Does it matter if the sum of all integers is -1/12?

In response to yesterday's post about the sum of all integers being -1/12 (or not), John Rimmer posted this comment:

"A grave disservice to numerical literacy" Good grief, do you think you could sound any more absurdly pompous if you tried? Most people's numerical literacy doesn't extend past high school mathematics, and doesn't need to. They certainly won't be harmed by having a confused understanding of infinite series. Given the amount of debate this video has generated, I'm not convinced mathematicians themselves understand infinite series.

Prompted by this comment I looked up the definition of "pompous". It means "affectedly and irritatingly grand, solemn, or self-important." But I stand by both the spirit and the letter of my original statement. I think the stakes are much higher than most people appreciate. In my opinion, this is not just a trivial geek quibble over math. This is a symptom of the general decline of critical thinking in our society.

Phil Plait, the Slate correspondent who wrote the original article that I was complaining about, posted a followup today.  It's not bad, but I still take issue with this bit:

The method used in the video to write out some series and manipulate them algebraically is actually not a great way to figure this problem out. It uses a trick that’s against the rules, so strictly speaking it doesn’t work. It’s a nice demo to show some fun things, but its utility is questionable at best.

Saying that "its utility is questionable at best" implies that manipulating non-convergent infinite series by standard algebraic rules might have some utility.  It doesn't.  It's simply wrong, full stop.  The reason that it's wrong is that it is inconsistent.  It leads to contradictory results, and from a contradiction you can conclude anything.

At the risk of stating the obvious, the whole point of reasoning is to distinguish true statements from false ones (relative to some set of assumptions of course).  But once you admit a contradiction into your thinking you can no longer make such a distinction (because you can prove anything).  Contradictions are much, much worse than simply being "of questionable utility."  They poison everything.

My real issue with the video, and Phil Plait's original uncritical endorsement, is that it endorses, and hence encourages people to accept, unsound reasoning.  The fact that the unsound reasoning in this particular case led to a conclusion that superficially resembles a conclusion that can also be arrived at by sound reasoning just makes it that much worse.  It encourages people to think: because this mode of reasoning led to a "correct" conclusion in that case, then it will probably lead to correct conclusions in other cases.

If the problem were confined to mathematics I might not make such a big deal out of it, but it's not.  The problem of people uncritically accepting conclusions drawn by unsound methods of reasoning pervades our society and causes real damage.  The best current examples are climate-change denialism and creationism, but there are many, many others.  (An example from the left side of the political spectrum: guns are bad, therefore the second amendment doesn't mean what it says.)

The real problem, as I see it, is that people put a lot more stock into the question of whether something is true than why it is true.  It doesn't so much matter that the sum of all the integers is -1/12 .  What matters is that it is possible to define the concept of an analytic continuation of a function, and under this definition (which does not lead to contradictions), the sum of all the integers is -1/12.

It is important to understand this sort of thing even if you don't understand what an analytic continuation is because it helps you detect bullshit.  Digging down through an argument to find the basic assumptions on which that argument rests is a crucial skill in today's society.  Without that skill you are an open invitation to demagoguery and flim-flammery, and it can cost you and your family dearly.

So, John Rimmer, if you don't think that concern is warranted about doing "a grave disservice to numerical literacy" I say you have not appreciated the gravity of the situation.

No, the sum of all the positive integers is not -1/12

I haven't been blogging recently because my new startup is taking up all my time, but someone needs to stand up and say the emperor has no clothes so it might as well be me.  About a week ago, two British mathematicians named Tony Padilla and Ed Copeland, who produce a video blog called Numberphile, posted a video that purports to prove that the sum of all the positive integers is -1/12.  It was making the usual geek rounds where I would have been content to let it circulate, but today the story was picked up by a naive and credulous reporter at Slate, where the story stands to do some real damage if not challenged.

(Aside for mathematicians: yes, I am aware that the claim is true under Ramanujan summation.  That is not the point.)

Let me start by recapping the argument.  Finding the flaw in the reasoning makes a nice puzzle:

Step 1: Let S1 = 1 - 1 + 1 - 1 ...

Then S1 + S1 = (1 - 1 + 1 - 1 ...) + (1 - 1 + 1 - 1 ...)

 = (1 - 1 + 1 - 1 ...) +
   (0 + 1 - 1 + 1 - 1 ...)

 = (1 + 0) + (1 - 1) + (1 - 1) ....

 = 1 + 0 + 0 ... = 1

So 2xS1 = 1.  So S1 must equal 1/2.

Step 2: Let S2 = 1 - 2 + 3 - 4 + 5 ...

So S2 + S2 = (1 - 2 + 3 - 4 + 5 ...) + (1 - 2 + 3 - 4 + 5 ...)

 = (1 - 2 + 3 - 4 + 5 ...) +
   (0 + 1 - 2 + 3 - 4 + 5 ...)

 = (1 + 0) + (1 - 2) + (3 - 2) + (3 - 4) + (5 - 4) + ...
 = 1 - 1 + 1 - 1 + 1...
 = S1 = 1/2

So S2=1/4.

Step 3: Let S = 1 + 2 + 3 + 4 + 5 ...

So S - S2 = (1 + 2 + 3 + 4 + 5 ...) - (1 - 2 + 3 - 4 + 5 ...)

 = (1 + 2 + 3 + 4 + 5 ...) -
   (1 - 2 + 3 - 4 + 5 ...)

 = (1 + 2 + 3 + 4 + 5 ...) +
  (-1 + 2 - 3 + 4 - 5 ...)

 =  0 + 4 + 0 + 8 + 0 + ...

 = 4 x (1 + 2 + 3 + 4 + 5 ...)

 = 4S

So S - S2 = 4S.  But S2 = 1/4.  So:

S - 1/4 = 4S

3S = -1/4

S = -1/12

Seems like an ironclad argument, doesn't it?  Like I said, finding the flaw in the reasoning (and there most assuredly is one) makes an interesting puzzle.  Here's a clue:

Let S3 = 1 + 1 + 1 + 1 ...

So S3 - S3 = (1 + 1 + 1 + 1 ...) - (1 + 1 + 1 + 1 ...)

 = (1 + 1 + 1 + 1 ...) - (0 + 1 + 1 + 1 ...)

 = (1 - 0) + (1 - 1) + (1 - 1) + ...

 = 1 + 0 + 0 + 0 ...

 = 1

But S3 - S3 must also equal 0, so we have just proven that 0=1.

The flaw in both cases is the same: the algebraic rules that apply to regular numbers do not apply to infinity.  Actually, it's more general than that: the algebraic rules that apply to regular numbers do not apply to non-converging infinite sums.  All of the sums above are non-converging infinite sums, so regular algebraic rules do not apply.  It is no different from using regular algebra when dividing by zero.  It doesn't work.

Now, there are ways to define the sums of non-converging infinite series so that they do not lead to contradictions.  The one that leads legitimately to the conclusion that 1 + 2 + 3 + 4 ... = -1/12 is called Ramanujan summation, which in turn is based on something called an analytic continuation.  But the problem is that the Numberphile video makes no mention of this.  They present the result as if it is legitimately derivable using high school algebra, and it isn't.  Telling people that it is does a grave disservice to the cause of numerical literacy.

UPDATE: Here's some more of the real math.