## Monday, January 20, 2014

### Even more on infinity and probability

Imagine a sequence of blindfolded electrically charged people, at positions 1,2,3,... on a line in some coordinate system. Let f be a permutation of the natural numbers with the property that f(n) is even if and only if n is divisible by four—this permutation maps the even numbers onto the numbers divisible by four and the odd ones onto the numbers not divisible by four. Suppose the person at position n has electric charge f(n). You're one of the people. And that's all the empirical data you have.

Question 1: What is the probability that your electric charge is even?

Obvious Answer: The people at positions on the line divisible by four have even electric charge. So the probability that you have positive electric charge equals the probability that your position is divisible by four. But every position has exactly one person on it, and so the probability that your position is divisible by four is 1/4. Hence that's also the probability that your electric charge is even.

Inobvious Answer: Instead of mentally arranging people by position, arrange them by electric charge. There is exactly one person per natural number arranged by charge. And so the probability that your number is even is 1/2.

The point here is that P(charge is even)=P(position is divisible by four). When we focus on the arrangement by position we are inclined to assign 1/4 to the right hand side, and hence we assign it to the left hand side, too. When we focus on the arrangement by charge we are inclined to assign 1/2 to the left hand side, and hence we assign it to the right hand side, too.

Which is right?

Maybe we can say that one of the two orderings is more relevant for calculating probabilities. I suspect that anybody who takes this route will take the positional arrangement to be that one.

But is that really right? Imagine an angel that is assigning positions and charges to an infinite number of people subject to the rule that the person at position n gets charge f(n). The angel might start by first holding an infinite lottery where each person first gets given a position number, and then will use that position number to calculate the charge f(n) for the person. Or the angel might start by holding an infinite lottery where each person first gets given a charge number m, and then the position number is calculated as f−1(m). In the former case, our ansewr might seem to be 1/4, while in the latter it seems to be 1/2. We have no idea which the angel is going to do if the information listed is all we have, nor any idea whether it is going to be an angel, or a natural process, or whatever. Maybe then we should average the two probabilities, and get (1/2+1/4)/2=3/8 as our probability? But that doesn't seem right, either.

I suspect the right answer is that in this scenario there just is no answer to the question. And if that is right, then where there is a simultaneous infinity of cases in a reference class—as in some multiverse scenarios—there are no probabilities.

But what if instead of spatial arrangement we have temporal arrangement? Then I have an intuition that the temporal arrangement takes priority over the charge arrangement for the calculation of probabilities (and would even take priority over the positional arrangement, I guess). I don't know if I should keep or abandon this intuition. It might offer an important disanalogy between space and time.