counter-intuitive solution.

Unfortunately, the world is generally more complicated than that and smart people need to be much more careful with how they think openly in public to avoid making fools of themselves.

• Max Kaufmann says:

@Will Let's apply a cost to your grammar: "TOO frequently..."

• Ian Kaplan says:

I was wrong. Those who wrote that Phil Plait is a weak thinker were right. I just read the defense he wrote in Slate of the original article. I'm not much of a mathematician, but even I can see that he's wrong in his analysis. Yet he admits nothing and writes at the end that he's "prima facie" correct. Sad, really sad.

So I take it all back. Send those slings and arrows Plait's way. He does deserve them.

• Casey Bowman says:

You kind of have a point, but it's not quite there. Unfortunately, in the realm of internet posts, there is no assurance that people who see the original incorrect post will EVER find out it is wrong. That is what is so dangerous about making posts like that in the first place. The damage done by incorrect statements far outweighs losing some engagement by the math-phobic in my opinion. And the worst part is that he's right about it being completely obvious just based on the fact that summing positives can't leave you with a negative. If people are taken in with that kind of garbage then I don't want to hear what they have to say about math.

• Joe Shelby says:

I agree. All of these corrections here and elsewhere, and I still expect the original video and incorrect numbers to make the "that is so cool..." rounds on Facebook within the next week.

If people looked ahead before forwarding stuff like this, snopes wouldn't be in business.

• Bill says:

NPR have picked up the story uncritically and posted it on facebook. Will no one ever ask a mathematician?

• jj says:

not knowing mathematics isn't stupid. you are right. but what is stupid is doing things you don't have a clue about, selling them as mathematics and then not even admitting that you didn't do "correct" mathematics (yes, there's only one kind of mathematics, correct). and i think that's what is being done by many people in support of this "wrong calculation" recently.
yet it might still be counter-productive to use the word "stupid" even in this case.

everyone can make mistakes, even mathematicians and they often do.
but in the nature of every mathematician should be integrity. integrity to realize a mistake and not be offended but be glad to have learned something new through your mistake. glad to have eliminated this false piece of information from your mind and replaced it by the a correct piece of information.

greetings.

• MarkCC says:

Thank you. I think that's a very good way of explaining things.

But at the end of the day, I am who I am. What I do on this blog is say what I think. If it's overly harsh at times, that's the way that I am. I think that trying to filter out my rough edges just ends up taking what I want to say, and stripping out everything that makes it worth saying. I don't mean that I don't have anything to say beyond harsh phrasing, but rather that I'm just not a good enough writer to know how to do it.

It was honestly not my intention to be insulting to Phil. In fact, I thought that I was being reasonably gentle! If you compare what I wrote to the things I said in the immediately preceeding post about Randell Mills, you'll see what I'm like when I want to be insulting!

That said, I still stand by what I wrote. It was, perhaps, harsher than it should have been. But I still think that the kind of mistake that Phil made was egregious, and that writing a post that explained why was a good thing to do.

• Bill says:

Only one kind of mathematics? Bzzzt. There are infinitely many kinds of "correct" mathematics, for some definition of correct. Any formal system capable of admitting the integers is either incomplete or inconsistent. Simply put, I can continue adding consistent axioms to any system forever once I have integers represented. This result is over 100 years old, people. It's time to start acting like it means something.

• To parse it a bit more finely, different formalisms (versions of set theory, geometries, whatever) are not different mathematics. Mathematics is not a statement of what "is true" or "is false", but is about "given these axioms, you can derive these conclusions". That idea of correct derivations is what makes the singular field of (correct) mathematics.

To tie it back: Numberphile and Plait pulled a bait-and-switch by setting you up to think they're talking about the usual standard axioms and definitions, but then switch to different ones without telling you.

• MarkCC says:

Actually, any formal system capable of doing Peano arithmetic is either incomplete or inconsistent. Important disntiction there: the theory of real numbers does include the integers - but it doesn't include Peano arithmetic. And real number theory is both complete and consistent!

In the case of this Zeta result, it's important because of consistency. If you accept the definition that the sum of the natural numbers is equal to -1/12, then you've got an inconsistent system. You can't add that as an axiom.

You can do all sorts of interesting things with infinite series. But it's important to be clear about what you're doing.

This goes back to one of my pet peeves, which I discussed in one of the followups to this post.

People get the idea that math is incomprehensible, difficult, and mysterious. Due to the terrible way we teach math, many people - perhaps even most people - come away with the idea that they can't understand math. That there's something almost magical about it. That it's completely beyond their reach.

If you take a result like this, and you say "The sum of the natural numbers is -1/12 and the result will carve out a piece of your soul and leave hollow space. Seriously, please, bear with this. I pounded my head on my desk enough to leave a dent in the wood and a welt in my forehead to figure this out, so please just stick with it" - you're giving people an interesting piece of information, but you're doing it in a way that says "you can't possibly make sense out of this".

If, however, instead of prefacing it with how insane and impossible it is, you say: look, here's this series. If you look at it normally, it obviously adds up to infinity. But in some places, instead of trying to add it up, you can use a neat mathematical trick to allow you to compare it to other infinite sequences, and according to that trick, you can use the number -1/12 instead of the sum.

Now, instead of telling people that it's impossible, and they can't hope to understand it, you've taken a fascinating mathematical subject, and turned it into something that non-mathematicians can make sense out of. Instead of teaching people that they can't understand it, you're teaching them that they can.

Telling people that math is beyond their reach, that a mathematical result that you could explain in three lousy sentences is something that will "carve out a piece of your soul and leave a hollow space" - that's a terrible thing to do.

• william e emba says:

So when you told everyone that anyone who thought 1+2+3+4+...--1/12 was stupid and obviously moronic, that was OK, since it could never convince anyone that the result was beyond comprehension?

Hypocrite.

• "Someone as allegedly intelligent and educated as Phil Plait should know that."

I thought this was a little unfair, then I scanned the article. Ugh. Addition is commutative, so could, with a little work, get any number we wish out of this voodoo. Depressing that Plait thinks -1/12 is significant.

Reddit.com/r/math would have demolished this in the time it takes to get a cup of coffee. I will be charitable and say that Plait doesn't have as much experience being foolish as I do, so he doesn't know all the avenues for soliciting sanity-checks, as I do.

• Jackson says:

Well, there goes that theory. I scanned the top posts and saw almost no skepticism. Epic fail, /r/math.

• Bill says:

Wrong. 14 days ago, /u/Trekky0623 said: "There's a supplementary video that proves the result using the Riemann zeta function. This first part without going into detail is basically just troll math until you understand why we can say this." So this scientopia article is also wrong and jumping the gun a bit. Phil Plait is just unclear in his definitions, he's not been duped.

• MarkCC says:

The Riemann Zeta function says that z(-1)=-1/12. It does not say that the sum of the natural numbers is -1/12.

To illustrate the issue, let's try a different solution for zeta.

Euler proved that for all integer values :

If I put s=1/2 into this equation, the resulting value z(1/2) will not be the correct value for zeta(1/2). Because this solution only works within its domain, and its domain is the set of integers. Since 1/2 is not an integer, this won't work.

In the Riemann zeta function, for some values, you can solve zeta using a series. That solution is undefined at s=-1, where it expands to the sum of the natural numbers. But just like the Euler solution doesn't work for s=1/2, the additive series solution doesn't work for s=-1.

Zeta(-1)=-1/12. That's not the same thing as saying that the additive series solution, which doesn't work at -1/12, actually is defined. Because it isn't.

• william e emba says:

You obviously do not have a clue about mathematics or physics. Every definition is a convention. The sum of an infinite series defined in terms of partial sums and limits is one convention. So is zeta-function regularization. Neither one is "correct". Both have their uses.

Physicists perform thousands of calculations like 1+2+3+4+...=-1/12 every day. The vast majority of them are way too esoteric to explain in any layman fashion. This sum, which goes back to Euler, is one of the few exceptions, perhaps the only one. Even more fascinating is that it actually shows up at the ground floor of string theory. Here's another reference: Green, Schwarz, Witten Superstring Theory Volume I (Cambridge 1987) p. 96.

• I agree completely that this result is wrong as stated.

But, is there not some sense that the idea that the -1/12 result is useful in the some ways? It's claimed on the numberphile video and in other places that this result is used in string theory, for example.

My understanding, which is limited, is that this is more a problem of terminology. That the summation is correct as long as you are not using standard definitions, but using Ramanujan summation.

• MarkCC says:

The series does not, in any way, shape, sense, or form equal -1/12. In a certain limited context that involves using a continuous form of a clasically integral problem (Riemann Zeta), certain non-integral parameter values produce a solution of -1/12, and other parameter values produce the natural number summation series.

• Bard Bloom says:

Physics is full of cases where very dubious mathematics produces useful and physically-correct results. Dirac delta functions, for example.

It sort of makes sense that reality sometimes agrees with Cesaro summation: photons can't just say "not defined", they have to do something (perhaps many things), even in a situation defined by a non-convergent summation.

Which implies exactly nothing about math. By the standard definition, the sum of the natural numbers is divergent and that is that. Cesaro summation and generalized functions are *different* mathematics, possibly interesting and possibly useful, but not to be confused with the basics.

• Chas says:

The Dirac delta is mathematically dubious only if it considered as a function. As a distribution it is just fine. Physicists and engineers like to play fast and loose with it, but in almost all ways it is used it can be made rigorous relatively easily. You reference generalized functions later so I assume you are familiar with how the Dirac delta is formalized, but I wanted to make sure that no one reading the comments got the wrong impression.

• Bill says:

In the words of a great president-mathematician, that depends on what your definition of is, is. Equals here doesn't mean equals in the normal sense. Yes the result is useful and not entirely arbitrary.

• Sizik says:

You can get the same result by methods other than the Cesaro sum.
S1 = 1 - 1 + 1 - 1 ...
-S1 = - 1 + 1 - 1 + 1 ...
1 + -S1 = 1 - 1 + 1 - 1 ... = S1
1 = 2S1
S1 = 1/2

• Which is still not justified, which should have been covered in second-semester calculus, and is definitely covered in undergraduate real analysis. These are more tricks; they're not actual mathematics.

• Jesse says:

Just out of curiosity, what is wrong with this supposed proof?

• Ben Weaver says:

For starters, a series sum is actually a sequence of partial sums. When the limit of a sequence exists, you can do arithmetic on it. However, if the limit does not exist, you can't. Here, the limit does not exist.

• Nico says:

Jesse, The proof of Sizik could just as well stop at the second line:
S1= 1 -1+1 -1 ...
-S1= -1 +1 -1 +1 ...
-S1= 1 -1 +1 -1... (rearranging two by two)
-S1=S1
therefore S1=0 (a = -a iff a=0)

• Dale says:

To put it simpler S1 does not exist so you can't use.

• Aaron says:

This is unnecessarily harsh and borderline wrong. There's a huge literature on 'summing' divergent series, and it's actually pretty important, both mathematically and physically. In some cases, the manipulations can be made rigorous.The fact that the 'sum' of the natural numbers is equal to -1/12 also shows up in any number of physics calculations leading to a meaningful (experimental!) result.

See, for example, Wikipedia:

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

and this mathoverflow question and references therein:

http://mathoverflow.net/questions/19201/summation-methods-for-divergent-series

Also, this article by Terry Tao explains why these numbers show up in physics (although you need to know some physics to understand how the math there translates into physics):

http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

• UserGoogol says:

It's an arbitrary matter of convention to say that the classical sum is what the series "really" adds up to and the Cesaro sum is "just" this other way of assigning numbers to series. They're both different ways of assigning numbers to infinite series.

• MarkCC says:

The important point is to understand that the method of assigning a measure to an infinite series is just that: an assignment, not an equality. You need to understand what the assignment means, and how you can work with values that result from it.

Cesaro sums for series don't support algebra on the series. Which should be glaringly obvious, because there is no such thing as overflow/rollover for real numbers.

• The important point is to understand that the method of assigning a measure to an infinite series is just that: an assignment, not an equality.

So the real question is: is there a summability method under which carrying out those manipulations on the assigned numbers on the LHS of the equation and on the sums directly on the RHS is valid? If so, then the axiomatic approach to summation implies that the manipulations are legitimate under at least one method for assigning numbers to infinite series. Most of the manipulations shown in the video require linearity; the exception is the shifting trick, which (I think?) requires stability. (And I assume there is such a method, or else the Wikipedia article wouldn't use stability to calculate the summation of divergent geometric series.)

The real bad math in the video is the part where the physicist starts talking about how you need to "keep going" to get the counter-intuitive result; this might give the impression that keeping going is sufficient. Keeping going is necessary but not sufficient to get the counter-intuitive result; the sufficient condition is just that the (implicit) summability method has the properties necessary to preserve assigned numbers under those algebraic manipulations.

• william e emba says:

The physicist's "keep going" language (half the time it's 0, half the time it's 1, etc) was a Cesàro summation done in simple English. It is neither good nor bad math. It will sound good to some viewers, it will not sound good to others. It is guaranteed to sound painful to anyone who does math in math, So what?

Let me phrase this a little more starkly. Physicists have been butchering math for a century now, including the endless number of fellows who have made the Internet possible, from quantum mechanics and transistors to chips and satellites. If you're on some mathematics purification crusade, I suggest you start by getting off the Internet completely. That way you won't look like a hypocrite and parasite.

• John Fringe says:

If you're on some blog purification crusade, I suggest...

• Antendren says:

"The important point is to understand that the method of assigning a measure to an infinite series is just that: an assignment, not an equality. You need to understand what the assignment means, and how you can work with values that result from it."

What do you think it means to say that an expression equals a number?

And as mentioned below, Cesaro sums are linear, so they do support certain amounts of algebra.

• Dale says:

Let's see if the "algebra" used is well defined. Try doing the shift argument on S1

1-1+1-1+1...
1-1+1-1...
-------------------
1+0+0+0+0+....

This gives us that each partial sum is 1, thus the Cesaro sum is 0.

So we 2S1 = 0.

Thus S1=0, but we had S1= 1/2. It seems to me we have a problem.

• Alex R says:

Sigh. Phil's post may have oversimplified the issues, but it's not nearly as boneheaded as Mark is making it out to be. The result that -1/12 "equals" 1 + 2 + 3 + 4 + ... has a long history, some of which is laid out in the Wikipedia article on the topic that Phil linked to: http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

Here's my own take on it: Let's start with just the real numbers, infinite sequences of real numbers, and ordinary arithmetic on real numbers -- addition, subtraction, etc. It's not 100% correct to say that that the sum of an infinite series "equals" any real number -- the addition operation takes 2 real arguments, not an infinite number. Extending addition to finite numbers of real arguments is straightforward, due to commutativity and associativity, but that still doesn't allow us to find the sum of all of the elements of an infinite sequence. In order to define a0+a1+a2+... one must make additional definitions. The first thing we did in pre-calculus when we encountered these series is to *define* the sum of the as the limit of the partial sums. But this is something new; it's not just addition -- it's addition plus extra stuff. The extra stuff causes this operation to behave differently. For example, strict commutativity no longer holds: 1 - 1/2 + 1/3 - 1/4 + 1/5 - ... does not equal 1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + 1/9 ... (In the second series, I've moved the negative terms over to the right so that every third term is negative instead of every other term. The first series sums to log(2), the second sums to 1.5 log(2).)

Most importantly, the definition of the series sum as the limit does not assign a result to every sum -- if the limit exists, we say the series converges, but the limit doesn't exist for every series. A natural thing for a mathematician to do, when she has function definition that fails for some subset of possible arguments, is to try to find new definitions that extend the function to more arguments. Thus we have methods such as Cesaro summation, Abel summation, Borel summation, zeta function regularization and so forth... More in the Wikipedia article on divergent series.

If you want to say that the claim that we can sensibly and consistently extend the concept of summing an infinite series to 1 + 2 + 3 + 4 + ... and get -1/12 is "mathematical stupidity", then you would have to apply that expression to Euler and Ramanujan....

• No, getting -1/12 is not what's stupid. What's stupid are the unjustified tricks that Plaid quotes from the Numberphile video. There is no mathematical justification for the manipulations presented; infinite sums simply do not work the way finite sums do, especially with respect to reordering terms (commutativity) and regrouping sums (associativity). Even Césaro, Abel, Borel, zeta, and other methods agree on this fact.

• Aaron says:

First of all, it's not true that Cesaro sums don't support algebra. They don't support *all* algebra, but there is a subset of manipulations that works (it's linear, for example).

But, more importantly, math is fun. The fact that you get sensible answers out of these formal manipulations should be a sign that something cool is going on, not that someone's an idiot for even trying them. Why not explain what the cool thing is?

And, it's not like divergent series are the only place where someone took what looked like nonsense and turned it into a cool theory. Look at the umbral calculus, for example:

http://en.wikipedia.org/wiki/Umbral_calculus

• George says:

Thanks to the commenters for your lucid explanations. Mark, I love your blog, but sometimes your anger gets in the way of imparting truth.

• anon4cecAnon says:

Yeah mate, your posts seem good but you need to settle down a little. After all we're all interested in the same field right.

• antiquark2 says:

In your next post you should explain why the Riemann zeta(-1) == -1/12. This seems to be a case of mathematical nonsense entering the real world!

• Siggy says:

There are an interesting set of contradictions you can get if you assume that summing two series causes their cesaro sums to add. However, I'm not sure that's what's going on here. It's shown that S_3 - S_2 = 4*S_3, but from this you can only prove that S_3 = -S_2/3 if you assume that S_3 converges in the first place. S_3 clearly doesn't converge.

A clearer way to put it: In some number systems, you can show that ∞ + 1 = 2*∞. But you cannot derive from this ∞ = 1.

• blu28 says:

I have to agree with some of the other commenters here, the criticism of Phil is a bit harsh. Obviously under any normal set of algebraic rules the sum of the natural numbers must be positive and infinitely large. I think that Phil knows that. But a little research into the proof presented in the video shows that it is neither erroneous (as are most of the standard "1 = 0" proofs) nor meaningless. In some sense it is correct and meaningful, but obviously not by the standard rules of algebra that most people are used to. That makes it an interesting video, worth viewing and worth passing on as Phil did. Perhaps few people watching the video would be able to understand when it would be meaningful, but few actually need to.

• Dale says:

Yes it is erroneous. It is not well defined in the sense that you can use his techniques to show that $S_1$ is equal to any number you want. Above you will find a couple of "proofs" that $S_1 = 0$ as well as equalling 1.

• Casey Bowman says:

A point to make is that while the natural numbers have a Ramanujan Sum of -1/12, that is not exactly what was being claimed in the original post. It was implied (maybe by omission, but still implied) that the natural numbers have a Cesaro sum of -1/12, which is not true. The natural numbers don't have a Cesaro sum at all. The original post claims the natural numbers "sum" to -1/12. By leaving out the fact that it is a Ramanujan sum is incredibly misleading, and quite dangerous for young math students. Ramanujan sums require knowledge far outside of most people's experience in order to comprehend them, and reducing the entire argument down to some misleading algebra and a YouTube video is a disservice to anyone truly interested in learning math. It is also being presented as if it were magic, which only serves to confuse people more when they start to really delve into the issue.

Speaking as someone who teaches math at the college level (with a masters degree), the original post was irresponsible, and DOES NOT help the cause of educating the public in mathematics.

• Jim Vernon says:

I'm not a math major, but it was pretty obvious to see that his trick was to make the claim using a common definition of "sum" that most people understand (1+2 = 3) and then, in his proof, swap it with a less common definition that few people understand. It seems dishonest at best.

• Pierre says:

Myself not being very knowledgeable beyond everyday math (high school), I can still appreciate the video for some advanced knowledge that I can't quite grasp the mechanics, but as a twist of the mind, a window to a different understanding of the world and universe (of math I guess) and new possibilities. Just like a kid could be fascinated by algebra (letters can add to numbers?!?)

Besides, Plait repeats how this is an oversimplified way to express the calculation and that there are some more rigorous ways to get this conclusion. As a non-mathematician (because being "good" in math here has nothing to do with being good to most posters on this page), I can discern that what is presented to me is kept at a level where I can remain curious about this "mysterious math" rather than confused and bored. (try explaining politics to a five-year-old: Not interested beyond "knowing" what it's about...).

So yeah, watching the video, here's what I learned: with "advanced math", you can come up with a value of -1/12, and this number is important elsewhere in physics, notably string theory (wich I know exist because of Big Bang theory (whop!). How far have I been mislead today?

• Dennis Towne says:

You've been mislead in two ways. First is that the "sum" he ends up with is something very, very different from the additive sum he presents. The "sum" is in no way an "addition of all terms in the sequence".

The second is far more worrying: the math he uses to get there is actually wrong, at a fundamental level. It's not that this is "advanced knowledge that I can't quite grasp the mechanics [of]", it's that it's actually bad math. You'll never be able to grasp the mechanics of it, because it's actually wrong.

At best, in the future, you'll be able to understand the mechanics of why it's wrong, and why you should never do what he has done in that explanation. You'll never be able to grasp the mechanics of why it's right, because it isn't.

• coolstar says:

The author here gives the Very Bad Astronomer WAY too much credit. there's a reason many of us professional astronomers call him the VBA: he makes errors just as silly in elementary astronomy all the time, and rarely, if every admits to them.

• Infoholic says:

My mother used to tell me: "It's not what you say but how you say it that matters." Your post could have taken two approaches to this problem; (1) "Oh look at that stupid Phil Plait. He made a mistake that anyone who can correctly sum two and two should never make. Watch me show you how much cleverer I am than Phil Plait.", or ; (2) "Hey Phil Plait it appears that you have misunderstood this mathematical concept. Let me show you the correct interpretation and some examples that might help you understand it better."

Option 1 (the approach you took) makes you look like a points-scoring, grandstander, while option 2 (the approach I suggest you should have taken) would have made you look like a knowledgeable professional who is trying to politely share knowledge with the world.

• MarkCC says:

You know, I'd take this concern trolling more seriously, except for two little things...

I've been writing this blog for very nearly 8 years - I started in March of 2006. In those years, I've written over a thousand posts. Somewhere around half of them have been critical of someone for saying something stupid. And I've been really harsh with my criticism all along - much worse than this tiny little slap at Phil. I've called people crackpots, cranks, idiots, morons, assholes. Just this week, I once again accused one of the free energy crowd of not just being wrong, but of being a deliberate fraud! And not a single complaint to be heard for any of it. The only time I've seen an uproar like this is the one time I called Richard Dawkins an arrogant ass. Which leads me to a simple conclusion. It's not that I said something nasty. It's that I said something critical of Phil.

That leads to the second point. Phil was dead wrong in his post. But he was far, far worse than just wrong. He was stupidly wrong. There's no nice way to say that: he didn't just make a reasonable mistake. He made a gigantic, incredibly stupid mistake. He made a mistake that he could have avoided by reading Wikipedia fer crissakes. And he didn't just make a stupid mistake: he took that stupid mistake, and expanded on it at length, talking about what it meant, when he hadn't bothered to take five lousy minutes to figure out what it really meant.

He went in front of a huge audience, as a person who is widely respected as an authority, and he did something incredibly, arrogantly, profoundly, and avoidably stupid.

I can easily cite another example of a case where I was really hard on someone who did that. Me. I've been wrong on this blog plenty of times. One time in particular, I blew it bigtime. Huge. And I called myself a Bozo, a Moron, and an Idiot for it. Because that what I deserved for making that kind of error.

Phil earned some criticism. I wasn't horribly harsh with him. But I don't pull punches: when someone says something stupid, I'll call it stupid, no matter who they are.

If you don't like that, you're very welcome to not bother to read my blog. After all, no one is forcing you. I'm not making any money here; I don't need to trawl for pageviews. (Scientopia loses money, and it all comes out of my pocket. The few ads we show don't actually come close to covering our monthly expenses.) I write the way I write; it's what I've been doing for a while. If you like it, I'm delighted. If you don't, I'm sorry. But not sorry enough to change the way that I write. This is what I do for fun, so I'm going to keep on doing it in the way that I enjoy - and that means no false diplomacy. So deal with it, or don't.

• Jack says:

What's even worse is that Lawrence M. Krauss in a debate repeated the same stupidity when discussing infinities.

• Jack says:

Actually, Marc, you are being too kind by only calling Phil stupid. I would have called him much worse. Your anger is called "righteous anger" and anyone with a PhD that can't grasp simple math and uses obfuscation to dumb down his/her audience deserve nothing better. Phil is supposed to debunk stupid ideas, not perpetuate them!

• Sunday Afternoon says:

@Jack: You say, and I agree with you, "Phil is supposed to debunk stupid ideas, not perpetuate them!"

The same can be said for the authors of the video too. Personally, I hold them as more responsible. Waffling at the end along the lines of "infinity is weird" is a bullshit explanation. They give the impression of having done their homework, but clearly they did not.

• OMF says:

To begin with, S1 does not equal 1/2. S1 is a non-converging series. It doesn't converge to 1/2; it doesn't converge to anything. This isn't up for debate: it doesn't converge!

This is exactly what is up for debate. It is up for debate because, at the core of the entire argument, is the notion of the "convergence" of an infinite series. We say that the series 1+1/2+1/4+1/8 + ... = 2 is convergent, but since it is an infinite series the notion of convergence itself must first be defined.

Ordinarily, we define the notion of convergence of a series a_0+a_1+a_2+... by defining s_n = sum_0^n a_n, and saying that if lim_{n->inf} s_n =s, then the series is convergent and its sum is s.

What must first be realized is that this definition of convergence is itself an arbitrary one.

The important thing to note here is that we are not saying that the Cesaro sum is equal to the series. We're saying that there's a way of assigning a measure to the series.

That is exactly what we are saying here. The Cesaro sum can be used as an alternative way of defining the convergence/sum of an infinite series. It's as valid a method as the ordinary limit definition (which it may well disagree with).

The book you want to read here is "Divergent Series" by G. Hardy. It goes through the "machines" for summing infinite series.

What makes this worse is that it's obvious. There is no mechanism in real numbers by which addition of positive numbers can roll over into negative. It doesn't matter that infinity is involved: you can't following a monotonically increasing trend, and wind up with something smaller than your starting point.

Actually that's exactly what you can do in the extended complex plane. positive and negative infinity are identified with a single point +inf which can be passed through like any other. I don't know if the -1/12 result can be shown in this framework, but the notion of going towards +inf and ending up in the negative numbers can be understood even by looking at the graph of the function 1/(1-x).

• Dale says:

It's an arbitrary way to assign a value to the series; yes. But the algebra used is not well defined. The use of an offset means his system is in no sense a legitimate group operation, let alone an algebra.

• Erik says:

You're barking up the wrong tree by talking about Cesàro sums. These are all consistent values of the divergent series as Ramanujan sums and as Riemann sums. No one but you claimed they were Cesaro sums.

• JSintheStates says:

Thank you! I saw that article the other day and thought, that is what's wrong with the internet, and self-publishing! (And that's also what's wrong with the education system in the US that there are too many college graduates out there, including teachers, who can't follow the math or the arguments you present here!) My sincere compliments!!!

• Jack says:

OMF says "Actually that's exactly what you can do in the extended complex plane."
So what? Marc is talking about "real numbers" in that quote to which you are replying to.

• william e emba says:

In other words, you support Mark for making up things about what Plait was saying, and using his made up version as grounds for insulting Plait. One can show that Mark's reasoning does not generalize, and hence cannot be assumed to apply to Plait, by providing counterexamples. One would think readers of a math blog would understand such a basic form of argumentation.

Of course the statement "1 + 2 + 3 + ... = -1/12" is not a theorem of "classical mathematics" (Zermelo-Frankel set theory plus the axiom of choice?). Nobody said it was. Of course the sum does not converge as convergence is classically defined. Nobody said it did.

The contention in MarkCC's post seems to be that any mathematics that is not "classical mathematics" is "bad mathematics." As someone who wrote a PhD dissertation years ago working with nonstandard axiom systems, I don't have a lot of sympathy for that contention.

I am about as certain as certain can be that both Phil Plait and the guy in the original video understand that the sum doesn't converge classically. They didn't mention it, because classical convergence is irrelevant to the system in which they are working. Cesaro sums are also irrelevant.

Now I don't have a good enough grasp of the system in which they are working to pass informed judgment on its merits, and I'm certainly not qualified to comment on the purported applications in physics. But here's the thing: it is painfully obvious that the author of this post doesn't have a good enough grasp of that system in order to judge it either, and yet he is doing so, using arguments that are utterly irrelevant.

I see that Phil Plait is preparing a follow-up post to address the objections he has heard. At the very least, Mark owes Mr. Plait an apology for charging that Mr. Plait does not understand something that I'm quite sure he does understand--and it wouldn't surprise me one bit if, at the end of the day, Mark ends up owing him a much bigger apology than that.

• MarkCC says:

No: the use of non-classical mathematics while asserting that it is classical mathematics is bad math.

• David Starner says:

1 + 2 + 3 + ... = 3. It's a simple finite sum of all the integers; 1 + 2 = 3 + 3 = 0 + 4 = 4 + 5 = 3. Did I not mention that I was working in Z6? I could also say the sum of all the integers is -32768, which is true in 16-bit two's complement arithmetic. Yes, I would say that to pass either of those off as an answer to the sum 1 + 2 + 3 ... without explaining the rules that I was using is bad mathematics.

In any case, the math in the video is as valid as saying that 16 / 64 = 1/4 because you can cancel the 6's. It might get the right answer in some sense, but if you try it on any other problem, you'll get the wrong answer.

• Dale says:

And did you even bother to see if the associated axioms are even well defined?

• blueollie says:

Interesting. There is a wonder chapter on "alternative definitions of convergence of sums" in Knopp's book "Infinite sequences and sums" which you can buy from Dover at a very reasonable price. Sure, 1-1+1-1....doesn't converge in the usual sense, but Euler had no problem writing this as 1/2; this even makes sense if you use the standard "geometric series sum formula for 1 + x + x^2 + x^3.... = 1/(1-x) and plug in x = -1. Yeah, I know; that is outside of the interval of convergence.

• vortexkd says:

hm. what about the other proof? (there's another video)
I'll confess i can't see what's wrong with this one.
S = 1+1-1+1-....
1-S = 1-(1+1-1+1-...)
1-S = 1-1+1-1+1-....
1-S=S
S=1/2

• Sizik says:

Well first of all, the series is 1-1+1-1... (alternating between 1 and 0), not 1+1-1+1... (alternating between 1 and 2)
Secondly, 1-(1+1-1+1-...) = 1-1-1+1-1+1-..., not 1-1+1-1+1-...
Thirdly, 1+1-1+1-... != 1-1+1-1+...

• vortexkd says:

whoops, sorry about that, i meant 1-1+1-1...

• Because between the first and the second lines you're using the associative rule that you're familiar with from finite sums as if it's appropriate for divergent infinite sums. Then you use the distributive rule likewise.

• vortexkd says:

Ah, thanks, i get it now ^^

• Tualha says:

In the same vein, here is a pseudo-proof that all triangles are equilateral:

http://www.mathematik.com/Isoscele/

First saw it in a story in Analog, "Paper Virus".

• Rahul Raj says:

The Maths was bad on Phil's link, but you are missing the point as well. The 'error' is not in ascribing the Cesaro sum to Grandi's series, it is muck later in ignoring the pwoer of Zero. I was looking for a good rebuttal, and ended up starting a page for this: http://rahulraj-says.blogspot.in/2014/01/the-power-of-zero.html

• Rahul Raj says:

The Maths was bad on Phil's link, but you are missing the point as well. The 'error' is not in ascribing the Cesaro sum to Grandi's series, it is much later in ignoring the power of Zero. I was looking for a good rebuttal, and ended up starting a page for this: http://rahulraj-says.blogspot.in/2014/01/the-power-of-zero.html (edit:typo)

• Roger Crew says:

1 + 2 + 4 + 8 + 16 + 32 + ... = -1

just in case you were wondering why twos-complement arithmetic works the way it does.

the 2-adic metric being the one where you treat the *rightmost* bits in the binary expansion as being the most significant rather than the leftmost bits as we're normally used to doing. That is, large powers of 2 are treated as insigificant as compared to the question of evenness vs. oddness, and having 1/2's and 1/4's in there matters even more, etc...

... in which case the above actually *is* a converging series.

To be fair to Mark, his statement "There is no mechanism in real numbers..." remains
correct since we're decidedly outside the realm of real numbers here (the 2-adics
contain all kinds of oddities, e.g., -1 and 2 don't have square roots but -7 does...).

But there is lots of weird shit out there.

• Brendan H says:

Chew on this:

http://en.wikipedia.org/wiki/Ramanujan_summation

• Rahul Raj says:

Wiki info is not the gospel truth. Also, even in the reference cited, Ramanujan has not shared any proof or derivation of the summation. Rare mistake from the great man possibly.

The maths behind this piece is not that complicated, non Maths majors might appreciate the link I posted earlier : rahulraj-says.blogspot.in/2014/01/the-power-of-zero.html

• Ramanujan didn't write down proofs or derivations for most of his ideas. The manuscript he sent to Hardy had none, as I recall; Hardy thought they must have been nonsense until he spot-checked a few and found they turned out to be correct. The proofs came later.

• Nullius in Verba says:

It kinda reminds me of those people who, on being told that the square root of minus one is something called "i", scream until they are blue in the face that "Negative numbers don't have square roots!!! What kind of moron are you?!! And what sort of 'number' is 'i'? Imaginary?! Imaginary!?!!! You even admit it!!! You're making it up!!! You can't go around making up imaginary mathematics!! Don't you realise the inconsistencies that would result in?! Bad, BAD mathematics!!!"

And then on being told that of course negative numbers don't have square roots among the real numbers; the complex numbers are an extension, they calmly say "Well, in that case, of course. Why didn't you say so to begin with?"

Incidentally, if you want a more 'intuitive' way to see how a sum of positive numbers could 'roll over' to become a negative one, there's a playful bit of mathematics in which you treat the decimal representation of an integer as the sum of a series of digits times powers of ten, and then ask what happens if you allow infinite sums in *both* directions?

The semi-infinite 'integer' in which every digit is a '9' ends ...999999999. What happens if you add 1 to it? Well, each digit turns into a zero and is carried to the left. So you end up with every digit being zero, which of course is zero, and means that the number we started with must be -1. You can of course extend this to numbers like ...999999997 which is the same as -3, and the repeating sequence ...857143857143857143 which is equal to 1/7. (Just try multiplying it by 7 and see what you get...) Try it! It's fun!

These things are related to a very important branch of number theory called the p-adic numbers (although in this case they're not quite p-adic, since 10 is not a prime), in which you redefine the absolute value to get an alternative definition of convergence. These 'infinite integers' do actually converge when considered as p-adic numbers, but not under the normal valuation.

There are more things in mathematics, Horatio, than are dreamed of in your philosophy. I think it's nice that people make the effort to introduce the public to some of the more interesting bits, and if they do so with a bit of mischievous mystification by leaving some of the caveats out, well at least that gets people arguing and thinking about it.

• Nullius in Verba says:

Sorry, slight typo there. The expansion of 1/7 is ...857142857142857143 with only the last digit being a 3.

• MarkCC says:

If you look at the archives of this blog, you'll find that I did a bunch of posts on the P-adics. They're fascinating and strange. But the thing is, if you present a conclusion from the P-adics as if it applies to the real numbers, you're an idiot. The P-adics aren't the real numbers, and if you want to talk about them, you need to be clear that that's what you're talking about.

• Nullius in Verba says:

How about if you present a result from the p-adics (or complex numbers) without specifying what number system you're using? Is it OK to assume they're Real without being told?

• MarkCC says:

Yes. It's all about clarity and precision. If you state a result that talks about a real sum of real numbers, and you don't bother to be clear about the fact that you're not talking about real numbers, then you don't get to complain when someone calls you an idiot. Because any reasonable reader, confronted with something like the sum of a series of real numbers being a specific real number is going to assume that - surprise! - you were talking about real numbers.

And this case isn't even that simple. It is not the case that the simple sum of the set of natural numbers considered as a subset of the complex numbers, performed in the complex number system, produce -1/16th.

• David Starner says:

If we can't assume the reals, then 1 + 2 + 3 + 4 ... = -1/12 means nothing. It's not surprising or worth noting; we could have simply defined + (a, b) to equal -1/12. We could have chosen pi, or 42 or anything.

Put another way, if I say I'm going to give you a gift, and hand you a bottle of bourbon, is it my fault or yours if you take a shot? I told you it was poison; you just assumed for some reason that when I said gift, I was using the English word and not the German.

• william e emba says:

We know 1+2+3+4+...=-1/12 means something. It's a useful identify for various applications. Which means there's a mistake in your logic here.

The challenge is to find a natural mathematical domain where this sum makes perfect sense in a way that is very close to the intuitive sense. Do it right, and you'll probably solve string theory and the Riemann hypothesis. My guess is categorization and K-theory and probably the Langlands programme to boot.

• Alex Palazzo says:

My understanding is that the original numberphile video is indeed based on Euler's proof and that Euler was wrong. However, if you use the Riemann zeta function to derive the answer, at some point there is an integration operation used and that this operation spans the complex plain ... thus you are actually finding the sum of all the positive integers OVER THE COMPLEX PLAIN, and for this the answer is indeed -1/12.

• MarkCC says:

It's not quite finding the sum over the complex plane, but that's a whole lot closer than what the original video said.

The fundamental point of math - of all of the formalisms and proofs, and all of the elaborate constructions - is precision. Math is about building abstractions and understanding what they mean.

When you talk about natural numbers and real numbers, you're talking about natural and real numbers. In the real number system, the positive integers do not add up to -1/12. That can only happen in a fundamentally different mathematical domain.

Making the distinction between the different mathematical domains isn't nitpicking - it's crucial.

If I'm talking about the natural numbers, then 1/2 doesn't exist, and I can't use it. In the natural numbers, subtraction isn't a closed operation, and addition doesn't have a closed inverse. If I present a proof about something in the natural numbers, and it relies on subtraction being closed, it's wrong. It doesn't matter that if I was talking about the integers, it would be correct.

If I'm working with the real numbers, then the square root of -1 doesn't exist. It doesn't matter that it exists in the complex number system: it doesn't exist in the reals.

In a logical domain containing pure real number theory, there is a complete logic. But Gödel proved that logic is incomplete. He also proved that it's complete in real number theory. There's no contradiction there: they're different domains. Incompleteness applies to any logical formalism that can express Peano arithmetic and first order predicate logic. Real number theory doesn't include Peano arithmetic, and therefore incompleteness doesn't apply.

The video, and Phil's extended explanation of it weren't talking about complex numbers. Claiming that he's correct about that sum because it works in a particular construction over the complex numbers is as valid as talking about natural numbers, and then claiming that mathematical logic is complete, because there's a complete real number theory.

• iyba says:

Here's my (admittedly naive) problem: when a sum such as 1+2+3+... arises, how does one determine what domain to consider the problem in? That, it seems to me, is the crux of the problem vis-a-vis any physical application or result.

• Martin Argerami says:

But "infinite addition of reals" does not belong to the reals. There are several ways to attach a meaning to a series of reals; you may like one method more than another, but it would still be arbitrary.

• iyba says:

Fair enough. But a physical result can't rely on a matter of taste. So then am I to infer that any regularization method must yield the same result in order to have physical implications? Otherwise, my question remains: how, as an engineer, to choose a method when confronted with a divergent series?

• MarkCC says:

By actually understanding the domain in which you work.

These "solutions" aren't just random noise that some jerk pulled out of his behind. They're actual solutions to problems that someone wanted to solve.

If you're an engineer working on a problem, and you expect to be successful, you have to actually understand the domain that you're working in, and what the correct solutions for that domain are.

The whole -1/12 nonsense comes because someone misunderstood the difference between a domain-specific shorthand for a complex solution, and absolute universal reality.

-1/12 is not really the sum of the natural numbers. But in string theory, people are doing a particular kind of work that involves an analytical continuation of a function that, for some inputs in its domain, produces the natural number as a divergent series; but in the analytical continuation, there are values outside the domain of the series equation, and at the place where the series produces the natural numbers (and thus diverges), the analytical continuation doesn't diverge; it produces a nice simple value like -1/12.

Of course, that's a hell of a mouthful to keep repeating. So they say: "in this domain, we say that the definition of this function is this series, and at -1, it's got the value -1/12".

• Rahul Raj says:

Can you point me to a derivation of the Riemann Zeta function giving -1/12?

• MarkCC says:

Give me a couple of days. I'm working on a post about the Reimann Zeta, but this is not my field of expertise; in fact, it's pretty much the area of math that I'm worst at, so it's not something I can throw together quickly.

• I, for one, am looking forward to it. I know the idea of analytically continuing around a pole, and that the result is uniquely defined, but I haven't seen this particular example worked out.

• william e emba says:

There was a link to one in Phil Plait's video.

• iyba says:

In the few examples of this sort that I've perused, I've noticed the following.

In heat-kernel methods, there seems to be a leading divergent term (for Casimir it's $frac1{epsilon}^2$); one then makes a physical argument for ignoring this divergence, leaving an $epsilon$-independent term which, as $epsilonrightarrow 0$, is claimed to be the value of the series S.

In Euler arithmetic-manipulation-of-infinite-sums methods, the difference between multiples of S always seems to occur, which has the effect of subtracting off a leading divergent term, leaving the same finite result. Aside from the dodgy-ness of taking the difference of divergent quantities, these two examples seem to indicate that the series S does, in fact, diverge -- but that the order $epsilon^0$ term is what we're claiming to be the sum.

How far off am I?

• william e emba says:

The heat-kernel method is discussed briefly in the [CABK] reference I gave, for motivation. The book goes on to rely on zeta-function methods. As a reminder, they are actually concerned with regularizing infinite products, like infinity-factorial, and each product is associated with its own zeta-function. It might be in Berline, Getzler, Vergne Heat Kernels and Dirac Operators (Springer, 1992), but it's been too long since I looked.

The Euler-style method is discussed by Tao in the blog page someone linked to. (And reprinted in one of his books.) Euler himself, when he proved 1+2+3+...=-1/12, used the zeta-function. Tao points out how an example of inconsistent arithmetic (subtracting a divergent series from itself to prove 0=1) becomes consistent when the error terms are written explicitly. Tao also shows this method is usually equivalent to the zeta-function method.

• iyba says:

Well if the difference between divergent quantities can be made consistent, then this would seem to be in concert with a leading divergent term in heat kernel methods -- and that the quoted sum (e.g., the infamous -1/12) is the remaining finite term. (Much the same way that the Euler-Mascaroni constant emerges from the harmonic series by subtracting off the logarithmic divergence.) But I'll go look up the Tao article.

• DrWm says:

Surely the line "So 2S2 = S1; therefore S2 = S1=2 = 1/4." is supposed to read "So 2S2 = S1; therefore S2 = [S1]/2 = 1/4." Clearly S2 cannot = 1 as stated, under any wild assumptions about the series involved.

• pmm says:

-1/12 is .083333 repeat. Isn't that infinity going in the small direction?
If they both reach it, aren't they the same , not in value but same as is dozen of this or that? maybe this relationship is not value but factor.

• Qihong Bao says:

Very informative! Thanks for that.

Just wanted to chime in and say that I came across the same assertion in an E/M physics book from back in University while studying for an engineering degree.

It did make me do a double-take, but I accepted it as mathematics voodoo.The assertion does seem to have uses in applied fields, though.

• Thanks for the post. What you've done, that impresses me most, is you've not hidden behind any bullshit. You've not said "here is a step I can't show you because it has a funny name and lots if symbols and you won't understand it, so accept it's true because I know this stuff". And everything you argue is very easy to understand and very clear. That there is a measure rather than a value makes sense. You have educated rather than shown off. Bravo, and thanks.

Whether fairly or not, the original Bad Astronomer piece greatly strengthened my gut instinct that string theory is almost certainly complete bullshit. At best some interesting techniques which are worth working on before new results come along to guide Occam's razor in its brutal prune of the theory of everything search tree. But mostly, hopelessly speculative phantasy.

• william e emba says:

All in all, I have to agree that the reaction to Plait is way way overdone, hysterical and not too intelligent. In brief, it has been downright stupid and embarrassing, with the half-educated posing as if they actually knew what they were talking about, inventing excuses to cover up their initial jackassery.

I refer to [CABK]: Soulé, Abramovich, Burnol, & Kramer, Lectures on Arakelov Geometry, Cambridge Studies in Advanced Mathematics 33 (1992). As Soulé writes in the book's foreward, the book is a fleshed-out version of the notes to a 1989 advanced graduate course he gave at the Harvard Mathematics Department. In chapter V, [CABK] proves that 1·2·3····=∞!=√(2π) by various methods, as a warmup to taking determinants of Laplacian operators.

The authors ask what is 2·3·5·7·11····? This was answered by E. Muñoz-García, R. Pérez Marco, "The Product Over All Primes is 4π²", Communications in Mathematical Physics 12/2007; 277(1):69-81.

Mark and others are, quite simply, working with a pathetic, ignorant view of mathematics. We are not constrained to working within a known, (believed-to-be-)consistent system. Taking leaps into unjustified formal calculations is the lifeblood of discovery. It's what Witten got his Fields medal for! I'd love to have students excited by Plait's calculation. I'd love even more to have one that finds the space in which the calculation is rigorous. That space is going to be infinitely more interesting than -1/12.

What we all know--and frankly, it's boring as hell to point it out--is that this space is not the reals with the standard topology. It's the real clueless idiots you think pointing this out is important, and worse, grounds for showing off their own minimal understanding of something, no matter how trivial, in mathematics.

I would, frankly, be very positively impressed if someone came up with derivations of the above infinite products using fairly trivial methods, a la Plait and -1/12. (My reaction, for the record, was wow, I only knew the zeta-function renormalization method, the kind I learned in String Theory 101.)

• I would, frankly, be very positively impressed if someone came up with derivations of the above infinite products using fairly trivial methods

And this is exactly the problem. Even Witten didn't take his formalisms as "correct" -- as anything other than suggestive tricks -- until someone came along and provided a rigorous framework within which they made actual sense. In fact, this program is ongoing, largely under the rubric of topological quantum field theory and Khovanov homology. Witten only won after early results from Reshetikhin and Turaev started trickling out, and it was more for his supergravity proof of the positive energy theorem anyway. Cf: people thinking Einstein won the Nobel Prize for relativity instead of for the photoelectric effect.

The problem is that these "fairly trivial methods" are not derivations. The zeta-function method is the actual rigorous derivation, but the producers of the video are willing to present the unjustified formalism as correct without pointing to the fact that they're leaving out a huge part of the story; Phil Plait is willing to reinforce this without disclaimer; and you, and many like you, are willing to swallow it without remembering that -- like Witten's results -- it's merely a suggestive trick until you back it up with a rigorous formalism.

• william e emba says:

It's not a problem. No one went around calling Witten stupid.

The Fields Medal for Witten was simply for his overwhelmingly brilliant use of mathematical physics over and over again. More than one of my friends were shocked at the time: "but he didn't prooove anything". (One of them changed his mind after the Seiberg-Witten equations. Again, he didn't prooove anything, but they were just so over the top significant.)

I repeat. This is not a problem.

Zeta-function regularization is not a derivation either. It is "simply" a far far superior technique to naive algebra. We still know very little about a wide range of problems that naively can be solved using naturally divergent sums. Similarly, the rigorous justification of much of Witten's work is non-existent. For example, only some of his knot invariants--the simplest ones--have rigorous TQFT proofs.

I am not interested in whether Plait was "misleading" or not. The bottom line is that 1+2+3+...=-1/12 is useful, and heuristics are always welcome. I have not "swallowed" anything. People take the geometry of the "field with one element" seriously because it is way way suggestive. People take "primes are random" seriously because it is way way suggestive. There is nothing wrong with merely suggestive.

Real mathematical research in general consists of nonsense shortcuts, flawed analogies, bogus definitions that time and again guide our puny brains to important results. If you and Mark and friends had been dominating physics departments this past century, we'd be 50 years behind where we are today.

PS: Einstein won for "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect", weasel words (chosen because there was one antirelativity idiot) which cover everything, including Brownian motion and relativity.

• David Starner says:

"Real mathematical research in general ... If you and Mark and friends had been dominating physics departments this past century"

I do not see what the connection between those two are. Had this video been discussing string theory, far be it for me to intrude into the subject. When it's discussing mathematics, that's a different question.

The video made a claim that held under rules that most people wouldn't be familiar with, and "proved" it with a completely invalid proof. I can do better then that; I can make the claim that 1 + 2 + 3 ... = -32768 under rules that most people wouldn't be familiar with, and yet prove it with a completely valid proof. Of course, viewers wouldn't be deceived into believing that these were the normal integers working in normal ways, like this video did.

• william e emba says:

The connection is that most physicists need to know a lot of mathematics, and part of the reason they know a lot of mathematics is because they actually liked it along the way for all sorts of reasons, and not because it was forced on them. And if part of what they liked along the way was all sorts of illegitimate calculations, which no one like you and Mark and all were present at to "rescue" them, I say fine and dandy. Large portions of 20th century physics has depended on people who are experts at illegitimate calculations.

For the record, 1+2+3+...=-1/12 is at the root of string theory. It shows up in the proof that the quantum bosonic string can only live in 26 dimensions.

That proof, of course, is not rigorous. Almost nothing is rigorous in string theory. Yet the rigorously proven mathematical takeaway, from deep results in enumerative algebraic geometry to first results in mirror symmetry are breathtaking.

Why people think it would have been a good idea to stomp this out from the beginning, and are trying their hardest to strangle analogous breakthroughs in the crib, is beyond me.

Meanwhile, the exact same lack of rigor is present throughout research mathematics, where it is normally carefully phrased (at least in the more pure domains).

• David Starner says:

I would like to see a comparable examples from the physics side. Like, would physicists be fine with videos claiming that it was okay to dip your hand in hot lead, along with faked videos showing people holding their hands in hot lead for a long time? What lies will physicists be okay with us telling about physics?

People should not be tricked by facile lies in the name of the greater good. We should not be applauding TV shows on alien influences in historical building for driving people's interests in history. We ask that when they claim to be doing nonfiction, they do nonfiction, they teach the truth, not some gussed-up version that mixes half-truths with falsehoods. That's what people deserve, the truth.

• Replying to emba, but too deeply nested:

The ultimate arbiter of truth in physics is agreement with experiment. Though this still leaves string theory up shit creek, the path integral formalism proved itself in making predictions before it was (in some cases) mathematicially justified.

But we're not talking about physics; the video glibly presented this as a mathematical result, and the ultimate arbiter of truth in mathematics is rigor. This is, as presented, not rigorous.

Even when good physicists like Feynman do bend the rules, they're very clear that their formalisms are not yet justified. That is: they're honest; Numberphile and Plait have not been.

• william e emba says:

You mention it would be nice to have a comparable example from the physics side, and then offer up something that is so obviously incomparable, you're just a cheat who has no argument and pretty much knows he has no argument. We all know that there is zero harm in having an intuitive non-rigorous understanding of 1+2+3+...=-1/12.

A closer example would be somebody citing the Tacoma Bridge collapse as due to resonance. In fact, for decades the textbooks all said this, quite gleefully in fact, since it comes with an amazing video, until someone studied it more deeply and realized it was due to flutter. Have there been any disasters caused by a few generations of engineers getting the wrong news? No.

I have several times over my years on-line seen physicists flame certain derivations and explanations of E=mc^2, pretty much self-justified in the style of Mark and others here. What they didn't know was they were essentially flaming Einstein and Wheeler. And what we're seeing here is pretty much the same thing: people flaming Newton and Euler.

• william e emba says:

My education in mathematics and physics was filled with lies and lies along the way. For example, I taught myself calculus in middle school from a 1920s textbook--it was older than my father--and it did not in any way shape or form hold me back. Quite the contrary, it was a supercharged kick in the pants. Sort of like discovering Dr Seuss after years of Dick and Jane.

Over time, I had to relearn probably every single page of that book. Was that a problem? No. Would it have been good to have somebody knowing the "truth" slap that archaic freewheeling adventure out of my hands and force me to stick to the modern professional treatments? No.

Really, lying is an essential part of good teaching. All good teachers know this. The bad ones just stick to a script, guaranteed safe. And boring. And totally unrelated to how people actually think.

• David Starner says:

A good teacher may simplify, but never lies. You don't tell someone that people can fly and then tell them that they do it by flapping giant mechanical wings. There are ways to get this answer, but the way they showed could get you any answer you wanted to the problem. This is not an intuitive way of understanding the problem; the way they showed to solve the problem gives you any answer you want.

Is it problematic that bridge-builders don't understand why bridges fall down? I'm going with yes. Is it good that people have a false picture of why resonance happens? Why?

Instead of lying about 1 + 2 + 3 ..., we should just advertise that mathematics will get you rich and laid. Since it doesn't matter the truth, just that it attracts people, right?

• william e emba says:

Actually, string theory has had numerous mathematical consequences. There are numerous reasons a great number of mathematicians are jumping in, despite the tidal wave of hand-waving lack of rigor.

Your description of string theory as being out of luck experimentally is popular, but not accurate. In the limit, string theory reproduces both quantum field theory and general relativity. That was one of the reasons it caught everyone's attention so quickly: it automatically included gravity.

And string theory has been one of the greatest triumphs of "experiments" in mathematical history. Amazing, bold conjectures and spin-off research programmes have come out of string theory. We've mentioned Witten's supersymmetry proof of the positive mass conjecture. There was Kontsevich's integral, and his use of it to resolve ancient enumeration problems in algebraic geometry. There was the construction of the moonshine module for the Monster group, using string-theoretic vertex operator algebra, and Borcherd's use of to proof the moonshine conjectures. There is the on-going development of rigorous mirror symmetry, as physicist-proved conjectures are slowly being replaced with mathematical rigor.

it's a good thing for mathematics that no one engaged in this Mark-style bashing at string theory's birth. Well, probably there were such people, but they're not exactly bragging about it. But we see people here who are talking smack about how it would have been a good thing to strangle all this in the crib.

• william e emba says:

Feynman did not "bend" the rules. He used mathematics, or perhaps I should call that "mathematics", as was necessary, no more, no less. As a result, he thought most mathematical research was just plain silly, and the idea of rigor a waste of time. Once, when someone in the early days of QFT announced in a Feynman seminar that he knew how to do Feynman's work "rigorously", Feynman flipped the fellow off.

I won't say I'm with Feynman, but I'm certainly not against Feynman.

For the record, I'm pretty certain I learned calculus from the same "lying" text he did, and while I later went on to learn the "truth" and he did not, I have no doubt whose career was more spectacular. lies and all.

• william e emba says:

Guess what? Simplification is a lie. Deal with it, instead of going off--again--with the extreme examples of lying as if that proves anything. Hint: to refute a statement of the form "some lies are acceptable", you do not come up with a statement of the form "some lies are unacceptable"

And you "guess" that the decades-long textbook lie about the Tacoma Narrows Bridge collapse has been "problematic"? That's it. Has it in fact caused trouble? Or has it gotten more people into engineering in the first place?

By the way, let's see you do this: 1 + 2 + 3 ... = -32768. Be as simple and intuitive as Plait.

By the way, is there someone reason you and Mark and others lie so much? Plait is quite explicit that there's something really weird going on, not "normal ways" as you lie above.

• David Starner says:

Accusing us of lying sort of loses its bite when you're endorsing lying.

1 - 1 + 1 - 1... = 1/2 is not intuitive. The partial sums can be checked by hand; they're 1, 0, 1, 0 ... It's obvious to even the most inexperienced that that will extend out forever and never equal or approach 1/2. I would have been confused and annoyed upon reading "Mathematics and the Imagination" in junior high to find that claim, and deeply offended to have found that what I was shown was a fake proof.

You can draw people in with the deep simple truths of Cantor's proof, and you can draw them in with the amazing complexities of the Four Color Theorem. But if you try and draw them in by imply the second is trivial through trickery, they will be rightfully frustrated.

• william e emba says:

Yet another reply to David Starner.

Now that I've bothered to see the video, I can tell you that your harping on wanting to see something analogous in physics is provably ridiculous. Plait makes it absolutely clear that he is relying on what his physics friends tell him, and even includes a cameo of Polchinski String Theory (a standard textbook), both the front cover and the page where 1+2+3+4+...=-1/12 is put to work. And having said repeatedly that the result just seems impossibly absurd, its appearance in physics is what Plait says convinces him it must be true.

• william e emba says:

Pointing out that those who are having silly little fits over (alleged) lies have been engaged in blatant lying simply reveals who is a scumbag hypocrite.

As it is, you continually lie, quite deliberately by this point, regarding the fact that I've defended some lies. I have not defended the kind of lies you and Mark engage in, nor the ones you've pointed out as obviously atrocious lies.

• william e emba says:

Addendum to an earlier comment of mine.

Kontsevich also used mirror symmetry in his celebrated solution to classical problems in enumerative algebraic geometry. More precisely, he extracted something totally rigorous that was only found by diving into the endless piles of "illegitimate" mirror symmetry theorizing.

Mirror symmetry is especially amusing in the context of this thread. Numerous string-theory friendly mathematicians, used to making rigorous a few bits and pieces of their physicist friends' whimsies, or at least expressing them as honest conjectures, were initially disbelieving the claims of mirror symmetry. "What, there's a pairing between Calabi-Yau manifolds we haven't heard about? Ridiculous!"

Of course, once the evidence grew too strong to ignore, the relevant mathematicians swallowed their pride, admitted their kneejerk reactions were incorrect, and joined the party. Some of them have even been caught trying to channel Euler.

• MarkCC wrote (Jan 17 2014):
> Using the equality of an infinite series with its Cesaro sum, I can prove that 0=1 [...]

Sure; since that would be incorrectly attributing some definite value to the "infinite series" as such, i.e. in distinction to "its Cesaro sum"

(or to "its sum" by any of various other "definite summation techniques").

Doing so, and cheerfully arguing along the lines by which
"S3 = -1/12"
was derived in the article, a "proof of 0 = 1" proceeds for instance as follows:

[1+1+1+1+1+1+1+...] +
[1-1+1-1+1-1+1-...]
===================
[2+0+2+0+2+...] =?= [2+2+2+...] =?= 2 * [1+1+1+...].

"Subtracting [1+1+1+...] on both sides" yields (*):

[1-1+1-1+1-1+1-...] =?= [1+1+1+1+...].

Also:

[1+1+1+1+1+1+1+...] +
[0+1-1+1-1+1-1+...]
===================
[1+2+0+2+0+2+...] =?= 1 + [2+0+2+0+2...] =?= 1 + 2 * [1+1+1+1+...].

"Subtracting [1+1+1+...] on both sides" yields:

[0+1-1+1-1+1-1+...] =?= 1 + [1+1+1+1+...], and "thus"

0 + [1-1+1-1+1-...] =?= 1 + [1+1+1+1+...].

Now substituting (*):

0 + [1+1+1+1+...] =?= 1 + [1+1+1+1+...].

Finally "subtracting [1+1+1+...] on both sides" yields:

0 =?= 1,

i.e. a contradiction which indicates that arguments along those lines are incorrect.

(Disclaimer: it does not therefore follow that arguments made by using the indicated "definite summation techniques" were always and necessarily correct, either.)

p.s.
The article still has the apparent typo, "So 2S2 = S1; therefore S2 = S1=2 = 1/4.", which was noted by DrWm (January 19, 2014 at 10:14 am) already.

• Frank Wappler wrote (
> a "proof of 0 = 1" proceeds for instance as follows:

Oh, well -- that was just a clunky (bad! 😉 version of

0 = 0 + 0 =?=

0 + ([1+1+1+...] - [1+1+1+...]) =?=

0 + (1 + [1+1+1+...] - [1+1+1+...]) =?=

0 + 1 + ([1+1+1+...] - [1+1+1+...]) =?=

0 + 1 + 0 = 1.

• william e emba says:

See the Tao blog page linked to elsewhere in the comments. He provides a "smoothed version with error terms" of these divergent sums. The upshot is that the various manipulations, both Plait's and yours, all work consistently.

• Chris L says:

// begin rant

I am a graduate mathematics student, and have to say I agree exactly with what the author wrote in this article. They clearly should have said this sum has a cesario value of 1/2, but them failing to do so implies its true for traditional infinite series. The problem is that tons of people will go around now assuming this is true because there is a youtube video going around from some "math guys" that propogates this. Can we just leave the math to the mathematicians? I mean either A. they knew this was BS but wanted to show an exciting result, or B. they don't understand even basic college level analysis. Either situation is unacceptable considering they feel that they know enough to produce math videos.

//end rant

• william e emba says:

Good news. Tons of people will not be going around assuming anything. Really.

The problem I'm concerned about is people like you who are going to stomp on people who get a little bit interested in math. Jerk.

Perhaps you should take your own brand of advice, and leave English to the English majors? I mean, do you really think it is unacceptable that there are people who write such monstrosities as "propogate"? Or "cesario" even? Sheesh.

• Chris L says:

Are you serious? First, I'm not producing videos on English, and I apologize, I was/am writing from a cell phone in a hurry. Wasn't expecting to be critiqued on my grammar. Second, there are people that believe this nonsense, as I had to fight the uphill battle of explaining what the author did to some students already. Third, it seems that anyone with any real mathematical training shares the same sentiment as I do. I am all about creating interest in math but not at the cost of throwing rigor and precision out the window. Yes, I'm the jerk. Good day

• william e emba says:

I have real mathematical training--several decades more than you--and I do not share anything close to your sentiment. I gave sources regarding infinity-factorial and infinity-primorial, these authors obviously disagree with you. It seems proven that you like to talk about things you really don't know anything about.

Why were you bothering to fight such an uphill battle? Perhaps you should share with us your refutation of infinity-factorial and infinity-primorial? I mean, if you think it's so important? Perhaps you can stomp out the very concept of infinite-dimensional determinants?

• MarkCC says:

I don't understand this attitude.

Why is it better to invest time and effort into teaching people that math is mysterious, incomprehensible, and that there's no point trying to understand it: just accept what the experts say, no matter how crazy it sounds?

It seems to me that that's what Phil's post really does. It takes a result form a specialized domain, presents it without any explanation beyond "look at this! it'll blow you mind, because math is weird and you can't hope to understand it!"

But somehow, pointing out that he's full of shit, that he got it wrong because he didn't bother to try to understand it, and that he could have understood it and explained it if he invested five lousy minutes reading freaking wikipedia fer cressakes - that's going to drive people away from trying to understand math?

• Rahul Raj says:

Exactly! That is what is required, not trying to turn math into some arcane mumbo jumbo that scares people (there *are* some scary places though, where I would never tread).

People would appreciate Math better if they found it more relatable, and not some alien artifact that changes the rules of the game on one whim.

• william e emba says:

Your response to Plait now boils down to what it seems to you he was doing? Sheesh, you're as bottom of the barrel scum pathetic as Rush Limbaugh, always knowing what's really going on in the minds of his targets.

How about putting a deliberately positive spin on his video instead of a deliberately negative spin? I, personally, can't stand 99% of all popularizations in math and science. But I normally don't criticize them, because it was exactly such popularizations--of the sort that I would never look at nowadays--that got me (and thousands of others) interested in math and science in the first place. Where the hell do you think the next generation of mathematicians and scientists are going to come from? From people whose eyes are carefuly shielded, or from people who are fired up?

I like to have students who are fired up. Even if it's over 1+2+3+...=-1/12 and an impossible proof. As I mentioned, best of all would be to find the space where the proof is in fact legitimate, just like we know how to rigorously prove 1+2+4+8+...=-1.

And extreme example is provided by Stephen Jay Gould _Wonderful Life_. On a technical level, the book was riddled with errors. Not just Gould backing wrong theories later disproven, but things peer-review should have killed in the first place. Yet Gould did far more than all the researchers actually studying the early Cambrian to convince people this stuff was great and amazing and important.

• […] shared on Facebook. Happily I discovered the root on Saturday and some of the criticism by both computer scientists and biologists along with string theorist defenders of the result. Fortunately, the defenders […]

• David Davis says:

"It seems to me that that's what Phil's post really does. It takes a result form a specialized domain, presents it without any explanation beyond "look at this! it'll blow you mind, because math is weird and you can't hope to understand it!"

Um, no, that's really not his post did... Look, speaking as a non-math guy, it seemed really obvious from the video that you didn't/couldn't actually sum up the series, but there were really interesting ways of working with that series that involved using an approximation that curiously turned out to be a -1/12, and that factor had implications in a number of diverse areas of study. So the take-away was that there are really interesting ramifications in even an apparently simple series, which is way cool. Sure they blew the explanation of why this might be so, and frankly I didn't find it particularly compelling, but it did lead me out to look up more info, including analytic continuation, which was fascinating. Incidentally, Terry Tao's post on the subject (http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/) was very informative and intriguing.

And of course it led me to this site and the comments/discussion...

“But somehow, pointing out that he's full of shit … that's going to drive people away from trying to understand math?”

Well, yeah, sorry to tell you, but speaking as one of the great math unwashed (and a member of that set supposedly that is terrified of that scary math) my take away from this site is that if someone like Phil, who in a goofy, ham-fisted, over-enthusiastic but essentially well-meaning way posts something he thinks is a cool result, and actually is a real outcome from a particular approach to working with the series, but he gets it somewhat to farcically, avoidably, wrong, the collected wrath of the math internet will jump. your. shit. because MATH! bitches.

I’m not sure a better message to ‘golly, math is cool’ is the message that: math is a difficult and rigorous subject that punishes even well-meaning error by calling the perpetrator stupid and full of shit.

Yikes.

Hard to believe I know that that might be off-putting to people interested in the subject... 🙂

Not looking to be an apologist for Phil, I don’t read him often, I just feel that the dire consequences, of his rather silly post, to the sanctity of math and its effect on the spread of ignorance to persons such as myself, are a tad patronizing and arrogant.

Entertaining discussion though, so thanks. Seriously.

• MarkCC says:

What kind of dire consequences?

Seriously: Phil has one of the most widely read blogs on the internet. He gets paid a generous sum for writing it. He's a happy, successful guy. He publicly screwed up, and got called out for screwing up. The callout consists of another blog saying: "Hey, dude, you screwed up in stupid way".

That's "dire consequences"?

In my world, consequences are serious things. You know, paying a price, losing a job, getting fined or penalized, having your pseudonym revealed to the world, getting reported to your company's HR department for unprofessional behavior, etc. Not having someone say "You did something stupid" after doing something stupid. (Note that I'm not saying that Phil deserved any of those things; he obviously doesn't. But when you do something stupid, and someone tells you that you did something stupid in response, that's doesn't come within miles of the bar for "dire consequences".)

Hell, if that's dire consequences, then I've faced dire consequences on this blog many times, and barely noticed it. Because honestly, I've made some dumb mistakes in my time. I've felt bad about it, because I hate screwing up. But I've never been angry or felt unfairly put-upon by the people who pointed it out, even when they did so in not particularly pleasant terms. Because, y'see, when I screw up, it's my fault, and I don't think it's unfair to point that out.

• william e emba says:

The "dire consequences" referred to are the ones you have been imputing to Plait's video. The ones his unwitting readers are now allegedly suffering from, and don't even know it. So dire, in fact, that there is apparently no cure short of warning in strong, shocking language just how awful exposure to Plait's video was to the unprotected mind.

• MarkCC says:

Once again with the exaggerations. Saying that he screwed up in a stupid way when, in fact, he screwed up in a stupid way is "strong shocking language"? Come on, give me a break!

He wrote a blog piece saying "Wow, cool, look how crazy this is", and I responded with a blog piece saying "Dude, you're wrong in a really dumb way".

If that's what passing for shocking language and dire consequences in your world, I wanna know where you live, so that I can move there!

• william e emba says:

No, there have been no exaggerations. Your response was much more than saying Plait screwed up in a stupid way. You went on in great detail in all the ways he missed out on the supposedly extremely obvious, and you have expanded repeatedly on his alleged sins.

Also, your response gets automatically jacked up a level of incompetence and rudeness, simply because Plait did not screw up in a stupid way, as you kept alleging rather incompetently. He did not write for a professional mathematical audience. Fine by me.

Second, you're being willfully dense again regarding "dire consequences". The "dire consequences" comment made above was regarding the alleged sufferings his readers have faced. There have been none, despite you and others trying to spell them out.

That's what's "shocking". I've also posted references to other counterexamples concerning even weirder infinite operations done by high-level professional mathematicians in extreme seriousness.

You're as pathetic as the Cantor cranks.

• John Fringe says:

[Facepalm]

• Katherine says:

You have a typo: "So 2S2 = S1; therefore S2 = S1=2 = 1/4.", should be S2=S1 /2=1/4

• Katherine says:

• JimV says:

I used to read this blog daily several years ago. Then I read a case where the author was clearly wrong and it was clearly described by a commenter and the author refused to change his mind or take back the insults he had applied. Then it happened again, and I haven't been back since, until now, on a link from Crooked Timber. This is the third time - not as egregious as the prior two, except for the lack of humility which I would have thought the prior instances might have taught. There are semantic points that could be made ("equal" vs. "assigned value"), but they don't justify the insulting tone. People who watch the video and read the links it gives will learn some interesting stuff which has real-world applications - without any name-calling.

The use of the Internet has expanded a lot since I stopped coming here, so I don't doubt this blog has more readers now than then (plus some people probably enjoy reading verbal abuse just as many enjoy watching violent sports). But in fact it has also lost readers who believe in civility (as in the basis of civilization). (I know - civility on the Internet - what a concept.)

• Rex Philpot says:

I admit I found this demonstration interesting, but also found it unbelievable. I am interested in the math experts commentary on this topic. As I understand it Ramanujan is regarded as a great mathematician in history and not just someone with a math degree posting on the internet today (see http://en.wikipedia.org/wiki/Srinivasa_Ramanujan).

Did Ramanujan in fact make the following statement in a letter to G.H. Hardy:

"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …" (see also http://en.wikipedia.org/wiki/Ramanujan_summation)

I don't imagine anyone posting here regards Ramanujan as an idiot who plays fast and lose with math, or that performs sloppy algebra etc. (insert various aspersions thrown about in posts above) nor do I imagine that the posters here see themselves as mathematically superior to Ramanujan. So I have 3 questions:

1) Is the information regarding Ramanujun I provided here accurate (you know Wikipedia can be unreliable at times)?

2) Is Phil Plait saying anything different than Ramanujan stated?

3) If so, where is the error and why should I not view the statements of Phil Plait regarding this calculation in the same light as those of Ramanujan?

Your thoughtful response is appreciated. Thank you.

• MarkCC says:

I think that taking one sentence out of a Ramanujan letter isn't ever going to bring clarity to much of anything. Ramanujan was a brilliant, brilliant man. But he wasn't known as a paragon of clarity!

In this, like in so many things about math, you need to be extremely clear about what you're talking about. In the case of Ramanujan, I'm pretty damned sure that he knew what he was talking about, and I would guess that if you actually read his entire letter, the domain would be clear.

In the normal context, the sum of the natural numbers doesn't converge. If what people understand by addition works, it can't. And once again, I'm pretty damned sure that if you asked Ramanujan, he would have said of course, the sum of the positive integers doesn't converge!

Is Phil Plait saying something different from Ramanujan? Damn right, yes. Just read the intro of the wikipedia page, which sets the context: "Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to infinite divergent series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined."

Ramanujan knew that the series didn't converge. And Ramanujan didn't pull the nonsense algebra games that Plait did. What he did do was create an alternative analytical framework for studying divergent series. That's quite different.

• Rex Philpot says:

Thank you for taking the time to reply. I appreciate it.

I am by no means a mathematician nor will I pretend to be but from what I've gathered here it seems that that using conventional summation this can't be done (tried last night on my calculator and stopped when blood started spewing from my finger tips). Thus, in order to derive an answer you must use an approach designed for divergent series. Now I have no idea if his proof is in line with what Ramanujan outlined (in fairness, to the lay person, it did seem that he knew he was over simplifying the presentation) BUT it doesn't seem like he was misleading in the general presentation.... The sum of an infinite number of terms of the series: 1+2+3+4... does equal -1/12 (as per Ramanujan). The fact that this requires an approach designed for a divergent series seems implicit to the problem, as this is a divergent series. It does not seem to me that this should require an explicit statement to be known.

As for the rest of it, if in fact this calculation is used in physics (don't know that that claim is true) and it produces results that can be verified experimentally, doesn't this validate the answer as real and not bad math or hocus pocus? I may be missing the point entirely but I still find the discussion interesting.

• David Starner says:

1 + 2 + 3 + 4 ... does not equal -1/12. If they're cardinals, counting numbers, the sum is infinite, aleph null to be precise. There are contexts where it is useful to call the sum -1/12, and ultimately there is no one answer for it.

And it's not really so much about the answer as how they got there. I don't know if there was anything legitimate in what they were doing; I tend to believe they had an answer, -1/12, and went through manipulations they knew were invalid to get the answer they wanted. Much like drawing up a horoscope based on Lincoln's time of birth and discovering that he was destined to become president. Maybe it was like taking 16 / 64 and canceling 6's, something a student could honestly do and get the right answer using methods that will fail for most cases.

You want to fuzz over the details of math, go ahead. I haven't seen a book on the Four-Color Theorem or Fermat's Last Theorem that actually dug into the details of the proof. You can even cover proofs and handwave large parts. Just don't claim to show how to solve a problem and make it up as you go.

• Rex Philpot says:

Thank you for taking the time to respond to my post. I feel like Wittgenstein needs to step in here and throw a poker into the fireplace or something. It seems like this boils down to semantics at this point. If in some context(s) 1 + 2 + 3 + 4 .... equals -1/12 then that means in OTHER context(s) 1 + 2 + 3 + 4 .. does not equal -1/12. It seems inappropriate to state the latter as a fact and then qualify that in some cases the former is true. They are both true within specified context. As I stated in an earlier post, it seems to me that the context for this calculation is defined by the problem itself, if the numbers are divergent and the goal is to calculate a finite value, then application of Ramanajun summation seems appropriate and implicit to the nature of the problem. The context is defined by the problem.

I would never argue as to the quality of the math demonstrated in the video, and the math experts can have at him all they want on this issue. I tend to waive that aspect off because I do not think the intent of the video was to teach appropriate calculations to students of math, but rather to try to simplify the process for lay people with the intent of showing that the answer is -1/12 (in some context).

As for it's application in physics, this is not Einstein's cosmological constant, the fact that this was calculated by Ramanajun prior to 1913, and perhaps by others prior to that, makes it very hard to believe that the answer was derived for the intent to resolve an issue with a quantum physics equation. The number was derived independently, for completely unrelated reasons, yet happens to work within these equations. That seems like evidence that this is a valid result with real world meaning and not math trickery or total BS.

• David Starner says:

It's about communication, which means it boils down to semantics. But it's only interesting if we presume that the watchers have some understood semantics about 1+2+3+4.... I can get 1+2+3+4 ... to be any sum k I want by defining +(x,y) to be k. Only when certain semantics are assumed does this become the least bit interesting.

1+1 can equal anything you want, and in fact 1+1 = 0 for an branch of mathematics important to computer science. And we can use the same methods in the video to prove that 1 + 1 = 0. The big difference between this video and one showing that 1+1=0 is that people would take the later to be a joke.

I don't understand the point in simplifying the process this way. The same "simplification" can be used to show that 1+2+3+4... is any value you want. It doesn't edify the audience at all. It's the trickery of a magician or conman, not the simplification of an educator.

• Rex Philpot says:

David,

Again, I am not a mathematician by any stretch of the imagination and do not pretend to be. I do find the response that several individuals have had to various questions here disturbing. Essentially that statement is "You can make anything equal anything else." OK, cool.... but what does that say about math as a field of study? When I hear this stated with such confidence and certainty from so many I wonder is there any point to math at all... heck I can make anything equal anything else without math, so what is the purpose of mathematics? Sounds like philosophy with numbers....

• David Starner says:

You brought up Wittgenstein, and math is all about worrying about semantics. "Cat" can mean a lot of things; but if someone's says that every man is in love with his "cat", then it's up to them to make it clear where they mean "feline" or "catamaran" and not conflate the two. There's no problem with anyone saying "1 + 1 = 0" because either it will be clear from context, it will be explicitly written that we're working in Z_2, or people will get to legitimately gripe at them.

• Dennis Towne says:

Rex, I think it's important to be very, very clear about what you're stating, and I think the syntax used here is really hindering understanding. With that in mind, consider the following:

1+2+3+4+... diverges. It has no sum.

Ramanujan_sum(1, 2, 3, 4, ...) = -1/12

People in this thread are using "1+2+3+4+..." to mean multiple, different things, and it would help everyone to explicitly note which form they are using in each instance.

• Ramanujan was also an autodidact, and in many cases derived his results without anything like the kind of rigor modern mathematics demands. It was in his collaboration with Hardy that actual proofs were introduced to back up the results he'd discovered.

Yes, formalism can be a fruitful source of inspiration, but it is not a substitute for proof.

• MarkCC says:

For gods sake, you've made your point, OK? You think I'm a despicable, stupid, idiotic, hypocritical moron, and that everything I say is hopelessly dumb. Reiterating that point over and over in the comments is tiresome, and it doesn't add anything to the conversation. If you hate my blog so much, don't read it. If you want to flame me, go ahead - write you own blog about how dumb I am.

• John Fringe says:

The Ramanujan example is also very interesting because, effectively, Ramanujan "played fast" and "lose" with math from time to time. Of course he was no idiot, and of course he was far better than us on math. But he was not rigorous on his proofs, and this led him to being wrong from time to time.

From what I read, there was always some tension between him and Hardy because the second understood the importance of proven results.

So this example illustrates the obvious: that you can go on with heuristics or loose procedures. Of course, ideas don't born mature, and playing with concepts is wonderful and necessary, and nobody here said the opposite. But it's also important to know when your results are not proven.

Because, without proofs, even Ramanujan can be wrong.

The error Mark pointed wasn't using results not rigorously proven, but presenting them without explaining this or the realm they were being used to a public not able to tell the difference. Which, I agree, is bad.

• Rex Philpot says:

John,

Thanks for the reply. As I stated in a previous post it seems pretty clear that the presenter is aware that he is playing fast and loose with the problem. I believe that that was intentional to keep the lay person from turning it off inside of 30 seconds. No disrespect to math gurus out there but expounding on the details of math has been demonstrated to be second only to propofol in its sedative effects on the lay person. If this is the basis for the criticism, I get it, but when evaluating a presentation one must always keep in mind the target audience and the intent of the presentation. I don't think this was intended as an educational video.

• It's also important to stress, "this is a trick; it doesn't really work quite like this, and I'm sweeping a lot of the details under the rug." A little smirk as you perform the trick isn't enough to count as honest.

• The "proof" is fast and loose yes because he is trying to explain a strange idea to non-mathematicians. In particle physics we often find times that the difference between two infinite sums (or integrals) is a finite and relevant physical number. You have to be very careful about how you subtract them or the difference can still be infinite, but when done right the difference is in fact measurable and surprisingly accurate.

What this sum really represents is how much extra this particular sum leaves when subtracted off from other sums with identical divergences. In this case, this sum's particular value, when you remove the infinity, is -1/12. But yes, it is technically divergent - 1/12.

• Mary Curran says:

I am an artist and admitted math moron. My own children stopped asking me for help with math when they were in the second grade. However, even with my limited confidence, my math BS meter went off watching the video in question. This discussion at least validates my intuition, which apparently still functions.

• MarkCC says:

It's clear: in your opinion, I'm an idiot, and my treatment of Plait was completely unacceptable. I disagree, but you're entitled to your opinion. But you've expressed it enough times here; I no longer feel any obligation to provide you with a forum to continue to shout. Find your own blog, or twitter, or whatever; you're not welcome here.

• annyingram says:

Nice one writing it is so informative and useful i suggest to start Learn Math Analysis

• Wow. I've been doing this interwebs thing for some time, and rarely have I seen people as smart as the lot of you appear to be go for the mud so quickly and with such desperation. I remember hearing an aphorism, "The fights in academia are so fierce because the stakes are so low." [And, yes, I'm sure you've heard your own version.] Strikes me that it's applicable here.

2) He wasn't suitably repentant, nor did he do enough to rectify his incorrect thinking.

Is this well and truly the End Times? I gotta say "No." Try being a bit nicer in your interactions, and a bit less willing to throw folks under the bus next time, eh? If you need me to, I'll happily go dig up some Netiquette rules from Usenet.

\$.02

Note: of course, this doesn't apply to everyone; some tried to remain rather civil. But jeepers, thems that didn't, well... didn't.

• MarkCC says:

It really, really continues to amaze me how shocked! utterly shocked! people claim to be, all because I had the audacity to point out that Phil pulled a dumb screwup.

Good lord, if I got this upset every time anyone said anything bad about me, I'd spend my whole life sulking.

I didn't accuse him of being a child molester. I didn't claim that he'd never said anything intelligent. I didn't even claim that he had a history of acting like an idiot. All I did was point out a case where he said something stupid. Big fat hairy fucking deal. Get over it!

• RaphM says:

After stumbling across those viral youtube videos, reading the various reactions and explanations there is one thing that puzzles me:

-> What came out pretty clear by you is that there is of course no "standard way" of assigning values to infinite divergent series...

-> ...yet there exist several approaches that are able to assign values to at least a subset of all possible infinite divergent series (e.g. Zeta function regularization and Ramanujan summation are able to assign a value to 1+2+3+..., but Cesaro summation cannot) - the important thing is that values obtained via such approaches do NOT represent sums in the traditional sense, BUT they may still be useful in certain contexts.

Now -1/12 can be obtained via two seemingly independent such approaches (Zeta function regularization and Ramanujan summation), also the experimentally verified results regarding the Casimir force indicate that this value is really meaningful in a practical context.

Now isn't the really astonishing part that the very "arbitrary entity" -1/12 and the very "natural entity" 1+2+3+... seem to be deeply related to each other, even via multiple different means?

• MarkCC says:

As I've repeated, over and over and over...

The zeta function is the basis of -1/12. Zeta is an incredibly important abstraction that is deeply tied to the fundamental structure of numbers. There are several approaches that come up with the value -1/12. But they're all different approaches to finding the value of the (unique) analytical continuation of the zeta series. The value of zeta(-1) is -1/12. No ifs, ands, or buts: the correct value of the Riemann zeta function at -1 is equal to -1/12.

The reason that the experimental results work when you use -1/12 as the value of the zeta function at -1 is because -1/12 is the value of the zeta function at -1.

Once you realize that it's all about this one, single function - Riemann zeta - then it's not remotely astonishing that zeta(-1)=-1/12. Nor is it remotely astonishing that any of the sideways routes at arriving at the correct value of zeta(-1) produce the same correct result. And once you understand what an analytical continuation is, it's not remotely surprising that the base series for zeta diverges at -1, but that zeta doesn't.

The reality of what's going on is less mysterious, but much more beautiful.

• rhaegar7 says:

Don't let them ruin your day, Mark - you're right on all counts. 🙂

• Pogsurf says:

S1 = 1 - 1 + 1 - 1 + 1 ...

So

S1 = 1 + 1 + 1 + ...
- 1 - 1 - 1 - ...

Do a shift of say three places

S1 = 1 + 1 + 1 + 1 + 1 + 1 + ...
- 1 - 1 - 1 - ...

Now start summing S1 and you get the following values: 1, 2, 3, 4, 3, 4, 3, ...

So now it seems S1 alternates between the value of 4 and 3 ad infinitum,

and so Sum(S1) = 3.5

According to the above, Sum(S1) can take any value of the form n.5 positive or negative depending on how big a shift is employed. In other words there is no "real" value for Sum(S1) it just depends on how you add it up.

I expect there is a similar problem when adding S2.

If you can't reliably sum S1 or S2 you can't reliably use them to calculate S3.

• Jonas says:

Isn't the Cesaro sum of a series defined as

"1/inf * sum to inf a_k" ?

So Cesaros middle of Grandis series is

[(sum 0 to inf (-1)^k)/inf]

and not as said in the article each part of the sum/(k at the sumpart)

or did i missunderstood something?
please if someone could clarify that for me would be awesome

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