Why is it not more likely that mathematicians need to reexamine

>>8773748
It is a better option

You're free to try to reinvent the wheel tho, no one is stopping you. Oh wait, it's fucking hard? That's the way the rest of them feel like.

Because it might be true even if it doesn't make any goddamn sense yet.

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Literally the best country in North America

>>8773764
Wrong thread lmao

>>8773758
No. It's not true. If I add one rock to a pile, then 2, then 3 is my pile getting bigger or smaller? When does my pile magically dissappear in negative rocks?

This is nothing more than proof we should have taken mathematicians lunch money at some point

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>>8773748
the statement is not "if you add up all the positive integers you get -1/12".
that's fucking stupid. you can't sum infinitely many things.
if you think that anyone with a proper mathematics education believes the above statement then you are delusional.

the real statement is that "there exists a method of assigning values to infinite series, which (a) agrees with the familiar limit-of-partial-sums when both are defined, (b) has great number theoretical significance, and (c) assigns the value -1/12 to the series whose summands are the positive integers."

please stop pretending that your deeply flawed understanding of pop math topics is accurate, it makes you look like a moron.
maybe read a fucking book. here's one: "introduction to analytic number theory" by apostol. it's big and yellow and has fun little squares on it you can look at when you're done pretending you can do big boy math.

>>8773770
It's an abuse of notation. The notion of "infinite sum" here is different then the usual.

>>8773748
Apparently, the proof makes sense under weird other methods of counting. I haven't bothered to look into that.

Obviously, the proof isn't true for the real world, but it doesn't matter, sense you can't have infinite objects in the real world anyways.

its more of a weird quirk of infinite sums than anything

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>>8773770
You have a geometric intuition/model for why [math] 1/2 + 1/4 + 1/8 + ... = 1 [/math] should be true and in analysis you learn a theory taylored to make the geometric intuition "true". And then you go on and require all notions of infinite sum follow some intuition of yours.

Just because something is true for the standard math of finite objects doesn't imply it's true for an infinite setting. This is a totally different context.

Consider the claim
>if a set B is bigger than some smaller set S (i.e. |B|>|S|), then the set of all subsets of B, i.e. the power set PB, is bigger than the PS, i.e. |B|>|S| implies |PB|>|PS|).
This is evident for finite sets but unprovable in general (like the continuum hypothesis)

You can write down a theory of infinite sums where the sum of all natural numbers can't be evaluated, and you can write down theories where it's -1/12. And the latter is a value that various seemingly independent theories agree upon. Besides, the theories where the sum equals -1/12 are none that many people learn about (e.g. with sums alla Ramanujan or defined via analytic continuation).

>>8773883
That language is a little sloppy.
Even if you e.g. don't believe in an infinite amount of objects in your space, in case you believe in the contiuum of space, and that i.e. e.g. makes sense to consider an unending sequence of parting e.g. 1 inch into two parts, you'll end up with the need for mathematical theories that involve unendinging aspects.
And besides, the -1/12 results is used in quantum electrodynamics, the theory well suited to describe lasers and electronics on a microscopic level.

>>8773908
Huh, TIL

Also isn't space not continuous due to Planck length shit, or is the Planck length just a limit to observation

>>8773748
[math] \displaystyle
\zeta \neq \Sigma
[/math]

>>8773770
It's easy to confuse yourself with this shit but it's quite simple.

All that the ramanujan summation stuff, cutoff and zeta regularization does, is look at the smoothed curve at x = 0.
What sums usually do is look at the value as x->inf.

It's just a unique value you can assign to a sum, really they have many such values.

https://en.wikipedia.org/wiki/File:Sum1234Summary.svg

>>8773923
You're mistaken, the Planck length (basically a unit) doesn't affect the continuous model of space (or spacetime) that underlies essentially all quantum theories/quantum field theories. Planck length as limit of observation doesn't apply either.

>>8773943
Did you write a bot that always prints this post?

>>8773748
Why is not more likely that OP is a shitposting high schooler who's talking out of his ass because he hasn't even taken Calc 1?

The point of logic is to value consistency and rigor over intuition.

>>8773748
The fuck kind of equation is this?

>>8773748
Mathematicians don't claim that this is true. The method of obtaining this answer involved using a valid method of computing the sum of series, which is valid when you plug in numbers between -1 and 1. If you plug in -1, the method technically doesn't work because the series then diverges but if you forget about that fact, the sum of the natural numbers would be -1/12.

The sum of the natural numbers diverges but if you try to sum it anyway, you get -1/12.

>>8773871
>the real statement is that "there exists a method of assigning values to infinite series, which (a) agrees with the familiar limit-of-partial-sums when both are defined, (b) has great number theoretical significance, and (c) assigns the value -1/12 to the series whose summands are the positive integers."

You forgot the clause "(d) satisfies the familiar properties and axioms of addition and sums, as implied by the use of the summation sign".

You will no doubt notice that this clause is false.

>please stop pretending that your deeply flawed understanding of pop math topics is accurate, it makes you look like a moron.
Honestly, I feel this one is the fault of the mathematicians. Deeply flawed notation leads to deeply flawed understanding.

>>8773748
Because this sum exists in nature too. Ever heard of the Casimir effect?

I just think it's fucking cool that if you do analytical continuation of the rieman zeta function for 0 and 1 you get the same value as you would if you use partial sums of 1+1+1... and 1+2+3....

>>8774069
There aren't so many symbols, just because [math] \sum [/math] for finite sums has some properties, doesn't mean all uses must do so too.

>You forgot the clause "(d) satisfies ...".
>You will no doubt notice that this clause is false.
That doesn't make sense.
Yes, it doens't behave like finite sums. But that's no argument.

For example, if you add a finite quantity of rational numbers, you get a rational number. But using the theory of limits, i.e. standard infinite sums,

[math] 4 \sum_{k=0}^\infty\frac{(-1)^k}{2k+1} = \pi [/math]

and [math] \pi [/math] isn't rational.

You don't want sums of positive numbers to be a negative numbers. But again, that's just because you have a physical intuition for the sum defined in the context that's tailored for real geometry.

>>8773748
This makes no sense.

>>8774310
>i dont understand it it must be wrong

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>>8774320

>>8774320
There should be a comma after that first it, brainlet.

>>8774331
The comma was omitted because I'm mocking them, using improper grammar to imply that they aren't smart. Also maybe see a proctologist for the stick up your ass.

>>8774360
>No comma after Also

>>8774444
Nice quads

>>8773770
You can never get to infinite rocks, so your point is not valid.

>>8774078
also the Lamb shift

http://physicsworld.com/cws/article/news/2016/dec/08/sonic-lamb-shift-detected-in-ultracold-atoms

>>8773877
>>8773877
>>8773877
>>8773877
>>8773877
Fucking this.

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>>8773943
>look at the smoothed curve at x = 0
wrong

>>8774679
You need more points

>>8773748
The moment you start writing 1 + 2 + 3 + 4 + ..., your sum STOPS MAKING ANY GOODMAN SENSE.

You can twist it in any algebra you want, it will still be utterly senseless. Slap any algebra into it and you might find literally everything.

Case in point with 1 - 1 + 1 - 1 + 1 - ...
This sum diverges. The writing above (with the dots) has no meaning.

I can "show" it has 3 different values.
(1 - 1) + (1 - 1) + (1 - 1) + ... = 0

1 - [(1 - 1) + (1 - 1) + ...] = 1

s = 1 - 1 + 1 - ... = 1 - (1 - 1 + 1 - 1 + ...)
s = 1 - s
s = 1/2

See ? That's why 1 + 2 + 3 + 4 + 5 + ... doesn't work : the number literally doesn't exist.

>>8773770

The sum of 1 + 2 + 3 + 4 + ... , is undefined (many such cases). However, plotting a graph of sums that are defined makes a nice picture of lines that end abruptly where sums are not defined.

Recall the function, f : y = 2x over the interval 0 <= x <= 1 using the cartesian coordinate system. This produces a line with endpoints at (0,0) and (1,2) . As a thought experiment, you could "evaluate" the function when x = 2 . Normally this wouldn't make any sense, 2 is not in the domain of x and hence the function is not defined when x = 2 . However, you could imagine using one of the points of the line: let us say (0,0) for simplicity, and the slope of the line: 2 , to "extend" the line through values of x for which the function is undefined, and would result in the point (2,4) . You can say that by continuing the line analytically you arrived at a reasonable solution outside of the domain and range of f . In this case, the solution is trivial and my example using an interval is cheating a bit to illustrate the basic point, but similar methods of analytic continuation are used in more complex instances to evaluate functions over domains and ranges where the function is undefined.

>>8774069
>You forgot
i said what i meant, you hubristic moron

you can't sum infinitely many things using the finitistic axioms of numbers. there are too many things to combine. as soon as you put an infinity on top of the sigma you toss your familiarity with anything out the window.

hence why i stressed that the shit you learn in calc is just one way of assigning values to the otherwise bogus symbology[eqn]\sum_{\text{whatever}}^\infty[/eqn]

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>>8773748
x = 1 + 2 + 3 + 4 + ...
y = 1 + 1 + 1 + 1 + ... = infinity
x + y = 2 + 3 + 4 + 5 + ...
so x - (x + y) = 1
i.e. -y = 1
But y = infinity
So infinity = -1
shantih shantih shantih

>>8775064
IQ points that is

>>8773871
The real problem is that they used the "=" symbol. They should have invented a new symbol that means something else. Saying those two quantities are equal was just asking for ridicule.

>>8773748
Same damn thread every time.
We need a banner for these nincompoops.
IT SOMETIMES ACTS LIKE -1/12 BUT NOT ALWAYS
-1/12 AND ANY OTHER SEEMINGLY STRANGE ANSWER LIKE 6^0=1 OR 0.5! = sqrt (pi)/2 (or whatever it was) IS BECAUSE OF PATTERNS FOUND THROUGH REGRESSIONS.

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>>8776495
This is a meme. Saying that the sum 1/2+1/4+1/8+ = 1 "deserves" the equal sign is bs. This result, while much more useful than OPs, is also just a theorem from SOME mathematical theory. Unless you're a mathematical platonist, there will be no reason to reserve this character for some particular theories.
E.g. in Peano arithmetic, 2+7=9. In modular arithmetic modulo 4, you have 2+7=1.
Both are theories helping you with some situations, and when you set up formal logic, you have some standard axioms for = and add your paritcular axioms, for =, on top of it. E.g. pic related 7 and 8, which are particular for nats, vs. the first few ones for =, such as a=b implies b=a, which is taken to hold for all theories that use equality.
Don't be a mathematical realist and then go on to choose your Platonism according to what you've found most useful.

>>8776521
No. If I have more things, the number is bigger. Always. There is no debate.

>>8774679
Step 0, a(x) = -0.8(3) = -1/12.

>>8776212
x = 1 + 2 + 3 + ... = infinity
y = 1 + 1 + 1 + ... = infinity
x + y = infinity
so x - (x + y) = infinity - infinity, which is undefined

>>8774679
>>8775064
How is this supposed to work, with two coefficients?

[math] f(x) = \frac{1}{12} \left( - 1 + a_1 \cdot x + a_2 \cdot x^2\right) [/math]

With two coefficients, you end up with

[math] f(x) = \frac{1}{12} \left( - 1 + \frac{15}{2} x + \frac{11}{2} x^2\right) [/math]

so that
[math] f(0) = -\frac{1}{12} [/math]

and

[math] f(1) = -\frac{1}{12} \left( 1 - \frac{26}{2} \right) = 1 [/math]

and

[math] f(2) = -\frac{1}{12} \left( -1 + \frac{37}{1} \right) = 1+2 [/math]

but then you don't have any more coefficients to fit anything into.

>>8776798
Dumbass, why don't you research why it's sometimes-1/12