After step one, you have a 50% chance of finding yourself the original; there is nothing controversial about this much. If you are the original, you have a 50% chance of finding yourself still so after step two, and so on. That means after step 99, your subjective probability of still being the original is 0.5^99, in other words as close to zero as makes no difference.
The way subjective probability is usually modeled, there is this huge space of possibilities. And there is a measure defined over it. (I’m not a mathematician, so I may be using the wrong terminology, but what I mean is that every ‘sufficiently nice’ subset of this set of possibilities has a number attached which behaves something like an area for that subset of the space.)
And then, in this model, the probability of some proposition is the measure of the subset where the proposition is true divided by the measure of the whole set. Numerator and denominator. And then each time you learn something, you throw away all of the points in that space that are no longer possible. So, you have typically decreased (never increased) both numerator and denominator. Do the division again and get the new updated probabilities. The space of all possibilities only loses points and measure, it never gains.
But I am not so sure this rule still applies when copying is involved. I think that each time you copy, you need to duplicate the subjective space of possibilities. The original space covered the measures of possibilities from one subjective viewpoint. At the point of copying, that space is duplicated because you now have two viewpoints. Initially, both original and copy are unsure which half of the space is theirs. But when they find out, they each throw out half of the doubled space. And then, as they learn more, possibilities are thrown away from one or the other of the spaces and each one updates to his own subjective probabilities.
So how does this apply to the copying scenario above? Start with one universe. Copy it when you copy the person. Produce a second copy when you produce the second copy of the person. Produce the 99th copy of subjective reality when you produce the 99th copy of the person. If at any stage, one of these persons learns for sure which copy is his, then he can prune his own subjective universe back to the original size.
So, if the protocol is that after each copying, the copy is told that he is a copy and the original is told that he is the original, then before any copying, the person should anticipate being told “You are original” N times, where N is between 0 and 99 inclusive. And he should attach equal probability to each of those events. That is, he should be 99 to 1 sure he will be the original the first time, 98 to one sure the second time, etc.
Forgive me if this is already known as one of the standard approaches to the problem.
Interesting! So you propose to model mind copying by using probabilities greater than 1. I wonder how far we can push this idea and what difficulties may arise...
The way subjective probability is usually modeled, there is this huge space of possibilities. And there is a measure defined over it. (I’m not a mathematician, so I may be using the wrong terminology, but what I mean is that every ‘sufficiently nice’ subset of this set of possibilities has a number attached which behaves something like an area for that subset of the space.)
And then, in this model, the probability of some proposition is the measure of the subset where the proposition is true divided by the measure of the whole set. Numerator and denominator. And then each time you learn something, you throw away all of the points in that space that are no longer possible. So, you have typically decreased (never increased) both numerator and denominator. Do the division again and get the new updated probabilities. The space of all possibilities only loses points and measure, it never gains.
But I am not so sure this rule still applies when copying is involved. I think that each time you copy, you need to duplicate the subjective space of possibilities. The original space covered the measures of possibilities from one subjective viewpoint. At the point of copying, that space is duplicated because you now have two viewpoints. Initially, both original and copy are unsure which half of the space is theirs. But when they find out, they each throw out half of the doubled space. And then, as they learn more, possibilities are thrown away from one or the other of the spaces and each one updates to his own subjective probabilities.
So how does this apply to the copying scenario above? Start with one universe. Copy it when you copy the person. Produce a second copy when you produce the second copy of the person. Produce the 99th copy of subjective reality when you produce the 99th copy of the person. If at any stage, one of these persons learns for sure which copy is his, then he can prune his own subjective universe back to the original size.
So, if the protocol is that after each copying, the copy is told that he is a copy and the original is told that he is the original, then before any copying, the person should anticipate being told “You are original” N times, where N is between 0 and 99 inclusive. And he should attach equal probability to each of those events. That is, he should be 99 to 1 sure he will be the original the first time, 98 to one sure the second time, etc.
Forgive me if this is already known as one of the standard approaches to the problem.
Interesting! So you propose to model mind copying by using probabilities greater than 1. I wonder how far we can push this idea and what difficulties may arise...