Multiverses and the Inverse Gambler’s fallacy

Many have thought that the constants of this universe are fine-tuned for life. Some think this provides support for an intelligent designer; others think this provides support for a multiverse. While some multiverse models are inferred empirically (e.g. inflationary cosmology), we will be considering multiverse models inferred on purely a priori probabilistic grounds.

Following Ian Hacking, Roger White argues that a probabilistic inference for a multiverse commits the Inverse Gambler’s fallacy.  To see why, we’ll need to lay down some groundwork on principles of confirmation, and define some terms.

  • Observation O confirms hypothesis H, given background knowledge K, if and only if P(H|O&K) > P(H|K) (“Confirm” does not mean prove; it’s more like “support” or “favors”.)
  • P1: P(H|O&K) > P(H|K) if and only if P(O|H&K) > P(O|¬H&K).
  • P2: P(H|O&K) = P(H|K) if and only if P(O|H&K) = P(O|¬H&K)

On White’s multiverse model, there are n distinct possible configurations of constants a universe can have, and each configuration of constants has a probability of 1/n, independent of the configuration of constants in other universes. (Note: not all multiverse models will fit White’s model, so consider those multiverses as outside our scope.)

Let’s define:

  • E’ = some universe is life-permitting
  • E = this universe is life-permitting
  • M = multiverse

It seems that P(E’|M&K) > P(E’|¬M&K); that is, the probability of some life-permitting universe is greater on a multiverse than not. Following the P1, that means P(M|E’&K) > P(M|K), which means that the fact that some universe is life-permitting confirms the existence of a multiverse. (Note that the fact that E’ is life-permitting isn’t important; E’ with any constants is more probable on M.)

Now consider E, that this universe is life-permitting. Based on White’s multiverse model, the odds of this universe being life-permitting is in 1/n, regardless of what happens in other worlds, since, by assumption, other universes are causally isolated and independent. In other words: P(E|M&K) = 1/n = P(E|¬M&K). Using P2, this entails that P(M|E&K) = P(M|K), which means that E does not confirm M. To sum up, while the fact that some universe is life-permitting confirms a multiverse, the fact that this universe is life-permitting does not confirm a multiverse.

Whether our evidence confirms a multiverse depends on whether we use E or E’.  Which should we use? White replies that based on the principle of total evidence we should use E instead of E’. The principle of total evidence says we should use the logically stronger evidence if it makes a difference to our calculations. In this case, E is logically stronger since E entails E’, but E’ does not entail E; therefore, we should use E, and our evidence does not confirm a multiverse. (White defends this principle with an example, but I’ll skip it because I think this principle is correct. Whether the principle is applied correctly, here, is up for debate. See chapter 1 of Elliott Sober’s Evidence and Evolution for a defense of the principle.)

I should note that we’ve sidestepped the more difficult question of whether to believe in a multiverse, since that depends on prior probabilities, and, more broadly, your view on Bayesianism.  The simpler question that concerns us is whether the fine-tuned constants provide any confirmation for a multiverse. If White is right, it does not.

Ian Hacking compares the fallacious reasoning with multiverses with fallacious reasoning involving a gambler. Consider a gambler who sees someone grab some dice, roll it, and hit a double six. The gambler reasons that given the rarity of double six, the dice must have been rolled many times, since it’s more likely to roll double six if the dice had been rolled many times. Ian Hacking calls this the Inverse Gambler’s fallacy, because the likelihood that this roll is double six is independent of any past rolls; there being many rolls in the past has no effect on how likely it is that this roll is double six. Analogously, the fact that there are many other universes has no effect on how likely it is that this universe is life-permitting.

Some have dubbed White’s objection the “This Universe” objection. In 2003, White replies that he regrets putting the issue in these terms; instead, there is a more important principle at work:

Observation principle: An observation I make gives me evidence for hypothesis H only if it is more likely given H that I would make that observation.

For example, it is not enough that people in distant rooms rolling dice make it more likely that someone will observe a double six. What is needed is that these rolls make it more likely that I will observe a double six.

The reasoning here is independent of the question of whether I know that this pair of dice landed double-six, on this roll, rather than just that there is a double-six. What we need is a probabilistic link between my experiences and the hypothesis in question. One way of establishing such a link in the present case is to suppose that I was once an unconscious soul waiting to be embodied in whichever universe produced a hospitable living organism. On this assumption the more universes there are, the more likely I am to observe one. This is not just a cheap shot. It is an illustration of the kind of story that we need to support the inference to multiple universes.

For a reply to this argument, see Darren Bradley’s A Defense of the Fine-Tuning Argument for a Multiverse and Multiple universes and observation selection effects

White, Roger.  Fine-Tuning and Multiple Universes.

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