I think that may be the right answer. This is a trickological argument to (fail to) justify the expected result, which is that within a certain context, modelling this series as -1/12 produces the same results that are obtained by doing the computation a more legitimate way.
That’s sorta like saying that 2*pi can be modelled as 0 on a plane because trig functions wrap around at that point – which is great if you’re talking about the rotation, lousy if you’re using it to compute the distance a wheel has rolled. It’s a false statement, but within a well-defined set of operations it’s a useful shortcut.
Indeed, Ms. Barbie is correct; math done properly is hard, but it can be fun too. Here, we can see that the sum of the first N integers goes as (N+1)*N/2, as the young Gauss worked out. Clearly, as all the numbers are positive, it is going to continue to increase rather than converge on a limit. And yet, in the video the same sum is expressed as a sum of other infinite series appears to converge. What is going on? Well, if you to ask Ms. Barbie, she would tell you that this video is you negative terms faster than the positive terms and that is why it appears to converge when laid out on a bit of paper in this way. But if you could lay out the convergent series in a way that it approached the sum of integers, then your negative numbers would be a lot more spaced out than the positive ones. It all shows you how careful you have to be when dealing with infinite series.
Get it? No? Oh, well, I expect if you have enough paper, you will see why eventually. Meanwhile, Ms. Barbie is going out shopping as they have the latest version of Mathematica at the mall, and she wants to test her theory about the Riemann Zeta function on her pink laptop…
I am not a real mathematician, but I think the sleight of hand is more intended to inform than conceal.
He is playing silly buggers - to an extent. Math undergrads who don’t respect the domain of convergence of a series will find their test papers coming back poxed with highly unphotogenic red Xs. But there is more going on here than meets the eye: Numberphile is saying that if you break the rules in a well-defined, consistent way, you get well-defined, consistent results, which may be highly unintuitive, but which may have actual physical applications.
I first thought of complex numbers when I saw this video too. I’m reminded of this quote, from Roger Penrose’s The Road to Reality:
Let us ask the following question: does the equation that we obtain by putting x = 2 in the above expression, namely
1 + 2^2 + 2^4 + 2^6 + 2^8 + … = (1-2^2)^-1 = -1/3
actually make sense? The great 18th-century mathematician Leonard Euler often wrote down equations like this, and it has become fashionable to poke gentle fun at him for holding to such absurdities, while one might excuse him on the grounds that in those early days nothing was properly understood about matters of ‘convergence’ of series and the like. Indeed, it is true that the rigorous mathematical treatment of series did not come about until the late 18th and early 19th century, through the work of Augustin Cauchy and others. Moreover, according to this rigorous treatment, the above equation would be officially classified as ‘nonsense’. Yet, I think that it is important to appreciate that, in the appropriate sense, Euler really knew what he was doing when he wrote down apparent absurdities of this nature, and there are senses according to which the above education must be regarded as ‘correct’.
In mathematics, it is indeed imperative to be absolutely clear that one’s equations make strict an accurate sense. However, it is equally important not to be insensitive to ‘things going on behind the scenes’ which may ultimately lead to deeper insights…For a pertinent example, let us recall the logical absurdity of finding a real solution to the equation x^2 + 1 = 0. There is no solution; yet, if we leave it at that, we miss all the profound insights provided by the introduction of complex numbers… In fact, it is perfectly possible to give a mathematical sense to the answer ‘-1/3’ to the above infinite series, but one must be careful about the rules…it may be pointed out that in modern physics, particularly in the area of quantum field theory, divergent series of this nature are frequently encountered.
So what’s the practical application for a Cesàro summation on a series that doesn’t converge? Keep in mind that the -1+1-1+1 series also doesn’t converge.
The problem with this way of thinking is that when you assuming the Universe computes like a von Neumann machine with binary memory, you automatically discount the set of other possible computation devices.
I too have to thank these guy for proving, at least to my satisfaction, that string theory is full of beans. I haven’t read any other comments yet that mention how the unseen speaker in the video says the results of this “proof” are used for real work in string theory. Very fishy.
And if the bad math (IMHO) isn’t enough to convince you, I think the fact that this is being drawn out on brown paper seals the bargain.
What Witten proposed in his talk was a method for consistently defining such path integrals by analytic continuation.
That would make his reformulation quite a bit like what I was saying in a previous post. The thing you are computing when you renormalize (or when you play with the series as in the video) is dependent on your sum being equivalent – in some formal way – to the analytic continuation of a function which actually makes sense.
Ok guys. We can all holler that these summations are playing hard and fast with the rules as we know them (which is another way of saying that in some fields the symbols we learned in middle school have different meaning)…
but let’s not make ourselves look silly by sitting in our armchairs and saying “ah ha, I always knew that this incredibly important theory about the microscopic nature of the Universe, which is being worked on by thousands of the very best minds in the world, was bollocks from the beginning, so I’m so glad I just studied underwater basket weaving instead.”
Until you understand at least as much about the theory as, say, a first year graduate student in the field, you really don’t understand enough about to rebut it. I’m sorry but it’s true. Whatever arguments you have against the pop-version of it (“10 dimensions?!? But my hand only goes through 3! Or… 4!”) honestly sounds as silly as a Fox News Reporter arguing against global warming.
I know, I know, this sounds incredibly elitist, but it’s true – string theory involves an incredible about of complicated math, and to sit back and say “well that doesn’t sound right to me, so it’s bollocks” without spending a few decades learning the math is insulting to the people who have.
There! We made a sum that would have taken hours to do on a calculator and just shuffled around the order that we did the operations in, and made it into something trivial.
The general point is that you can shuffle around adding terms as much as you like (so long as you follow rules). a + b + c - d + e is exactly the same as c + a - d + e + b.
When you have one series on top of the other, it’s perfectly fine to “shift” one of them along, as they did in the video, or even to completely flip a series as I did above. You still have to add together all the terms, you just do it in a different order.
What cool is that it takes human intuition to work out how to shuffle things around in such a way that we can make a sum that’s very difficult into one that’s trivial.
Sure, I’ll take the bait! There are a significant number (say, greater than (1-2^2)^-1) of mathematicians and physicists who have ever so many decades of education in the fields who find string theory to be little more than a toy for cats. Very intelligent gents, so I’m told, and some lovely ladies too. So between them and the folks who you accuse us of insulting, one group is utterly wrong, despite their bravura bona fides.
Them’s the breaks when you are discussing theories so early in development that there are no practical proofs or applications. On the other hand, things like man-made climate change (global warming is a terrible term) and Darwinian evolution have tons of evidence behind them. Other forms of math have been used to solve a million real problems and put men on the moon, etc. So when they use string theory to turn my Coke to Pepsi, give me a call.
That’s a bad general point, because you can’t do that for infinite series in general, even for some nicely convergent series. A classic example is taking:
Finding a way to define answers for problems that didn’t have them is an important part of mathematics; it’s where we get things from negative numbers to differentials. But here we’re allowing so much we get incompatible answers, which makes it useless.
There are different ways of assigning values to infinite series, but even the ones that give the value of -1/12 will have limitations on what operations you can do to make sure that doesn’t happen.
Which is fair, but it’s still kind of insulting to imagine one is so utterly wrong it would be obvious to anyone who understands basic calculus. Nothing here really reflects on physics; it’s only a legitimate thing in mathematics that has been obscured with an illegitimate description.
An intelligent person faced with a prima facie absurdity that seems to call a well-accepted model into question has a dilemma.
Assuming the person doesn’t have the time to devote themselves to studying every problem to the point where they themselves are an expert, they can either stifle their doubts and just assume the experts know what they’re talking about - a very difficult and unsatisfying thing to do, cognitively - or they can be skeptical.
The intelligent person only has access to their own thoughts on the subject, and doesn’t see the many lines of thought that have been tried and discarded. When presented with counterarguments one doesn’t fully understand, the natural impulse is to resist what might be seen as an attempt at obfuscating away a critical weakness in the model’s armour. The problem of sophomoric skepticism is especially bad in disciplines where the arguments seem most accessible (if you think nonexperts opining about math is a nuisance, come over to philosophy and try to explain the paradox of induction to a layman some time.)
Well the good news is it also seems to be useful in quantum mechanics. If I’m reading other references correctly, various famous physicists weren’t happy with this sort of thing even if it managed to yield results that could be verified.
You know how when everyone is standing around bullshitting and then one person says something so profoundly correct and sensible that everyone shuts up and silently nods their head?
My hat is off to you, sir. 19 “hearts of love” and not a single dissenter or even a comment. Something of a rarity, surely, for a thread on Boing Boing with nearly 60 replies.
