Counter-intuitive Summation

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Counter-intuitive Summation

#1  Postby Paradoxica » Jan 10, 2014 9:31 am

Not sure if this has been discussed already... But the sum of all natural integers is -1/12. See proofs in videos below.

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Re: Counter-intuitive Summation

#2  Postby Greyman » Jan 10, 2014 10:06 am

Nice try, but the very first premise is blatantly erroneous leading to a mathematical fallacy.

1-1+1-1+1-1+..... =\= 1/2

The sum is indefinite, because the series does not converge. It does not equal to one half. You do not just take the average of the partial sums like that.


It's the equivalent of hiding a divide by zero step to prove 2=1.

1. Let a and b be equal non-zero quantities
a = b

2. Multiply through by a
a^2 = ab

3. Subtract b^2
a^2 - b^2 = ab - b^2

4. Factor both sides
(a - b)(a + b) = b(a - b)

5. Eliminate the common term
a + b = b

6. Observing that a = b (step 1)
b + b = b

7. Combine like terms on the left
2b = b

8. Divide by the non-zero b
2 = 1

QED


[EDIT:--correction noted--]
Scratch that. Apparently you can take the arithmetic mean to obtain the Cesàro sum, which does indeed lead to such paradoxical equations.

And I now have a new area of revision for the weekend.
[/EDIT]
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Re: Counter-intuitive Summation

#3  Postby Paradoxica » Jan 10, 2014 10:11 am

Greyman wrote:Nice try, but the very first premise is blatantly erroneous leading to a mathematical fallacy.

The sum is indefinite because the series does not converge. You do not just take the average of the partial sums.

It's the equivalent of hiding a divide by zero step to prove 2=1.

1. Let a and b be equal non-zero quantities
a = b
2. Multiply through by a
a^2 = ab
3. Subtract b^2
a^2 - b^2 = ab - b^2
4. Factor both sides
(a - b)(a + b) = b(a - b)
5. Divide out (a - b)
a + b = b
6. Observing that a = b
b + b = b
7. Combine like terms on the left
2b = b
8. Divide by the non-zero b
2 = 1
QED


No, the thing is, regular laws of mathematics tend to break down at infinity. If we do not approach these ideas carefully, we end up with the fallacy you posted. But the natural numbers summation is different. You need to look into it further, and the fact it is used in quantum mechanics and string theory gives the summation some significance. You cannot approach it with standard mathematics. It requires far more advanced mathematics than the standard human is expected to know.
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Re: Counter-intuitive Summation

#4  Postby Sendraks » Jan 10, 2014 10:12 am

Paradoxica wrote: You cannot approach it with standard mathematics. It requires far more advanced mathematics than the standard human is expected to know.


Non-standard humans? What are we talking about here?
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Re: Counter-intuitive Summation

#5  Postby Paradoxica » Jan 10, 2014 11:39 am

Sendraks wrote:Non-standard humans? What are we talking about here?

Mathematicians and Quantum Physicists.
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Re: Counter-intuitive Summation

#6  Postby Sendraks » Jan 10, 2014 11:46 am

Paradoxica wrote:Mathematicians and Quantum Physicists.


And non-standard humans. What are they? How do they fit into this?
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Re: Counter-intuitive Summation

#7  Postby twistor59 » Jan 10, 2014 11:54 am

Twas discussed at length here
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Re: Counter-intuitive Summation

#8  Postby VazScep » Jan 14, 2014 4:26 pm

Smells like bullshit to me.

I'll agree that ζ(-1) = -1/12, but I can't see anyone giving a purely mathematical reason why this should be taken as a sensible value for 1 + 2 + 3 + ....
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Re: Counter-intuitive Summation

#9  Postby CdesignProponentsist » Jan 14, 2014 5:05 pm

As someone who hates math, I love the Numberphile. They explain everything they cover in language even I can understand and make sense of.
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Re: Counter-intuitive Summation

#10  Postby CdesignProponentsist » Jan 14, 2014 5:09 pm

VazScep wrote:Smells like bullshit to me.

I'll agree that ζ(-1) = -1/12, but I can't see anyone giving a purely mathematical reason why this should be taken as a sensible value for 1 + 2 + 3 + ....


I would smell bullshit if there was a sensible answer. Dealing with infinite series rarely ever results in sensible answers.
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Re: Counter-intuitive Summation

#11  Postby VazScep » Jan 14, 2014 5:28 pm

CdesignProponentsist wrote:I would smell bullshit if there was a sensible answer. Dealing with infinite series rarely ever results in sensible answers.
There's plenty of sense to be found in the standard treatment of infinite series, where you just say it's the number you eventually get as close as you like to by adding more and more of the summands.

Sometimes, there is no such number for a given series, but you can come up with silly alternatives if you pretend it exists and then start doing algebra. For instance:

S = 1 + 2 + 4 + 8 + 16 + ...
2S = 2 + 4 + 8 + 16 + ...
2S + 1 = 1 + 2 + 4 + 8 + 16 + ... = S
S = -1/2.

However, this is just bullshit.

The videos above look like a combination of bullshit algebra and then a bullshit claim that values of the zeta function are generally sums of an infinite series. There might be some reason from physics to get away with this with a wink and a grin, but I can't see any serious mathematical reason to do this.
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Re: Counter-intuitive Summation

#12  Postby Pulsar » Jan 14, 2014 7:19 pm

VazScep wrote:S = -1/2

Actually, S =-1.

And no, this is not bullshit at all. Calling it bullshit is like saying that the square root of -1 is bullshit, or that a triangle for which the sum of the angles isn't 180° is bullshit. Sure, for those who've never heard of complex numbers and non-Euclidean geometry, it looks like nonsense. But that's only because they haven't learned the more advanced stuff.

All mathematical statements are derived from some set of axioms. These axioms can be changed, from which new statements can be derived. All that matters is that the results are logically consistent.

In this case, one can change the traditional criterion of convergence, and derive these results. In particular, it has to do with Zeta function regularization.
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Re: Counter-intuitive Summation

#13  Postby VazScep » Jan 14, 2014 7:47 pm

Pulsar wrote:
VazScep wrote:S = -1/2

Actually, S =-1.
D'oh. Yeah.

And no, this is not bullshit at all. Calling it bullshit is like saying that the square root of -1 is bullshit, or that a triangle for which the sum of the angles isn't 180° is bullshit. Sure, for those who've never heard of complex numbers and non-Euclidean geometry, it looks like nonsense. But that's only because they haven't learned the more advanced stuff.
There are principled reasons for allowing non-Euclidean geometries and permitting a square root of -1.

All mathematical statements are derived from some set of axioms. These axioms can be changed, from which new statements can be derived. All that matters is that the results are logically consistent.
All that matters? No. Mathematics isn't just derivations from arbitrary sets of consistent axiom systems and random definitions.

In this case, one can change the traditional criterion of convergence, and derive these results. In particular, it has to do with Zeta function regularization.
So this is getting to some sort of justification, though as I said above, I'd like to see some purely mathematical justifications, rather than one's motivated by physics.
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Re: Counter-intuitive Summation

#14  Postby CdesignProponentsist » Jan 14, 2014 11:16 pm

VazScep wrote:
CdesignProponentsist wrote:I would smell bullshit if there was a sensible answer. Dealing with infinite series rarely ever results in sensible answers.
There's plenty of sense to be found in the standard treatment of infinite series, where you just say it's the number you eventually get as close as you like to by adding more and more of the summands.

Sometimes, there is no such number for a given series, but you can come up with silly alternatives if you pretend it exists and then start doing algebra. For instance:

S = 1 + 2 + 4 + 8 + 16 + ...
2S = 2 + 4 + 8 + 16 + ...
2S + 1 = 1 + 2 + 4 + 8 + 16 + ... = S
S = -1/2.

However, this is just bullshit.

The videos above look like a combination of bullshit algebra and then a bullshit claim that values of the zeta function are generally sums of an infinite series. There might be some reason from physics to get away with this with a wink and a grin, but I can't see any serious mathematical reason to do this.


I'm not a mathematician but from what they said in the video, this isn't just mathematical trickery, but what they provided was an actual proof.
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Re: Counter-intuitive Summation

#15  Postby VazScep » Jan 14, 2014 11:25 pm

CdesignProponentsist wrote:I'm not a mathematician but from what they said in the video, this isn't just mathematical trickery, but what they provided was an actual proof.
It isn't. You need definitions in place to prove something like this, and you need to motivate those definitions. All of that is absent from the video.

These aren't proofs. And from what I've read of twistor's linked thread, this looks to me, at best, a case of "if you squint at this hard enough through a string theorist's glasses, 1 + 2 + 3 + ... = -1/12." Weak arguments like this do come up in mathematics, where there is some motivation for an idea, but you end up admitting that it is so abusive that you can't take it too seriously.
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Re: Counter-intuitive Summation

#16  Postby twistor59 » Jan 15, 2014 8:08 am

Or, if the idea of application of a regulator is a bit distasteful, you can think of it as a Ramanujan sum instead of a sum....as wikipedia says:

Ramanujan summation essentially is a property of the partial sums, rather than a property of the entire sum, as that doesn't exist.


Image

where the notation <squiggly R> indicates Ramanujan summation.

Although of course this sidesteps the original question, which was about a sum. But I personally find that attaching a famous mathematician's name to something justifies not worrying too much about rigour. :lol:

Oh I just saw this. LOL at Ramanujan's letter to Hardy:

"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …"[
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Re: Counter-intuitive Summation

#17  Postby Thommo » Jan 15, 2014 8:45 am

I watched most of the first clip and stopped, what he calls a proof isn't a proof, he performs operations without defining them or showing they have the properties he relies on (e.g. providing the definition for taking the arithmetic mean of the two values as the sum of S1 or showing that placewise addition of terms displaced by one position as used to derive S2 represents addition of sequences - i.e is commutative, associative, has an identity and inverses) which are nonstandard for the notation he uses. It may well be that they have the properties and the result is valid (although confusing as the notation matches that for divergent sequences and summation via epsilon definitions) but there's no possible way to conclude that from the video. Presumably the idea is to inspire people to go and learn some new mathematics, but personally I'm turned off by the claim of proof and bypassing of the treatment via Riemann zeta function which would make a lot more sense.
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Re: Counter-intuitive Summation

#18  Postby Calilasseia » Jan 15, 2014 10:14 am

Right, first of all, the original definition of the Riemann zeta function was as follows:

ζ(s) = sum(n = 1 to ∞ ; n-s)

For s>1, this definition produces converging series. The function is actually a special case of a class of series known as Dirichlet series, which converge absolutely provided that s is greater than a certain value.

However, when this function is extended to the complex plane, and s is a complex variable, this is where life becomes interesting. Because on the complex plane, there exists a process known as analytic continuation. In elementary terms, analytic continuation consists of the following steps:

[1] Let the region of the complex plane in which a function f(z) is defined, and within which f(z) is an analytic function, be labelled U.

[2] Let there exist another region, V, such that V contains points both inside and outside of U.

[3] Let another function, g(z), be defined within V.

[4] If a function g(z), defined and analytic within V, can be found, such that g(z)=f(z) for all z lying within U, then g(z) is said to be the analytic continuation of f(z) into V.

This process can be performed as often as desired, in order to extend a function across the entire complex plane, provided that suitable regions V and suitable functions g(z) can be found in each case.

Now of course, the rigor of this procedure hinges upon the definition of 'analytic function', which is a function that can be represented within the region of definition by a convergent power series. An elementary and trivial example is f(z)=ez, which happens to be representable by a convergent power series over the entire complex plane. The inverse function, f(z)=ln(z), also admits of a convergent power series, but not over the entire complex plane, and when analytic continuation is applied to this function to extend it over the entire complex plane, the resulting function is actually multi-valued, and one has to select a principal value from the infinitely many options. Basically, we have:

z=re

ln(z) = ln(re) = ln(r) +ln(e)

= ln(r) + iθ

However, on the complex plane, θ + 2nπ = θ for all θ, therefore:

ln(z) = ln(r) + i(iθ+ 2nπ) (n is any integer)

Therefore for the complex logarithm, one has to select a particular value of n and choose that as the principal value (usually, the choice n=0 is made).

Now, in the case of the Riemann zeta function, life gets interesting, because not only is analytic continuation possible (and has been done for this function), but when that analytic continuation has been performed, the result is a function whose value ζ(s) for values of s<1 is finite. In particular, ζ(-1) = -1/12. But, referring back to the original definition of the Riemann zeta function, ζ(-1) is the sum of the natural numbers, which from an elementary standpoint is divergent (and hence infinite). But of course, the analytically continued version of the Riemann zeta function departs from the original power series definition outside the region s>1, because the analytically continued function is built from pieces involving other power series. At least, that's my understanding of the issue.

Of course, a proper treatment of this takes one rapidly into the wonderful world of sheaf cohomology, which is beyond my remit I'm afraid. :)
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Re: Counter-intuitive Summation

#19  Postby Pulsar » Jan 15, 2014 9:31 pm

VazScep wrote:There are principled reasons for allowing non-Euclidean geometries and permitting a square root of -1.

There are principled reasons for these series too, see the posts from Twistor and Cali.

VazScep wrote:Mathematics isn't just derivations from arbitrary sets of consistent axiom systems and random definitions.

Sure it is. People didn't sit down and decide to invent non-Euclidean geometry or complex analysis. Discoveries are made in more haphazard ways, by trying out things and seeing what happens. Motivations are filled in after the fact.
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Re: Counter-intuitive Summation

#20  Postby scott1328 » Jan 15, 2014 9:43 pm

Thommo wrote:I watched most of the first clip and stopped, what he calls a proof isn't a proof, he performs operations without defining them or showing they have the properties he relies on (e.g. providing the definition for taking the arithmetic mean of the two values as the sum of S1 or showing that placewise addition of terms displaced by one position as used to derive S2 represents addition of sequences - i.e is commutative, associative, has an identity and inverses) which are nonstandard for the notation he uses. It may well be that they have the properties and the result is valid (although confusing as the notation matches that for divergent sequences and summation via epsilon definitions) but there's no possible way to conclude that from the video. Presumably the idea is to inspire people to go and learn some new mathematics, but personally I'm turned off by the claim of proof and bypassing of the treatment via Riemann zeta function which would make a lot more sense.

I agree completely about the first video, the second video is much better about this though.
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