Because another poster specifically asked they laymen to provide their intuition and the way they work through the problem. It allows the more technically/mathematically inclined people to see what specifically is causing problems with understanding the concept. And since the thread started with a layman trying to understand something complicated and technical, saying they should just stay out of all math discussions seems a little rude.
Nonsense. YanWo gave an answer that is clearly factually wrong. It is a common misunderstanding that pops up regularly in math threads and is the cause of much anguish in the numerous and never-ending does .99999… = 1? threads. Assuming - even insisting - that your lay understanding of math is as good and as meaningful as a mathematician’s understanding is not equivalent to not getting the concept.
There is a huge difference between the OP’s “I can’t see how this happens,” which is the proper approach to getting help and learning new concepts, and YanWo’s assertion that adding infinities is impossible, which simply kills discussion. This is true for any and every subject on this board, from math to physics to economics to cars to guns to digestion to tiddlywinks. If you have zero technical knowledge of the field don’t insist to the professionals that you know better than they do. Try to learn something instead.
To give YanWo the benefit of the doubt, when he gave his “layman’s response” he might have meant “This is how it appears to me, but of course I’m just a layman, so what I’m saying shouldn’t be regarded as official or correct.”
If that’s not what he meant, then your rebuke (and Itself’s) is entirely appropriate. Anyone who claims that something is meaningless because it’s physically impossible doesn’t understand how mathematics works.
By the way, the book you recommended is on my to-be-read shelf, but I agree that Ian Stewart is one of the best writers of popular math books working today.
Everything can be explained to a five-year-old. It’s just that for some things, the explanation takes 15 or 20 years.
Old threads give us all the more time to think about it.
When this first came up, I decided to try it for myself, the old-fashioned simple-minded (grade-school arithmetic) way, by simply adding all the terms 1 + 2 + 3 + 4 + . . . , one term at a time, to see for myself what it all adds up to.
I’m still working on it . . . .
This is a possibility, and I might at one time have thought it to be true but I’ve read 500 threads with this kind of statement in it and that possibility hasn’t occurred yet.
It could in the future, but my hunch is that it will happen no sooner than the appearance of an odd perfect number, so it will require a minimum of 10[sup]1500[/sup] threads.
This is correct. Indistinguishable specifically asked (three years ago) what the idea of an infinite sequence of addition means to a layman, for the express purpose of having a starting point to explain what they mean in the context of formal mathematics. Exapno and others were wrong to jump down YanWo’s throat for answering that question.
If you’re referring to post #38, I don’t think it’s a reasonable answer to his comments there, and certainly not reasonable if you read further in the thread.
If I missed a different reference, please cite it specifically.
I’m referring to YanWo’s first (and thus far only) post, in which he quotes and replies to Indistinguishable from post #62, first paragraph.
Whether it’s appropriate to bump a three-year-old thread to answer a post from one page ago is another discussion, but I am annoyed because it seems like a few people here were unnecessarily rude and dismissive to a new member who was probably trying to learn something.
Post #62 is a restatement of post #38 so I missed it in a quick scan through the thread. I apologize for that. But the question was not only a direct one to Darth Panda but the answer was given - again - as part of that same post.
We only know what people mean by the words they type. Darth Panda was engaging with the other posters in a dialog. YanWo, IMO, was not. Maybe he or she meant to but my reading of that post saw an example of perhaps the biggest challenge to “fighting ignorance;” one that we see all too much of.
If YanWo posts again I’ll interpret that post as I read it and not on the basis of this one. Just as I do to your posts. You made a legitimate objection; I gave my answer. That’s a dialog. Everybody gets to judge who they agree with.
The Straight Dope Message Board will be having a lot more Infinite Crises before that happens
I came here looking for an answer to the OP’s question, and found it. I very much appreciate those who took the time to write detailed and comprehensive responses.
I had more to say, but decided not to in the hope of avoiding a flame war.
Yes, all we have are the guy’s words. But you did not just respond to just his words. You deliberately assumed bad faith on his part. You didn’t even just assume he was ignorant, as the response to that would have been to teach. You assumed he was someone who could not possibly learn and scolded him for daring to bother saying anything. You told him that, because of his basic assumption, he was unqualified to discuss the question. You didn’t try to fight his ignorance, you tried to shoo him away.
If you actually knew anything about math, you’d know that his interpretation is correct, too. The entire reason why we are using analytic methods is that the original method is impossible. We are doing an impossible type of addition here. I suggest you do some reading on the subject. Start with the wiki article on mathematical philosophy. Not everyone embraces the idea of mathematical fictionalism.
See how insulting that is? See how it makes you mad rather than helps you? Yeah, why do that?
Flame wars? Not at all. This is the *polite * forum. We have a number of experts in many fields who share their expertise, which makes it one of most fascinating places on the net. I hope you stay and poke around a bit.
I’m glad that you found some enlightenment in this thread. Math concepts are the hardest to put across in comprehensible language. I have read The Music of the Primes, the one in the OP, and yet when I just read Ian Stewart’s Visions of Infinity I felt he did a better job on the Zeta function and its uses. That counting to infinity is physically impossible is true, but thereby rejecting it as a mathematical concept simply blocks all further understanding in the lay reader. I hope you can get past it. Infinities are found everywhere in math; properly manipulating them is crucial.
I have no idea what you think ficitionalism has to do with this, nor am I at all interested in finding out. But what you seem to be missing is that this notion of assigning sums to divergent series is completely in line with the activities of mainstream mathematics. No one who actually knows anything about modern math–including Exapno, who gets this stuff as well as any non-specialist can–would even bat an eyelash at it.
And yes, Exapno’s response may have been a bit snippy. But the post he was responding to is the start of a fruitless back-and-forth that we’ve had at least a hundred times before. Forgive us if we’re not looking to rehash it again.
Edit: I think Exapno is not actually a mathematician. Please correct me if I’m mistaken.
I’m not technically anything at all. I’m one of the last living generalists. But I do thank you for the compliment.
I dunno, I would certainly batt an eyelash at assigning a value to a divergent series. I’m not completely opposed to the notion, and I can be talked into an interpretation where some value or another is meaningful, but it’s at least worthy of comment.
Assigning value to a convergent series, though, that’s totally mainstream and just taken for granted, and it’s quite an extraordinary claim that it can’t be done, just because it’s infinite.
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.
And now Thing #2: I’ll give, for those who are interested in such things, a nifty alternative perspective on this summation which occurred to me upon re-reading this thread:
The fact that there are other manipulations very similar to those above which will yield different values for such series may be troubling; it may not be clear how to generalize these results to other series, except in an ad hoc manner. Thus, I will present, for the reader of moderate mathematical sophistication, a particular fixed summability method which yields all these results and more. [Of course, there are still other summability methods with other results… Such is the world.]
Given a series a[sub]1[/sub] + a[sub]2[/sub] + a[sub]3[/sub] + a[sub]4[/sub] + …, we can form the associated function f(h) = a[sub]1[/sub] e[sup]-1h[/sup] + a[sub]2[/sub] e[sup]-2h[/sup] + a[sub]3[/sub] e[sup]-3h[/sup] + a[sub]4[/sub] e[sup]-4h[/sup] + …, such that the desired sum is the value f(0).
It may happen that f(h) converges in the standard sense for small positive h but not for h exactly equal to 0. Our goal, then, will be to find a way to extrapolate a value of f(0) from the values of f(h) for small positive h; any such extrapolation method will yield a corresponding series summation method.
The most obvious way to do so is to reason as though f were continuous at 0, taking f(0) to be the limit, as h approaches 0 from above, of f(h). This summation method is already enough to conclude 1 - 1 + 1 - 1 + … = 1/2 and 1 - 2 + 3 - 4 + … = 1/4; it is often called Abel summation, as Abel proved it agrees with the standard sum for every series which converges in the standard sense.
But even that may not be enough to yield a finite value for some series of interest [like the titular one]. So we may apply stronger extrapolation techniques based on stronger smoothness assumptions for f. In particular, if f is n times continuously differentiable, then f(0) = the limit, as h goes to 0, of the nth derivative of h[sup]n[/sup]/n! f(h). [Go ahead; check this for yourself. It’s straightforward to see if you think in terms of Taylor expansions]. We may use this to extrapolate a finite value for f(0) even when f(0) would ordinarily “blow up”; I’ll call this the degree n sum of the original series.
The degree 0 case is just Abel summation, as discussed before. But as one chooses larger degrees, more and more series are assigned finite sums via this extrapolation. [Luckily, if a series has a finite sum at some degree, then it has the same sum at any larger degree]. Let us declare the sum of an infinite series to be the finite value, if any, given by its sum at sufficiently high degree.
This fixed summation method yields all the results mentioned in this thread; in particular, we will get 1 + 1 + 1 + 1 + … = -1/2 at degree 1, and 1 + 2 + 3 + 4 + … = -1/12 at degree 2. Continuing in the same vein, we get values for each 1[sup]n[/sup] + 2[sup]n[/sup] + 3[sup]n[/sup] + 4[sup]n[/sup] + … at degree n + 1. [These are easy to calculate for yourself once you observe that the function associated to 1 + 1 + 1 + 1 + … is f(x) = e[sup]x[/sup]/(1 - e[sup]x[/sup]), and that the function associated to the sum of {i * a[sub]i[/sub]} is the derivative of the function associated to the sum of {a[sub]i[/sub]}]. Hooray!
[In fact, for any complex number n, this method will produce a finite value for 1[sup]n[/sup] + 2[sup]n[/sup] + 3[sup]n[/sup] + 4[sup]n[/sup] + … at precisely those degrees at least as large as the real component of n + 1, except for the case of n = -1 [the harmonic series], which continues to be unsummable by this method. Bundling all these values together, we will get… the Riemann zeta function. And thus this technique is actually the same thing as zeta function regularization, but presented in a systematic, rather than ad hoc way. (Specifically, this technique turns out to be equivalent to assuming linearity for series summation + assuming every result given by Abel summation + assuming the values given to 1[sup]n[/sup] + 2[sup]n[/sup] + 3[sup]n[/sup] + 4[sup]n[/sup] + … for natural number n by the Riemann zeta function)]
Finally, a note on alternative values for series such as 1 + 2 + 3 + 4 + …: While the summation technique outlined here has a great many good properties, it doesn’t always have the property that shifting a series’ terms down and adding extra zero terms to the beginning leaves the series’ value unaffected [it has this property just for those series which are already Abel summable]. So by changing what one considers the starting index of a series, one can change the resulting value (for example, if we considered the starting index of 1 + 2 + 3 + 4 + … to be 2 rather than 1, with each term being one less than its index rather than equal to it, we would get a sum of 17/12, rather than of -1/12. But I think we can agree that its natural to take the terms of this series to match their indices, rather than be offset from them).
(On preview, I see you are addressing this. I’ll leave this as a concrete example of an alternate derivation, and go read your latest post.)
I’m not opposed to the shifting over by 1, but I don’t like the (zeros interleaved with 4 * Z) step.
Here’s an alternate derivation that gets a different answer:
Z = 1 + 2 + 3 + 4 + …
Z + Z shifted over 1 = 1 + 3 + 5 + 7 + …
Z + Y = 2 + 0 + 6 + 0 + 10 + … = 2 * (zeros interleaved with (Z + Z shifted)) = 4Z
So 4*Z = Z + Y, or 3Z = Y or Z = +1/12
So the same magnitude, but opposite sign.