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