Atheistic Versus Theistic Explanations of the Beginning of Spacetime

by Quentin Smith

 

1. The Explanatory Poverty of Traditional Atheism

  Why does time exist? In the context of the "space-time theories" of special or general theory of relativity, this question should be more appropriate phrased as "Why does spacetime exist?" In the context of the philosophy of religion, there are two familiar responses to this question. The theist says the question has an answer and that this answer is that God created spacetime. The standard response of the atheist is to say that there is no answer to this question; the existence of spacetime is brute fact or has no explanation.

    In this essay, I propose to reject both the standard atheist response and the theist response. I shall argue that there is a causal, atheistic explanation of the beginning of spacetime that has greater prior probability and greater posteriori probability than theism. I shall argue that the big bang singularly is a timelessly existing cause of spacetime ("timeless" in the sense that it exists outside of the time of general relativity) and that it has greater prior and posterior probability of existing than a timeless deity (who is timeless in the same sense) who causes spacetime to begin to exist. (It is not my concern in this essay to discuss other theories or senses of "time" than that postulated in the general theory of relativity.)

I shall adopt as an a priori, necessary truth that the simpler something is, then, all else being equal, the greater its prior probability. Since this principle is in fact used to choose the "correct" scientific theory on each topic in science, if it were false or truth-valueless, there would be no true (or even approximately true) scientific theories or else no justification for believing any of them (if the justification for choosing the "best" one was based a truth-valueless principle of simplicity). Nor can this be an inductively acquired principle, since any such induction presupposes the prior principle of simplicity for choosing among the infinite number of hypotheses compatible with the observed data (this has been effectively argued by Schelesinger in [  ]).

One problem is the lack of clarity about the meaning of principles of simplicity, and I shall make the relevant principles clearer by my manner of employment of the principles in this essay. I shall argue that the big bang singularity is the simplest possible timeless particular and that God is the most complex possible timeless particular, and therefore that the existence of a causal explanation of spacetime in terms of a singularity has far greater prior probability than a causal explanation in terms of a deity. For this and other reasons, it also has far greater posteriori probability.

    I am treating causal explanations of the beginning of spacetime, since there has been nearly universal consensus among physical cosmologists since the mid to late 1960s that spacetime had a beginning (the most commonly given age of spacetime being 15 billion years ago).

   My view in this essay departs from my previous discussions of big bang cosmology, for there I treated the big bang singularity as both uncaused and as the initial temporal state or boundary of spacetime, and implied that if the singularity (=earliest boundary of spacetime) is uncaused, then spacetime's beginning to exist is uncaused [e.g. Craig and Smith, 1993; and Smith, 1997]. The difference is that I here treat the singularity as a timeless, spatial particular rather than as a spatiotemporal particular. This way of treating the singularity is more physically justified than the temporal treatment, since time in general relativity requires a metric and (as all agree) the singularity has no metric (the metric is "undefined on the singularity") Secondly, I here treat the singularity as a cause of spacetime, even though the singularly itself is an uncaused, timeless particular.

The thesis of this essay is that it is more likely that a timeless singularity causes spacetime to begin to exist than that a deity causes spacetime to begin to exist. This thesis implies that atheism is probably true, since it belongs to the essence of God to be the originating cause of spacetime if spacetime begins to exist. There is no possible world in which it is the case both that God exists and in which spacetime is caused to begin to exist by a timeless singular point. More important than this atheistic conclusion (a purely negative ontological conclusion, a conclusion that a certain entity--God--probably does not exist) is the positive conclusion that there is an atheistic causal explanation of the existence of spacetime. This counters traditional atheism as much as it counters theism.

I begin by arguing that the big bang singularity is the simplest possible timeless particular, that God is the most complex possible timeless particular, and therefore that the big bang singularity has a higher prior probability than God. (Recall that by "timeless" I mean exists outside of the time of General Relativity.)

 

2. A Priori Ontological Simplicity

      The hypothesis that spacetime is caused by a timeless point cannot be used to explain anything if it has a zero degree of confirmation. The timeless-point hypothesis h has a zero degree of confirmation if its prior probability is zero. If the hypothesis h prior probability is zero, then h's posterior probability is zero, regardless of how much evidence there is for big bang cosmology. This follows from Bayes' theorem. p(h) is the prior probability of a hypothesis h and p(h/e) is its posterior probability, its probability given all the relevant evidence e. If p(h) = 0, then p(h/e) = 0. In Bayes' 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 0 x 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 suggests that it is essential to our argument to show that the prior probability of our point-cause hypothesis is a positive, nonzero number.

By "probability" I mean the degree of belief of a logically possible, perfectly rational finite mind. If an hypothesis is 80% probable, that means a possible, perfectly rational, finite mind would believe the hypothesis to the degree 80% 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 attempting to approximate the judgements of a (logically possible) perfectly rational, finite mind when we formulate probability judgements. There are other legitimate senses of "probability", e.g., frequency, propensity, etc., but the sense of "probability" pertinent to my argument is the one I defined, which is sometimes called the "personalist theory of probability" or the "normative subjective theory of probability". This is a kind of objective, mind-independent probability, since it is defined counterfactually as what a perfectly rational finite mind would believe, 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 truth-makers of the relevant counterfactuals are merely possible worlds in which the specified perfectly rational, finite minds and their beliefs exist. Possible worlds are understood here in approximately the Adams-Plantinga sense.

   Since the timeless-point hypothesis h is about explaining why spacetime begins to exist, the "prior probabilities" of h are a priori probabilities; they are probabilities determined prior to taking into account any a posteriori evidence e that consists of the existence of our universe and any observed or inductively derived causes of its existence, if there are such causes.

   The a priori probability of something existing is a function of its simplicity. The simpler something is, the greater its prior probability ceretis paribus. This principle of inductive logic is used in all the sciences and is employed to form the core of "common sense", where " common sense means something largely similar to what Moore defined it as being in his "A Defence of Common Sense".

An entity x or property F is simpler than an entity y or property G if x or F has less parts, less (logically independent) properties and lower degrees of properties (in case of degree properties, such as being more or less large) than y or G. An object's color is logically independent of its shape (neither logically entails the other), but an object's being in motion is not logically independent of its occupying space, since its movement logically entails that it occupies space. In the case of comparing items with infinite parts or properties, we count the number of distinct classes of parts or distinct classes of properties. The word "simple" is ambiguous and I am only using "simple" in one of its senses in my partial definition, namely, its ontological sense, where simplicity is a property of a concrete particular or a relation ("simpler than") between concrete particulars. Ontological simplicity is a supervenient property, whose base or subvenient property is the quantity of a concrete particular's parts, logically independent properties, and degrees of degreed properties. My definition is partial since it merely states a sufficient condition ("if") and not all the necessary conditions. For the purposes of this paper, I need merely discuss a sufficient condition to argue that a timeless point is ontologically simpler than God.

    If we are talking about merely possible objects, then "simplicity is a property of the object x" is analyzed as meaning that simplicity is one of the contingently unexemplified conjunction of properties that would all be exemplified by x if the object x were to actual.

    One other sense of "simplicity" is the sense in which laws are simple. This is nomological simplicity. One law L1 is simpler than another law L2 if L1 has fewer independent variables (such as the temporal variable t and the position variable p) and theoretical constants (e.g. the gravitational constant N and velocity of light constant c) than L2. And one theory T1 is simpler than another theory T2 if T1 has less independent axioms than T2 and T1 has fewer theorems than T2 (or fewer classes of theorems, if both T1 and T2 have an infinite number of theorems based on each axiom). This is theoretical simplicity. Other senses of "simpler" (postulating fewer kinds of entities or postulating fewer instances of these kinds) are analyzable into a combination of two or more of these three senses of simplicity (ontological, nomological and theoretical).

   "Simplicity" is often used, or mis-used, to express other concepts, such as explanatory power, nonarbitrariness, symmetry, fit with background knowledge, conservativeness, etc. But this (mis-)usage of the word "simplicity" introduces needless imprecision in one's terminology and it is best to use different words for these different concepts. Increase in explanatory power may bring with it increase in theoretical simplicity (but not in the case in the transition from Newton's physics to general relativity!), but it does not follow from this that "explanatory power" expresses the same concept as "theoretical simplicity".

However, I am only talking about the simplicity of concrete particulars, ontological simplicity, which is defined in terms of the number of their parts, independent properties (n-adic properties, so we include both monadic properties and relations), and degrees of degree-properties. There is some historical and theological pedigree for this definition, even though for me "conforming to a theological tradition" does not carry any intrinsic evidential weight, since I am not doing revealed theology but metaphysics or philosophy of cosmology. Aquinas writes: ". . . we must discuss His simplicity, whereby we remove composition from Him" (p. 28 in Introduction to St. Thomas Aquinas, edited by A. Pegis; Summa Theologica I, Q. III, "The Simplicity of God"). This suggests the simplest thing has no parts, a result that will emerge from our analysis. But we cannot accept the details of Aquinas' theory, for reasons that will emerge later in our discussion.

 3. The Prior Probability of a Timeless, Singular Point

Leibniz, Swinburne, Parfit, Quinn and many others have suggested that nothing is simpler than something. Here the domain over which "something" ranges is the class of all concrete particulars. Some particular can be temporal but not spatial (e.g., an intentional act or a mental substance on a substance dualist philosophy of mind), some can be spatial but not temporal (e.g. a spatial point that exists timelessly) and many are both spatial and temporal (electrons, organisms and planets).

If nothing is simpler than something, then what is next simplest after nothing is the simplest possible thing (particular). If nothing is simpler than something, then the simplest possible thing is the thing nearest to being nothing at all (or the thing that is more similar to nothing than any other thing).

   But perhaps this Leibnizian way of talking is a bad way to start, since it unnecessarily saddles us with the myriad of objections that could be make to the logical coherency of the sentence "nothing is simpler than something". For example, it might be objected that nothingness cannot stand in the relation to something of being simpler than it and cannot have the property of being simple. For if nothing is truly nothing at all, what could stand in this relation or possess this property? I think these objections can be answered. For instance, one can analysis his sentence as meaning that a possible world in which no things exist is simpler than a possible world in which some things exist. Nonetheless, I shall adopt a more logically clear way of talking.

    The most logically clear way to introduce the notion of the ontologically simplest thing is to characterize such a thing in terms of degreed properties. The simplest concrete particular has degreed properties that all have the degree zero. Being temporally extended is a degreed property in the sense that x has a higher degree of temporal extension than y if x endures for a longer period of time than y. Something is temporally simpler than something else if it is less temporally extended than it. If something exists for an instant only, it has zero temporal dimensions. It has the degree zero of the degreed property of being temporally extended. A concrete particular that is even simpler than a temporally instantaneous particular lacks even the property of being instantaneous. It has zero temporal extension, but not because it is instantaneous; rather, it has zero temporal extension because it is timeless. It does not exist "for one instant only" but timelessly.

Philip Quinn [1997] has aptly noted that this characterization is unsatisfactory as it stands since it fails to take into account metrical conventionalism. However, if metrical realism ("time has one, intrinsic metric") is true, then this characterization need not be modified to account for conventionalism. I think realism is true, but nonetheless the account can be easily modified to accommodate conventionalism. We need merely say that relative to the metric convention C, x endures for a longer period than y, but relative to a different metric convention C', x endures for a shorter period than y. This does away with much of the intrinsic ontological significance of an entity's temporal simplicity, but it still leaves the notion of the simplest temporal particular with an intrinsic ontological notion. If x has a zero degree of temporal extension, it has a zero degree of temporal extension on every possible way of conventionally metricating time. A conventional metrication of time pertains only to the lengths of extended intervals and leaves untouched unextended times or points. Thus, regardless of whether metrical conventionalism or realism is true, it remains the case that the temporally simplest entity has a zero degree of temporal extension.

    Suppose something exists timelessly. If it is a concrete particular rather than an abstract particular (such as the null set) it must either be a timeless spatial particular or a timeless mental particular. Mental particulars shall be discussed in a later section. For now, let us discuss timeless spatial particulars.

    The simplest spatial particular is something with a zero degree of spatial extension and yet is a spatial point, a zero-dimensional spatial entity.

    Apart from considerations of shape (and considering only size), something is spatially simpler than something else if it is less spatially extended than it. The simplest way for a thing to exist spatially is to have zero spatial dimensions, i.e., to be a spatial point (as distinct from a line, square, cube, etc.). Since a spatial point has no shape, spatial size is all that is relevant to the simplest spatial thing. An instantaneously existing line, square or cube is a more spatially complex existent than an instantaneous point, just as a spatial point that endures for one hour or one billionth of a second is a more temporally complex particular than a point that has zero duration. Likewise, a timelessly existing spatial point is a simpler concrete particular than an instantaneously existing spatial particular.

Some philosophers, such as Swinburne, have argued that spatial particulars must exist in three dimensions, and others, such as Chisholm, have argued that physical objects must be spatially extended. I have elsewhere maintained that Swinburne's argument is invalid [Craig and Smith, 1993] and Chisholm's thesis pertains to physical things in a more narrow sense ("everyday objects" such as trees, tables, etc.) whereas I am talking about "things" in a wider sense, where any spatial particular (as opposed to a universal) counts as a thing. Furthermore, one cannot simply assert, without argument, that a spatial point is "ontological equivalent to nothing", as Craig writes [Craig and Smith: 1993 ]. Considering that Newtonian physics, Einstein's Special Theory of Relativity, Einstein's General Theory of Relativity, and Quantum Field Theory (the unification of Quantum Mechanics with the Special Theory of Relativity) all explicitly state that mass points exist, one would need some rather strong argument that all of physics since the 17th century is fundamentally false. (Craig's statement is false in any case, since if something has the property of being a point, it is something, not nothing, since nothing does not have any properties, including the property of being pointlike.).

    If it is objected that "spatial particulars must exist in time", I have two responses. First, what justifies the claim that a spatial particular must exist in time? Obviously, a changing spatial particular must exist in time. But why cannot an unchanging spatial point exist timelessly? If someone's belief that a point cannot exist timelessly is based on a "linguistic intuition", "modal intuition" or "metaphysical intuition" or the like, I would respond in three ways. First, what is the person's evidence that what she calls an "intuition" has any epistemic value and is not merely a psychological "hunch" (for a good critique of contemporary philosophers use of the epistemically laden word "intuition", and the charge that without further argument these should be considered nonepistemic "hunches", see Hintikka's article in Journal of Philosophy, March 1999.). Second, I would respond that I do not share any such "intuition" and, if we are trading in so-called "intuitions", I have the intuition that a spatial point can exist outside outside of the temporal dimension of general relativistic spacetime. However, I would prefer to junk pairs of philosophies that conflict merely because they are based on conflicting "intuitions" (with no other evidence to decide between them), so I will pass to a third response. The belief, hunch or intuition that a spatial particular must exist in time has already gone the way of the old post-Kantian belief that it is a synthetic a priori truth that every event has a sufficient cause. Both scientists and philosophers have now generally agreed that since the development of quantum mechanics in the late 1920s, such "causal intuitions" are in contradiction with contemporary physics. (We need the caveat that on a de Broglie-Bohm interpretation of quantum mechanics, events do have sufficient causes. But this is an a posteriori argument based on inductive criteria about which interpretation of quantum mechanics has the greatest explanatory power, conservativeness, fit with background knowledge, etc.) As for a purported "intuition" that a spatial particular cannot exist timelessly, this intuition has gone by the boards since the early 1980s. The two sorts of quantum gravity "wave functions of the universe", the Hartle-Hawking wave function [1983] and the Vilenkin wave function [1988] both imply that there exists a timeless four dimensional space that is topologically connected to our spacetime (for discussion, see [Smith, 1997].) If I have the "intuition" that the Earth is flat, am I epistemically entitled to reject contemporary astronomy? If I am rational, I engage in the metalevel judgement that my "intuition" that the Earth is flat is epistemically flawed. The same goes for "hunches" about timelessly existing spatial particulars.

   But more needs to be said about the simplicity of a timeless spatial point. What of its monadic properties other than its atemporality and spatially pointlike features?  The simplest possible concrete particular has no positive, non-trivial monadic properties". A trivial property of a thing is being something or being self-identical. A non-trivial property is being spatially extended or being rational.  A positive monadic property is any property that is not reductively definable in terms of a lack of a property or the possession of a zero-degree property. The property of being a zebra is a positive monadic property. But the simplest possible concrete particular lacks such properties and has only zero-degree monadic properties. It has the monadic property of being timeless, but this is analyzable into possessing the property of having a zero-degree of the degreed property of being temporally extended and into the lack of the positive monadic property of being temporally instantaneous. It has the monadic property of being spatial, but its spatiality is analyzable into the zero-degree property of having zero spatial extension. Thus, the simplest possible concrete particular does not have any positive, non-trivial monadic properties. For example, if this timeless point had the property of being a person, of having a mind, it would have a positive, essential, non-trivial essential property and thus would be more complex than a point that lacked this property.

The simplest thing is also compositionally the simplest possible concrete particular; that is, it has no parts. This follows immediately from the fact that it is a timeless, spatial point with no mental parts or properties.

If x stands in more independent relations to other things than does y, then x is more relationally complex than y. In terms of trivial relations, each particular has an infinite number of trivial relations such as being different than the number one, being different than the number two, and so on. Relevant to our case of a timeless cause of spacetime's beginning are non-trivial relations to grounds of something's existence. Since we are interested in the simplest cause of spacetime, we include in our discussion the point's contingent relation of being an efficient cause of something else's existence, namely, spacetime's. But we wish to exclude the point's existence having a ground in something else's existence. A point x is simpler than a point y if x does not have positive, relational properties of having grounds for its existence. It is uncaused, has no purpose, and is not dependent on the existence of any other concrete particular. Each concrete particular is dependent on the number three in the trivial sense that in each possible world in which the particular exists, the number three also exists. Other such trivial dependencies also obtain; I am concerned with non-trivial dependencies, such as being caused by a mind, being able to exist only as a part of a human, being able to exist only as a whole composed of three quarks, and the like.

 A caused point is more causally complex than an uncaused point, since a caused point has the relational property of being the effect of something other than itself. A timeless point that exists for a purpose is more teleologically complex than a purposelessly existing point, since it has an additional, positive relational property, namely, existing for the sake of realizing the purpose F. Further, being dependent on something else x is a positive, relational property that would make a timeless point more complex in this respect than a point that did not possess this positive property of being dependent on something.

However, since we are theorizing about the simplest cause of the existence of spacetime, our domain is confined to concrete particulars that have the relational property of being the ground of something else's existence, specifically, the property of being the cause of the existence of spacetime. This is the only positive, non-trivial, relational property possessed by the simplest particular in the class of particulars we are examining. But this relational property is contingent and so the class of essential, non-trivial, positive relational properties is empty for the simplest thing, the timeless point.

    The timeless point contingently causes spacetime to exist in David Lewis's sense of sufficient cause. Lewis's definition of causation is applicable since it allows for singularist causation, does not require temporal precedence or spatiotemporal contiguity, transfer of energy or the conditions that are part of other definitions of causality (see Smith, 1996). 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 timeless point c exists and spacetime's beginning e both exist, and spacetime would not have begin if there were no singular point. According to the Friedman 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 the singular point we have postulated. 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' definition of sufficient causation), if e had not occurred, c would have occurred but would have failed to cause e. In some physically possible worlds, the singular point c exists timelessly but does not cause spacetime to begin to exist. Spacetime requires a causal pointlike singularity to begin to exist, but the timeless point does not need to cause spacetime in order to exist. The point only is a "boundary" or "edge" of spacetime (to use these terms in the technical senses they have in general relativity) in cases where the point causes spacetime. The point has no nomological structure that requires it to cause spacetime to begin to exist or to exist as a boundary of spacetime. "A singularity is a place [a spatial point] where the classical concepts of space and time down as do all the known laws of physics" [Hawking, [1976: 2460].; the point's causation of spacetime is a physically accidental or contingent relation in which the point stands to spacetime. On the other hand, spacetime's beginning to exist as an effect of the singular point is a physically necessary relation in which spacetime stands to the singular point (according to the Hawking-Penrose singularity theorems and Friedman's equations).

Notice that in this case of causation there is no requirement of temporal precedence, spatiotemporal contiguity or instantiation of a law. Indeed, in our case we have a case of singularist causation. There are some similarities here to what would be required of a deity that existed outside of the time of general relativity, but a full analysis of the notion expressed by "divine causation" requires more space than I have here (see Smith [1996]; Vallicella [1999], Deltete [1999].)

   In the third section of this essay, we have discussed "The Prior Probability of a Timeless, Singular Point". We aim to treat the prior probability of a divine cause as well, and then to determine their respective posteriori probabilities. In the next section, I will show that, contrary to the theistic tradition from Plato to Swinburne and Craig, the prior probability of a deity is significantly lower than that of a nonmental, spatial point.

4. The Prior Probability of a Deity

(a) The Complexity of God   

What is the simplest possible cause of spacetime? Is it a timeless, unmetricated spatial point? Or is it a timeless mind? Is a mind simpler than a spatial point?

   Does a timeless mental cause have no positive, nontrivial properties and does it have only zero-degree properties that are nontrivial? Note that the spatial point has the nontrivial property of being an (unmetricated) spatial point and being timeless. Both of these properties are zero-degree properties; the spatial property is being spatially extended to a zero degree or having zero spatial dimensions. The spatial point is unextended but it does not have the property of not being spatial. Its property of being spatial is the same property as being spatial extended to a zero degree. This is a positive, nontrivial, zero-degree property. The point's lack of metrication is its failure to instantiate a basic law of general relativity. The timelessness is being temporally extended to a zero degree and in addition having the negative property of being temporally unextended by virtue of not being temporal.

   But a mind has many properties that are positive, nontrivial and not zero-degree. Since we are comparing the timeless point with the deity postulated by Judeo-Christian-Islamic monotheism, the mind in question is omniscient, omnipotent, perfectly good and free. These make God much more complex than the timeless point. But what of the traditional theological dictum that "God is supremely simple"?

   Let us begin with a contemporary theory of divine simplicity. Richard Swinburne argues that the fact that God is infinite powerful, wise, etc. makes him simple, since Swinburne holds that zero and infinity are equally simple. Swinburne has not discussed the simplicity of the singularity,  but he claims that zero and infinity are equally simple, which would suggest there could be two simplest possible things, a zero-degree thing and an infinite-degree thing. But I think Swinburne's theory can be shown to be mistaken. Swinburne is talking about the simplest possible simplest concrete thing in general, and he claims that since God is infinite there is nothing that is simpler than God. However, if we can refute his claims about infinity, we can show that an infinitely degreed thing is not equally as simple as a zero-degreed thing.

    Swinburne says zero and infinity are equally as simple and that something with a positive, finite number of characteristics is more complex. He claims this is the criterion of simplicity used by scientists. But I think Swinburne is conflating the principle of simplicity with the principle of nonarbitrariness. All else being equal, it would be more arbitrary to postulate a world with 37 stars than to postulate a world with zero stars or an infinite number of stars. Zero and infinity are less arbitrary quantitative features than some positive, finite number. But to claim that zero and infinity are equally as simple is a mistake. Zero is simpler than one, one is simpler than two, and infinity is infinity more complex than any finite number. Intuitively, it would be rather bizarre to think that if a solar system has a star with zero planets (and only asteroids), the next or equally simplest solar system is a star with an infinite number of planets. If the star does not have zero planets and if scientists wanted to postulate a solar system of the next simplest sort, they would postulate a solar system that has one planet. Whether or not this postulate is more or less arbitrary than some other postulate is a different issue, since "being nonarbitrary" and "being simple" are neither synonymous nor logically equivalent predicates. Swinburne's account of simplicity has further problems I have discussed elsewhere [Smith, 1998a].

 

(b) Second Order Logic and the Thesis that God's Attributes are Identical with Each Other

 

   "God is the simplest possible being" is usually associated with the Identity Thesis of philosophers from Augustine to the late scholastics, where God is identical with his attributes and his existence, and thus (by transitivity of identity) God's attributes are identical with each other. This doctrine has been revived in contemporary times mainly by Stump and Kretzmann, Brian Leftow and William Vallicella. As Leftow puts it, "if God is simple, it is false that God is the subject of a distinct attribute, goodness. Rather, if God is simple, 'God is good' is true because of precisely the same SOA [state of affairs] making 'God is God' true, and this 'SOA' contains no components or complexity of any sort." [1991: 67]. By the same token, "God is powerful" is true because of precisely the same partless state of affairs that makes "God is good' true. It seems to me that the natural response to this doctrine is that it is an explicit contradiction in second order logic. Being powerful is a distinct attribute than being knowledgeable, since a person can be knowledgeable to a very high degree and yet powerful to a very low degree; likewise, someone can be powerful to a very high degree but knowledge to a very low degree. One and the same property cannot both be of a high degree and a low degree in the same respect at the same time. Now the Identity Theory of God holds that these attributes are identical at their maximal degree--being all powerful, being all knowing, etc. Now it is a theorem of second order logic that if property F (being powerful) is different than property G (being knowledgeable), then degrees of F are different than the degrees of G, regardless of the degree. If F is different than G, then being all-F (all-powerful) is different than being all-G (all good). Usually, second order logic is semantically interpreted as quantifying over classes, but I align myself with the logicians who semantically interpret it as quantifying over properties. Nominalists about properties prefer to interpret second order logic as quantifying over classes, whereas realists (Aristotelian realists or Platonic realists) prefer to interpret this logic as quantifying over properties. I am realist about properties and I interpret second order logic in accordance with this realism.

We have the formalism:

 

F is a predicate constant, expressing the degreed property of being powerful.

G is a predicate constant, expressing the degreed property of being knowledgeable.

F_n is a degree of the property expressed by F and _GN is a degree of the property expressed by G.

Let F_n-max and _GN-max be the highest degrees of F and G.

 

(1) ($(F Ù G)) Ù  (F = G).

(2) F_n É F.  

(3) _GN É G.

Therefore,

(4) F_n = _GN

Therefore

(5) F_n-max = _GN-max.

 

Since the Identity Thesis implies the negation of (5), it is a negation of a theorem of second order logic. It is logically impossible for the divine attributes to be identical with each other. The medieval and early modern concept of divine simplicity is logically inconsistent.

    A defender of the Identity Thesis may respond that second order logic not only is consistent with the Identity Thesis but even includes it as a theorem. This is because second order logic has for its axioms all the axioms of first order predicate logic with identity, plus two additional axioms. The additional axioms are the axiom of extensionality and the axiom of comprehension.

In my statement of these axioms, Π (pie) and φ (phi) are metavariables ranging over predicates, and z₁, etc, are individual variables. "º" means material equivalence and "=" means identity (of some sort, e.g. extensional identity). The axiom of extensionality is that:

 (z₁) , , , (z_n)Π(z₁. . . z_n) º φ (z₁ . . . z_n)) É Π = φ.

 

This implies that Π is identical with φ if everything z is such that it has Π if and only if it has φ. I will discuss the relevance of this axiom shortly. The axiom of comprehension is

 (Eφ)(z₁) . . . (z_n)(φ(z₁ . . . z_n) º A).

 

This axiom implies that there is a property φ such that everything z is such that it has φ if and only if the sentence A is true. A is a sentence implying that everything that satisfies the conditions mentioned in the sentence has the property φ.

   Now how can the theist demonstrate the Identity Thesis in second order logic? The axiom of extensionality says that Π = φ if and only if each individual z that has Π also has φ. But there is only one individual z₁ that has the property F_n-max and there is only one individual z₂ that has the property G_n-max, and z₁ = z₂. This individual is God. Therefore, F_n-max =  G_n-max. The Identity Thesis is a theorem of second order logic.

   But in order to accept this proof, the theist has to accept the axiom of extensionality. But no theist can accept this axiom, since it is inconsistent with theism. Many examples show this. For example, it is plausible that each morally active human has done at least one morally wrong action E and also has done at least one morally right action R. It follows that something z possesses the property E of humanly performing a morally wrong action each if and only if z also possesses the property R of humanly performing a morally right action. Since the biconditional states a material equivalence in the extensionality axiom, it follows that E = R. This is inconsistent with theism, since God approves of exemplifications of R and disapproves of exemplifications of E, and God cannot do this if E = R. It is a contradiction for God to approve and disapprove in the same respect of the same exemplification, the exemplification of the single property E = R.

    Second order logic can be done without the axiom of extensionality, and indeed this axiom is rejected by modally oriented theists and others who countenance intensional entities and identity relations strong than extensional identity. But this implies that the proof for the Identity Theory based on the axiom of extensionality is not valid, and that the disproof of the Identity Theory I earlier offered is valid. My disproof is based on a rejection of the axiom of extensionality. I accept only the axiom of comprehension, which states that there is a property (e.g., being knowledgeable to some degree) such that everything has that property if and only if the sentence A is true, where A is a sentence stating that everything that fulfills the conditions mentioned in that sentence has the property. For example, A may be a sentence that states that everything meets all the individually necessary and jointly sufficient conditions of being knowledgeable to some degree has the property G of being powerful to some degree.

(c) Nonarbitrariness As Well as Simplicity Determines the A Priori Probability of God

With the god of the Identity Theorists being excluded by second order logic, we left with the usual conception of God accepted by contemporary theists, such as Plantinga, Craig and Swinburne, namely, that God is distinct from his attributes. But now we simply return to our comment that such a God is infinitely complex, and infinity-- contra Swinburne --is not as equally simple as zero.

    Thus, by a principle of inductive logic, namely, that the simpler something is, then (ceteris paribus) the higher its prior probability, it follows that the prior probability of a timeless point is higher than the prior probability of the deity hypothesized by monotheism.

But how should we understand the ceteris paribus clause? Suppose one grants that the timeless point is simpler than God. Suppose one further grants the principle of inductive logic, that, all else being equal, the simpler of two items has greater prior probability. The theist may still fasten on to the qualifying phrase "all else being equal". Simplicity is not the only principle determining prior probabilities in inductive logic. There is another principle that is called by one word by physicists and another word by a few philosophers. Physicists very frequently offer as an a priori justification for postulating a basic law or the initial boundary conditions that they are the most "natural". For example, this is precisely how cosmologists (Hartle and Hawking, Vilenkin, Linde, Pagels and many others) defend a priori their postulation of the most fundamental laws and initial conditions of the universe. Physicists know perfectly well what concept the word "natural" expresses in their talk about prior probabilities, but (due to differences in standard terminology) philosophers are likely to be baffled by (or dismiss as "metaphysical nonsense") the idea that one fundamental law of nature or set of boundary conditions has greater a priori probability than another law or set because it is "more natural". What physicists mean by this word is what philosophers would call "less arbitrary". The degree of naturalness of L = the degree of nonarbitrariness of L. Nonarbitrariness is partly determined by maxima and minima: this has been seen clearly by Parfit [1999], by Swinburne under a different name ("simplicity", as we have seen), and by Unger [1981]. Parfit and Unger agree that the least arbitrary ontological theory is one that postulates the maximum (all possible worlds are actual) or the minimum (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 nonarbitrary (or "simple" as he mistakenly called it).

Although maxima and minima are clear examples of nonarbitrary state of affairs, they are only some of the kinds or ways of being nonarbitrary. A partial definition is that an entity (concrete particular) x is less arbitrary than a particular y if x has the maximum or minimum degree of its degreed properties and y has non-maximal and non-minimal degrees of its degreed properties. Thus, a zero dimensional space is equally as nonarbitrary as an infinite dimensional space, and an event (e.g. in the sense of Special or General Relativity) that is instantaneous is equally as nonarbitrary as an infinitely long process of the order type ω * + ω (. . . -2, -1, 0, 1, 2,. . . .). These are not subjectively relative and epistemically unreliable "intuitions", as Quinn might suggest. Rather, they are logically implied by the definition I offered of nonarbitrariness and my definition has the epistemic property that it is evident of itself to a sufficiently informed person. This is epistemically partly similar to the inductive principle of the uniformity of nature or the reliability of induction; these principles are either evident of themselves or else are truth-valueless "conventional stipulations". But if they are the latter, then all a posteriori beliefs are truth-valueless, since justification is truth-value preserving relation (and thus preserves truth-valuelessness if the basic principles of empirical "knowledge" are themselves neither true nor false.) As far as I can see, conventionalism and relativism are self-referentially unjustified doctrines [see Smith, Ethical 1997: pp. ] and the admission of self-evident propositions is unavoidable if knowledge is possible at all. (But I certainly would allow there are degrees of self-evidence; see T.D. Sullivan [1994]) and Plantinga [1993].)

    Cannot we say that God has the maximal degree of his degreed properties and qua maximal is nonarbitrary? Certainly. But we also can say that the point has the minimal degree of its degree properties and qua minimal is nonarbitrary. The point and God are equally nonarbitrary and thus have equal a priori probability considering only the principle of nonarbitrariness. But since the principle of simplicity also determines a priori probability, the greater simplicity of the timeless point makes its over-all a priori probability higher than God's.

This does not end the matter, since the maximum/minimum distinction is only one of the species of nonarbitrariness. Some entity x is more arbitrary than some entity y if and only if (all else being equal) x's existence has an explanation and y's existence does not. Craig uses the phrase "explanatorily simpler" to express this second species of nonarbitrariness. Craig writes of my earlier theory in [Craig and Smith, 1993] that the singularity exists in time, for one instant at t = 0, that my atheistic theory of this singularity "is not explantorily simpler [less arbitrary] than theism" [Craig and Smith, 1993: 273]. 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 on a part 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." [Craig and Smith, 1993: 272-273].

 

Suppose the singularity exists in time and exists for an instant only. It still would have a (noncausal) reason for happening, namely, that it is the simplest possible thing and the simpler something is, the more likely it is to exist unexplained (see [Craig and Smith, 1993: 249-251; Smith, 1997a] for a detailed defence of this claim]. But for reasons articulated in Smith [1999],I now accept that the disproofs of the c-bundle construction and g-bundle construction (for attaching the singularity to the spacetime manifold as the earliest instant t = 0) justify the belief that the singularity exists outside of the time postulated by the general theory of relativity. It can be added to our spacetime manifold (but not to many other kinds of spacetime) as a "place" on the boundary of the spacetime manifold for reasons given by Wald [1984]. This will be discussed more below.

    If the singularity exists timelessly, it is not subject to the criticisms levelled in Craig's passage. Further, Craig is wrong when he says of God and the singularity that "both can be said to be without cause or reason" [Craig and Smith, 1993: 272]. They are both timeless and uncaused, but both have a significant a priori probability of existing and if they exist, this a priori probability is a probabilistic and noncausal reason why they exist. Not all explanations or reasons are causal; I agree with the many philosophers who object to the Salmon-Humphreys position that to explain something is to find its cause. Many explanations are causal, but there are also noncausal explanations, such as explanations in terms of prior probabilities of existing uncaused (see [Smith, 1990; 1994; 1997b; 1998a]).

Thus, Craig is left with only one alleged reason why God is less explanatorily arbitrary than the timeless point. But I think Craig states this reason in an erroneous way; he says "God is without a reason for His existence in the sense that His existence is metaphysically necessary." [Craig and Smith, 1993]. But if God exists in every metaphysically possible, 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 does a square have four sides is "Because it belongs to the essence of a square to have four sides", so the answer to "why does God exist?" is "Because it belongs to God's essence to exist".

However, Craig has given no justification whatsoever for his belief that God's existence is metaphysically necessary. If he has an "intuition" about this, this so-called intuition (for all Craig has said) amounts to what Hintikka has called an epistemically valueless "hunch". And even if, contrary to fact, there is an "intuition" that God is metaphysically necessary that has epistemic weight relative to certain "intuiters", to coin a word, then another group of "intuiters" could have an equally weighty "intuition" that God is not metaphysically necessary and that the timeless point is metaphysically necessary. Just as God is said to exist in every world but to cause spacetime only in some worlds, so the point exists in every world but causes spacetime only in some worlds. The transworld identification of this point can be intuitively made by theoretically ostending (or having in mind) "dthat point" (to use Kaplan's rigidfying functor) and saying of it that it exists in every possible world. Earman's [1990] argument that points are individuated by their relation to other spacetime points is irrelevant to the case at hand, since our timeless point is not a spacetime point and Earman argued only that this individuation condition applies to a certain class of points, spacetime points. But if we grant the relevant "intuitions (hunches) of metaphysical necessities" to the respective groups of "intuiters", is there any objective, interpersonally valid and "non-hunchlike" epistemic grounds that could decide which of the group of "intuiters" is more likely to have reliable "intuitions"? The foregoing arguments imply that the a prior probability of the timeless point is higher than God's, since while the point is equally nonarbitrary as God, the point is simpler. This gives reason to think that the people who have the "intuition" that God is not metaphysically necessary but that the timeless point is necessary are more probably right on a priori grounds. However, there is no need to think the timeless point is metaphysically necessary for the explanation in this essay to work. All we need to establish is that it exists in all physically possible worlds and can exist in some worlds where there is no spacetime. There can be many metaphysically possible worlds in which the timeless point does not exist. If the theist falls back on her "hunch" that God is metaphysically necessary, and rejects the point-cause hypothesis on this ground, then I would respond that she has abandoned natural theology and is doing revealed theology. Given this, I have no philosophical interest in her hunches; I am interested only in explanations of spacetime that rely on deductive logic, inductive logic and observational evidence. Mystical experiences deserve evaluation, but I think the conclusion of such evaluation is that they are evidentially counterbalanced by experiences of God's nonexistence [Smith, 1986: ], proving at best an agnosticism if experiences or "revelations" are considered as our only data.

The natural theologian may be object that God and the timeless point are not equally nonarbitrary in respect of their causally explaining the existence of spacetime. It may be said that God causes spacetime to exist for a reason, whereas the timeless point causes it to exist for no reason. It is not an arbitrary or brute fact that God causes spacetime, but it is an arbitrary or brute fact that the timeless point causes spacetime. However, the state of affairs consisting of God or the timeless point causing spacetime to exist obtains for no reason, and so the maximal mental particular (God) and the minimal physical particular (the point) are equally arbitrary in this respect. The state of affairs, God's causing or performing her act of creating spacetime for the reason that it is good that spacetime exists is uncaused and unexplained, representing a brute fact about God.

Some theists, such as Robert Deltete, do not seem to understand adequately the fundamental arbitrariness that belongs to the state of affairs postulated by theism. They misunderstand the my account of its arbitrariness since they mistakenly attribute it to the divine act of creation, where it does not belong. Thus, Deltete says my account of the arbitrariness of the theistic state of affairs implies the divine free acts are "arbitrary, since there are no reasons for them" [Deltete, 1998: 494]. Deltete argues against me that God's creation of spacetime is not "merely arbitrary" but has a reason. Deltete briefly asserts that anyone who claim that the singularity and God are equally arbitrary in respect of the state of affairs consisting of their causing spacetime has a "woefully inadequate" understanding of divine agency. In fact, it is theists such as Deltete who do not correctly understand divine agency, both in the respect I have in mind and in other respects (cf. [Smith, 1996], but also [Vallicella, 1999] and Deltete [1999] for interesting and intelligent responses to which I hope to reply elsewhere).

    It seems to me that there are four "woeful inadequacies", so to speak, in Deltete's critique and in his own theory of divine agency: