String theory likes this math because it assumes that there is a curve to the sum, it will get smaller eventually. This helps make the "vibration" part of sting theory work.
But just because you can prove something with math doesn't mean it is "real".
We all know that if you add any number of positive integers you get a positive integer. This is very "provable".
The two are in contradiction. The sum of all natural numbers can't be 1/12th if the sum of any two positive integers is another positive integer.
We all know that if you add any number of rational numbers you get a rational number. Yet the sum of an infinite number of rational numbers can be irrational (eg equation 2 in the article). Contradiction?
Not sure that's any less intuitive than getting a rational number from an infinite sum of integers, or a negative number from an infinite sum of positives. It is indeed a paradox until you rigorously define what you mean by "infinite sum", which is the whole point of the article :)
Wonderful article, though I got slightly distraught once he got through all that hard math only to state essentially that it would get more interesting below the fold.
I used to understand some of that (Taylor and Maclaurin series). I think the "Integral Test" had been my high-water mark, it's amazing to see how much further the mathematical concepts can be carried.
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.
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.
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?
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.
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.
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.
I didn't read or listen to the OP, but
1+2+3+ ... = infinity, using infinity + k = infinity, then k=0 for any k, then -1/3 = 0 = Whatever, then like Bertrand Russel said, I am the Pope since 2=1 and the Pope and I are two people.
But just because you can prove something with math doesn't mean it is "real".
We all know that if you add any number of positive integers you get a positive integer. This is very "provable".
The two are in contradiction. The sum of all natural numbers can't be 1/12th if the sum of any two positive integers is another positive integer.