[This article was originally published in Analysis 57.4, October 1997, pp. 295-304.]

Quantum Cosmology's Implication of Atheism (1997)

Quentin Smith



'In principle, one can predict everything in the universe solely from physical laws. Thus, the long-standing 'first cause' problem intrinsic in cosmology has finally been dispelled.' Fang and Wu, (1986):3)

1. The quantum cosmology developed by Hawking and his collaborators gives an unconditional probability for the existence of our universe that is inconsistent with theism. This was argued in Smith 1994, but William Lane Craig (1997, this issue) responds that the Hartle-Hawking cosmology purports to offer no unconditional probabilities and Ned Markosian (1995) and Alvin Plantinga (1996) respond that even if it does offer such probabilities, these probabilities are consistent with theism. In the following three sections, I argue these responses are unsound.

2. Craig argues that the Hartle-Hawking cosmology does not purport to explain the initial conditions of our universe and purports to give only conditional probabilities, such as the probability that some 3-space exists given the initial conditions. A 3-space is 'the universe at one time', i.e., a three dimensional spacelike slice of a four dimensional spacetime. The universe's initial conditions are its physical boundary conditions, the curvature of spacetime and the amount and distribution of matter belonging to the universe's initial state. Craig writes: 'The question of why these initial conditions exist or any space at all exists is not a question addressed by the H-H model.' (1997:294) According to Craig, the explanation of these boundary conditions is not a task for physics, but of religion or theology.

Craig's interpretation of the H-H model is directly contradicted by Hawking. Hawking writes that:


many people [such as Craig!] would claim that the boundary conditions are not part of physics but belong to metaphysics or religion. They would claim that nature had complete freedom to start the universe off any way it wanted. That may be so, but it could also have made it evolve in a completely arbitrary and random manner. Yet all the evidence is that it evolves in a regular way according to certain laws. It would therefore seem unreasonable to suppose that there are also laws governing the boundary conditions.(1984b:258)

Craig further claims that 'in quantum cosmology one is not concerned with the initial conditions which play a role in classical cosmology' (1997:292, my emphasis). Pace Craig, a main goal of quantum cosmology is to explain the initial conditions that classical cosmology (the standard hot big bang model or the inflationary model) take as unexplained givens. These are the conditions that give rise to the inflationary expansion and subsequent formation of galaxies, some of which contain intelligent life. Halliwell and Hawking criticize the results of the inflationary model for precisely this reason: 'One cannot regard these results as a completely satisfactory explanation of the origin of structure in the Universe because the inflationary model does not make any [explanatory] assumption about the initial or boundary conditions of the Universe.' (1985:1777)

Quantum cosmology makes an advance over standard hot big bang cosmology and inflationary cosmology in that it predicts that the initial conditions of the universe comprise a ground state (a state of minimum excitation) with small-scale fluctuations that leads to inflationary expansion and galaxy formation. Halliwell and Hawking state:


The boundary conditions imply that these modes [the curvature and matter field modes] start off in the ground state. ... Thus the proposal that the quantum state of the Universe is defined by a path integral over compact four-metrics seems to be able to account for the origin of structure in the Universe: it arises, not from arbitrary initial conditions, but from ground-state fluctuations that have to be present by the Heisenberg uncertainty principle.(1985:1778)

These predicted fluctuations occur at the beginning of our universe, the Planck era, and involve virtual black holes which appear and disappear in about a Planck time, 10^-43 second (Hawking 1982:572).[1]

Craig's associated claim that the H-H model does not give us a probability amplitude for the universe's beginning to exist is also in flagrant contradiction with the model. Hartle and Hawking assert quite explicitly that the wave function gives the probability 'amplitude for the Universe to appear from nothing' (1983:2961).

One reason Craig does not understand the H-H probability amplitude is that he confuses the probability amplitude for a 3-space, such as the initial 3-space of our spacetime, with the probability amplitude for an entire 4 dimensional spacetime. In (1994) I discussed psi[hij, phi], which is the probability amplitude for the initial 3-space. Craig says that 'Smith repeatedly errs in speaking of this 3-space as "the initial state of the universe" and hence of "the probability that the universe will begin to exist with the metric hij and the matter field phi" (1997:293). According to Craig, psi[hij, phi] 'has immediate reference to spacetime as a whole, rather to any 3-dimensional cross-section of it.' (293)

But the error is Craig's, since there is a distinction between two wave functions of the universe, the function psi[hij, phi] and the function psi[guv, phi] Craig needs to pay attention to the fact that guv is the metric of spacetime as a whole and hij is the metric of a given 3-space.

The function psi[hij, phi] is the probability 'amplitude for the occurrence of a given spacetime [as a whole]' (Hartle and Hawking 1983:2962); this function has immediate reference to spacetime as a whole, rather than to any 3-dimensional cross-section of it. See Hawking 1984a: 355-56 for more details.

The function psi[hij, phi] is the probability 'amplitude psi for the given three-geometry hij and the matter field configuration phi' (Hawking 1984a:357). See Hartle and Hawking 1983:2962 for a similar statement. This wave function does have immediate reference to a 3-dimensional cross-section of a spacetime.

The probability amplitude psi[hij, phi] for the initial state of our universe is derived from the probability amplitude psi[guv, phi] for our universe as a whole (Hartle and Hawking 1983:2962 and Hawking 1984a:355-56). Since psi[hij, phi] includes components that determine where and how the initial state fits into our universe as a whole, this function is a probability amplitude for the beginning to exist (the initial state) of our universe. This is why Hartle and Hawking can correctly say that psi[hij, phi] gives the probability 'amplitude for the Universe to appear from nothing' (1983:2961). Accordingly, Craig's claim that the Hartle-Hawking model does not give an unconditional probability for the existence of our universe is doubly flawed; we can derive a probability of this sort from either psi[guv, phi] or psi[hij, phi].

Craig can, perhaps, be forgiven for a further mistake he makes, namely, that Hartle and Hawking believe that the probability amplitude psi[hij, phi] is for 'our 3-space' to evolve from an unexplained given, viz., a zero three-geometry, and therefore that the probability is conditional upon the existence of this zero three-geometry. In the relevant passage, Hartle and Hawking are metaphorically talking of nothingness as a 'zero-three geometry'. They write: 'one can interpret the functional integral over all compact four-geometries bounded by a given three-geometry as giving the [probability] amplitude for that three-geometry to arise from a zero three-geometry, i.e., a single point. In other words the ground state [probability amplitude] is the [probability] amplitude for the Universe to appear from nothing.' (1983:2961). Hartle has since expressed regret for including the misleading metaphorical statement about nothing being a 'zero-three geometry' and has retracted it (1990). Properly speaking, the universe appears from literally nothing, which is only metaphorically a zero three-geometry. Indeed, the wave function implies there is no original or isolated zero three-geometry; any point that exists is a part of some three or four dimensional region of the manifold (see Smith 1997).

But can Craig be forgiven for his next mistake, a fallacy of equivocation on 'condition'? One of Craig's arguments is that the Hartle-Hawking wave function cannot give us unconditional probabilities for the universe's beginning to exist, since there are mathematical boundary conditions for the computation of the wave function of the universe. (For example, one mathematical boundary condition is the axiom that the possible metrics summed over in the path integral are metrics of possible finite universes.) Craig's argument is invalid since it switches between two senses of 'condition'. In (1994), I used 'the unconditional probability that a universe begins to exist with the metric hij and matter field phi' to mean a probability that is not dependent upon the existence of any concrete thing or event. The probability is dependent only on a mathematical 'abstract object', the wave function psi[hij, phi], since the probability value is the square of the modulus of the amplitude, |psi[hij, phi]|^2. This probability is, of course, conditional upon the mathematical boundary conditions that belong to the wave function, but it is a fallacy of equivocation to infer from this that the probability is conditional upon concrete things or events (or upon physical boundary conditions).

3. If quantum cosmology gives us unconditional probabilities for the universe's beginning to exist, why should that bother the theist, apart from Craig's concern about cosmology trespassing on theological territory? It does not bother Plantinga (1996); and Markosian (1995), although an atheist, thinks the theist should not be bothered.

But the theist should be bothered. The Hartle-Hawking wave function psi[hij, phi] provides a high probability that is less than one (we will call it .95) that a universe shall begin to exist with a three dimensional space that has a certain matter field phi and metric hij. If God wills that a Hartle-Hawking universe shall begin to exist, the probability of its beginning to exist is not 95% but 100%, since God's willing is omnipotent. But it cannot be true both that the Hartle-Hawking law obtains, such that the probability that a Hartle-Hawking universe shall begin to exist is 95%, and that God wills that a Hartle-Hawking universe shall begin to exist, such that the probability that a Hartle-Hawking universe shall begin to exist is 100%. As I argued in (1994), the unconditional probability derived from quantum cosmology makes this theory logically incompatible with theism. Since quantum cosmology is confirmed by the observational evidence, this incompatibility should be troubling to anybody who believes theism is a rationally acceptable world-view.

Ned Markosian thinks any incompatibility between the Hartle-Hawking wave function and theism can be resolved by matching up the probability values, so that both are 95%. He described an allegedly possible scenario where the unconditional probability of a Hartle-Hawking universe beginning to exist is 95% and yet that God creates this universe. Markosian asks us to suppose there are 20 possible universes that are tied for best in an intrinsic value-ranking and that 19 of them are Hartle-Hawking-type universes. According to Markosian, since God is omnipotent, God could see to it that, for each of these universes, there is a 5% chance that he will create (on a whimsy) that universe. It follows that there is a 95% probability that a Hartle-Hawking universe will be created by God. As it happens (Markosian's scenario continues), God does will that a Hartle-Hawking universe exist. Markosian thinks this scenario makes theism consistent with Hartle's and Hawking's cosmology.

But it does not, for the wave function of the universe implies that the natural-mathematical properties of the relevant possible universes make it 95% probable that a Hartle-Hawking universe begins to exist uncaused. This probability statement is not consistent with the theistic assertion that there is 0% probability that a Hartle-Hawking universe begins to exist uncaused.

Furthermore, Markosian's scenario implies the 95% probability of a Hartle-Hawking universe obtains only because it is derived from supernatural considerations. According to theism, if a universe is to have any probability of existing, this probability is dependent upon God's beliefs, desires and creative acts. But the Hartle-Hawking probability is not dependent on any supernatural considerations; Hartle and Hawking do not sum over anything supernatural in their path integral derivation of the probability amplitude.

Moreover, even apart from these problems, the counter-argument Markosian presents involves mistaking a conditional probability for an unconditional probability. The Hartle-Hawking probability is 'unconditional' in the sense that it is not conditional upon any concrete event or object that exists. But the 95% 'unconditional probability' that Markosian purports to derive in his theistic scenario is not in fact unconditional. First, it is conditional upon God having decided to create something rather than not create anything. Second, it is conditional upon God deciding to create a universe, rather than just disembodied spirits. Third, it is conditional upon God deciding to create one of the 20 universes that are tied in value-ranking (19 being Hartle-Hawking universes and one not of this type). So there is no sense in which Markosian has given us what he purports to have given us: an unconditional probability of a Hartle-Hawking universe beginning to exist.

In addition, Markosian's theistic scenario is internally inconsistent. He supposes that God leaves it to a 'whimsy' as to which of the 20 universes he will eventually create, thereby making it 95% probable that the universe he will whimsically create is a Hartle-Hawking universe. Markosian supposes there is time before creation, with God first deciding that he will whimsically create one of the twenty universes, and later creating one of them. But God has foreknowledge of what he shall do, and he foreknows which universe he will create. Therefore, there is never a time at which a Hartle-Hawking universe has a 95% probability of existing; God always knows or foreknows that the universe he creates is a Hartle-Hawking universe, and thus there is always a 100% probability that a universe of this type will exist. So the probability values do not even match up.

As if this were not enough, Markosian is supposing there is time before the universe. But the Hartle-Hawking wave function gives us a probability of 95% that time begins to exist. Time begins to exist with the universe, as the temporal dimension of spacetime. The wave function of the universe is an abstract objection that timelessly exists and it timelessly entails a 95% metric hij in the probability amplitude psi[hij, phi]. But Markosian's scenario that God creates a matter field phi that comes into existence at a certain time in a pre-existent time. Discussing this cosmological scenario in effect amounts to changing the subject; we are no longer talking about the Hartle-Hawking theory that matter and spacetime begin to exist.

I conclude that Markovian's arguments fail to justify the belief that the Hartle-Hawking wave function of the universe is logically consistent with theism. But there is another argument by Alvin Plantinga that these two theories are consistent.

4. Plantinga claims that the relevant unconditional probability is:


the proportion of possible worlds in which the universe has the characteristics [specified by the H-H wave function]. (Of course the figure of proportions of possible worlds here is just that -- a figure; we have no reason to think possible worlds occupy something like a space, and no reason to think that there are at most continuum many possible worlds.) So the absolute probability of there being such a universe is, say, .95. But according to theism, God's existence is a necessary truth; so the probability that there be such a universe on the existence of God is the same as its probability on any necessary truth, which is just its absolute probability. So where's the inconsistency? Of course the probability that there is such a world, given that God wills that there be such a world, is 1. But that's not an absolute probability, but a probability conditional on the (contingent) truth that God wills there be such a world.(1996)

Craig (1997:292, fn.2) believes Plantinga's argument for the consistency of quantum cosmology and theism is logically valid. However, I believe there are at least three reasons to think Plantinga's argument is invalid.

First, the argument that theism and quantum cosmology are consistent is invalid according to the principles of relevance logic. Let p be the complex proposition that states the Hartle-Hawking theory. For any conjunction of the Hartle-Hawking proposition p with any necessary truth q, p by itself will entail (in the sense of relevance logic) the statement r of the probability value. The proposition r is:


(r) The probability that a universe begins to exist with the matter field phi and metric hij is .95.

But if theism is true, p does not relevantly entail r. It must be a theistic proposition q1 that relevantly entails r, since the probability of a universe existing solely based on natural-mathematical truths and without divine causation is zero. Thus, quantum cosmology and theism will differ as regards to which conjunct in the conjunctive proposition, p and q1, relevantly entails r, which prevents the two theories from being consistent according to the relevance logic.

The second problem is that there is no candidate for the theistic necessary truth q1. Since the theist cannot allow that p, in the conjunction p and q1, relevantly entails r, the theist must find some necessary truth of theism that entails r. Nor does the theistic necessary truth any universes that exists is created by God. We could introduce a theistic proposition of the sort Markosian mentions, but (even apart from the problems I have already noted about Markosian's propositions) Markosian-type propositions have their truth values contingently.

Third, there is an inconsistency even according to standard propositional logic between theism and quantum cosmology. I have been using 'conditional probability' to mean a probability that is dependent on the existence of some concrete things or events. Let us know use 'conditional probability' to refer instead to any probability of the form c(h/e & k), where c is the probability value, h a contingent hypothesis, e a contingent evidence statement, and k the 'background knowledge' of necessary truths. An 'unconditional probability' now refers to probabilities of the form c(h/k), which can be abbreviated as c(h) to highlight their unconditional nature (they are not conditional on any contingent proposition). Let us assign the following values to these letters:

h = there is some Hartle-Hawking universe.
e = there obtains the wave function of the universe psi[hij, phi].
k = elm trees are trees, and ..., etc. (the conjunction of all necessary truths).

The proposition c(h/e & k) = .95 is true if Hawking's quantum cosmology is true and it is no part of Plantinga's argument to argue this cosmology is false. But if classical theism is true, k will include some truths that are incompatible with c(h/e & k) = .95, since it is a necessary truth of classical theism that for any possible universe U, the conditional probability that U exists is zero unless the conditions include some positive, contingent truths about divine states or acts. A positive, contingent truth about divine states or acts is any truth of the form, God exists and contigently is in the state S or performs the act A. If theism is true, c(h/e & k) = 0, since e includes no positive, contingent truths about divine states or acts. Thus if quantum cosmology and theism are both true, it follows both that c(h/e & k) = .95 and that it is not the case that c(h/e & k) = .95. Thus, we need not rely on relevance logic to show that quantum cosmology and theism logically inconsistent.

I conclude that Craig, Markosian and Plantinga have given no satisfactory answers to Hawking's famous question, namely, if quantum cosmology is true, 'what place, then, for a creator?' (1988:141).[2]

Western Michigan University
Kalamazoo, MI 49008


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[1] The wave function of the universe predicts that this initial state is topologically connected to both the present day expanding universe and to a small, closed, four dimensional space, a hypersphere. This hypersphere is a Euclidean spacetime with a positive definite metric (++++), which means in effect that it has four spatial dimensions but no temporal dimension. Craig (1997:292) asserts that the H-H probability is conditional upon this hypersphere, and that this hypersphere is taken by Hartle and Hawking as an unexplained given. Craig's assertion is false, since the H-H wave function predicts the existence of this hypersphere. See Smith 1997, which supersedes my interpretation in Craig and Smith 1993, Chapter XII; also see Deltete and Guy 1996 for a good critical discussion of my interpretation in Craig and Smith 1993, Chapter XII.

The wave function also predicts that several universes, in addition to our universe are topologically connected to this hypersphere. For the sake of simplifying the discussion of probabilities, I refer only to our universe in the present paper and in (1994).

It is worth noting that a probabilistic explanation of the existence of our universe is possible even on cosmologies where spacetime is not quantized (but where the matter fields are quantized). See Smith 1995 for details.

[2] A longer and different version of this paper was read at Notre Dame University on December 1, 1995. I am grateful to the philosophy and science departments at Notre Dame for a plethora of stimulating criticisms. Whether I emerged from that debate alive is still a matter of contention. I should also like to thank Arthur Falk for his extensive and insightful criticisms of Markosian's article (1995), which he has refuted (in my opinion) by original arguments that are logically independent of the arguments in the present paper.

Research for this paper was supported by an American Council of Learned Societies Fellowship for 1996.