> Obviously it doesn't make sense to add all the positive integers (the series doesn't converge), but if you squint and ignore this, and just do the arithmetic a certain way, you get -1/12.
I'm not a pure-maths type person but in my experience, if you get one answer by following simple, well understood maths (like "the sum of two positive integers is a positive integer") and another answer by "squinting and ignoring it", this doesn't mean the simple answer is wrong, it means you did something else wrong (like the hidden divide-by-zero present in your typical "proof that 1 = 2"). Paradoxes point to an error in the formulation of the question.
I'm not a pure-maths type person but in my experience, if you get one answer by following simple, well understood maths (like "the sum of two positive integers is a positive integer") and another answer by "squinting and ignoring it", this doesn't mean the simple answer is wrong, it means you did something else wrong (like the hidden divide-by-zero present in your typical "proof that 1 = 2"). Paradoxes point to an error in the formulation of the question.