HN Theater @HNTheaterMonth

You are right and I am glad I didn't write 'because Maths'. The sum you're quoting from is monotonically growing but at an ever shrinking clip. However, the series does not describe the kind of growth that our economy seemingly needs and our politicians like to promise, nor does it AFAICS describe the OP's idea of continued population growth given his recommendation to shun shrinking populations with fertility below replacement rate; theirs is a constant or in the midterm increasing growth and notwithstanding [minority opinions](https://www.youtube.com/watch?v=w-I6XTVZXww) claiming that 1 + 2 + 3 + 4 + 5 + ... equals -1/12 such a series cannot have a finite sum, hence is physically unfeasible to represent.

⬐ superasn

Sorry my original title was

1 + 2 + 3 + 4 + 5 + ...

Not sure how it got changed

This is a good video explaining it: https://www.youtube.com/watch?v=w-I6XTVZXww

Although it relies on a particular interpretation of the left side of the equation, it's used in some areas of physics.

⬐ crvdgc

I find a response video to it more enlightening. It's often found as the number one related video.

https://youtube.com/watch?v=YuIIjLr6vUA

It's referencing a theorem from string theory, as discussed in this Numberphile video: https://www.youtube.com/watch?v=w-I6XTVZXww

The catch is that this isn't valid in what we consider standard mathematics, and you can find many discussions of this online, but this one is fairly short and straightforward: http://curiouscheetah.com/BlogMath/infinity-and-string-theor...

Few days back I stumbled upon a video about Ramanujan [0] and the: 1 + 2 + 3 + 4 ..(to infinity) = -1/12.(Turns out the proof in the video is incorrect or at least not concise enough, because you can't do that with divergent series). So that got me curious and I wanted to understand what's going on. How can 1 + 2 .. etc equal to -1/12 and not infinity. So I went on small binge watching videos to get an intuition and understanding about this. This led me to Riemann Hypothesis and Euler's identities and complex numbers etc. Now I have a list of books that I want to skim/read to get a historical context around these problems. I don't take any notes just yet, because I don't know what is important and what is not, I just jump into the stuff and hope that when I come out of the other end I will at least have a understanding why it equals -1/12. It's like panning out gold, there might be some gold nuggets. And maybe, just by pure chance, I come up with some ridiculous ideas how to disprove Riemann Hypothesis ;P (I think I have to revisit Gödel, Escher, Bach and Gödel's Incompletness Theorems to prove that it's not provable).

- [0] https://www.youtube.com/watch?v=w-I6XTVZXww

EDIT: Sorry to all facepalming mathematicians who happen to read this comment.

⬐ ganafagol

I'd recommend you study some real math for this. Popular science articles and books are great entertainment but are distinctly different from the real deal.

You don't become a surgeon by watching Dr. House or a historian by watching the history channel. Don't expect to understand math by reading a wikipedia article. To really get it, you need to dive deep and get your hands dirty.

⬐ sabas123

To add to this, people underestimate how long it might take understand material on a proper level.

My favorite quote in Axler's linear algebra done right was that if you take less than an hour per page, your doing something wrong. I agree with him.

Don't get me wrong, learning math is definitely worthwhile, but please don't think you can skim some books and gain profound insights without hard work.

Here's a Numberphile video on this

https://www.youtube.com/watch?v=w-I6XTVZXww

I learned about it through this Numberphile video: https://www.youtube.com/watch?v=w-I6XTVZXww

There's also a nice write-up from Terry Tao: https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri...

https://youtu.be/w-I6XTVZXww

There are a lot. I thought this one was really funny [0]. This one I found very interesting [1].

Somewhat different, is the Hello Internet [2] podcast, which is by Grey and Brady Haran who you might know from the Numberphile youtube channel [3]. It's basically the 2 of them chatting about random stuff, but I find it very entertaining.

[0] https://www.youtube.com/watch?v=LO1mTELoj6o

[1] https://www.youtube.com/watch?v=LrObZ_HZZUc

[2] http://www.hellointernet.fm/

[3] https://www.youtube.com/watch?v=w-I6XTVZXww

I thought that -1/12 was amazing, even to Vulcan scientists. -https://youtu.be/w-I6XTVZXww -https://youtu.be/0Oazb7IWzbA

⬐ mrob

-1/12 is the result of applying zeta function regularization or Ramanujan summation to the sum of the positive integers. It's arguably interesting, but hardly amazing to the vast majority of people who have never heard of those techniques. But the thing that really annoys me is all the people presenting it as the finite limit of a divergent series (this is the default meaning of "=" after an infinite series, if you're using a non-standard meaning you have to specify that!). The first of those videos does this! It's nonsense, and this kind of sloppy approach only encourages contempt for mathematics.

If they make the limit infinite, you'd end up being paid -1/12 of a cent.

https://www.youtube.com/watch?v=w-I6XTVZXww

⬐ barsonme

Good Lord, I watched the second video -- the one featuring Ed Copeland -- and I think I just fried my brain for the rest of the day.

(Also, he has fantastic penmanship.)

⬐ rdancer

Tony Padilla sure can sell that sleight of hand :-)

⬐ serge2k

Only if you're an Indian Mathematician.

or a physicist.

⬐ owenversteeg

Nope! http://goodmath.scientopia.org/2014/01/17/bad-math-from-the-...

⬐ openasocket

Incorrect, there is actually some well-established mathematics dealing with divergent series as summing to finite values via analytic continuation. These results have to be treated with care, because they are inconsistent under certain algebraic manipulations, but the techniques are "real" enough that their results are used in particle physics.

https://en.wikipedia.org/wiki/Regularization_(physics)

⬐ owenversteeg

"The sum of the series 1+2+3+4+5+6... = -1/12" is patently false, without a previous assertion that we have assumed the Cesàro sum of a series is equal to the series.

Even mathematicians working with Cesàro sums surround such statements with "this holds only if we interpret the infinite sum defining Z to be the Cesàro sum..." [0]

Precisely none of the times I've heard the "1+2+3+4...=-1/12" bullshit has the person stating it prefaced their statement with "this holds only if we interpret the infinite sum defining Z to be the Cesàro sum..."

If you say that "1+2+3+4...=-1/12" without stating your prior assumptions, you suddenly allow anyone to make any assumption whatsoever, no matter how obscure it is. In your imaginary world, someone could walk into a store and claim that "this 95 cent pack of gum is free" because they just made the unstated assumption that all non-integers do not exist, and seconds later they could return it for a full refund of $0.95 after making the unstated assumption that in fact the rational numbers do exist. Numbers, and in fact the entire system of mathematics fail to work at all once you allow arbitrary, unstated assumptions no matter their obscurity. And in fact, the assumption that non-integer numbers do not exist is made far, far more frequently than the assumption that the infinite sum defining the sequence is the Cesàro sum.

The only difference is that assuming the non-integer numbers do not exist is a defensible assumption in many, many scenarios... but Cesàro summations are only invoked about twelve times a year, in pure math or advanced physics papers.

[0] Madras, Neal. "A Note on Diffusion State Distance." arXiv preprint arXiv:1502.07315 (2015).

I love the idea, but I am concerned that if run by and for physicists, it will perpetuate the lack of rigor and "shorthands" physicists use to abuse mathematics [1]. I would propose an additional feature: that entries be highlighted in an additional way if they are mathematically false as written (or misleadingly overloaded, or just a huge abuse).

[1] e.g. the sum of natural numbers being -1/12, perpetuated with reckless abandon in this video https://www.youtube.com/watch?v=w-I6XTVZXww to the dismay of mathematicians everywhere. I also have personal gripes with the way physicists use delta functions, and hear many physicists say (about some topic in mathematics), "I don't want to actually do the mathematics here, I just want to get intuition about it."

⬐ ColinWright

This has been discussed a lot, but there is something real and genuine going on here that is, unfortunately, masked by the obvious nonsense.

Briefly let me introduce the main ideas.

When -1<x<1, sum_{n=0}^{n=oo} x^n = 1/(1-x). We might be tempted, then, to say that

    1 + x + x^2 + x^3 + x^4 + ... = 1/(1-x)

But that's obvious nonsense when, say, x=2. The LHS clearly makes no sense, even though the RHS exists and has the very sensible value of -1. It therefore cannot be said that:

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

However, underlying all this is the concept of Analytic Continuation which says that under certain circumstances, this is a reasonable thing to be doing.

Sort of.

Here it is being done properly:

https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri...

Here is a Hacker News discussion of that:

https://news.ycombinator.com/item?id=7078744

If that has whetted your appetite for more you might like to follow some of the discussions and links given in previous submissions:

https://news.ycombinator.com/item?id=7078489 : 53 comments

https://news.ycombinator.com/item?id=7073976 : 29 comments

https://news.ycombinator.com/item?id=7057049 : 19 comments

https://news.ycombinator.com/item?id=7038809 : 6 comments

https://news.ycombinator.com/item?id=7081885 : 4 comments

https://news.ycombinator.com/item?id=8843219 : 4 comments

Other submissions:

https://news.ycombinator.com/item?id=7033444

https://news.ycombinator.com/item?id=7074820

https://news.ycombinator.com/item?id=7079921

https://news.ycombinator.com/item?id=7096326

https://news.ycombinator.com/item?id=7176024

https://news.ycombinator.com/item?id=7602317

https://news.ycombinator.com/item?id=7638147

I think the quality of numberphile videos vary quite considerably. The one you linked is a rather honest explanation of the idea of comparing the size of sets through bijections and a good explanation of Cantors diagonal argument. As a mathematician I'm quite fine with it.

The one linked in the submission is more of an obvious statement hidden by obscure use of language. Slightly silly, but not very bad.

The worst offender with "magic" is the one with the Riemann zeta function [1], which went viral a while ago. The problem here is that they get people started off on the wrong foot, confusing them with wrong arguments and hidden definitions. Now, if people really want to understand why this can be made meaningful, you first have to explain that the better portion of the video is absolutely wrong, and only then can you explain what is actually going on.

[1] http://youtu.be/w-I6XTVZXww

There's a wonderful proof of this on Numberphile's channel here: https://www.youtube.com/watch?v=w-I6XTVZXww

⬐ mturmon

I will add to the comments nearby and say that I found that video appalling. They write

  S = 1 - 1 + 1 - 1 + ... = 1/2,

and justify this nonsense statement with some hand-waving, and then proceed to use this "fact" to deduce several other "facts" through manipulation of infinite sums. (Sure, the statement is true under some interpretations of "...", "+", and "=", but they never tell the viewer that they are using a particular, and non-standard, interpretation.)

Then, at the end, they really ice the cake by appealing to "physics" and "string theory" to affirm that it is all true.

It's mystical, and abuses the appeal to authority. It's the reverse of what math should be.

Additionally, for someone who suffered through real analysis, it's galling to have an issue that legitimately confused mathematical geniuses in the 1800s (convergence and the nature of the "passage to the limit"), and was then figured out, used to troll people in the 2000s.

Phil Plait ("Bad Astronomer") used this in his column, and the results were disastrous. This is the only time I've seen him make such a bad mistake, he's usually both fun and correct.

⬐ j2kun

Actually, I think these notes were posted in response to the extremely bad mathematics that has been coursing through the internet due to this video.

⬐ None

None

⬐ nilkn

I'm pretty sure this was posted precisely to explain the great amount of confusion which that video has created.

⬐ lisper

https://news.ycombinator.com/item?id=7078489

⬐ jfarmer

Warning, prolix rant ahead.

I actually think that video is awful and it certainly doesn't illustrate what Terry is illustrating in his blog post. Rather, the video goes through a bunch of non-rigorous symbolic manipulations which ends with the author writing down the sequence of symbols "1 + 2 + 3 + ... = -1/12".

However, unless one says precisely what one means by "+", "...", and "=" — which Terry does — then we have no way of really saying whether the steps taken to reach the so-called "conclusion" are valid or not. What's more, just because the same sequence of symbols appear in both the video and Terry's blog post doesn't mean they represent the same thing or have anything to do with each other at all.

Put another way, if you and I reach the same conclusion, but you do so rigorously and I do so speciously, that doesn't mean I've proven the same thing as you have.

For example, in the video, why are we allowed to add together two infinite series term by term in the way they describe? It seems "natural," I know, but if that's natural, why can't we also, say, group the addition differently or rearrange the terms? After all, a + (b + c) = (a + b) + c and a + b = b + a, no matter what a and b are. Why can't we write

  S = 1 - 1 + 1 - 1 + ...

as

  S = (1 - 1) + (1 - 1) + ...

If we permit ourselves to do that, well, suddenly the sum "appears" to be 0 and not 1/2. BTW, if you want a somewhat-more rigorous reason for why the sum "should be" 1/2...

      S = 1 - 1 + 1 - 1 + ...

  So then
      
  1 - S = 1 - (1 - 1 + 1 - 1 + ...)
        = 1 - 1 + 1 - 1 + ...
        = S

  which implies S = 1/2

We're not proving that S = 1/2 here, though. We're proving this statement: if it makes sense to talk about S and we're permitted to do the things we just did to S then S = 1/2.

Terry knows all this, of course, which is why he says, "If one formally applies (1) at these values of {s}..." That word "formally" is key here. To a mathematician "formally" means "in a purely symbolic manner without considering whether there's a sensible or consistent way of interpreting these symbols."

So, Terry is saying, "If we treat these sums as purely symbolic entities then when we substitute in s = -1 we get a the purely symbolic statement 1 + 2 + 3 + ... = -1/12." He then goes on to illustrate ways we might make sense of this purely symbolic (formal) sum.

The video, however, is no "proof" of anything at all. It's just a shell game with symbols on a page, relying on people's vague intuition about what we're allowed to do with numbers. Just because the symbols in the video include those we typically take to represent numbers and addition doesn't mean they actually do.

⬐ chii

that video isn't about proving anything - it's a pop science channel (but for maths), meant for the laymen.

⬐ jfarmer

Nevertheless, that seems to be how people understand the video. See, e.g., the person whose comment I was replying to.

⬐ pavpanchekha

An intelligent "layman" might watch that video and declare, "Math proves that 1 + 2 + 3 + 4 + … = -1/12; yet this is obviously wrong. Those mathematicians don't know what they're doing." Now, I do not know such a layman, but you must acknowledge this might happen. And this is the cost of the little lie permitted in the video you've linked.

If I wrote a "how to program" tutorial in which the code did not compile, you would likely berate my inept tutorial. Please allow mathematicians the same grace.

⬐ johnbm

If you're worried laymen might come away thinking mathematics is completely alien to human interest, start with regular textbooks first.

⬐ Smaug123

> An intelligent "layman" might watch the video and declare, "Math proves that 1 + 2 + … = -1/12"

Indeed, I have seen this exactly happen, with a secondary-school student. It took me some time to explain why it's rubbish-at-a-school-level.

⬐ camperman

Thanks for the correction. I'm most definitely a layman when it comes to infinite series. A couple of things gave me confidence in this video: the result is presented in the string theory text and there's another video demonstrating the same result using Riemann Zeta functions (so it must be legit :)).

I sympathize with your frustration at the lack of rigor but isn't this kind of like taking pot shots at a middle school physics textbook for not covering Lagrangian mechanics?

⬐ gus_massa

No. The problem is that for a very large range of applications 1+2+3+4+...=infinity. That’s the standard definition and the result is quite intuitive, even for a layman (infinity = verrry biiig).

For other applications, it’s sensible to define 1+2+3+4+...=-1/12(R) with an (R) to denote that you are not using the standard definition, but the Ramanujan definition. (You can drop the (R) one you are sure all the public has enough technical background.) The problem is that the Ramanujan definition doesn’t have many of the intituive properties of the standard definition. For example, in this article, eq. (8) and eq.(9) say that 0+2+3+4+... != 2+3+4+5+...

This is not similar to not discussing Lagrangian mechanics in a secondary school book. It’s more similar to mix Newtonian mechanics with the properties of the Higgs boson, and mix the density of water and the fact that electron really don’t have mass, and even say that the R and L electrons are different particle in spite the gravity force cancels the centrifugal force of the Moon (in a no Newtonian reference frame). It’s confusing, and mixing the theories can produce a paradox and be unintelligible.

If you mix them correctly and use just a little of the properties of theory inside the other, you can produce a convincing almost intelligible explanation that produce a paradox. The important point is to hide the technical problems in seemingly obvious properties, like in magic. The standard examples are mixing results of special relativity and Newtonian mechanics, or Quantum mechanics and Newtonian mechanics.

For the layman, I prefer an explanation that start saying that 1+2+3+4+...=infinity, then explain that there are other definitions, then a Ramanujan photograph, then some magic and handwaving to show 1+2+3+4+...= -1/12 (R), then enumerate some applications of this new definition, then show that 1+2+3+4+... != 0+1+2+3+4+... , so you must be very careful with this new summation.

It’s impossible to explain all the technical details to a layman, but it’s important to explain that they are hidden there, and why sometime there is necessary to make definitions that are not intuitive.

⬐ jfarmer

No. While it's true that ζ(-1) = -1/12 and that the ζ function plays an important role in physics, your reasoning is fallacious.

If X implies Y and we know that Y is true, that does not mean X is true. So just because the video reached a "correct" conclusion does not mean that the means by which they reached that that conclusion are sensible or even consistent.

If I threw a dart at a dartboard labeled "What is 1 + 2 + 3 + ...?" and it landed on the section marked "-1/12", would you believe my answer? Would the fact that it happened to land on "-1/12" and also agreed with the ζ function lend credibility to my dart-throwing method of proof?

Indeed, if I encapsulated the methods used in that video, I could use those methods to have 1 + 2 + 3 + ... turn out to be any number I choose. This is the problem with specious reasoning — one can use it to reach any conclusion.

this infinite sum was featured in a numberphile video a few days ago with a entraining and simple "proof": http://www.youtube.com/watch?v=w-I6XTVZXww

and a more advanced explanation & proof in the bonus-video: http://www.youtube.com/watch?v=E-d9mgo8FGk

you should definitely subscribe to numberphile (http://www.youtube.com/user/numberphile) and their sister channel computerphile (http://www.youtube.com/user/computerphile) on youtube :)

⬐ DougBTX

Here's a demonstration of how the "simple" proof breaks down: http://www.quora.com/Mathematics/Theoretically-speaking-how-... (via http://kottke.org/14/01/the-sum-of-all-positive-integers)

Those of us of a less mathsy persuasion may wish to watch a more layman-friendly explanation:

https://www.youtube.com/watch?v=w-I6XTVZXww

⬐ j2kun

Except that this video is the cause of the recent widespread misunderstanding. 1 + 2 + 3 + … is actually NOT -1/12 in the usual sense of sums.

Watch this entertaining Numberphile video about this topic:

http://www.youtube.com/watch?v=w-I6XTVZXww

Apparently it's used in many areas in Physics.

⬐ deletes

I watched a lot of their videos, and until now they were pretty honest. This video is trying to be astonishing by explaining high level math to regular people without clarifying that holds true under different circumstances. That will make people confused over what the sum really is and feels deceptive, since they don't explain that if you are not in a special field of math/physics, the sum is divergent(inf).

⬐ abc_lisper

Properties of numbers change at infinity. It is not conceivable to me, we can use normal arithmetic operations on quantities tending to infinity. For example, consider this...

   9999999999........... infinity
  -9999999999........... infinity
   -------------------------
   00000000000............ infinity
   -------------------------

Now, subtracting infinite numbers from infinite numbers should give a infinite result. All we have is infinite zeros here, which cannot be inifinity.

⬐ kazagistar

You cannot understand "infinite numbers" without understanding how equality is defined (bijection).

⬐ abc_lisper

Ok.. My post was a bait :).. Please tell me more or can you suggest me some books i can read?

⬐ krapp

>All we have is infinite zeros here, which cannot be infinity.

Ok, this is probably a dumb question but how is an infinite set of zeros not an infinite result?

⬐ ivan_ah

Oy mate! Don't you have a pint of cider to drink in a cozy pub somewhere instead of producing videos of such bollocks?

The series $\sum_{n=1}^\infty n$ is divergent, so you can't say anything about its sum. For more info on the 1 - 1 + 1 - 1 +1 ... see: http://en.wikipedia.org/wiki/History_of_Grandi's_series This quote from the page is telling:

   G. H. Hardy dismisses both of these as "little more than nonsense."

⬐ mathnoob

It is not totally nonsense but the use of the benign sign = associating the analytic continuation of a series where it converges to a place where it does not without huge warning is not very rigorous.