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By making the a priori assumption that we are equally likely to be anywhere along the chronological list of humans, the Doomsday argument implies that our position n is correlated with the future total number of humans, N.

But if the many-worlds interpretation of quantum theory is correct then all futures will exist with all values of N. In that case our position n is not correlated with any particular future with a particular value of N.

Thus, as it seems reasonable that we cannot predict the future of the human race, can we use the failure of the Doomsday argument as evidence for the many-worlds interpretation?

More detail

Using Bayes's Theorem the probability of the total population N given our position n, P(N|n) is P(N|n) = P(n|N)P(N) / P(n).

If we assume prior complete ignorance of n and N then we should use the improper priors P(N)=1/N and P(n)=1/n.

Assuming a uniform probability for our position n given a particular total population size N, P(n|N)=1/N, we find that Bayes's theorem says

P(N|n) = n / N^2.

This is the Doomsday argument prediction. It implicitly assumes only one future with some particular total population N with prior probability P(N)=1/N.

But in the many-worlds scenario we would have many futures with many values of total population N weighted by the function W(N)=1/N. The probability of our position n would then be given by the sum

P(n) = Sum[N=n to infinity] P(n|N) W(N)

P(n) = Sum[N=n to infinity] 1/N * 1/N = 1 / n

As mentioned above the probability P(n)=1/n and W(N)=1/N implies total ignorance. Therefore in the many-worlds scenario if we lived until the end of the human race our birth position n would not tell us anything about the final population size N that any particular version of ourselves will experience.

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    My hunch is that when you run the maths for both together, you end up where you began... where your prediction of future population is purely contingent on how you evaluate the conditions of the current word you're in. I'm a little skeptical about the usefulness of knowledge gained by statistical tricks applied to variations of the anthropic principle. – Ask About Monica Nov 27 '17 at 17:33
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    Suppose you have some phenomenon X which is a priori unlikely and a theory T that makes it more likely, can X be considered evidence for T? In a very weak sense, yes. Very weak because there could be plenty of other theories making X more likely, and the usual approach is to survey them to select the "best" one according to some criteria. In this case there is a long list of alternative explanations, so it does not really work as "evidence". – Conifold Nov 27 '17 at 20:07
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This does not follow at all. If you buy the "Doomsday Theory", which argues that statistically speaking, there are likely to roughly be as many people born in the future as have already lived, then you are accepting the idea that we can make accurate predictions of this nature purely on the basis of statistics.

It might be false for many reasons, including that we don't have enough information to determine a proper reference class. However, the fact that we don't have the means to independently confirm its predictions is merely denying the premise that we can make such predictions by this method. It doesn't thereby provide significant support (let alone proof) for an unrelated unproven theory.

Interestingly enough, the title "Doomsday Theory" is a complete misnomer, based on the popular belief that there are more people alive today than have ever lived and died, which would place us right at the end of human history, by the theory. But in fact, the actual number of people who have ever lived dwarfs the number of people who are alive now. So the theory is actually quite reassuring, it implies we have quite a long distance to go before extinction, even at current rates of population growth (which are unlikely to be sustained).

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    I sincerely hope you are right, but I don't see how you can draw the conclusion that we have quite a long way to go before extinction. I was put in a rather gloomy mood by this book; will we have enough fresh water and food? Title: The world according to Pimm : a scientist audits the Earth Author Pimm, Stuart L. (Stuart Leonard) Publisher:McGraw-Hill,Pub date 2001. – Gordon Nov 28 '17 at 13:05
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    @Gordon - The point isn't my personal opinion --or anyone else's-- on whether extinction is likely, it's that this so-called "Doomsday Theory" actually predicts we are a minimum of 1000 years away from extinction, which is considerably longer than many of us might otherwise expect. – Chris Sunami Nov 28 '17 at 14:19
  • @gordan: I think there's room for hope, I read recently that Portugal ran for whole eight hours on renewable energy over the summer. – Mozibur Ullah Dec 1 '17 at 10:30
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That's an interesting argument, but it seems that you're merely resisting the idea of calculating p(N|n) on the ground that p(n|N) would not make sense if there are many futures. However you should be careful about the interpretation of probabilities you are assuming. Some authors (Deutsch and Wallace) have argued that the best interpretation of probabilities in the context of MWI is one based on rational decision theory. In this context, calculating p(N/n) makes sense: it amounts to "bet" on the set of futures with a specific N. And it seems to me that you can still arrive at the conclusion of the doomsday argument.

Or to say it differently, I'm not sure it's true that in MWI "our position n is not correlated with any particular future with a particular value of N" because all future branches have a different weight, hence different probabilities given the present.

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The doomsday argument as explained in your post is defeated by using assumptions that make no sense. For example, if you have complete ignorance of something you can't make probability estimates about it. Probabilities are specific numbers you can't get specific numbers from ignorance. Probability estimates come from knowledge of explanations. For example, the probability of getting spin up or spin down when measuring an electron is a result of knowing its state, not of ignorance:

https://www.youtube.com/watch?v=wfzSE4Hoxbc.

There are several ways in which the doomsday argument could be wrong. For example, the number of people could be bounded above at some number close to the present number of humans because we will soon have cheap technology that will keep us all alive indefinitely, e.g. - medical technology advocated by Aubrey de Grey. See also "The Beginning of Infinity" by David Deutsch: look up probability in the index.

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The "doomsday argument" fails on its own terms, as a misapplication of Bayesian reasoning. This is explained in detail in O'Neill (2014). The "doomsday argument" asserts a "Bayesian shift" in beliefs based on observation of birth-order, and this paper argues that this is a misapplication of Bayes' theorem, leading to a foregone conclusion.

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