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>Under the normal rules which hold for direct use obviously the answer is positive infinity just like you would be expect, you're not stupid and you could be a

>Under the normal rules which hold for direct use obviously the answer is positive infinity just like you would be expect, you're not stupid and you could be a mathematician if you wanted to. There is no answer for an infinite sum. It is impossible to sum an infinite quantity of integers. The answer is definitely not positive infinity, as that is not a number and the sum of integers must be an integer. The article points out a few times that such a "sum" is undefined.

mrob 10y ago
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: https://en.wikipedia.org/wiki/Non-standard_analysis https://en.wikipedia.org/wiki/Non-standard_analysis
- jsprogrammer 10y ago
  > 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.
  - andrepd 10y ago
    >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.
    - Dylan16807 10y ago
      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.
  - eximius 10y ago
    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)
  - Dylan16807 10y ago
    >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).
    - jsprogrammer 10y ago
      >That same argument gives you Zeno's paradox. I believe Zeno's paradox is on the rationals, not integers. >You can sum a pattern of numbers in O(1) time if you use logic instead of brute force. There may be closed solutions for finite summation patterns, but infinite summation patterns of integers have no solution (by definition).
dmfdmf 10y ago
> There is no answer for an infinite sum. It is impossible to sum an infinite quantity of integers. I agree. Infinity is a process that can yield a number but is not an actual number and, Cantor et. al. notwithstanding, there is no such thing as a "completed infinity" other than terminating it at a finite step. If you are careful and in certain contexts you can use the "limit" of an infinite converging process but you must make that assumption explicit to avoid errors. All these bizarre math tricks rest on treating it as a number when its undefined. Its like those puzzles I read as a kid that "prove" 1=0 and they typically depend on an implicit division-by-zero step which is also undefined just like infinity. Once you start working with the undefined you have to very careful and even Gauss made errors when he was laying the groundwork for infinite series. To the degree that this math has ANY validity it is in the context of some esoteric and specialized area of math and it is NOT appropriate to foist it on the general public as a general result. The motive in such attempts is to impress or intimidate or destroy math (nihilism) which I find despicable.

mrob 10y ago

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: https://en.wikipedia.org/wiki/Non-standard_analysis https://en.wikipedia.org/wiki/Non-standard_analysis

jsprogrammer 10y ago
> 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.
- andrepd 10y ago
  >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.
  - Dylan16807 10y ago
    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.
- eximius 10y ago
  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)
- Dylan16807 10y ago
  >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).
  - jsprogrammer 10y ago
    >That same argument gives you Zeno's paradox. I believe Zeno's paradox is on the rationals, not integers. >You can sum a pattern of numbers in O(1) time if you use logic instead of brute force. There may be closed solutions for finite summation patterns, but infinite summation patterns of integers have no solution (by definition).

dmfdmf 10y ago

> There is no answer for an infinite sum. It is impossible to sum an infinite quantity of integers. I agree. Infinity is a process that can yield a number but is not an actual number and, Cantor et. al. notwithstanding, there is no such thing as a "completed infinity" other than terminating it at a finite step. If you are careful and in certain contexts you can use the "limit" of an infinite converging process but you must make that assumption explicit to avoid errors. All these bizarre math tricks rest on treating it as a number when its undefined. Its like those puzzles I read as a kid that "prove" 1=0 and they typically depend on an implicit division-by-zero step which is also undefined just like infinity. Once you start working with the undefined you have to very careful and even Gauss made errors when he was laying the groundwork for infinite series. To the degree that this math has ANY validity it is in the context of some esoteric and specialized area of math and it is NOT appropriate to foist it on the general public as a general result. The motive in such attempts is to impress or intimidate or destroy math (nihilism) which I find despicable.