Resolved: Mathematicians are insane

[pokerfan]

Exhibit:

1 + 2 + 3 + 4 + 5 + ... = -1/12 ??!!

[NoOneMan]

1. cool

[1monster]

2. Interesting....

[getting old in mke]

3. I think the insanity is more on the physicists usage...

A mathematician likely to tell you that trying to operate on divergent series, you deserve any answer you get.

[pokerfan]

4. String Theory...

Leonard: At least I didn't have to invent 26 dimensions just to make the math come out.
Sheldon: I didn't invent them. They're there.
Leonard: In what universe?
Sheldon: In all of them, that is the point!

[salib]

5. Yep

For me, as a physicist also, it is at best a compelling but controversial conclusion. On the face of it I can prove easily that the series 1+2+3+4... is > 1, let alone -1/12. A Simple mathematical induction proof will do that.

It is simply the "and then a miracle occurs" of re-Normalization. That was Feinman's contribution. Very important and very valuable.

It is not really clear that one can treat renormalized results as part of an Abelian group, I.e., apply simple algebraic operations as was done here. Thus, this is really "on the sly."

For me, indeed, there is no compelling reason to assume the Abelian nature of these series, unless one is conforming (renormalizing infinitesimal for example) to experimental reality. It is not the case here, and unfortunately seems a bit like a wishful, rather than compelling, requirement.

[muriel_volestrangler]

8. It worries me a bit that they use this in string theory

It does strike me as an interesting 'trick' that, as you say, shows you should have limits on what you apply certain operations to, rather than something you can take as a result and then use elsewhere, as that string theory textbook seems to do.

[Jim__]

12. Mathematical induction doesn't actually extend to infinity.

It bothered me that there is a rigorous mathematical proof for this, but your claim about an inductive proof seemed valid. I found an article that says you can't extend an inductive proof to infinity:

Here is a quote from B. Russell's Introduction to mathematical philosophy, pages 27-28, that I think describes well this limitation of induction:

Mathematical induction affords, more than anything else, the essential characteristic by which the finite is distinguished from the infinite. The principle of mathematical induction might be stated popularly in some such form as "what can be inferred from next to next can be inferred from first to last." This is true when the number of intermediate steps between first and last is finite, not otherwise. Anyone who has ever watched a goods train beginning to move will have noticed how the impulse is communicated with a jerk from each truck to the next, until as last even the hindmost truck is in motion. When the train is very long, it is a very long time before the last truck moves. If the train were infinitely long, there would be an infinite succession of jerks, and the time would never come when the whole train would be in motion. Nevertheless, if there were a series of trucks no longer than the series of inductive numbers..., every truck would begin to move sooner or later if the engine persevered, though there would always be other trucks further back which had not yet begun to move.

There are contexts in which a statement P(n) can be proved for all n?N by induction, and has a counterpart P(? ) that is false. In other contexts, P(? ) may be true. But even then, induction on N does not prove the P(? ) case.

...

[Locut0s]

6. This :D nt.

[phantom power]

9. exactly, their 'S1' does not equal 1/2. that sequence doesn't have a well-defined sum.

Their entire "proof" is built on a bogus premise. It surprises me that they'd make this error, I generally think of physicists as being exceptionally good at math.

[jakeXT]

7. What's up with those clerks?

[struggle4progress]

10. Of course, the manipulations shown in the video are entirely unjustifiable. HOWEVER

that does not imply that the claimed conclusions are necessarily meaningless: it merely means that the video has actually provided no clue about the meanings of the conclusions

There are all manner of notions of sequence convergence, and some of those notions are more general than others. And there's no obvious reason that every computation-based theory, that requires evaluation of the limit of an infinite series, must use only the simple limit notion many of us learned in first-year calculus: the appropriate limit notion could ultimately depend on the actual phenomena one is trying to model

Just to make the point more clearly, I'd point out that there are ways to transform sequences, that preserve the limit when the sequence has a limit in the ordinary sense but that transform some non-convergent sequences into sequences that do converge. Here's a simple example (due, I think, to Euler): given a sequence $a_n$, define a new sequence $b_n = \sum_{0<=j<=n} C_{n,j}a_j 2^{-n}$ where the $C_{n,j}$ are binomial coefficients. One then says the original sequence $a_n$ converges, in the sense of Euler, to the limit b, if the transformed sequence $b_n$ converges, in the ordinary sense, to the limit b. There are two important facts to note now: the first is that if $a_n$ converges, in the ordinary sense, to some particular limit then also $a_n$ converges, in the sense of Euler, to that same limit; the second is that the transformation, applied to some sequences that do NOT converge in the ordinary sense, can actually produce sequences that DO converge in the ordinary sense; and thus the notion of Euler converge really extends the ordinary notion of convergence non-trivially. As an example, note that the sequence 1,0,1,0,1,0,... does NOT converge in the ordinary sense but that it DOES converge in the sense of Euler to the limit that Leibnitz wanted it to have, namely, 1/2

[Jim__]

11. Apparently there is a rigorous proof for this.

On this video John Baez, a physicist, first goes through Euler's proof (not acceptable today) and then gives a quick overview of an acceptable proof using Reimann's Zeta function. The proof ends at about 15 minutes in - the proof itself begins at about 7 minutes in, but the intro is worth watching.

[SwissTony]

13. This surely is a pisstake.

But I just can't believe the formula is given in a book - OK, maybe Martin Gardner, but he liked to play with us.

I used to amuse my students with the series S1 = 1 - 1 + 1 - 1 + 1 ...

I did the usual...

S1 = (1 - 1) + (1 - 1) + (1 - 1) ...
= 0 + 0 + 0... = 0

But then rearrange the brackets...

S1 = 1 + (- 1 + 1) + (- 1 + 1) ...
= 1 + 0 + 0...= 1

But we all know S1 = 6

S1 = 1 + 1 + 1 + 1 + 1 +1 + (1 - 1) + (1 - 1) + ...
= 6 + 0 + 0+ .... = 6

But of course, we could rewrite S1 as

S1 = 1 + 1 + 1 + 1 + 1 +1 + (1 - 1) + (1 - 1) + ...
S1 = 1 + 1 + 1 + 1 + 1 +1 + S1 = 6 + S1
Therefore, 6 = 0.

Of course, I could "prove" S1 = -1 or -6 or whatever.

But, I'm really worried about that book.

[mathematic]

14. The proof is garbage but the results are true*

*For the appropriate definitions of summability/convergence

S1 does not converge with the usual definition but it does have an obvious intuitive answer (1/2). You can generalize the summation of an infinite sequence with the notion of Cesaro summability, which is the limit of the averages of the partial sums of the series.

On the contrary, to do the operations on S1 that you list out requires a more strict definition of convergence called absolute convergence, which requires that the series with each term Ak replaced with |Ak| also converge.

1+2+3+... requires even more heavy duty mathematics to deal with but more disturbingly the result seems to be utterly devoid of intuitive meaning. I mean seriously, -1/12? Wtf, universe. There's obviously a breakdown somewhere in the chain from mathematical definitions to our senses/brains.

[DetlefK]

17. A series with two convergence-limits and he arbitrarily picks one...

I stopped watching after 1 minute.

[jakeXT]

15. Plait got himself a bit into trouble

http://www.slate.com/blogs/bad_astronomy/2014/01/18/follow_up_the_infinite_series_and_the_mind_blowing_result.html

[jakeXT]

16. In the End, It All Adds Up to – 1/12

...

Even the makers of the video, Brady Haran, a journalist, and Ed Copeland and Antonio Padilla, physicists at the University of Nottingham in England, admit there is a certain amount of “hocus-pocus,” or what some mathematicians have called dirty tricks, in their presentation. Which has led to some online grumbling.

But there is broad agreement that a more rigorous approach to the problem gives the same result, as shown by a formula in Joseph Polchinski’s two-volume textbook “String Theory.”

So what’s going on with infinity?

“This calculation is one of the best-kept secrets in math,” said Edward Frenkel, a mathematics professor at the University of California, Berkeley, and author of “Love and Math: The Heart of Hidden Reality,” (Basic Books, 2013), who was in town recently promoting his book and acting as an ambassador for better math education. “No one on the outside knows about it.”

http://www.nytimes.com/2014/02/04/science/in-the-end-it-all-adds-up-to.html

[agnostikos]

18. String theory is a cult

Last edited Wed Feb 12, 2014, 08:31 PM - Edit history (1)

The first mistake is to think that there are infinite numbers in the series. Why not? One starts with a finite number (1) and its successor is also finite, as is its successor, all the way. This is, if I recall, a crude example of 'the genetic method' in some branch of number theory. The list of integers is limitless, therefore infinite- but no number on the list is infinite. A paradox, but I think the limitation is linguistic rather than mathematical.
To put it another way, there is no finite number x such that x+1=infinity. There are other errors in the 'proof'. Infinite sums are notoriously tricky to handle, and the best advice I've ever heard is to simply ignore the nonsensical results.
Why is string theory a cult? As in, say, scientology, an outsider is not deemed worthy to ask questions about the faith; an adept is, but wouldn't dream of questioning the faith!
So why does the factor -1/12 occur so often in string theory? Because one is always bumping into (3,2)- or (2,3)-valent antisymmetric tensors.
I can't help but think these fellows are having a good laugh at our expense. "Pitiful... laymen. Heeheehee!"

[btrflykng9]

19. KnR