1
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. 1 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.2
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,
2
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. 3
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.4 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.5 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
3
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
4
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
5
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’)
6
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 lifepermitting, 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
7
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
8
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. 6
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
9
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,7 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 finetuned 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 finetuned would provide us with evidence, on the background knowledge White assumes,
that there is a large number of universes.8 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
10
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. 9
Endnotes
1
See (Rees 2000).
2
For a defense of both of these points of view, see (Leslie, 1989).
3
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.
4
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).
5
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).
6
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.
7
White does not say this explicitly, but it seems clear that this is what he intends.
8
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.
11
9
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)