You Don’t Count Your Universes From a Physicist’s Table:



The Multiverse and the Inverse Gambler’s Fallacy:

A Response to Nick Bostrom on Behalf of Roger White

by Kenneth Boyce

Last Revised 4/11/12

Abstract: Many have claimed that the apparent fine-tuning of the universe is evidence that it is but one of numerous (perhaps infinitely many) universes. This claim has been forcefully challenged by Roger White. White, developing an objection originally put forward by Ian Hacking, compellingly argues that those who claim that the apparent fine-tuning of our universe provides evidence for there being a large number of universes commit what Hacking dubbed “the inverse gambler’s fallacy.” In this paper, I defend White’s thesis from an objection that has been raised by Nick Bostrom.

Introduction

One of the most astonishing discoveries in modern cosmology is that the window of values in which the fundamental constants of nature had to fall in order for the universe to be life-sustaining appears to be an extremely narrow one. Had things been ever so slightly different, it seems, the universe would not have been a place where life could have developed. [i] The universe is apparently ‘fine-tuned’ for the existence of life. Some have taken this discovery as providing evidence that our universe was designed. Others, however, have taken the apparent fine-tuning of the universe as evidence for the hypothesis that ours is but one of numerous, perhaps infinitely many, universes.[ii]

The reasoning behind the latter view often goes something in the neighborhood of what follows: Since it is fairly unlikely that any particular universe would be fine-tuned, it is fairly unlikely that there would be a life-permitting universe like ours unless numerous other universes exist. If there were numerous universes and the constants of those universes varied randomly, however, mostly likely, at least some universes would, by sheer chance, have constants that fall within the life-permitting window. And it would be no surprise that we find ourselves in such a universe, since, after all, the only kind of universe that we could have possibly existed in is one that is life permitting. And these facts give us at least some reason to believe that a large number of universes exist. [iii]

The claim that the apparent fine-tuning of our universe provides evidence for the existence of a large number of universes has been forcefully challenged, however, by Roger White. White, developing an objection originally put forward by Ian Hacking, compellingly argues that those who claim that the apparent fine-tuning of our universe provides evidence for a large number of universes commit what Ian Hacking dubbed “the inverse gambler’s fallacy.”

It is my view that White’s argument for this thesis is sound, though I will not argue directly for that conclusion here. And though various objections can be and have been raised against White’s thesis, I will not reinvent the wheel by offering responses that have already been provided by White and others.[iv] Instead, I want to add to the litany of such responses by offering one of my own to an objection that, to my knowledge, has not yet received a direct reply. The objection comes from Nick Bostrom.[v] I describe it below, after providing a brief summary of White’s argument.

1. Roger White’s Attack on the Multiverse

Ian Hacking introduced what he called “the inverse gambler’s fallacy” by way of the following example:

[The gambler] enters the room as a roll is about to be made. The kibitzer asks, ‘Is this the first roll of the dice, do you think, or have we made many a one earlier tonight?’ The gambler takes a quick look around the room, and determines his own personal or subjective probability distribution as to his judgment among the possibilities that this is the first roll, the thirty-sixth roll, or the hundredth roll, and so forth. But slyly, he says, ‘Can I wait until I see how this roll comes out, before I lay my bet with you on the number of past plays made tonight?’ The kibitzer, no fool, agrees, although charging a slight fee for allowing this extra ‘information’. The roll is a double six. The gambler foolishly says, ‘Ha, that makes a difference – I think there have been quite a few rolls.’ (1987, 333)

Obviously, as Hacking points out, the gambler is reasoning fallaciously. Whether this particular roll comes up a double six is probabilistically independent of whether there have been a large number of rolls. Hacking charges various advocates of multiple universe hypotheses with committing this very same fallacy. In particular, he charges with committing this fallacy those who take the fine-tuning of our universe as evidence for the hypothesis that ours is but one of an infinite number of successive universes whose features are causally independent of one another. As Hacking puts it, according to such models, the successive universes “have no memories”. They are like independent rolls of the dice. (337-38)

Roger White (2000, 2003) takes Hacking’s charge to apply more generally to those who take fine-tuning as evidence for a large number of universes (whether successive or not) and develops it, in some detail, as follows:

For simplicity, let us suppose that we can partition the space of possible outcomes of a Big Bang into a finite set of equally probable configurations of initial conditions and fundamental constants: {T1, T2,…, Tn} (think of the universes as n-sided dice, for a very large n). Let the variable ‘x’ range over the actual universes. Let a be our universe and let T1 be the configuration that is necessary to permit life to evolve. Each universe instantiates a single Ti, i.e. (∀x)(∃i)Tix. Let m be the number of universes that actually exist, and let

E = T1a = a is life-permitting

E’= (∃x)T1x = some universe is life-permitting; and

M = m is large (the multiple-universes hypothesis)

It is important to distinguish E from the weaker E’. For while E’ is more probable given M than it is given ~M, M has no effect on the probability of E. First let us consider E’. In general,

(P((∃x)Tix/m = k) = 1 – (1 – 1/n)k for any i.

So

P((∃x)Tix/M) > P((∃x)Tix/~M) for any i.

So

P((∃x)T1x/M) > P((∃x)T1x/~M)

That is

P(E’/M) > P(E’/~M)

E, on the other hand, is just the claim that a instantiates T1, and the probability of this is just 1/n, regardless of how many other universes there are, since a’s initial conditions and constants are selected randomly from a set of n equally probable alternatives, a selection that is independent of the existence of other universes. The events that give rise to universes are not causally related in such a way that the outcome of one renders the outcome of another more or less probable. They are like independent rolls of a die. That is,

P(E/M) = P(T1a/M) = 1/n = P(T1a/~M) = P(E/~M)

… So … P(M/E) = P(M), i.e. the fact that our universe is life-permitting does not confirm the multiple-universe hypothesis one iota. (2003, 231-32)

2. Nick Bostrom: Defender of the Multiverse

Bostrom (2002) has offered an interesting objection to the above argument. Before I consider that objection, however, I want to briefly consider, for the purpose of setting them aside, two other objections that Bostrom raises.

The first of these objections is that White asserts that “the events that give rise to universes are not causally related in such a way that the outcome of one renders the outcome of another more or less probable,” an assertion for which there is “no empirical warrant” (21). I believe this objection is misguided, however, since White is charitably read as responding to probabilistic arguments that take the fine-tuning of our universe to provide evidence for the existence of multiple universes regardless of what the causal connections among those universes and ours are like. And given such a dialectical context, White is perfectly justified in assuming, for the sake of argument, that the features that different universes have are causally independent of one another.

The second of these objections is that “even if we consider the hypothetical case of a multiverse model where the universes bear no causal relations to one another … it does not follow that the outcomes of those events are uncorrelated in one’s rational epistemic probability assignment” (22). Certainly, Bostrom is correct about this. We might have had good reason, for example, to believe that God had revealed to us that if our universe is fine-tuned, then there are many universes that bear no causal relations to our own. But it is also not clear that this is something White denies. White is both charitably and plausibly read as tacitly assuming (and not without good reason!) that our actual background knowledge contains no such non-causal information concerning the existence of other universes. Bostrom himself would say, in this connection, that White is ignoring the impact that observation selection effects should have on our rational epistemic probability assignments (35-39). But White (2003, 235-238) responds to this charge. And, as I said in the introduction, I do not intend to reinvent the wheel by responding to objections to White’s thesis that have already been addressed.

Instead, I want to focus my attention on a different objection that Bostrom (2002, 20-21) puts forward. Borrowing from White’s way of framing the issue, Bostrom invites us to suppose that α, β1, … , βm-1 are all the actually existing universes (α being our universe). He further stipulates that ‘Ei’ (where i = α, β1, … , βm-1) will stand for the proposition that if some universe is life permitting, i is life permitting. Bostrom notes that White is committed to each of the following:

P(M/E’) > P(M)

P(M/E) = P(M)

He then points out that since, by definition, P(M/E’Eα) = P(M/E), the above claims imply that

(*) P(M/E’Eα) < P(M/E’)

But, Bostrom argues, we should also maintain that “P(M/E’βj) = c [where c is some constant], for every βj, for no ground has been given for why some of the universes βj would have given more reason, had it been the fine-tuned one, for believing M, than would any other βj similarly fine-tuned.” Furthermore, Bostrom notes that since E’ implies the disjunction E’Eα v E’Eβ1 v E’Eβ2 V … v E’Eβm-1, “this together with (*) implies:

P(M/E’Eβj) > P(M/E’) for every βj”

“In other words,” Bostrom concludes, “White is committed to the view that, given that some universe is life permitting, then: conditionalizing on α being life-permitting decreases the probability of M, while conditionalizing on any of the β1, … , βm-1, increases the probability of M.” Bostrom then argues that this consequence of White’s thesis is unacceptably counterintuitive:

But that seems wrong. Given that some universe is life-permitting, why should the fact that it is this universe that is life permitting, rather than any of the others, lower the probability that there are many universes? If it had been some other universe instead of this one that had been life-permitting, why should that have made the multiverse hypothesis any more likely? Clearly, such discrimination could be justified only if there were something special that we knew about this universe that would make the fact that it is this universe rather than some other that is life permitting significant. I can’t see what sort of knowledge that would be.

In the next section I argue that this objection fails to undermine White’s thesis.

3. The Inverse Gambler’s Fallacy Strikes Back

One thing to note about Bostrom’s argument is that, when we consider it in light of Hacking’s original example, it appears to prove too much. Suppose that, unlike Hacking’s original gambler, a highly successful gambler, one who is adept at probabilistic reasoning, walks into the room. The bookie invites him to place his bet on how many rolls of the dice there have been and offers “as a friendly gesture” to let the gambler see how the next roll comes out. It comes out a double six. But our probabilistically adept gambler, unlike Hacking’s, informs the bookie that he does not consider it any more likely that there have been several rolls, given that this roll has come up a double six.

Suppose the bookie then responds by offering the gambler the following argument:

Let ‘M*’ stand for the proposition that there have been a large number of rolls of the dice. Suppose α*, β*1, … , β*m-1 are all the rolls of the dice that have occurred. Let ‘E*’’ stand for the proposition that some roll comes up a double six. Let ‘E*’ stand for the proposition that this roll of the dice, α*, comes up a double six. Let ‘E*i’, where i = α*, β*1, … , β*m-1, stand for the proposition that if some roll came up double six, i came up double six. Now you would concede, I imagine, that discovering some roll or other came up a double six gives evidence for M*, even though you deny that your learning this roll came up a double six does. So you are committed to each of the following:

P(M*/E*’) > P(M*)

P(M*/E*) = P(M*)

… [The bookie continues to fill out Bostrom’s argument, mutatis mutandis] …

So you are committed to the claim that while P(M*/E*’E*α) < P(M*/E*’), P(M*/E*’E*βj) > P(M*/E*’) for every β*j. But what could justify such discrimination among the rolls of the dice? Your view commits you to an unacceptably counterintuitive result.

Most of us, I suspect, are inclined to believe that if the gambler falls for this argument, he will have been successfully conned. We suspect that the bookie’s argument must have a subtle flaw in it somewhere. But where?

The flaw lies in the bookie’s premise that the gambler doesn’t have any information available to him that justifies his discriminatory treatment. To see why this is so, suppose the gambler’s situation had been different from the way in which we originally envisioned it. Suppose that the gambler had simply been waiting outside the room when the bookie came out to meet him to ask if he wanted to place a bet on how many rolls there had been. Let’s also suppose that somehow the gambler had available to him a catalog naming all of the possible rolls that there could have been that night. [vi] Suppose further that, after the gambler has set his subjective credences, the bookie informs him that α* came up a double six. How should the gambler update his credences in this case?

One thing that the gambler learns in the above case is that α* occurred. And in learning this, the gambler should increase the amount of credence that he lends to the hypothesis that there have been many rolls. Why? Because it is more likely that α* has occurred on the hypothesis that there have been many rolls than it is on the hypothesis that there have been only a small number of rolls. The more rolls that have occurred, the more likely it is that α* would be among them. So, in this case, the gambler should not maintain that P(M*/E*’E*α) = P(M*). He should maintain, rather, that P(M*/E*’E*α) > P(M*). And similarly, by the very same reasoning, he should maintain that P(M*/E*’E*βj) > P(M*) for every β*j. In this case, then, there is no asymmetrical treatment of different rolls of the dice.

Why, then, is there such asymmetrical treatment in the original case? The reason is that it is already part of the gambler’s background knowledge that α* occurs, prior to his learning that α* comes up a double six (recall that he was already waiting to see how the “next roll” – i.e. α* – would come out). Since, in the original case, the fact that α* occurs is already part of his background knowledge, it is not among what the gambler learns when he finds out that α* comes up as a double six. And since that α* occurs is not among what he learns, it is not available to serve as evidence for the hypothesis that there have been many rolls. For any of the other β*j’s, however, that it has occurred is not part of the gambler’s background knowledge. If he were to learn that a particular one of the β*j’s came up a double six, he would also thereby learn that that particular β*j had occurred, and that would provide him with some evidence that there has been a large number of rolls. So, in this case, his treating the evidential bearing of other rolls of the dice coming up double six differently from the way he treats the evidential bearing of α* coming up a double six is justified.

How does the above bear on Bostrom’s original argument against White? It should be noted, as Kai Draper, Paul Draper and Joel Pust do in another context, that

White takes the existence of our universe to be part of the background information … Further, he does this presumably for the good reason that he is interested in evaluating a fine-tuning argument for [there being a large number of universes]. He does not address the question of whether the mere existence of our universe confirms [the multiple universes hypothesis]… (2007, 296)

White takes the relevant background knowledge to include the claim that α exists,[vii] but presumably not the claim that any of the other βj’s exist. Since the existence of α is already part of the background knowledge that White presupposes, it isn’t available as evidence to confirm the hypothesis that there is a large number of universes. And, given the fact that this universe exists, whether it is fine-tuned is probabilistically independent of whether there is a large number of universes. So the fact that this universe is fine-tuned doesn’t provide us with additional evidence, on the background knowledge that White stipulates, that there is a large number of universes.

But presumably, if we had a catalog of all the possible universes, then our learning that some particular one of them other than α (i.e. some particular βj) is fine-tuned would provide us with evidence, on the background knowledge White assumes, that there is a large number of universes.[viii] For to learn that particular βj is fine-tuned is also to learn that it exists. And it is more likely on the hypothesis that there is a large number of universes that that βj exists than on the hypothesis that only a small number of universes exist. Note, however, that it would be our learning of the existence of that particular βj, rather than its being fine-tuned per se, that would be doing all of the evidential work for us.

Thus, the asymmetric treatment we afford to each of these cases is justified, given the background knowledge that White’s argument presupposes. Consequently, Bostrom’s objection to White’s argument fails. [ix]

-----------------------

Endnotes

[i] See (Rees 2000).

[ii] For a defense of both of these points of view, see (Leslie, 1989).

[iii] For example, Leslie’s (1989, 69-70) summary of why fine-tuning data supports the hypothesis that there are multiple universes goes along these lines.

[iv] See (White 2003, 243-48) for a response to several such objections. See (Draper et. al. 2007) for responses to the objections raised in (Holder 2002) and (Manson and Thrush 2003). For another response to (Holder 2002), see (Rota 2005).

[v] The closest I’ve seen to a reply to the sort of criticism from Bostrom that I will discuss is White’s reply to “Objection 1” in his 2003 paper (see 243-44).

[vi] I will consider rolls of the dice to be events and I will take events to be states-of-affairs-like abstract objects that can either occur or fail to occur in the way that states of affairs can obtain or fail to obtain. “Possible events” on this way of conceiving things, are events that can occur. “Actual events” are events that do occur.

[vii] White does not say this explicitly, but it seems clear that this is what he intends.

[viii] I’m loosely speaking as if a possibilist metaphysic, according to which there are merely possible universes that don’t actually exist, is correct. Were I speaking more strictly, I would speak of various individual essences of universes either being instantiated or failing to be instantiated, rather than of possible universes that may or may not exist.

[ix] Acknowledgements.

References

Bostrom, N. (2002). Anthropic bias: Observation selection effects in science and

philosophy. (New York: Routledge)

Draper, K., Draper, P. & J. Pust. (2007). Probabilistic arguments for multiple universes. Pacific Philosophical Quarterly, 88, 288-307

Hacking, I. (1987). The inverse gambler’s fallacy: the argument from design. The anthropic principle applied to Wheeler universes. Mind, 96, 331-340

Holder, R. D. (2002). Fine-tuning, multiple universes and theism. Nous, 36, 295-312.

Leslie, J. (1989). Universes. (New York: Routledge)

Manson, N.A. and M.J. Thrush. 2003. Multiple universes and the ‘this universe’ objection. Pacific Philosophical Quarterly, 84, 67-83

Rees, M. (2000). Just six numbers: The deep forces that shape the universe. (New York: Basic Books)

Rota, M. (2005). Multiple universes and the fine-tuning argument: a response to Rodney Holder. Pacific Philosophical Quarterly, 86, 556-576

White, R. (2000). Fine-tuning and multiple universes. Nous, 34, 260-67

White, R. (2003). Fine-tuning and multiple universes. (In N. A. Manson (Ed.), God and design: The teleological argument and modern science (pp. 229-250). New York: Routledge)

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