Never. With Cory? He would have just closed down the comments. I don’t think I’ve ever seen Cory retract something after reading the comments.
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I used to be on the fence about String Theory, but now I’m even more convinced that it’s fundamentally flawed.
Ok, I’m not a mathematician, so I have to live in the real world sometimes, but when he added two completely different sums together and shifted them around to get a result, I went “are there even rules here?” I’m sure this is a well respected form of mathematics, but I’m not sure how much it relates to the real world. He talked about it mapping to physics, but that physics was String Theory, so it’s really still just math. I was also dubious that the +1 -1 sum is 1/2, that seemed like a bit of magic.
Overall this video felt like one of those “Proof that 1 = 0” things you see on the internet from time to time.
I can’t help but feel like if I did a technique that yielded -1/12 from a simple sum of positive integers, I would suspect a flaw in my technique.
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The sum of any number of positive numbers will be positive. Any other result is just fucking around with the rules. And he did it right off the bat with averaging his first result to 1/2.
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This stuff is pretty cool, but it’s only really true if you take “sum” to mean something different than what most people mean.
Also,
“string theory”
“real-world, useful stuff”
Not sure I can follow where you’re taking me.
A retraction is definitely called for here.
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f***ing integers - how do they work?
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what are those fucking asterisks supposed to mean?
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Perhaps, but there are things like that going back before string theory. Well-established quantum field theory doesn’t work without renormalization, where you basically take infinite quantities on either side of the equations and cancel them out. If you know some math, you know you shouldn’t be able to do that and get a sensible answer, and I believe Dirac thought it mean his theory was fundamentally flawed.
But it turns out it’s not as bad as that; if renormalization looks absurd at face value, there are reasons it works where it’s applied. I’m no expert, but I think it has to do with being able to treating our universe as a limit of not-quite-solutions, adding more and more virtual particles to your diagrams until they converge on something that works.
I would suspect that’s why this result applies too, since as others have explained, -1/12 is not a solution to the infinite series but is something certain other not-quite-the-same infinite series converge on. The real trick would be knowing what justifies using those not-quite-the-same series and not others, though.
It’s a shame this wasn’t kept straight, as if you would want it to look like a 0=1 proof. I would rather the miraculous be made understandable than the other way around.
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And that’s NumberWang!
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Agreed! That is a good explanation.
If you were inclined to be slightly more charitable while making the same point, I guess you could also say he’s using the usual meanings of + and =, but he’s changing the value of “…” in a way which is, apparently, the right thing to do in string theory. But it’s still wrong to do this in mathematics without at least saying what you’re doing. On the internet there’s more people who know a little bit of mathematics than a little bit of string theory. So the video comes off as a honeypot for LOLWUTs, for most of us.
Here’s what’s even worse though. I don’t see how to arrive at 1/2 = 1 + 0 + 1 + 0 + … even if “…” means “interpret the preceding stuff as the simplest possible weighted average of values of the Riemann Zeta function”. If dude would explain that in a youtube video I would be much mollified.
In order to support his claims about the sum of all positive integers, he leaps immediately to a series that contains negative integers. Nothing questionable about that, surely.
IANAM (I am not a mathematician) but, I believe it’s axiomatic that the sum of 1 or more integers must be an integer. And that the sum of 1 or more positive integers cannot be negative.
Finally, we could observe that for any number n, the sum of the first n positive integers is greater than the sum of the first (n-1) positive integers. Or, approaching from the other end, the sum of the first (infinity - 1) positive integers must be smaller than the sum of the first infinity positive integers. Ergo, the sum of the first infinity positive integers cannot be less than the sum of the first 1 positive integers, i.e. 1.
His ‘proof’ looks more like mathematical sleight of hand, and we should be looking closely for the misdirection that allows him to distract us from what he’s really doing and produce this counterintuitive ‘answer’.
As I was reading on Quora the other day, the type of summation referred to here is not the type of sum people use when adding up their monthly bills. It’s a special kind of sum that only makes sense to mathematicians.
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All numbers are 42 if you multiply them by zero and add 42.
PhD, please.
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When he doubles S2, adding a second series to the first, he says (at 3:09) “but I’m going to shift it along a little bit”. How do these “maths” of yours work, again?
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Yes, he did break the rules there, but not by averaging. What he claimed was that
S1-1 = 0-1+1-1+1…
S1+(S1-1) = (1-1+1-1…)
(+0-1+1-1+1…)
=0
–>2S1-1=0–>S1=1/2
Definitely breaking the rules for infinite series (i.e. if I let myself rearrange the order in which I write down the natural numbers, I can “prove” that only every billionth number is odd), but not by averaging.
There are reasons why calling these series 1/2, 1/4, and -1/12 can be useful in certain contexts, but as described the explanation is incorrect.
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“Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn’t belong in mathematics.” - Gauss
The problems start right away with 1 + -1 + 1 … with partial sums 1, 0, 1, 0 … and we learn at this stage to try and find a limit which doesn’t exist. Instead the video proceeds to say with this new series start taking averages of the first n terms, so 1, 1/2, 2/3, 2/4, 3/5 and NOW take a limit and one gets 1/2. This is still somewhat reasonable for certain purposes, a Cesaro mean. But the purpose determines first that we do an averaging procedure and when we can finally take the limit.
The title grabs attention because the timing of the limit occurs in an unnatural way. In the end probably only limits which converge absolutely to a finite number are immune to these games.
So the sum of all positive integers, which are all by themselves greater than Zero, is a number that is not only not an integer, but is less than Zero?
I definitely do not understand 
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This reminds me of the complex analysis class I took in grad school. I never really got complex numbers until I took this course. Of course, I could do trigonometry and algebra with complex numbers, and I (sort-of) understood Fourier and Laplace transforms, but once I saw the gears and pulleys underlying complex numbers, it all made sense.
I think the comparison to complex numbers is apt - of course you can’t obviously multiply a number by itself to get a negative number. But it’s a useful thing to do. So let’s just invent a way to do it and work the logic out, even if your accountant can’t see how it’s useful in balancing a ledger. String theory isn’t just a matter of balancing a ledger either. Maybe we need to invent new mathematics to talk about it.
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This is ridiculous. Everyone knows the sum of all positive integers is 42.
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