*Posted January 12, 2004*

Contents:

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.
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 General Relativity and Quantum Cosmology, abstract.

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 FAQ argument 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 government work.]

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) and

P(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 Jefferys |

Statistical Research Division | Department of Astronomy |

Bureau of the Census | University of Texas |

Washington 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.

David Kwon has posted a web page 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.

[**Note added 040109:** Kwon's original article has disappeared from the web. The above link is to the last version of his article archived by the Internet Wayback Machine via Makeashorterlink.com.]

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 an

a 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 The Ekpyrotic Universe: Colliding Branes and the Origin of the Hot Big Bang).
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.]

**Note added 040109:** Kwon has never responded to our criticisms; his web page disappeared when he apparently finished his career as a Berkeley graduate student. It is archived and can be obtained courtesy of the Internet Wayback Machine via Makeashorterlink.com.]

This article was first posted at Bill Jefferys' Home Page. .

Location of this article: http://www.talkreason.org/articles/super.cfm