5 ms·

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

j2kun 13y ago
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.
- deleted 13y ago
  [deleted]
jfarmer 13y ago
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 13y ago
  that video isn't about proving anything - it's a pop science channel (but for maths), meant for the laymen.
  - jfarmer 13y ago
    Nevertheless, that seems to be how people understand the video. See, e.g., the person whose comment I was replying to.
  - pavpanchekha 13y ago
    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.
    - Smaug123 13y ago
      > 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.
    - johnbm 13y ago
      If you're worried laymen might come away thinking mathematics is completely alien to human interest, start with regular textbooks first.
- camperman 13y ago
  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?
lisper 13y ago
https://news.ycombinator.com/item?id=7078489 https://news.ycombinator.com/item?id=7078489
mturmon 13y ago
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.
nilkn 13y ago
I'm pretty sure this was posted precisely to explain the great amount of confusion which that video has created.

j2kun 13y ago

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.

deleted 13y ago
[deleted]

jfarmer 13y ago

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 13y ago
that video isn't about proving anything - it's a pop science channel (but for maths), meant for the laymen.
- jfarmer 13y ago
  Nevertheless, that seems to be how people understand the video. See, e.g., the person whose comment I was replying to.
- pavpanchekha 13y ago
  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.
  - Smaug123 13y ago
    > 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.
  - johnbm 13y ago
    If you're worried laymen might come away thinking mathematics is completely alien to human interest, start with regular textbooks first.
camperman 13y ago
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?

lisper 13y ago

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

mturmon 13y ago

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.

nilkn 13y ago

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