Monday, March 1, 2010

Multiverses: Is more better?

I'm sure you've all heard of the multiverse hypothesis, the idea that many different universes exist besides our own.  Perhaps you've also heard this argument in favor of the multiverse hypothesis:
If there exist multiple universes, humans are that much more likely to exist.  Therefore, we are much more likely to find ourselves in a multiverse than a lone universe.
This argument is similar to the fine-tuning argument.  But let's call it the "More is Better" argument.  I disagree with it.

Many Multiverses and Many Worlds

But first, we cannot call it "the multiverse hypothesis".  If I simply stated that there are multiple universes, this would provide no explanations and make no predictions, because I have said nothing about the nature of these multiple universes.  Literally anything is possible.  To formulate a useful hypothesis, we need to be more specific about the nature of the universes.  There are in fact many different multiverse hypotheses, each with different specifics.

There are a bunch of questions we can ask about any multiverse hypothesis.  Where does the idea come from?  Are the universes such that there is a unique way to count them?  If so, how many are there?  What things are possible in these universes?  Are we equally likely to find ourselves in any of the universes with humans?

For example, consider the Many Worlds hypothesis, which could be considered a multiverse hypotheses, but is unusual in some ways.  Many Worlds doesn't come from cosmology, but from certain interpretations of quantum mechanics.  Contrary to the title, there is in fact only one universe in Many Worlds.  However, this one universe is in a superposition of many quantum possibilities, and each of those possibilities might be considered an independent "universe" of sorts.

The unusual thing about Many Worlds is that it explicitly states that not every universe is equally likely.  Just because two universes contain me in it does not mean I am equally likely to find myself in either of those universes.  For example, take the following thought experiment.
Flip a coin.  If it's heads, make a quantum measurement which has 1000 possible outcomes.  If it's tails, don't do anything.  Now there are 1000 universes where you got heads, and 1 universe where you got tails.  Does this mean you were 1000 times more likely to get heads than tails?
There's very little point in counting the number of "universes" in Many Worlds, because each universe is not equally likely.  It would be like counting the pips on each side of a dice.  Just because one side has six pips and another side has only one does not mean that we are six times as likely to roll a 6 as we are to roll a 1.

Multiverses and Priors

Let's say we are comparing two hypotheses:
Hypothesis 1: There are ten universes.
Hypothesis 2: There are a hundred universes.
Let's suppose that each of these universes contains humans like ourselves.  There exist ten universes in which hypothesis 1 is true.  There exist a hundred universes in which hypothesis 2 is true.  Does this mean that hypothesis 2 is ten times more likely?  Not necessarily, I think.

It depends on your choice of prior probabilities.  Usually, because prior probabilities are based on complete ignorance, you want to give equal likelihood to each possibility.  But how do you count the possibilities?  Do you say, "There are 110 possible universes, and I am equally likely to land in any of them"?  Or do you say, "There are 2 possible multiverses (one with 10 universes, one with 100), and I am equally likely to land in either of them"?

In statistical analysis, it's common practice to try a few different prior probabilities, and see what results you get.  If you get completely different results from reasonably chosen priors, then that means the results are inconclusive.

What if there is only a 1% chance that any given universe contains humans like ourselves?  Does this change things?  Under hypothesis 1, there is a 90% chance that there are no humans at all.  Under hypothesis 2, there is a 37% chance that there are no humans at all.  Clearly, if the hypothesis predicts no humans, then it must be false.

But I can think of three ways to analyze it, with very different results:
  1. Assign all universes equal prior probabilities.  Then calculate the conditional probability for each universe, given the condition that humans exist.  Hypothesis 2 is 10 times more likely than hypothesis 1.
  2. Assign all multiverses equal prior probabilities.  Then calculate the conditional probability for each multiverse, given the condition that humans exist.  Hypothesis 2 is about 3.7 times as likely as hypothesis 1.
  3. Assign all multiverses equal prior probabilities.  Then calculate the conditional probability for each universe, given the condition that humans exist.  Hypothesis 1 and 2 are equally likely.
The fact that we get three different results with three reasonable methods of analysis suggests that we need more evidence to reach a conclusion.

A better kind of evidence

If you want to argue about Many Worlds, I want to hear about interpretations of quantum mechanics, not about the sheer number of worlds proposed by it.  There are an absurd number of worlds in Many Worlds.  Supposedly, the number increases exponentially with entropy.  But this does mean that it is absurdly more likely than the alternative quantum interpretations?

There's a class of multiverse hypotheses that proposes independent universes with different initial conditions beyond our observable horizons.  This is plausible, but not because I think more is better.  It's because we suspect the universe is infinite in size, though we may only see part of it due to the finite speed of light and finite age.

Another kind of multiverse is the one where new universes are created as "bubbles" in the space-time of older universes.  I don't care how many bubbles it supposes.  If I wanted more evidence for this kind of multiverse, I would look to the theory of cosmological inflation, from which the idea arises.

I don't have anything against the idea of a multiverse.  I just prefer that they are supported by scientific theory, not philosophical arguments.  If that means being inconclusive, then so be it.


Eduard said...

I agree. I see some similarity between actual cosmologies and astrology, a NEW-astrology.

Chris said...

Hey, Miller,

I commented on your blogpost about the fine-tuning argument a while back (January maybe?). Sorry for not responding - have only just thought to come back.

I agree there's a difficulty (perhaps impossibility?) in establishing the priors for multiverse hypotheses (or, indeed, pretty much any hypothesis). However, in establishing the conditional probabilities, I think it's mistaken to calculate them for each universe. You should calculate them for each multiverse.

The reason I say this is that the existence of human beings will be observed as long as they exist somewhere in the multiverse, and not in any one particular universe. We shouldn't look at the likelihood that this universe contains life given H1 or H2; rather, we should look at the likelihood that at least one universe contains life given H1 or H2.

For a demonstration of the idea I'm trying to drive at (I hope it's clear, but if not, let me know): A computer is set up to pick a random integer from 1 to 1,000,000. If it picks #1, you're saved. If it fails to pick #1, you're killed. A man tosses a coin. If it comes up heads, the computer goes through the process ten times (and as long as it picks #1 once, you're fine). If it comes up tails, the computer goes through the process ten million times.

You are told all these rules, but you do not get to observe the coin toss or the computer. After a while, you're set free, and not killed. Did the coin land heads or tails?

Even if you can't know the priors (which maybe you could in this case), you can certainly know the conditionals. The probability of the observation of surviving given heads or tails is not calculated by looking at the probability of each random number being 1 given heads or tails, but rather by looking at the probability that at least one of the random numbers was a 1, given heads or tails. Consequently, p(live|H) < p(live|T).

Much the same should apply in the multiverse case. Our observation that there is life in universe alpha will only be observed if there is life somewhere in the multiverse. Neither H1 nor H2 increases the likelihood of universe alpha sustaining life, but H2 does increase the likelihood of some universe sustaining life. The conditional probability should take the whole multiverse into account, not just individual universes. (Though I think I'm right in saying you could take both into account without it making any difference.)

I hope I'm being clear. If not, my apologies. Say so and I'll try to rectify.

Regarding the usefulness of multiple universe theories, as they stand, I can see two uses: one, opposing design hypotheses, and two, if physicists accept MU hypotheses, they're more likely to incorporate them into their theories; if MU hypotheses are correct, this is good, and maybe this will ultimately lead to increased understanding of reality than would otherwise occur without the belief in MU hypotheses. Maybe.


miller said...

That is a very plausible scenario, and I think some multiverse theories might look like that. Analysis method #2 is best for that scenario.

However, I can think of another scenario where analysis #3 seems more appropriate.

A mad philosopher has a machine that generates a random integer between 1 and 1,000,000. If it picks anything other than 1, then it kills whatever's inside.

The mad philosopher flips a coin, but does not let you, his Frankenstein monster, see its result. If it is heads, then he puts you and nine other monsters through the machine. If it is tails, then he puts you and 9,999,999 other monsters through the machine.

If at least one monster survives, then it is most likely that the coin was tails. However, if all you know is that you survived, then there is no reason to think the coin was heads or tails.

Does this example make sense?