Janos2
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Re: whose CEV?
I'm certain this was explained in an OB post (or in the CEV page) at some point, but the notion is that people whose visions of the future are currently incompatible don't necessarily have incompatible CEVs. The whole point of CEV is to consider what we would want to want, if we were better-informed, familiarized with all the arguments on the relevant issues, freed of akrasia and every bad quality we don't want to have, etc.; it seems likely that most of the difference between people's visions of the future stems from differing cultural/memetic backgrounds, character flaws, lack of information and time, etc., and so maybe the space of all our CEVs is actually quite small in configuration-space. Then if the AI steered towards this CEV-region in configuration space, it would likely conform to many people's altruism, and hence be beneficial to humankind as a whole.
Ben:
Well, that depends on your number system. For some purposes +infinity is a very useful value to have. For instance if you consider the extended nonnegative reals (i.e. including +infinity) then every measurable nonnegative extended-real-valued function on a measure space actually has a well-defined extended-nonnegative-real-values integral. There are all kinds of mathematical structures where an infinity element (or many) is indispensable. It's a matter of context. The question of what is a "number" is I think very vague given how many interesting number-like notions mathematicians have come up with. But unquestionably "infinity" is not a natural number, or a real number, or a complex number.
Probability theory, on the other hand, would have... (read more)
Eliezer:
I'm not sure what an "infinite set atheist" is, but it seems from your post that you use different notions of probability than what I think of as standard modern measure theory, which surprises me. Utilitarian's example of a uniform r.v. on [0, 1] is perfect: it must take some value in [0, 1], but for all x it takes value x with probability 0. Clearly you can't say that for all x it's impossible for the r.v. to take value x, because it must in fact take one of those values. But the probabilities are still 0. Pragmatically the way this comes out is that "probability 0" doesn't imply impossible. If... (read more)
I suspect Eliezer would object to my post claiming that I'm confusing map and territory, but I don't think that's fair. If there's a map you're trying to use all over the place (and you do seem to), then I claim it makes no sense to put a little region on the map labelled "maybe this map doesn't make any sense at all". If the map seems to make sense and you're still following it for everything, you'll have to ignore that region anyway. So is it really reasonable to claim that "the probability that probability makes sense is <1"?
Utilitarian:
Measure theory gives a clear answer to this: it's 0. Which is fine.... (read more)
I agree with cumulant. The mathematical subject of probability is based on measure theory, which loses a ton of convergence theorems if we exclude 0 and 1. We can agree that things that are not known a priori can't have probability 0 or 1, but I think we must also agree that "an impossible thing will happen soon" has probability 0, because it's a contradiction. An alternate universe in which the number 7 (in the same kind of number system as ours, etc.) is prime is damn-near inconceivable, but an alternate universe in which impossible things are possible is purely absurd.
If our mathematical reasoning is coherent enough for it to be meaningful... (read more)
Agreed re: the bashing of mainstream math in PT:TLOS. AFAIK, his claims that mainstream math leads to paradoxes are all false; of course trying to act as though various items of mainstream math meant what an uneducated first glance says they mean can make them look bad. (e.g. the Banach-Tarski paradox means either "omg, mathematicians think they can violate conservation of mass!" or "OK, so I guess non-measurable things are crazy and should be avoided") It's not only unnecessary and annoying, but also I think that using usual measure theory would clarify things sometimes. For instance the fact that MaxEnt depends on what kind of distribution you start with, because a probability... (read more)