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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 num

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"

munchler 4y ago
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.)
- karpierz 4y ago
  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.
- gmadsen 4y ago
  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.
- karpierz 4y ago
  > 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.

munchler 4y ago

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.)

karpierz 4y ago
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.
gmadsen 4y ago
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.
karpierz 4y ago
> 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.