Nick Bostrom showed that either position in Extreme Sleeping Beauty seems absurd, then gave a third option. I argued that his third option seems worse than either of the original pair. If I am right there that the case for Bayesian conditioning without updating on evidence fails, we have a choice of disregarding Bayesian conditioning in at least some situations, or distrusting the aversion to extreme updates as in Extreme Sleeping Beauty. The latter seems the necessary choice, given the huge disparity in evidence supporting Bayesian conditioning and that supporting these particular intuitions about large updates and strong beliefs.
Notice that both the Halfer and Thirder positions on Extreme Sleeping Beauty have very similar problems. They are seemingly opposed by the same intuitions against extreme certainty in situations where we don’t feel certain, and extreme updates in situations where we hardly feel we have any evidence. Either before or after discovering you are in the first waking, you must be very sure of how the coin came up. And between ignorance of the day and knowledge, you must change your mind drastically. If we must choose one of these positions then, it is not clear which is preferable on these grounds alone.
The Presumptuous Philosopher
39It is the year 2100 and physicists have narrowed down the search for a theory of everything to only two remaining plausible candidate theories, T1 and T2 (using considerations from super-duper symmetry). According to T1 the world is very, very big but finite, and there are a total of a trillion, trillion observers in the cosmos. According to T2, the world is very, very, very big but finite, and there are a trillion, trillion, trillion observers. The super-duper symmetry considerations are indifferent between these two theories. Physicists are preparing a simple experiment that will falsify one of the theories. Enter the presumptuous philosopher: “Hey guys, it is completely unnecessary for you to do the experiment, because I can already show to you that T2 is about a trillion times more likely to be true than T1 (whereupon the philosopher […] appeals to SIA)!”
The Presumptuous Philosopher is like the Extreme Sleeping Beauty Thirder because they are both in one of two possible worlds with a known probability of existing, one of which has a much larger population than the other. They are both wondering which of these worlds they are in.
Is the Presumptuous Philosopher really so presumptuous? Analogous to the Extreme Sleeping Beauty Halfer then shall be the Unpresumptuous Philosopher. When the Unpresumptuous Philosopher learns there are a trillion times as many observers in T2 she remains cautiously unmoved. However, when the physicists later discover where in the cosmos our planet is under both theories, the Unpresumptuous Philosopher becomes virtually certain that the sparsely populated T1 is correct while the Presumptuous Philosopher hops back on the fence.
The Presumptuous Philosopher is often chided for being sure the universe is infinite, given there is some chance of an infinite universe existing. It should be noted that this is only as long as he cannot restrict his possible locations in it to any finite region. The Unpresumptuous Philosopher is uncertain under such circumstances. However she believes with probability one that we are in a finite world if she knows her location is within any finite region. For instance if she knows the age of her spatially finite universe she is certain that it will not continue for infinitely long. Here her presumptuous friend is quite unsure.
It seems to me that as the two positions on Extreme Sleeping Beauty are as unintuitive as each other, the two philosophers seem as presumptuous as each other. The accusation of inducing a large probability shift and encouraging ridiculous certainty is hardly an argument that can be used against the SIA-Thirder-Presumptuous Philosopher position in favor of the SSA-Halfer-Unpresumptuous Philosopher side. Since the Presumptuous Philosopher is usually considered the big argument against SIA, and not considered an argument against SSA at all, an update in favor of SIA is in order.