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I read them. Cali's post just outlines how to define the zeta function by analytic continuation without addressing the crucial question of what this has to do with defining 1 + 2 + 3 + 4 ....
My emphasis.Calilasseia wrote: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.
A bunch of guys working on solving cubic equations in the 16th century sat down and started messing around with imaginary numbers. The work was justifiably regarded as convenient bullshit until William Rowan Hamilton showed how to exhibit the complex numbers as pairs of reals, and is now even more justified when we think of complex numbers as the algebraic completion of the reals. Similarly, a bunch of guys in the 18th century sat down and starting messing around with geometry whilst denying the parallel postulate. Their work was justifiably regarded as mere bullshit until a model of non-euclidean geometry was exhibited, and folk like Riemann gave a general account of geometries.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.
Lubos Motl wrote:a [right-wing] as*hole
Oh, good. There’s this claim going around that the sum of all natural numbers (1+2+3+4+5…) converges on the value -1/12. I saw that and said to myself that it’s obviously wrong, but saw the smooth patter and rapid-fire use of mathematical jargon and infinities, and no mathematician myself, couldn’t see where the error slithered in.
Thommo wrote:Clearly instead of the approach they used they could instead have assigned S1 a value by algebraic manipulation:-
S1 = 1 - 1 + 1 - 1 + 1 - ...
1 - S1 = 1 - (1 - 1 + 1 - 1 + 1 - ...) = 1 - 1 + 1 - 1 + 1 - ... = S1
Thus 2S1 = 1, S1 = 1/2
twistor59 wrote:OK, more from the physics point of view, but Lubos Motl, who isLubos Motl wrote:a [right-wing] as*hole
has just posted on this.
Pulsar wrote:As simple as the proof is, many people clearly don't understand infinite series. Many commenters on that video think they can disprove the result, and fail completely (despite getting massively upvoted; one comment has 91 votes as I write this). I've never seen a Dunning–Kruger effect on such a massive scale. Fascinating.
This is very nice, but it seems to be about something other than integers and addition, I assume this is what VazScep was getting at. You need to say something about what your symbols and operations are, otherwise your result appears to be false by contradicting the well ordering of the integers.Pulsar wrote:There's something quite pleasing about the fact that
S = 1 + 2 + 4 + 8 + 16 + ... = -1
Look at how S would be written in binary:
1 = ...000001
1 + 2 = ...000011
1+ 2 + 4 = ...000111
...
S = ...111111
So S looks like an infinite string of '1s'. What would S + 1 be? The rightmost digit would become 0. But the digit next to it would also become 0, and so forth
So S + 1 = ...000000 = 0,
therefore S = -1.
Both those infinities are countable, so they're the same size.JSS wrote:Isn't it true that "all infinities are not alike"?
If you do anything to change the number of terms in an infinite series, you can still claim that it is infinite, but not identically infinite, not "equal to" the original.
[1+1+1+...+1] =/= [1+1+1+...+1] + 1
JSS wrote:How do you measure "size" such that they are the same size?
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