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Comment by munchler | Hacker News Reader

It's unsound because it assumes that a divergent series has a finite value. It's a kind of fanciful fiction, no matter who came up with it.

In other words, if 1 + 2 + 3 + ... had a finite value, then that value would be -1/12. But it doesn't have a finite value, because it diverges.

It's like saying that if pigs had wings, then they could fly. That's not something I'm happy to base physics on.

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There is solid reasoning behind it though. Divergent perturbation series can be summed in a meaningful way depending on context. Eg https://en.wikipedia.org/wiki/Borel_summation

The mathematically sounds footing for this is unfortunately routinely not taught in undergrad physics.

its only unsound if you are using standard summation. That is not the only kind, nor is it the only useful kind. The same can be said about zero or negative numbers, or the real number line. Those are also "fanciful fictions"

Divergent series have no finite sum, by definition. It's self-contradictory to proceed as though they have a sum anyway, even if it gives "meaningful" results. I think that's different from zero, negative numbers, real numbers, etc., which aren't self-contradictory.

I mean, do you really think the sum of the positive integers is -1/12? It's very easy to prove that it isn't. (The sum of any two positive integers is guaranteed to be greater than either one of them. Therefore, no sum of positive integers can be negative.)

> Divergent series have no finite sum, by definition. It's self-contradictory to proceed as though they have a sum anyway, even if it gives "meaningful" results.

It's self-contradictory in the definition that you're using for sum. It'd be like saying:

1 apple + 1 orange != 2 fruits because addition only works across the same units.

Also, if finite sums of integers have a property, that does not mean that infinite sums of integers have the same property. Ex:

1. The sum of any two integers is guaranteed to be bounded.

2. Therefore, all sums of integers are bounded.

one thing to keep in mind, is that infinity as a physical entity may not even exist. we have constructed infinity from ZFC, and deduced consequences of those axioms.

Many many examples in Math go against natural intuition but they are still useful

for example, you say that the real number isn't "self-contradictory". Look at the banach-tarski paradox, that certainly contradicts intuition. A large part of mathematics is foregoing natural intuitive, and just follow the logical deductions of axioms. That is how we got hyperbolic geometry, which is essential to special relativity.

It is not unsound, and it does not assume that a divergent series has a finite value.

It is true, that on real numbers addition is an RxR->R function, and you can only add 2 numbers, and you can't add 3 or even infinitely many numbers. But then you can define an another function that can take a whole series as an argument, and call it addition too.

You may forbid mathematicians naming another function 'addition', and talk about infinite sums, but then who cares. You can even die on this hill. Again, who cares.

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