The sum of positive integers and quantum physics

[Atla]

Would be interesting to learn more about this, I suspect that it hints to a deeper philosophical understanding of reality, I haven't been able to work out what it is.

If you have any insights please share.

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The sum of positive integers is positive infinity:
1+2+3+4+5+6+... = ∞

To me, this seems to be sort of a "normal" equation, where both sides of the equation are made of the same kind of "classical" numbers, and so the equals sign between them is warranted. (I know that infinity is not really a number.)

But, applying the Riemann Zeta function, which seems to assign a meaningful value to this infinite series, we get -1/12.
1+2+3+4+5+6+... ~ -1/12

To me, this seems to be sort of a "mixed" equation, where the two sides are made of different kinds of numbers, and so I put a tilde instead of an equals sign.

We all learned that the first one is correct, but quantum physics tells us that, at least in the quantum realm, the second one is correct. Substituting -1/12 actually gives the correct answers in some quantum phenomena like the Casimir effect. -1/12 is also an integral part of string theory.

So it seems to be a fundamental part of reality. But what is the nature of the connection? What is it really saying?

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I've only managed to come up with some batshit crazy, almost certainly wrong speculation so far, where I split -1/12 into -1/2 * 1/6.

Here 1+2+3+4+5+6+... would refer to the "positive" conceptualization of our apparent universe and -1/12 would refer to the "negative" conceptualization of our apparent universe, saying that they are equivalent.

In other words, there might be a matter "realm" and an antimatter "realm", and they go together, and the entirity of our universe may be made of 6 such pairs. We only directly see one-sixth or one-twelth of what's really going on. That would explain a lot about missing dark matter/dark energy/apparent information loss at the Big Bang/impossibly big ancient black holes etc.

6 is a number that seems to come up again and again, it's sort of a magical minimalistic number of nature. Things are the mostly densely packed this way.

And so the 6 realms of the universe might be one on the quantum level but appear separate on the classical level.

Anyway this sounds crazy to me too so forget my speculation. Any better insights?

[Science Fan]

The claim that adding up all the positive integers equals -1/12 is based on a bogus claim that a non-converging series should be treated as if it actually converges. Moreover, nothing in physics, or in any empirical discipline, could ever answer the question of what all the positive whole numbers add up to. For one thing, math is non-empirical, and, for a second thing, no one can ever add an infinite series of numbers that do not converge in the real world to see what they total.

[Atla]

The claim that adding up all the positive integers equals -1/12 is based on a bogus claim that a non-converging series should be treated as if it actually converges.

Which is why I gave two results, infinity and -1/12. Both seem to be correct within their own context. You seem to be mixing them together. I don't think you read my comment.

Moreover, nothing in physics, or in any empirical discipline, could ever answer the question of what all the positive whole numbers add up to. For one thing, math is non-empirical and, for a second thing, no one can ever add an infinite series of numbers that do not converge in the real world to see what they total.

Yet -1/12 seems to work in physics, how do you explain that.

Math is a system of abstractions, one umm "layer" of math seems to help describe one layer of physical reality, another layer of math seems to help describe another layer of physical reality. Or something like that. How are the two layers related, what does it mean.

[Science Fan]

The answer -1/12 is wrong. The claim is based on an error in mathematical reasoning. One cannot assign a value to a series that does not converge, and that is how the so-called proof for the -1/12 claim comes about. It's completely inaccurate.

Physics can never answer the question, any more than curved spacetime refutes Euclidean geometry.

This claim about adding up an infinite number of positive whole numbers and ending up with -1/12 is just another example of bullshit being spread across the globe through the internet. No positive number being added to another positive number could ever cause one to cross into the negative side of the real number line. The fact so many people believe this nonsense does not speak well for the intelligence of the human race. It does speak well for our gullibility, but not for our brain power.

[Atla]

Ok

[Atla]

Dismissing Quantum Field Theory and String Theory, and dismissing the Riemann zeta function / Ramanujan Summation, without an explanation other than that the human race is stupid, is a bit over the top from a science fan, imo.

But denying the Casimir effect is a bit even more over the top. Because it is an experimentally established fact that substituting -1/12 gives the correct results, and only this gives the correct results.

[Atla]

Another way of looking at it (again this is major speculation) might be that: this kind of summation technique happens to resemble how a quantum superposition naturally behaves. An infinity of possible states within the superposition, being one. (Maybe this behaviour is best conceptualized by the Feynman path integral, the infinite possibilities being summed together.)