Tl;dr: No it doesn't. Under some unintuitive definition for infinite summation that is useful in some physical calculations it does which is surprising.
Under the normal rules which hold for direct use obviously the answer is positive infinity just like you would be expect, you're not stupid and you could be a mathematician if you wanted to.
edit: For fun, a short story:
If Muhammed is on top of a strangely shaped mountain that with every step down gets one step wider. The mountain is so high he can't see the bottom yet Muhammed wants to move this mountain. So Muhammed starts fetching horses and ties them to the mountain with ropes to move it. That's a direct use of this equality, and you can't stand from afar and look at the scene and say "my, I think that's about -1/12 horses Muhammed is fetching". You'll see Muhammed taking an infinite amount of time fetching an infinite number of horses, and you'll definitely seem him do it more than once.
It appears in some quantum field theory calculations, in particular for Casimir Effect.
In simple words Casimir Effect consists of a force that emerges between two conductor planes that are parallel to each other. The force is proportional to sum of energies of all possible standing electromagnetic waves between the planes. In calculations for this force a divergent series of sum of all natural numbers (or their powers) appears and physicists use 1 + 2 + ... = -1/12 to calculate it (or continuation of zeta function in other points if appropriate).
I've never seen a physics book that treats this using the zeta function, except popular articles that try to present this calculation as mysterious. In practice, in my QFT class we needed to compute the sum
that is the series was multiplied by a decaying exponential function with a rate of decay that goes to zero. This sum can easily be evaluated for small epsilon takes the form
sum = 1/epsilon - 1/12 + O(epsilon).
The 1/epsilon term (which goes to infinity) drops out of the final physical result when you do the calculation properly.
Well I'm not a physicist, but I think at the very least we perceive time (locally) as proceeding in a linear fashion infinitely.
My mountain example obviously couldn't happen in the physical world. I suppose in that case you might as well substitute the infinite value for an arbitrary large one. Which is not really what infinite values are about in mathematics, as they are more about describing the (imaginary?) limit of a divergent series.
I guess my point is more that for a mathematician it would probably be obvious that when you talk about the limit of a divergent series it could be any imaginary or intermediary value. But for a layman infinite values and infinite series are interpreted as larger than any value you can come up with, and more than any repetition you can write down. So any explanation for this equality should, I think, involve first deconstructing that.
Are you suggesting that time extends infinitely into the past? Because if it doesn't, there will never be an infinite amount of time, so there won't be any point in time when this -1/12 becomes relevant to physics in this sense.
> I guess my point is more that for a mathematician it would probably be obvious that when you talk about the limit of a divergent series it could be any imaginary or intermediary value.
I'm a mathematician and I might agree (not entirely sure what you mean). But -1/12 is a concrete value so it doesn't apply here.
Well it might, but in what calculation would the fact that time extends infinitely be a factor? You can say all past time resulted in the current moment, but as far as I know there's no such thing flowing from the future into now. So any physics calculation you do will deal with finite amounts of time.
I won't pretend to know every possible calculation that might be desired, but I can imagine any sort of prediction or simulation about the future may want to use such an infinite view of time going forward.
We don't have to get anywhere near the Plank scale. If you trace the 'edges' of atoms you'll end up with a finite multiple of a meter-scale or kilometer-scale measurement. And that's the end of the fractal.
It doesn't mean that there are an infinite number of real things that add up, it means that the mathematics we use to model physics comes up with infinite sums of abstract things.
Nevertheless, there are infinite sums of "real" things in physics too. I have put "real" in scare quotes because it turns out they aren't real :)
In quantum electrodynamics, the charge of an electron turns out to be infinite. And it turns out that in Real Life, the charge of an electron is indeed infinite. Ish.
... but we know it isn't, right?
So what happens is that the real electron gets surrounded by positively charged "virtual" particles. Virtual particles are basically quantum probabilities of a particle appearing out of nowhere with its antiparticle (among other things). So you can say that with some probability, that particle is there. Since there's an electron nearby, the positively charged particle is attracted to the electron, while the negatively charged antiparticle is repelled. This screens the electron charge. With an infinite number of virtual particles, the electron's charge is screened enough to become finite again. Basically, we subtracted two infinities and got something finite. The subtraction done here is called renormalization -- and a similar thing is being done in the -1/12 sum. While mathematics tells us that divergent series can be rearranged to get any "sum", this trick is often used in physics -- provided you can justify that rearrangement.
In fact, if you probe an electron hard enough (by bombarding it with other charged particles with tons of energy), its apparent charge increases since the particles used to measure its charge "pierce" the shielding.
Of course, this is all really a fancy way of saying that charge itself is energy-dependent, and what we call charge is actually the 0-energy charge.
But for modelling purposes, virtual particles work better, and thinking about things in those terms gives a physicist a cleaner abstraction boundary to deal with. You get infinities everywhere, though.
This is basically an example of the pattern I'm talking about. Abstractions in the model may have all kinds of infinities popping up. In the real world, these don't really manifest themselves because they're not directly linked to observables. You can apply your model to your detection mechanism to get values for non-observables and say "hey, look, an infinity", but that's really circular logic. The "Real Charge" of an electron isn't something we see. Virtual particles aren't something we see; unless we make them into real particles, but you can't do that to the infinite virtual particles around, so you'll never see an infinity.
I wouldn't assume my definition is a better one than any other. But I am just like the articles author confounded by the idea that you can communicate this equation on a public forum without making it absolutely clear that you are interpreting the equation in a way that is different from the way an undergraduate or just regular person would interpret it.
I get how it's fun for intellectual people to come up with new definitions for things that expand our knowledge and understanding of the universe, and tease people who have the old simple understanding with seemingly impossible quandaries. But I would like the emphasize that I think this teasing is counterproductive. It leads people to believe mathematicians operate in a field that is fully disconnected from reality. Either take the time to explain how you define your construct, so they can truly appreciate the usefulness and ingenuity of the equation, or use different symbols and present it as a valuable yet opaque contribution to practical mathematics by showing its applications. Both cases will yield the much deserved admiration.
What you're actually saying: "Make math simple enough for regular people to understand"
Problem with what you're saying: Hundreds and Hundreds of years of history and tradition and the sheer knowledge that's been built up in the current way we do Maths.
Another problem: It's not teasing if the ideas are actually complicated. Which this one most certainly is.
Another problem: Not all Math is practical and applied, some of it almost certainly operates in a field that is "disconnected from reality".
Another problem: Your solution of using different symbols has the same problem as XKCD standards.
Another problem: "Regular people" is vague. People fear math for a reason. They can suck at it.
Another problem: This idea is also fairly old. So your part about "old simple understanding" also doesn't hold true.
Only thing we can agree on: Math can suck at communicating ideas if you're not used to it and/or don't have the context and knowledge.
I think you're missing one important consideration, here: they're saying "make math simple enough for regular people to understand when your audience is regular people."
Scientific American is targeted at educated laypeople. The author's job is to communicate with that target audience. And it absolutely is a tease to use a headline that, to that audience, will come over as a bizarre indication that they're actually clueless.
He didn't say "bad definition". He said "unintuitive" and I agree. The video is obviously meant for an audience that is not intimately familiar with infinite series or their applications in string theory. I'm sure most people found it most unintuitive as it is very much in conflict with how sums behave in everyday life. The fact that the video failed to discuss the important details is well worth criticizing, as the article explains.
the definitions are identical to a number line
+ is a move to the right
- is a move to the left
The point is, in "the universe" most of "infinity" are actually finite circles that you can travel along "infinitely" - because you eventually arrive back where you started.
and -1/12 just means 1/12 to the left before you get back where you started moving right.
This is normally called the integers modulo N, and this is not a useful way to look at this infinite sum. One way to tell is that if you didn't already know the answer, there's no way to get it from the construction.
Er, if you're working in the reals modulo a number, then all numbers that differ by a multiple of the modulus are regarded as equal (or equivalent). So in the reals modulo 2, it is indeed the case that 1.2 is equal to -0.8.
Why do you think otherwise?
But now you've moved away from talking about the integers mod N, which is what zeroer was talking about[0] which was in response to you talking about the numbers wrapped around a circle[1]. Their response to you seemed reasonable, but I don't understand why you leaped to talking about real numbers, nor why you claim that 1.2 is not equal to -0.8 when working modulo 2.
> ... to get 0.5 for the sum of
> 1-1+1-1+.... requires real numbers.
Actually it doesn't, it only requires the rationals.
But it's clear now that you're not really talking about maths at all, so the comment about existing, established theory about modular arithmetic doesn't really help. You seem to be doing something, well, different.
And regardless, in the long-established theory of modular arithmetic, 1.2 is equal to -0.8 mod 2, regardless of you claiming that it's nonsense.
So at this point I have no idea what you're talking about.
meh, I tend to skip thinking about the rational numbers when I move onto this hardcore theory stuff :p
There are two ways of constructing a number line from -inf to +inf
The first, is that "nothing exists" to the left of -inf or to the right of +inf
The other, more useful, is that -inf=+inf+1 and +inf=-inf-1, (or -inf=+inf, never remember which is the more useful) and they form a loop.
Such as that constructed by a signed integer.
e.g.
e.g. with an 8 bit number
127+1 = -127
This has nothing to do with modulo afaik. (but all the basic construct stuff is related) and is more to do with every dimension being curved in another (meaning they always form such loops)
>Under the normal rules which hold for direct use obviously the answer is positive infinity just like you would be expect, you're not stupid and you could be a mathematician if you wanted to.
There is no answer for an infinite sum. It is impossible to sum an infinite quantity of integers. The answer is definitely not positive infinity, as that is not a number and the sum of integers must be an integer.
The article points out a few times that such a "sum" is undefined.
The main point of this calculation is tricking laypeople by sneakily changing definitions. It's a bit like a school kid asking you to deny something embarrassing then informing you that it's Opposites Day. But there's no need to accept their definition. If they don't specify then the best definition is the commonly used intuitive one, under which it's indeed possible to sum infinite positive integers, resulting in positive infinity.
This number system was used to invent calculus, and worked just fine for over 150 years despite theoretical unsoundness. And it turns out that it's possible to formally define a provably consistent number system that obeys our intuition, the hyperreal numbers. See:
> If they don't specify then the best definition is the commonly used intuitive one, under which it's indeed possible to sum infinite positive integers, resulting in positive infinity.
I disagree that it is possible to sum an infinite amount of integers. It would take infinite time and space to perform the calculation. There is literally no end to the integers, so the calculation would never complete.
I also still claim that the result cannot be positive infinity due to the definition of addition on integers. The result of addition of integers must be another integer and positive infinity is not an integer.
I do agree with the article however, that one can take the limit of a well-defined infinite series; but the limit is not the sum, only the bound that will never be exceeded no matter how long you are able to continue adding numbers for.
I completely agree with you and the article that 1+2+3+...=-1/12 is a sneaky trick and that the definition should be rejected.
>I disagree that it is possible to sum an infinite amount of integers. It would take infinite time and space to perform the calculation. There is literally no end to the integers, so the calculation would never complete.
That same argument gives you Zeno's paradox.
You can sum a pattern of numbers in O(1) time if you use logic instead of brute force. It doesn't matter if physically spending O(n) time on something is impossible when you only need O(1).
Reread the parent comment. In the hyperreals, the reals are extended by infintesmals and positive and negative infinity. Two things happen there: 1. You no longer have +:Z -> Z, you have +:R'->R', which means that plus can be closed under the hyperreals.
Also, yes, you can't literally compute an infinite sum but among any crowd that has likely taken calculus 1, you can place implied limits. :P (which, arent actually needed in the hyperreals because it HAS infinity, but whatever)
>I completely agree with you and the article that 1+2+3+...=-1/12 is a sneaky trick and that the definition should be rejected.
This not something you can agree or disagree on. You can make physical calculations with this result and get a prediction that you can measure and confirm. This result is sound.
You can definitely agree or disagree on whether a definition should be rejected. If I tell you I want to replace 'five' with 'fish', even though my new system can be used to calculate things, you should tell me it's a terrible idea.
In this case you might find the -1/12 useful, but have the opinion that they really should not be using '=' as a shorthand for what they're doing with the zeta function.
> There is no answer for an infinite sum. It is impossible to sum an infinite quantity of integers.
I agree. Infinity is a process that can yield a number but is not an actual number and, Cantor et. al. notwithstanding, there is no such thing as a "completed infinity" other than terminating it at a finite step. If you are careful and in certain contexts you can use the "limit" of an infinite converging process but you must make that assumption explicit to avoid errors.
All these bizarre math tricks rest on treating it as a number when its undefined. Its like those puzzles I read as a kid that "prove" 1=0 and they typically depend on an implicit division-by-zero step which is also undefined just like infinity. Once you start working with the undefined you have to very careful and even Gauss made errors when he was laying the groundwork for infinite series. To the degree that this math has ANY validity it is in the context of some esoteric and specialized area of math and it is NOT appropriate to foist it on the general public as a general result. The motive in such attempts is to impress or intimidate or destroy math (nihilism) which I find despicable.
Under the normal rules which hold for direct use obviously the answer is positive infinity just like you would be expect, you're not stupid and you could be a mathematician if you wanted to.
edit: For fun, a short story:
If Muhammed is on top of a strangely shaped mountain that with every step down gets one step wider. The mountain is so high he can't see the bottom yet Muhammed wants to move this mountain. So Muhammed starts fetching horses and ties them to the mountain with ropes to move it. That's a direct use of this equality, and you can't stand from afar and look at the scene and say "my, I think that's about -1/12 horses Muhammed is fetching". You'll see Muhammed taking an infinite amount of time fetching an infinite number of horses, and you'll definitely seem him do it more than once.