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Aseity from the Latin a se “by itself” is God’s selfexistence or independence. God does not merely exist in every possible world as great as that is but, even more greatly, he exists in every world wholly independently of anything else. Moreover, God is unique in his aseity all other things exist ab alio through another.1 In virtue of his aseity, God is, in Brian Leftow’s words, “the sole ultimate reality.”2 4.1 Biblical Data Concerning Divine Aseity The conception of God as the sole ultimate reality is firmly rooted in Scripture, Church tradition, and perfect being theology.3 The biblical witness to God’s sole ultimacy is both abundant and clear. Undoubtedly one of the most important biblical texts, both theologically and historically, in this regard is the prologue of the Gospel of John. Speaking of the preincarnate Christ as the Logos or Word 1.14, John4 writes, In the beginning was the Word, and the Word was with God, and the Word was God. He was in the beginning with God. All things came into being through him, and without him not one thing came into being 1.1–3. “All things” panta connotes all things taken severally, not simply the Whole. Of course, God is implicitly exempted from inclusion in “all things,” since he has already been said to have been ēn in the beginning en archē v. 1. God and the Logos are not the subject of becoming or coming into being, but of being simpliciter. They simply were in the beginning. Everything other than God and the divine Logos “came into being” egeneto through the Logos. The verb is the aorist form of ginomai, whose primary meaning is “to become” or “to originate.” V. 3 thus carries the weighty metaphysical implication that there are no eternal entities apart from God. Rather everything that exists, with the exception of God himself, is the product of temporal becoming. The verb ginomai also has the meaning “to be created” or “to be made.” This meaning emerges in v. 3 through the denomination of the agent di’ autou responsible for things’ coming into being. The preposition dia genitive indicates the agency by means of which a result is produced. The Logos, then, is said to be the one who has created all things and brought them into being. A second, equally significant metaphysical implication of v. 3 thus emerges only God is selfexistent everything else exists through another, namely, through the divine Logos. God is thus the ground of being of everything else. Jn 1.3 is thus fraught with metaphysical significance, for taken prima facie it tells us that God alone exists eternally and a se. It entails that there are no objects of any sort, abstract or concrete, which are coeternal with God and uncreated by God via the Logos. Partisans of uncreated abstract objects, if they are to be biblical, must therefore maintain that the domain of John’s quantifiers is restricted in some way, quantifying, for example, only over concrete objects.5 The issue is a subtle one, easily misunderstood. The question is not did John have in mind abstract objects when he wrote panta di’ autou egeneto Probably not. But neither did he have in mind quarks, galaxies, and black holes yet he would take such things and countless other things, were he informed about them, to lie within the domain of his quantifiers. The question is not what John thought lay in the domain of his quantifiers. The question, rather, is did John intend the domain of his quantifiers to be unrestricted, once God is exempted It is very likely that he did. For not only is God’s unique status as the only eternal, uncreated being typical for Judaism,6 but John himself identifies the Logos alone as existing with God and being God in the beginning. Creation of everything else through the Logos then follows. The salient point here is that the unrestrictedness of the domain of the quantifiers is rooted, not in the type of objects thought to be in the domain, but in one’s doctrine of God as the only uncreated being. But was John, in fact, ignorant of the relation between abstract objects and divine creation when he wrote vv. 1–3, as we have assumed It is, in fact, far from clear that the author of John’s prologue was innocent concerning abstract objects and their relation to the Logos. For, as we have seen,7 the doctrine of the divine, creative Logos was widespread in Middle Platonism,8 and the similarities between John’s Logos doctrine and that of the Alexandrian Jewish philosopher Philo 20 B.C.–A.D. 50 are numerous and striking.9 Of particular interest is the role of the Logos as the instrumental cause of creation. The use of dia genitive to express instrumental creation is not derived from Wisdom literature but is an earmark of Middle Platonism indeed, so much so that scholars of this movement are wont to speak of its “prepositional metaphysics,” whereby various prepositional phrases are employed to express causal categories.10 Philo identifies the four Aristotelian causes by these prepositional phrases, stating that “that through which” to di’ hou expresses creation by the Logos.11 The similarities between Philo and John’s doctrines of the Logos are so numerous and close that most Johannine scholars, while not willing to affirm John’s direct dependence on Philo, do recognize that the author of the prologue of John’s Gospel shares with Philo a common intellectual tradition of Platonizing interpretation of Genesis chapter one. Now while John does not tarry to reflect on the role of the divine Logos causally prior to creation, this precreation role features prominently in Philo’s Logos doctrine. Recall that for Middle Platonists, the intelligible world kosmos noētos served as a model for the creation of the sensible world kosmos oratos. But for a Jewish monotheist like Philo, the realm of Ideas does not exist independently of God but as the contents of his mind. The intelligible world may be thought of as either the causal product of the divine mind or simply as the divine mind itself actively engaged in thought.12 Especially noteworthy is Philo’s insistence that the world of Ideas cannot exist anywhere but in the divine Logos. Just as the ideal architectural plan of a city exists only in the mind of the architect, so the ideal world exists solely in the mind of God. On Philo’s doctrine, then, there is no realm of independently existing abstract objects. According to Runia, while not part of the created realm, “the κόσμος νοητός, though eternal and unchanging, must be considered dependent for its existence on God.”13 Preoccupied as John is with the incarnation of the Logos, he does not linger over the precreatorial function of the Logos, but given the provenance of the Logos doctrine, he may well have been aware of the role of the Logos in grounding the intelligible realm as well as his role in creating the realm of temporal concrete objects. However this may be, reflection on Jn 1.1–3 leads to the conclusion that the author of the prologue of John’s Gospel conceives of God as the creator of everything apart from himself. There are no uncreated, independently existing, eternal objects, for God exists uniquely a see. Consider as well the Pauline witness. The same Hellenistic Judaism, epitomized by Philo, that forms the background of John’s prologue also finds echoes in Paul’s statements on God’s being the source of all things. Consider the following representative Pauline texts there is one God, the Father, from whom are all things and for whom we exist, and one Lord, Jesus Christ, through whom are all things and through whom we exist 1 Cor 8.6 NRSV. For just as woman came from man, so man comes through woman but all things come from God 1 Cor 11.12 NRSV. For from him and through him and to him are all things Rom 11.36 NRSV. He Christ is the image of the invisible God, the first born of all creation for by him all things in heaven and on earth were created, things visible and invisible, whether thrones or dominions or rulers or powers – all things have been created through him and for him Col 1.15–16 NRSV. Commenting on the background of Rom 11.36, Douglas Moo observes, “The concept of God as the source ek, sustainer dia, and goal eis of all things is particularly strong among the Greek Stoic philosophers. Hellenistic Jews picked up this language and applied it to Yahweh and it is probably, therefore, from the synagogue that Paul borrows this formula.”14 Stoic thought is the more distant progenitor, however more immediately we find here variations on the prepositional metaphysics of Middle Platonism that Philo adopted. Noting how unusual for Paul such prepositional formulations are, Richard Horsley has argued that a Philonic provenance for Paul’s expressions is especially evident in Paul’s Corinthian correspondence.15 He shows that “numerous passages in Philo’s writings provide an analogy for nearly every aspect of the Corinthians’ religious language and viewpoint.”16 He comments, I Cor. 8.6 is an adaptation of the traditional Hellenistic Jewish form of predication regarding the respective creative and soteriological roles of God and SophiaLogos, which Philo or his predecessors had adapted from a Platonic philosophical formula concerning the primal principles of the universe. What was already a fundamental tenet of the Hellenistic Jewish religion expressed in the book of Wisdom appears in more philosophical formulation in Philo that God is the ultimate Creator and final Cause of the universe, and that SophiaLogos is agent and paradigm of creation or the instrumental and formal cause.17 Notice that whereas God is regarded as the efficient and final cause of the universe of created things, SophiaLogos serves as the instrumental cause and the formal cause or paradigm of creation, this latter role being specifically the source or ground of the kosmos noētos or ideal world. Paul’s innovation is that he substitutes Christos for SophiaLogos, having “Christ take over what were the functions of Sophia, according to the gnosis of the Corinthians.”18 In ascribing to Christ the creative role of the Logos, Paul is affirming that everything apart from God and his Logos has been created by God through Christ. The domain of Paul’s quantifiers is unlimited everything other than God is the result of divine creation. Rom 11.36 is thus, in Moo’s words, “a declaration of God’s ultimacy.”19 Dunn concurs “Where the focus is so exclusively on the supreme majesty and selfsufficiency of God, the Stoic type formula provides a fitting climax he is the source, medium, and goal of everything, the beginning, middle, and end of all that is.”20 The biblical witness to divine aseity and God’s being the sole ultimate reality is thus impressive. God is affirmed to exist independently of everything else and to be alone eternal in his being. Everything apart from God is said to belong to the realm of temporal becoming and to have been created by God through Christ, the divine Logos. 4.2 Testimony of the Church The conviction that God is the sole ultimate reality eventually attained credal status at the Council of Nicaea. In language redolent of the prologue to the fourth Gospel and of Paul, the Council affirmed I believe in one God, the Father Almighty, Maker of heaven and earth and of all things visible and invisible And in one Lord, Jesus Christ, the only Son of God, begotten of the Father before all ages, light from light, true God from true God, begotten not made, consubstantial with the Father, through whom all things came into being. The phrase “Maker of heaven and earth and of all things visible and invisible” is Pauline Col 1.16, and the expression “through whom all things came into being” Johannine Jn 1.3. At face value the Council seems to affirm that God alone is uncreated and that everything else was created by him. An examination of anteNicene theological reflection on divine aseity confirms the prima face reading. At the heart of the Arian controversy which occasioned the convening of the Council of Nicaea lay a pair of terminological distinctions prevalent among the Church Fathers agenētosgenētos and agennētosgennētos. 21 The word pair agenētosgenētos derives from the Greek verb ginomai, which means to become or to come into being. Agenētos means unoriginated or uncreated, in contrast to genētos, that which is created or originated. The second word pair agennētosgennētos derives from a different verb gennaō, which means to beget. That which is agennētos is unbegotten, while that which is gennētos is begotten. These distinctions allowed the Fathers to hold that while both God the Father and God the Son are agenētos, only the Father is agennētos. Like their Arian opponents, the anteNicene and Nicene Church Fathers rejected any suggestion that there might exist agenēta apart from God alone.22 According to patristic scholar Harry Austryn Wolfson,23 the Church Fathers all accepted the following three principles 1. God alone is uncreated. 2. Nothing is coeternal with God. 3. Eternality implies deity. Each of these principles implies that there are no agenēta apart from God alone. But lest it be suggested that abstracta were somehow exempted from these principles, we should note that the anteNicene Church Fathers explicitly rejected the view that entities such as properties and numbers are agenēta. The Fathers were familiar with the metaphysical worldviews of Plato and Pythagoras and agreed with them that there is one agenētos from which all reality derives but the Fathers identified this agenētos, not with an impersonal form or number, but with the Hebrew God, who has created all things other than himself ex nihilo. 24 If confronted by a modernday Platonist defending an ontology which included causally effete objects which were agenēta and so coeternal with God, they would have rejected such an account as blasphemous, since God is the sole and alloriginating agenētos. Given this background to the Nicene Creed, it is virtually undeniable that the Creed means to affirm God as the Creator of everything other than himself, the sole uncreated reality.25 4.3 Requirements of Perfect Being Theology In addition to the biblical and patristic witness to God’s status as the sole ultimate reality, the requirements of sound systematic theology include the affirmation that God is the source of all things apart from himself. For divine aseity is a fundamental requirement of perfect being theology. As a perfect being, the greatest conceivable being, God must be the selfexistent source of all reality apart from himself. For being the foundational cause of existence of other things is plausibly a greatmaking property, and the maximal degree of this property is to be the cause of everything else that exists.26 God would be diminished in his greatness if he were the cause of only some of the other things that exist. Seen in this light, divine aseity is a corollary of God’s omnipotence, which certainly belongs to maximal greatness.27 For if any being exists independently of God, then God lacks the power either to annihilate it or to create it. An omnipotent being can give and take existence as he sees fit with respect to other beings. God’s power would thus be attenuated by the existence of independently existing abstract objects. 4.4 The Challenge of Platonism The doctrine of divine aseity confronts a serious challenge from one of the oldest and most persistent of philosophical doctrines Platonism, which holds that in addition to concrete objects like people and planets there also exist abstract objects like numbers and sets and propositions and properties.28 Platonists maintain that such objects, though abstract and usually held to exist beyond time and space, are nonetheless every bit as real as the familiar physical objects of our daily experience. They exist necessarily, for it is inconceivable that there should exist, for example, a possible world lacking in numbers or propositions, even if that world were altogether devoid of concrete objects other than God himself. Moreover – and this is the crucial point – they exist a se. There is no cause of the existence of such entities they each exist independently of one another and of God. It is this feature of Platonism, more than any other, which has troubled many Christian theists. Not only is there an infinite number of such objects there is an infinite number of natural numbers alone, but there are higher and higher orders of infinities of such objects, infinities of infinities, so that God is utterly dwarfed by their unimaginable multitude. God finds himself amidst infinite realms of uncreated beings which exist just as necessarily and independently as he. The dependence of physical creation upon God for its existence becomes an infinitesimal triviality in comparison with the existence of the infinitude of beings that exist independently of him. Platonism thus entails a metaphysical pluralism which is incompatible with God’s being the sole ultimate reality. In the contemporary debate over Platonism,29 there are principally two arguments lodged against Platonism and one argument in its favor. The two objections usually urged against Platonism are the socalled epistemological objection and the uniqueness objection.30 The major consideration weighing in for Platonism is the socalled Indispensability Argument.31 Whether Platonists can successfully defeat the two principal objections lodged against their view may remain a moot question here. Rather our concern is with the Indispensability Argument, for if the Indispensability Argument is sound, it will constitute a defeater of biblical theism. 4.4.1 The Indispensability Argument Today there are a variety of Indispensability Arguments on tap. Philosopher of mathematics Mark Balaguer nicely epitomizes such arguments as follows I. If a simple sentence i.e., a sentence of the form ‘a is F’, or ‘a is Rrelated to b’, or. . . is literally true, then the objects that its singular terms denote exist. Likewise, if an existential sentence is literally true, then there exist objects of the relevant kinds e.g., if ‘There is an F’ is true, then there exist some Fs. II. There are literally true simple sentences containing singular terms that refer to things that could only be abstract objects. Likewise, there are literally true existential statements whose existential quantifiers range over things that could only be abstract objects. III. Therefore, abstract objects exist.32 Premise I states a metaontological criterion of ontological commitment. It does not tell us what exists, but it does claim to tell us what must exist if a sentence we assert is to be true. The claim is that singular terms such as proper names and definite descriptions and existential quantification are devices of ontological commitment. Premise II states that the denotations or referents of certain singular terms in literally true, simple sentences like, for example, “2 2 4,” cannot plausibly be taken to be concrete objects of any kind. Premise II likewise states that there are literally true existentially quantified statements involving quantification over abstract objects like, for example, “There is a prime number between 2 and 4.” Given the proffered criterion of ontological commitment, anyone who asserts such truths finds himself committed to the reality of abstract objects. 4.4.2 Responses to the Indispensability Argument Fortunately, there is a wide range of options open to the Christian thinker in response to Platonism’s challenge. The contemporary debate over Platonism finds its epicenter in the philosophy of mathematics with respect to the ontological status of mathematical objects like numbers, sets, and so forth. Taking such objects as paradigmatic abstract objects, we may lay out some of our options as follows Looking at Figure 1, we see that the various options can be classed as realist mathematical objects exist, arealist there is no fact of the matter concerning the existence of mathematical objects, or antirealist mathematical objects do not exist. Further, there are two brands of realism about mathematical objects views which take them to be abstract objects and views which take them to be concrete objects. Of realist views which consider mathematical objects to be abstract, there is in addition to Platonism a sort of modified Platonism called absolute creationism, which holds that mathematical objects have, like concrete objects, been created by God. Concretist versions of realism are antiPlatonist views which take mathematical objects to be either physical objects or mental objects, the latter either in human minds or in God’s mind. The most promising concretist view is some sort of divine conceptualism, the heir to the view of the Church Fathers. In between realism and antirealism about mathematical objects is arealism, the view that there just is no fact of the matter about the reality of mathematical objects. The classic version of arealism was the conventionalism of Rudolf Carnap, who held that talk of mathematical objects makes sense only within a conventionally adopted linguistic framework. Although conventionalism was motivated by a now defunct verificationism, there are contemporary philosophers who are not conventionalists but nonetheless embrace ontological arealism, according to which certain ontological questions like “Does the number 2 exist” just have no objective answers. When we turn to antirealist options, we find a cornucopia of different views. Neutralism holds that the use of singular terms and existential quantification is neutral, so that we are not ontologically committed to the existence of numbers by asserting obvious truths like “2 3.”33 Free logic, on the other hand, takes existential quantification to be ontologically committing but denies that the use of singular terms is a device of ontological commitment. NeoMeinongianism denies that existential or particular quantification is ontologically committing, though it affirms that there are objects referred to by abstract singular terms like “2” – with the caveat that these objects are nonexistent Fictionalism holds that the use of singular terms and existential quantification is ontologically committing to the objects referred to or quantified over but denies that mathematical statements like “2 2 4” are true. Ultima facie strategies hold that mathematical discourse may be taken as true without ontological commitment because it can be either taken figuratively or paraphrased so as to avoid reference to or quantification over mathematical objects. Pretense theory considers mathematical discourse to be a species of makebelieve, prescribed to be imagined true, so that mathematical objects are no more real than fictional characters. And so on Of these options, only absolute creationism looks on the Indispensability Argument with insouciance.34 For the absolute creationist affirms that mathematical objects are abstract objects existing in some sense apart from God, though causally dependent upon him. Accordingly, this option has appealed to some Christian theists as the best way to safeguard divine aseity. Unfortunately, absolute creationism appears to involve a vicious circularity which has come to be known as the bootstrapping objection. The problem can be simply stated with respect to the creation of properties. In order to create properties, God must already possess properties. For example, in order to create the property being powerful God must already possess the property of being powerful, which involves a vicious circularity. The only plausible way to avoid the bootstrapping problem, it seems, is to affirm that God can be powerful without having the property of being powerful. But such a solution removes any motivation for realism about properties. One might try to escape the bootstrapping problem by appealing to the doctrine of divine simplicity, according to which God transcends the distinction between a thing and its properties. A strong doctrine of divine simplicity would avoid the vicious circularity threatening absolute creationism by denying that God has any properties rather God is a simple being identical with his existence. Unfortunately, the doctrine of divine simplicity, given its controverted status, cannot provide a plausible escape from the problem.35 With their backs apparently to the wall, some absolute creationists have conceded that God does not create his own properties but they insist that God has created all other properties. This answer, however, will either sacrifice divine aseity on the altar of Platonism or else remove any adequate rationale for Platonism. On the one hand, if the absolute creationist affirms that God’s essential properties are exemplified by him logically prior to his creation of all remaining properties and so are not created by him, then we have sacrificed the doctrine of divine aseity in favor of Platonism, for this just is the theologically unacceptable position that in addition to God there exist other uncreated entities.36 On the other hand, if the absolute creationist affirms that explanatorily prior to his creation of properties, God has no properties but is as he is without exemplifying properties since they have not yet been created, then we have abandoned the Platonistic ontological assay of things with respect, at least, to God. Such a move either collapses into the implausible doctrine of divine simplicity or else concedes to the antiPlatonist that in order to be, for example, powerful, God need not exemplify the property of being powerful. Perhaps the absolute creationist will say that only logically posterior to his creation of properties does God come to acquire properties prior to his creation of properties, God is, for example, powerful even though the property of being powerful has not yet been created. But then properties are not doing any metaphysical work, since God is already powerful before coming to exemplify being powerful. 37 This is to deny the standard Platonist ontological assay of things. But then the absolute creationist seems to have lost any rationale for positing the existence of such abstract objects. They are not doing any metaphysical work, as they do in the usual Platonist scheme of things. As Moltmann says, they simply enable reference for reifying expressions like “the property wisdom” or “the number 7.” Such is, in fact, Peter van Inwagen’s view. Van Inwagen rejects constituent ontologies, which ascribe to particular things an ontological structure.38 He therefore repudiates the Platonist’s ontological assay of things, denying that properties are ontological constituents of things. Although van Inwagen is not an absolute creationist, nonetheless an absolute creationist who endorses van Inwagen’s favored ontology could hold to the reality of abstract objects without falling prey to the bootstrapping objection because van Inwagen’s favored ontology denies the typical Platonist ontological assay. So a Christian Platonist like van Inwagen ought to find absolute creationism to be an attractive option. Nevertheless, to my knowledge no one has adopted this viewpoint. Van Inwagen reports, “I am the only proponent of the Favored Ontology I am aware of,”39 and he is not an absolute creationist Neither do I personally find his favored ontology attractive. To my mind, having to include such metaphysically idle entities as are countenanced by van Inwagen’s favored ontology ought to prompt us to call into question the premises of the Indispensability Argument that would force such ontological commitments upon us. In light of the metaphysical idleness of such abstract entities, it therefore seems to me that theists would be welladvised to look elsewhere than absolute creationism for a solution to the challenge posed by Platonism to the doctrine of divine aseity. To be successful, proponents of the Indispensability Argument have to show that of all the options delineated, Platonism alone is plausible, an order so tall that probably no contemporary philosopher thinks that it can be filled. Let us consider briefly the prospects of each alternative to Platonism.40 4.4.2.1 Responses to Premise II Consider, first, responses to the Indispensability Argument which dispute Premise II. These responses come in realist, arealist, and antirealist options. 4.4.2.1.1 Realist Responses Realist versions of antiPlatonism dispute that in true mathematical sentences singular terms refer to and existential quantifiers range over things that could only be abstract objects. NonPlatonic realists hold that various objects normally thought to be abstract are in fact concrete. These may be taken to be either physical objects, such as marks on paper, or mental objects or thoughts, either in human minds or in God’s mind. Nineteenth century philosopher of mathematics Gottlob Frege subjected the views that mathematical objects are physical objects or human thoughts to such withering criticism, however, that such views are scarcely taken seriously today.41 Frege’s objections to formalism and psychologism – such as the intersubjectivity, necessity, and plenitude of mathematical objects – do not, however, touch divine conceptualism. With the late twentieth century renaissance of Christian philosophy divine conceptualism is once more finding articulate defenders. Alvin Plantinga, for example, locates himself in the Augustinian tradition “in thinking of numbers, properties, propositions and the rest of the Platonic host as divine ideas.”42 Unfortunately, conceptualism is not worryfree. Conceptualism requires that in virtue of his omniscience God be constantly entertaining actual thoughts corresponding to every proposition and every state of affairs.43 This may be problematic for the theist. Graham Oppy complains that “it threatens to lead to the attribution to God of inappropriate thoughts bawdy thoughts, banal thoughts, malicious thoughts, silly thoughts, and so forth.”44 I think that the Christian theist should take this worry very seriously. If God has the full range of thoughts that we do, then he must imagine himself, as well as everyone else, to be engaged in bawdy and malicious acts, and, moreover, rather than putting such detestable thoughts immediately out of mind as we try to do, he keeps on thinking about them. What about banal and silly thoughts Why in the world should we think that God is constantly thinking the nondenumerable infinity of banal and silly propositions or states of affairs that we can imagine Take the thought that for any real number r, r is distinct from the Taj Mahal. Why would God retain such inanities constantly in consciousness Or consider false propositions like for any real number r, r is identical to the Taj Mahal. Why would God hold such a silly thought constantly in consciousness, knowing it to be false Obviously, the concern is not that God would be incapable of keeping such a nondenumerable infinity of thoughts ever in consciousness, but rather why he would dwell on such trivialities. It is a non sequitur to infer that in virtue of his omniscience everything God knows he is actually thinking about. Yet another worry for conceptualism is that concrete objects like God’s thoughts do not seem suitable to play the roles normally ascribed to abstract objects. Consider properties, for example. The chief rationale behind construing properties as abstract universals rather than particulars is the supposed need for an entity that can be wholly located in diverse places. The difficulty, then, for conceptualism is that God’s thoughts, as concrete objects, are not universals but particulars and so cannot be wholly present in spatially separated objects. Perhaps in response the conceptualist could say of divine thoughts what the Platonist says of abstract universals particulars stand in some sort of relation to them in virtue of which particulars are the way they are. An immediate problem for the conceptualist is that if properties are God’s thoughts, then particulars must exemplify God’s thoughts. But a concrete object does not seem to be the sort of thing that is exemplifiable any more than it can be a universal, since concrete objects are particulars and particulars are not exemplifiable but rather exemplify. Accordingly, God’s thoughts cannot be properties. But perhaps the conceptualist could say that divine thoughts can play the role of properties. In substituting God’s thoughts for properties, Plantinga has suggested that particulars stand to God’s thoughts in a relation analogous to exemplification.45 He appeals to Frege’s notion of “falling under a concept” as the relation in which particulars stand to God’s thoughts. Thus, all brown things fall under God’s thought of brown. Things which are brown resemble each other in virtue of falling under the same concept. Intriguing as this suggestion might be, it is problematic. In the first place, concepts are not plausibly construed as concrete objects, for they are shared by multiple thinkers in a way that thoughts are not. Concepts seem to be part of the content of our thoughts. As such they are plausibly abstract objects, if they exist at all. Moreover, as mental states, thoughts are characterized by intentionality – being about things – not by things’ falling under them. My thought of redness is about redness it is not itself redness, nor do things fall under it. This problem can be generalized. God’s thought of the number 2 is about 2. But then his thought is not 2 but something distinct from 2. 2 is what he is thinking about. But he is not thinking about his thought he is thinking about 2. Therefore, his thought cannot be 2. Furthermore, substituting the notion of falling under a concept for exemplifying a property seems to get the explanatory order backwards.46 Things are not brown because they fall under God’s concept brown, in the way that things are brown because they exemplify brownness rather they fall under God’s concept brown because they are brown. Thus, the relation of falling under a concept cannot do the work of exemplification. If this is right, then the conceptualist who wants God’s thoughts to play the role of properties still has plenty work cut out for him, if his view is to commend itself as an attractive option for theists. Or consider the suitability of divine thoughts for playing the role of mathematical sets. Plantinga suggests that sets be taken to be God’s mental collectings. But if sets are really particular divine thoughts, then how do we have any access to sets The question here is not whether I have a causal connection with sets. Rather it is that sets, the real sets, are locked away in God’s private consciousness, so that what we talk about and work with are not sets at all. When I collect into a unity all the pens on my desk, that set is not identical, it seems, with the set constituted by God’s collecting activity. Since we have two mental collectings and since sets are God’s particular collectings, the “set” I form is not identical to the set of all the pens on my desk. But if sets are determined by membership, how could they not be identical, since they have the same members While by no means knockdown objections to conceptualism, these worries should motivate the theist to look seriously at the wide variety of nonrealist alternatives to Platonism before acquiescing too easily to a realist viewpoint. 4.4.2.1.2 Arealist Responses Consider now the arealist response to Premise II. As mentioned, Carnap held that once we have adopted a linguistic framework involving terminology for abstract objects like numbers, internal questions like “Is there a prime number greater than 100” are meaningful.47 For someone who is outside the framework, the question is meaningless. No contemporary philosopher would defend Carnap’s verificationism but his conventionalism does find an echo today in what is sometimes called ontological pluralism.48 According to pluralists, certain ontological questions, though meaningful, do not have objective answers. Mark Balaguer and Penelope Maddy, for example, would deny that the question “Do mathematical objects exist” has an answer that is objectively true or false.49 On arealism there just is no fact of the matter whether or not mathematical objects exist. Now at first blush arealism might seem a quick and easy solution to the challenge posed by Platonism to divine aseity. If there is no objective truth about whether or not mathematical objects exist, then the use of mathematical terminology is devoid of ontological significance. Internal questions about the existence of certain sets or numbers may be answered affirmatively but there just is no answer to the external question of the existence of such entities. In that case one cannot truthfully assert that there are objects which God did not create. Alas, however, there is no succor for the theist here. For given God’s metaphysical necessity and essential aseity, there just is no possible world in which uncreated mathematical objects exist. Hence, there most certainly is a fact of the matter whether uncreated, abstract objects exist they do not and cannot exist. Therefore, arealism is necessarily false, as is conventionalism about existence statements concerning mathematical and other abstract objects.50 4.4.2.1.3 AntiRealist Responses 4.4.2.1.3.1 Fictionalism We turn now to what is perhaps the boldest option in response to Premise II fictionalism. Fictionalists flatly deny that mathematical sentences are true, period. Statements involving quantification over or reference to abstract objects are false or at least untrue. Abstract objects are merely useful fictions that is to say, even though no such objects exist, it is useful to talk as though they did. Hence, the name fictionalism. The most evident objection to fictionalism is that some mathematical statements, like “2 2 4,” are just obviously true. Indeed, they seem to be necessarily true. But Hartry Field reacts to the claim that a mathematical assertion like “2 2 4” must be true simply as a consequence of the meaning of its terms by saying that this claim cannot be right because analytic truths cannot have existential implications.51 Field rightly contends that we cannot infer the existence of things from merely definitional truths, an insight gained from discussions of the ontological argument. But it is only his unquestioned presupposition of the proffered criterion of ontological commitment that leads Field to think that a statement like “2 2 4” has existential implications. Deny that criterion and one is not forced to into the awkward position of denying that 2 2 4. In short, one’s attitude toward the objection from the obviousness of elementary arithmetic is probably going to depend on one’s attitude toward the proffered criterion of ontological commitment. If with the fictionalist we are convinced that quantification and singular terms are devices of ontological commitment, then we shall find upon reflection that the sentences of elementary mathematics, if taken literally, are anything but obvious. For we shall come to see that statements which we have unhesitatingly accepted as true since childhood are, in fact, radical ontological assertions about the existence of mindindependent abstract objects. As such, they are not at all obviously true. We come to realize that we have, in fact, misunderstood them all these years we literally did not understand what we were asserting. On the other hand, if we find sentences of elementary arithmetic to be obvious because we do not take them to be ontologically committing, then we shall be led to reject the proffered criterion of ontological commitment which would saddle us with such commitments. This seems to me the obvious course to take. Indeed, the antiPlatonist has a quick and easy argument to that end 1. If the proffered criterion of ontological commitment is correct, then 22 4. 2. 2 2 4. 3. Therefore, proffered criterion of ontological commitment is not correct. After all, the sentences of elementary mathematics are much more obviously true than any criterion of ontological commitment and so should be more tenaciously held and less quickly surrendered than the proffered criterion of ontological commitment. 4.4.2.1.3.2 Ultima facie Strategies Ultima facie interpretive strategies are a diverse group of antirealist responses to indispensability arguments united by the conviction that true mathematical sentences are capable of being reformulated or interpreted without prejudice to their truth in such a way as to avoid any ontological commitments to abstract objects. Proponents of the Indispensability Argument themselves do not take us to be committed ontologically to all the things referred to or quantified over in ordinary language. For example, if I say, “There are deep differences between Republicans and Democrats,” I do not mean to commit myself Aseity 83 to the existence of objects called differences some of which are deep, despite my use of the informal quantifier “there is.” In a case like this, I can reformulate my original claim so as to serve the same purpose without such ontological commitments on my part, e.g., “Republicans differ deeply from Democrats.” Proponents of the Indispensability Argument demand such paraphrases of mathematical sentences from antirealists, confident that they cannot be provided. In fact, however, there are a number of strategies for reformulating or interpreting mathematical sentences which preserve their truth without ontological commitment to mathematical objects. The strategies of Geoffrey Hellman’s modal structuralism, Charles Chihara’s constructibilism, and Stephen Yablo’s figuralism come to mind.52 It is noteworthy that the mathematical adequacy of the paraphrases of mathematical sentences offered by ultima facie strategists has gone largely unchallenged. So what complaint is there, from the Platonist’s perspective, with various ultima facie strategies The most common complaint in the literature is that such strategies either grossly misrepresent mathematical discourse or advocate without sufficient justification a radical revision of mathematical discourse.53 The implication of this objection for ultima facie strategies is that the proffered reinterpretations should be rejected in favor of the prima facie interpretation, which is taken to be the default interpretation. None of the prominent ultima facie strategists, however, advocates a revision of mathematical discourse. On the contrary, they see no reason that mathematicians should not continue in the use of their Platonistic language. Rather they are offering merely an undercutting defeater of Premise II of the Indispensability Argument that mathematical sentences cannot be reformulated or interpreted so as to be truthpreserving but without ontological commitment to abstract objects. Neither are Hellman and Chihara offering a hermeneutical claim about how practitioners understand mathematical discourse, since they say they have no idea what the majority of mathematicians and scientists think about these questions. While Yablo does espouse figuralism as a hermeneutical thesis, there is no reason that the antirealist has to present it as such. Like Chihara and Hellman, he can remain agnostic, in the absence of linguistic and sociological studies, about hermeneutical questions and present the figurative interpretation as one reasonable way of understanding abstract object talk without commitment to abstract objects. If such an interpretation is reasonable, then the Indispensability Argument has been defeated. 4.4.2.1.3.3 Pretense Theory Pretense theory takes no position with respect to the truth value of discourse involving quantification over or reference to abstract objects. The pretense theorist’s essential point, rather, is that whether or not such sentences are true, we are invited to imagine that they are true. Abstract discourse is fictional and therefore does not commit us to the reality of abstract objects. Contemporary theories of fiction draw much of their inspiration from the brilliant, pioneering work of Kendall Walton.54 Prescribed imagining lies at the heart of Walton’s theory of fiction. Fictional propositions are propositions which in certain contexts we are to imagine to be true.55 Walton emphasizes that truth and fictionality are not mutually exclusive. Some of the propositions prescribed to be imagined by a historical fiction like War and Peace, for example may be true. Even if all the sentences in a novel about the future like George Orwell’s 1984 turned out to be true, that novel still remains fiction.56 What is essential to fictionality is not falsehood but a prescription to be imagined. Mary Leng is an antirealist philosopher of mathematics who embraces Walton’s theory of fiction in order to deal with the alleged commitments of our best scientific theories to mathematical objects.57 Leng regards the mathematical objects appearing in our scientific theories as merely useful fictions. In support of the plausibility of her view, Leng provides two nonmathematical examples of the use of fictions in scientific theorizing idealizations like ideal gases and theoretical entities like certain fundamental particles. These two cases illustrate the point that even in our best scientific theories a reason to speak as if an object existed is not always a reason to believe that that object exists. Leng thinks that nothing in our current, best scientific worldview gives us good reason to believe in mathematical objects as anything more than useful fictions. Hence, we have no reason to think that such objects exist. An important objection to pretense theory is that mathematics is not of the genre of fiction. Mathematicians and scientists take mathematics to be a body of knowledge and a realm of discovery, not invention. To regard mathematical statements as fictional is alleged to distort the nature of this discipline. In weighing this objection, we need to keep in mind that the pretense theorist is not defending a hermeneutical claim about how professional mathematicians or scientists in fact understand mathematical sentences. In order to undercut the Indispensability Argument for Platonism, the pretense theorist need show only that mathematical sentences can be reasonably taken to be fictional. In fact, there are some features of mathematical discourse that make it seem a prime candidate for a fictional interpretation. For example, axiomatization of mathematical theories naturally invites a pretense theoretical interpretation. Take the Axiom of Infinity in standard Zermelo–Fraenkel set theory Axiom of Infinity There exists a set x having the empty set ∅ as a member, and for any member y of x the union of y and y is also a member of x. It is striking that this is the only axiom of standard set theory with existential implications. On a pretense theoretical approach, the Axiom of Infinity is something we are prescribed to imagine true. We are to make believe that there is an infinity of these things called sets and then are free to explore the fictional world of our imagination. This will certainly be a journey of discovery, which will issue in a great deal of knowledge of the mathematical world determined by the axioms. Such an attitude toward the axioms of set theory is not uncommon among mathematicians and philosophers of mathematics. For example, postulationalism, which treats the axioms of competing set theories as postulates whose consequences may be explored, invites us, in effect, to make believe that the axioms are true without committing ourselves to their objective truth.58 In fact, the philosopher of mathematics Stewart Shapiro observes, “The strongest versions of working realism are no more than claims that mathematics can or should be practiced as if the subject matter were a realm of independently existing, abstract, eternal entities.”59 This characterization would make working realists into pretense theorists But is it plausible to take the Axiom of Infinity as something prescribed to be imagined rather than as a straightforward metaphysical assertion I think the answer is obviously affirmative. It is universally admitted that the Axiom of Infinity is not intuitively obvious. Its lack of intuitive warrant was one of the heavy stones that helped to sink logicism, an early twentieth century attempt to derive set theory from logic alone. Since the axiom lacks intuitive warrant, the Axiom of Infinity is adopted by contemporary mathematicians for reasons that are variously called “pragmatic” or “regressive” or “extrinsic,” reasons which do not justify its truth, but its mathematical utility. Moreover, as Potter points out, there are multiple versions of the Axiom of Infinity which postulate different sets. A defense of the axiom on pragmatic grounds “does not directly give us a ground for preferring one sort of axiom of infinity over another,” he says, since any of them will work.60 As a result, says Potter, Platonists “have frustratingly little they can say” by way of justification for their preferred version of the axiom.61 As a serious assertion of ontology, the Axiom of Infinity is a breathtaking claim that utterly outstrips our intuitions. So much more can be said on this head, but axiomatic set theory seems to be a perfect candidate for a pretense theoretical approach. This conclusion has sweeping significance because set theory is typically regarded as foundational for the rest of mathematics, since the whole of mathematics can be reductively analyzed in terms of pure sets. In the words of philosopher of mathematics Penelope Maddy, “our muchvalued mathematical knowledge rests on two supports inexorable deductive logic, the stuff of proof, and the set theoretic axioms.”62 If the latter involve no ontological commitments, neither does the whole of mathematics. 4.4.2.2 Responses to Premise I The foregoing strategies all accept or, at least, do not challenge, the criterion of ontological commitment that lies at the heart of the Indispensability Argument. More fundamental challenges to the Indispensability Argument call into question that criterion and, hence, Premise I. Some of these versions of antiPlatonism ask why true sentences may not contain irreferential singular terms or why successful reference must involve the existence of an object in the world. Others challenge the criterion by demanding why use of the existential quantifier should be interpreted to assert the existence of a real world object which is the value of the variable bound by the quantifier. 4.4.2.2.1 Free Logic Free Logics are logics which are free of existential import with respect to singular terms but whose quantifiers are taken to be devices of ontological commitment. Free logic will avoid the unwanted commitments which the proffered criterion would engender by scrapping classical logic’s inference rules of Existential Generalization EG and Universal Instantiation UI. So, for example, from the truth of “Sherlock Holmes is the most famous detective of English fiction” we cannot infer that ∃x x the most famous detective of English fiction. By the same token, from the arithmetic truth that 3 5 we cannot infer that ∃x x 5. A view that asks us to abandon EG and UI would seem to require very powerful motivations. It is perhaps surprising how powerfully motivated some of the claims of free logic are. Karel Lambert, a pioneer of free logic, complains that although modern logic in the late nineteenth century shed itself of various existence assumptions implicit in Aristotelian logic with respect to the use of general terms,63 modern logic remains infected with existence assumptions with respect to the use of singular terms, assumptions that ought not to characterize a purely formal discipline.64 For we have at the deepest level “a primordial intuition that logic is a tool of the philosopher and ideally should be neutral with respect to philosophical truth. . . . So if there are preconditions to logic that have the effect of settling what exists and what does not exist, they ought to be eliminated because they corrupt the ideal of logic as a philosophical tool.”65 These existence assumptions regarding singular terms surface dramatically in the way in which standard modern logic handles identity statements. For such statements cannot be true, according to standard logic, unless the referents of the singular terms employed in such statements exist. In other words, identity statements are ontologically committing for him who asserts them. But it seems bizarre to think that from a seemingly tautologous truth of the form t t, where t is some singular term, it follows that the thing denoted by t actually exists. Nevertheless, this is what standard logic requires. Lambert takes this ontological implication of mere identity statements to be absurd. For it would follow from the fact that “Vulcan Vulcan” that there is some object identical with Vulcan, that is to say, that Vulcan exists Standard logic avoids this untoward result by restricting the terms in true identity statements to those designating existing objects. For example, standard logic must regard a statement like “Vulcan Vulcan” as false, even though it appears to be a tautology that is necessarily true. Standard logic cannot therefore distinguish the truth value of identity statements like “Zeus Zeus” and “Zeus Allah.” Yet the first seems necessarily true and the second obviously false. Nor can standard logic affirm the truth of “Aristotle Aristotle,” since Aristotle no longer exists and so there is no thing with which he can be identified. It would be the height of ontological presumption, I think, to claim that the truth of such a statement implies a tenseless theory of time, according to which all moments and things in time are equally existent. Such an inference would only underscore the free logician’s claim that modern logic is still infected with inappropriate existence assumptions. As a result of limiting truths of identity to those whose singular terms denote existing objects, standard logic becomes limited in its application to certain inferences and does not permit us to discriminate between inferences where the referentiality of the terms is crucial and those where it is not. For example, we are prohibited from inferring, “Lincoln was the Great Emancipator Lincoln brooded therefore, the Great Emancipator brooded,” an inference whose obvious validity should not be dependent on Lincoln’s existing. Proponents of free logic therefore propose to rid logic of all existence assumptions with respect to both general and singular terms. Free logic has thus become almost synonymous with the logic of irreferential or nondenoting, vacuous, empty singular terms. Thus, unlike neoMeinongianism to be discussed below free logic need not presuppose that the referents of such terms are nonexistent objects rather, there just are no referents. Advocates of socalled positive as opposed to negative or neutral free logic maintain that certain sentences can be truly asserted even though they contain irreferential singular terms. This feature of positive free logic strikes me as wellmotivated and eminently plausible. The truth of identity statements involving vacuous singular terms is of a piece with the assumed truth of many sentences which feature vacuous singular terms. Michael Dummett cites as an illustration the following paragraph from a London daily Margaret Thatcher yesterday gave her starkest warning yet about the dangers of global warming caused by air pollution. But she did not announce any new policy to combat climate change and sea level rises, apart from a qualified commitment that Britain would stabilize its emissions of carbon dioxide – the most important ‘greenhouse’ gas altering the climate – by the year 2005. Britain would only fulfill that commitment if other, unspecified nations promised similar restraint.66 Dummett then observes, “Save for ‘Margaret Thatcher,’ ‘air’ and ‘sea,’ there is not a noun or noun phrase in this paragraph incontrovertibly standing for or applying to a concrete object is a nation a concrete object, or a gas.”67 Obviously not but then is a nation or a gas an abstract object No. Taking the singular terms of this paragraph to be ontologically committing would commit its user to such strange entities as commitments, dangers, and the climate. If we are not ingenious enough to find acceptable paraphrases for such sentences, are we really committed to such bizarre entities, on pain of the falsehood of our discourse Free logic’s denial that use of singular terms in sentences we take to be true is ontologically committing for their user is, I think, quite plausible and constitutes a step in the right direction. But because free logic takes the existential quantifier of firstorder logic to carry existence commitments, it cannot avoid the Platonistic commitments of much mathematical talk. Instead, one will have to have recourse to some other antirealism like fictionalism in order to avoid such unwelcome commitments – unless, that is, one interprets logical quantifiers to also be ontologically neutral. Such a neutral logic will not, technically speaking, be a free logic, but as Alex Orenstein remarks, such terminology may be misleading, for “Isn’t a logic which disassociates the quantifiers from existence a paradigm of a logic that is free of existence assumptions, indeed freer of existence assumptions than Lambert’s variety”68 The viability of these solutions will occupy us below. 4.4.2.2.2 NeoMeinongianism Are there things that do not exist Notoriously, the Austrian philosopher Alexius Meinong thought so, and there has been among contemporary analytic philosophers in recent decades a remarkable resurgence of Meinongian thinking with respect to nonexistent objects. Since neoMeinongians typically include abstract objects among the things that do not exist, neoMeinongianism is a congenial option for the antiPlatonist. With respect to the customary devices of ontological commitment, neoMeinongianism holds that successful reference does, indeed, involve a relation to an object of some sort but denies that quantification over various objects is ontologically committing. With respect to singular terms, Richard Routley, the dean of contemporary neoMeinongians, inveighs against what he calls “the Reference Theory” as “the fundamental philosophical error” of the customary semantics.69 That theory states, RT. All primary truthvalued discourse is referential, where “referential” is understood to mean that the subject terms of that discourse have as their referents existing objects.70 Nevertheless, Routley accepts wholeheartedly the assumption of RT that successful reference is a relation in which words stand to certain objects and even goes beyond RT in holding that no singular terms fail to stand in such a relation. What he denies is that all of the objects which are the referents of singular terms are existing objects some, rather, are nonexisting objects. For Meinong and his followers, singular terms in true sentences do refer to objects – they are not irreferential – but the objects they refer to may be nonexistent. On the other hand, neoMeinongianism holds that quantification over various objects is not ontologically committing. Like free logicians, Routley contends that the central truths of logic should be prior to and independent of claims of particular metaphysical theories. But while Routley applauds free logic’s denial that use of singular terms commits us to existing objects as the referents of those terms, he faults free logic for retaining the existentially loaded quantifiers of classical logic. Because free logic permits quantification over only what exists, it proves to be “an unsatisfactory halfway house” on the road to an adequate theory, which is a truly neutral quantification logic.71 One may retain entirely the formalism of classical logic, including UI and EG, by removing any ontological commitments in pure logic. To signal the difference in interpretation, Routley substitutes the particular quantifier “P,” to be read “for some. . .,” for the existential quantifier “∃,” and the universal quantifier “U” for the traditional “∀”. By adding an existence predicate “E” we can symbolize “Some things do not exist” as Px ¬Ex. Routley thus arrives at a neutral logic, whose virtues he unfolds at length.72 Meinong himself held enigmatically that mathematical objects have being even though they do not exist. NeoMeinongians reject the distinction between being and existence and so regard abstract objects as belonging to the realm of objects which do not have being, since they do not exist. Classical mathematics can be retained without reservation once a neutral quantificational logic is in place. Accordingly, neoMeinongians think that there are objects, including mathematical objects, that do not exist and, moreover, that we frequently refer to such objects in literally true sentences. The neoMeinongian, then, would undercut the Indispensability Argument for Platonism by rejecting the customary criterion of ontological commitment. Quantificational logic is taken to be neutral with respect to ontological commitment, and even though abstract singular terms are referential, their referents do not exist. Therefore mathematical discourse can be true without committing its user ontologically to existent objects. What might be said by way of assessment of neoMeinongianism Consider first its treatment of singular terms. As we have seen, Routley assumes reference to be a relation between words and objects rather than an activity of persons. So what Routley objects to in RT is that the objects which stand in relation to certain words must be existing objects. It seems to me, therefore, that Routley’s critique of RT is not as radical in the sense of getting to the root of the matter as it ought to be. J. N. Findlay indicts Meinong’s Theory of Objects primarily on these grounds. He writes, Meinong assumes throughout his treatment that an object is in some sense a logical prius of a conscious reference or intention for there to be a conscious reference or intention there must in some wide sense be something which that reference or intention is ‘of’. . . . Since this is obviously not true in an ordinary sense in the case of some conscious intentions, there must, Meinong thinks, be a subtle sense or senses in which there are objects of such conscious intentions. . . . What should be seen is that this whole line of argumentation is wrong . . . what our usage shows is that ‘thinking’ and its cognates are not relational expressions like ‘above’, ‘before’, ‘killing’, ‘meeting’, c., nor can they be said to express relations. . . . We cannot therefore validly take an object of thought out of its objectposition in a statement and make it an independent subject of reference from ‘X thinks of a Y as being Z’ it does not follow that there is a Y which is being thought of by X, nor even that a thoughtof Y really is Z. . . . That intentionality is not a relation but ‘relationlike’ relativliches is, of course, an insight of Brentano’s Meinong, who frequently surpassed his master, in this respect certainly lagged behind him. He could only conceive intentional references in terms of objects logically prior to them, on which they necessarily depended hence, the many absurdities of his theory of objects.73 Meinong’s Theory of Objects should be accompanied by a more radical critique of RT, so as to call into question the assumption that reference is a wordobject relation. Routley does not, to my knowledge, directly engage the BrentanoHusserlinspired construal of reference as an intentional property of agents, nor have I encountered such an engagement in other neoMeinongian thinkers.74 Launching such a fundamental critique of RT would, however, be to abandon neoMeinongianism for neutralism, the next and final option we shall consider. With respect to existential quantification, it seems to me that neoMeinongians have argued persuasively that a neutral quantificational logic is preferable to a logic with existentially loaded quantifiers. As I have argued elsewhere, the use of existentially loaded quantifiers results in an inadequate treatment of important philosophical issues with respect to intentionality, temporal becoming, modal discourse, and mereology.75 These considerations certainly suffice to show at least that the existentially loaded interpretation of the existential quantifier is far from incumbent upon us and that therefore we may plausibly reject it. Alex Orenstein wisely reminds us that “There are fashions in philosophical explications,” one of which is today taking metaphysically heavy existence assertions to be expressed by the firstorder existential quantifier.76 It is perfectly reasonable to buck the fashion trend in this regard. So doing opens the door for affirming truths like “There is a prime number between 2 and 4” without commitment to the existence of mathematical objects. 4.4.2.2.3 Neutralism Neutralism, like free logic and neoMeinongianism, rejects certain key assumptions about ontological commitment which constitute the common ground shared by fictionalism, ultima facie strategies, and pretense theory with Platonism. The neutralist holds that neither existential quantification nor the use of singular terms is a device of ontological commitment. He is therefore unfazed by the Platonist’s conviction that various statements quantifying over or featuring singular terms referring to mathematical objects are, without qualification, true.77 In our discussion of neoMeinongianism and free logic we have already alluded to the disadvantages of taking firstorder quantifiers and singular terms to be devices of ontological commitment. Here we ask, what reasons might given for adopting the Platonist’s proffered criterion of ontological commitment Consider first existential quantification. Jody Azzouni challenges Quine’s assumption that the existential quantifier of firstorder logic does or should carry the burden of expressing ontological commitment. Quine thought it obvious that the existential quantifier of firstorder logic is ontologically committing. 78 Azzouni takes Quine’s argument for the triviality of the criterion to be that in ordinary language “there is” carries ontological commitment and that this idiom is straightforwardly regimented as the existential quantifier in firstorder logic. The existential quantifier naturally reproduces the ontological commitments carried by “there isare” in the vernacular. The problem for the Quinean is that in ordinary language informal quantificational phrases do not seem to be ontologically committing. Consider Thomas Hofweber’s list of some of the things we ordinarily say there are • something that we have in common • infinitely many primes • something that we both believe • the common illusion that one is smarter than one’s average colleague • a way you smile • a lack of compassion in the world • the way the world is • several ways the world might have been • a faster way to get to Berkeley from Stanford than going through San Jose • the hope that this dissertation will shed some light on ontology • the chance that it might not • a reason why it might not.79 It would be fantastic to think that there are real objects answering to these descriptions. There is no evidence that the ordinary language speaker labors under the delusion that there are. NeoQuineans may simply prescribe taking “there isare” and “exists” in such a way as to indicate ontological commitments. When we do, then we must correct the ordinary language speaker by either denying the truth of his assertion or offering a paraphrase that avoids the unwanted commitments. Azzouni takes a very dim view of this sort of move on the part of the neoQuinean, characterizing it as a kind of philosophical chicanery The literal onticist may also claim one can compel the ordinary person into trying to paraphrase. ‘There are fictional mice that talk,’ one says, during a discussion about talking fictional animals. ‘Oh, so you believe fictional mice really exist’ the philosophical trickster responds. ‘I didn’t say that,’ one responds. ‘Yes, you sure did – you said “there are fictional mice that talk”. What do you think “there is” means’ One may now try to paraphrase. But what’s happened is that the philosophical trickster has implicitly switched on usage – now one is speaking so that ‘there is’ at least for the time being does convey onticity and the ordinary person isn’t sophisticated enough to see how he or she has been duped.80 The trick is coopting the evident truth of the ordinary language expression in order to imply the truth of a metaphysically heavyweight assertion. This is illegitimately borrowed capital. As we saw in our discussion of fictionalism, when the alleged ontological commitments of a statement are made plain by the neoQuinean, then it is often not at all obvious that the statement so interpreted is true and its denial false. Theodore Sider is more candid than the trickster “Ontological realism should not claim that ordinary quantifiers carve at the joints, or that disputes using ordinary quantifiers are substantive. All that’s important is that one can introduce a fundamental quantifier, which can then be used to pose substantive ontological questions.”81 When one does so, one is no longer speaking ordinary English, but “a new language – ’Ontologese’ – whose quantifiers are stipulated to carve at the joints.”82 If one stipulates that “there isare” is being used to indicate ontological commitments, then, of course, the neutralist who is an antirealist will regard a statement like “There are odd numbers” as false, and the neutralist who is a Platonist will regard the statement as true. But neutralism has not thereby been defeated or shown to be irrelevant, for it is a claim about how we do commit ourselves ontologically in ordinary English, not in a foreign language like Ontologese. Do we, in fact, commit ourselves ontologically through formal and informal quantifiers, as Quineans have asserted The answer is plausibly, no. In fact, the neutralist should protest the suggestion that there even is such a language as Ontologese which is spoken in the metaphysics seminar. All there really is is ordinary language, and we can employ various linguistic and rhetorical devices to make it clear when we are speaking in a metaphysically heavy sense. But nobody – not metaphysicians, not ontologists – actually speaks the language of Ontologese, for there just is no such language. Metaphysical discussions would quickly grind to a halt if every participant had to be able to furnish on demand paraphrases of ordinary language sentences involving unwanted ontological commitments when construed as sentences of Ontologese. Of course, one may simply stipulate that formal and informal quantifiers are to be taken in a metaphysically heavyweight sense. But stipulationists trivialize the debate by simply requiring, in effect, that we must choose between realism and antirealism about various purported objects. We already knew that.83 What they cannot do is use the obvious truth of ordinary language locutions involving quantificational phrases as justification for accepting the truth of sentences in Ontologese stipulated to carry ontological commitments.84 The point remains that without a refutation of neutralism, the Indispensability Argument, which is usually based on how we actually do use formal and informal quantifiers, cannot get off the ground. But what about neutralism’s view of the use of singular terms It seems to be a datum of ordinary language that we frequently assert true statements which contain singular terms which do not denote existent objects. Consider the following examples • The weather in Atlanta will be hot today. • Sherrie’s disappointment with her husband was deep and unassuageable. • The price of the tickets is ten dollars. • Wednesday falls between Tuesday and Thursday. • His sincerity was touching. • James couldn’t pay his mortgage. • The view of the Jezreel Valley from atop Mt. Carmel was breathtaking. • Your constant complaining is futile. • Spassky’s forfeiture ended the match. • He did it for my sake and the children’s. It would be fantastic to think that all of the singular terms featured in these plausibly true sentences have objects in the world corresponding to them. Examples like these are legion. In fact, I suspect that singular terms which refer to real world objects may actually be the exception rather than the rule in ordinary language. How could it be that we are able to assert truths by means of sentences with empty singular terms In order to get at this question, we first need to address the question, do vacuous singular terms refer And in order to answer that question, we need to ask what it is to refer, or what is the nature of reference This question is largely neglected by contemporary theorists. Almost all contemporary theories of reference are actually theories about how to fix reference rather than theories about the nature of reference itself. The unspoken assumption behind most contemporary theories of reference is the presupposition that reference is a wordworld relation, so that terms which refer must have real world objects as their denotations. It therefore behooves us to look more deeply into the nature of reference. It is an experiential datum that referring is a speech act carried out by an intentional agent.85 Words in and of themselves engage in no such activity. Lifeless and inert, words are just ink marks on paper or sounds heard by a percipient. Absent an agent, shapes or noises do not refer to anything at all. If, for example, an earthquake were to send several pebbles rolling down a hillside which randomly came to rest in the configuration JOHN LOVES SUSIE, the names – if we would even call them names – would not refer to anybody. As John Searle says, “Since sentences . . . are, considered in one way, just objects in the world like any other objects, their capacity to represent is not intrinsic but is derived from the Intentionality of the mind.”86 An interpreting agent uses his words as a means of referring to something. Referring is thus an intentional activity of persons, and words are mere instruments. It is the great merit of Arvid Båve’s deflationary theory of reference that he takes truly seriously the fact, given lip service everywhere, that it is persons who refer to things by means of their words, so that words at best refer only in a derivative sense, if at all.87 Båve’s significant contribution to our understanding reference is not so much that his theory is deflationary, as helpful as that may be, but that he furnishes a central schema for reference formulated in terms of the referring activity of agents. Båve proffers the following deflationary schema for reference R a refers to b iff a says something which is about b, where “a” always stands for a speaker. Though formulated in terms of agents rather than words, this account is truly deflationary because it does not attempt to tell us anything about the nature of reference itself. It leaves it entirely open whether reference is a relation as Frege and Meinong assumed or whether it is an intentional property of a mind as held by Brentano and Husserl. Given R, we now ask, what does it mean to say that a says something “about” b, as stipulated on the right hand side of the biconditional R Båve proposes to “analyse the expression ‘about’, and then explain ‘refer’ in terms of it.”88 He offers the following schema as implicitly defining “about” A That S t is about t, where S is a sentence context with a slot for singular terms. Again, Båve’s account of aboutness is extraordinarily deflationary. It does not tell us what aboutness is but simply provides a schema for determining what a thatclause containing a singular term or, presumably, terms is about. So, for example, that Ponce de Leon sought the Fountain of Youth is about Ponce de Leon and about the Fountain of Youth because the singular terms “Ponce de Leon” and “the Fountain of Youth” fill the blanks in the sentence context “ sought.” Now as a deflationary schema, it is very difficult to see how A provides an analysis of aboutness or serves as an explanation of reference, as Båve claims. Taken as an explanation of reference, A seems to get things exactly backwards. The reason why That S t is about t is because “t” is used by some agent to refer to t. It is natural, then, to provide an account of aboutness that is cashed out, not in terms of linguistic expressions, but in terms of the speaker’s intentions. What is wanted in place of A is something along the lines of A´ a says something S about b iff in saying S a intends b, where “a intends b” means something like “a has b in mind.” In accordance with R I refer to b because I say something about b, and in line with A´ I say something about b because in making my utterance I intend b. On such an account what some theorists have called “speaker’s reference” becomes paramount.89 By construing reference in terms of the referring activity of agents and characterizing aboutness in intentional terms, we make an advance over neoMeinongianism to arrive at a more satisfactory neutralist account. This account is consistent with antirealism because successful singular reference does not require that there be objects in the world which stand in some sort of relation to a speaker’s words. Of course, sometimes objects answering to the designations we use may exist. But in a surprisingly large number of cases, as our earlier illustrations showed, there are no such objects. That does not stop us from talking about them or referring to them, for these activities are, at least in such cases, purely intentional activities. The neutralist who is also an antirealist will thus stand in the tradition of Brentano and Husserl in thinking of reference, not as a wordworld relation, but as an intentional activity of agents which may or may not have a correlative real world object.90 Brentano insisted upon the uniqueness of mental phenomena as objectdirected or intentional. The objectdirectedness of mental reference does not imply that intentional objects exist in the external world “All it means is that a mentally active subject is referring to them.”91 For Husserl intentional objects are real world objects about which one is thinking but not all intentional activity has intentional objects associated with it. Rejecting Meinongianism, Husserl held that when no object exists, then the intentional activity exists without any object.92 I can say things about nonexistents like Pegasus, the accident that was prevented, holes, or numbers without committing myself to there being objects of which I am speaking. Thus, the nonexistence of mathematical objects does not preclude our talking about or referring to them. 4.5 Concluding Remarks The Bible teaches that God is the sole ultimate reality, the Creator of everything apart from himself. This biblical doctrine of divine aseity is confirmed by various arguments of natural theology and the demands of perfect being theology. The principal objection to the biblical doctrine of divine aseity comes from contemporary Platonism, which maintains that uncreated abstract objects exist. The principal reason for thinking that such abstracta exist is epitomized in the Indispensability Argument based upon our ineluctable quantification over and reference to abstract objectsWe have seen, however, a plethora of philosophically defensible antiPlatonist alternatives to Platonism on offer today. Free logic, neoMeinongianism, and neutralism constitute formidable challenges to the criterion of ontological commitment that comes to expression in Premise I of the Indispensability Argument. Coupled with the formidable challenges to Premise II of that argument posed by various realist and antirealist alternatives such as absolute creationism, conceptualism, fictionalism, ultima facie strategies, and pretense theory, these considerations go to defeat the Indispensability Argument, the centerpiece of contemporary Platonism. In my studied opinion, both its premises are plausibly false. But in order to defeat the Indispensability Argument, so bold an opinion need not be defended. It suffices merely to undercut the warrant for one of its premises. Given how controversial the debate over contemporary Platonism is, no view can plausibly commend itself philosophically to the exclusion of all others. In order to undercut the Indispensability Argument, the Christian theologian need not defend any one of these views but can recognize the viability of a plurality of viewpoints. I conclude that the challenge posed by Platonism to the doctrine of divine aseity can be met successfully. The biblical doctrine that God is the sole ultimate reality is eminently reasonable. in sentences we take to be true.