Monday, March 16, 2015

Internal time, external time, probability and disagreement

Suppose that Jim lives a normal human life from the year 2000 to the year 2100. Without looking at a clock, what probability should Jim attach to the hypothesis that an even number of minutes has elapsed from the year 2000? Surely, probability 1/2.

Sally, on the other hand, lives a somewhat odd human life from the year 2000 to the year 2066. During every even-numbered minute of her life, her mental and physical functioning is accelerated by a factor of two. She can normally notice this, because the world around her, including the second hands of clocks, seems to slow down by a factor of two. She has won many races by taking advantage of this. An even-numbered external minute subjectively takes two minutes. Suppose that Sally is now in a room where there is nothing in motion other than herself, so she can't tell whether this was a sped-up minute or not. What probability should Sally attach to the hypothesis that an even number of minutes has elapsed from the year 2000?

If we set our probabilities by objective time, then the answer is 1/2, as in Jim's case. But this seems mistaken. If we're going to assign probabilities in cases like this—and that's not clear to me—then I think we should assign 2/3. After all, subjectively speaking, 2/3 of Sally's life occurs during the even-numbered minutes.

There are a number of ways of defending the 2/3 judgment. One way would be to consider relativity theory. We could mimic the Jim-Sally situation by exploiting the twin paradox (granted, the accelerations over a period of a minute would be deadly, so we'd have to suppose that Sally has superpowers), and in that case surely the probabilities that Sally should assign should be looked at from Sally's reference frame.

Another way to defend the judgment would be to imagine a third person, Frank, who lives all the time twice as fast as normal, but during odd-numbered minutes, he is frozen unconscious for half of each second. For Frank, an even numbered minute has 60 seconds' worth of being conscious and moving, while an odd numbered minute has 30 seconds' worth of it, and external reality stutters. If Frank is in a sensory deprivation chamber where he can't tell if external reality is stuttering, then it seems better for him to assign 2/3 to its being an even-numbered minute, since he's unconscious for half of each odd-numbered one. But Frank's case doesn't seem significantly different from Sally's. (Just imagine taking the limit as the unconscious/conscious intervals get shorter and shorter.)

A third way is to think about time travel. Suppose you're on what is subjectively a long trip in a time machine, a trip that's days internal time long. And now you're asked if it's an even-numbered minute by your internal time (the time shown by your wristwatch, but not by the big clock on the time machine console, which shows external years that flick by in internal minutes). It doesn't matter how the time machine moves relative to external time. Maybe it accelerates during every even-numbered minute. Surely this doesn't matter. It's your internal time that matters.

Alright, that's enough arguing for this. So Sally should assign 2/3. But here's a funny thing. Jim and Sally then disagree on how likely it is that it's an even-numbered minute, even though it seems we can set up the case so they have the same relevant evidence as to what time it. There is something paradoxical here.

A couple of responses come to mind:

  • They really have different evidence. In some way yet to be explained, their different prior life experiences are relevant evidence.
  • The thesis that there cannot be rational disagreement in the face of the same evidence is true when restricted to disagreement about objective matters. But what time it is now is not an objective matter. Thus, the A-theory of time is false.
  • There can be rational disagreement in the face of the same evidence.
  • There are no meaningful temporally self-locating probabilities.


Heath White said...

We could address the problem this way. Think of asking the question “Has it been an even or odd number of minutes since 2000?” as taking a sample. We are picking this instant, however long it takes to raise this question mentally, and asking whether it happens in an even or odd minute. The trick is that Sally can take twice as many samples in even minutes as in odd minutes (just as she can run twice as fast, etc.) This is a perfectly objective difference between Frank and Sally, so it is not wrong to say that, for Sally, it is twice as likely that she is sampling an even minute as it is for Fred.

Alexander R Pruss said...

Very nice.

This make me wonder if we don't need to specify how the question was raised. Suppose I find out that this minute Sally happens to ask herself: "Has it been an even number of minutes since 2000?" Well, that's an odd question to ask oneself (pun not initially intended). Sally asks twice as many odd questions during an even-numbered minute. So the fact that Sally asked herself this odd question is actually evidence that it's an even-numbered minute. In that case, if Jim finds out that Sally asked herself this question, that gives *him* reason to assign 2/3 as well. After all, 2/3 of the questions Sally asks herself fall within even-numbered minutes.

On the other hand, if an ordinary third party asked Sally the question (and did so using a modality where she couldn't tell the speed--perhaps by email), then Sally should go for 1/2. For half of the questions asked by ordinary third parties fall in even-numbered minutes.

It initially sounds weird to think that the answer should depend on who asked the question (given that the question doesn't have a personal pronoun in it), but the fact that someone asks a question sometimes provides us with some evidence relevant to an answer.

But what if Sally and Jim ask themselves the question in the same minute and both find out that the other also asked the question in the same (external) minute? The case seems to be like the following. Sally tosses a biased coin that lands heads 2/3 of the time and Jim a fair coin. A buzzer goes off iff the coins land the same way. The buzzer goes off. How likely is it to be heads? Easy exercise (just work out the probabilities of the four cases): 2/3. So Sally doesn't learn from Jim while Jim learns from Sally. (In particular, they should not split the difference.) This isn't counterintuitive, because Jim's thoughts don't discriminate between even and odd numbered minutes, while Sally's do, so Sally's thoughts are evidence.

So it seems that there is no paradox. Both should go for 2/3 if they learn that they are both asking the question of their own accord.

Alexander R Pruss said...

I suspect that in this case it makes no sense to ask "What should Sally's credence be?" when Sally isn't asking herself the question. This could have interesting implications for epistemology more broadly: there need be no answer as to what someone should think about something when the question hasn't come up.