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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.

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How does an infinite sum of integers appear in nature (physics)?

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).

https://en.wikipedia.org/wiki/Casimir_effect#Derivation_of_C...

and

https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_ef...

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

lim \epsilon -> 0+ ( \sum_{n=1}^\infty n e^{- \epsilon n} + ...),

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.

Why can't time begin at a point in the past, but extend infinitely into the future?

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.

Length of the coast of Britain?

That assumes details can be arbitrarily small, which Planck refuted.

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.

Also, at an atomic scale, the word "coast" doesn't really make sense any more...

Refuted? I thought that hypothesis was far from proven.

Not Planck, but by a few others after him. https://en.wikipedia.org/wiki/Quantum_spacetime

Fourier transform, which is used in classical wave theory (optics and electricity) and in quantum mechanics

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.

> No it doesn't. Under some unintuitive definition for infinite summation that is useful in some physical calculations it does which is surprising.

It's not surprising, you can pretty much get any result you want like this.

not checked this myself. but. my assumption is that this also applies when defining the numberline as a finite circle.

for example on a circle of circumferance 15, 14 is also equal to -1, 13 is equal to -2 etc.

which says the sum of 1+2+3... averages to -1/12

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.

Nonesense. Since when did 1.2 modulo 2 equal -0.8

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.

[0] https://news.ycombinator.com/item?id=12106633

[1] https://news.ycombinator.com/item?id=12106409

because to get 0.5 for the sum of 1-1+1-1+.... requires real numbers. I never restricted it to integers.

The point of the construct is it applies to a circle of arbitrary/unknown length 14,12.2, 500,000 million billion point 6 light years.

and the average of the sum of 1+2+3+4+... will be/tend to -1/12

and its easy to test, just use a signed x bit number.

  > ... 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:

https://en.wikipedia.org/wiki/Non-standard_analysis

> 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).

>That same argument gives you Zeno's paradox.

I believe Zeno's paradox is on the rationals, not integers.

>You can sum a pattern of numbers in O(1) time if you use logic instead of brute force.

There may be closed solutions for finite summation patterns, but infinite summation patterns of integers have no solution (by definition).

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.

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