In precisely the same sense that we can write 1 + 1⁄2 + 1⁄4 + … = 2, despite that no real-world process of “addition” involving infinitely many terms may be performed in a finite number of steps, we can write 1 + 2 + 3 + 4 + 5 + … = −1/12
I think this is overstating things (which is fair enough to make the point you’re making).
The first is simply a shorthand for “the limit of this sum is 2”, which is an extremely simple, general definition, which applies in almost all contexts, and matches up with what addition means in almost all contexts. It preserves far more of the properties of addition as well—it’s commutative, associative, etc. In most cases where you want to work with the sum of an infinite series, the correct value to use for this series is 2.
The second is a shorthand for something far more complex, which applies in a far more limited range of cases, and doesn’t preserve almost any of the properties we expect of addition. It’s not linear or stable. In most cases where you want to work with sums of infinite series, the correct sum for this series is infinity. Only very rarely would you want −1/12.
The first is simply a shorthand for “the limit of this sum is 2”,
It doesn’t need to be! It can be more generally something like “the unique value that matches patterns,” where what counts is extended first beyond integers, and then to infinite series, and then to divergent infinite series.
You run into the trouble of having to defend why your way to fit the divergent series into a pattern is the right one—other approaches may give different results.
The claim is that they don’t: any pattern that points to a finite result points to the same one. If you want proof, then you need a more rigorous formalism.
Now the problem is this pattern leads to a contradiction because it can equally prove any number you want. So we don’t choose to use it as a definition for an infinite sum.
So you need to do a bit more work here to define what you mean here.
I think this is overstating things (which is fair enough to make the point you’re making).
The first is simply a shorthand for “the limit of this sum is 2”, which is an extremely simple, general definition, which applies in almost all contexts, and matches up with what addition means in almost all contexts. It preserves far more of the properties of addition as well—it’s commutative, associative, etc. In most cases where you want to work with the sum of an infinite series, the correct value to use for this series is 2.
The second is a shorthand for something far more complex, which applies in a far more limited range of cases, and doesn’t preserve almost any of the properties we expect of addition. It’s not linear or stable. In most cases where you want to work with sums of infinite series, the correct sum for this series is infinity. Only very rarely would you want −1/12.
It doesn’t need to be! It can be more generally something like “the unique value that matches patterns,” where what counts is extended first beyond integers, and then to infinite series, and then to divergent infinite series.
You run into the trouble of having to defend why your way to fit the divergent series into a pattern is the right one—other approaches may give different results.
The claim is that they don’t: any pattern that points to a finite result points to the same one. If you want proof, then you need a more rigorous formalism.
Sure, but you’re just claiming that, and I don’t think it’s actually true.
That’s clearly not true in a general sense. Here’s a pattern that points to a different sum:
1 + 2 + 3 + … = 1 + (1 + 1) + (1 + 1 + 1) + … = 1 + 1 + 1 + … = − 1⁄2
Now the problem is this pattern leads to a contradiction because it can equally prove any number you want. So we don’t choose to use it as a definition for an infinite sum.
So you need to do a bit more work here to define what you mean here.