Does Fine-Tuning Prove God Exists?

There is a version of the Argument from Design, one of the traditional metaphysical arguments for the existence of God, that has caught some popularity recently. It figures prominently in the work of theistic public intellectuals like Dinesh D'Souza, and it is (allegedly) what enabled some unscrupulous people to take advantage of Anthony Flew in his dotage. It's usually called the "Fine-Tuning Argument," for reasons that will shortly become apparent, and what makes it both salient and insidious in our political scene is that it pidgins the discourse of science, mathematics, and philosophy well enough to appeal to people who fancy themselves intellectuals, and at the same time provides the basic argumentative structure to propaganda on behalf of the peasant revolt against knowledge known as the Intelligent Design movement.

The Fine-Tuning Argument goes something like this: The laws of nature areGod: Still not proven specified not only in terms of variables like force, mass, charge, spin, color, flavor, etc., but also in terms of fundamental constants, real number values that are (at least presently) irreducible from physics. These constants are measured to an extraordinary degree of precision: the fine structure constant is 7.297352570(5) x 10^-3; Planck's constant is 6.62606893(33) x 10^-34 J⋅s; and so on (the numbers in parentheses are uncertainties of the last digits). Suppose each constant is set or tuned by some vast cosmic dial. If any dial were turned just a little bit --- and "a little bit" here means by magnitudes far smaller than anything human beings can consciously comprehend --- the formation of the universe would have been radically different from what it turned out to be, and in particular, there would have been no life in the universe.

Here's where the proponent of fine-tuning comes in. (The term itself, obviously, suggests an anthropomorphism.) Only a tiny range of values for the fundamental physical constants, a range smaller than any human imagination can conceive, permits the existence of life in the universe. Yet there is life in the universe --- look around. With that background established, the proponent of fine-tuning can now deliver her decisive blow: "Sure, maybe the fundamental constants just randomly all happened to settle on values conducive to life rather than the vastly larger range of values that would not have supported life. But isn't it infinitely more probable, given the apparent fine-tuning of the universe and the vanishingly small probability of the universe randomly fine-tuning itself, that some intelligence deliberately fine-tuned the physical constants so that they would support life? From a purely rational perspective, therefore, doesn't the fine-tuning of the universe warrant belief in a Fine-Tuner?"

At the extremes of the debate, this argument doesn't tend to move many people. Theists are already in the position the fine-tuning argument wants to take them to. Atheists, on the other hand, are far more likely to think there's something fishy about the argument than to be persuaded, but are seldom in a position to say just what's wrong with it. However, in the broad ecumenical center where those who "just know there's something out there" reside, an argument like fine-tuning that doesn't explicitly contradict evolutionary theory (indeed, it's a means by which religious believers can be Darwinists) and instead maintains the trappings of scientifically-informed discourse has great potential to shore up people's faith. It also --- and this is not the intent of all its proponents --- shores up the reasoning that supports Intelligent Design theory. It's a truly ingenious little argument.

But in addition to being ingenious, it's a bad argument. There are at least three fatal objections to it, which recur in one way or another in debates over Intelligent Design --- hence understanding them is a key to understanding how ID proponents mislead their audiences. The first objection undermines Fine-Tuning on its own premises, so I'll dwell on it a little more than the others (bear with me). In order:

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1) The Fine-Tuning argument gets off the ground by convincing us that randomly selecting life-conducive values of the physical constants is improbable to the point of impossibility. For that claim to be persuasive, the probability of selecting a value of the physical constants has to be calculated in the ordinary way we do probability calculations, i.e., by dividing the number of desired outcomes of an event by the total number of outcomes. (For example, the probability of rolling a 2 on a fair die is 1/6, the one 2 divided by the six possible outcomes of the roll; the probability of rolling an even number is 1/2, the three even numbers divided by the same six possible outcomes.)

Let's imagine some arbitrary fundamental physical constant k, and suppose that if k is between 1 and 3 (inclusive), there will be life in the universe, and if k is anything else, there won't be life. Now if k could only take an integer as a value, the probability of randomly selecting a life-conducive value of k would be 3 divided by infinity, which is 0. But k, by hypothesis, can take any real number as a value (at the very least! perhaps it can take complex numbers as values too, which only bolsters my point). How many real numbers are there --- i.e., what is the divisor for the probability calculation if it's the sort of probability calculation the Fine-Tuning argument foists on us? The size of the set of all real numbers (its 'cardinality') is not only infinite, it's a very big infinity, much bigger than the set of all rational numbers, natural numbers, or integers, which are countably infinite; the real numbers, on the other hand are uncountably infinite. To spare the reader a bit of background mathematics (you can read about it here), the cardinality of the real numbers, which is equal to the cardinality of the real number line (also known as 'the continuum'), is 2^ℵ₀, where ℵ₀ is the cardinality of a countable infinity like the natural numbers.

That cardinal number --- 2^ℵ₀ --- is the total number of possible outcomes of a selection of a value for a physical constant like k. Now how many desired outcomes are there, i.e., how many real numbers between 1 and 3 are there including 1 and 3? The answer, again to spare the reader the background mathematics, is given by the Cantor-Bendixson theorem: any closed subset of the real number line has the cardinality of the continuum, i.e. 2^ℵ₀. (Give yourself a vivid demonstration of this point. Pass your arm horizontally across your body. How many points has your hand passed through? Uncountably infinitely many.) With luck, even the reader who abhors math can see where this is going. There are as many numbers between 1 and 3 as there are real numbers. To jibe with intuition, the Fine-Tuning Argument treats the relevant probability calculations as if they involved ordinary finite numbers; unfortunately, infinite numbers don't behave the same way.

There is an escape hatch for the proponent of Fine-Tuning, however. Although there are as many points between 1 and 3 as there are on the real number line, measure theory can settle on unequal measures of subsets of the real line (e.g., the measure of the interval [1,3] is 2). That produces the result the Fine-Tuning proponent was after, namely, a probability of randomly selecting a life-conducive value for k that approaches zero. But there's a catch: this only works if the Fine-Tuning proponent adopts something called the Axiom of Choice. I'll again spare the reader of the details; suffice to say, the Axiom of Choice is ubiquitous across many varied disciplines of mathematics, and it gives the Fine-Tuning Proponent the discrepant measures she requires. Because the cardinalities we're dealing with are uncountable, weaker axioms (such as the Axiom of Countable Choice) won't do the trick, so there is no avoiding the Axiom of Choice if the Fine-Tuning Argument is to keep going from this point.

Now the Axiom of Choice, because of its usefulness, is effectively a default positionThe Banach-Tarski Paradox: If this doesn't look like common sense to you, you don't get to claim Fine-Tuning is common sense for people working in set theory, mathematics, and mathematical logic (indeed, scrutinizing it is doing philosophy of math, which many mathematicians are loath to do). But it has some very, very strange consequences, the most prominent of which is the Banach-Tarski paradox.

Given the Axiom of Choice, Banach-Tarski shows, it is possible to take a solid three-dimensional sphere, break it apart into finitely many pieces, and simply by moving those pieces around, rotating and rearranging them --- i.e. doing nothing like stretching them that would affect their volume or surface area --- it is possible to reassemble them into two spheres of precisely the same proportions as the original. That's fucking weird. (The way this works is that the pieces themselves have a weird property whose existence the Axiom of Choice entails.)

The upshot is threefold. First, no metaphysical argument for the existence of God that relies on the Axiom of Choice has any claim to the mandate of common sense. Second, the existence of God should be independent of the Axiom of Choice. For all we know, the Axiom of Choice is false. If mathematicians suddenly adopted the Axiom of Determinacy (which contradicts Choice), it's hard to see how deists and theists would become atheists as a result. But if Fine-Tuning is the basis of one's belief in God, then one's theism or deism is as fragile as the consensus supporting Choice.

Third, and most importantly, anybody who tries to foist the Fine-Tuning Argument on you self-evidently doesn't understand the mathematics she is relying on to try to persuade you of the existence of God (nor do most people, since it's really, really complicated). So take her claims with a grain of salt.

2) Even if the probability calculations the Fine-Tuning Argument trades on were sensible, the overall structure of the argument makes no sense. It relies on the assumption that the laws of nature themselves are necessary truths, but the values of the physical constants are contingent. And there is simply no reason to think this assumption is true, except for a prior commitment to the existence of God and to proving it by means of Fine-Tuning. In particular, if God can set the physical constants at any value, why couldn't He (or She) establish whatever laws of nature He (or She) wants to?

In classical mechanics, for example, the force of gravity between two objects is given by F=(G⋅m₁⋅m₂)/r², where F is the force, G is the universal gravitation constant (another physical constant), m₁ and m₂ are the masses of the objects, and r is the distance between their centers of gravity. So the force of gravity is inversely proportional to the square of the distance between masses. Well, why shouldn't it be inversely proportional to the distance itself? Why not inversely proportional to the distance cubed? Why not proportional (not inversely) to the distance? Why couldn't an omnipotent being make it that way, for any conceivable combination of physical properties and real number exponents of them? And why, for that matter, should an omnipotent being be constrained in Creation by the fact that only certain physical constants and laws of nature are conducive to life in the actual world? That constraint exists only according to the actual laws of nature themselves --- but in initially designing the laws, an omnipotent being is presumably constrained only by logical consistency (if by anything) and could attach the property of being life-conducive to any set of laws at all. And why, finally, would an omnipotent being be limited to designing a universe by distributing the fundamental physical properties of the actual world; couldn't He or She instead make use of what metaphysicians call "alien properties," properties that are not even describable in the language of actual world physics? Allowing for the possibility of alien properties --- which an omnipotent being would surely have at His or Her disposal --- the array of possible laws of nature that support life, as well as the array of possible laws that don't support life, are literally not even limited by an idealized imagination.

In other words, the inference that the universe is finely-tuned only follows from assuming the laws of nature that actually exist are the only possible laws of nature; but in that case, the universe is necessarily finely-tuned for life, and couldn't be any other way, so there is no need to posit a Fine-Tuner in the first place. Contrarily, if the laws of nature are contingent, there are at least as many possible laws of nature as there are possible worlds (at least ℵ₂ according to the metaphysician David Lewis, which is an even bigger uncountable infinity than the cardinality of all real numbers).

Now whether or not any great cosmic intelligence exists, that intelligence either does or doesn't make the laws of nature conducive to life. If the cosmic intelligence doesn't make laws conducive to life (either because there is no such intelligence or because it won't or can't), then the conduciveness or non-conduciveness to life of any set of possible laws of nature is a property internal to the set of laws, and there are as (uncountably) many possible life-conducive laws as possible non-life-conducive laws. Hence the fact that our world has life-conducive laws is neither unlikely nor has any other special significance. On the other hand, if it takes a cosmic intelligence to make any set of laws life-conducive, then the foundational premise of the Fine-Tuning Argument is false --- i.e., it's false that only a small range of physical constants could produce life in the universe. What's true is that according to the actual laws of nature, other laws of nature won't produce life; but according to (uncountably many) non-actual possible laws, those same possible laws do produce life and the actual laws don't. In that case, our actual physical constants, as well as our actual physical laws, have no special significance, so there is no argument for deism or theism to be drawn from them.

3) The Fine-Tuning Argument is an appeal to a particular kind of probability, namely conditional probability. E.g., perhaps for some event A, the probability of A is quite small, but the probability of A given B is even money. The problem for Fine-Tuning is that it is an instance of Prosecutor's Fallacy, namely an invalid identification of one conditional probability with another, different one. (This point is originally due to Michael Ikeda and William Jefferys, to give due credit).

Here's how the sleight-of-hand works: The probability of a universe fine-tuned for life given atheism is (according to the premises of Fine-Tuning) quite small. But life clearly does exist --- look around and confirm it. So the probability of life arising given any condition at all just isn't that interesting. The question is, what is the probability of atheism given that (ex hypothesi) the universe is fine-tuned for life, an entirely different probability with no inherent relationship to the probability of fine-tuning for life given atheism. And we just have no way of determinately settling the former probability, though the points in (1) and (2) ought to suffice to show that there's nothing particularly unlikely about atheism being true given fine-tuning.

Clearly, something is responsible for the existence of life, since the emergence of life is an event (or rather, set of events) in a causal chain that stretches backward to the origins of the universe. God could be the thing responsible for life, or it could be the deterministic outcome of the initial conditions of the universe, or it could be the indeterministic quantum coin-flips of every moment since the beginning of time. We really don't need to concern ourselves with whether life might not exist if any of those hypotheses is true. Life exists! Unless there is some reason to suspect that atheism is logically inconsistent with the existence of life --- and not only have proponents of Fine-Tuning not provided any such reason, their argument readily concedes that atheism is perfectly consistent, logically and nomologically, with the existence of life --- it's on no worse explanatory footing than theism in describing the origin of a universe finely-tuned for life.

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There's a moral to this story. The Fine-Tuning Argument, like the Argument from Design of which it is an instance, and like the other traditional metaphysical arguments for the existence of God, relies on a series of equivocations and fallacies which are very easy to miss unless you're familiar with them and on the lookout. Thus, the great achievement of such arguments is also their fatal shortcoming: when delivered with enough rhetorical agility, they can appear outwardly persuasive, but their persuasiveness can never go beyond appearances. The positive arguments for Intelligent Design are exclusively (pseudo-)philosophical arguments, never scientific, and they always piggyback on the Fine-Tuning argument. Consequently, they inherit its misleading logic and confusions of discrepant probabilities.

Now, there's always the possibility that the flaws of all the metaphysical arguments for theism and deism that have so far been given don't generalize to all possible metaphysical arguments for theism and deism. But I doubt it. It's been an awfully long time, and, as the Fine-Tuning case shows, "new" metaphysical arguments for theism are never really new, but merely recapitulate the errors of the old arguments in new terminology.

For non-believers, the clear lesson is that any argument from first principles for the existence of God or an Intelligent Designer or a Fine-Tuner or whatever is virtually guaranteed to go off the tracks somewhere, no matter how superficially fluent in philosophical, mathematical, or scientific language the proponent of such an argument might be. If you find yourself being pulled in by such an argument, recall Hume's maxim about miracles. It's a hell of a lot more likely that there is a flawed step in the argument that you can't manage to identify than that your interlocutor has succeeded where all philosophical theists have failed and proven the existence of God --- or even increased the rational warrant for belief in God by an infinitesimal degree.

For believers on the other hand, the lesson is to grow a little backbone and stop hoping that the metaphysical argument to prove your faith is waiting around the next corner. It isn't. If you can't sustain your faith without resort to fallacious arguments, it's time to reconsider your faith. If you can, then stop making fallacious arguments, and stop apologizing for those who do --- it doesn't do your faith any credit. Believe by virtue of the absurd, as Kierkegaard had it, or don't believe. You can, of course, be a religious believer who also believes in contemporary biology and physics; but not because belief in modern science is logically consistent with any particular rational argument for theism. Rather, since there is no valid rational argument for theism, theism can only be adopted irrationally, and so adopting it makes it consistent with anything at all: the laws of physics, evolutionary theory, and outright contradictions.