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mbkennel
reply to post by ImaFungi
Why -1/12?
Because that's the answer you get by considering the sum as only one example of an analytical process.
en.wikipedia.org...
en.wikipedia.org...
en.wikipedia.org...
So go further and tell us what happens. I don't see how it will be any different.
ImaFungi
Im asking what is not arbitrary about the choice to only use those specific partial sums? If you went further would that answer be different?
Arbitrageur
So go further and tell us what happens. I don't see how it will be any different.
ImaFungi
Im asking what is not arbitrary about the choice to only use those specific partial sums? If you went further would that answer be different?
Arbitrageur
reply to post by mbkennel
The wikipedia link shows this graph which illustrates a y-axis intercept of -1/12, which is pretty clear to me:
I don't have any problem saying -1/12 is a "zeta regularized sum", but it doesn't make sense to me to call -1/12 a "sum".
What seems to be going on in the graph and the math is something like this:
1 + 2 + 3 + 4 +.... = y + (infinite expression)
where y= -1/12
-1/12 is what you get if you toss the infinite expression you want to get rid of, but to express the equation as
1 + 2 + 3 + 4 +.... = y
disregards the infinite expression which was discarded, and is misleading at best.edit on 23-2-2014 by Arbitrageur because: clarification
Soylent Green Is People
One overriding proof of this would be that adding only positive integers together can only result in a positive sum. Therefore, that right there is evidence that something is wrong with the -1/12 answer. That "something wrong" is the fallacious idea that you can simply "average out" the answer to the 1-1+1-1+1-1... question.
That "something wrong" is the fallacious idea that you can simply "average out" the answer to the 1-1+1-1+1-1... question.
ImaFungi
I am stating that -1/12 is completely arbitrary and it might as well be -1/2 -1/3 -1/4 -1/5 etc. You can give no absolute reason why the mode of averaging was stopped at the number it was, that in turn provided the -1/12.
mbkennel
...It works like this:
a) ok, these things don't have limits under classical definition X.
b) so what do we do with them?
a) how about this, in an analogy to other circumstances we can define a different kind of transformation and limiting process in which they do
b) alright, but why should I choose this one instead of another
a) because in other cases, the classical and new ones merge very well, and this permits us to use new tools to reason about whole classes of things we previously had to give up on
b) alright, cool new math, let's hunt theorems!
Soylent Green Is People
The statistician could tell me "Well, the science of probability tell us that it is 50% chance for heads and 50% for tails", but I really don't care about what his science called "mathematics" may tell us. I know it is 100% certainty that I will get one or the other; that's what matters.
It's helpful to clarify using different notation that two expressions which look similar are not actually the same. For example, look at the notation Wikipedia uses to distinguish a "Ramanujan summation":
mbkennel
It's not misleading if you understand the mathematics, where the symbolic notation is not literally confined to mathematics invented before the 1700's and taught in high school (or elementary school if you were smart and russian).
I don't see where calculus summations distort the meaning of the equals symbol like summations of divergent series do. Divergent series aren't really sums, as explained above and below:
mbkennel
More advanced mathematics universally progresses by analogy and extension. All of abstract algebra extends e.g. "addition" and "multiplication" to less elementary domains. Calculus extended "sum" as well, and this is just another one.
You may have noticed that all through this post, I have avoided writing “This series equals -1/12,” or “the value of the sum of the series is -1/12.” This is due to my conversation with Ellenberg, which was fascinating to me. We talked about different methods, different rules, how new concepts were not accepted at first, and that things we think are simple now (like using fractions) were at one point in history heatedly debated as to their reality and usefulness. He put it very well:
"It's not quite right to describe what the video does as “proving” that 1 + 2 + 3 + 4 + .... = -1/12. When we ask “what is the value of the infinite sum,” we've made a mistake before we even answer! Infinite sums don't have values until we assign them a value, and there are different protocols for doing that. We should be asking not what IS the value, but what should we define the value to be? There are different protocols, each with their own strengths and weaknesses. The protocol you learn in calculus class, involving limits, would decline to assign any value at all to the sum in the video. A different protocol assigns it the value -1/12. Neither answer is more correct than the other."
Why do you suppose he says "Thus we may jokingly say that 1 + 2 + 3 + .... = -1/12"?
One reason bosonic string theory works best in 26 dimensions is that
1 + 2 + 3 + .... = -1/12
and 2 x 12 = 24. Of course, this explanation is unsatisfactory in many ways. First of all, you might wonder what the above equation means! Doesn't the sum diverge???
Actually this is the least unsatisfactory feature of the explanation. Although the sum diverges, you can still make sense of it. The Riemann zeta function is defined by
ζ(s) = 1-s + 2-s + 3-s + ....
whenever the real part of s is greater than 1, which makes the sum converge. But you can analytically continue it to the whole complex plane, except for a pole at 1. If you do this, you find that
ζ(-1) = -1/12.
Thus we may jokingly say that 1 + 2 + 3 + .... = -1/12.
Heisenberg's uncertainty principle has nothing to do with coin tosses.
saneguy
I guess you don't accept Heisenberg's uncertainty principle then.
saneguy
Soylent Green Is People
The statistician could tell me "Well, the science of probability tell us that it is 50% chance for heads and 50% for tails", but I really don't care about what his science called "mathematics" may tell us. I know it is 100% certainty that I will get one or the other; that's what matters.
I guess you don't accept Heisenberg's uncertainty principle then.
Soylent Green Is People
mbkennel
...It works like this:
a) ok, these things don't have limits under classical definition X.
b) so what do we do with them?
a) how about this, in an analogy to other circumstances we can define a different kind of transformation and limiting process in which they do
b) alright, but why should I choose this one instead of another
a) because in other cases, the classical and new ones merge very well, and this permits us to use new tools to reason about whole classes of things we previously had to give up on
b) alright, cool new math, let's hunt theorems!
I agree that it can be "cool new math" and it's fun to consider these things, but the answer to 1-1+1-1+1-1+1-1... is still EITHER -1 or 0. it is never -1/2, unless mathematicians decide they will allow themselves to play a bit "fast and loose" and allow for accounting tricks, such as saying the answer is NEITHER -1 NOR 0, but is actually the average between the two.
I'm all for mathematicians coming up with these interesting proofs and digging down deep into the way they like their science to manifest itself to the world, but the "real" answer to 1+2+3+4+5+6+7... can never be negative.
If I decided I would flip a coin, a statistician would tell me that I hand a 1/2 chance (50%) of getting a heads, and a 1/2 chance of getting tails. But the reality of the situation is that I will get EITHER a heads OR a tails; one or the other....
...Therefore, it's not relevant what the odds are for flipping a heads or a tails. The odds don't matter to me, because I will truly get one or the other; that's all I care about, not the odds.
The statistician could tell me "Well, the science of probability tell us that it is 50% chance for heads and 50% for tails", but I really don't care about what his science called "mathematics" may tell us. I know it is 100% certainty that I will get one or the other; that's what matters.
Similarly, When I add together 1-1+1-1+1-1+1-1..., depending on where I stopped, I will get a -1 or 0 100% of the time. There is no other answer. The proof that tells me I get a -1/2 is simply an approximation
Arbitrageur
I don't see where calculus summations distort the meaning of the equals symbol like summations of divergent series do. Divergent series aren't really sums, as explained above and below:
Why do you suppose he says "Thus we may jokingly say that 1 + 2 + 3 + .... = -1/12"?
boards.straightdope.com...
This thread having been bumped recently, I thought I would do two things.
Thing #1: I'll recap the simplest, most layperson-friendly answer to the titular question (which, although present in the previous discussion, might not ever have been made as explicitly clear as possible):
Specifically, the sense in which 1 + 2 + 3 + 4 + ... = -1/12 is this:
First, consider X = 1 - 1 + 1 - 1 + .... Note that X + (X shifted over by one position) = 1 + 0 + 0 + 0 + ... = 1. Thus, in some sense, X + X = 1, and thus, X = 1/2.
Now consider Y = 1 - 2 + 3 - 4 + ... . Note that Y + (Y shifted over by one position) = 1 - 1 + 1 - 1 + ... = X. Thus, in some sense, Y + Y = X, and thus, Y = X/2 = 1/4.
Finally, consider Z = 1 + 2 + 3 + 4 + ... Note that Z - Y = 0 + 4 + 0 + 8 + ... = (zeros interleaved with 4 * Z). Thus, in some sense, Z - Y = 4Z, and thus, Z = -Y/3 = -1/12.
In contexts where the above reasoning is applicable to what one wants to call summation, we have that 1 + 2 + 3 + 4 + ... = -1/12. In other contexts, we don't.
Soylent Green Is People
I agree that it can be "cool new math" and it's fun to consider these things, but the answer to 1-1+1-1+1-1+1-1... is still EITHER -1 or 0. it is never -1/2, unless mathematicians decide they will allow themselves to play a bit "fast and loose" and allow for accounting tricks, such as saying the answer is NEITHER -1 NOR 0, but is actually the average between the two.
mbkennel
To repeat. Every finite partial sum is -1 or 0, which is undisputed. The question is the varying meanings of infinite limits.
You seem to feel that something which you don't understand is immoral or unethical, but it isn't. It's just strange.
Soylent Green Is People
I'm all for mathematicians coming up with these interesting proofs and digging down deep into the way they like their science to manifest itself to the world, but the "real" answer to 1+2+3+4+5+6+7... can never be negative.
mbkennel
Mathematics isn't about "real". In any case, this sort of mathematics has been useful and used for making nontrivial physical predictions in quantum field theory, where the appropriate treatment of infinities---meaning considering the finite residuals disputed above as the "real" answer---did turn out to give the physically useful ones. So there's a data point in favor of indulging mathemeticians for a while.
Soylent Green Is People
If I decided I would flip a coin, a statistician would tell me that I hand a 1/2 chance (50%) of getting a heads, and a 1/2 chance of getting tails. But the reality of the situation is that I will get EITHER a heads OR a tails; one or the other....
...Therefore, it's not relevant what the odds are for flipping a heads or a tails. The odds don't matter to me, because I will truly get one or the other; that's all I care about, not the odds.
mbkennel
The philosophical underpinnings of probability have been examined for hundreds of years. You're not going to invent a new idea.
WeAre0ne
So let me get this straight...
If you give me $1, and someone else gives me $2, then another person gives me $3, and other people keep giving me money in the same pattern, forever (infinitely), and I don't spend any of it...
I would end up $-1/12 in debt?
That is the most idiotic thing anyone has ever postulated.
Words have meanings, as do symbols. Jokingly means one thing. Absurd means something else. Unexpected means something else.
mbkennel
Jokes are about the absurd and unexpected. And of course the guy knows it's an extension to the elementary definitions.
I'm not sure if you're aware of the ruckus that occurred after the first video numberphile was posted, but part of the reason a second video was posted is that the first video wasn't right, and that straight dope except has the same problem, it promotes mathematical illiteracy:
Anyway I found this on another message board about this very same question:
The second numberphile video that was posted was intended to explain the result more rigorously, and not using the type of "math" that can "prove" that 0=1 like the first video did.
The flaw in both cases is the same: the algebraic rules that apply to regular numbers do not apply to infinity. Actually, it's more general than that: the algebraic rules that apply to regular numbers do not apply to non-converging infinite sums. All of the sums above are non-converging infinite sums, so regular algebraic rules do not apply. It is no different from using regular algebra when dividing by zero. It doesn't work.
Now, there are ways to define the sums of non-converging infinite series so that they do not lead to contradictions. The one that leads legitimately to the conclusion that 1 + 2 + 3 + 4 ... = -1/12 is called Ramanujan summation, which in turn is based on something called an analytic continuation. But the problem is that the Numberphile video makes no mention of this. They present the result as if it is legitimately derivable using high school algebra, and it isn't. Telling people that it is does a grave disservice to the cause of numerical literacy.
WeAre0ne
sum = 0;
x = 0;
while(true)[
x++;
sum += x;
]
Tell an experienced programmer that the value of "sum" will become a signed decimal at some point during runtime, and not remain an unsigned integer, and he will have a good laugh in your face.edit on 24-2-2014 by WeAre0ne because: (no reason given)