On Sun, Mar 8, 2020 at 7:46 PM Russell Standish <[email protected]> wrote:
> On Sun, Mar 08, 2020 at 06:50:52PM +1100, Bruce Kellett wrote: > > On Sun, Mar 8, 2020 at 5:32 PM Russell Standish <[email protected]> > wrote: > > > > On Fri, Mar 06, 2020 at 10:44:37AM +1100, Bruce Kellett wrote: > > > > > That is, in fact, false. It does not generate the same strings as > > flipping a > > > coin in single world. Sure, each of the strings in Everett could > have > > been > > > obtained from coin flips -- but then the probability of a sequence > of > > 10,000 > > > heads is very low, whereas in many-worlds you are guaranteed that > one > > observer > > > will obtain this sequence. There is a profound difference between > the two > > > cases. > > > > You have made this statement multiple times, and it appears to be at > > the heart of our disagreement. I don't see what the profound > > difference is. > > > > If I select a subset from the set of all strings of length N, for > example > > all strings with exactly N/3 1s, then I get a quite specific value > for the > > proportion of the whole that match it: > > > > / N \ > > | | 2^{-N} = p. > > \N/3/ > > > > Now this number p will also equal the probability of seeing exactly > > N/3 coins land head up when N coins are tossed. > > > > What is the profound difference? > > > > > > > > Take a more extreme case. The probability of getting 1000 heads on 1000 > coin > > tosses is 1/2^1000. > > If you measure the spin components of an ensemble of identical spin-half > > particles, there will certainly be one observer who sees 1000 spin-up > results. > > That is the difference -- the difference between probability of 1/2^1000 > and a > > probability of one. > > > > In fact in a recent podcast by Sean Carroll (that has been discussed on > the > > list previously), he makes the statement that this rare event (with > probability > > p = 1/2^1000) certainly occurs. In other words, he is claiming that the > > probability is both 1/2^1000 and one. That this is a flat contradiction > appears > > to escape him. The difference in probabilities between coin tosses and > > Everettian measurements couldn't be more stark. > > That is because you're talking about different things. The rare event > that 1 in 2^1000 observers see certainly occurs. In this case > certainty does not refer to probability 1, as no probabilities are > applicable in that 3p picture. Probabilities in the MWI sense refers > to what an observer will see next, it is a 1p concept. > > And that 1p context, I do not see any difference in how probabilities > are interpreted, nor in their numerical values. > > Perhaps Caroll is being sloppy. If so, I would think that could be > forgiven. > Yes, I think the Carroll's comment was just sloppy. The trouble is that this sort of sloppiness permeates all of these discussions. As you say, probability really has meaning only in the 1p picture. So the guy who sees 1000 spin-ups in the 1000 trials will conclude that the probability of spin-up is very close to one. That is why it makes sense to say that the probability is one. The fact that this one guy sees this is certain in Many-worlds (This may be another meaning of probability, but an event that is certain to happen is usually referred to as having probability one.). The trouble comes when you use the same term 'probability' to refer to the fact that this guy is just one of the 2^N guys who are generated in this experiment. The fact that he may be in the minority does not alter the fact that he exists, and infers a probability close to one for spin-up. The 3p picture here is to consider that this guy is just chosen at random from a uniform distribution over all 2^N copies at the end of the experiment. And I find it difficult to give any sensible meaning to that idea. No one is selecting anything at random from the the 2^N copies because that is to how the copies come about -- it is all completely deterministic. The guy who gets the 1000 spin-ups infers a probability close to one, so he is entitled to think that the probability of getting an approximately even number of ups and downs is very small: eps^1000*(1-eps)^1000 for eps very close to zero. Similarly, guys who see approximately equal numbers of up and down infers a probability close to 0.5. So they are entitled to conclude that the probability of seeing all spin-up is vanishingly small, namely, 1/2^1000. The main point I have been trying to make is that this is true whatever the ratio of ups to downs is in the data that any individual observes. Everyone concludes that their observed relative frequency is a good indicator of the actual probability, and that other ratios of up:down are extremely unlikely. This is a simple consequence of the fact that probability is, as you say, a 1p notion, and can only be estimated from the actual data that an individual obtains. Since people get different data, they get different estimates of the probability, covering the entire range [0,1]; no 3p notion of probability is available -- probabilities do not make sense in the Everettian case when all outcomes occur. This is the basic argument that Kent makes in arxiv:0905.0624. The difference from the deterministic coin tossing situation is that in that case, only one outcome occurs in any trial, so the sequence of N trials generates a single bit sting of length N, indicating a particular value of the probability for success on any toss. The situation could not be more different from the case in which all outcomes always occur. Bruce -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. 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