Where the Conflict Really Lies: Science, Religion, and Naturalism (27 page)

Nevertheless, the McGrew objection is certainly based on genuine intuitions. Suppose we have finitely many mutually exclusive and jointly exhaustive equi-probable possibilities: the relevant measure will assign each possibility the same probability; it will be additive; and the probabilities of these possibilities will sum to 1. We have non-discrimination, normalizability, and additivity. There are fifty-two cards in the deck; in a random draw, each
card has the same probability (1/52) of being selected; the sum of these probabilities is 1. But things go awry when we move to infinite magnitudes. Suppose we have an infinite deck of cards: now we run into the difficulty noted above: we can no longer assign each card the same probability of being drawn in such a way that these probabilities sum, given countable additivity, to 1. As before, if we assign the same finite probability to each, the sum of these probabilities will be infinite; if we assign zero probability, the sum of the probabilities will be zero. We can’t have all of non-discrimination, countable additivity and normalizability. We can preserve normalizability by asserting that the probability (given a drawing) that one or another of the cards will be drawn is 1, while the probability with respect to each card that it will be drawn is zero; but then of course we lose (countable) additivity. Or we can preserve countable additivity by assigning probabilities to the various cards in accord with some series that sums to 1: assign the first card a probability of 1/2, the second 1/4, and so on. But then we lose non-discrimination: we are not assigning them the same probability. We can also preserve countable additivity by assigning each of them together with their (infinite) disjunction a probability of zero: but then of course we lose normalizability.

Formally similar problems arise when we try to understand and generalize to the infinite case, our homely notions of length, area, and volume. Probability theory is a branch of measure theory, which grew out of attempts to deal satisfactorily with these geometrical notions. The history of measure theory is the history of attempts to come to an account of measure that deals properly with sets of infinite magnitude and is also intuitively satisfactory.
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As it turns out, no wholly satisfactory account is possible. Thus H. L. Royden:

Ideally, we should like
m
(the measure) to have the following properties: that m is defined for every set of real numbers, that the measure of an interval is its length, that the measure is countably additive, and that it is translation invariant. Unfortunately, as we shall see … it is impossible to construct a set function having all of these properties.
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These problems with probability and infinite magnitudes arise at a more basic level. Suppose we think about logical probability in terms of possible worlds. Clearly, if there are only finitely many possible worlds, there’s no problem: the logical probability of a proposition
A
will be the proportion of
A
worlds; the conditional probability of a proposition
A
on a proposition
B
will be the proportion of
A
worlds among
B
worlds, that is, the quotient of the number of worlds in which both
A
and
B
are true by the number of worlds in which
B
holds. If there are infinitely many possible worlds, however, there will be infinitely many mutually exclusive propositions (for any possible world
W
, for example, there will be the proposition that
W
is actual). And now problems rear their ugly heads. For example, suppose propositions form a countable set; each of the possible worlds is presumably as likely (on no contingent evidence) to be actual as any other; but (given countable additivity) clearly it won’t be possible to assign each proposition of the form
W is actual
the same nonzero probability in such a way that these probabilities sum to 1.
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And of course problems are only exacerbated if there are more than countably many possible worlds. Here as elsewhere infinity presents serious problems. One possibility, obviously, is to follow Leopold Kronecker and a host of finitary mathematicians and stoutly declare that there aren’t any actual infinities. There may be quantities that approach infinity as a limit, but there aren’t and couldn’t be any actually infinite quantities. Given the various paradoxes of infinity (for example, Hilbert’s hotel), this has a certain ring of sense. Whether it is actually true, however, is of course a monumentally contentious question.

With respect to our current topic, the McGrew et al. objection to the FTA: if we don’t reject infinite magnitudes, perhaps the most sensible way to proceed is to give up countable additivity. The velocity of light could fall within each of infinitely many mutually exclusive and jointly exhaustive small intervals; the
probability that it falls within any particular one of these intervals is zero, but of course the probability that it falls within one or another of them is one. This seems to fit well with intuition, or at any rate as well or better than any other proposed solution. For example, suppose space is in fact infinite, and suppose it is divided into infinitely many mutually exclusive and jointly exhaustive cubes one cubic mile in volume. Suppose you know that exactly one of them contains a golden sphere of radius one-half mile. You will of course assign a probability of one to the proposition that one of them contains that golden sphere; but you won’t assign any finite probability to the proposition, with respect to any particular cube, that it contains that sphere. No matter what the odds, you won’t place a bet on the proposition, with respect to any particular cube, that it contains the sphere.
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But surely it is not the right remedy to follow McGrew et al. in accepting infinitary mathematics while refusing to countenance probabilistic arguments where the values of the quantities in question fall into intervals that are unbounded. With respect to the FTA, we can sensibly think of the matter as follows. Let C be the comparison range for the value of some parameter P—the strength of gravity, for example—and let L be the life-permitting range of values for P. The larger the ratio between C and L, the greater the fine-tuning of P. Say that P is fine-tuned to degree d (where d is a positive integer greater than 1) if C/L is greater than or equal to d. As C goes to infinity (given L finite), so does the degree to which P is fine-tuned; add that if C is actually infinite, P is maximally fine-tuned. McGrew et al. and others point out (“the coarse-tuning argument”) that if we follow this course, we’ll have to take any parameter with an infinite comparison range but a finite life-permitting range as maximally fine-tuned—even if the finite life-permitting range is very large; they propose this as a
reductio
of fine-tuning arguments with infinite comparison ranges. But better to reverse the argument: these coarse-tuning arguments are also good arguments, despite our initial distrust; this is just one more area where our intuitions get severely bent when we think about infinite magnitudes.

One of the most interesting responses to the FTA goes as follows: perhaps there are very many, even infinitely many different universes or worlds; the cosmological constants and other parameters take on different values in different worlds, so that very many (perhaps all possible) different sets of such values get exemplified in one world or another. If so, however, it’s likely or inevitable that in some worlds these parameters take on values permitting life, and of course we would find ourselves in such a world. There are several ways to develop this thought. According to the inflationary “multiverse” suggestion, for example, in the very early history of the universe, an enormous number of subuniverses formed, these subuniverses displaying different values for those parameters. Another suggestion (surprisingly similar to the ancient stoics “palingenesia”): there is an eternal cycle of “big bangs,” with subsequent expansion to a certain limit and then subsequent contraction to a “big crunch” at which those cosmological values are arbitrarily reset.
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Under either scenario, it isn’t at all surprising that in one or another of the resulting universes, the values of the cosmological constants are such as to be life-permitting. (Nor is it at all surprising that the universe in which we find ourselves has life-permitting values: how could we find ourselves in any other sort?) But then the FTA fails: if there are all those other universes, it is very likely that at least one of them should be fine-tuned, and of course we could only find ourselves in one that
is
fine-tuned.

What are these alternative universes supposed to be like; what sort of beast are they? First, we aren’t talking about possible worlds in the usual philosopher’s sense. On the most common way of thinking about possible worlds, they are abstract objects—maximal possible
states of affairs, or propositions, perhaps.
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The many universes of the many worlds objection are not, of course, abstract; they are not like propositions or states of affairs; they are concrete objects (or perhaps heaps of concrete objects). They are therefore much more like the possible worlds of David Lewis—spatiotemporally maximal concrete objects (that is, concrete objects that are spatiotemporally related only to themselves and their parts).
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On the big bang/big crunch scenario, the many universes of the many worlds objection, like Lewis worlds, are also related, spatiotemporally, only to themselves and their parts.
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On the inflationary scenarios, the universes may be spatiotemporally related to each other in that they form a branching structure, so that any two such universes share an initial segment. But of course the main thing to see here is that these universes are concrete, not abstract.

Here is one response to this many-worlds objection. True, given many universes displaying different sets of parameters, the probability that
one or another
of them will be fine-tuned, display a life-permitting set of parameters, is high. Perhaps it is as high as the probability that our universe, the one we find ourselves in, is fine-tuned, given theism. But how does that affect the probability that
our
universe,
this particular
universe is fine-tuned? Return to the Old West: I’m playing poker, and every time I deal, I get four aces and a wild card. The third time this happens, Tex jumps up, knocks over the table, draws his sixgun, and accuses me of cheating. My reply: “Waal, shore, Tex, I
know
it’s a leetle
mite suspicious that every time I deal I git four aces and a wild card, but have you considered the following? Possibly there is an infinite succession of universes, so that for any possible distribution of possible poker hands, there is a universe in which that possibility is realized; we just happen to find ourselves in one where someone like me always deals himself only aces and wild cards without ever cheating. So put up that shootin’ arn and set down’n shet yore yap, ya dumb galoot.” Tex probably won’t be satisfied; this multi-game hypothesis, even if true, is irrelevant. No doubt
someone
in one of those enormously many poker games deals himself all the aces and a wild card without cheating; but the probability that
I
(as opposed to someone or other) am honestly dealing in that magnificently self-serving way is very low. (In the same way, it is not probable that
I
will live to be 110 years old, although it is very likely that
someone or other
will.) It is vastly more likely that I am cheating; how can we blame Tex for opening fire? And doesn’t the same go for the many-worlds objection to FTA? The fact, if it is a fact, that there are enormously many universes has no bearing on the probability (on atheism) that
this
universe is fine-tuned for life; that remains very low.

But suppose theism is true, and that there are very many universes: doesn’t that mean that the probability that this universe is fine-tuned for life is small, perhaps as small as its probability on the atheistic many-universe hypothesis? After all, while it makes good sense to suppose God would want there to be life and indeed intelligent life, why think he would be especially interested in there being life in this particular universe?
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I think there is a reasonably good reply here. Ever since the sixteenth century, many believers in God have supposed that the universe is teeming with life; if so, it is unlikely that Earth is the only place where God has created life. If theism is true and there is only one universe, the chances are
that intelligent life is to be found in many places in that universe. But in the same way, if theism is true and there are many universes, the chances are that a significant proportion of those universes contain life. So the sensible thing to think is that if theism is true and there are many universes, the proportion of universes that contain life is fairly high—much higher than the proportion of universes that contain life if the atheistic many-universe hypothesis is true. If so, the probability, with respect to any particular universe that it is fine-tuned, is greater given the theistic many-universe hypothesis than given the atheistic many-universe hypothesis. Hence the objection fails.

I said above that on the many-worlds hypothesis, it is likely that
some world or other
is fine-tuned for life, but no more likely that
this
world is thus fine-tuned: that remains as unlikely as ever. That response seems right in the above Old West poker scenario; but does it also work with those many other universes? Maybe not. As Neil Manson puts it,

the “This Universe” objection helps itself to some non-obvious metaphysical assumptions, the most important of which is that the Universe could have taken different values for its free parameters…. whether the values of its free parameters are among the essential properties of a universe will depend, we think, on what a given multiverse theory says a universe is.
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