Not For Science Non-Believers: 1+2+3+4+infinity = -1/12

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    Feb 04, 2014 3:30 AM GMT
    You might think that if you simply started adding the natural numbers, 1 plus 2 plus 3 and so on all the way to infinity, you would get a pretty big number. At least I always did.

    So it came as a shock to a lot of people when, in a recent video, a pair of physicists purported to prove that this infinite series actually adds up to ...minus 1/12.

    http://www.youtube.com/watch?feature=player_embedded&v=w-I6XTVZXww

    http://www.nytimes.com/2014/02/04/science/in-the-end-it-all-adds-up-to.html?ref=science?src=dayp
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    Feb 04, 2014 3:41 AM GMT
    That's actually fascinating. I think the reason it relates to chaos or string theory is that mathematically these theories are based on probability laws. Like the professor says in the video, if you stop the series you get a large positive number. It's only when the series is permitted to go to infinity that you get -1/12. Sort of the same idea behind the description of electrons in an atomic "shell" according to quantum theory.
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    Feb 04, 2014 3:43 AM GMT
    Exactly! Gosh, you have brain and of course brawn!
  • MikeW

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    Feb 04, 2014 3:59 AM GMT
    Not all agree…

    Sum.png

    Despite a lot of controversy this video seems to be generating lately, I don't see a good explanation of what exactly is wrong with this proof among the top Google results, so I'm going to try to make it clear here.

    Main points

    The derivation uses "..." to denote something different in the case of computing S1 and something else in the case of S2, hence the entire derivation is invalid.

    There is no interpretation of "..." which, if used consistently alongside sums of natural numbers, would lead to S = 1 + 2 + 3 + ... = -1/12.

    Now, mathematical models may exist which use similar notation to the one used for integers and sums, but model phenomena other than natural numbers. Such models may abide by different rules and you may get results which don't carry over to the natural number world. This is probably what the String Theory book is doing (you can check it out at Google Books here). But these mathematical models would still have to be internally consistent in order to be useful, unlike the Numberphile framework which could be used to derive contradictory results, hence has no application other than entertainment.

    Details

    The first part or the proof concludes that:
    S1 = 1 - 1 + 1 - 1 + 1 - 1 ... = 1/2

    The argument behind this is that if the sum were cut short at any given point: if this point were odd, the sum would be 1 and if it were even, the sum would be 0. Tony goes on to say: "Do we stop at an odd or even point? We don't know, so we take the average of the two."

    Note that this whole argument relies on the assumption that we do stop the series at some point. We just don't know if this point is odd or even, but not stopping at all is not an option. This is an essential distinction.

    The traditional mathematical definition of infinite sums, assumes that you never stop. Since you never stop, you never stop at a point which is odd or even, hence the sum is undefined (we'd call this a divergent series).

    So what Tony is really computing here, is not the infinite sum, but the expected value of a finite sum stopped at a random place.
    In more verbose notation we would write what Tony said as:
    S1 = 1 - 1 + 1 - 1 + 1 - 1 ... (n times)
    S1 = {0 when n is even; 1 when n is odd}
    P(n is even) = 1/2
    P(n is odd) = 1/2
    E(S1) = 1/2 * 0 + 1/2 * 1 = 1/2

    There would be nothing wrong with using the infinite sum notation to denote the expected value of a finite sum. In fact, the above definition is called the Cesàro sum and is a well-known mathematical concept. The problem is that having chosen the Cesàro sum as the interpretation of "...", Tony would have to stick to this definition for the remainder of the proof. But he does not do that. In fact, for the rest of the proof, he uses the same notation to mean the actual infinite sum, which never stops.

    This is clear, for example, in his derivation of S2. The derivation of S2 is based on adding the series to a version of itself shifted by one, like so:
    S2 = 1 - 2 + 3 - 4 + 5 - 6 ...
    2 * S2 = 1 - 2 + 3 - 4 + 5 - 6 ...
    + 1 - 2 + 3 - 4 + 5 …

    This is a perfectly valid operation when you are dealing with actual infinite series, because those go on forever. So even if you shift them, they can still align perfectly all the way to infinity. But it does not work under the expected value definition, since the expected value definition assumes that the series stops at some point, in which case the shifted version of S2 would have an extra term at the end which would not align with the un-shifted version.

    The above from this page: http://www.databonanza.com/2014/01/why-sum-of-all-natural-numbers-is-not.html
  • Posted by a hidden member.
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    Feb 04, 2014 4:05 AM GMT
    The video is interesting to watch but I didn't buy it either.
    I'll watch it again tomorrow when I'm completely awake.
    (And I think you wrote the subject line incorrectly.)
  • Posted by a hidden member.
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    Feb 04, 2014 4:08 AM GMT
    You need to think in terms of probability to sum the sequence. In fact, it is a text book formula as evidenced in the video.
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    Feb 04, 2014 4:13 AM GMT
    ^^^Exactly.
    It's a difficult leap for people to make, especially as in classical math there are certainties and we always instinctively look for them. But in quantum (and string/chaos) theories, behaviors are described according to the likelihood of their existing at any given point at any given time. As I said above, the way quantum physics describes electrons not as existing anywhere but rather as having a percentage chance of existing at any given point, even though we can be 100% certain they are somewhere.

    This is why the "extra term" at the end of the series is irrelevant, and also where you stop the S1 sequence is irrelevant too. Because the point here isn't the actual sum, it's the value. The value of the term is 1/2 (or in S, -1/12). The actual sum does depend on where you stop the sequence, but the value of the sequence depends on it never stopping.

    Difficult to wrap our minds around, as I said.
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    Feb 04, 2014 4:17 AM GMT
    tumblr_mvmibiAsQh1rmwfgao1_500.gif
  • Posted by a hidden member.
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    Feb 04, 2014 4:22 AM GMT
    Actually, I think Mike's highlighted text is the somewhat blatant logical error that disproves it.
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    Feb 04, 2014 4:25 AM GMT
    The NYT said it 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.”
  • MikeW

    Posts: 6061

    Feb 04, 2014 4:26 AM GMT
    I say FIDDLE STIX:

    Ideal_68_Piece_Fabulous_Fiddlestix_Setz1


    The PROBLEM IS we assume whole integers REPRESENT ACTUAL THINGS IN THE NATURAL UNIVERSE.

    This is not so. There is no ONE THING to be found anywhere in nature and, consequently, there is no 1+1 thing much less a 1+2+3 … thing. One can say the apple sitting on the table in front of me is one thing. But the "oneness" of the apple is solely conceptual, useful for identifying and counting "things" but not factual in the absolute sense.

    Why? How can I say that?

    Well, the apple is made up of many things: Skin, pulp, core, seeds, stem… to name but a few. Each of these things, in turn, is made up of more things, and each of those is made up of still others on smaller and smaller scales, etc., ad infinitum (so far as we know). Similarly, working the other way around, no "apple" can exist independently of the tree that bore it, nor the genome of which the tree was apart, nor the environment which was home to the genome, onward and outward through time to and through the whole evolutionary process of the physical universe.

    No ONE *thing* exists independently of all others, EXCEPT as a concept. All these *things* are conceptual fiddle stix.

    What DOES exist in the natural universe is not whole numbers but ratios and relationships. Thus, for example, π (what is called an "irrational" number… well, they are "irrational" from the POV of conceptual 'whole' numbers because they are incapable of expressing them) IS THE SAME (so far as we know) *at every scale of the universe*. Thus, from the POV of what actually IS… π has a higher 'existence quotient' than the whole number 1.

    I suggest what we're really dealing with here is a form of this:

    800px-Fibonacci_spiral_34.svg.png
    A proportion or ratio that is implicit in sum addition and explicit in the multidimensional forms of the known universe at every scale of its existence.

    Of course I'm stoned and haven't a clue what I'm talking about but w/e, this is RJ right?



  • Posted by a hidden member.
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    Feb 04, 2014 4:27 AM GMT
    ^Not correct. Nature as shown by Einstein is quantum and wave like.
  • MikeW

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    Feb 04, 2014 4:28 AM GMT
    woodsmen saidThe NYT said it 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.”

    As my page points out, THAT is a context other than the one presented in the video. See my post above with the "here" link to the book page.
  • MikeW

    Posts: 6061

    Feb 04, 2014 4:29 AM GMT
    woodsmen said^Not correct. Nature as shown by Einstein is quantum and wave like.

    Of course it is but π (for example) is still evident at each scale is it not?
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    Feb 04, 2014 4:33 AM GMT
    At this point, I feel like we're all playing Mornington Crescent and must bow out...
    But the subject line is still incorrect. icon_wink.gif
    G'Nite.
  • MikeW

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    Feb 04, 2014 4:38 AM GMT
    JohnSpotter saidAt this point, I feel like we're all playing Mornington Crescent and must bow out...
    But the subject line is still incorrect. icon_wink.gif
    G'Nite.

    alice30a.gif
  • Posted by a hidden member.
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    Feb 04, 2014 5:58 AM GMT
    Aristoshark said^^^Exactly.
    It's a difficult leap for people to make, especially as in classical math there are certainties and we always instinctively look for them. But in quantum (and string/chaos) theories, behaviors are described according to the likelihood of their existing at any given point at any given time. As I said above, the way quantum physics describes electrons not as existing anywhere but rather as having a percentage chance of existing at any given point, even though we can be 100% certain they are somewhere.

    This is why the "extra term" at the end of the series is irrelevant, and also where you stop the S1 sequence is irrelevant too. Because the point here isn't the actual sum, it's the value. The value of the term is 1/2 (or in S, -1/12). The actual sum does depend on where you stop the sequence, but the value of the sequence depends on it never stopping.

    Difficult to wrap our minds around, as I said.


    Wait just a darned minute here...

    Is this how your daughter wins at poker?
  • Posted by a hidden member.
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    Feb 04, 2014 6:31 AM GMT
    ^ it is probabilistic!
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    Feb 04, 2014 8:12 AM GMT
    Let f(x) = 1 / (1-x)^2. Use algebra to check that f(-x) = f(x) - 4x * f(x^2). Plug x=1 to obtain f(-1) = -3 * f(1). But f(-1) = 1/4, so f(1) = -1/12.

    What does this have to do with anything? Well, remember from calculus the following power series expansion:

    f(x) = 1/(1-x)^2 = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + ....

    Ta dah!

    The catch here is that the power series expansion doesn't converge unless x is strictly between -1 and 1. So plugging in x=-1 or x=1 is not allowed. But if you could define the power series to make sense outside of the region of convergence, the "obvious" value would be f(x) = 1/(1-x)^2, the "obvious" value for 1-2+3-4+5-6+... would be 1/4, and the "obvious" value for 1+2+3+4+5+6+... would have to be -1/12. icon_razz.gif

    In fact there are legitimate ways to argue that the sum of the natural numbers equals -1/12, but you have to define carefully what you mean by that. The guys in the video are deliberately skirting the hard math and instead are using gee-whiz trickery that most people will want to disbelieve. It's this kind of trickery that turns people off of math---makes it look like math is full of arcane rules that you can invoke arbitrarily to prove nonsensical results.
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    Feb 04, 2014 3:07 PM GMT
    I can't really see how infinity has a solution, seeing as the definition "having no limits".

    From another perspective, you could say that numbers have a limit in terms of distinctiveness, since base numbers are 0-9 and everything after that is a combination of base numbers in an ascending sequence, which gets higher when you begin to move the decimals and add on 0s. A person could set out to count to infinity in natural numbers (counting numbers) and never reach a solution in their lifetime. Infinity could have a solution in theory, like this theory suggests, but in reality, it doesn't have one.
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    Feb 04, 2014 3:34 PM GMT
    blkapollo saidI can't really see how infinity has a solution, seeing as the definition "having no limits".

    Right, it does seem counter intuitive.
    Adding up whole numbers and ending with a fraction doesn't ring the bell of truth either.
  • FuerteC

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    Mar 31, 2014 12:24 AM GMT
    It works in string theory and other areas of physics
  • FuerteC

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    Mar 31, 2014 12:26 AM GMT
    woodsmen said^Not correct. Nature as shown by Einstein is quantum and wave like.


    That's wave particle duality. Can you think of an experiment that demonstrates light is a particle and a wave at the same time??
  • Posted by a hidden member.
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    Mar 31, 2014 12:38 AM GMT
    Not all of these apply but the conclusion of this logician is that math can be used to prove something that isn't true.
    http://listverse.com/2010/05/28/11-brain-twisting-paradoxes/
  • Posted by a hidden member.
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    Mar 31, 2014 12:48 AM GMT
    all this math stuff is like a trip down undergrad memory lane. i remember some of it, but not all of it. :/