2) Some of the arguments against Pascal’s Wager and Pascal’s Mugging don’t depend on the probability. For instance, Pascal’s Wager has the “worshipping the wrong god” problem—what if there’s a god who prefers that he not be worshipped and damns worshippers to Hell? Even if there’s a 99% chance of a god existing, this is still a legitimate objection (unless you want to say there’s a 99% chance specifically of one type of god).
That argument is isomorphic to the one discussed in the post here:
“Hmm...” she says. “I hadn’t thought of that. But what if these equations are right, and yet somehow, everything I do is exactly balanced, down to the googolth decimal point or so, with respect to how it impacts the chance of modern-day Earth participating in a chain of events that leads to creating an intergalactic civilization?”
“How would that work?” you say. “There’s only seven billion people on today’s Earth—there’s probably been only a hundred billion people who ever existed total, or will exist before we go through the intelligence explosion or whatever—so even before analyzing your exact position, it seems like your leverage on future affairs couldn’t reasonably be less than a one in ten trillion part of the future or so.”
Essentially, it’s hard to argue that the probabilities you assign should be balanced so exactly, and thus (if you’re an altruist) Pascal’s Wager exhorts you either to devote your entire existence to proselytizing for some god, or proselytizing for atheism, depending on which type of deity seems to you to have the slightest edge in probability (maybe with some weighting for the awesomeness of their heavens and awfulness of their hells).
So that’s why you still need a mathematical/epistemic/decision-theoretic reason to reject Pascal’s Wager and Mugger.
What you have is a divergent sum whose sign will depend to the order of summation, so maybe some sort of re-normalization can be applied to make it balance itself out in absence of evidence.
Actually, there is no order of summation in which the sum will converge, since the terms get arbitrary large. The theorem you are thinking of applies to conditionally convergent series, not all divergent series.
Strictly speaking, you don’t always need the sums to converge. To choose between two actions you merely need the sign of difference between utilities of two actions, which you can represent with divergent sum. The issue is that it is not clear how to order such sum or if it’s sign is even meaningful in any way.
Without discussing the merits of your proposal, this is something that clearly falls under “mathematical/epistemic/decision-theoretic reason to reject Pascal’s Wager and Mugger”, so I don’t understand why you left that comment here.
That argument is isomorphic to the one discussed in the post here:
Essentially, it’s hard to argue that the probabilities you assign should be balanced so exactly, and thus (if you’re an altruist) Pascal’s Wager exhorts you either to devote your entire existence to proselytizing for some god, or proselytizing for atheism, depending on which type of deity seems to you to have the slightest edge in probability (maybe with some weighting for the awesomeness of their heavens and awfulness of their hells).
So that’s why you still need a mathematical/epistemic/decision-theoretic reason to reject Pascal’s Wager and Mugger.
What you have is a divergent sum whose sign will depend to the order of summation, so maybe some sort of re-normalization can be applied to make it balance itself out in absence of evidence.
Actually, there is no order of summation in which the sum will converge, since the terms get arbitrary large. The theorem you are thinking of applies to conditionally convergent series, not all divergent series.
Strictly speaking, you don’t always need the sums to converge. To choose between two actions you merely need the sign of difference between utilities of two actions, which you can represent with divergent sum. The issue is that it is not clear how to order such sum or if it’s sign is even meaningful in any way.
Without discussing the merits of your proposal, this is something that clearly falls under “mathematical/epistemic/decision-theoretic reason to reject Pascal’s Wager and Mugger”, so I don’t understand why you left that comment here.