Time Began With a Timeless Point

Published in: Gegansall and Woodruff (eds.), God and Time. New York. Oxford University Press, 2001..

Why does time exist? In the context of the “spacetime theories” of the special or general theory of relativity, this question should be more appropriately phrased as “Why does spacetime exist?” I will narrow the question further and adopt the results of contemporary general relativistic cosmology, namely, that spacetime began to exist about fifteen billion years ago. Accordingly, my question will be “Why did spacetime begin to exist?”

There are two familiar, contemporary responses to this question. The theist says that the question has an answer and that this answer is that God caused spacetime to begin to exist. The standard response of the atheist is to say that there is no answer to this question; spacetime’s beginning to exist is a brute fact or has no explanation. This standard atheist response seems to give theism a prima facie theoretical superiority to atheism; theists offer a detailed explanatory hypothesis about why spacetime begins to exists, and standard atheists are content to leave spacetime’s beginning to exist unexplained.

I reject standard or traditional atheism and side with theism on this issue. A theory that includes an explanatory hypothesis about some observational evidence e, such as spacetime’s beginning to exist, is ceteris paribus epistemically preferable to any theory of the observational evidence e that does not include such an explanatory hypothesis. No atheist has ever provided a proof that the existence of spacetime is a brute fact and, consequently, standard atheism remains, in this respect, an unjustified hypothesis.

My agreement with theism runs deeper: I agree that there is a cause of spacetime’s beginning to exist. Further, I agree with many theists that a simple being caused space- time, where “simple” means here “has no parts.” (Note that “simplicity” is used in many different senses and that I later use it in a different sense, a sense where “simplicity” expresses a property of hypotheses, not of concrete particulars.) I also agree with some theists, such as Brian Leftow, that the cause of spacetime exists timelessly.[1] And I agree with theists that the cause of spacetime is essentially uncaused and exists a se (i.e., is not dependent upon any concrete object). And I agree with Plantinga and most other contemporary analytic theists that the cause of spacetime has a metaphysically necessary existence.[2] I also share the view of these theists that this metaphysically necessary being contingently causes spacetime to exist (such that this necessary being exists in some merely possible worlds where it does not cause spacetime). I further agree with theists that the act of causation is a case of singularist causation that relates a necessary being to a contingent being, spacetime. Moreover, I agree with theists that the cause of spacetime is transcendent, at least in the sense explicated in this passage from Plantinga: “Perhaps we can also give an explanation of what it is for a being to be transcendent: such a being transcends the created universe; and a being transcends the created universe if it is not identical with any being in that universe (if it is not created) and if it depends on nothing at all for its existence.”[3] Note that a being is transcendent, by this definition, only if it transcends the created universe, such that a being can possess the property of being transcendent only in the possible worlds where there is a created universe. Thus, the property of being transcendent is possessed contingently by the necessary being that causes spacetime, but it can be possessed by a being that does not cause spacetime, such as an abstract object, assuming abstract objects do not depend on God for their existence.

However, I argue that the timeless, uncaused, simple, independent, necessary and transcendent being that causes spacetime’s beginning to exist is not God but a spatially zero-dimensional point. This point contingently has the property of being the big bang singularity from which our maximal spacetime (“the Friedmann universe”) exploded and began to expand. This point is also the big bang singularity postulated by quantum gravity cosmologies, as I shall show below. (The idea that quantum gravity cosmology eliminates the big bang singularity is a popular myth, as we shall see.) Since the point is posited by both classical general relativistic cosmology and by the new quantum gravity cosmology, there are two independent empirical avenues to its existence. General relativistic cosmology and quantum cosmology cannot both be true, but since the point’s existence is posited by both cosmologies, the hypothesis that the point exists has empirical warrant regardless of which cosmology is true.

The timeless point that, in the actual world, has the property of being the big bang singularity is “timeless” in the sense that it exists outside of the time of general relativity and quantum gravity cosmology. In this essay, I use “time” to mean the time postulated by general relativity and quantum gravity cosmology and “timeless” to mean existing outside of the time postulated by these theories.[4] Note that this allows many theists to agree with my thesis that “God is timeless,” for they could agree that “God exists in time” in some sense, even though God does not exist in the temporal dimension of the four-dimensional spacetime of general relativity or quantum cosmology.

My arguments and theses in this essay depart from my previous discussions of big bang cosmology, for there I treated the big bang singularity as a temporal being, as metaphysically contingent, as dependent upon spacetime (and thus lacking aseity), as contingently uncaused, and as immanent (rather than transcendent) in the sense that I identified it with the initial temporal boundary of spacetime that exists for an instant at t = 0.[5] I asserted that since the singularity is uncaused, spacetime’s beginning to exist is uncaused. One crucial difference is that I here treat the singularity as a timeless point rather than as a spatiotemporal point that exists at the first instant t = 0. This way of treating the singularity is more physically justified than the spatiotemporal treatment, since a spacetime point in general relativity requires a metric and (as all agree) the singularity has no metric (the metric is “undefined on the singularity”). Time can not be extended backward to the singularity at a hypothetical first instant t =0; rather, there is no first instant t = 0 and each interval is half-open in the earlier direction.[6] Second, I here treat the singular point as a metaphysically necessary and independent being, rather than as a metaphysically contingent being that depends for its existence on the existence of spacetime. This suggests that my present arguments against theism are not merely “negative” atheistic arguments (e.g., “the existence of spacetime is not caused by God”), but rather are positive arguments that God has a “competitor,” so to speak, that shares many of God’s properties (being metaphysically necessary, being essentially uncaused, being timeless, simple, existing a Se, causing spacetime to begin to exist, being transcendent, being the uncaused cause of all contingent beings and thus as being “the ultimate ground of being”). My positive argument is that this cornpetitor—the timeless point—fares better than God in a theory of the most likely causal explanation of why spacetime begins to exist.[7]

This point is a concrete object if only for the reason that it has causal powers; unlike John Leslie, I follow the received view that “x has causal powers” entails “x is concrete.” The concreteness of this point makes the following conditional true: if this essentially uncaused and metaphysically necessary point is the being that causes spacetime, then there does not exist the deity of perfect being theology (a metaphysically necessary, omnipotent, omniscient, disembodied, free, and perfectly good person who causes every concrete being other than himself in each possible world in which there are concrete beings other than himself).

These clarifications of my notions of the point and God enable me to answer a pertinent question raised by David Woodruff (private communication) about whether God could cause the point to exist and also cause the point to have the properties that would result in there being a spacetime in which intelligent life develops. I would say that perhaps it is formally logically possible that there is some other point that is such a God-caused point, but that the different point I am talking about has the essential property of being uncaused. Woodruff also raises the question: is there is a possible world in which there is no spacetime and in which both God and the timeless point exist? My definition of perfect being theology implies the answer is negative: the concrete, timeless point is essentially uncaused, and God causes every concrete being other than himself in each possible world in which he exists.

I shall make the case that the proposition the above-described point is the cause of spacetime‘s beginning has a significantly higher probability of being true than the proposition the god of perfect being theology is the cause of spacetime‘s beginning. From the viewpoint of metaphysics or philosophical cosmology, the negative conclusion that the god of perfect being theology does not exist, which merely tells us what does not exist, is less important than the positive thesis that spacetime’s beginning has a cause, namely, a point that has the various properties I have mentioned. If we wish to call this atheistic theory a type of atheism that differs from “traditional atheism,” we may call it “explanatory atheism.”

Since my argument is probabilistic, I need to assess the comparative prior and postenor probabilities of the two relevant hypotheses, namely,

(h₁) There exists a metaphysically necessary, essentially uncaused, timeless, and independent (“a se”) point that, if spacetime begins to exist, is the transcendent cause of spacetime’s beginning to exist.

(h₂) There exists the god of perfect being theology who, if spacetime begins to exist, is the transcendent cause of spacetime’s beginning to exist.

It is a familiar idea that it can be more or less epistemically probable whether or not a hypothesis is metaphysically necessary, even though in other senses of “probability” a metaphysically necessary hypothesis h by definition has the probability one, p(h) = 1.[8] In this essay I use “probability” only to mean epistemic probability, which I shall identify with a type of personalist probability.

I first examine the issue of whether h₁ or h₂ has greater prior probability; in the later parts of this essay, I determine which has greater posterior probability conditional upon the observational evidence e about spacetime’s beginning to exist. This entails that “my argument that the point hypothesis is more probable than the theistic hypothesis” requires a restricted interpretation. I am not arguing that the point hypothesis is more probable than the theistic hypothesis conditional on all the relevant observational evidence, but only on a part of this evidence, the evidence e about spacetime’s beginning to exist. Roughly speaking, I am arguing that given contemporary cosmogonies (confirmed theories of the beginning of spacetime), the probability of theism is lower than the probability of the point hypothesis. Despite this limited scope, my argument (if successful) nonetheless refutes perfect being theology, for this theology implies that there is no atheistic hypothesis conditional upon the observational evidence e of spacetime’s beginning to exist that is more probable than the theistic hypothesis conditional upon the observational evidence e of spacetime’s beginning to exist. Even more strongly, the observational evidence e turns out (as we shall see) to falsify a prediction implied by the theistic hypothesis (in conjunction with the auxiliary proposition that spacetime begins) and thereby falsifies theism.

There is a limitation upon the domain of my argument. The theistic hypothesis and the point hypothesis are not the only two hypotheses about possible causes or explanations of spacetime, and consequently my conclusion in this essay is limited to showing that the point hypothesis (h is significantly more probable than the theistic hypothesis (h (regardless of how probable the point hypothesis is relative to other explanatory hypotheses, be they religious hypotheses about Brahman, Tao, Hegel’s Absolute Spirit, the Godhead of God postulated by Meister Eckhardt, or various atheistic hypotheses).[9]

Some may say that first in the order of things is to ask if the hypotheses of a metaphysically necessary point and a metaphysically necessary God have even a possibility of being true. If we adopt the plausible assumption of theists such as Plantinga, Craig, and Leftow that S5 is the system for metaphysical modalities, then by the “possibility of being true” we cannot mean metaphysical possibility, for at most only one of these hypotheses can be metaphysically possible. If the point hypothesis is meta physically possible, there is some metaphysically possible world Win which this hypothesis is actually true. But if the point hypothesis (which implies the point exists of metaphysical necessity) is actually true in W, then the point exists in every metaphysically possible world, and thus is metaphysically necessary. In this case, the theistic hypothesis would be metaphysically impossible (for reasons given in the previous section).

How shall we decide which of these two hypotheses is metaphysically possible (if one of them is metaphysically possible)? The standard or widely accepted procedure to decide such issues has been to rely on one’s so-called modal intuitions. But Hin tikka has correctly noted that in the absence of an epistemological theory of modal “intuitions,” appeals to modal “intuitions” have an unknown epistemic value, if any, and that until and unless such an epistemological theory is developed, these so-called intuitions should be called “hunches.” If we treat these individually variable hunches about the metaphysical necessity or impossibility of theism as having epistemic weight by virtue of the fact that the hunches[10] seem self-justifying to the person who has the hunch, then what Paul Moser calls “substantive relativism about justification” would apply to these hunches: “the view that whatever one takes to be justified is actually justified . . . [which entails] an ‘anything goes’ attitude toward justification and evaluative assessment.”[11] I think substantive relativism about justification is self- referentially incoherent: one could take to be justified that whatever one takes to be justified is not actually justified. It would then follow, given substantive relativism about justification, that whatever one takes to be justified is both actually justified and not actually justified.

We can avoid a mutually question-begging debate, an epistemically valueless trading of hunches, and a self-contradictory substantive relativism about justification if we interpret “possible” as meaning formal logical possibility, which is governed by S5. If the point hypothesis is true in some formally logically possible world W then it is actually true in some formally logically possible world W1. This means it is a for mal logical possibility that the point exists of metaphysical necessity, which the theist can accept without giving up at the outset her belief that God exists of metaphysical necessity. The theist can consistently conclude, if the arguments go in her favor, that the formally logically possible world W1 in which the point is metaphysically necessary is a metaphysically impossible world. (Metaphysically possible worlds are a proper subclass of formally logically possible worlds.) Likewise, the explanatory atheist can conclude, if the arguments go in her favor, that God exists of metaphysical necessity in some formally logically possible worlds that are not metaphysically possible. Thus the theist and atheist can begin with the same premise: there is at least one formally logically possible world W2 in which God, but not the point, is metaphysically necessary and, second, there is at least one formally logically possible world W1 in which the point, but not God, is metaphysically necessary. What they both are trying to find out by means of logical argumentation, and without mutually question-begging hunches, is whether W1 or W2 is metaphysically possible.

Formal logical possibility includes propositional logic and predicate logic with identity. It includes not only first order predicate logic with identity, but also second order predicate logic with identity, third order predicate logic with identity, and predicate logic of order omega.[12] Mathematics is definable in terms of set theory, and set theory is definable in terms of predicate logic of order omega; thus mathematical and set-theoretical truths are formally logically necessary truths.

It may be objected that we have all learned on our mother’s knee that first order predicate logic with identity is about “narrowly logical necessities,” that mathematics and set theory are about “broadly logical necessities,” and that “broadly logically necessary” means metaphysically necessary.[13] I grant that we learned that this is Plantinga‘s philosophy of logic, but I would note the following theses. (a) Mathematics is definable in terms of set theory. (b) If first order predicate logic with identity is narrow logic, then, by strict analogy, second and higher order predicate logics with identity are narrow logics (Plantinga offers no reason to deny this inference; in fact, he does not discuss the subject). (c) One of the two axioms of second order predicate logic with identity is the axiom of extensionality,[14] which is also a basic axiom of set theory. (d) This and other axioms of higher order logics enable set theory to be defined in terms of predicate logic of order omega. (e) Since mathematics is definable in terms of set theory, and the latter in terms of predicate logic of order omega, mathematics and set theory are not about metaphysical necessities but about transfinite predicate logic necessities, i.e., narrowly or formally logical necessities. For this reason, I will include set theory, and thus mathematics, in the formally logically possible worlds in which the theistic and point hypotheses are possibly true. (I will use “formal logical possibility” since “narrow logical possibility” is Plantinga’s technical phrase and my formally logically necessary/metaphysically necessary distinction is different from the distinction marked by his widely adopted uses of the phrases “narrowly logically necessary” and “broadly logically necessary,” which seems to me to represent an arbitrary, or rather, mistaken, distinction.)

What is the probability that the theistic and point hypotheses are not merely logically possible, but also metaphysically possible (and thus necessary)? The probability we are talking about is an epistemic probability, not a statistical or “actual frequency” probability, for by definition the statistical or “actual frequency” probability of the theistic or point hypothesis is zero or one (assuming the dubious thesis that the notion of statistical probability can coherently be applied to these hypotheses). Henceforth, by “probability” I mean epistemic probability, which I identify with personalist probability,

The point hypothesis (h₁) cannot be known to be probably metaphysically possible (and thus probably metaphysically actual and necessary) if it has a zero degree of prior epistemic probability. The hypothesis h₁ (or equivalently, the hypothesis that h₁ is metaphysically possible) has a zero degree of posterior probability if its prior probability is zero. If the point hypothesis’s prior probability is zero, then the point hypothesis’s posterior probability is zero, regardless of how much evidence there is for general relativistic or quantum cosmology. This follows from Bayes’s theorem: p(h) is the prior probability of a hypothesis h, and p(h/e) is its posterior probability, its probability given the evidence e. If p(h) 0, then p(h/e) 0. In Bayes’s theorem, p(h/e)=p(h) x p(e/h), divided by p(e). If p(h) = 0, then the numerator of the equation is zero. In other words, if p(h) =0 and zero is multiplied by the probability n of e/h, we get zero since Ox n 0, for any number n. If we divide this by the probability n of e, we get 0/n, which is 0, since zero divided by any number n equals 0. This shows that it is essential to our argument to show that the prior probability of our timeless point hypothesis is a positive, non-zero number (and the same for the theistic hypothesis [h₂]).

If the hypothesis of a metaphysically necessary point or deity has a formally logically possible truth but a prior probability that is a positive, non-zero infinitesimal number (in the sense of Abraham Robinson’s theory of non-standard real numbers)[15] or a negligible standard real number, the posterior probability will remain infinitesimal or negligible regardless of how much evidence e there is for the hypothesis. How can we justifiably obtain non-negligible prior probabilities?

The solution to the problem involves, in part, adopting Shimony’s “tempered condition” for prior personalist probabilities.[16] This condition implies that the prior probability p(h) be sufficiently high to allow the possibility that h be preferred to all rival “seriously proposed hypotheses” as the result of the envisaged observations relevant to formulating the observation statement e. I think (contra Shimony) that this tempered condition can be more than a stipulated axiom required for the theory of person alist probability to be a useful theory. I think it can be justified (at least in our explana tory project) by defining the set S of “serious hypotheses” as the small, finite set of hypotheses (i) that have a very high degree of nonarbitrariness (in the sense defined in a later part of this essay on non-arbitrariness) and that (ii) exclude the hypotheses that are less parsimonious than a parsimonious hypothesis of the first degree.[17] Generalizing upon Schlesinger’s mathematical characterization of degrees of parsimony, we can say that any hypothesis that is parsimonious to the second degree or higher is formulated by virtue of adding explanatorily idle conditions to a parsimonious hypothesis of the first degree. A hypothesis is parsimonious to the first degree if and only if it includes the least number of explanatory conditions required to explain the data e. Thus we exclude from our set S of hypotheses about spacetime’s beginning (a) arbitrary hypotheses such as that 8,617 disembodied finite minds jointly formed a total cause of spacetime’s existence and (b) nonparsimonious hypotheses such as that a three-centimeter line (rather than a point) caused spacetime, which is a hypothesis obtained by adding the explanatorily useless condition to the point hypothesis that this point plus the additional points on a three-centimeter line formed the total cause of spacetime. With these two conditions, nonarbitrariness and first degree parsimonious, determining our set S of (formally logically) possible explanatory hypotheses, we can ensure that the point hypothesis and the theistic hypothesis each have a prior probability that is both non-negligible and is sufficiently high to allow the possibility that the hypothesis be preferred (once the evidence is obtained) to all rival hypotheses in the set S. (In partial analogy to Parfit’s theory,[18] we can include the standard atheistic theory that spacetime has no explanation in our set S of possible explanatory hypotheses. We may call it the “null explanatory hypothesis,” reminding us that “for no reason” counts as an answer to “Why did spacetime begin to exist?”) Accordingly, we can say that the set S includes only hypotheses that are parsimonious and non-arbitrary possible answers to the question “Why did spacetime begin to exist?”

The same considerations apply to the probability of the evidence e. In Bayes’s theorem, e is more fully expressed as e/k, that is, the evidence e conditional upon “background knowledge” k. In our case, k is the proper class of logical truths. However, Bayes’s theorem implies that both theism and explanatory atheism are false if p(e/k) = 0, for in this case the probability of theism (or explanatory atheism) given the evidence e and the background logical truths k would equal a mathematically undefined expression, namely n/0, where n is the numerator and zero the denominator, This is an unacceptable result, since it is mathematically meaningless to divide by zero. I think the solution of the problem is the application of the criteria of nonarbitrariness and first degree parsimony to give us a finite class of possible evidence propositions e’. As we will see later, one of the two most non-arbitrary ways for spacetime to begin to exist is a maximum (maximal order) or a minimum (minimum order), which gives the evidence e I shall discuss later in this essay a significant prior probability.

By “probability” I mean the degree of belief of a logically possible, perfectly rational finite mind. If a hypothesis is 80 percent probable, that means a possible, perfectly rational, finite mind would believe the hypothesis to the degree 80 percent if that mind understood that hypothesis and all the relevant considerations that would be considered by a mind that is perfectly rational and finite. Just as scientists constructing theories are attempting to “approximate the truth,” so humans are implicitly attempting to approximate the judgments of a (logically possible) perfectly rational, finite mind when we are more or less confident in some hypothesis, i.e., when we believe it to some degree.[19]

Personalist probability (with which I have identified epistemic probability) is a kind of objective, mind-independent probability, since it is defined counterfactually as what a perfectly rational finite mind would believe to a certain degree, if there were such a mind and belief. This does not require the factual existence of such a mind (this is the sense in which it is mind-independent). The truthmakers of the relevant counterfactuals are possible worlds in which the perfectly rational, finite mind and its beliefs exist. This is how I shall understand personalist probability. Clearly, philosophers of a nominalist, physicalist, extensionalist, and verificationist bent would prefer to say that the expression “degrees of belief’ has meaning (semantic content) only insofar as it refers to belief-tokens of mental organisms in the actual world. Since I have argued elsewhere that nominalism, physicalism, extensionalism, and verificationism are false, it seems to me there are grounds for rejecting these theories.[20]

The Background Truths upon Which the Prior Probabilities of Theism and Explanatory Atheism Are Conditional

Since the theistic and timeless point hypotheses involve explaining why spacetime exists, the “prior probabilities” of these hypotheses are a priori probabilities; they are probabilities determined prior to taking into account any a posteriori evidence that consists of the existence of spacetime or anything else that exists and is knowable only posteriori. The prior probability p(h) is definable in terms of a conditional probability p(h/k), where the background truths k relative to which the probability p(h) is conditional are a proper class (not a set)[21] of a priori truths, the truths comprising formal logic, as I explained previously. The fact that some mathematical truths (truths of predicate logic Of order omega), say, can be learned by actually existent intelligent organisms in an a posteriori manner is consistent with saying that such truths are known a priori by a perfectly rational finite mind in some possible world.[22] For our purpose of determining a priori personalist probabilities, we are only interested in what is known a priori by such a mind. If we say that “p(h) is an unconditional probability” this shall mean that it is not conditional upon any merely a posteriori evidence e or upon any thing else but formal logic; thus, p(h) =p(h/k).

The class k of truths of formal logic does not include all a priori truths. Some a priori necessary truths, such as l am here now[23] and If something is red, it is colored,[24] are not truths of formal logic. Although many metaphysically necessary truths are know able only a posteriori, some are knowable a priori, and the examples about indexicals and colors are examples of a priori metaphysical necessities.[25]

The most important implication of my definition of the background class of truths k is that if something is a metaphysical necessity, but is not a truth of formal logic, it does not belong to k. The exclusion of merely metaphysical necessities from k enables us to avoid begging the question in favor of explanatory atheism or perfect being theology. I am not denying that some metaphysical necessities, such as the a priori truth If anything is red all over at t, it is not green all over at t, are self-evident to all normal humans. The problem is that most of the alleged metaphysical necessities that pertain to our explanatory project, such as necessarily, no disembodied mind can exist or necessarily, no point can exist except as a part of a spacetime manifold, evoke different “hunches” in different people, and there is no epistemological theory (that has yet been formulated) that enables us to decide which of these “hunches” has epistemic rather than merely psychological significance.[26]

When Plantinga says that the atheists’ and theists’ probability judgments are individually relative to different background beliefs, specifically to classes k₁ and k₂ of rationally acceptable metaphysical theses, this statement is premised upon his inclusion of (alleged) metaphysical necessities in his background classes and his use of individually relative “metaphysical intuitions” as the method of access to these alleged necessities.[27] We avoid this substantive relativism about justification by using only formal logical truths in k. By the time of his three books on warrant, Plantinga’s epistemological position seems closer to the methodology of the present essay, and he now seems to reject substantive relativism about justification or warrant.[28] His new non-relativistic position enables him to say that it is warranted to be a theist and not warranted to be an atheist, rather than merely that theism is warranted for a theist and atheism is warranted for an atheist.

Whether or not the alleged metaphysical necessities h and h the point hypothesis and the theistic hypothesis, are metaphysical necessities is not determined by one’s “hunches,” but is determined by (1) assessing their comparative prior probabilities conditional upon the background class k of formal logical truths and (2) assessing their comparative posterior probabilities conditional upon the observational evidence e. Since the criteria used in these assessments belong to inductive logic, the assessment is logical, not an appeal to “hunches.”

The Criteria for Assessing the Prior Probabilities of Theism and Explanatory Atheism

Apart from delimiting the background knowledge k, there is another issue that needs to be addressed. What criteria does a perfectly rational, finite mind use for assessing the comparative prior probabilities p(h₁/k) and p(h₂/k) where h and h are the point and theistic hypotheses respectively? (These will also be the criteria that we—actual human organisms—use to approximate the assessments of this possible mind in our actual degrees of belief.)

Criteria for determining a priori personalist probabilities, conditional only upon the background knowledge k, are the hypotheses’ degree of conservativeness, symmetry, simplicity, and non-arbitrariness, criteria that are also used in the empirical sciences to decide a priori among observationally equivalent hypotheses. I reject verificationism, conventionalism, positivism, Millian-Quinian empiricism, and other anti-a priorist or anti-realist epistemological or metaphysical theories, and hold that conservativeness, symmetry, simplicity, and non-arbitrariness are a priori criteria that can determine which of two observationally equivalent hypotheses is more likely to be true. Our present task is to determine the prior probabilities of the timeless point hypothesis and the theistic hypothesis, conditional upon k. Whether or not these two hypotheses are observationally equivalent is addressed at the end of this essay on the comparative posterior probabilities of these hypotheses.

Explanatory power predictive success, predictive comprehensiveness, and predictive novelty are criteria for assessing the probabilities of hypotheses relative to the observational evidence, and thus are not employed in the first (prior probability) stage of our inquiry. I shall now begin evaluating the prior probability of the point hypothesis and the theist hypothesis conditional upon k by assessing which of the two is more conservative, symmetrical, simple, and non-arbitrary.

(h₁) There exists a metaphysically necessary, essentially uncaused, timeless, and independent (“a se”) point that, if spacetime begins to exists, is the transcendent cause of spacetime’s beginning to exist.

(h₂) There exists the god of perfect being theology that, if spacetime begins to exist, is the transcendent cause of spacetime’s beginning to exist.

Whether the point hypothesis or the theistic hypothesis has greater prior probability is partly determined by which is more conservative relative to k, which is the extent to which each is similar to what is already known, namely, the proper class k of all truths of formal logic that are knowable by a perfectly rational, finite mind. Roughly speaking, if the entity, God, postulated by the theistic hypothesis is radically different than the abstract objects of formal logic, then the postulation that God exists is a radical, non-conservative departure from what is already known to exist. But if the entity, the point, is much more similar than God to what is already known to exist, the formally logical abstract entities, then the point hypothesis is more conservative than the theistic hypothesis and thus, if all else is equal, is a priori more likely to be true. This can be spelled out more precisely and in more detail.

The hypothesis h is more conservative than h’ if and only if h conserves more of what is already known than h’ and h introduces fewer novel properties than does h’. In our special case of the theistic and point hypotheses and the background logical truths in k, we can use a more precise definition. The point hypothesis is more conservative than the theistic hypothesis if and only if the point hypothesis conserves more of the properties essentially possessed by all of the abstract objects pertinent to formal logic, and ascribes fewer novel essential properties, than does the theistic hypothesis. By a “property” that is novel or conserved, I do not mean whatever can be expressed by a linguistic predicate, but a kind that is intrinsic to reality or a universal that is determinative of such a kind. I would say “natural kind,” but given that I am talking about supernatural properties, non-natural properties (of abstract objects), and natural proper ties, a more general term is needed. “Real kind” is the appropriate phrase here, which contrasts with “artificial kind.”

Both the point hypothesis and the theistic hypothesis conserve at least four proper ties in the sense that they ascribe four properties to the point, or God, such that these four properties are essentially possessed by all the abstract objects of formal logic:

(C1) being timeless (existing outside of the time postulated by general relativity)

(C2) aseity (existing non-dependently on the existence of spacetime or any other concrete beings, if there are such existents)

(C4) metaphysical necessity, i.e., existing in all the metaphysically possible worlds

Note that the property of being a point is conserved, since there are abstract points, e.g., the points in the abstract topological space postulated by point-set topology. The point hypothesis introduces the novel idea (novel relative to the existentially quantified truths in k) that there is a point that is concrete rather than abstract. Some thing is concrete if and only if it is (a) mental and/or has (b) extended or unextended spatiality or temporality and (c) is able to cause something and/or is able to be causally affected by something. The concrete, timeless point that is postulated by (h₁) has, like all the concrete spatial points and mass points that belong to our universe, a zero-dimensional spatiality, and thus it is concrete by virtue of instantiating (N1). Having property (N2) also implies it is concrete. Unlike the spatial and mass points of our universe, the timeless point has no metrical properties, but having metrical properties is not a necessary condition of being concrete. (I am here assuming substantivalism about spacetime for ease of exposition and because I think it is true; my argument could be reformulated with only minor modifications in terms of relationalism [reductionism] about spacetime.)

The property (N2) is a dispositional property, like the divine property of omnipotence, and it determines the kind of entity the timeless point is. The properties our hypotheses ascribe can be “occurrent” or “dispositional.”

Since the point hypothesis is more conservative than the theistic hypothesis, relative to the abstract objects in the proper class k, it follows that if all other things are equal, the point hypothesis has a higher prior probability than the theistic hypothesis.

The argument from conservativeness is one way of interpreting some of David Woodruff’s plausible remarks about the theistic and point hypotheses, although Woodruff does not agree with all my theses. Woodruff does not mention the criterion of conservativeness, but I believe his following statements (with which I concur) can be interpreted in terms of this criterion. Woodruff writes:

In assessing degrees of [a priori] probability for contingent things (or the degree of probability that a thing is truly necessary) we would no doubt include simplicity as one factor, but again it seems to me that this would only be one thing and not even the first or most important thing in the assessment. The first thing we would assess here would be its likelihood given what else we think necessarily exists... Suppose that if either exists (the point or God), that they are necessary. Then ask given what we would accept as necessarily existing apart from these things, say numbers or sets, properties, relations and so forth, which one of these (if either) is more likely to exist. Well, the theist thinks that God is personal (and concrete. . . ) and that is radically different from the rest of the necessary things. Whereas, the point is not personal (and I think not concrete) and thus far more similar to the rest of the necessarily existing things. To me this suggests that the prior probability of the point, given what we normally accept as necessarily existing, is higher than the prior probability of God.[29]

I am not sure how much weight we should put on my claim that the point is concrete and Woodruff’s claim that it is not, since we may just define “concrete” differently; Woodruff and I are both talking about a point that is hypothesized to cause spacetime, and if Woodruff allows non-concrete objects to be causes, perhaps this is a mere terminological difference. The main issue, I think, concerns Woodruff’s true statement that God is personal and the point is not personal and that this makes the point “far more similar to the rest of the necessarily existing things.” Woodruff is a theist, but I think his plausible remarks here indicate that the sort of considerations I have adduced about conservativeness can be accepted as intersubjectively justified by both theists and atheists. It seems likely that Woodruff or some others who think theism is true will be looking at the “other things” in the “if all other things are equal” modifier that essentially belongs to the relationship between a hypothesis’s conservativeness and the extent to which this conservativeness affects the hypothesis’s prior probability of being true.

This brings us to the question “Are all other things equal?” Maybe the theistic hypothesis is more symmetrical, simpler, and more non-arbitrary than the point hypothesis, such that these “other things” make theism more a priori probable than the point hypothesis, despite the fact that theism is much less conservative than the point hypothesis.

Symmetry is a criterion relevant to a hypothesis’s a priori probability. I shall take my point of departure from Van Fraassen’s Laws and Symmetry since this work contains the most philosophically sophisticated discussion of symmetry.[30] However, we will (in due course) need to offer a different, more comprehensive, and stronger definition of symmetry than the one Van Fraassen and others use.

The theistic hypothesis contains more a priori symmetry (i.e., symmetry with k) than does the point hypothesis, and thus (considered only in respect to the symmetry criterion) has a greater prior probability than the point hypothesis. But what is this critenon? As a first approximation, I shall quote Van Fraassen’s example: “The paradigm of symmetry is the mirror image. I and my image are a symmetric pair.”[31] The theistic hypothesis exhibits symmetry with k by virtue of postulating an omniscient being. Omniscience can be mapped onto the elements of k by an injective function that preserves exact similarities with respect to truth. To see this, consider the definition of symmetry that Van Fraassen and others use. Symmetries are injective functions that map each element in a domain onto some element in the codomain, such that the relevant essential properties possessed by the element in the domain are also possessed by the element in the codomain to which the element in the domain is related.[32]

The theistic hypothesis ascribes the property of omniscience, which entails that the hypothesized person knows (at least) each formally logical truth. Let our domain be the proper class of God’s intentional acts (in Brentano’s sense) that are God’s “knowings” of the formally logical truths in the proper class k, such that for each such truth p, there is a distinct knowing that p. Let our respect be truth and let the symmetry functional relation associate each divinely known p with each theorem p, such that all the elements in the domain (each divinely known p) are exactly similar to the elements in the codomain (each theorem pink) in respect of the property being true. Each divinely known p has the essential property of being true, and each theorem p in k has the es sential property of being true.

Note that this functional relation is bijective, giving us a stronger symmetry than Van Fraassen’s merely injective function (Van Fraassen calls these injective functions “one-to-one functions,”[33] but in the precise language of set theory “the graph of an injective mapping is said to be one-to-one.”[34] A function relates each element in the do main to some element in the codomain. A bijective function relates each element in the domain to a unique element in the codomain, such that no two elements in the domain can be related to the same element in the codomain and there is no element in the codomain that is not related to an element in the domain. An injective function relates each element in the domain to some member(s) in the codomain, such that all the elements in the domain can be related to the same element in the codomain. Further, there is a third functional relation that is more symmetrical than an injective relation and less symmetrical than a bijective relation, namely, a surjective functional relation, whereby at least one element in the codomain is uniquely related to some element in the domain.

Thus, the theistic hypothesis, by virtue of including omniscience in its set of ascribed properties, contains a domain that is maximally (bijectively) symmetrical with k.[35] How does the set of properties ascribed by the point hypothesis compare with the ism in regard to the criterion of symmetry?

The point hypothesis’s conserved properties (being timeless, etc.) are also con served by the theistic hypothesis, so its distinctive degree of symmetry will be exhibited in its set of novel properties, namely, the properties of being a spatially zero- dimensional point and being able to cause spacetime’s beginning. The property of being a spatially zero-dimensional point is the only distinctive property that bears a symmetry relation. The domain is the property of being a spatially zero-dimensional point, and the respect is being pointlike , i.e., being a concrete or abstract point. The do main, consisting only of the property of being spatially zero-dimensional point-like, is exactly similar only to the abstract points in the codomain k in respect of point-like ness, giving us merely an injective symmetry, the lowest degree of symmetry. The atheistic hypothesis (h₁) fares poorly with respect to the criterion of symmetry.

The theistic hypothesis is considerably less conservative (with respect to k) than the point hypothesis by virtue of postulating a person, but is considerably more sym metrical (to k) by ascribing omniscience to this person. This is a conclusion with which both atheists and theists can agree.

There is no reason to think that, ceteris paribus, (h₁) greater conservativeness raises the prior probability of (h₁) to the same degree of prior probability as (h₂) greater symmetry raises the prior probability of (h₂) But there is also no reason to sup pose that considerations of conservativeness and symmetry make one hypothesis have an enormously higher priorprobability, ceteris paribus, than the other hypothesis. There is no defeater for the rational belief that their comparative prior probabilities, considering only conservativeness and symmetry, are approximately equal.

Will our two remaining criteria, simplicity and non-arbitrariness, give one of the two hypotheses a significantly higher prior probability conditional upon k?

According to Eliot Sober, simplicity is informativeness.[36] According to dozens of other philosophers, simplicity is something else, such as explanatory power, or predictive comprehensiveness, or predictive novelty, or non-ad-hocness, etc. Physicists often say that simplicity is a hypothesis’s beauty or its naturalness, but make no attempt to explain what the phrase “beautiful hypothesis” or “natural hypothesis” could possibly mean. In response to the philosophers’ theories of simplicity, I agree that such properties as informativeness, explanatory power, predictive comprehensive ness, predictive novelty, and non-ad-hocness are theoretical virtues that can make one hypothesis more probable than another if all else is equal, but these virtues are conditional upon observational evidence e and cannot help us in determining the probability of our two hypotheses (_h1) and (h₂) conditional only upon k. “Simplicity,” which perhaps wins the prize for being the most equivocally used word in the philosophical and scientific literature, is used in the present part of this essay to express a virtue of hypotheses (in more familiar language, a “theoretical virtue”) distinct from the above-mentioned ones, namely, a virtue that makes one hypothesis more a priori probable, ceteris paribus, than another hypothesis, conditional only upon k.

Since my discussion is restricted to the comparative probabilities of the point hypothesis and theism, I will define the kind of simplicity that is pertinent to such hypotheses. I will define “simplicity” in terms of properties rather than parts since neither the timeless point nor God is a whole composed of parts and thus the two hypotheses are equally “simple” in the compositional sense of implying that God and the point have no parts. The relevant comparison concerns properties or attributes, not parts, and thus we may call our simplicity “attributive simplicity” as distinct from“ compositional simplicity.”

The hypothesis h ascribes an attributively simpler set of essential n-adic properties o x than does h if and only if the set S containing all and only the non-trivially essential properties ascribed by h contains fewer logically independent kinds of properties than the corresponding set S of properties ascribed by h’. The two kinds of properties and G are logically independent if and only if, for any possible existent x, “x exemplifies F” does not logically imply “x exemplifies G” and “x exemplifies G” does not logically imply “x exemplifies F.”

Kinds of properties are being mental, being physical, being animate, being abstract, etc. Kinds of properties can have subkinds that are also kinds of properties; for example, being animate has several subkinds, such as being a plant and being an anima!. By contrast, being the number one, being the number two, and being the number three are properties of the same kind, namely, being a positive who!e number. Trivial essences, such as being-self identical and being something, are possessed by every thing (where “thing” is understood broadly to range over concrete objects and abstract objects) and thus are not relevant to the comparative degrees of simplicity of our two hypotheses.

The set S1 of logically independent kinds of non-trivially essential properties ascribed by h₁ is:

S1: {being a spatial point, being metaphysically necessary, timelessness, aseity (i.e., existing independently of any other concrete object, if there are other concrete objects), being able to cause spacetime, being transcendent if there is a created spacetime}.

Why is the property of being metaphysically necessary logically independent of the property of aseity? Because it is logically consistent for the point to exist in every metaphysically possible world and to be dependent on some concrete object in each of these worlds, and, second, it is logically consistent for the point to exist nondependently on any other concrete object and yet to exist in only some metaphysically possible worlds.

There are many truths about what non-trivial essential properties the point does not possess; it is true that the point does not exemplify the properties of being animate, having a metric, being a part of a spacetime manifold, having parts, being caused, being mental, etc., but this point possesses only six logically independent kinds of non-trivially essential properties. If someone argues that being essentially uncaused, being temporally unextended, and being partless are also logically independent kinds of non-trivially essential properties, then she may add them to the atheist set Si and to the theist set S2, which will not affect the comparative simplicity of theistic and point hypotheses. Of course, one may object to this; for example, does not being a zero-dimensional spatial point entail the property of being partless, such that these two properties are not logically independent? And does not aseity entail being uncaused, and does not being timeless entail being temporally unextended?

The set S2 of logically independent kinds of non-trivially essential properties as cribed by h₂ is:

S2: {being personal, being omniscient, being omnipotent (and thus being able to cause spacetime), having libertarian free will, being perfectly good, being a spirit, being metaphysically necessary, aseity, timelessness, being transcendent if there is a created spacetime }.

Neither being omniscient nor being a spirit (i.e., being an unembodied mind) logically implies being personal. Some logically possible beings, such as Brahman (specifically, the impersonal, omniscient spirit postulated by the Hindu philosopher Shankara) or sunyata (“emptiness”), the impersonal, omniscient spirit postulated by the Mahayana Buddhist Asvaghosha, are examples of such logically possible beings. Furthermore, omnipotence does not logically imply aseity, since there is some logically possible being that cannot exist without creating other concrete objects (some Leibniz scholars interpret his god in this way), such that this being cannot exist unless other concrete objects exist.

Many theists hold that the divine attributes are not logically independent from one another. But they mean by this that they are not broadly logically or metaphysically in dependent from one another, and my definition of attributive simplicity uses “logically independent” to mean formally logically independent, as I have indicated. (I should add that I agree with Plantinga, Craig, Swinburne, and most contemporary an alytic theists that the “Identity Theory” of God is false.)[37]

The set Si of logically independent kinds of non-trivial essential properties ascribed by h₁ contains fewer members than the set S2. (It contains infinitely fewer members if “omniscience” designates the transfinitely numerous kinds of knowledge God possesses and “omnipotence” the transfinitely numerous kinds of abilities to act in different ways.) It follows that h has greater a priori attributive simplicity than h1 and thus, if all other things are equal, has greater prior probability. The point hypothesis is more conservative and more attributively simple than the theistic hypothesis, but the theistic hypothesis exhibits greater symmetry with k. Nothing clearly decisive has emerged from our application of these criteria; there is no defeater for a perfectly ra tional, finite mind’s belief that the prior probabilities of these two hypotheses do not diverge to a significant degree relative to these three criteria.

Perhaps the issue of prior probabilities could be decisively resolved if we could show, for example, that the theistic hypothesis alone has a high degree of non-arbitrariness and that this is sufficient to make the theistic hypothesis have a significantly greater prior probability than the point hypothesis. It could be that theism is highly non-arbitrary and that the point hypothesis is extremely arbitrary. Is this so?

Swinburne, Unger, and Parfit have all noted that degrees of non-arbitrariness are at least partly determined by maxima and minima.[38] Swinburne uses the word “simplicity” to express the property of non-arbitrariness.[39] He holds that zero and infinity are equally as simple (non-arbitrary). Postulating a universe with zero electrons is equally as non-arbitrary as postulating a universe with an infinite number of electrons (if all other things are equal), and both postulates are less arbitrary than postulating a universe with 874 trillion electrons (if all other things are equal). Zero members of a certain kind and infinite members of a certain kind are less arbitrary numbers of members of that kind than some positive, finite number of members of that kind, if all other things are equal.

The fact that non-arbitrariness is partly determined by maxima and minima has been noted (under that name) by Parfit and Unger, although they use the maxima criterion in a broader sense than does Swinburne. Parfit and Unger argue that the least ar bitrary ontology of concrete objects is one that postulates the maximum of possible concrete objects (all possible worlds exist, in David Lewis’s sense) or the minimum of possible concrete objects (there is “nothingness,” or, a world containing no concrete objects is actual). This is very close to Swinburne’s idea that zero and infinity are equally non-arbitrary (or “simple” as he calls it), except Swinburne’s “infinite being” does not refer to all possible worlds but to a being with unlimited knowledge, power, etc., in one world. Other examples of equally non-arbitrary postulates of maxima and minima are postulates that there is a zero-dimensional space (a point) or an infinite- dimensional space (assuming all other things are equal). Cannot we say that God is the maximal concrete being (that is able to cause spacetime) in the sense pertinent to perfect theology, namely, in the sense that he has the maximal compossible degrees of the essential properties of being powerful, free, good, knowledgeable, etc., that the metaphysically greatest possible concrete being could possess, and that in this respect theism is a non-arbitrary hypothesis? Certainly. God is “infinite” in the sense of being unlimited in his power, knowledge, goodness, etc. But we also can say that the point is the minimal concrete being (that is able to cause spacetime) in the sense that it has the minimal compossible degrees of the essential properties that concrete objects (physical or mental objects) could possess. The point has a zero degree of mentality, a zero degree of physical massiveness, a zero degree of spatiality, a zero degree of temporality, a zero degree of metrication, and so on. (The minimal concrete being must be distinguished from the worst concrete being, which is a being that has the maximal degrees of evilness, power, knowledge, freedom, etc. This being has maximal degrees of many metaphysically great-making properties, such as omniscience and omnipotence, which enable it to be more evil than any other being, and thus the metaphysically worst being is distinct from the metaphysical minimal being.)[40]

Just as God exhibits “infinity” in the above-explained sense, so the point exhibits “zeroness” in the above-explained sense. In this respect the point hypothesis is a non-arbitrary hypothesis. The point hypothesis and the theistic hypothesis are equally non-arbitrary in this respect and thus have equal prior probability considering only the maxima/minima aspect of the criterion of non-arbitrariness. It seems so far, then, that the point and theist hypotheses do not significantly diverge in respect of their prior probabilities with respect to conservativeness, symmetry, simplicity, and non-arbitrariness.

But this statement may need qualification, since the maximum/minimum distinction is only one of the kinds of non-arbitrariness. One hypothesis is less arbitrary than a second hypothesis if (all else being equal) the first hypothesis postulates an entity x whose existence has an explanation and the second hypothesis postulates an entity y whose existence has no explanation. William Craig uses the phrase “explanatorily simpler” to express this second species of non-arbitrariness. Craig writes that, according to my earlier theory, the singularity existing in time for one instant at t = 0 “is not explanatorily simpler [less arbitrary] than theism.”[41] He writes at more length:

The sense in which God is unexplained is radically different from the sense in which the initial cosmological singularity is unexplained. Both can be said to be without cause or reason. But when we say that God is uncaused we imply that He is eternal, that He exists timelessly or sempiternally. His being uncaused implies that He exists permanently. But the singularity is uncaused in the sense that it comes into being without any efficient cause. It is impermanent, indeed, vanishingly so. These hypotheses can therefore hardly be said to be on a par with each other. Moreover, God is without a reason for His existence in the sense that His existence is metaphysically necessary. But the singularity’s coming to be is without a reason in the sense that, despite its contingency, it lacks any reason for happening.[42]

If the singularity exists timelessly (as I now believe, due to the failure of the B-boundary and G-boundary constructions)[43] and exists necessarily, it is not subject to the criticisms leveled in Craig’s passage. Further, Craig is wrong when he says of God and the point that “both can be said to be without cause or reason.”[44] They are both timeless and uncaused, but the hypotheses h₁ and h₂ also ascribe metaphysically necessary existence. It is a mistake to say with Craig that “God is without a reason for His existence in the sense that His existence is metaphysically necessary.”[45] If God exists in every metaphysically possible world, then it belongs to God’s essence to exist, since “x’s essence includes its existence” means x exists in every metaphysically possible world. Contra Craig, the fact that x’s essence is to exist is a sufficient reason for x to exist and sufficiently explains why x exists. Just as the answer to the why-question “Why is yellow a color?” is “Because it belongs to the essence of yellow to be a color,” so the answer to “Why does God exist?” is “Because it belongs to God’s essence to exist.” The same holds for the metaphysically necessary point postulated by the point hypothesis (h₁).

However, Craig has given no justification whatsoever for his belief that God’s existence is metaphysically necessary. If he has a “metaphysical intuition” about this, this so-called intuition (for all Craig has said) amounts to what Hintikka has called an epistemically valueless “hunch.” Another group of “intuiters” could have an epistemically similar “hunch”[46] that God is metaphysically impossible and that the timeless point is metaphysically necessary. If we are to have indefeasibly justified beliefs about the matter, then we would have to argue that the criteria of conservativeness, symmetry, simplicity, and non-arbitrariness provide the theistic and point hypotheses with a significant prior probability conditional upon the proper class of background truths k, and that other criteria, such as predictive success and explanatory power, provide one or the other hypothesis a greater posterior probability conditional upon the observational evidence e. (I think this is the proper avenue to pursue an epistemological theory of how we know metaphysical necessities and possibilities; “modal intuitions” [hunches] are junked in favor of the application of criteria belonging to inductive logic.)

The Posterior Probability of the Point Hypothesis Conditional upon General Relativistic Cosmology

The conclusion of previous parts of this essay is that the theistic and point hypothesis are equally non-arbitrary, that the point hypothesis is more simple and conservative, but that the theist hypothesis is more symmetrical. Given this, it is reasonable to conclude that their prior probabilities are approximately equal or at least not significantly different. The consequence of this more or less approximate equality of h₁‘s and h₂’s prior probabilities is that the contribution of the observational evidence e to the comparative posterior probabilities of these two hypotheses will be the overwhelmingly decisive factor in determining which hypothesis deserves a greater degree of rational belief. If p(h₁/k) » p/h₂/k then our theism-versus-atheism argument is going to turn on the extent to which p(h₁/e) is greater or less than p(h₂/e).

I think general relativistic big bang cosmology shows that the posterior probability of the point hypothesis is much higher than the posterior probability of the theistic hypothesis, all else being equal. One reason for this is that the timeless point I have been discussing has the contingent property of being the big bang singularity postulated by contemporary physical cosmology.

The attributively used definite description (in Donnellan’s sense of “attributively used”) “the metaphysically necessary, timeless, uncaused, simple, independent, and transcendent point-cause of spacetime” has a different sense than the attributively used definite description “the big bang singularity,” but the evidence from physical cosmology shows they have the same referent. According to general relativistic big bang cosmology, the universe began about 15 billion years ago with a big bang singularity. The big bang singularity is temporally and spatially zero-dimensional. The metric tensor, which is defined on each point in spacetime, is not defined on the big bang singularity, which thereby is not a spacetime point. The metric describes the curvature of spacetime, and since the singularity is an isolated point, it has no curvature; that is to say, the notion of curvature is undefined on the singularity, which is what physicists mean when they say the point has “infinite” curvature. Methods of attaching the singularity to spacetime and defining the metric tensor on the singularity as the first instant of time, t = 0 (rather than as a metrically undefined timeless point), have run into the problem of being counterexampled. As I have indicated earher, the two procedures for attaching the singularity as the first instant of time, the B-boundary procedure and the G-boundary procedure, have been shown to be unacceptable definitions of a singularity attached as a first instant of time.[47] But this failure still leaves one able to attach the singularity as an unmetricated spatial point that is topologically attached to certain types of spacetime, most notably, the type of spacetime that general relativists believe we occupy, a Robertson-Walker-Friedmann spacetime. This was first noticed by one of the general relativists who argued that the B-boundary and G-boundary methods fail, Robert Wald. Wald notes that adding an unmetricated singular point to an otherwise metrically well defined spacetime “would allow one to talk in precise terms of a singularity as a ‘place’ even though the metric is not defined there. However, while this could be done ‘by hand’ in a few simple cases like the Robertson-Walker or Schwarzschild spacetimes, severe difficulties arise if one tries to give a meaningful general prescription for defining a singular boundary.”[48]

John Earman’s recent argument that the big bang singularity in no sense “exists” is invalid.[49] He believes that counterexamples to the definition of a general relativistic singularity, counterexamples that consist in possible spacetimes acquiring physically impossible features if an existent singularity is added to them, suffice to show that our universe does not have an existent singularity. However, Earman’s conclusion that singularities are not existents, since there is no adequate definition (either the B-boundary definition or the G-boundary definition) that applies to all general relativistic singularities, is a non sequitur. If there is no adequate definition of a game that applies to all games, it does not follow that there are no chess games. There may be no defining essence that is common to all singularities, and “singularity” may not have a univocal meaning. In the case at hand, there can be a big bang singularity in a Robertson-Walker-Friedmann spacetime (which is the spacetime in which we live, according to general relativists) even if there is no adequate definition that applies to all singularities. As I mentioned above, Robert Wald, one of the physicists who showed there is no definition that applies to all singularities, argues that one can nonetheless have a singularity in some cases, one of the cases being our Robertson-Walker-Friedmann spacetime. Since the equations of big bang cosmology predict an initial singularity, the prediction of the singularity requires a realist interpretation if the equations are given a realist interpretation. Physicists correctly note that “the initial singularity is a consequence of the equations of general relativity.”[50] A justification for deciding to interpret all of the consequences of the equations of general relativity realistically but this one consequence would be that this consequence has no coherent realist interpretation. But if the Friedmann equations, which general relativists believe describe our spacetime, have an incoherent consequence, how could they even be possibly true, let alone actually true (or highly confirmed)? The singularity has the spatial topology of a point (even though it has no spatial metric and does not exist in time), and it is topologically connected to the metrically well defined spacetime.

I also indicated that the timeless point, by virtue of being essentially attributively simple, need not instantiate any laws, such as the laws of general relativity. The big bang singularity is lawless. As Stephen Hawking writes: “A singularity is a place where the classical concepts of space and time break down as do all known laws of physics because they are formulated on a classical space-time background.”[51] The big bang explosion is the effect of the timeless point-cause, and the big bang explosion is the earliest phase of the universe’s existence (perhaps the first 10^-43 seconds). There is a contingent, singular causal relationship between the point and the explosion; the explosion occurs at a temporal interval t, but it is timelessly the case that the point causes the explosion to occur at the temporal interval t.

The timeless point causes spacetime to exist in David Lewis’s sense of sufficient cause.[52] Lewis’s definition of causation is applicable since it allows for singularist causation and does not require temporal precedence or spatiotemporal contiguity, transfer of energy, or other conditions that are part of other definitions of causality and that preclude the timeless point from being a cause.[53] According to Lewis’s definition, c causes e just in case both c and e exist and e would not have existed had c not have existed. (The modality of the subjunctive is a physical rather than a metaphysical or logical modality.) The timeless point c and spacetime’s beginning e both exist, and spacetime would not have begun if there were no singular point c. According to the Friedmann equations and Hawking-Penrose singularity theorems that determine the basic physical laws of our universe, spacetime must begin in a singularity, i.e., as an “explosion” of a singular point. Thus, in all physically possible worlds in which c does not exist, e does not exist. (If e exists and c does not, the Hawking-Penrose singularity theorems are violated.) By contrast (and in accordance with Lewis’s definition of sufficient causation), if e had not occurred, c would have occurred but would have failed to cause e. In some possible worlds, th point c exists timelessly but does not cause spacetime to begin to exist. It is not governed by any law that implies it causes spacetime if it exists. Spacetime requires a causal point-like singularity to begin to exist, but the timeless point does not need to cause spacetime in order to exist. The point is an unmetricated, timeless, topological “boundary” or “edge” of spacetime (to use these terms in the technical senses they have in general relativity) only in the possible worlds where the point causes spacetime.

Let us consider some objections that are pertinent to my theses that the big bang singularity is a partless point and that the point hypothesis h₁ has the sort of compositional and attributive simplicity I ascribed to it. If the big bang singularity is not the partless point described by the point hypothesis (h₁) then the empirical evidence for big bang cosmology and the big bang singularity is not evidence for the point hypothesis (h₁) I am advocating in this essay.

Robert Deltete states that the big bang singularity is a “complex entity” since “literally everything is concentrated in a geometrical point.”[54] If this statement is true, then the big bang singularity is not the timeless point I have been discussing, since the timeless point is partless and thus is not composed of anything (let alone “literally everything”). However, Deltete’s statement is necessarily false, since neither Socrates nor Mount Everest nor any other three-dimensional spatial object can occupy a zero-dimensional singularity. These objects exist only in the spacetime that is caused to exist by the singular point,

Deltete also implies that the hypothesis that there is a big bang singularity lacks the compositional and attributive simplicity I attributed to the point hypothesis (h the singularity has “infinite space-time curvature and infinite mass-energy density—surely a complex entity.”[55] Thomas Sullivan makes a similar sort of objection: “If the big bang singularity is pointlike in its simplicity, it is nonetheless infinitely dense.”[56] I respond that these objections represent a misunderstanding of these concepts, although there is no doubt that the common surface appearance of the way physicists describe the situation lends itself to just this misunderstanding. For example, Michael Berry writes: the “matter and radiation [are] packed into zero initial proper volume; this ‘point,’ however, includes the whole of space—there is nothing ‘outside.”[57] For another example, Hawking says: “all the matter and energy that was contained in that spherical volume of space will be compressed into a single point, or singularity. . . . [T]he entire observable universe is considered to have started out compressed into such a point. . . . Because of the infinite compression of matter and energy, the curvature of spacetime is infinite at the Friedmann singularities too. Under these circumstances the concepts of [curved, three-dimensional] space and time cease to have any meaning.”[58] What does this mean? The singular point has “infinite mass-energy density” in the sense that the measure of its density has zero for its denominator, not in the sense that it has aleph-zero density. Density is the ratio of mass-energy to unit volume, e.g., grams per cubic centimeter. Given the conservation of mass-energy at the singularity, there would be a large number of grams per zero cubic centimeters, since the singularity has zero volume. However, n/0 is a mathematically meaningless expression since division by zero is impermissible. The expression “the singularity is infinitely dense” means that the concept of density is inapplicable at the singularity. Likewise “the singularity has infinite curvature” means the concept of curvature is inapplicable, since a point cannot be curved,

The Posterior Probability of the Point Hypothesis Conditional upon Quantum Gravity Cosmology

Is not my theory of a timeless causal point as outdated as the classical, general relativistic theory of a big bang singularity? If we are talking about empirical confirmation, do we not need to ensure our point-cause hypothesis is consistent with the quanturn gravity research program?

Most philosophers believe that the hypothesis of the big bang singularity is inconsistent with quantum gravity cosmologies. However, contra Deltete, Sullivan, Craig, and most other philosophers of physics, quantum cosmologies do postulate a big bang singularity. Deltete says that “a key feature of quantum-gravity proposals ... is that they deny the existence of an initial singularity.”[59] Pace Deltete, these proposals predict there is a big bang singularity. For example, Barrow and Tipler’s quantum gravity cosmology is based on a functional law (a wave function of the universe) that predicts both the existence of a big bang singularity and the explosion of this singularity in a “big bang” that evolves into our present-day universe. There is a singularity at R = 0, which means the radius R of the universe is zero (i.e., there exists only a point). One of two quantum laws of nature “tell us what happens to wave packets when they hit (i.e., are mathematically related to) the singularity at R = 0. It should be emphasized that n either case, the singularity is a real entity which influences the evolution of the Universe (or more precisely, its wave function) at all times via the boundary conditions at the origin. In the classical universe, the singularity is present only at the end and at the beginning of time, so in a sense the singularity is even more noticeable in quantum cosmology than in classical cosmology.”[60] For the reasons I explained above, “the singularity is present at the beginning of time” needs to be interpreted as meaning the singularity timelessly exists and causes the beginning of time if Barrow and Tipler’s theory is to take into account the failure of the B-boundary and G-boundary constructions. Although the current observational evidence is that time is endless, the phrase “singularity at the end of time” is interpreted as referring to some point other than the point hypothesized by the point hypothesis (h₁) The wave-functional law governing the timeless, causal singularity encodes all the information about the probabilities for the evolution of the universe and explains why there is a big bang explosion caused by the singularity.

There is also a big bang singularity in the better-known quantum gravity cosmology developed by James Hartle and Stephen Hawking.[61] In their cosmology, there is a wave function of the universe that gives an unconditional probability “for the occurrence of a given spacetime” as a whole.[62] A probability for the existence of the big bang singularity is obtained from this functional law by means of the derived or more specified wave function.[63] “The wave functions which result from this specification [of the ground state wave function] will not vanish on the singular, zero-volume three geometries which correspond to the big bang singularity,” and thus there is a functional law of nature that implies there is a probability that the singularity explodes in a big bang. (For a given universe, such as the actual universe, “the wave function can be finite and nonzero at the zero three-geometry,” such that the big bang singularity of the single, actual universe does not consist of many zero-volume three-geometries.) In this case as well we have functional laws that predict with some degree of probability the existence of a universe, a big bang singularity, and the temporal evolution of the spacetime caused by this singularity.

Quantum gravity does not require us to add attributive complexity to the set of kinds of essential properties ascribed to the point by the point hypothesis (h₁) The reason for this is that (if quantum gravity cosmology is true) the timeless point contingently exemplifies the quantum gravity laws in the actual world, and the timeless point’s attributive simplicity is defined in terms of the kinds of properties it essentially possesses. If the timeless point instantiates quantum gravity laws (“the wave function of the universe”) in the actual world, there are some possible worlds in which it does not instantiate these laws.

As we will see in the next section, these considerations show that quantum gravity cosmology and general relativistic big bang cosmology confirm the point hypothesis to a much higher degree than they confirm the theistic hypothesis (h₂).

First, we shall consider the comparative posterior probability of theism conditional on classical, general relativistic cosmology. Let us see if theism can meet the criteria of posterior probability (predictive success and explanatory power) as well as they are met by the point hypothesis. Consider the point hypothesis and e. If the timeless point caused spacetime to begin to exist, we can predict what sort of state would constitute the beginning state (the first state of some small temporal interval such as 10^-43 seconds) that would be the effect of this cause. The state that would be the effect of a timeless, partless, attributively simple, and totally lawless cause is a completely unstructured state, i.e., a state of utter chaos or maximal disorder. It is exactly a state of this sort that is found to occur by contemporary big bang cosmology. For example, Hawking expresses the common view when he says that particles were emitted from the singular point in random microstates, which resulted in an overall macrostate state of maximal disorder.[64] This a posteriori scientific discovery constitutes our evidence e. Since the point has no nomological structure or any other structure that could determine or influence some ordered configuration of particles to emerge (e.g., a Garden of Eden), we are left with a chaotic outpouring from the singularity; the singularity “would thus emit all configurations of particles with equal probability.”[65] Now this observation statement e corresponds to the prediction derivable from the hypothesis that a nomically ungoverned point caused the beginning of spacetime. Numerous observed phenomena, ranging from the cosmic background radiation to the behavior of elementary particles described by the so-called standard model, significantly confirm this prediction of a chaotic singularity. Further, the hypothesis (h₁) explains the maximal chaos that obtains at the earliest era of spacetime. The reason that there is maximal disorder is that the hypothesis about the timeless point is compositionally and attributively simple to the extent that it implies the point lacks the structure that could enable its act of singular causation (the initiation of the “big bang explosion,” to use the metaphor of big bang cosmologists) to bring about an ordered effect.

But this is not what a perfectly rational, as so far finite mind would predict a perfectly good, powerful, and knowledgeable person to cause; the finite, rational mind would predict that spacetime begins in a way that an all-powerful, all-knowing, and good person would begin spacetime, namely, in a highly ordered way. It is perfectly reasonable to expect a very good, wise, and powerful person to begin his creation in a very beautiful and magnificent way that exhibits an admirably high degree of naturally good order. “Complete chaos is just ugly,”[66] and a perfectly rational finite mind would predict that ugliness is not the very first thing that a good, all-powerful person would want to create. This expectation is so natural and obvious that the belief that the early universe contained the Garden of Eden persisted in Jewish and Christian thought for nearly two thousand years, requiring extensive scientific evidence to be falsified.

This is why the current observational evidence that the beginning of spacetime is a state of maximal chaos falsifies theism. The theistic hypothesis is predictively unsuccessful and is explanatorily valueless, since “Because God created it” is not an explanatorily informative answer to “Why is the first state of spacetime totally chaotic rather than ordered in a very beautiful and admirably good way?”

This is not the argument I have presented previously.[67] Max Jammer, in his recent book Einstein and Religion: Physics and Theology, sums up my earlier argument as follows:

Smith challenged the theistic interpretation of the Big Bang cosmology not only on the grounds that it has a viable competitor in a nontheistic interpretation, but, more importantly, because, in his view, it is inconsistent with this cosmology. His argument, in brief, runs as follows. According to the Big Bang cosmology, there exists an earliest state, E, of the universe, which, by Hawking’s so-called principle of ignorance, does not guarantee to evolve into an animate state. In the theistic interpretation, E is created by an omniscient, omnipotent, and perfectly benevolent God and must therefore evolve into an animate state; for an animate state is better than an inanimate one.[68]

Jammer does not disagree with my argument. Naturally, the above-quoted passage does not include all the details of my earlier argument, such as the requirement that E is an animate state or else E necessarily or probably evolves into an animate state. (The detail about probability was also missed by Swinburne in his critique.)[69] Jammer’s passage also does not reproduce my explanation of why my use of the premise “an animate state is better than an inanimate one” cannot be rebutted by the “there is no best possible world” argument.[70] I will only note here that my comparison of the predictive success of theism versus the point hypothesis does not mention whether or not the first state E is an animate state or will (probably or necessarily) evolve into an animate state. Rather, I am here only comparing the point hypothesis’s predictive success and explanatory power in predicting and explaining the total lack of order of the caused first state, and the theistic hypothesis’s predictive and explanatory failure with regard to the observational evidence of utter chaos. It is caused order or caused chaos, not the different issue of caused animinateness or uncaused inanimateness, that my present argument is about.

Theists such as Craig, Swinburne and Robert Deltete (and neoplatonists such as John Leslie) have addressed the general issue of the chaotic nature of spacetime’s beginning.[71] They attempt to save theism by replacing the falsified prediction (that spacetime begins to exist in a highly ordered state) that is derived from the conjunction of theism with the auxiliary proposition that spacetime begins to exist, with a new prediction that is not derived from theism and the auxiliary proposition, but is postulated solely for the purpose of making theism consistent with the newly discovered observational data e. In short, they turn theism into an ad hoc hypothesis, and thereby make it an inductively illogical hypothesis.

Craig offered the most detailed attempt to save theism from the falsifying evidence e discovered by general relativistic cosmologists. Instead of predicting the initial stage of spacetime by deriving it from the theistic hypothesis and the auxiliary proposition that spacetime begins to exist, Craig first learns through contemporary cosmology about the nature of this initial stage and then (after this new, falsifying observational evidence is acquired) modifies the theistic hypothesis solely for the purpose of enabling it to predict this newly acquired evidence. Craig redefines perfect being theology so that it now includes the proposition God delights in fashioning a highly ordered, good, and beautiful universe out of maximal chaos. Craig writes, “Moreover, what if His goals include, not merely the having of a created order, but the divine pleasure of fashioning a creation [out of disordered ‘raw material’ God first created]?”[72] Craig seems to commit here a further fallacy: by positing the mere epistemic possibility that God takes pleasure in organizing totally disorganized states, the added assumption cannot be used to derive a predication or explanation of the first state. It is a fallacy in epistemic logic to derive p is known from p is consistent with what is known. By direct analogy, one cannot derive the prediction that an ordered, beautiful, and good state such as the Garden of Eden will be the effect of the point from the assumption that it is epistemically possible that a state such as the Garden of Eden is the effect of the uncaused point. (Moreover, the assumption that these propositions are even epistemically possible remains unjustified.)

The ad hoc modification to theism is even more blatantly demonstrable in Swinburne’s work. Swinburne’s recent discussion is a modification of his earlier theory in that it is designed solely for the purpose of making his theism consistent with the ob servational evidence e of which he was (apparently) unaware earlier.[73] His earlier theory reads:

We saw that God has reason, apparently overriding reason, for making, not merely any orderly world (which we have been considering so far) but a beautiful world—at any rate to the extent to which it lies OUTSIDE the control of creatures. (And he has reason too, I would suggest, even in whatever respects the world does lie within the control of creatures, to give them experience of beauty to develop, and perhaps also some ugliness to annihilate.) So God has reason to make a basically beautiful world, although also reason to leave some of the beauty or ugliness of the world within the power of creatures to determine; but he would seem to have OVERRIDING REASON NOT to make a basically ugly world BEYOND THE POWERS OF CREATURES TO IMPROVE.[74]

The “complete chaos” of which the entire world consisted at the big bang explosion, which is “just ugly,”[75] is beyond the power of creatures to improve (unless they have backward causation, which Swinburne denies they do). Thus, the observational evidence e directly contradicts Swinburne’s original theistic hypothesis, for e implies there is evidence that there is chaos and ugliness beyond the powers of creatures to improve. But once Swinburne learned that the world initially was in a state of complete chaos, he changed his theistic theory to include the thesis that God does have overriding reason to create the early state to be chaotic and ugly; he adopted Craig’s ad hoc hypothesis that God delights in ordering chaos (a chaos beyond the control of creatures to improve).

Can middle knowledge save the inductive validity of theism? Craig adds to theism a thesis about middle knowledge designed to make theism consistent with the theistically unexpected observational evidence e that turned up. He added the proposition It is metaphysically necessary that God has the middle knowledge that the beginning of spacetime, in the actual world W, would be a state of maximal disorder that would eventually lead to an orderly state of the universe where free, rational, moral agents exist and act. This is obviously ad hoc. This thesis cannot be derived from the classical creationist hypothesis (from Molina to Suarez to Plantinga and others), but is an addition to this hypothesis that is made solely for the purpose of making this hypothesis consistent with the newly found observational evidence. If this is not a purely ad hoc modification to theism, then why did not Molina, Suarez, and other defenders of the middle knowledge theory of divine omniscience derive this prediction from the theistic hypothesis and the auxiliary proposition that spacetime begins to exist? Why could not Craig derive it before he began reading books and articles on physical cosmology?[76]

Returning to the non-ad-hoc point hypothesis (h₁) quantum gravity cosmology also supports the prediction of a chaotic first state that is derived a priori from the point hypothesis (h₁) in conjunction with the auxiliary proposition that spacetime begins to exist. The wave function of the universe predicts that there is a big bang singularity, but since the singular point does not have enough structure to determine or constrain its effect to have a certain order, the wave function predicts that a maximally chaotic state will follow from the big bang singularity.[77]

How can the evidence e of the nature of spacetime’s beginning be logically sufficient to falsify the theistic hypothesis (h₂)? Even if Swinburne’s and Craig’s theories are inductively invalid, cannot the theist maintain that there is an inductively valid version of theism? Specifically, does not the requirement of total relevant evidence require that we take into account the existence of the present beauty and orderliness in nature and the existence of free, intelligent, morally capable humans that came into existence billions of years after the beginning of spacetime? The answer is that we have taken into account the total relevant evidence for the prediction of how spacetime would begin, and the failure of this prediction is sufficient to render the theistic hypothesis false.[78] This can be most clearly demonstrated by exhibiting the logical structure of the theistic and timeless point arguments in the following deductive formats. Regarding the timeless point hypothesis (h₁) we have this argument (A1):

(1) There exists a metaphysically necessary, essentially uncaused, timeless and independent (“a se”) point that, if spacetime begins to exists, is the transcendent cause of spacetime’s beginning to exist. (Axiom h₁).

In this essay, we derived the conclusion that this axiom has a certain attributive and compositional simplicity that enables the following theorem to be deduced:

(2) The point is structureless and will have an unordered effect if it causes spacetime’s beginning to exist. (Theorem T1)

(4) Spacetime begins to exist in an unordered way. (Prediction derived from 1-3)

(6) There exists the god of perfect being theology that, if spacetime begins to exist, is the transcendent cause of spacetime’s beginning to exist. (Axiom h₂)

In this essay, we derived the conclusion that this axiom has a certain attributive cornplexity that enables the following theorem to be deduced:

(7) God is highly structured in terms of goodness, power, and knowledge and will have a highly ordered effect if he causes spacetime’s beginning to exist. (Theorem T2)

(8) Spacetime begins to exist in a highly ordered way. (Prediction derived from the conjunction of 6, 7, and 3)

Noting that the observational evidence e falsifies the prediction (8) and thus falsifies the theistic hypothesis (h₂) someone like Craig and Swinburne who wishes to retain theism in the face of the scientific evidence will modify the hypothesis (h₂) after the fact for the specific purpose of enabling a prediction of the observational evidence e to be derived. Instead of the prediction of e being derived from the theistic hypothesis and the auxiliary proposition that spacetime begins, we have the reverse situation (A3):

(9) There exists the god of perfect being theology who, given (5), delights in ordering total chaos and, if spacetime begins to exist, is the transcendent cause of spacetime’s beginning to exist. (Ad hoc modification of [h₂] that enables it to predict e)

(11) Spacetime begins to exist in an unordered way. (Prediction derived from the conjunction of 5, 9, 10 and 3)

Since the observational evidence statement (5) asserting e is one of the premises from which the prediction (11) of e is derived, the argument is inductively invalid; specifically, it commits the fallacy of being ad hoc. Since the original, non-ad-hoc prediction (8) is falsified by the observational evidence e, the theistic hypothesis (h from which (8) is derived (in conjunction with the relevant premises) is also falsified.[79] We are led to the conclusion that p(h2/e & k) = 0 and, on a more metaphysically positive note, that p(h₁/e & k) >> p(h₂/e & k).

Ä I am grateful to David Woodruff for extensive, stimulating, and very helpful comments on an earlier draft of this essay. I also thank Dean Zimmerman, for his influential criticisms of an even earlier draft that he offered to me twenty-five thousand feet above the earth.

[1] Brian Leftow, Time and Eternity (Ithaca: Cornell University Press, 1991).

[2] See Alvin Plantinga, The Nature of Necessity (Oxford: Clarendon Press, 1974), and Plantinga, “World and Essence,” Philosophical Review 79(1970), 466-73.

[3] Alvin Plantinga, Warranted Christian Belief (New York: Oxford University Press, 2000), 6.

[4] I am assuming the theory of time postulated in the standard interpretation of general relativity and quantum cosmology. This is for the purpose of showing how contemporary cosmology, on its standard interpretation, can explain why time exists. A theory based on the theory of time in my “Absolute Simultaneity and the Infinity of Time,” in Questions of Time and Tense, ed. Robin Le Poidevin (Oxford: Clarendon Press, 1998): 135-83, would be a different project.

[5] This earlier theory is articulated in many of my articles, including “Simplicity and Why the Universe Exists,” Philosophy 72 (1997), 125-32; “Did the Big Bang Have a Cause?” British Journal for the Philosophy of Science 44(1995), 649-68; “Stephen Hawking’s Cosmology and Theism,” Analysis 54(1994), 236-43; “Can Everything Come to Be without a Cause?” Dialogue: Canadian Philosophical Review 33 (1994), 325-35; “Anthropic Coincidences, Evil, and the Disconfirmation of Theism,” Religious Studies 28 (1992), 347-50; “Atheism, Theism, and Big Bang Cosmology,” Australasian Journal of Philosophy (1991); “A Natural Explanation of the Existence and Laws of Our Universe,” Australasian Journal of Philosophy 68 (1990), 22-43; and “The Uncaused Beginning of the Universe,” Philosophy of Science 55 (1988), 39-57. See also William Lane Craig and Quentin Smith, Theism, Atheism, and Big Bang Cosmology (Oxford: Clarendon Press, 1993).

[6] For a discussion of the relevant arguments, see my “Problems with John Earman’s Attempt to Reconcile Theism with General Relativity, Erkenntnis 52 (2000), 1-27.

[7] If this is the case, this may suggest to some people that an emotion like Schleiemacher’s “feeling of dependence” is appropriately directly upon the only existent transcendent cause of spacetime, the timeless point, and is not appropriately directed upon the hypothetical being postulated by perfect being theology. The theory of this transcendent, creative point is neither monotheism, nor pantheism, nor neoplatonism, nor polytheism, nor animism, but (if one wishes to coin a neologism) pointism. Since I argue that theism is probably false, I am not a theist; in this sense I am an atheist. However, qua defender of the theory in this essay, I may also call myself a pointist, to coin a second neologism. An atheist is free to appreciate emotionally other intentional objects (in Brentano’s sense) of possible feeling-acts, in addition to the point-cause. The atheist’s beliefs about the positive nature of reality as a whole, about what does exist, allow her to appreciate emotionally the world-whole, the existence of the universe, presentness, the conjunction of all concrete objects, and perhaps other intentional objects of feeling-acts as well. On this appreciation, see my The Felt Meanings of the World: A Metaphysics of Feeling (West Lafayette: Purdue University Press, 1986); “An Analysis of Holiness,” Religious Studies 24 (1988), 511-27; Language and Time (New York: Oxford University Press, 1993); and Ethical and Religious Thought in Analytic Philosophy of Language (New Haven: Yale University Press, 1997). Given Bouchard’s study of identical twins (Thomas Bouchard, “Whenever the Twain Shall Meet,” Sciences, September/October 1997, 52-57), from which he plausibly concludes that religiosity and non-religiosity are genetically inherited, and given the steadily in creasing encroachment of science upon monotheism, it is my guess that in the distant future such “felt meanings” (the point-cause, the world-whole, etc.) will become the established tar gets of humankind’s religious emotions.

[8] 8. See Alvin Piantinga, Warrant and Proper Function (New York: Oxford University Press, 1993), 163.

[9] Could there be any other sort of atheistic explanatory hypotheses? I think there are many others, such as the three explanatory hypotheses discussed in my “The Reason the Universe Exists Is That It Caused Itself to Exist,” Philosophy 74 (1999), 579-86; a fourth hypothesis discussed in my “Simplicity and Why the Universe Exists”; a fifth hypothesis discussed in my “A Defence of a Principle of Sufficient Reason,” Metaphilosophy 26 (1995), 97-106, and “World Ensemble Explanations,” Pacific Philosophical Quarterly 67 (1986), 73-86; and a sixth hypothesis discussed in my “A Natural Explanation of the Existence and Laws of Our Universe.” Peter Unger’s ”Reducing Arbitrariness,” Midwest Studies in Philosophy (1984), involves an explanatory hypothesis that is of interest, namely, that all possible universes (David Lewis’s possible worlds) exist. However, this hypothesis presupposes a reductionist, physicalist, nominalist, and tenseless theory of time and can be undermined on this basis. See my Language and Time, esp. 242.

[10] Jaakko Hintikka, “The Emperor’s New Intuitions,” Journal of Philosophy 96 (1999), 127-47.

[11] Paul Moser, Philosophy after Objectivity (New York: Oxford University Press, 1993), 8.

[12] See W. Marciskwski, Dictionary of Logic (The Hague: Nijhoff, 1981), 283-84, for a brief exposition of predicate logic of a transfinite order.

[13] See Plantinga, The Nature of Necessity, and Plantinga, “World and Essence.”

[14] Second order predicate logic with identity is first order predicate logic with identity plus the axioms of extensionality and comprehension. The axiom of extensionality is that:

[15] 15. See Quentin Smith, “Why Stephen Hawking’s Cosmology Precludes a Creator,” Philo, 1(1998), 75-93, for a discussion.

[16] A. Shimony, “Scientific Inference,” in Pittsburgh Studies in the Philosophy of Science, ed. R. Colodny (Pittsburgh: Pittsburgh University Press, 1970), 92.

[17] See George Schlesinger, “Confirmation and Parsimony,” in Induction, Probability, and Confirmation, ed. G. Maxwell and R. Anderson (Minneapolis: University of Minnesota Press, 1975), 324-42.

[18] Derek Parfit, “The Puzzle of Reality: Why Does the Universe Exist?” in Metaphysics: The Big Questions, ed. P. van Inwagen and D. Zimmerman (Oxford: Blackwell, 1998), 418-27.

[19] To say with Plantinga (Warrant and Proper Function, 8) and David Woodruff (private communication) that there are no degrees of belief, but that epistemic probability involves a full belief that a proposition has the probability n of being true, is not to define but use the word “probability.” In Plantinga’s case, we are left with no satisfactory idea of what a proposition’s epistemic probability is, if it is not a degree to which it is believed. In fact, when Plantinga explains the normative component of epistemic probability, he typically characterizes it in such phrases as “the claim that the reasonable or rational degree of confidence, in the judgement in question, is r” (163). “Rational degree of confidence” is left undefined and (as far as I can tell) means rational degree of belief or has no discernible meaning at all.

The other component Plantinga claims belongs to epistemic probability, namely, the objective component, is characterized as the logical probability of a proposition h on an evidence proposition e plus propositions such as simpler theories are more likely to be true than complex ones (162). This fails because it is circular (“more likely” means more objectively probable, which is the very concept Plantinga is trying to explain). Furthermore, Plantinga’s argument against the thesis that probability is degree of belief (8, n. 10) is invalid, since “if I have no idea at all whether the proposition in question is true” it does not follow (pace Plantinga) that I be lieve it to degree 0.5. What follows is that I merely entertain or comprehend the proposition, without any assertoric force (belief of any degree, be it zero or 0.5, etc.) at all. This blocks the contradiction Plantinga tries to deduce from the theory that probability is a degree of belief.

Accordingly, I think that the thesis that there are degrees of belief is defensible and that the definition of epistemic probability in terms of degrees of belief does not suffer the problems confronting Plantinga’s theory of epistemic probability.

[20] See my various works cited in this essay.

[21] Set-theoretic paradoxes arise if we define all truths of mathematical logic in terms of a set. Rather, they must be definable in terms of a proper class. For a plausible argument for this claim, see Christopher Menzel, “On Set Theoretic Possible Worlds,” Analysis 46 (1986), 2.

[22] 22. See Saul Kripke, “Naming and Necessity,” in Semantics of Natural Language, ed. D. Davidson and G. Harmon (Dordrecht: Reidel, 1972), 265.

[23] David Kaplan, “Demonstratives,” in Themes from Kaplan, ed. J. Almog (New York: Oxford University Press, 1989), 508-09.

[24] Panayot Butchvarov, Skepticism in Ethics (Bloomington: Indiana University Press, 1989), 62-64.

[25] See Plantinga, “World and Essence”; Ruth Barcan Marcus, “Essential Attribution,” Journal of Philosophy 68 (1971), 187-202, and “Discussion of the Paper of Ruth B. Marcus,” Synthese 14 (1962), 132-43; and Kripke, “Naming and Necessity.” For a definition of metaphysical necessity in terms of the semantics for quantified modal logic, see Quentin Smith, “A More Comprehensive History of the New Theory of Reference,” in The New Theory of Reference: Kripke, Marcus, and Its Origins, ed. James Fetzer and Paul Humphreys (Kluwer Academic Synthese Library Series, 1998), 235-83.

[26] Note that Stephen Yablo’s “Is Conceivability a Guide to Possibility,” Philosophy and Phenomenological Research 53 (1993), 1-42, addresses some relevant epistemological issues, but his essay assumes rather than proves that metaphysical “intuitions” (he aptly uses the word “conceiving” rather than “intuiting”) have epistemic value, as does George Bealer’s insightful essay “The Limits of Scientific Essentialism,” in Philosophical Perspectives, vol. 1, Metaphysics, ed. J. Tomberlin (Atascadero: Ridgeview, 1987). Brian Leftow, Time and Eternity, 14-17 does not presuppose this assumption in his outline of an epistemological theory of meta physical modal claims, and his outline suggests he is on the right track for the development of such an epistemological theory.

[27] Alvin Plantinga, “The Probabilistic Argument from Evil,” Philosophical Studies 35 (1979), 1-53.

[28] Alvin Plantinga, Warrant: The Current Debate (New York: Oxford University Press, 1993); Plantinga, Warrant and Proper Function; Plantinga, Warranted Christian Belief

[29] David Woodruff, private correspondence.

[30] Bas Van Fraassen, Laws and Symmetry (Oxford: Clarendon Press, 1989), 233-348. See also H. Weyl, Symmetry (Princeton: Princeton University Press, 1952); J. Rosen, Symmetry Discovered (Cambridge: Cambridge University Fress, 1975); and G. E. Martin, Transformational Geometry (New York: Springer-Verlag, 1982).

[31] Van Fraasen, Laws and Symmetry, 233.

[32] I am using the terminology of set theory, specifically that found in Michael Potter, Sets (New York: Oxford University Press, 1990), since Van Fraassen makes things a little unclear by using words that sometimes have the sense they have in set theory and sometimes have the sense they have in first order predicate logic with identity. For example, he uses “domain” in the sense of predicate logic (when we talk about quantified variables ranging over a domain of values), but “range” is used to have the set-theoretic meaning of “codomain.” (See Van Fraassen, Laws and Symmetry, 243.

[33] Ibid.

[34] Potter, Sets, 32. The emphasis of “the graph” is mine.

[35] This symmetry argument is one of the (necessary but insufficient) reasons I think the “conceptualist argument for God’s existence” is more plausible than the traditional arguments for God’s existence, such as the cosmological, teleological, and ontological arguments. See my “The Conceptualist Argument for God’s Existence,” Faith and Philosophy 11(1994), 38-49.

[36] Eliot Sober, Simplicity (Oxford: Clarendon Press, 1975).

[37] The Identity Theory held by analytic philosophers—namely, William Mann, “Divine Simplicity,” Religious Studies 18 (1982) 451-71; Eleonore Stump and Norman Kretzmann, “Absolute Simplicity,” Faith and Philosophy 2(1985), 353-82; Brian Leftow, “Is God an Abstract Object?” Nous 24 (1990), 594-96, and Time and Eternity; William Vallicella, “On Property Self-Exemplification: Rejoinder to Miller,” Faith and Philosophy 11(1994), 478-81, and “Divine Simplicity: A New Defense,” Faith and Philosophy 9 (1992), 508-25; and Barry Miller, From Existence to God (London: Routledge, 1992)—implies that God is identical with his essential properties and that each of his essential properties is identical with each of his other essential properties. Perhaps some progress has been made in showing how the proposition the individual, God, is identical with his essential properties is not logically incoherent (see especially Vallicella, “Divine Simplicity: A New Defense”), but I have seen no demonstration that the proposition God’s essential properties are identical with each other is logically coherent. The latter proposition, in fact, is a negation of a theorem of second order predicate logic with identity. This can be briefly illustrated as follows.

Let F be a predicate constant, expressing the degreed property of being powerful. G is a predicate constant, expressing the degreed property of being knowledgeable. Fn is a degree of the property expressed by F, and G_n is a degree of the property expressed by G. Let F_n-max and G_n-max be the highest degrees of F and G or at least the degrees of F and G possessed by God. We can then formulate this argument:

Since the Identity Theory implies the negation of (5), it is a negation of a theorem of second order predicate logic with identity.

[38] Richard Swinburne, The Existence of God, 3d ed. (Oxford: Oxford University Press, 1991); Unger, “Reducing Arbitrariness”; Parfit, “The Puzzle of Reality: Why Does the Universe Exist?”

[39] This is demonstrated in my “Swinburne’s Explanation of the Universe,” Religious Studies 34(1998), 91-102.

[40] I argued in “Anthropic Coincidences, Evil, and the Disconfirmation of Theism” that if an omnipotent person created spacetime, it is more likely to be an omnimalevolent being than a perfectly morally good being. (For considerations relevant to this argument, see the post-Mackie “logical argument from evil” in my Ethical and Religious Thought in Analytic Philosophy of Language, 137-57, and the “probabilistic argument from evil natural laws” in my “An Atheological Argument from Evil Natural Laws,” International Journal for the Philosophy of Religion 29 (1991) 159-74.

[41] Craig and Smith, Theism, Atheism, and Big Bang Cosmology, 273.

[42] Ibid., 272, 273.

[43] See my “Problems with John Earman’s Attempt to Reconcile Theism with General Relativity.”

[44] Craig and Smith, Theism, Atheism, and Big Bang Cosmology, 272; my emphasis.

[45] Ibid.

[46] Hintikka, “The Emperor’s New Intuitions.”

[47] See my “Problems with John Earman’s Attempt to Reconcile Theism with General Relativity.”

[48] Robert Wald, General Relativity (Chicago: University of Chicago Press, 1984), 213.

[49] John Earman, Bangs, Crunches, Whimpers, and Shrieks (New York: Oxford University Press, 1995).

[50] Michael Berry, Principles of Cosmology and Gravitation (Cambridge: Cambridge University Press, 1989), 156.

[51] S. W. Hawking, “The Breakdown of Predictability in Gravitational Collapse,” Physical Review 14(1976), 2460.

[52] David Lewis, “Causation,” Journal of Philosophy 70(1973), 556-67.

[53] See my “Causation and the Logical Impossibility of a Divine Cause,” Philosophical Topics 24(1996), 169-91, and “The Concept of a Cause of the Universe,” Canadian Journal of Philosophy 23 (1993), 1-23.

[54] Robert Deltete, “Simplicity and Why the Universe Exists: A Reply to Quentin Smith,” Philosophy 73 (1998), 493.

[55] Ibid.

[56] T. D. Sullivan, “On the Alleged Causeless Beginning of the Universe: A Reply to Quentin Smith,” Dialogue: Canadian Philosophical Review 33 (1994), 332.

[57] Berry, Principles of Cosmology and Gravitation, 156.

[58] Stephen Hawking, “The Edge of Spacetime,” in The New Physics, ed. Paul Davies (Cambridge: Cambridge University Press, 1989), 61-69.

[59] Deltete, “Simplicity and Why the Universe Exists: A Reply to Quentin Smith,” 492.

[60] 60. See John Barrow and Frank Tipler, The Anthropic Cosmological Principle (New York. Oxford University Press, 1986). These laws are numbered 7.40 and 7.41 in Barrow and Tipler’s work.

[61] James Hartle and S. W. Hawking, “The Wave Function of the Universe,” Physical Review D28 (1983), 2960-75.

[62] The wave function is equation 2.1, ibid.

[63] This equation is numbered 1.11, ibid.

[64] Hawking, “The Breakdown of Predictability in Gravitational Collapse.”

[65] Ibid., 2460.

[66] Swinburne, The Existence of God, 146.

[67] See my “Atheism, Theism, and Big Bang Cosmology”; “A Big Bang Cosmological Argument for God’s Nonexistence,” Faith and Philosophy 9(1992), 217-37; and Theism, Atheism and Big Bang Cosmology, 195-217.

[68] Max Jammer, Einstein and Religion: Physics and Theology (Princeton: Princeton University Press, 1999), 262.

[69] Richard Swinburne, “Review of William Craig’s and Quentin Smith’s Theism, Atheism, and Big Bang Cosmology,” Philosophical Review 104 (1995): 337-39.

[70] Craig and Smith, Theism, Atheism, and Big Bang Cosmology, chs. 7, 9.

[71] Craig in Craig and Smith, Theism, Atheism, and Big Bang Cosmology; Swinburne in “Review of William Craig’s and Quentin Smith’s Theism, Atheism, and Big Bang Cosmology”; Deltete in “Simplicity and Why the Universe Exists: A Reply to Quentin Smith”; Leslie in “Review of William Craig’s and Quentin Smith’s Theism, Atheism, and Big Bang Cosmology,” Zygon 31(1996), 345-49.

[72] Craig and Smith, Theism, Atheism, and Big Bang Cosmology, 267.

[73] Swinburne’s most recent discussion is found in “Review of William Craig’s and Quentin Smith’s Theism, Atheism, and Big Bang Cosmology.” His earlier discussion is found in The Existence of God (1st ed., Oxford: Oxford University Press, 1979), 145-51, and the third edition of this same work (1991), 145-51.

[74] Swinburne, The Existence of God 1st ed., 150, my emphases and capitalization.

[75] Ibid., 146.

[76] A separate problem with Craig’s argument that I considered is that his (and Swinburne’s) concept of God taking aesthetic delight in fashioning order out of chaos is a logically self-contradictory concept, that middle knowledge of the big bang is also a logically incoherent concept, and that the concept of divine causation is logically incoherent. In order to produce an argument logically independent of my above mentioned arguments, I am here granting, for the sake of argument, to the theist her assumption that these concepts are logically coherent. See my

“Simplicity and Why the Universe Exists,” “Atheism, Theism, and Big Bang Cosmology,” “Causation and the Logical Impossibility of a Divine Cause,” and “Stephen Hawking’s Cosmology and Theism.” The mentioned theistic concepts are defended (unsuccessfully, I believe) in Craig and Smith, Theism, Atheism, and Big Bang Cosmology; J. McCleland and R. Deltete, “Divine Causation,” Faith and Philosophy 17 (2000), 1-27; and William Vallicella, “God, Causation, and Occasionalism,” Religious Studies 35, 3-18.

[77] See, for example, J. Halliwell and S. W. Hawking, “The Origin of Structure in the Universe,” Physical Review D31 (1985), 1777-91.

[78] Nonetheless, I have elsewhere addressed “other evidence” or theistic arguments and found them wanting. See the articles of mine cited in this essay.

[79] Of course, (h₂) will not be falsified if one rejects science and its auxiliary proposition (3). But by this rejection, one puts oneself beyond the bounds of “natural reason” and is thereby excluded from the intended audience of this essay.