I think that when working on countable additivity, supertasks, infinite utility sequences and the like, it's really important to remember that infinite sums are not sums. Infinite summation is a limiting procedure that goes from an infinite sequence of numbers to a number, satisfying some of the properties of summation. There is nothing absurd about a process where in the first half-second you walk half a meter, in the next quarter-second you walk a quarter of a meter, in the next eighth-second you walk an eighth of a meter--and at the end of the whole second you're a mile away. This is all basically a point from Benacerraf or Thomson.
What this means is that in philosophical contexts where summing up an infinite sequence comes up, one needs to justify the idea that the right way to sum up the sequence is to use this limiting procedure. Sometimes, as in my example of walking, while it's not absurd that you would end up a mile away, you can assume a principle of continuity that gives a more natural answer.
The most risky cases are where the sum is only conditionally convergent, as in Nover and Hajek's Pasadena Game, where with probability 2-n you win -(-2)n/n. If what is being "added up" are things where the ordering does not matter--e.g., utilities--then the idea that you're "adding" is dubious. (This does not affect Nover and Hajek's use of the game, and indeed it's basically their point.)
"Infinite sums are not sums" - OUCH! My brain hurts!
ReplyDelete"The most risky cases are where the sum is only conditionally convergent, as in Nover and Hajek's Pasadena Game, where with probability 2-n you win -(-2)n/n. If what is being "added up" are things where the ordering does not matter--e.g., utilities--then the idea that you're "adding" is dubious." - Is this why us Michigan Wolverines lost more Rose Bowls than we won? Is this the cause of the phantom touchdown against us by USC in 1979 Rose Bowl? It must be that Pasadena Game.
P.S. I'd love to see a Michigan/Baylor match up in a bowl game. I'm sure you'd want a rematch too.
"If what is being "added up" are things where the ordering does not matter--e.g., utilities--then the idea that you're "adding" is dubious."
ReplyDeleteNitpick: for comparisons of utility not only does ordering matter, it's the only thing that does. That's the reason summing them can be dubious.
James:
ReplyDeleteSuppose I have an hour of pain, then my memory of it is wiped, and then I have an hour of pleasure, and then my memory of it is wiped. Do I get any different amount of utility if the order is swapped?
If yes, then order does matter--but then it's not additive.
Alex:
ReplyDeleteIn a way the order does matter. By suffering the hour of pain first, you at least get the bad part out of the way even if you don't remember it. It's good to get the nasty stuff over first, then it's smooth sailing even if you don't remember that too. At least I'd like to have it that way, even if I don't remember it. Some might say, if you can't remember it why care? I think while I'm in that pain state, I very much do care.