Wednesday, March 30, 2016

A tale of four multiverses

Start with these two countably infinite multiverses:

  1. Before any universe of multiverse 1 exhibits life, a fair die is cast in each universe. If the die shows six, a diamond core forms on an uninhabited moon of one earthlike planet. Otherwise, an iron core forms there. Then, completely independently of the die roll and the core of the moon, a single person comes into existence on that earthlike planet in each universe of that multiverse. The person is then informed of all of the above facts, as well as of the fact that infinitely many dice showed sixes and infinitely many did not (that fact was very likely, but now the person is sure).
  2. Multiverse 2 is just like multiverse 1, except that when the die shows six, an iron core forms; otherwise, a diamond core forms.

Question: What probability should the persons in the multiverses assign to the proposition that the moon of their planet has a diamond core?

Intuitively, in multiverse 1, the probability should be 1/6, and in multiverse 2, it should be 5/6. But now notice this. At the time that the person is asked to assign the probability, the two multiverses are exactly alike in all relevant features, and the persons know that. In each multiverse, there are infinitely many earthlike planets containing a person and having an uninhabited moon with a diamond core, as well as infinitely many where the moon has an iron core. The history of the two multiverses is different, but why should the history matter when the outcomes are relevantly the same? This line of thought suggests that the persons in the two multiverses should assign the same probability to their moon having a diamond core. Presumably that same probability will be neither 1/6 nor 5/6, but may be 1/2 or an interval or just plain undefined. This is a counterintuitive conclusion, but it is hard to avoid.

Now let's consider two more multiverses:

  1. The same as multiverse 1, except that the dice are cast and the moon cores formed after the persons come into existence.
  2. The same as multiverse 2, except that the dice are cast and the moon cores formed after the persons come into existence.
Suppose now that the person in multiverses 3 and 4 is asked to predict the composition of the moon care. Surely the probability of a diamond core in multiverse 3 is the same as in 1, and in 4 is the same as in 2. We should assign the same probabilities to unknown past die rolls as to unknown future ones. There is one tricky little thing here: The person in multiverses 3 and 4 doesn't know for sure that infinitely many cores will be diamond and infinitely many will be iron (it might be that all the dice will show sixes). But I ignore this, since the unknown thing has probability 1, and learning a fact that has probability 1 shouldn't make a difference to one's probability assignment.

So, by the counterintuitive conclusion about multiverses 1 and 2, the people in multiverses 3 and 4 should make the same probability assignments about their moons' compositions. In other words, their probabilistic reasoning is undercut.

If this is right, however, then if it turns out that we live in an infinite multiverse, our normal probabilistic reasoning is undercut. In particular, any science based on probabilistic reasoning--and all modern science is like that--that concludes to an infinite multiverse has undercut itself.

Objection: Being informed that there are infinitely many iron cores and infinitely many diamond cores in multiverses 1 and 2 violates causal finitism: the informant's words need to depend on infinitely many events.

Response: It shouldn't matter whether one is or is not informed of the fact, since the fact has probability 1, and learning a probability 1 fact shouldn't change one's probabilities.


William said...

The Copernican principle in cosmology, that we as living persons on our particular star, planet and galaxy are not a low-probability case, which might let us reason probabalistically based on assuming our position as an average one, does seem to break down with some types of multiverse.

Outside of cosmology, the question of whether or not a given phenomenon being studied is unique to our planet or not makes little difference in the science that is done: studying antibiotic resistance does not depend upon whether other worlds have bacteria, for example. Fortunately, that kind of science will never be expected to conclude to a multiverse.

Alexander R Pruss said...

If the argument I give is correct, reasoning about dice breaks down if we are an infinite multiverse. Antibiotic resistance is relevantly like dice.

William said...

Are the multiverse arguments against the accuracy of induction in the case of a empirical result of repeated measurement yielding a probability (like that the dice are fair) similar to other arguments against induction? If so, can we put them aside pragmatically just as with the other skeptical concerns about induction?

IanS said...

I don’t see the problem. Multiverses 1 and 2 are relevantly similar, but also relevantly different. Suppose you knew only that there were infinitely many “diamond” universes and infinitely many “iron” ones. You would indeed assign no probability (or perhaps 1/2, if you believe in indifference) that your universe was a diamond one. But if you also knew the history – that your universe became diamond or iron according as a die rolled in your universe had showed 6 or non-6 – you would assign probability 1/6 to diamond.

If your argument worked, wouldn’t it apply equally to a single spatially infinite universe? Imagine a single infinite universe with a countably infinite number of galaxies. In each galaxy there is an earth-like planet with an uninhabited moon etc. Do you think that your argument would undercut probabilistic reasoning in this universe?

Alexander R Pruss said...


The present temporal slices of multiverses 1 and 2 could be *exactly* alike. Given that the judgment concerns the present slice, why should the history matter?

(Like you, I feel a pull to thinking it *should* matter, but I can't justify it.)

Yes, the argument would apply in a single spatially infinite universe.

IanS said...

Probability theory is about using what you know. This can include history. The present states of multiverses 1 and 2 could not be exactly alike – the people in them have different knowledge. This justifies their different probability assignments. It could be (with some suitable correspondence) that corresponding moon-cores are alike. But if the people don’t know it, the mere possibility should not affect their probability judgements.

Here is a finite case. You are one of 6 prisoners on death row. You hear that the Governor has pardoned one of you. What is the chance that it is you? You don’t think indifference is reasonable: your crime was worse than the others. So you either assign no probability, or you go through subjective Bayesian contortions to get some probability less than 1/6. Now suppose you hear that the pardoned prisoner was selected by rolling a die, one face for each prisoner. You assign probability 1/6. The history – how the Governor came to pardon the prisoner – is relevant, even though the pardon has already been signed.

Alexander R Pruss said...


But what about this intuitive line of thought? In both M1 and M2 you have infinitely many universes of type A and infinitely many universes of type B, each containing one denizen. Moreover, the denizens are all on par, and you're no more likely to be one of them than another. So why would you have different probabilities in the two cases?

Well, the more I think about it, the more I think the paradoxical appearance is due to the fact that there is an infinite fair lottery behind the story--the lottery that "decides" which of the infinitely many universes you're in.

Alexander R Pruss said...

By the way, the reason I formulated the argument initially in terms of multiverses rather than single big universe is that a single universe will have some natural orderings derived from space. For instance, we could order the persons by distance from some fixed point. And then cases 1 and 2 would typically look different. In case 1, as we go along such a fixed order, we would encounter a diamond core typically one out of six times (law of large numbers), while in case 2, we would encounter it typically five out of six times. But I was thinking that in a multiverse there isn't any non-arbitrary ordering between the universes, so M1 and M2 are just exactly alike in the relevant respects.

IanS said...

[This is a response to your 8:59 AM comment.]

That seems right. The lottery intuition drives the appearance of paradox. But the intuition is misleading. Whichever universe “you” are in, “your” moon-core is tied to “your” die-roll. So in M1, your probability of a diamond-cored moon is 1/6. [And in M2, “your” moon-core is tied to “your” die roll, but differently. This (I’m arguing) is the relevant difference between M1 and M2]

The 1/6 probability is a straightforward application of Bayes – condition on infinite diamond-cored moons and infinite iron-cored moons. It is interesting to see how it works in a finite case. Take your original setup, but suppose that M1 is finite, with 100 universes. You are in M1. An angel tells you that, as it turned out, M1 contains 90 diamond-cored moons and 10 iron-cored ones. You are surprized, but you know the angel is trustworthy (and, to be pedantic, that the angel would have told everyone the number of diamond-cored moons, no matter how the universes had turned out). What is the probability that your moon is diamond cored? Answer: 9/10. This, too, is straightforward Bayes.

Why the lottery-like answer in the finite case but not in the infinite one? Why did the 1/6 vanish in the finite case but not in the infinite one? There is no paradox here. In the infinite case, the condition had probability 1. In the finite case, it had low probabiliy. Bayes is more subtle that it seems.