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Warp <war### [at] tag povrayorg> wrote:
> scott <sco### [at] scottcom> wrote:
> > There's no grouping like you did in the original proof. Which part of
> > the original proof assumes the length of the summation is anything other
> > than infinite?
>
> This part of the proof is grouping elements in pairs and summing them up:
>
> (s-4s) = 1+2+3+4+5+ 6+...
> -4 -8 -12-...
> -3s = 1-2+3-4+5-6+...
>
> > Welcome back BTW :-)
>
> I had some problems with the dreaded "can't get fully qualified domain
> name" error (which was incidentally solved by ticking one checkbox in
> an obscure system setting. But damned it was hard to figure that out.)
>
> --
> - Warp
As my knowledge of mathematics only goes up to differential equations (and is
more than somewhat spotty around the edges) I have no idea what zeta functions
are, so I had to actually do some research on why this astoundingly bad
wikipedia article is also astoundingly wrong. Or as they say on TV Tropes
(Warning: Timesink) Not Even Wrong.
https://plus.maths.org/content/infinity-or-just-112
Firstly, the article title is confusing as hell. Secondly, it opens by stating
that the sum of the natural numbers is equal to a value lower than the smallest
term in that sequence. It then purports to offer proof of this concept which is
akin to taking a true statement and tacking on another true statement which
leads to a logical result that is mathematically correct, given the whole, but
which has nothing at all to do with the original premise of the foundational
statement.
So it turns out that the result is mathematically valid, and relates to the
Casimir Effect. The problem I have with it is the astounding level of
intellectual irresponsibility in conflating the sum of Natural Numbers with a
result that clearly requires advanced analytical mathematics that few of the
people reading it will know about. Was this done intentionally in order to make
people research the topic? If so, then it's still quite unethical, since not
everybody will research it properly, and those with only a vague understanding
of the first principles will be confused by it. It's bad enough that the
youtube video presented it the way it did, but the fact that it's now on
wikipedia where the first line of the article can be taken as fact without
considering the rest of the article, and the underlying math and physics
principles, is inexcusable.
Regards,
A.D.B.
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>> There's no grouping like you did in the original proof. Which part of
>> the original proof assumes the length of the summation is anything other
>> than infinite?
>
> This part of the proof is grouping elements in pairs and summing them up:
>
> (s-4s) = 1+2+3+4+5+ 6+...
> -4 -8 -12-...
> -3s = 1-2+3-4+5-6+...
This one is different, as there is no assumption that the "..."
(infinite list) has any other properties other than "it goes on
forever". So long as both parts of the sum "go on forever" then there
will always a pair for each item.
However if you try and group elements like:
s = 1-1+1-1+1-1+1-...
s = (1-1)+(1-1)+(1-1)+(1-1)+...
s = 0+0+0+...
s = 0
Then the "..." in the 3rd (and perhaps 2nd) line makes the assumption
that there are an even number of terms, that the series ends in a "-1".
That (I think) is a wrong assumption, an infinite list/sum doesn't have
any concept of a "last" item.
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scott <sco### [at] scottcom> wrote:
> However if you try and group elements like:
> s = 1-1+1-1+1-1+1-...
> s = (1-1)+(1-1)+(1-1)+(1-1)+...
> s = 0+0+0+...
> s = 0
> Then the "..." in the 3rd (and perhaps 2nd) line makes the assumption
> that there are an even number of terms
No, it doesn't. It simply makes the assumption that you can choose
each odd-placed and even-placed number in the series (which is true)
and sum them together (which is also true). This can be done forever.
--
- Warp
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>> s = 1-1+1-1+1-1+1-...
>> s = (1-1)+(1-1)+(1-1)+(1-1)+...
>> s = 0+0+0+...
>> s = 0
>
>> Then the "..." in the 3rd (and perhaps 2nd) line makes the assumption
>> that there are an even number of terms
>
> No, it doesn't. It simply makes the assumption that you can choose
> each odd-placed and even-placed number in the series (which is true)
> and sum them together (which is also true). This can be done forever.
Yes, I just realised that you could also write s as:
s =+1+1+1+1+1+...
-1-1-1-1-1-...
= 0+0+0+0+0+...
Which makes no such assumptions.
But then you could probably just as validly (which might not be valid at
all) write s as:
s = +1+1+1+1+1+1+...
-1-1-1-1-...
= 1+1+0+0+0+0+...
So essentially you could "prove" any value you like for s.
Funnily enough if you use the standard formula for the infinite sum of
geometric progressions, you also get 1/2.
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>> https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
>
> You may also like the proof that all triangles are equilateral:
>
> https://youtu.be/Yajonhixy4g
Doesn't that come under a different category of just being a trick/hoax
though (a bit like all the 1=2 type "proofs")? As opposed to this
assuming 1+2+3+...=-1/12 thing is actually useful in other areas of
maths and science.
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Le 2015-07-27 05:19, scott a écrit :
> Maybe I'm a bit late to the party here, probably because I'm an Engineer
> rather than a Mathematician, but this seemed a pretty crazy "proof" of
> what you get if you sum all the natural numbers up:
>
> s= 1+2+3+4+5+6+...
>
> 4s= 4+8+12+16+...
>
> (s-4s) = 1+2+3+4+5+ 6+...
> -4 -8 -12-...
> -3s = 1-2+3-4+5-6+...
^
TYPO +2. Not -2.
Likewise for +6, +10, +14...
So:
-3s = 1+2+3+(4-4)+5+6+7+(8-8)+9+10+11+...
-3s = 1+2+3+(0)+5+6+7+(0)+9+10+11+...
> -3s-3s = 1-2+3-4+5-6+...
> +1-2+3-4+5-6+...
> -6s = 1-1+1-1+1-1+1-...
No.
-6s = 2+4+6+10+12+14+18+20+22...
Then the rest is wrong.
>
> 1-(-6s)= 1-(1-1+1-1+1-1+1-...)
> = 1-1+1-1+1-1+1-...
> = -6s
> 1+6s = -6s
> 12s = -1
>
> s = -1/12
>
> Crazy huh?
>
> https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
--
/*Francois Labreque*/#local a=x+y;#local b=x+a;#local c=a+b;#macro P(F//
/* flabreque */L)polygon{5,F,F+z,L+z,L,F pigment{rgb 9}}#end union
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/* gmail.com */}camera{orthographic location<6,1.25,-6>look_at a }
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scott <sco### [at] scottcom> wrote:
> >> https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
> >
> > You may also like the proof that all triangles are equilateral:
> >
> > https://youtu.be/Yajonhixy4g
>
> Doesn't that come under a different category of just being a trick/hoax
> though (a bit like all the 1=2 type "proofs")? As opposed to this
> assuming 1+2+3+...=-1/12 thing is actually useful in other areas of
> maths and science.
No professor I've ever met would accept this statement as true without the
intermediate theorems which would show (if I'm understanding correctly) that the
intent is to subtract infinity from the sum of all natural numbers.
As written, this is a false premise.
The people that made the video -knew- that they were oversimplifying the
premise. They did this to create a mystery where there was no mystery in order
to "Engage the wider public". The idea was that people would research the
topics more fully in order to understand how this could be, but the problem is
that this whole topic is useless unless you're working with quantum mechanics,
in which case you have a great deal more knowledge about mathematics, and this
becomes a carny trick.
There are plenty of interesting areas of mathematics that can be showcased to do
what they were attempting to do without resorting to mathematical slight of
hand.
Regards,
A.D.B.
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Le 04/08/2015 22:46, Francois Labreque a écrit :
>>
>> (s-4s) = 1+2+3+4+5+ 6+...
>> -4 -8 -12-...
>> -3s = 1-2+3-4+5-6+...
> ^
> TYPO +2. Not -2.
> Likewise for +6, +10, +14...
>
>
> So:
>
> -3s = 1+2+3+(4-4)+5+6+7+(8-8)+9+10+11+...
> -3s = 1+2+3+(0)+5+6+7+(0)+9+10+11+...
You are on something.
It was not a typo per itself, but the intent to make the -4s part more
dense than the s part (so as to remove the 4s every 2 terms of s,
instead of nullifying every 4 terms).
Of course, such intent is dishonest when dealing with infinite number of
terms. Is ((s -2s) -2s ) more honest ?
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> The people that made the video -knew- that they were oversimplifying the
> premise. They did this to create a mystery where there was no mystery in order
> to "Engage the wider public". The idea was that people would research the
> topics more fully in order to understand how this could be, but the problem is
> that this whole topic is useless unless you're working with quantum mechanics,
> in which case you have a great deal more knowledge about mathematics, and this
> becomes a carny trick.
It worked though - I only actually found the video after a friend at
work sent me the "proof" and started researching further. Learning a bit
more maths is never a bad thing :-)
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scott <sco### [at] scottcom> wrote:
> Doesn't that come under a different category of just being a trick/hoax
> though (a bit like all the 1=2 type "proofs")? As opposed to this
> assuming 1+2+3+...=-1/12 thing is actually useful in other areas of
> maths and science.
I'm still not sure how valid it is to use "1+2+3+..." here, even when
talking about physics.
The so-called "Euler zeta function" is the infinite sum, where n goes
from 1 to infinity, of 1/n^s, where 's' is a real number. This infinite
sum converges to a finite value for any value of s > 1. For any value
of s <= 1 the sum converges to infinity (and thus is undefined).
Bernhard Riemann had an epiphany about said function when he was
studying it (something about it being related to the density of
prime numbers), and he extended it for all complex values of s.
The infinite sum still converges to a finite value when the real
part of s is > 1 (the imaginary part can be anything), and to infinity
when the real part is <= 1 (and thus is undefined.)
There is a way, however, to extend such functions to cover the entire
complex plane in such a manner that the result is still the same for
all Real(s)>1, but defined for all the remaining complex values of s
as well (except for the single singularity at s=1+0i, which remains
undefined).
This so-called analytical continuation of the Euler zeta function is
the so-called Riemann zeta function. Said function gives the exact
same values as the former for all Real(s)>1. However, the function
is rather different from the much simpler Euler zeta function. It's
not the same function.
It turns out that the Riemann zeta function gives a value of -1/12
when s = -1. If you were to plug s = -1 into the Euler zeta function,
you would get the infinite sum 1+2+3+4+5... (try it to see.)
However, the Euler zeta function is *not* the Riemann zeta function.
They give different results for all Real(s) <= 1. When you plug s=-1
into the Riemann zeta function, you are *not* getting 1+2+3+4+5+...
You are getting something completely different (something that results
in -1/12).
Why they are somehow considered "equal", I don't understand. (I'm not
a mathematician.)
--
- Warp
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