/sci/ - Science & Math

>>14886311
Does anyone have that "Mathematicians have taken us for absolute fools" infographic?

>>14886317
This one?

>>14886334
Yes, Thanks anon!

>>14886311
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

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

https://youtu.be/sD0NjbwqlYw?t=10m

>>14886334
He has point. How many apples would that be?

>>14886311
OP, let me explain it to you as someone who fell for the 1 + 2 + 3 + ... = -1/12 meme, i'm not a mathematician but i think i figured it out (an actual mathematician might correct me here)
This sum above is not convergent, anyone with a brain can see that, any "proof" you see with someone letting S = 1 + 2 + 3 + ... and then doing some tricks to show that S = -1/12 is an invalid proof because the order in which the terms are summed matters in an infinite series (you have to read how an infinite seires is defined to understand why)
When someone says 1 + 2 + 3 + ... = -1/12 then they are either dumb or they mean that when you define [eqn] Z:(1,\infty)\rightarrow \mathbb{R}^+ \\ Z(s) = \sum_{k=1}^\infty \frac{1}{n^s} [/eqn]Then take the analytic continuation of [math] Z [/math] and call it [math] \zeta [/math] then what we really mean by the above statement is that [math] \zeta(-1) = -1/12 [/math] which is not the same as saying [math] Z(-1) = -1/12 [/math] because these are not the same function but we often use the latter as a shorthand for the former.

>>14886443
>How many apples would that be?
"1 Real projective plane embedding into [math]\mathbb{R}^3[/math]" Apples

>>14886494 (me)
>analytic continuation
And to be honest here, i don't know what analytic continuation means, but i know it's some kind of operation or transformation that takes in a function and outputs a function, somewhat akin to taking the derivative where you get a new function from another function

>>14886494
>>14886504
>OP, let me explain it to you as someone who fell for the 1 + 2 + 3 + ... = -1/12 meme, i'm not a mathematician but i think i figured it out (an actual mathematician might correct me here)
I think "mathematician" is a term for college kids looking for an identity, but as someone with a math phd I'll just comment: your explanation is perfect, also
>And to be honest here, i don't know what analytic continuation means
An analytic function is a function that can be described entirely by its taylor series. An analytic continuation is where you take a function that is defined on some subregion (say [math](1, \infty)[/math] in this case), take its taylor series expansion there, and then start plugging in values that were not in the original domain, like in this case [math]-1[/math]

>>14886504
>i don't know what analytic continuation means
It just means taking a function of one set and expanding it to work in a different set.
e.g. The factorial function ( x! = x* (x-1) * (x-2)... * 1 ) only deals with integers.
But you can continue the idea over to the gamma function ( f(x) ≈ (2πx)^.5 * (x/e)^x ) to get further probability related applications without being limited to integers.

>>14886529
Wouldn't the only reason to get a phd be that your identity is the phd subject matter?
I don't think it'd make much sense to dedicate that much time and money towards doing something if it were anything less than what you were staking your identity to.

>>14886586
That's what a continuation is, but your explanation doesn't talk about what's specific about an analytic continuation vs other types of continuation.

>>14886592
>Wouldn't the only reason to get a phd be that your identity is the phd subject matter?
>I don't think it'd make much sense to dedicate that much time and money towards doing something if it were anything less than what you were staking your identity to.
Math is something you do, not something you are. I got a Math PhD because I love math.

>>14886504
>>14886494
It's actually relatively easy to understand the sum 1+2+3+...--1/12, and you don't need to know about analytic continuation. You can probably understand it in like 10 minutes if you already understand the concept of convergences. I recommend this article if you have the time
https://www.cantorsparadise.com/the-ramanujan-summation-1-2-3-1-12-a8cc23dea793

You are obviously correct that the sum is not convergent, but surprisingly this alleged sum doesn't involve any weird tricks that aren't valid like in the types of proofs where someone supposedly shows that 1+1=3 or something like that. It's not logically invalid or anything like that. However, what it does rely on is a non-standard notion of convergence. There's actually a whole subfield on non-standard notions of convergence. One of them yields this sum. In fact, the topic of convergence and limits and the various non-equivalent attempts at defining and understanding these concepts was an important part of what led to the demand for "rigorous" foundations in mathematics and ultimately the foundational crisis.

>>14886605
You are correct, and I'm not the same anon, but the only thing different about analytic continuation is that it involves an analytic function, which is simply a complex differentiable function. In complex analysis this forms an important topic concerning essential and removable discontinuities.

If you don't know complex analysis, the best way to think of analytic continuation is just as finding a making a differentiable function continuous on a larger set of values. The formal definition literally just states that you have an analytic function f defined on a subset V and analytic continuation of f is just any analytic function f' defined on a larger subset V' containing V such that
f(z)=f'(z) for all z inside of V.