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Some notes on R. Collins’ indifference principle
2010 Jul 30
In June of 2009, Wiley-Blackwell published The Blackwell Companion to
Natural Theology, an impressive collection of ten carefully-crafted Christian
apologetic treatises. In the third of these, philosopher Robin Collins cites the
“fine tuning” of physical constants at which cosmologists occasionally express
marvel, and incorporates related scientific observations into a rigorous argument
for the existence of God. I do not intend this paper to address that argument in
its entirety. Instead, I mean only to reject one specific element of his case, called
the principle of insufficient reason, or, more commonly, the principle of indifference (hereafter PoI). The following, then, is a very rough sketch of rebuttals
to Collins’ defense of the PoI, as well as some of my own arguments against it.
The PoI has been discussed in some fashion or another as early as the eighteenth century, and perhaps even before then.1 In modern language, and in
its broadest form, it states that if, for some random variable Y , we have only
knowledge of its sample space, and no other relevant information, then we ought
to assign to Y a uniform distribution. In recent years philosophers and statisticians have occasionally revisited the PoI in academic literature, attempting to
find some consistent and defensible formulation; of the textbooks I’ve encountered, however, most are either silent on the issue,2 or in some cases dismiss it
as problematic and controversial.3 In Blackwell, Collins attempts to justify his
own version, which he calls the restricted principle of indifference, and which
1 Hacking, Ian. “Jacques Bernoulli’s Art of Conjecturing,” The British Journal for the
Philosophy of Science, vol.22, no.3, (Aug. 1971), pp209-229.
2 A sampling of textbooks which are silent includes DeCoursey, Stat. & Prob. for Eng.
App. w. Mic. Excel (2003), Grinstead, Intro. to Prob., 2nd rev. ed. (1997), Khuri, Adv.
Calc. w. App. in Stat., 2nd ed. (2003), Montgomery, App. Stat. & Prob. for Eng., 3rd ed.
(1997), Mukhopadhyay, Prob. & Stat. Inf. (2000), Ross, Intro. to Prob. and Stat. for Eng.
and Sci., 3rd ed. (2004), Rowe, Multiv. Bayes. Stat., Models for Source Separ. & Signal
Unmix. (2003), Ryan, Mod. Eng. Stat. (2007), Soong, Fund. of Prob. & Stat. for Eng.
(2004), Wackerly, Math. Stat. w. App., 7th ed. (2008), Wasserman, All of Stat., A Concise
Course in Stat. Inf. (2003).
3 For example, one textbook remarks on the PoI thusly: “The principle of insufficient reason sounds fine. But there are real problems. Would you say it is equally probable that
a car is red or not red? Or that it is red or blue or green or some other color? The
principle quickly leads to a lot of paradoxes.” Hacking, Ian. An Introduction to Probability and Inductive Logic, 2001, Cambridge University Press, ISBN 0-521-77501-9, p143.
when we have no reason to prefer any one value of a variable p over
another in some range R, we should assign equal epistemic probabilities to equal ranges of p that are in R, given that p constitutes a
Collins offers a three-fold defense of his principle: First, it is widely used;
second, it is often our only means of assigning probabilities; and third, physicists
have empirically confirmed it. While all three arguments seem superficially
plausible, I regard them as ultimately unpersuasive, for various reasons which
I shall elucidate here. I therefore address each of them in turn, adopting the
same order in which they are presented in Blackwell, for the purpose of casting
doubt on the principle. Afterward, I will briefly outline three counter-arguments
against it. By the end of the paper, I hope to have drawn attention to what I
see as serious problems in Collins’ restricted PoI.
We henceforth begin with his first supporting argument: If statisticians already use some form of the PoI in order to work with probabilities, then we may
feel comfortable in perpetuating that tradition by applying it to the fine-tuning
argument. Collins cites philosopher Roy Weatherford for support, who insists
that “the Principle of Indifference is the major theoretical justifications [sic] for
the equiprobability of alternatives in dice games and is therefore the principal
source of initial probability in such cases.”5 Yet I find Weatherford’s suggestion highly implausible. Are we really unable to justify uniform distributions
in dice games without appealing to some recondite philosophical principle? I
readily grant that our intuition may play some significant role in motivating
our exploration of uniform distributions in certain probability models, but it
cannot and indeed does not justify any eventual decision to assign such distributions. Rather, the accuracy of uniform distributions is verified empirically, by
observing, to extend Weatherford’s illustration, the results of dice games, and
our high rates of prediction of the outcomes thereof. Now, nobody can deny
that many successful probability models have been constructed using uniformly
distributed random variables, but we ought not misinterpret those assignments
as being justified by a generalized principle based on intuition. So I disagree
with Collins that the predictive success of uniform distributions in particular
contexts (e.g. dice games) justifies using uniform distributions whenever we’re
not sure what else to do. In short, I can find no empirical support for treating
the uniform distribution family as a catch-all for situations where we have only
Yet one might point out that the recognition of key common features of
such contexts where uniform distributions are successfully applied is precisely
what enables us to generalize the PoI. In the second prong of his defense of his
restricted version, Collins provides an example case regarding a twenty-sided die.
4 Collins, Robin. “The Teleological Argument: An Exploration of the Fine-tuning of the
Universe,” The Blackwell Companion to Natural Theology, ed. William Lane Craig and J.P.
Moreland, ISBN 978-1-405-17657-6, p234.
5 Weatherford, Roy. Philosophical Foundations of Probability Theory, 1982, ISBN 0-71009002-1, p30. http://books.google.com/books?id=B809AAAAIAAJ
Suppose, he argues, that our real-world experience is limited to dice with fewer
than twenty sides, such as six-sided cubic or four-sided pyramidal dice. In that
case, we still would feel justified in assigning an equal probability p(i) = 1/20 to
each side i of a twenty-sided die such that, when it is thrown for the very first
time, we have a 1/20 probability that side i will show face up. He further posits
that the only justification we can muster for such an assumption will employ
some version of the PoI. However, I am quite able to articulate an alternative
justification: For in the case of the twenty-sided die, our intuition is grounded
not in a probabilistic abstraction, but in first-hand experience with and nthhand knowledge of physical systems, and the mathematical laws which govern
them. This position is not unique; it has been acknowledged, for example, by
Frank Ramsay, who noted that “we sometimes really assume a theory of the
world with laws and chances” (emphasis original).6 In particular for the case
of the twenty-sided die, we appeal to our understanding of symmetry, which is
determined, it seems to me, in large part if not entirely by real-world experience.
In other words, we know something about how falling objects behave, and this
knowledge is gleaned from applying induction to our past observations.
A defender of a PoI might observe that these two accounts are not mutually exclusive; that is, if one appeals to experience with symmetrical systems,
one may still require additional assumptions, among them the PoI. I sympathize with this notion, which perhaps is true in some limited sense. Indeed,
there are many assumptions required when acting in the world at all, which
when formalized often seem arcane and almost always have some initial appearance of arbitrariness. A classic example of this is the problem of induction, a
universally-agreeable solution to which remains indefinitely elusive. Yet we all
accept induction in some form, and it has made possible the generation an even
greater, in my judgment, body of enormously useful results than Collins claims
for his restricted PoI. We may in similar fashion consider codifying some form
of the PoI, and classing it, as Collins implies that we must, among those necessary assumptions for navigating our human experience. Yet while I would be
amenable to attempting such an exercise, I am forced to regard Collins’ restrictions on the PoI, for the reasons already given, as insufficient to this task. For
as we have seen, in pointing out that, for example, dice games may lack direct
empirical justification, he overlooks the usefulness of indirectly-related (e.g. the
relationship of experience with four- and six-sided dice to the predicted behavior of twenty-sided dice) similar past experience to inform assignments of prior
probability distributions. Furthermore, whatever principle we formulate must
withstand the direct arguments against the PoI, which I will present later in
It is important at this point to keep in mind that Collins is aiming his effort
not at justifying the assumptions of statisticians, but rather generalizing those
assumptions into a single principle. Such a general rule will only be useful in
cases whereby assigning a uniform distribution is not already agreeably appro6 Ramsay,
Philosophical Papers (1990), ed.
D.H. Mellor, ISBN 0-521-37621-1, p97.
priate. So, for example, in the illustration of throwing the twenty-sided die,
where our intuition uniquivocally demands we assign 1/20 probability values
evenly to each outcome, that we ought to use a uniform distribution is evident
without appealing to Collins’ restricted PoI. If we wish to gather empirical support for it, we should look to situations where a uniform distribution is not
obviously appropriate, and test whether or not the restricted PoI should then
prove useful. For if we could justifiably construct probability models without
the application of a PoI, instead evaluating each model individually and independently, then we would have no need to appeal to a generalization. In those
cases for which it is not independently justifiable to use a uniformly-distributed
random variable, Collins implies we have no choice but to either admit that
we have no justification at all—an absurd prospect, we might suppose—or else
adopt some form of the PoI, which he claims is “the only justification we have
for assigning probability.”7
In contrast, I contend that we have no business constructing a probability
model in the first place, unless we have sufficient data, whether gleaned from
controlled collection or subjective judgment, to inform our construction. Collins
does not directly address this position, perhaps because he believes his restricted
PoI is required for all probability models, regardless of our background knowledge. However, Paul Castell, who eleven years earlier than Collins argued for
his own “consistent” version of the PoI, did take time to address what he called
the “abstention position.”8 In order to challenge it, he recalls an illustration of
Ramsay, who tells us the following: There are eight people in a room, exactly
one of whom has a name beginning with the letter ‘A’, but of whom we know
nothing else of relevance. Ramsay suggests that it is “reasonable” to have a degree of belief equal to 1/8 that the name of each person begins with ‘A’, “unless
we felt there was something else relevant.”
The Ramsay illustration is quite powerful. For who can deny that our intuition tells us the probability, in the absence of additional information, of some
particular man in the room having a name beginning with ‘A’ ought to be assigned as equal to every other man? As Castell observes, “it is hard to see how
the inference could be justified without employing some version of PoI.”9 Yet
even Ramsay seemed to understand that this intuitive approach does not come
to us without difficulties. He wrote, almost as if changing his mind from before,
that “to introduce the idea of ‘reasonable’ is really a mistake” because “what
is reasonable depends on what is taken as relevant.”10 Instead, I suggest, less
controversially than it might appear at first, that we refuse to assign probabilities at all until such time that we are able to construct a useful and justifiable
In order to explain this counter-intuitive suggestion, we may begin by noting that Castell anticipated the abstainer’s objection, and offered two argu7 Blackwell,
Paul. “A Consistent Restriction of the Principle of Indifference,” The British
Journal for the Philosophy of Science, Vol. 49, No. 3 (Sep., 1998), pp. 387-395.
9 Castell, p389.
10 Ramsay, pp100-101.
ments against it. First, he considers the activities “coin tossing, die casting and
horse racing” and claims that, in analyzing those games, statisticians make use
of uniform distributions which lack justification under an abstention position.
However, I have already suggested that we use our understanding of physical
systems, and not numerical intuition, to justify probability assignments for die
casting; indeed, the same sort of empirical justification may be applied to coin
tossing, outcomes of which are likewise informed by our knowledge of physics,
whether folk or formal.
To address the example of horse racing, we consider now Castell’s second
argument: We may encounter situations, he tells us, in which we are compelled,
for whatever reasons, “to bet on” which person will turn out to have a name
beginning with the letter ‘A’. Similarly, we may encounter circumstances in
which, if we wish to bet on a horse to win a race, we must do so without any
relevant knowledge to inform our decision. Clearly, he reasons, the probability
of our success in correctly choosing the person whose name begins with ‘A’ is
1/8; a similar methodology applies to horse betting. Yet I submit that we are
able to assign such probabilities because we observe particular rates of success
for arbitrary predictions, and that gives us a meaningful interpretation of the
probabilities with respect to the context. Without that interpretation, that
is, without a connection to some real-world application, then the abstention
position would apply here, as well.
Fortunately, such applications are often easy to imagine under a frequency
interpretation of probability, which in this context involves anticipating proportions of successes to trials given hypothetical repetitions. This interpretation is
drawn from real-world experience, and thus subject to empirical justification.
It allows us to assign uniform distributions responsibly, for use with various
applications, including, for instance, evaluating the rationality of betting decisions and communicating degrees of belief. It informs the house which odds will
result in profit, and to what extent. None of this requires appealing to intuition, but it is all firmly grounded in real-world experience. In the end, then, we
do have justification for assigning 1/8 probabilities as Ramsay suggested, not
due to intuition or some form of the PoI, but rather to our knowledge of past
performance by uninformed players in guessing games.
In the third prong of his defense of his restricted PoI, Collins points to a
physical constant called the gyromagnetic ratio, the value of which was accurately measured by modern scientists. He argues that his principle could have
been used to accurately predict the value of the ratio, and that this constitutes
a further “powerful” reason to accept the restricted PoI. Now, I do not deny
a PoI may be employed with success in many different situations, but is it appropriate to do so? Can the PoI actually constitute justification for assigning
a uniform distribution to a random variable, or does it simply happen to often
agree with independently-justifiable assignments? I see no reason to assume the
former; after all, it disagrees quite often, as well, unless selectively restricted.
Moreover, I am unaware of any link between the PoI and the gyromagnetic ratio
as described in the physics literature, despite Collins’ suggestions that the two
are related. I interpret the physicists’ silence to mean that, even if it is the
case that they could have successfully employed the PoI, it is highly unlikely
they actually did so. So, it does not appear that Collins’ evidence from physics
substantially if at all confirms his restricted PoI.
So it is that I must reject all three of Collins’ arguments, and I now shall
proceed to suggest three positive reasons of my own for rejecting his restricted
version of the PoI. Firstly, and most obviously, it leads to paradoxical conclusions. This is true for every version of the PoI which I have thus far encountered,
and which can be applied to infinite cases.11 Indeed, logicians have long recognized the paradoxes inherent in the PoI. Castell agrees that “in order to avoid
inconsistency, one must find some way to restrict the allowed sets of propositions
to which PoI may be applied.”12 John Maynard Keynes in his famous Treatise
on Probability attempted to do just that, and thereby provide his own consistent
version, but it has been rejected, for example, by Ramsay, who remarked, “it
is fairly clearly impossible to lay down purely logical conditions for its validity,
as is attempted by Mr Keynes.”13 Nevertheless, I will follow Keynes in his suggestion that “the plausibility of the principle will be most easily shaken off by
an exhibition of the contradictions which it involves.”14 One of these paradoxes
we therefore describe presently.
Let a, b denote two unknown statements regarding physical systems. Then
exactly one of the following is true: a ∧ b, a ∧ ¬b, ¬a ∧ b, or ¬a ∧ ¬b. Now let
A, B denote random variables, with A = 0 denoting the event whereby a is true,
A = 1 denoting the event whereby ¬a is true, B = 0 denoting the event whereby
b is true, and B = 1 denoting the event whereby ¬b is true. Next we let c denote
the statement “A and B are independent random variables,” and define C as a
random variable such that C = 0 denotes the event that c is true, and C = 1
denotes the event that ¬c is true. Applying the restricted PoI to C, we have
P (C = 0) = P (C = 1) = 1/2. Applying the restricted PoI to A and B, however,
yields P (A = 0, B = 0) = P (A = 0, B = 1) = P (A = 1, B = 0) = P (A = 1, B =
1) = 1/4 and P (A = 0) = P (A = 1) = P (B = 0) = P (B = 1) = 1/2.
Now we can see the paradox we have generated. If we apply the restricted PoI
to A and B, then we conclude that c is true, and therefore that ¬c is false. Yet
if we apply the restricted PoI to C, then we have P (C = 0) = P (C = 1) = 1/2,
which according to Castell means it is no less rational to bet that ¬c is true.
To avoid this, we might deny that C is a natural variable, but this is a highly
subjective matter, and not at all clear. One might be intuitively inclined to
suppose A and B are natural variables, but is not a single variable such as C
that Castell’s “consistent” version of the PoI evades this particular criticism by,
among other precautions, restricting its application to finite condition sets. So, it is useless
to Collins, who requires a means of applying a continuous probability distribution function to
a real interval.
12 Castell, p390.
“Truth and Probability” (1926),
simpler than a parsing of that variable? Of the three, only C seems to meet
Collins’ criterion of “simplest formulation” for natural variables,15 but this too
requires subjective judgment. To avoid this problem, we could deny that any
of the three variables are natural by Collins’ criteria, and so the restricted PoI
cannot be applied to them. This solution, however, prevents the restricted PoI
from being invoked for a large class of random variables, since the method of
exposing the paradox applies to any two complementary sentences for which
evidentiary support is equally lacking.
Second, if observed frequencies or expected propensities are essential to justifying subjective degrees of expectation in competing outcomes, as I believe
they are, then the restricted PoI leads us in the case of fine-tuning to absurd
results which we must not tolerate. This argument echoes my earlier criticism
of Collins’ and Castell’s charges that some version of the PoI is required in order to justify our very use of probability theory. I submit instead that even
if we interpret probabilities by means of subjective confidence levels, they will
nevertheless reflect empirical conclusions regarding frequencies or propensities,
because those considerations form the basis for justifiable predictions.
To show that Collins’ view is incompatible with this claim, suppose we do
not have access to the physics research used by him to support his fine-tuning
argument. In this hypothetical situation, we aren’t able to apply uniform distributions to the physical constants, because we have no knowledge of the existence
or role of those constants in physics models. In absence of this information, we
may proceed to ask the same sort of question demanded by Collins in Blackwell : At the earliest infancy of our universe, should we expect it to support
life apart from divine intervention? Applying the restricted PoI, we assign
equiprobabilities to each of LPU|NSU & k0 (where k0 is our aforementioned
hypothetical background information) and its negation, such that P (LPU|NSU
& k0 ) = P (¬LPU|NSU & k0 ) = 1/2. Of course, this isn’t the situation actually
facing us. In fact, Collins claims we do have data regarding the relationship
between the values of physical constants and the ability of the universe to support life. Suppose, then, we acknowledge all the alleged physics data from his
argument. In that case, we re-apply the restricted PoI to yield, according to
Collins,16 P (LPU|NSU & k0 ) 1. However, this leads us to a curious problem;
for we notice that the information used to revise our probability assignment
informs us neither to the frequency nor the propensity of any theoretical set of
initial conditions to generate life-supporting universes. However, if subjective
degrees of confidence should follow only from observed frequencies or expected
propensities, then we have no good reason to revise our expectations, which
means that our application of the restricted PoI must have been in some way
Thirdly, I submit that, if we are to generalize our intuition into some kind of
PoI, then that intuition must supercede the generalization when assessed on a
case-by-case basis. To clarify this point, suppose we formulate a PoI as follows:
We observe that we have a clear and agreeable intuition to assign uniform distributions to the members of some set S of random variables modeling real-world
situations. We therefore seek out common attributes among the members of S,
and by doing so construct a list of generalized conditions which appear to be
sufficient for membership in S. Suppose further that we encounter a random
variable X ∈
/ S which satisfies all the conditions in our list, but upon which our
intuition is either silent or else disagreeable. In that case, although we might
strongly suspect that a uniform distribution might serve us if assigned to X, I
contend we would not be justified in assuming that our assignment is appropriate solely on the basis of X satisfying the conditions in our list, because I
regard the implied inductive case for using X as too weak apart from evidentiary confirmation. In particular, if our intuition is disagreeable, as opposed to
merely silent, then that counts as evidence against the assignment of a uniform
distribution to X.
As a final closing thought, I wish to make it clear that the controversy
surrounding the PoI is only one of multiple problems I see with the fine-tuning
argument as presented by Collins and others. From my perspective, we have
four outstanding issues which remain unresolved: First, and most seriously in my
judgment, I’m extremely hesitant to accept the apologist’s picture of the initial
conditions of the universe; for although he claims it is supported by science, I
have grave doubts the scientists in question would condone his interpretation of
their work, much less the physics community in general. Second, the fine-tuning
argument depends on theological analysis regarding how we should expect God
to behave—namely, what reasons have we for thinking that God would create
for himself a life-permitting universe?—yet I find theology inherently dubious
on account of its disconnect between reason and the physical world by which
we may empirically verify results. Third, even if we accept the conclusion of
Collins’ argument, which is that certain physical observations count as evidence
against “the hypothesis that there is only one universe, the existence of which is
an unexplained, brute given”17 we must assess the significance of that evidence,
weighing it against alternative evidence favoring the naturalistic single universe
conclusion. Fourthly, we must also acknowledge multiverse hypotheses, and
compare their explanatory power to that of theism. Collins and other apologists
seem to be aware of all these issues, and have already written much in their
defense. However, their arguments have not met with consistent approval, and
demand close skeptical scrutiny. While that is outside the topical bounds of this
paper, we ought to remember this controversial context of our discussion.
I have now provided rebuttals to all three of Collins’ defenses for his restricted
PoI, as well as three positive counter-arguments of my own construction, and
four additional concerns regarding the remaining elements of his case for finetuning. Please recall that I promised in my introduction only an overview of
these points. Certainly a great deal more can be (and has been) said regarding
the problems with various forms of the PoI, but I composed this brief critique
in order to succinctly articulate some of the most obvious objections to the
principle. In the future, I may expand on these issues; for now, I believe they
help to reveal serious deficiencies in Collins’ restricted version as employed in
his fine-tuning argument.