That adding positive and negative infinity is undefined may be true mathematically, but you have to decide one way or another.
Right; or if you don’t decide exactly, at least you have to do (believe or not believe) one or the other.
I would say that the model breaks down. Mathematics (or at least the particular mathematical model being used) is not capable of describing this situation, but that doesn’t make the situation itself meaningless. (That would be a version of the map/territory fallacy.)
Defined or not, it is perfectly obvious that you should choose the second door.
Here I disagree with you. I would say that you have not given enough information. It is as if you gave the same problem statement but with the word ‘infinite’ removed (so that we only know whether the utilities are positive or negative). It may seem as if you have given all of the information: the probabilities and the utilities. But the mathematics which we use to calculate everything else out of those values breaks down, so in fact you have not given all of the information.
One important missing piece of information is the ratio of the first positive utility to the second. That and two other independent ratios would be enough information, if they’re all finite. (If not, then we might need more information.)
And don’t tell me that these ratios are undefined; the mathematical model that calculates the ratios from the information given breaks down, that’s all. In fact, there is an atlernative mathematical model of decision which deals only in ratios between utilities; if you’d followed that model from the beginning, then you would never have tried to state the actual utilities themselves at all. (For mathematicians: instead of trying to plot these 4 utilities in a 4-dimensional affine space, plot them in a 3-dimensional projective space.)
It may be true that some people couldn’t make themselves believe in God, but only in belief, but that would be a problem with them, not with the argument.
Right; the proper conclusion of the argument is not to believe, but to try to believe. And if you buy the argument, then you should try very hard!
I agree with everything you’ve said here, including that in the two door situation the decision could go the other way if you had more information about the ratio of the utilities. Still, it seems to me that what I said is right in this way: if you are given no other information except as stated, you should choose the second door, because your best estimate of the ratios in question will be 1-1. But if you have some other evidence regarding the ratios, or if they are otherwise specified in the problem, your argument is correct.
Right; or if you don’t decide exactly, at least you have to do (believe or not believe) one or the other.
I would say that the model breaks down. Mathematics (or at least the particular mathematical model being used) is not capable of describing this situation, but that doesn’t make the situation itself meaningless. (That would be a version of the map/territory fallacy.)
Here I disagree with you. I would say that you have not given enough information. It is as if you gave the same problem statement but with the word ‘infinite’ removed (so that we only know whether the utilities are positive or negative). It may seem as if you have given all of the information: the probabilities and the utilities. But the mathematics which we use to calculate everything else out of those values breaks down, so in fact you have not given all of the information.
One important missing piece of information is the ratio of the first positive utility to the second. That and two other independent ratios would be enough information, if they’re all finite. (If not, then we might need more information.)
And don’t tell me that these ratios are undefined; the mathematical model that calculates the ratios from the information given breaks down, that’s all. In fact, there is an atlernative mathematical model of decision which deals only in ratios between utilities; if you’d followed that model from the beginning, then you would never have tried to state the actual utilities themselves at all. (For mathematicians: instead of trying to plot these 4 utilities in a 4-dimensional affine space, plot them in a 3-dimensional projective space.)
Right; the proper conclusion of the argument is not to believe, but to try to believe. And if you buy the argument, then you should try very hard!
I agree with everything you’ve said here, including that in the two door situation the decision could go the other way if you had more information about the ratio of the utilities. Still, it seems to me that what I said is right in this way: if you are given no other information except as stated, you should choose the second door, because your best estimate of the ratios in question will be 1-1. But if you have some other evidence regarding the ratios, or if they are otherwise specified in the problem, your argument is correct.