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:
> 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).
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.
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