Math is “true” and “real” because it can be proved. It simply doesn’t matter that if it happens to have applications in engineering or physics or other disciplines. That happens quite unexpectedly often, though, and entire disciplines within math once thought to be completely theoretical have turned out to be crucial in the oddest places. Primes in encryption; graph theory in communications networks, tensor calculus in relativity.
Whether this particular summation of this particular series is useful elsewhere is irrelevant. It’s a special case of a much larger group of summations that are everywhere in math. This one is called out in Wendall Wagner’s first link:
So the simple answer is yes. But even if there wasn’t a quick answer, the more meaningful answer is also yes. The question isn’t that dissimilar to asking if complex numbers are useful in math and physics. Yes, they certainly are. It doesn’t mater if 2823818282 + 3838383i is useful or not. The entire class is.
I agree with this. If the sum stops at 1,000,000 we have a definite, positive answer. If it stops at 1 google-plex we have a definite and larger answer.
The answer to the equation 1+2+3+4…+infinity must equal infinity or else the concept of infinity as we know it does not map real world results and needs to be reexamined.
This is not merely bizarrely wrong and totally a-mathematical in general, but I just provided a real world example for which that statement is false.
As I wrote earlier:
Why should anyone, mathematician or layman, pay attention to your opinion? How do you learn anything if you state a priori that all the experts in the world don’t know anything about their own subject?
Intuitively, what you’re saying seems right to me, but if mathematicians and physicists unanimously say that it’s actually -1/12 then I figure it must be -1/12 and I simply don’t understand, because who am I to disagree with such expert unanimity? Otherwise we start sounding like climate change denialists or the people who insist that 0.999… isn’t equal to 1.
Infinity (or actually, the various infinities) don’t act like you would expect. You haven’t examined the issue enough to be taken seriously on this subject. The concept of infinity isn’t intuitive, nor should it be.
I won’t claim to be a mathematician at this level. But my question is how many people are mathematicians at the appropriate level that we’re discussing here? How many people can really follow the proofs that are given and honestly say they understand them enough to say whether they’re true or false? Versus how many people are just following along and trusting the person who’s telling them that what they’re seeing is true?
I’m not saying Tony Padilla and Ed Copeland are lying to me in the Numberphile video linked in the OP. But they tell me the series 1 - 1 + 1 - 1 + 1 - 1 + … = .5 and they provide an explanation that I can follow and which seems reasonable.
Now to me, my answer seems as reasonable as theirs. If they had told me that sequence of numbers added up to zero and shown me the steps I just posted, I’d have believed them.
But they used a different procedure and came up with a different answer. Obviously I can’t add up the sequence to see the right answer. I have to just use one of the procedures and assume it’s giving me the right answer without being able to check.
I’d be happy if somebody could explain to me why one procedure is right and the other is wrong. That would constitute proof and we’d be in the realm of science. But if the explanation is “the explanation is so complicated you wouldn’t understand it so you just have to trust me” then it seems more like I’m being asked to have faith.
Just amongst professional theoretical physicists, I’d guess a couple of thousand at a minimum who are currently active in the field and able to effortless follow the proof, almost without thinking about it. Factor in former examples of the like, not to mention mathematicians, and the group of people able to understand the point readily expands to hundreds of thousands of people. Actually easily millions of people.
How is this not true for virtually everything in every field in every time and every place? How much of everything people say on the Dope have you personally verified? Do you go to your doctor and say, prove to me that everything you know about medicine is true? Do you stop and drag an engineer to a bridge before you drive over it and demand proof that the stress forces on the substructure are lower than the failure point? Society simply does not function without the assumption that experts in their fields are in fact experts in their fields. Why is math - of all disciplines, the only one that requires absolute proof - the only one you make such demands of?
The quote from Indistinguishable already established that there are many distinct methods of summation in math, and that different methods may produce different results. Whether or not you “understand” this, he laid out the differences and the consequences clearly and succinctly. What more would you have him say without taking responsibility upon yourself to learn some math to follow along?
Isn’t the fact that these proofs are hundreds of years old, and therefore have been studied by students in every country in the world for those hundreds of years, sufficient to conclude that the number of people who can understand math past the high school level is in the millions or tens of millions or, collectively, even more?
And not only do they understand, some have taken the time, in that other thread, to attempt to explain the meaning of these summations, exactly as you are asking for - without any evidence that you yourself took the time and effort to read through them. I said before that I don’t understand this, and I now understand it even less than before.
Let’s be clear 1+2+3+4+…+n+… is divergent and in particular the limit of its partial sums as n goes to infinity is infinity.
Whilst the limit of the sequence partial sums is usually what we mean when we say that a series converges to some value L, there are other limits associated with the series that also converge. For example if a[sub]1[/sub]+a[sub]2[/sub]…+a[sub]n[/sub]+… is an infinite series that converges to L then it is easy to see that the infinite series b[sub]1[/sub]+b[sub]2[/sub]…+b[sub]n[/sub]+…; where b[sub]1[/sub]=a[sub]1[/sub]+a[sub]2[/sub], b[sub]2[/sub]=a[sub]3[/sub]+a[sub]4[/sub] and b[sub]n[/sub]=a[sub]2n-1[/sub]+a[sub]2n[/sub]; also converges to L. Conversely however if Sum(b[sub]n[/sub]) converges to L it does not mean that Sum(a[sub]n[/sub]. For example if Sum(a[sub]n[/sub]) is Grandi’s series: 1+(-1)+1+(-1)…+(-1[sup]n-1[/sup])+… it is divergent, but Sum(b[sub]n[/sub]) converges to 0.
If wanted to extend the definition of summation to include some divergent series we could use the above example, however it actually turns out to be a not very useful way of extending summation [because as Little Nemo notes there is an element of arbitrariness in choosing to extend summation in this way] . Why did I even bring it up then? My point is that for convergent series there are limits we can associate with the series that are guaranteed to have the same value as the limit of the partial sum, but will also be finite for some divergent series.
So all we are saying when we say 1+2+3+…+n+… = -1/12 is that there is a limit associated with an infinite series which for a convergent series has the same value as the limit of partial sums, but for the series 1+2+3+…+n+… is -1/12. So there’s nothing magical or bogus going on.
All then we have to contend with is whether the method we choose to extend summation is arbitrary, but I believe there are certain theorems as to the uniqueness of this extension.
As Exapno Mapcase notes these methods are used for calculations in physics. However I would say that in this case what is being ‘summed’ are not physically observable quantities, but rather quite abstract mathematical entities and the individual terms in the ‘series’ don’t have any physical meaning outside of the series. So again it’s not quite as magical as it may seem.
I’m willing to admit when I’m ignorant on a subject. My understanding is that the earliest proofs of this fact were developed by Bernhard Riemann. But he worked in the mid-19th century.
So what proofs are you talking about that existed hundreds of years ago?
I’ve been doing some checking on this subject in order to try to understand it. One mathematician seemed to be saying that it wasn’t really true to say that the sum of the series 1 + 2 + 3+ 4 + … added up to -1/12. He said that the series was divergent.
He seemed to say that -1/12 is the equivalent of the infinite series. And by this he meant that when you have a problem which contains the infinite series 1 + 2 + 3 + 4 + …, you can substitute the value -1/12 for the infinite series. This makes what was an insolvable problem easily solvable and the answer you get using the value -1/12 is the same answer you get using the value 1 + 2 + 3 + 4 + …
What I’m getting reading this is that we’re talking about different definitions of “sum”. Beyond that I still don’t understand where -1/12 comes from but I accept it.
Well you can’t go around substituting -1/12, every time you see 1+2+3+4+… . The fact that this particular series is divergent is actually a very important property of the series and very much of the time knowing that the Ramanujan sum of that series is -1/12 won’t be of any obvious usefulness. But there will be some times when it is useful to know.
The version we’re talking about here depends on work done by Jacob Bournelli, first published in a posthumous 1713 book, with the actual proof date unknown but obviously earlier. Riemann extended the formula to the more general case of complex numbers.
Let’s make sure that I see the equation correctly. I understand it to be
1+2+3+4+…=-1/12
We are taking positive whole integers, correct? And we are summing them all as they become infinitely and positively large, correct?
To say again, there are no negative numbers in this series, right?
If I have this right, then after the first two numbers in the set, I have 3 which is greater than - 1/12. If there is only addition and no negative numbers in the remainder of the set, then I cannot have less than 3.
If all of the experts think that I am wrong on this, then they are not using the same number system that we learned in school.
If different methods produce different results, then all of those methods, except for possibly one of them, are wrong. When you put forward a proposition that is so clearly wrong based upon observable facts, you need to explain it better than simply having a snarky attitude and resting upon an “experts agree and they are smarter than you” argument. Because if different methods produce different results to that equation, then the experts do not agree.
I realize that we cannot define infinity in precise terms, but in this equation we are defining it as “positive integers increasing by one forever and ever.” No summation of that series would ever be a negative number by its own terms.
If we are talking about different definitions of the word “sum” then there needs to be a different word. I can’t make some broad declaration about trees and then when you disagree with it, I get all shitty about it and say that I am not talking about those green leafy things, but something that only smart people like me would understand.
Nah, this comes up a lot in math. For instance, lambda calculus has ambiguous syntax that can drastically affect the practical evaluation of an expression. We’ve mostly settled on one or two conventions, but that’s for entirely arbitrary reasons (many involving how easy they are to reason about and/or how easy it is for computers to interpret them).
Higher level and theoretical math regularly dives into territory where things aren’t certain or can be completely counterintuitive. Often there are multiple different, completely correct, ways to doing things, that all yield different answers. Usually they’re used for different things because the different interpretations are useful in different contexts, and it’s the same here.
[QUOTE=Wikipedia]
In order to transform the series 1 + 2 + 3 + 4 + · · · into 1 − 2 + 3 − 4 + · · ·, one can subtract 4 from the second term, 8 from the fourth term, 12 from the sixth term, and so on. The total amount to be subtracted is 4 + 8 + 12 + 16 + · · ·, which is 4 times the original series.
[/QUOTE]
I have a problem with this step…
4+8+12+16+… is NOT 4 times larger than the original series. The first series sums to infinity and has no distinct value. This second series sums to infinity and has no distinct value. Why do we assume the second is 4 times larger? 4 times infinity is still infinity.
Trouble with this refutation is that you are already assuming your result. That isn’t a valid argument.
You assert that 1 + 2 + 3 + 4 … = infinity. But if you do this you are already assuming the result is not -1/12. So of course you can’t make any progress. You need to suspend evaluating the result and allow the rest of the manipulations to occur.
Same with 4 + 8 + 12 + 16 … - if you assume it sums to infinity you can’t make progress, doing so means you are assuming an answer.
The fact that ∞ * 4 = ∞ is already a clue that you can’t reason with infinite things in quite the way our intuition holds. You can throw your hands in the air the moment ∞ appears in any form and cease further discourse, or you can allow that there are manipulations that you can make valid by a set of appropriate carefully chosen dispensations.
Look at renormalisation and see how to skate over over unwanted infinite terms to finally, and incredibly get one of the most extraordinary accurate answers.