"This result is used in many areas of physics..."
*opens up a string theory book*
Maybe those areas of physics are wrong?
It could be the 'proof' is wrong, even obviously so, and it's up to a competent mathematician to show how it's wrong? Maybe one shouldn't do arithmetic on series like that in special cases?
Didn't Godel use logic to 'prove' the existence of god - not to show how strong logic was but to show that if something is convoluted enough you could prove anything you wanted to prove?
No, Godel never did that. Godel, most famously, proved that not all mathematical facts can be proven from a consistent set of axioms expressed in a language suitable for expressing all the mathematical facts.
He did have, unpublished, a version of the ontological proof for God's existence, but that is because he thought that argument worked.
Bertrand Russel worked out the proof that 1+1=2, much to the consternation of David Mitchell that wonders what would've happened if it hadn't been proven:
I think you are thinking of Pascal's Wager.
Wherein Pascal said that if you want to gamble a finite set (your years on earth) on either an infinite set of your years in heaven, or a null set of you just dying and being worm food, then you would be stupid if you didn't go all or broke.
What pisses me off about the video is not the result. Yes, in some sense, the series "equals" -1/12. Rather, it's the way they force it to be "counterintuitive." The only reason it runs against our intuition is because they are switching up what they mean by "summation" without telling us.
The series DOES equal "infinity" the way we expect it to, in the same way that 1 + 1/2 + 1/4 + 1/8 + ... = 2. The intuition is not wrong. They're just using other definitions that allow us to assign meaningful values to series that diverge under the "intuitive" definition.
The other thing is, in his attempt at a simple proof, he uses unjustified steps that are not allowed in general. You can't just add or subtract any two divergent series with shifts like that. If you could do that in general, you could subtract 1 + 1 + 1 + ... with a shift, and you'd get 0 = 1.
That should say "subtract 1 + 1 + ... from itself with a shift".
It puts me to mind of imaginary numbers. You start by defining the impossible, sqrt(-1) = i. Then you can do all sorts of interesting and at times very useful operations using i ( hey I use complex number a lot! ). And some counter intuitive things arise, but only because at the heart of it is this impossible thing that we've put off by defining it in a way that makes it amenable to said manipulation.
In fact, why not use the same approach? ( mathematicians here, feel free to call me an idiot and explain... ).
So I define the Wildcationry number o, such that the sum of S1 is now o rather than 1/2. Then our sum S is a more intuitive o/6 rather than 1/12. Intuitive in the sense that it's clear we've done some redefinition of things to achieve the result.
Any calculator will tell you that infinity looks something like this:
8=======D ------ ( ͡° ͜ʖ ͡°)
(While I'm a mathematician, I'm only a topologist, so take this with a grain of salt.) The reason 1/2 makes sense for S1 is one of analytic continuation. The infinite series 1 + x + x^2 + x^3 + . . . converges to the function 1/(1-x) as long as |x| < 1. But 1/(1-x) is still defined for any x other than 1. So we can smoothly extend the identity 1 + x + x^2 + x^3 + . . . = 1/(1-x) for all x not equal to 1. In particular, x = -1, which gives us 1 - 1 + 1 - 1 + ... = 1/2.
The 1 + 2 + 3 + ... = -1/12 comes from the continuation of a different series, the Riemann zeta function 1^(-s) + 2^(-s) + 3^(-s) + ... for all (complex) s not equal to 1. When s = -1, you get the series in question.
Presumably you mean |x|>=1? But I get the idea; that's a more compelling argument for 1/2 than the clip presents. I am also happy we have drug imaginary numbers into this with the Riemann Zeta function, because I'm just that kind of idiot (grin).
No, I meant |x| < 1 (If that is the place you're referring to). The geometric series only converges if -1 < x < 1.
|Oscar Wildcat |
It's that first sum where all the action is taking place. If you buy that, the rest is just manipulation. S1's result is not the only possible result; just the one we've all decided is the "best" possible one for what amounts to an impossible problem.
If only we had such flexibility in the physical world (grin).
I'm not going to say whether they're right or wrong, or try to find holes in the logic they used to reach their conclusion. I'm perfectly fine with it, if it works for them. I DO think this is a perfect example of how the average person doesn't really understand the concept of infinity, and how hard it is for a mathematician to explain it. It's one of the most counter-intuitive ideas in existence.
It's really late, so I can't watch this without waking room mates up.
But, I want to know, which infinity are they referring to? The countable or uncountable kind?
|infinite zest |
Goddamnit. I'm having flashbacks to a Philosophy grad course I took back in undergrad where String Theory and Popper's theory of Falsifiability were used to prove that things like Newton's and Einstein's theories were only as true as they could be disproved. I asked why the definition of "Infinity" is a "known" quantity but cannot be proved with any more unfalsiability than a theologian can prove that there is a heaven or hell, much less what it looks like.
I fucking hated that class.
Ad hoc is Ad hoc. If I say the sky is always blue, I can't disprove someone who's seen the sun go down.
"I can't understand this, so it must not make sense"
- Several poeTV users.
"I can't understand this, so it must make sense"
- Several other poeTV users.
So we define a sum S such that the summation of an infinite number of users opinions is undecided (1/2).
That makes sense to me.
I'm going to go with
"I can't understand this because I have an eighth grade education, so maybe I'll just keep my mouth shut." XD
Can you perhaps answer something? That first series (1 - 1 + 1 ...) sure seems like it could benefit from the "clone, shift, and add" process, where you would quickly reach 2S = 1 and then S = 1/2, without the fishy recourse to "eh, let's split the difference". Is there some reason he didn't do that? It's not like he wasn't going to trot out that technique a minute later anyway.
Oh, good. One of the commenters cites that effing "What the BLEEP do we know?" video series.
The answer is "Not much if you watch that."
I was lucky enough to see exactly one scene from that video. A friend gave it to me in a WMV file, and I fast forwarded to a random scene -- the one where Armin Shimerman (DS9's Quark) is saying to Marlee Matlin: "If thoughts can do that to water, imagine what our thoughts can do to us."
That is everything I could have hoped to get out of that video, and more.
Given he also appeared in the Atlas Shrugged movies, I'm beginning to thing Alan Shimerman isn't too many degrees away from starting his own Art Bell/Alex Jones podcast.
The whole thing is contingent upon accepting that S1 really sums to a number that the sequence never converges toward.
This is stupid. If an infinite sequence never converges toward a single number then it doesn't have a sum. Saying that it really sums up to the average of its possible outcomes is the mathematical equivalent of saying that the average human is a hermaphrodite.
|Scrotum H. Vainglorious |
I think I understand his more than I do Hearts of Iron 3.
Mathematics is amazing, and so is physics.
I wish this video included real physics.
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