Suppose I am sluggish mentally in the morning and function fast mentally in the afternoons. I find myself in a dark room with no information about what time it is. Consider the two hypotheses MORN, that it is between 7am and 8am and AFT, that it is between 1pm and 2pm. Which is more likely, or are they equally likely?
It seems obvious that if I function mentally the same way in the morning and afternoon, MORN and AFT should be equally likely.
But now suppose I live in a world where every morning all processes in the solar system slow down by a factor of two, returning to the normal speed in the afternoon. Then, surely, for all practical purposes, the period from 7am to 8am is half an hour long! And so it seems right to say P(MORN) = (1/2)P(AFT). But the case of my own mental sluggishness seems to be just a more localized version of the solar system slowing down. Thus, in my original story we should say that likewise P(MORN) < P(AFT).
Here is one way to imagine mental functioning slowing down: my thinking is divided into discrete moments (like a computer’s internal clock, where basic operations take a clock cycle), and in the morning there are fewer of these discrete moments of thinking. If so, then it seems reasonable to say that the probability that the current time is between x and y is proportional to the number of discrete moments of thought between x and y, and hence P(MORN) < P(AFT), there being fewer moments of thought in the morning.
But I have a hard time getting good intuitions about the continuous case.
6 comments:
The general relativity thought experiments can be used for the continuous time case. If the solar system moves at 1/2 light speed in the PM's and at 0 velocity relative to our outside observer in the AM's, time will seem the same for us in AM and PM (so P(MORN)= P(AFT) for us) but the outside observer will see us as if time slows for us in the AM (so for the observer P(MORN) < P(AFT)).
That's a good point. It seems very intuitive that in a relativistic context, the observer should use their own "proper time" for this. The probability of its being between t1 and t2 should be proportional to the proper time between t1 and t2 for this observer.
This means we will have disagreement between observers as to probabilities. To one observer, P(MORN)=P(AFT). To another P(MORN)<P(AFT). (I am assuming MORN and AFT are defined absolutely.) SInce these are self-locating probabilities, this is not too surprising, though it could be kind of weird.
Did you remove the book you're writing from Github?
Shortly before submitting to publishers.
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