`It has recently been claimed, most
prominently by Dr. Hugh Ross on his web site`

`that the so-called "fine-tuning"
of the constants of physics supports a supernatural origin of
the universe. Specifically, it is claimed that many of the constants
of physics must be within a very small range of their actual values,
or else life could not exist in our universe. Since it is alleged
that this range is very small, and since our very existence shows
that our universe has values of these constants that would
allow life to exist, it is argued that the probability that our
universe arose by chance is so small that we must seek a supernatural
origin of the universe.`

`In this article we will show that
this argument is wrong. Not only is it wrong, but in fact we will
show that the observation that the universe is "fine-tuned"
in this sense can only count against a supernatural origin of
the universe. And we shall furthermore show that with certain
theologies suggested by deities that are both inscrutable and
very powerful, the more "finely-tuned" the universe
is, the more a supernatural origin of the universe is undermined.`

`[ Note added 020106: We have
learned that the philosopher of science, Elliott Sober, has made
some similar points in a recent article written for the Blackwell
Guide to Philosophy of Religion. A draft copy can be obtained
from his website: http://philosophy.wisc.edu/sober/black-da.pdf.
We have some small differences with Professor Sober (in particular,
we think that his condition (A3) is too strong, and that a weaker
version of (A3) actually gives a stronger result), but he has
an excellent discussion of the role that selection bias plays
where the bias is due to self-selection by sentient observers.]`

`Our basic argument starts with a few
very simple assumptions. We believe that anyone who accepts that
the universe is "fine-tuned" for life would find it
difficult not to accept these assumptions. They are:`

a) Our universe exists and contains life.

b) Our universe is "life friendly," that is, the conditions in our universe (such as physical laws, etc.) permit or are compatible with life existing naturalistically.

c) Life cannot exist in a universe that is governed solely by naturalistic law unless that universe is "life-friendly."

`In this FAQ we will discuss only the
Weak Anthropic Principle (WAP), since it is uncontroversial and
generally accepted. We will not discuss the Strong Anthropic Principle
(SAP), much less the Completely Ridiculous Anthropic Principle
:-)`

`According to the WAP, which is embodied
in assumption (c), the fact that life (and we as intelligent life
along with it) exists in our universe, coupled with the assumption
that the universe is governed by naturalistic law, implies that
those laws must be "life-friendly." If they were not
"life-friendly," then it is obvious that life could
not exist in a universe governed solely by naturalistic law. However,
it should be noted that a sufficiently powerful supernatural principle
or entity (deity) could sustain life in a universe with laws that
are not "life-friendly," simply by virtue of that entity's
will and power.`

`We will show that if assumptions (a-c)
are true, then the observation that our universe is "life-friendly"
can never be evidence against the hypothesis that
the universe is governed solely by naturalistic law. Moreover,
"fine-tuning," in the sense that "life-friendly"
laws are claimed to represent only a very small fraction of possible
universes, can even undermine the hypothesis of a supernatural
origin of the universe; and the more "finely-tuned"
the universe is, the more this hypothesis can be undermined.`

`There are a number of traditional
arguments that have been made against the "fine-tuning"
argument. We will state them here, and we think that they are
valid, although our main interest will be directed towards some
new insights arising from a deeper understanding of probability
theory.`

`1) In proving our main result, we
do not assume or contemplate that universes other than our own
exist (e.g., as in cosmologies such as those proposed by A. Vilenkin
["Quantum creation of the universe," Phys Rev D
Vol. 30, pp. 509-511 (1984)], André Linde ["The
self-reproducing inflationary universe," Scientific American,
November 1994, pp. 48-55], and most recently, Lee Smolin [Life
of the Cosmos, Oxford University Press (1997)], or as in some
kinds of "many worlds" quantum models). One argument
against Ross has been to claim that there may be many universes
with many different combinations of physical constants. If there
are enough of them, a few would be able to support life solely
by chance. It is hypothesized that we live in one of those few.
Thus, this argument seeks to overcome the low probability of having
a universe with life in it with a multiplicity of universes. A
recent technical discussion of this idea by Garriga and Vilenken
can be found at http://xxx.lanl.gov/abs/gr-qc/0102010.`

`2) Others have argued against the
assumption that the universe must have very narrowly constrained
values of certain physical constants for life to exist in it.
They have argued that life could exist in universes that are very
different from ours, but it is only our insular ignorance of the
physics of such universes that misleads us into thinking that
a universe must be much like our own to sustain life. Indeed,
virtually nothing is known about the possibility of life in universes
that are very different from ours. It could well be that most
universes could support life, even if it is of a type that is
completely unfamiliar to us. To assert that only universes very
like our own could support life goes well beyond anything that
we know today.`

`Indeed, it might well be that a fundamental
"theory of everything" in physics would predict that
only a very narrow range of physical constants, or even no range
at all, would be possible. If this turns out to be the case, then
the entire "fine-tuning" argument would be moot.`

`While recognizing the force and validity
of these arguments, the main points we will make go in quite different
directions, and show that even if Ross is correct about "fine-tuning"
and even if ours is the only universe that exists, the
"fine-tuning" argument fails.`

`In this section, we will introduce
some necessary notation and discuss some basic probability theory
needed in order to understand our points`

`First, some notation. We introduce
several predicates, (statements which can have values true or
false).`

`Let L="The universe exists and
contains Life." L is clearly true for our universe (assumption
a).`

`Let F="The conditions in the
universe are 'life-Friendly,' in the sense described above."
Ross, in his arguments, certainly assumes that F is true. So will
we (assumption b). The negation, ~F, would be that the conditions
are such that life cannot exist naturalistically, so that if life
is present it must be because of some supernatural principle or
entity.`

`Let N="The universe is governed
solely by Naturalistic law." The negation, ~N, is that it
is not governed solely by naturalistic law, that is, some non-naturalistic
(supernaturalistic) principle or entity is involved. N and ~N
are not assumptions; they are hypotheses to be tested. However,
we do not rule out either possibility at the outset; rather, we
assume that each of them has some non-zero a-priori probability
of being true.`

`Probability theory now allows us to
write down some important relationships between these predicates.
For example, assumption (c) can be written mathematically as N&L==>F
('==>' means logical implication). In the language of probability
theory, this can be expressed as`

P(F|N&L)=1

`where P(A|B) is the probability that
A is true, given that B is true [see footnote 1 for a formal mathematical
definition], and '&' is logical conjunction.`

`Expressed in the language of probability
theory, we understand the "fine-tuning" argument to
claim that if naturalistic law applies, then the probability that
a randomly-selected universe would be "life-friendly"
is very small, or in mathematical terms, P(F|N)<<1. Notice
that this condition is not a predicate like L, N and F; Rather,
it is a statement about the probability distribution P(F|N),
considered as it applies to all possible universes. For this reason,
it is not possible to express the "fine-tuning" condition
in terms of one of the arguments A or B of a probability function
P(A|B). It is, rather, a statement about how large those probabilities
are.`

`The "fine-tuning" argument
then reasons that if P(F|N)<<1, then it follows that P(N|F)<<1.
In ordinary English, this says that if the probability that a
randomly-selected universe would be life-friendly (given naturalism)
is very small, then the probability that naturalism is true, given
the observed fact that the universe is "life-friendly,"
is also very small. This, however, is an elementary if common
blunder in probability theory. One cannot simply exchange the
two arguments in a probability like P(F|N) and get a valid result.
A simple example will suffice to show this.`

Example

Let A="I am holding a Royal Flush."

Let B="I will win the poker hand."

It is evident that P(A|B) is nearly 0. Almost all poker hands are won with hands other than a Royal Flush. On the other hand, it is equally clear that P(B|A) is nearly 1. If you have a Royal Flush, you are virtually certain to win the poker hand.

`There is a second reason why this
"fine-tuning" argument is wrong. It is that for an inference
to be valid, it is necessary to take into account all known
information that may be relevant to the conclusion. In the present
case, we happen to know that life exists in our universe
(i.e., that L is true). Therefore, it is invalid to make inferences
about N if we fail to take into account the fact that L, as well
as F, are already known to be true. It follows that any inferences
about N must be conditioned upon both F and
L. An example of this is seen in the next section.`

`The most important consequence of
the previous paragraph is very simple: In inferring the probability
that N is true, it is entirely irrelevant whether P(F|N) is large
or small. It is entirely irrelevant whether the universe is "fine-tuned"
or not. Only probabilities conditioned upon L are relevant to
our inquiry.`

`Richard Harter <cri@tiac.net>
has suggested a somewhat different interpretation of the "fine-tuning"
argument in E-mail (reproduced here with permission). He writes:`

This takes care of the WAP; if one argues solely from the WAP the FAQargument is correct. However the "fine tuning" argument is not (despite what its proponents say) a WAP argument; it is an inverse Bayesian argument. The argument runs thusly:

P(F|~N) >> P(F|N)

ergo

P(~N|F) >> P(N|F)

Considered as a formal inference this is a fallacy. None-the-less it is a normal rule of induction which is (usually) sound. The reason is that for the "conclusion" not to hold we need

P(N) >> P(~N)

[This is not the full condition but it is close enough for governmentwork.]

`There are two fallacies in this form
of the argument. The first is the failure to condition on L, mentioned
above. This in itself would render the argument invalid. The second
is that the first line of the argument, P(F|~N) >> P(F|N),
is merely an unsupported assertion. No one knows what the probability
of a supernatural entity creating a universe that is F is! For
example, a dilettante deity might never get around to creating
any universes at all, much less ones capable of supporting life.`

`[ Note added 010612: Since this
was written, we have proved that if You, knowing as a sentient
observer that L is true, adopt an a priori position that
is neutral between N and ~N, i.e., that P(~N|L) is of the same
order of magnitude as P(N|L), then when You learn that F is true
and that P(F|N)<<1, You will conclude that P(F&L&~N)<<1.
See Appendix I (Reply to Kwon) at the end of this essay for the
proof. This observation is problematic for Harter's argument.
For under these assumptions we have`

P(F&L&~N)=P(L|F&~N)P(F|~N)P(~N)<<1.

`Thus under these assumptions it follows
that at least one of P(L|F&~N), P(F|~N) or P(~N) is quite
small. A small P(L|F&~N) says that it is almost certain that
the supernatural deity, having created a "life-friendly"
universe, would make it sterile (lifeless). A small P(F|~N) says
that it is highly unlikely that this deity would even create
a universe that is "life-friendly". Both of these undermine
the usual concepts attributed to the deity by "intelligent
design" theorists, although either would be consistent with
a deity that was incompetent, a dilettante, or a "trickster".
A small P(F|~N) is also consistent with a deity who makes many
universes, most of them being ~F, with many of these ~F universes
perhaps containing life (that is, ~F&L universes, as we discuss
below). A small P(~N) says that it is nearly certain that naturalism
is true a priori and unconditioned on L, so that Harter's
"escape" condition P(N)>>P(~N) in fact holds.`

`Please remember that if You are a
sentient observer, You must already know that L is true, even
before You learn anything about F or P(F|N). Thus it is legitimate,
appropriate, and indeed required, for You to elicit Your
prior on N versus ~N conditioned on L and use that as Your starting
point. If You then retrodict that P(~N)<<1 as a consequence,
all You are doing is eliciting the prior that You would have had
in the absence of Your knowledge that You existed as a sentient
observer. This is the only legitimate way to infer Your value
of P(~N) unconditioned on L.]`

`Having understood the previous discussion,
and with our notation in hand, it is now easy to prove that the
WAP does not support supernaturalism (which we take to be the
negation ~N of N). Recall that the WAP can be written as P(F|N&L)=1.
Then, by Bayes' Theorem [see footnote 2] we have`

P(N|F&L) = P(F|N&L)P(N|L)/P(F|L)= P(N|L)/P(F|L)>= P(N|L)

`where '>=' means "greater
than or equal to." The second line follows because P(F|N&L)=1,
and the inequality of the third line follows because P(F|L) is
a positive quantity less than or equal to 1. (The above demonstration
is inspired by a recent article on talk.origins by Michael Ikeda
<mmikeda@erols.com>; we have simplified the proof in his
article. The message ID for the cited article is <5j6dq8$bvj@winter.erols.com>
for those who wish to search for it on dejanews.)`

`The inequality P(N|F&L)>=P(N|L)
shows that the WAP supports (or at least does not undermine) the
hypothesis that the universe is governed by naturalistic law.
This result is, as we have emphasized, independent of how large
or small P(F|N) is. The observation F cannot decrease the probability
that N is true (given the known background information that life
exists in our universe), and may well increase it.`

`Corollary: Since P(~N|F&L)=1-P(N|F&L)
and similarly for P(~N|L), it follows that P(~N|F&L)<=P(~N|L).
In other words, the observation F does not support supernaturalism
(~N), and may well undermine it.`

`The thrust of practically all "Intelligent
Design" and Creationist arguments (excepting the anthropic
argument and perhaps a few others) is to show ~F, since it is
evident, we think, that if ~F then we cannot have both life and
a naturalistic universe. We evidently do have life, so the success
of one of these arguments would clearly establish ~N. In other
words, given our prior opinion P(N&L), where 0<P(N&L)<1
but otherwise unrestricted (thus we neither rule in nor rule out
N initially), arguments like Behe's attempt to support ~F so as
to undermine N:`

P(N|~F&L)<P(N|L).

`But the "anthropic" argument
is that observing F also undermines N:`

P(N|F&L)<P(N|L).

`We assert that the intelligent design
folks want these inequalities to be strict (otherwise there would
be no point in their making the argument!)`

`From these two inequalities we readily
derive a contradiction, as follows. From the definition of conditional
probability [see footnote 1], the two inequalities above yield`

P(N&~F&L)<P(N|L)P(~F&L), P(N& F&L)<P(N|L)P( F&L)

`Adding,`

P(N&L)= P(N&~F&L)+P(N&F&L)< P(N|L)(P(~F&L)+P(F&L))= P(N|L)P(L)=P(N&L),

`a contradiction since the inequality
is strict.`

`If we remove the restriction that
the inequalities be strict, then the only case where both inequalities
can be true is if`

P(N|~F&L)=P(N|L) and P(N|F&L)=P(N|L).

`In other words, the only case where
both can be true is if the information that the universe is "life-friendly"
has no effect on the probability that it is naturalistic
(given the existence of life); and this can only be the case if
neither inequality is strict.`

`In essence, we see that the intelligent
design folks who make the anthropic argument are really trying
to have it both ways: They want observation of F to undermine
N, and they also want observation of ~F to undermine N. That is,
they want any observation whatsoever to undermine N! But the error
is that the anthropic argument does not undermine N, it
supports N. They can have one of the prongs of their argument,
but they can't have both.`

`[ Note added 010612: Some people
have objected to us that Behe is not making the argument ~F, but
is only making a statement that it is highly unlikely that
certain of his "IC" structures could arise naturalistically.
Our reading of Behe that he is making an argument that it is impossible
for this to happen (a form of ~F as we understand it), but even
if we are wrong and he is not making this argument, the point
of our comments in this section is that making the argument that
the universe is F or is "fine-tuned" (P(F|N)<<1)
does not support supernaturalism; the argument that should be
made is that the universe is ~F, since this manifestly supports
supernaturalism by refuting naturalism. See Appendix I (Reply
to Kwon) at the end of this essay.]`

`Ross' argument discusses the case
where the conditions in our universe are not only "life-friendly,"
but they are also "fine-tuned," in the sense that only
a very small fraction of possible universes can be "life-friendly."
We have shown that regardless how "finely-tuned" the
the laws of physics are, the observation that the universe is
capable of sustaining life cannot undermine N.`

`As we have pointed out above, others
have responded to the claim of "fine-tuning" in several
ways. One way has been to point out that this claim is not corroborated
by any theoretical understanding about what forms of life might
arise in universes with different physical conditions than our
own, or even any theoretical understanding about what kinds of
universes are possible at all; it is basically a claim founded
upon our own ignorance of physics. To those that make this point,
the argument is about whether P(F|N) is really small (as Ross
claims), or is in fact large. The point (against Ross) is essentially
that Ross' crucial assumption is completely without support.`

`A second response is to point out
that several theoretical lines of evidence indicate that many
other, and perhaps even an infinite number of other universes,
with varying sets of physical constants and conditions, might
well exist, so that even if the probability that a given universe
would have constants close to those of our own universe is small,
the sheer number of such universes would virtually guarantee that
some of them would possess constants that would allow life
to arise.`

`Nevertheless, it is necessary to consider
the implications of Ross' assertion that the universe is "fine-tuned."
Suppose it is true that amongst all naturalistic universes, only
a very small proportion could support life. What would this imply?`

`We have shown that the WAP tends to
support N, and cannot undermine it. This observation is independent
of whether P(F|N) is small or large, since (as we have seen) the
only probabilities that are significant for inference about N
are those that are conditioned upon all relevant data at our disposal,
including the fact that L is true. Therefore, regardless of the
size of P(F|N), valid reasoning shows that observing that F is
true cannot decrease the probability that N is true, and may increase
it.`

`We believe that the real import of
observing that P(F|N) is small (if indeed that is true) would
be to strengthen Vilenkin/Linde/Smolin-type hypotheses that multiple
universes with varying physical constants may exist. If indeed
the universe is governed by naturalistic laws, and if indeed the
probability that a universe governed by naturalistic laws can
support life is small, then this supports a Vilenkin/Linde/Smolin
model of multiple universes over a model that includes only a
single universe with a single set of physical constants.`

`To see this, let S="there is
only a Single universe," and M="there are Multiple universes."
Let E = "there Exists a universe with life." Clearly,
P(E|N)<P(F|N), since it is possible that a universe that is
"life-friendly" could still be barren. But, since L
is true, E is also true, so observing L implies that we have also
observed E.`

`Then, assuming that P(F|N)<1 is
the probability that a single universe is "life-friendly,"
that this probability is the same for each "random"
multiple universe as it would be for a single universe, and that
the probability that a given universe exists is independent of
the existence of other universes, it follows that`

P(E|S&N) = p = P(E|N) < P(F|N) < 1 (and for Ross, P(F|N)<<1);

P(E|M&N) = 1 - (1-p)^{m}, where m is the number of universes if M is true; This is less than 1 but approaches 1 (for fixed p) as m gets larger and larger. Since all the Multiple-universe proposals we have seen suggest that m is in fact infinite, it follows that P(E|M&N)=1. (If one postulates that m is finite, then the calculation depends explicitly on p and m; this is left as an exercise for the reader.)

`Since`

P(S|E&N) = P(E|S&N)P(S|N)/P(E|N) andP(M|E&N) = P(E|M&N)P(M|N)/P(E|N),

`with these assumptions it follows
by division that`

P(M|E&N) 1 P(M|N)-------- = --- x ------,P(S|E&N) p P(S|N)

`which shows that observing E (or L)
increases the evidence for M against S in a naturalistic universe
by a factor of at least 1/p. The smaller P(F|N)=p (that is, the
more "finely-tuned" the universe is), the more likely
it is that some form of multiple-universe hypothesis is true.`

`The next section is rather more speculative,
depending as it does upon theological notions that are hard to
pin down, and therefore should be taken with large grains of salt.
But it is worth considering what effect various theological hypotheses
would have on this argument. It is interesting to ask the question,
"given that observing that F is true cannot undermine N and
may support it, by how much can N be strengthened (and ~N be undermined)
when we observe that F is true?"`

`It is evident from the discussion
of the main theorem that the key is the denominator P(F|L). The
smaller that denominator, the greater the support for N. Explicitly
we have`

P(F|L)=P(F|N&L)P(N|L)+P(F|~N&L)P(~N|L)

`But since P(F|N&L)=1 we can simplify
this to`

P(F|L)=P(N|L)+P(F|~N&L)P(~N|L).

`Plugging this into the expression
P(N|F&L)=P(N|L)/P(F|L) we obtain`

P(N|F&L) = P(N|L)/[P(N|L)+P(F|~N&L)P(~N|L))]= 1/[1+P(F|~N&L)P(~N|L)/P(N|L)]= 1/[1+C P(F|~N&L)],

`where C=P(~N|L)/P(N|L) is the prior
odds in favor of ~N against N. In other words, C is the odds that
we would offer in favor of ~N over N before noting that the universe
is "fine-tuned" for life.`

`A major controversy in statistics
has been over the choice of prior probabilities (or in this case
prior odds). However, for our purposes this is not a significant
consideration, as long as we don't choose C in such a way as to
completely rule out either possibility (N or ~N), i.e., as long
as we haven't made up our minds in advance. This means that any
positive, finite value of C is acceptable.`

`One readily sees from this formula
that for acceptable C`

(1) as P(F|~N&L)-->0, P(N|F&L)-->1;(2) as P(F|~N&L)-->1, P(N|F&L)-->1/[1+P(~N|L)/P(N|L)]=P(N|L),

`where '-->' means "approaches
as a limit" and the last result follows from the fact that
P(N|L)+P(~N|L)=1.`

`So, P(N|F&L) is a monotonically
decreasing function of P(F|~N&L) bounded from below by P(N|L).
This confirms the observation made earlier, that noting that F
is true can never decrease the evidential support for N. Furthermore,
the only case where the evidential support is unchanged is when
P(F|~N&L) is identically 1. This is interesting, because it
tells us that the only case where observing the truth of F does
not increase the support for N is precisely the case when
the likelihood function P(F|x&L), evaluated at F, and with
x ranging over N and ~N, cannot distinguish between N and ~N.
That is, the only way to prevent the observation F from increasing
the support for N is to assert that ~N&L also requires
F to be true. Under these circumstances we cannot distinguish
between N and ~N on the basis of the data F. In a deep sense,
the two hypotheses represent, and in fact, are the same
hypothesis. Put another way, to assume that P(F|~N&L)=1 is
to concede that life in the world actually arose by the operation
of an agent that is observationally indistinguishable from naturalistic
law, insofar as the observation F is concerned. In essence, any
such agent is just an extreme version of the "God-of-the-gaps,"
whose existence has been made superfluous as far as the existence
of life is concerned. Such an assumption would completely undermine
the proposition that it is necessary to go outside of naturalistic
law in order to explain the world as it is, although it doesn't
undermine any argument for supernaturalism that doesn't rely on
the universe being "life-friendly".`

`So, if supernaturalism is to be distinguished
from naturalism on the basis of the fact that the universe is
F, it must be the case that P(F|~N&L)<1. Otherwise, we
are condemned to an unsatisfying kind of "God-of-the-gaps"
theology. But what sort of theologies can we consider, and how
would they affect this crucial probability?`

`To make these ideas more definite,
we consider first a specific interpretation that is intended to
imitate, albeit crudely, how the assumption of a relatively powerful
and inscrutable deity (such as a generic Judeo-Christian-Islamic
deity might be) could affect the calculation of the likelihood
function P(F|~N&L).`

`We suggest that any reasonable version
of supernaturalism with such a deity would result in a value of
P(F|~N&L) that is, in fact, very small (assuming that only
a small set of possible universes are F). The reason is that a
sufficiently powerful deity could arrange things so that a universe
with laws that are not "life-friendly" can sustain life.
Since we do not know the purposes of such a deity, we must assign
a significant amount of the likelihood function to that possibility.
Furthermore, if such a deity creates universes and if the "fine-tuning"
claims are correct, then most life-containing universes
will be of this type (i.e., containing life despite not being
"life-friendly"). Thus, all other things being equal,
and if this is the sort of deity we are dealing with, we would
expect to live in a universe that is ~F.`

`To assert that such a deity could
only create universes containing life if the laws are life-friendly
is to restrict the power of such a deity. And to assert that such
a deity would only create universes with life if the laws
are life-friendly is to assert knowledge of that deity's purposes
that many religions seem reluctant to claim. Indeed, any such
assertion would tend to undermine the claim, made by many religions,
that their deity can and does perform miracles that are contrary
to naturalistic law, and recognizably so.`

`Our conclusion, therefore, is that
not only does the observation F support N, but it supports it
overwhelmingly against its negation ~N, if ~N means creation by
a sufficiently powerful and inscrutable deity. This latter conclusion
is, by the way, a consequence of the Bayesian Ockham's Razor [Jefferys,
W.H. and Berger, J.O., "Ockham's Razor and Bayesian Analysis,"
American Scientist 80, 64-72 (1992)]. The point
is that N predicts outcomes much more sharply and narrowly than
does ~N; it is, in Popperian language, more easily falsifiable
than is ~N. (We do not wish to get into a discussion of the Demarcation
Problem here since that is out of the scope of this FAQ, though
we do not regard it as a difficulty for our argument. For our
purposes, we are simply making a statement about the consequences
of the likelihood function having significant support on only
a relatively small subset of possible outcomes.) Under these circumstances,
the Bayesian Ockham's Razor shows that observing an outcome allowed
by both N and ~N is likely to favor N over ~N. We refer the reader
to the cited paper for a more detailed discussion of this point.`

`Aside from sharply limiting the likely
actions of the deity (either by making it less powerful or asserting
more human knowledge of the deity's intentions), we can think
of only one way to avoid this conclusion. One might assert that
any universe with life would appear to be "life-friendly"
from the vantage point of the creatures living within it, regardless
of the physical constants that such a universe were equipped with.
In such a case, observing F cannot change our opinion about the
nature of the universe. This is certainly a possible way out for
the supernaturalist, but this solution is not available to Ross
because it contradicts his assertions that the values of certain
physical constants do allow us to distinguish between universes
that are "life-friendly" and those that are not. And,
such an assumption does not come without cost; whether others
would find it satisfactory is problematic. For example, what about
miracles? If every universe with life looks "life-friendly"
from the inside, might this not lead one to wonder if everything
that happens therein would also look to its inhabitants like the
result of the simple operation of naturalistic law? And then there
is Ockham's Razor: What would be the point of postulating a supernatural
entity if the predictions we get are indistinguishable from those
of naturalistic law?`

`In the previous section, we have discussed
just one of many sorts of deities that might exist. This one happens
to be very powerful and rather inscrutable (and is intended to
be a model of a generic Judeo-Christian-Islamic sort of deity,
though believers are welcome to disagree and propose--and justify--their
own interpretations of their favorite deity). However, there are
many other sorts of deities that might be postulated as being
responsible for the existence of the universe. There are somewhat
more limited deities, such as Zeus/Jupiter, there are deities
that share their existence with antagonistic deities such as the
Zoroastrian Ahura-Mazda/Ahriman pair of deities, there are various
Native American deities such as the trickster deity Coyote, there
are Australian, Chinese, African, Japanese and East Indian deities,
and even many other possible deities that no one on Earth has
ever thought of. There could be deities of lifeforms indigenous
to planets around the star Arcturus that we should consider, for
example.`

`Now when considering a multiplicity
of deities, say D _{1},D_{2},...,D_{i},...,
we would have to specify a value of the likelihood function for
each individual deity, specifying what the implications would
be if that deity were the actual deity that created the
universe. In particular, with the "fine-tuning" argument
in mind, we would have to specify P(F|D_{i}&L) for
every i (probably an infinite set of deities). Assuming that we
have a mutually exclusive and exhaustive list of deities, we see
the hypothesis ~N revealed to be composite, that is, it
is a combination or union of the individual hypotheses D_{i}
(i=1,2,...). Our character set doesn't have the usual "wedge"
character for "or" (logical disjunction), so we will
use 'v' to represent this operation. We then have`

~N = D_{1}v D_{2}v...v D_{i}v...

`Now, the total prior probability of
~N, P(~N|L), has to be divvied up amongst all of the individual
subhypotheses D _{i}:`

P(~N|L) = P(D_{1}|L) + P(D_{2}|L) + ... + P(D_{i}||L) + ...

`where 0<P(D _{i})<P(~N|L)<1
(assuming that we only consider deities that might exist, and
that there are at least two of them). In general, each of the
individual prior probabilities P(D_{i}|L) would be very
small, since there are so many possible deities. Only if some
deities are a priori much more likely than others would
any individual deity have an appreciable amount of prior probability.`

`This means that in general, P(D _{i}|L)<<1
for all i.`

`Now when we originally considered
just N and ~N, we calculated the posterior probability of N given
L&F from the prior probabilities of N and ~N given L, and
the likelihood functions. Here it would be simpler to look at
prior and posterior odds. These are derived straightforwardly
from probabilities by the relation`

Odds = Probability/(1 - Probability).

`This yields a relationship between
the prior and posterior odds of N against ~N [using P(N|F&L)+P(~N|F&L)=1]:`

P( N|F&L) P(F| N&L) P( N|L)Posterior Odds = --------- = ---------- x -------P(~N|F&L) P(F|~N&L) P(~N|L)= (Bayes Factor) x (Prior Odds)

`The Bayes Factor and Prior Odds are
given straightforwardly by the two ratios in this formula.`

`Since P(F|N&L)=1 and P(F|~N&L)<=1,
it follows that the posterior odds are greater than or equal to
the prior odds (this is a restatement of our first theorem, in
terms of odds). This means that observing that F is true cannot
decrease our confidence that N is true.`

`But by using odds instead of probabilities,
we can now consider the individual sub-hypotheses that make up
~N. For example, we can calculate prior and posterior odds of
N against any individual D_i. We find that`

P( N|F&L) P(F| N&L) P( N|L)Posterior Odds = --------- = --------- x -------P(D_{i}|F&L) P(F|D_{i}&L) P(D_{i}|L)

`This follows because (by footnote
2)`

P(N |F&L) = P(F| N&L)P( N|L)/P(F|L),P(D_{i}|F&L) = P(F|D_{i}&L)P(D_{i}|L)/P(F|L),

`and the P(F|L)'s cancel out when you
take the ratio.`

`Now, even if P(F|D _{i}&L)=1,
which is the maximum possible, the posterior odds against D_{i}
may still be quite large. The reason for this is that the prior
probability of ~N has to be shared out amongst a large number
of hypotheses D_{j}, each one greedily demanding its own
share of the limited amount of prior probability available. On
the other hand, the hypothesis N has no others to share with.
In contrast to ~N, which is a compound hypothesis, N is a simple
hypothesis. As a consequence, and again assuming that no particular
deity is a priori much more likely than any other (it would
be incumbent upon the proposer of such a deity to explain why
his favorite deity is so much more likely than the others), it
follows that the hypothesis of naturalism will end up being much
more probable than the hypothesis of any particular deity
D_{i}.`

`This phenomenon is a second manifestation
of the Bayesian Ockham's Razor discussed in the Jefferys/Berger
article (cited above).`

`In theory it is now straightforward
to calculate the posterior odds of N against ~N if we don't particularly
care which deity is the right one. Since the D_{i}
form a mutually exclusive and exhaustive set of hypotheses whose
union is ~N, ordinary probability theory gives us`

P(~N|F&L) = P(D_{1}|F&L) + P(D_{2}|F&L) + ...= [P(F|D_{1}&L)P(D_{1}|L) + P(F|D_{2}&L)P(D_{2}|L) + ...]/P(F|L)

`Assuming we know these numbers, we
can now calculate the posterior odds of N against ~N by dividing
the above expression into the one we found previously for P(N|F&L).
Of course, in practice this may be difficult! However, as can
be seen from this formula, the deities D _{i} that contribute
most to the denominator (that is, to the supernaturalistic hypothesis)
will be the ones that have the largest values of the likelihood
function P(F|D_{i}&L) or the largest prior probability
P(D_{i}|L) or both. In the first case, it will be because
the particular deity is closer to predicting what naturalism predicts
(as regards F), and is therefore closer to being a "God-of-the-gaps"
deity; in the second, it will be because we already favored that
particular deity over others a priori.`

`Some make the mistake of thinking
that "fine-tuning" and the anthropic principle support
supernaturalism. This mistake has two sources.`

`The first and most important of these
arises from confusing entirely different conditional probabilities.
If one observes that P(F|N) is small (since most hypothetical
naturalistic universes are not "fine-tuned" for life),
one might be tempted to turn the probability around and decide,
incorrectly, that P(N|F) is also small. But as we have
seen, this is an elementary blunder in probability theory. We
find ourselves in a universe that is "fine-tuned" for
life, which would be unlikely to come about by chance (because
P(F|N) is small), therefore (we conclude incorrectly),
P(N|F) must also be small. This common mistake is due to confusing
two entirely different conditional probabilities. Most
actual outcomes are, in fact, highly improbable, but it does not
follow that the hypotheses that they are conditioned upon are
themselves highly improbable. It is therefore fallacious to reason
that if we have observed an improbable outcome, it is necessarily
the case that a hypothesis that generates that outcome is itself
improbable. One must compare the probabilities of obtaining
the observed outcome under all hypotheses. In general,
most, if not all of these probabilities will be very small, but
some hypotheses will turn out to be much more favored by the actual
outcome we have observed than others.`

`The second source of confusion is
that one must do the calculations taking into account all
the information at hand. In the present case, that includes
the fact that life is known to exist in our universe. The possible
existence of hypothetical naturalistic universes where life does
not exist is entirely irrelevant to the question at hand, which
must be based on the data we actually have.`

`In our view, similar fallacious reasoning
may well underlie many other arguments that have been raised against
naturalism, not excluding design and "God-of-the-Gaps"
arguments such as Michael Behe's "Irreducible Complexity"
argument (in his book, Darwin's Black Box), and William
Dembski's "Complex Specified Information," as described
in his dissertation (University of Illinois at Chicago). We conclude
that whatever their rhetorical appeal, such arguments need to
be examined much more carefully than has happened so far to see
if they have any validity. But that discussion is outside the
scope of this article.`

`Bottom line: The anthropic argument
should be dropped. It is wrong. "Intelligent design"
folks should stick to trying to undermine N by showing ~F. That's
their only hope (though we believe it to be a forlorn one).`

Michael Ikeda Bill JefferysStatistical Research Division Department of AstronomyBureau of the Census University of TexasWashington DC 20233 Austin TX 78712

`Please E-mail comments on this proposed
FAQ to Bill Jefferys (bill@clyde.as.utexas.edu).`

`Michael Ikeda's work on this article
was done on his own time and not as part of his official duties.
The authors' affiliations are for identification only. The opinions
expressed herein are those of the authors, and do not necessarily
represent the opinions of the authors' employers.`

`Copyright (C) 1997-2002 by Michael
Ikeda and Bill Jefferys. Portions of this FAQ are Copyright (C)
1997 by Richard Harter. All Rights Reserved.`

`[1] By definition, P(A|B)=P(A&B)/P(B);
it follows that also P(A|B&C)=P(A&B|C)/P(B|C).`

`[2] We use Bayes' Theorem in the form`

P(A|B&K)=P(B|A&K)P(A|K)/P(B|K)

`which follows straightforwardly from
the identity`

P(A|B&K)P(B|K)=P(A&B|K)=P(B|A&K)P(A|K)

`(a consequence of footnote 1) assuming
that P(B|K)>0.`

APPENDIX I: Reply to Kwon (April 30, 2001)

`David Kwon has posted a web page at`

`in which he claims to have refuted
the arguments in our article. However, he has made a simple error,
which we detail below, along with comments on some of his other
assertions.`

`Kwon's Equation (3) reads as follows:`

P(N|F&L) = P(N&F&L) / {P(~N&F&L) + P(N&F&L)}

`This is an elementary result of probability
theory and we agree with it. Kwon then goes on and assumes what
he calls the "fine-tuning" condition P(F|N)<<1
from which he correctly derives Equation (8), the important part
of which reads`

P(N&F&L) << 1

`From these two results (3 and 8) Kwon
derives`

P(N|F&L)<<1 unless P(~N&F&L)<<1

`Unfortunately, nothing in Kwon's "proof"
shows that P(~N&F&L) is not <<1, so he cannot assert
unconditionally that P(N|F&L)<<1 as a consequence of
his assumptions. He asserts`

"The only way not to come to this conclusion [that P(N|F&L)<<1] is to start with ana prioriassumption of P(~N&F&L)<<1. In other words, the only way to hold on to naturalism is by assuming that theism is virtually impossible to begin with."

`This, however, is incorrect, and here
the "proof" falls apart. Kwon apparently recognizes
that according to his Equation (3), the value of P(N|F&L)
is not governed by the actual size of P(N&F&L), but instead
by the relative sizes of P(N&F&L) and P(~N&F&L).
In particular, if P(N&F&L)<<P(~N&F&L) then
P(N|F&L) will be close to zero; if P(N&F&L) is approximately
equal to P(~N&F&L), then P(N|F&L) will be of order
one-half; and if P(N&F&L)>>P(~N&F&L), then
P(N|F&L) will be nearly unity. Therefore, we need to look
at the ratio R = P(N&F&L)/P(~N&F&L) to see what
factors govern its size and what assumptions this entails.`

`We obtain:`

R = P(N&F&L) / P(~N&F&L) = {P(F|N&L) P(N&L)} / {P(F|~N&L) P(~N&L)} (A) = P(N&L) / {P(F|~N&L) P(~N&L)} (B) >= P(N&L) / P(~N&L) (C) = {P(N|L) P(L)} / {P(~N|L) P(L)} (D) = P(N|L) / P(~N|L) (E)

`Here, (A) and (D) follow from the
definition of conditional probability, (B) by the WAP--which Kwon
says he accepts--and which asserts that P(F|N&L)=1, (C) because
the probability P(F|~N&L) in the denominator is <=1, and
(E) by cancellation of P(L) in numerator and denominator.`

`Thus we see that in fact the ratio
R cannot be small unless P(N|L)/P(~N|L) is also small. Therefore
we cannot conclude that P(N|F&L)<<1 unless P(N|L)/P(~N|L)<<1--regardless
of the size of P(N&F&L). But what is P(N|L)/P(~N|L)? Why,
it is just the prior odds ratio that You assign to describe Your
relative belief in N and ~N before You learn that F is true. Thus,
although Kwon is correct in noting that the only way to keep P(N|F&L)
from being very small is to have P(~N&F&L)<<1, this
does not represent a prior commitment to naturalism as he asserts.
Indeed, a prior commitment to naturalism would be to assume that
P(N|L)/P(~N|L)>>1, and as (E) shows, if we assume P(N|L)/P(~N|L)
of order unity, which reflects a neutral prior position between
the N and ~N, and not a prior commitment to naturalism, we will
end up being at least neutral between N and ~N after observing
that F is true, regardless of the size of P(N&F&L) and
P(F|N).`

`Indeed, it requires a prior commitment
to supernaturalism to get P(N|F&L)<<1, because
You would have to presume a priori that P(N|L)<<P(~N|L).
Kwon has it exactly backwards.`

`So the absolute size of P(N&F&L)
and P(F|N) do not tell us anything about P(N|F&L); this is
a confusion between conditional and unconditional probability.
The only thing that counts is the ratio R. Kwon's calculation
in his steps (4-8) is simply irrelevant to the final result. Indeed,
we have the following theorem:`

Theorem: If p(F|N)<<1 and You are exactly neutral between N and ~N before learning F, then P(~N&F&L)<<1.

Proof: Under the assumptions we have P(F&N&L)=P(N|L)P(L)<<1; but if we are exactly neutral between N and ~N before learning F we have P(N|L)=0.5=O(1) so the unconditional probability P(L)<<1. But by standard probability theory P(~N&F&L)<=P(L)<<1. QED.

`Thus, far from reflecting a prior
commitment to naturalism as Kwon claims, the result P(~N&F&L)<<1
is a consequence of the fine tuning condition together with the
adoption of an at least neutral prior position on N versus ~N.
It is due to the fact that P(N&L&F) and P(~N&L&F)
both have P(L)<<1 as a factor when they are expanded using
the definition of conditional probability.`

`Furthermore, it is even possible for
P(~N|F&L) to be very small (and therefore P(N|F&L) close
to unity), without making a prior commitment to naturalism. For
example, suppose we adopt the neutral position P(N|L)=P(~N|L)=0.5;
then from (B) we find that R = 1/P(F|~N&L), and if P(F|~N&L)<<1
then R>>1 and P(F|N&L) is close to unity. But what does
P(F|~N&L)<<1 mean? Is this a "prior commitment
to naturalism?" No, a prior commitment to naturalism would
involve some conditional probability on N, not some conditional
probability on F. The condition P(F|~N&L)<<1 actually
means that it is likely that an inhabitant of a supernaturalistically
created universe would find that it is ~F: a universe where life
exists despite the fact that it could not exist naturalistically,
for example as a consequence of the suspension of natural law
by the supernatural creator. We discussed this extensively in
our article. Indeed, without psychoanalyzing the Deity and analysing
its powers and intentions, it is a priori quite likely that the
Deity might create universes that are ~F&L, for such universes
are not excluded unless we know something about this Deity that
would prevent it from creating such universes. An example of such
a universe would be Paradise, and it seems unlikely that enthusiasts
of the "fine-tuning" argument would be willing to say
that the Deity would not create anything like Paradise. But the
only way for them to escape from P(F|~N&L)<<1 would
be for them to assert that the Deity would only, or mostly, create
universes that, if they contain life, are F, and we see no justification
for such an assumption.`

`Kwon makes some other incorrect statements
later in his web article. He says that our argument "incorrectly
attributes significance to P(N|L)." Kwon here appears to
have missed the fact that we are talking about Bayesian probabilities.
The probability P(N|L) refers to our universe, and is Your Bayesian
prior probability that N is true, given that You know that L is
true (which must be the case since it is a condition of reasoning
that You be alive), but before You learn that F is true. It is
a reflection of Your epistemological condition or state of knowledge
at a particular moment in time. Thus, P(N|L) has a perfectly definite
meaning in our universe, although the value of P(N|L) will differ
from individual to individual because every individual has different
background information (not explicitly called out here but mentioned
in our article).`

`Furthermore, Kwon is incorrect when
he states that "P(N|L) is irrelevant to our universe for
the same reason that P(N|F) is irrelevant." We never said
that P(N|F) is irrelevant, only that it is irrelevant for inference.
The reason why P(N|F) is irrelevant for inference is that no sentient
being is unaware of L as background information. Every sentient
being knows that he is alive and therefore knows that L is true;
thus every final probability statement that he makes must be conditioned
on L. This is not true of F. There are sentient beings in our
universe, indeed in our world, that do not yet know that F is
true. Most schoolchildren do not know that F is true, although
they know that L is true. Probably most adults do not know that
F is true. Thus, Kwon errs in drawing a parallel between P(N|L)
and P(N|F).`

`Kwon started with the perfectly reasonable
proposal that "fine tuning" is best defined by P(F|N)<<1,
and attempted to derive his result. That he was unable to do this
comes as no surprise to us, because one of us [whj] spent the
better part of a year trying to get useful information from propositions
such as P(F|N)<<1, without success. All such attempts were
fruitless, and the reason why is seen in our discussion. For example,
suppose we were to assume in addition that P(F|~N)=1. Even then,
no useful result can be derived, for from this we can only determine
the obvious fact that P(F&L&~N)<=1, which gives no
useful information about the crucial ratio R. The inequality goes
in the wrong direction! Thus, "fine tuning"--P(F|N)<<1--tells
us nothing useful, which is why in our article we concentrated
instead on finding out what "life friendliness"--F--and
the WAP can tell us.`

`Kwon says, "We have always known
that F is true for our universe..." This is false. In fact,
the suspicion that F is true is relatively recent, going only
back to Brandon Carter's seminal papers in the mid-1970's. Earlier,
physicists such as Dirac had in fact speculated that the values
of some fundamental physical constants (e.g., the fine structure
constant) might have been very different in the past, which would
violate F, and somewhat later other scientists (for example Fred
Hoyle in the early 1950s) have used the assumption that F is true
in order to predict certain physical phenomena, which were later
found to be the case. Had those observations NOT been found to
be true, F would have been refuted, and we would seriously have
to consider ~N. Even today we do not know that our universe
is F--"life-friendly"--in the sense that we use the
term in our article. We strongly suspect that it is true, but
it is conceivable that someone will make a WAP prediction that
will turn out to be false and which might refute F.`

`Kwon incorrectly asserts that the
idea that there may be other universes is "simply unscientific."
Certainly many highly respected cosmologists and physicists like
Andrei Linde (Stanford), Lee Smolin (Harvard) and Alexander Vilenkin
(Tufts) and Nobel laureate Stephen Weinberg (Texas) would disagree
with this statement. Kwon claims that the hypothesis of other
universes "cannot be tested." While we might agree that
testing the hypothesis of other universes will be difficult, we
do not agree that the hypothesis is untestable, and neither do
scientists that work in this area. Some specific tests have been
suggested. For example, David Deutsch has proposed specific tests
of the Everett-Wheeler interpretation of quantum mechanics commonly
known as the "Many-Worlds" hypothesis. And recently
an article that proposed another way that other universes might
be detected was published ( Science, Vol. 292, p. 189-190,
original paper archived as http://arXiv.org/abs/hep-th/0103239).
Regardless, our argument is not dependent on the notion that there
are many other universes. It stands on its own.`

`Kwon misunderstands the point of the
"god of the gaps" argument. The problem isn't that the
gap is being filled by a god, the problem is what happens if the
gap is filled by physics. Then the god that filled the gap gets
smaller. This is a theological problem, not an epistemological
or scientific problem. We agree with Kwon that there are gaps
in our physical explanation of the universe that may never be
filled; but it is hoping against hope that we will never fill
any of the gaps currently being touted by "intelligent design
theorists" as proof of supernaturalism. Some of them are
certain to be filled in time, and each time this happens, the
god of the intelligent designers will be diminished. (In fact,
some of them were in fact filled even before the recent crop of
"ID theorists" made their arguments--this is true of
some of Michael Behe's examples, for which evolutionary pathways
had already been proposed even before Behe published his book).`

`As to Kwon's last point, that we incorrectly
claim that "intelligent design theorists" incoherently
assert both F and ~F. We believe that it is a correct statement
that at least some are arguing ~F. It is our impression, for example,
that Michael Behe is arguing that it is actually impossible, and
not just highly unlikely, for certain "irreducibly complex"
(IC) structures to evolve without supernatural intervention, and
that is a form of ~F. Regardless, even if no one is attempting
to argue from ~F to ~N, our point still stands. Attempts to prove
~N that argue from either F or P(F|N)<<1 or both do not
work. But attempts to prove ~N by showing ~F would work. Thus,
people making anthropic and "fine tuning" arguments
have hold of the wrong end of the stick. They should be trying
to show that the universe is not F. It is clear that showing that
the universe is not F would at one stroke prove ~N; it follows
that showing that the universe is F can only undermine ~N and
support N; this is an elementary result of probability theory,
since it is not possible that observations of F as well as ~F
would both support ~N. Since it is trivially true that observing
~F does support ~N, observing F must undermine it. Put another
way, it seems to us that Michael Behe--if we understand him--is
making the right argument from a logical and inferential point
of view, and Hugh Ross is making the wrong argument. If it turns
out that Behe is not making the argument we think he is, then
it is still the case that Hugh Ross is making the wrong argument.`

`Kwon makes some remarks about "nontheists"
that seem to indicate that he thinks that only "nontheists"
would argue as we have. This is not the case. The issue here is
whether the "fine tuning" argument is correct. It is
exactly analogous to the centuries of work done on Fermat's last
theorem. It is likely that most mathematicians thought that the
theorem was true for most of that time, yet they continued to
reject proofs that had flaws in them. They rejected them not because
they thought Fermat's last theorem was false, but because the
proofs were wrong. They even rejected Wiles' first attempt at
a proof, because it was (slightly) flawed. In the same way a theist
can and should reject a flawed "proof" of the existence
of God. Our argument is that the fine tuning arguments are wrong,
and no one should draw any conclusions about our personal beliefs
from the fact that we say that these arguments are wrong.`

`Conclusion: Kwon's "proof"
is fatally flawed. He incorrectly asserts that the only way to
keep P(N|F&L) from being very small is to assume naturalism
a priori. Quite the contrary, the only way to make P(N|F&L)
small is to assume supernaturalism a priori. Kwon apparently
does not understand the significance of some of the Bayesian probabilities
we use; this is forgiveable in a sense since Bayesian probability
theory is still misunderstood by most people, even those with
some training in probability theory...but it means that Kwon should
withdraw these comments until he understands Bayesian probability
theory well enough to criticize it. Kwon's assertion that we have
always known that our universe is F is false; his assertion that
the existence of other universes is untestable is also false,
and in any case is not relevant to our main argument. Finally,
he mistakenly thinks that the god-of-the-gap argument somehow
tells against science. It does not, since it is purely a theological
conundrum, not a scientific one.`

`Nonetheless, we thank David Kwon for
his serious and attentive reading of our article and for his comments.
He is the first to attempt a mathematical rather than a polemical
refutation of our argument. His argument fails because, as we
show here, it isn't possible to derive anything useful from the
fine-tuning proposition P(F|N)<<1. When all factors are
taken into account, it is clear that the only way to end up with
a final result that P(N|F&L)<<1 is to assume at the
outset that supernaturalism is almost surely true, thus begging
the question.
M. I.
W. J.
April 30, 2001`

`[ Note added 010613: When we
posted this response, we informed Mr. Kwon, so that he could either
respond to our criticisms or withdraw his web page. We regret
to say that up to now he has done neither.]`

`All materials at this website Copyright (C) 1994-2002 by
William H. Jefferys. This webpage Copyright (C) 1997-2002 by Michael
Ikeda and William H. Jefferys. Portions of this webpage Copyright
(C) 1997 by Richard Harter. All rights reserved.`

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