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Introduction

The attribute of divine aseity is defined. God alone is self-existent and therefore uncreated, the sole ultimate reality. Platonism’s challenge to the doctrine of divine aseity is introduced. Platonism posits the existence of uncreated abstract objects, so that God is not the sole ultimate reality. The distinction between abstract and concrete objects is expounded. A distinction between heavyweight and lightweight Platonism is drawn. It is explained that only heavyweight Platonism challenges divine aseity and so will be the subject of discussion. The term ‘nominalism’ is shunned because of the ambiguity and misleading connotations of the term. Realism, arealism, and anti-realism about mathematical objects are introduced, realism holding that such objects do exist, anti-realism denying their existence, and arealism holding that there is no fact of the matter.

Central to the Judaeo-Christian concept of God is the notion that God is a self-existent being. That is to say, God is not dependent upon any other being for His existence; rather, He exists independently of everything else. Were everything else magically to disappear, God would still exist. God has the property or attribute of self-existence.

Divine Aseity

This attribute of God is called aseity. The word derives from the Latin a se, which means of itself or from itself. God does not exist through another or from another. He just exists in and of Himself, independent of everything else. In other words, He is a self-existent being.

Protestant scholastic theologians typically distinguished between the communicable and incommunicable attributes of God. Communicable attributes are those which God shares with created things, though to different degrees. For example, knowledge and power are possessed by both God and created things, by created things to a finite degree, but by God, who is omniscient and omnipotent, to an infinite degree. Incommunicable attributes, on the other hand, are those which are unique to God. Nothing else has these attributes.

Aseity is traditionally held to be one of the incommunicable attributes of God. God alone is self-existent; everything else is dependent for its existence upon something else. Thus, the doctrine of divine aseity is closely related to the doctrine of creation. According to that doctrine, everything that exists (other than God) has been created by God. So everything that exists other than God is a created thing. Such things are therefore not self-existent, but are dependent for their (p.2) existence upon God, their Creator. Even God could not create a self-existent being, for a created, self- existent being is as logically incoherent as a round triangle or a married bachelor. To be self-existent is to be uncreated. So anything apart from God is a created being and therefore not self-existent. Aseity is thus an incommunicable attribute of God.

So on the traditional conception, God is what the philosopher Brian Leftow calls ‘the sole ultimate reality’,1 the

pinnacle of being, so to speak. For all other beings have been created by Him and therefore depend on Him for their existence, whereas God depends upon nothing else for His existence and is the source of existence of everything else. In the next chapter, I shall say something further about the biblical and theological motivations for the traditional doctrine of divine aseity, but now I want to introduce the most formidable challenge to the coherence of the doctrine.

Platonism

The strongest challenge to the coherence of the traditional doctrine of divine aseity comes from the philosophy of Platonism. Plato (429–347 BC) held that there exist uncreated entities other than God. These are not part of the physical world, which God has created, but are part of a transcendent, conceptual realm comprising what Plato called Ideas or Forms. They include mathematical objects like numbers and geometrical shapes, such as the perfect circle or triangle, which are not to be found in the physical realm. Plato held that, far from being created by God,

these transcendent realities served as God’s model or pattern after which He fashioned the physical

world.2 Although we might be inclined to look upon such objects as having at best a ghostly sort of existence, for Plato the objects of this transcendent realm were actually more real than the objects of the physical world, which are like mere shadows of these transcendent realities. (p.3) Plato held that these transcendent, ideal objects are uncreated, necessary, and eternal. God is therefore not the sole ultimate reality.

Contemporary Platonism differs vastly from classical Platonism in various respects;3 but both views are united in

holding that there exist uncreated entities—for example, mathematical objects—other than God. Contemporary Platonists call such entities ‘abstract’ objects in order to distinguish them from concrete objects like people, planets, and chairs. Insofar as these abstract objects are taken to be uncreated, necessary, and eternal, contemporary Platonism also comes into conflict with the traditional doctrines of divine aseity and creation.

Abstract and Concrete Objects

How should we understand the distinction which contemporary philosophers draw between abstract and concrete objects? Although the distinction is commonplace, it remains a matter of dispute just how to draw that distinction. Many philosophers have simply given up the job of supplying a criterion to distinguish abstract from concrete objects, choosing instead simply to point to examples which serve as paradigms of each type of object. For example, physical objects are universally taken to be examples of concrete objects. On the other hand, mathematical objects like numbers, functions, and sets are regarded as paradigmatic examples of abstract objects. Only slightly more controversially, properties (universal qualities which are exemplified by particulars), propositions (the information content of sentences), and possible worlds (ways reality might have been) are taken by most philosophers to be paradigmatic abstract objects. Most philosophers would agree that, if there are such things, then they are abstract rather than concrete objects. So a discussion of the reality of abstract objects is usually able to proceed on the basis of such shared examples, even in the absence of a clearly enunciated criterion distinguishing abstract from concrete objects. Indeed, any proposed criterion will be assessed by how well it categorizes such paradigm examples.

(p.4) On the basis of the examples thus far considered, one might be tempted to think that the distinction between the concrete and the abstract is the same as between the material and the immaterial. But a moment’s reflection shows that that cannot be correct. For if immaterial agents like souls or angels exist, they would indisputably fall in the class of concrete, not abstract, objects. Everyone recognizes that there is a world of difference between such causally active agents and things like numbers, propositions, and possible worlds. So even if, as it seems, all abstract objects are immaterial, not all immaterial objects are abstract. There could be objects which are both concrete and immaterial, so that the abstract/concrete distinction cannot be equated with the immaterial/material distinction.

One might then think that concrete objects, whether material or immaterial, are all spatiotemporal objects, while abstract objects are without exception non-spatiotemporal objects, that is, objects which transcend space and time. Is that how the distinction between abstract and concrete ought to be drawn? Again, the answer is, no. For God, if He exists, is plausibly thought to exist beyond space and time (at least sans the universe), being the free Creator of space and time. Yet, as a causal agent who has created the world, God would be a paradigmatic concrete object.

Moreover, while some abstract objects—most notably, certain mathematical objects like numbers—would seem to exist beyond space and time, if they exist at all, that is not the case for many other kinds of abstract object. Take properties, for example. Particular things are constantly changing in their properties, acquiring some and losing others at different times. Properties must therefore exist in time, if not in space, since they are constantly changing in their relation to temporal things, being exemplified by a particular object at one time and no longer exemplified by it at a later time. Such relational change is sufficient for being in time. Or consider propositions. They are even more clearly temporal, since many of them seem to change not just relationally, but intrinsically, over time. For many propositions—for example, George Bush is the President of the United States—are plausibly variable in their truth value, being sometimes true and sometimes false. If the propositional content of tensed sentences includes the sentences’ tense, then the propositions expressed by such sentences must exist in time, since they undergo intrinsic change with respect to their truth value.

There are even more evident, if recherché, examples of abstract objects which, if they exist at all, exist spatiotemporally. For example, (p.5) the Equator is a geometrical line which girdles the Earth and therefore exists in space. You can actually step over this abstract object! Moreover, it depends on the Earth for its existence and so exists only so long as the Earth exists. It is thus an abstract, spatiotemporal object. Or consider the centre of mass of the solar system. This is a point whose spatial location is constantly changing as the planets revolve around the Sun.

You could actually enclose this abstract object in the hollow of your hand—though not for long, since it would pass right through your hand to another location! One cannot therefore equate concrete objects with spatiotemporal objects and abstract objects with non-spatiotemporal objects.

The last two examples also serve to show that the distinction between the abstract and the concrete is not equivalent to the distinction between the metaphysically necessary and the metaphysically contingent.4 While numbers, propositions, properties, and possible worlds do seem to be metaphysically necessary, if they exist, that is not the case for all abstract objects, as the examples of the Equator and the centre of mass of the solar system show.

Moreover, many contemporary Platonists think that literary and musical compositions are abstract objects, not to be identified with any particular exemplar of those works. For example, Beethoven’s Fifth Symphony cannot plausibly be identified with some printed score, lest we be compelled to say that, if that score were destroyed,

Beethoven’s Fifth Symphony would no longer exist! But since literary and musical works are plausibly the creations of their respective authors, most Platonists hold them not to be metaphysically necessary in their existence.

Moreover, in mathematics, so-called ‘impure’ sets, that is to say, sets which have non-sets—for example, people— as their members, do not exist necessarily, since if their members do not exist, the set does not exist either. For example, the set {Ronald Reagan, Gerald Ford} would not exist in a world in which those two men never existed. In fact, this set no longer exists if Reagan and Ford have perished.

Neither are all concrete objects metaphysically contingent, since God is usually considered to exist necessarily, if He exists at all, and (p.6) yet, as I have said, God is a paradigmatic concrete object, being a personal agent endowed with causal powers and causally active in the world, interacting with other concrete objects.

Perhaps here we have a clue as to how the distinction between abstract and concrete objects is best drawn. It is very widely held among philosophers that abstract objects, in contrast to concrete objects, are causally impotent and so are not related to other objects as causes to effects. Moreover, their causal impotence seems to be an essential feature of abstract objects. The number 2, for example, does not just happen to be causally effete. It seems inconceivable that 2 could possess causal powers. Abstract objects’ causal impotence entails that they are immaterial, for if they were material objects, they would exist in time and space and so could come into contact with other things, thereby affecting those things. No wonder, then, that some thinkers have too hastily concluded that the abstract/concrete distinction just is the immaterial/material distinction! Perhaps the reason abstract objects are causally effete is precisely because they are neither material objects nor personal agents. Be that as it may, the criterion of essential causal impotence seems to delineate effectively abstract from concrete objects.

Fortunately, little hangs upon a successful delineation between abstract and concrete objects, for what is theologically problematic about such objects is not their abstractness, but their uncreatability, along with their necessity and eternality. Any object, whether concrete or abstract, which is uncreated will fatally compromise God’s being the sole ultimate reality. The theist can happily admit the existence of created, contingent, transitory abstract objects like the Equator or Beethoven’s Fifth. What he cannot allow is the existence of things which are as ontologically ultimate as God. The reason abstract objects are at the centre of this controversy is simply because they are the most—perhaps only—plausible candidates for uncreated, necessary, eternal objects apart from God Himself.

Terminological Clarifications

Platonism/Anti-Platonism and Realism/Anti-Realism

Before proceeding further, it will be helpful—indeed, almost crucial if we are to avoid misunderstanding—to clarify some terminology. (p.7) Platonism is the view that abstract objects exist, while anti-Platonism is the view that abstract objects do not exist. Sometimes these two views are equated with realism and anti-realismrespectively, but this equation is misleading. For, as we shall see, there are anti-Platonists who believe in the reality of mathematical objects, propositions, and so on, but who think that these objects are concrete, not abstract. Thus, some anti- Platonists are realists and some are anti-realists about said objects. I propose, therefore, that we take realism to be any view according to which mathematical objects, properties, possible worlds, and so on, exist and anti-realism to be any view that such objects do not exist. (Obviously, someone might be a realist about some objects, say, numbers, but an anti-realist about others, for example, possible worlds.) So a realist about some such objects might be either a Platonist or an anti-Platonist; but since the anti-realist about such objects holds that they do not exist, he is obviously an anti-Platonist.

Nominalism

In the literature, both anti-Platonism and anti-realism, as I have defined them, have been called nominalism. Since anti-Platonism and anti-realism are two different views, this label is very confusing. Moreover, there are two further reasons to shun the use of this label. First, ‘nominalism’ is a term which is used in two different philosophical debates to denominate very different views.5 The first is the age-old dispute over the existence of universals. In this

debate, nominalism is the view that universals do not exist, that everything that exists is a particular. The second debate is a very recent discussion, centred in the philosophy of mathematics, that has arisen only since the publication of the German mathematician Gottlob Frege’s Foundations of Arithmetic(1884). In this debate, the word ‘nominalism’ is often used as a synonym for anti-Platonism about abstract objects (abstract mathematical objects, for example, do not exist).

The problem is that a person who is a nominalist in one debate may not be a nominalist in the other debate. For example, in the old (p.8) dispute over universals, one type of nominalism is called class nominalism, according to which similar objects are those included in a certain class. Since classes are abstract objects akin to sets, however, such a thinker is not a nominalist in the second debate, but rather a Platonist. Similarly, a person who identifies universals as thoughts in someone’s mind is a nominalist in the second debate (since thoughts are concrete, not abstract objects), but not a nominalist in the first debate (since he takes thoughts to be real and therefore universals to be real). The tendency of some philosophers to blur the lines of these two debates by use of the word ‘nominalism’ has therefore been a source of confusion.

The second reason I think it advisable to avoid the word ‘nominalism’ is because of the very negative theological connotations—such as that God is not essentially good6—which the term acquired as a result of the first debate. These are utterly foreign to nominalism as defined in the second debate, being a development which has had wings

only since the publication of Hartry Field’s groundbreaking book Science without Numbers (1980). It would be theologically prejudicial to call positions in this second debate nominalist.

So in this book I shall refer to the position that mathematical objects (or propositions or properties or what have you) exist as realism with respect to such objects, and the position that they do not exist as anti-realism. As Figure 1 (p.

xii) illustrates, anti-realism comprises a diverse range of specific views; there are many different perspectives which count as anti-realist. Realism comprises any view that holds that things like mathematical objects exist, whether concretely or abstractly. Clearly, then, realism also comprises a diversity of specific views, some Platonist and some anti-Platonist.

Platonism

There is one more important terminological clarification that needs to be made, and that concerns the term ‘Platonism’ itself. For there are today two very different views on offer, both claiming the label of ‘Platonism’. One is a sort of ‘heavyweight’ Platonism which takes abstract objects to be just as real as the physical objects which make (p.9) up the world. For this sort of Platonist, numbers are just like automobiles, only more numerous, abstract, and eternal.7 Such a comparison makes us smile; but it serves to underline the seriousness of the heavyweight Platonist’s ontological commitment to abstract objects. As Michael Dummett says, ‘The mathematician is, therefore, concerned, on this view, with the correct description of a special realm of reality, comparable to the physical realms described by the geographer and the astronomer.’8 For the heavyweight Platonist, our ontological inventory of the world must include numbers, along with concrete objects.

The metaphysician Peter van Inwagen of the University of Notre Dame is doubtless the most prominent heavyweight Christian Platonist on the contemporary philosophical scene. Van Inwagen divides reality into two exclusive and exhaustive categories, the abstract and the concrete, and says that the objects belonging to each category exist in precisely the same sense.9 He has argued specifically for the existence of abstract objects like properties, shapes, and fictional characters.10 He rejects creationist views of such objects as well as anti-realist views, holding such objects to be, like God, uncreated things.11 He admits that such a viewpoint makes him uncomfortable, but he feels rationally obliged to concede the existence of such uncreated objects.

By contrast, there is also a sort of ‘lightweight’ Platonism whose ontological commitment to abstract objects is much more obscure. (p.10) For these thinkers, abstract objects seem to be merely semantic objects: they are what we are talking about when we use abstract terms like ‘3’ or ‘the square root of 9’. They need be no more real than grammatical objects. Something can be grammatically the direct object of a sentence without being a really existing object, as in ‘The Press Secretary knew the whereabouts of the Prime Minister.’ Similarly, ‘the whereabouts of the

Prime Minister’ can be semantically a term we use to talk about his whereabouts, that is, the term refers to his whereabouts, without implying that there is some really existing object which is the Prime Minister’s whereabouts. Lest you think this a bizarre example, the Prime Minister’s whereabouts is precisely one of the examples the Platonist philosopher Bob Hale uses to illustrate the abstract objects which serve as the semantic referents of certain terms. With regard to the question whether such objects exist, Hale says bluntly,

If it is taken as invoking the everyday notion of object, the question whether there are abstract objects is devoid of philosophical interest; its answer is quite certainly that there are not, but that is trivial—a great many kinds of thing beside those whose title to be recognized as abstract objects has been taken seriously by philosophers fail to count as objects in that sense. Vague though the common notion is, it is evidently outrageous to suggest that numbers, classes, directions and shapes, say, are objects in that sense. But the same goes for hurricanes, speeches (i.e., the actual historical events) and holes in the ground.12

Hale’s remarks are quite puzzling. He grants that abstract objects are not objects in the ordinary sense of the word and so in that sense do not exist. But he is unperturbed by this admission because many other things—such as hurricanes, speeches, and holes—do not exist in that sense either. Hale’s remarks are puzzling because his examples

of things that do not exist in the sense that ordinary objects exist are precisely things which many ontologists deny do exist, period.13 (p.11) Hurricanes and speeches are not things, but events, and many ontologists deny that events exist. Holes are probably the favourite illustration which ontologists use of something that we commonly talk about

but which does not exist. So if numbers, classes, and other mathematical objects have no more reality than holes and hurricanes, Hale ought to count as an anti-realist, not a Platonist.

John Burgess, another prominent lightweight Platonist, considers the question of the existence of mathematical objects in a theological light:

One very traditional sort of way to try to make sense of the question of the ultimate metaphysical existence of numbers would be to turn the ontological question into a theological question: Did it or did it not happen, on one of the days of creation, that God said, ‘Let there be numbers!’ and there were numbers, and God saw the numbers, that they were good? According to Dummett, and according to Nietzsche—or my perspective on Nietzsche—this is

the only way to make sense of questions of ontological metaphysics.…I myself believe, like Russell, that analytic atheism [the thesis that theological language is meaningless] is false, and suspect, contrary to the Australians, that the Nietzsche-Dummett thesis is true. If as I believe the theological question does make sense, and if as I suspect it is the only sensible question about the italics-added real or capital-R Real existence of numbers, then I would answer that question in the negative; but then I would equally answer in the negative the question of

the Real existence of just about anything.14

Burgess rejects what he calls ‘capital-R Realism’ in favour of a much weaker ‘realism’.15 This weak realism does not presume to tell us ‘just what God was saying to Himself when He was creating the universe’.16 The fact that Burgess thinks that very few things exist in the metaphysically heavy sense merely goes to show that he agrees with certain metaphysicians that composite material objects do not (p.12) exist.17 For Burgess, very few kinds of things exist—perhaps only fundamental particles—and abstract objects are not among them. Like Hale, he actually seems to be an anti-realist about abstract objects.

These lightweight Platonists—who are among the most ardent defenders of Platonism today—thus seem to be committed to abstract objects only in the sense that they are semantic objects. As Burgess’s statement implies, such a lightweight Platonism is not incompatible with God’s being the sole ultimate reality and the Creator of everything that exists other than Himself.

The focus of our investigation is therefore heavyweight Platonism, for this is the only kind of Platonism that is in conflict with the doctrine of divine aseity. From now on, then, whenever I refer to Platonism, it is metaphysically heavyweight Platonism that I have in mind. We shall want to understand what grounds there are for affirming heavyweight Platonism and how one might respond to it. As we shall see, there is one argument in favour of Platonism that dominates the contemporary discussion. We shall then embark on a wide-ranging exploration of various responses to that argument, with a view to assessing their credibility and utility to the classical theist. I hope to show that there is a cornucopia of viable responses available to the theist, many of which contemporary Christian philosophers have scarcely begun to explore.

But before we embark on our journey, we need to examine more closely the biblical and theological foundations of the doctrine of divine aseity.

Notes:

(15) Ibid., 19.

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