God and Abstract Objects

Platonism

What is platonism? Platonism is the ontological thesis there exist mind-independent, abstract entities. This characterization raises the question of what it means to be abstract. Metaphysicians take the distinction between concrete and abstract to be exclusive and exhaustive and typically provide paradigm examples of each kind of object: if such things exist at all, people, electrons, mermaids, and planets would be concrete objects, whereas mathematical objects (like numbers, sets, and functions), properties, and propositions would be abstract objects. Can we say something more about the definitive characteristics of an abstract object? It is frequently asserted that concrete objects just are spatio-temporal objects and that therefore any existing entity which is not spatio-temporal is an abstract object. But this cannot be right, for God, if He exists, is usually taken to transcend space and time and yet is a paradigm example of a concrete object, being a personal agent who effects things in the world. Perhaps that provides a clue to distinguishing between concrete and abstract entities. It is virtually universally agreed that abstract objects, if they exist, are causally impotent, that is to say, they do not stand in cause-effect relations. Numbers, for example, do not cause anything. More than that, their causal impotence seems to be an essential feature of abstract objects. The number seven, for example, does not just happen to lack all causal effects; there is no possible world in which seven could effect something. Their essential causal impotence serves to distinguish abstract objects from any entities which just happen to be causally isolated in the world, but which could have had effects, and from God, who could have refrained from creating and so could have stood in no causal relations.

Our characterization of platonism also raises the question of what it means to be mind-independent. Mind-independence serves to distinguish platonism from conceptualism, which ascribes to abstracta a sort of ideal existence only. Intuitively, one would like to say that mind-independent objects are those that would still exist even if there were no minds. But in a theistic context, it is metaphysically impossible that no minds exist, since God exists necessarily. Even if we allow counterfactuals with impossible antecedents to have non-trivial truth values, it avails us nothing, for the distinction between mind-dependent and mind-independent objects then collapses, since in the absence of God, nothing would exist. Even objects normally taken to be mind-independent, like physical objects, turn out to be mind-dependent. When the platonist ascribes mind-independence to abstract objects, he means that such objects exist as entities in the external world. They are not just the contents of consciousness, either of human or of divine minds. Even if the products of divine intellectual activity, they nonetheless exist extra Deum. They are objects existing in the world, even if not in the spatio-temporal realm.

I think it is fair to say that it is hard to believe that such queer objects really exist. Indeed, even many would-be platonists embrace what has aptly been called a “lightweight platonism” which looks suspiciously very much like conceptualism or even nominalism (see Linnebo 2009; Craig forthcoming). They admit that abstract objects are not objects in the ordinary sense of the word but just in the sense that they are the referents of certain abstract singular terms. On this view it would seem fair to say that Wednesday, for example, is an object, since it may be referred to in true sentences like “Today is Wednesday.” If abstract objects have no more reality than Wednesdays, then the affirmation that they exist may have no significance for ontology.

The Indispensability Argument for Platonism

So why should we go beyond lightweight platonism and affirm the existence of objects so strange as abstract objects? The answer given by contemporary platonists is that the truth of our best scientific theories demands it. Reference to and quantification over abstract objects, particularly mathematical objects, is simply indispensable to natural science, and therefore the truth of those theories requires that the abstract objects referred to and quantified over exist. The truth of mathematical statements, at least those that find application in scientific theories, is guaranteed by the truth of those theories. Customary semantics requires that if singular terms (like names, definite descriptions, and demonstrative expressions) fail to refer to any object and if quantifying expressions like “all” or “some” do not appropriately delimit the range of objects to which the predicate is ascribed, then the sentences containing such expressions cannot be true. Since the mathematical sentences comprised by our best scientific theories are true, it follows that the objects referred to and quantified over must exist. Anyone who regards scientific theories as true must therefore be a platonist.

What might be said in response to the Indispensability Argument? Both of its central claims have been vigorously challenged.

Challenge to the Truth of Mathematical Statements: Fictionalism

Consider first challenges to the truth of sentences referring to or quantifying over abstract objects. Fictionalists accept the customary semantics and so agree that that such sentences cannot be true unless abstract objects exist. Since abstract objects do not exist, it follows that the sentences in question cannot be true. Fictionalism treats abstract objects as more or less useful fictions. Sentences referring to or quantifying over abstract objects are akin to statements of fictional discourse. Just as “Hamlet was a Danish prince” is not true because the name “Hamlet” fails to refer to an object, so “2+2=4” is not true because there are no objects which are the referents of the names “2+2” and “4.” What can be truly said is that according to Shakespeare’s play Hamlet was a Danish prince, so that we may characterize the statement “Hamlet was a Danish prince” as fictionally true. Analogously, mathematical sentences, while not literally true, can be said to be fictionally true, e.g., 2+2=4 according to the standard model for arithmetic based on the Peano axioms.

In dealing with the Indispensability Argument, there are two routes open to the Fictionalist. One route, taken by Hartry Field, is to challenge the assumption that mathematics is indispensable for science and to provide a nominalized science in its place. Field adopts a paraphrastic strategy for re-writing scientific theories so that no reference to or quantification over mathematical objects occurs. The second route, adopted by Mark Balaguer (1998), is to admit that reference to mathematical entities cannot be paraphrased away but to maintain that however indispensable mathematics may be for scientific practice, it contributes nothing of content to our knowledge of the world. Although Balaguer disagrees with the consensus view that Field’s nominalization program is a failure, he nevertheless prefers to concede that mathematics is inextricably woven into empirical science and to maintain that while the nominalistic content of empirical science is for the most part true, its platonistic content is fictional. If, per impossibile, all the abstract objects in the mathematical realm were to disappear, there would be, given their causal isolation, no impact whatsoever on the physical world. Therefore, even if all mathematical objects did not exist, the nominalistic content of science would remain true. The platonistic content of science is something that science says incidentally in its effort to say what it really wants to say, namely, what is contained in its nominalistic content. On Fictionalism, then, scientific theories containing mathematical sentences are literally false.

Challenges to the Customary Semantics for Mathematical Discourse

Fictionalists accept the customary semantics for singular terms and existential quantification. More fundamental challenges to the Indispensability Argument call into question the customary semantics which it presupposes. One form of this challenge is to accept the customary semantics for non-mathematical discourse but to provide a different semantics for mathematical discourse.

Constructibilism. For example, Charles Chihara (1990) has developed a semantics for mathematical statements called Constructibilism which preserves the truth of mathematical statements without committing us ontologically to mathematical objects. This is achieved by re-writing ordinary Zermelo-Fraenkel set theory by replacing the existential quantifier with what Chihara calls a constructibility quantifier, so that existence claims are replaced by claims about what is constructible. The primitive constructibility quantifier Cx is to be understood as asserting, “It is possible to construct an x such that. . . .” What is constructible on Chihara’s theory are certain open sentence tokens, that is to say, sentence tokens containing unbound variables, and assertions of set membership are re-written as assertions about some individual’s satisfying an open sentence. Chihara does not claim that his semantics represents how mathematicians actually understand their language or that it should replace standard mathematical language but claims merely that it shows how mathematical statements may be regarded as true without any commitment to abstract objects.

Deductivism. A second proposed nominalistic semantics for mathematical sentences is counterfactual If-Then-ism or Deductivism. Traditional Deductivism interpreted mathematical statements to be disguised conditionals to the effect that if certain axioms are true, then certain theorems are true. So “2+2=4” is the assertion that if the Peano Axioms are true, then 2+2=4. The problem with this view is that on Nominalism the Peano Axioms are in fact false, given their existence assertions about mathematical objects, and so the statements of arithmetic are vacuously true. It is on this view equally vacuously true that “2 +2=5,” since that claim is also materially implied by the false antecedent of the conditional. This problem can be solved, however, by interpreting the relevant conditionals not as material implications but as counterfactual conditionals to the effect that if the Peano Axioms were true, then the theorems of standard mathematics would be true. Since counterfactuals are not truth functional, there is no implication that if the Peano Axioms were true, then would it would be the case that 2+2=5. Geoffrey Hellman’s Modal Structuralism (1989) would appear to be a variation on this approach. Hellman takes arithmetic statements like 2+2=4 to be statements about possible structures to the effect that if a certain structure were to exist, then it would be the case, say, that the object in the position named “2+2” in that structure is the object in the fourth place in that structure. Counterfactual Deductivism need not commit one to the contingency of mathematical objects and their existence in some possible world if one is willing to countenance non-trivial counterfactuals with impossible antecedents, as seems plausible.

Figuralism. Finally, a third proposal for a semantics of abstract object talk is Figuralism. Stephen Yablo (2000) is impressed with the similarities between abstract object talk and figurative talk such as we find in understatement, hyperbole, metonymy, and metaphor. An assertion like “It’s raining cats and dogs!” is literally false, but to stop there is to miss the whole point of such language. When a speaker uses figurative language, the literal content is not what the speaker is asserting. There is what Yablo calls a “real content” to figurative statements which may well be true. This is not to say that figurative statements can always be successfully paraphrased into expressions of their real content. Numbers may be indispensable as representational aids for the expression of the real content of mathematical language. The real content of mathematical statements is logical truths, which is why mathematics seems necessary and a priori. For example, the real content of “2+3=5” is the logical truth employing numerical quantifiers:

[∃2x(Fx) & ∃3y(Gy) & ¬∃z(Fz & Gz)] ⊃ ∃5u(Fu ∨ Gu).

Yablo extends his analysis to include other sorts of abstract object talk as well. For example,

The entities on the right are not what the expressions on the left are really about. We simulate belief, perhaps quite unconsciously, that the entities on the right exist, but they are mere figures of speech which are vehicles of the real content. Figurative speech may be true—herein lies the difference between Figuralism and Fictionalism—but the representational aids it employs are not ontologically committing.

Challenges to the Customary Semantics in General

The above strategies all accept the customary semantics in general but seek to develop a special semantics for abstract object talk in particular. Even more fundamental challenges to the Indispensability Argument call into question the customary semantics tout court.

Defense of Irreferential Terms: Free Logic. One such alternative to the customary semantics is provided by Free Logic. Free Logics are logics whose quantifiers remain ontologically committing but whose general and singular terms are devoid of existential import. Karel Lambert (2003) complains that modern logic still retains existence assumptions that ought not to characterize a purely formal discipline. These assumptions surface in standard logic’s treatment of statements of identity. According to standard modern logic, identity statements presuppose existence assumptions, that is to say, their truth requires the existence of the objects referred to in the identity statement. This is evident in the fact that from t=t, where t is some singular term, it follows that ∃x(x=t). For if we substitute the predicate “=t” for “P” in Pt ⊃ ∃x(Px), a theorem of modern logic, we have t=t ⊃ ∃x(x=t). But, then, absurdly, it would follow from “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. As a result 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.

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 non-denoting, vacuous, empty) singular terms. It does not presuppose (like Meinongianism) that the referents of such terms are non-existent objects; rather there just are no referents. Because Free Logic retains the existential force of the quantifiers of standard logic, it is consistent with the customary view that one is committed by existential quantification to the objects quantified over in true statements. But Free Logic avoids gratuitous commitments by modifying Existential Generalization and Universal Instantiation. EG now becomes ∃x(x=t) ⊃ [Pt ⊃ ∃x(Px)], and UI is replaced by ∀y(∀x(Px) ⊃ Py).

The Free Logician will point out that the Indispensability Argument must rely crucially on EG if it is not to be question-begging. When a scientific statement Pt includes a singular term t, we shall not be ontologically committed to a referent of t unless we can infer “Therefore, there is something that is P,” or ∃x(Px). But in Free Logic such existential generalization is not valid. We should also need to know that ∃x(x=t), which to assume is question-begging. If t is an irreferential term like “3,” as the nominalist believes, then the truth of “2<3<4” does not commit us to the reality of 3. Should the platonist simply begin with an existential assertion, e.g., “3 is a number between 2 and 4,” he begs the question. The nominalist Free Logician will, like the Fictionalist, regard this statement as false.

Reference to Non-Existent Objects: Neo-Meinongianism. Proponents of Free Logic have challenged the customary semantics for general and singular terms, but they have taken over unchallenged the customary semantics’ view of the existential quantifier. Other nominalist schools of thought have been more critical. Perhaps the most radical is neo-Meinongianism, which, in contrast to Free Logic, retains the referentiality of singular terms but rejects the existential quantifier of traditional logic. According neo-Meinongianism, singular terms in true statements do refer to objects, but these objects may not exist. When Meinong affirmed that “There are things that do not exist,” no contradiction was involved because for Meinong es gibt (“there is/are”) is not, as neo-Meinongian Richard Routley puts it, “existentially loaded” (Routley 1979, 76). The expression does not imply that something exists. Routley takes the quantifiers of classical logic to be existentially loaded and therefore proposes a reform of classical logic by replacing it with a neutral quantified logic. Like the Free Logician, Routley faults the traditional scheme of EG: Fa ⊃ (∃x)Fx. His rejection of traditional EG is the result of the neo-Meinongian repudiation of what Routley calls the Ontological Assumption, to wit, the assumption that a statement has the value true and is about something only if the subject of the statement refers to an existent object. The correct scheme of EG will involve the use of an existence predicate E: Fa & Ea ⊃ (∃x)Fx. In neutral quantification logic the existential quantifier will be replaced by a quantifier of particularization P to be interpreted as “for some item.” So “Some things do not exist” is symbolized (Px)(¬Ex). EG will be replaced by a Principle of Particularization Fa ⊃ (Px)Fx, that is, for some item, Fx.

Routley, in contrast to Meinong, who thought that abstract objects subsist, takes abstract objects to be items that do not exist and so discourse about them to be properly formalized by a neutral quantification logic (Routley 1979, 45). By replacing the existentially loaded quantifier of classical logic with a neutral logic featuring a quantifier of particularization, we may affirm, in contrast to the Fictionalist, that it is true that “There is a number 4” without committing ourselves to the reality of mathematical objects.

Existentially Neutral Quantifiers: Neutral Logic. The appeal to neutral logic is independent of Meinongianism. Advocates of Neutralism like Jody Azzouni (1998, 2004, 2007, 2010) do not advocate a reform of classical logic to replace the existential quantifier but challenge the assumption that the quantifier of classical logic is ontologically committing. The purpose of the existential quantifier is simply to facilitate logical inferences. It can carry out that function without making ontological commitments to objects, existent or non-existent. Why, Azzouni asks, should we think that this quantifier has any different meaning or carries any more ontological force than “there is/are” in ordinary language, which is clearly non-committing?

Philosophers typically discriminate between two interpretations of the existential quantifier: the objectual (or referential) and the substitutional. The objectual interpretation of the quantifier conceives it as ranging over a domain of objects and picking out some of those objects as the values of the variable bound by it. The substitutional interpretation takes the variable to be a sort of place-holder for particular linguistic expressions which can be substituted for it to form sentences. The substitutional interpretation is generally recognized not to be ontologically committing. But Azzouni maintains that even the objectual interpretation of the quantifier is not ontologically committing until one so stipulates. The claim that it must be ontologically committing overlooks the fact that the quantifiers of the metalanguage used to establish the domain of the object language quantifiers are similarly ambiguous. Whether the items in the domain D of the object language quantifier actually exist will depend on how one construes the “there is” of the metalanguage establishing D. Even referential use of the quantifier in the object language need not be ontologically committing if the quantifiers in the metalanguage are not ontologically committing. If, when we say that there is an element in D, we are using ordinary language, then we are not committed to the reality of the objects in D which we quantify over. There is no reason to think that one cannot set up as one’s domain of quantification a wholly imaginary realm of objects. D is then non-empty, but objectual quantification in the object language of the domain will not be ontologically committing. It will be contextual factors that will tip us off to whether locutions are being used in ontologically committing ways.

Reference without Ontological Commitment: Deflationary Reference.Neutral Logic will obviate any automatic ontological commitments thought to arise from existential quantification. This takes us back to questions of reference. The neo-Meinongian, like the Free Logician, continues to assume with the customary semantics that singular terms, when used referentially, must designate some mind-independent object. But why should we think that? It is an experiential datum that referring is a speech act carried out by an intentional agent. Absent an agent, ink marks on paper or noises do not refer to anything at all. Referring is an intentional activity of persons, and words are mere instruments. We need to take 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.

As obvious as this point is, theorists of reference remain strangely oblivious to the fact. Reference continues to be very widely construed as a relation obtaining between words and objects in the world. By contrast, Arvid Båve’s (2009) new deflationary theory of reference features a central schema for reference formulated in terms of the referring activity of agents:

(R) a refers to b iff a says something (which is) about b,

where “a” always stands for a speaker. This account is 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). Taking reference to be a relation between a speaker and some object makes (R) ontologically committing to either existing or non-existing objects. But Båve’s theory is ontologically neutral when it comes to the question of whether there must be objects corresponding to the singular terms we use successfully to refer. On his account, if I assert “1+1=2,” then I have said something about 2; it follows from (R) that I have thus referred to 2. But it does not follow that there is some such object, existent or non-existent, as the number 2. One has the option of avoiding the inference to “There is something to which I have referred” by restricting, with the Free Logicians, EG, or the option of granting the inference but rendering it harmless by denying, with the Neutral Logicians, that the so-called existential quantifier is ontologically committing. Hence, Båve recognizes the neutrality of his theory for the debate between nominalists and realists.

So what does it mean to say that a says something “about” b, as stipulated on the right hand side of the biconditional (R)? Båve 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 that-clause 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____.”

There is nothing in the deflationary schema (A) that entails that aboutness is a relation between propositions and objects. So if I assert, “Ponce de Leon sought the Fountain of Youth,” I have said something which is both true and about the Fountain of Youth (as well as about Ponce de Leon); but we are not entitled to infer with Meinong that there are non-existent objects like the Fountain of Youth which this sentence is about. I can say things about Pegasus, the accident that was prevented, or numbers without committing myself to there being objects of which I am speaking.

Theological Objection to Platonism

So much for responses to the case for platonism. What reasons are there for rejecting platonism? The two objections usually urged against platonism are the epistemological objection and the uniqueness objection (Benacerraf 1965, 1973). Whether platonists can successfully defeat these objections may remain a moot question here. My concern is one that is scarcely ever broached in the philosophical literature: that platonism is theologically untenable. If this contention is correct, then, given the truth of classical theism, it will defeat all forms of platonism, even versions crafted to avoid the epistemological and uniqueness objections. The reason for platonism’s theological unacceptability for orthodox theists is that it is incompatible with the doctrine of creatio ex nihilo and so fundamentally compromises divine aseity.

In the prologue of the Gospel of John, the evangelist presents a vision of God as the Creator of all things: “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” (Jn 1.1-3). God through his Word is responsible for the existence of literally everything other than God himself. Apart from God every existent belongs to the creaturely realm, the class of things which have come into being (geneta), and so owe their existence to God’s creative Word or Reason (logos), who is later identified as Christ (Jn 1.14-18). Jn 1.1-3 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 which are co-eternal with God and uncreated via the Logos by God.

God’s unique status as the only eternal, uncreated being is typical for Judaism (Copan and Craig 2004, 29-145). John himself identifies the Logos alone as existing with God (and being God) in the beginning. Everything else is then created through the Logos. It is who or what God is that requires the domain of John’s quantifier to be unrestricted, whatever beings might be found to lie in the domain. Indeed, given the striking similarities of John’s Logos doctrine to that of the Alexandrian Jewish philosopher Philo (20 B.C.-A.D. 50), it is not all implausible that John thought that the intelligible realm of what we would today call abstract objects was contained, as Philo held, in the divine Logos (see Leonhardt-Balzer 2004, 309-10, 318-319). Everything else has been created by God.

The evangelist’s conviction that God is the Creator of everything that exists aside from God himself eventually attained credal status at the Council of Nicaea. In language redolent of the prologue to the fourth Gospel (Jn 1.3) and of Paul (Col 1.16), the Council affirmed:

At face value the Council affirms that God alone is uncreated and that all else was created by him.

An examination of ante-Nicene theological reflection on divine aseity confirms the prima face reading. Like the Arian heretics, the ante-Nicene and Nicene Church Fathers rejected any suggestion that there might exist ageneta apart from God alone. [1] According to patristic scholar Harry Austryn Wolfson, the Church Fathers all accepted the following three principles (Wolfson 1970, 414):

Each of these principles implies that there are no ageneta other than God.

Lest it be suggested that abstracta were somehow exempted from these principles, we should note that the ante-Nicene Church Fathers explicitly rejected the view that entities such as properties and numbers are ageneta. The Fathers were familiar with the metaphysical worldviews of Plato and Pythagoras and agreed with them that there is one agenetos from which all reality derives; but the Fathers identified this agenetos, not with an impersonal form or number, but with the Hebrew God, who has created all things (other than himself) ex nihilo.  [2] If confronted by a modern-day platonist defending an ontology which included causally effete objects which were ageneta and so co-eternal with God, they would have rejected such an account as blasphemous, since such an account would impugn God’s aseity by denying its uniqueness and undermine creatio ex nihilo by denying that God is the universal ground of being. The Fathers could not therefore exempt such objects from God’s creative power, since he is the sole and all-originating agenetos.

Can an accommodation between platonism and classical theism be reached? The easiest and most obvious proposal is Absolute Creationism, which maintains that God has created any abstract objects that exist. Unfortunately, Absolute Creationism faces two difficulties, the first troublesome and the second truly serious. First, Absolute Creationism misconstrues either the scope or nature of creation. From a biblical perspective creation is an inherently temporal concept implying a temporal beginning of existence for any created thing (Copan and Craig 2004, chaps. 1-4); yet it is plausible that many types of abstract objects, if they exist, exist necessarily and so have no beginning of existence. So if we think of abstract objects as part of the order of dependent beings existing apart from God, then the scope of creatio ex nihilo becomes miniscule. The overwhelming bulk of things is merely sustained in being but not, properly speaking, created by God. If, to avoid this difficulty, we expand the meaning of “creation” so as to make any dependent being the object of God’s creation, then we have radically subverted God’s freedom with respect to creating. The vast majority of being flows from him with an inexorable necessity independent of his will. Thus, the ontology of Absolute Creationism is incompatible with the doctrine of creatio ex nihilo, attenuating either God’s freedom or the scope of creation.

The second and more serious problem with Absolute Creationism is that it appears to be logically incoherent. Simply stated, the problem is that the creation of certain abstract objects presupposes the existence of those objects, so that a vicious explanatory circle is formed (Bergmann and Brower 2006). For example, God cannot create the property of being powerful unless he already has the property of being powerful. (If one maintains that God can be powerful without exemplifying the property of being powerful, then one has thereby ceded the palm of victory to the nominalist, who denies that the truth of “God is powerful” entails the existence of a property.) The challenge facing Absolute Creationists is to find a way out of this explanatory circle. [3] Indeed, I should say that the chief problem posed by the existence of abstract objects to classical theism stems not from their necessary existence but from, in certain cases, their uncreatability. It is not the existence of abstract objects as such that poses a serious challenge to divine aseity but rather the putative existence of uncreatables.

Conclusion

Given the failure of Absolute Creationism, the Christian theist cannot consistently embrace platonism because such a doctrine compromises creatio ex nihilo and divine aseity by its postulation of uncreatables. Fortunately, there are abundant responses to the Indispensability Argument for platonism. The Indispensability Argument is at root an exercise in metaontology, not simply ontology. Behind it lie certain metaontological assumptions about the nature of reference and quantification that must be surfaced and examined. A deflationary theory of reference and neutral logical interpretation of first-order quantification are intuitive, defensible, and effective strategies for undercutting the Indispensability Argument. Nominalism or conceptualism remain open options for the Christian theist.

WILLIAM LANE CRAIG

References

Azzouni, Jody. “On ‘On what There Is’.” Pacific Philosophical Quarterly 79 (1998): 1-18.

_______. Deflating Existential Consequence: A Case for Nominalism. Oxford: University Press, 2004.

_______. “Ontological Commitment in the Vernacular.” Nous 41 (2007): 204-26.

_______. “Ontology and the Word ‘Exist’: Uneasy Relations.” Philosophia Mathematica 18 (2010): ***

Balaguer, Mark. Platonism and Anti-platonism in Mathematics (New York: Oxford University Press, 1998.

Båve, Arvid. “A Deflationary Theory of Reference.” Synthèse 169 (2009): 51-73.

Benacerraf, Paul. “What Numbers Could Not Be.” Philosophical Review 74 (1965): 47-73.

_______. “Mathematical Truth.” Journal of Philosophy 70 (1973): 661-679.

Bergmann, Michael and Brower, Jeffrey. “A Theistic Argument against Platonism (and in Support of Truthmakers and Divine Simplicity).” Oxford Studies in Metaphysics 2 (2006): 357-86.

Chihara, Charles. Constructibility and Mathematical Existence. Oxford: Clarendon Press, 1990.

Copan, Paul and Craig, William Lane. Creation out of Nothing. Grand Rapids, Mich.: Baker, 2004.

Craig, William Lane. “Why Are (Some) Platonists so Insouciant?” Philosophy (forthcoming).

Hellman, Geoffrey. Mathematics without Numbers. Oxford: University Press, 1989.

Lambert, Karel. Free Logic. Cambridge: University Press, 2003.

Leonhardt-Balzer, Jutta. “Der Logos und die Schöpfung: Streiflichter bei Philo (Op 20-25) und im Johannesprolog (Joh 1, 1-18).” In Kontexte des Johannesevangelium, Ed. J. Frey and U. Schnelle. Tübingen: Mohr-Siebeck, 2004.

Linnebo, Øystein. Stanford Encyclopedia of Philosophy, s.v. “Platonism in the Philosophy of Mathematics,” July 18, 2009, http://plato.stanford.edu/entries/platonism-mathematics/.

Routley [Sylvan], Richard. Exploring Meinong’s Jungle and Beyond. Canberra: Australian National University Research School of Social Sciences, 1979.

Wolfson, Harry Austryn. The Philosophy of the Church Fathers, vol.: I: Faith, Trinity, and Incarnation, 3rd ed . rev. Cambridge, Mass.: Harvard University Press, 1970.

_______. “Plato’s Pre-existent Matter in Patristic Philosophy.” In The Classical Tradition. Ed. L. Wallach. Ithaca, NY: Cornell University Press, 1966.

Yablo, Stephen. “A Paradox of Existence.” In Empty Names, Fiction, and the Puzzles of Non-Existence, 275-312. Ed. A. Everett and T. Hofweber. Stanford: Center for the Study of Language and Information, 2000.

Further Reading

Azzouni, Jody. Deflating Existential Consequence: A Case for Nominalism. Oxford: University Press, 2004.

Balaguer, Mark. Platonism and Anti-platonism in Mathematics (New York: Oxford University Press, 1998.

_____. Stanford Encyclopedia of Philosophy, s.v. “Platonism in Metaphysics,” April 7, 2009 http://plato.stanford.edu/entries/platonism/

Båve, Arvid. “A Deflationary Theory of Reference.” Synthèse 169 (2009): 51-73.

Linnebo, Øystein. Stanford Encyclopedia of Philosophy, s.v. “Platonism in the Philosophy of Mathematics,” July 18, 2009, http://plato.stanford.edu/entries/platonism-mathematics/.

• [1]

  Justin Dialogue 5; Methodius On Free Will 5; Irenaeus Against heresies 4.38.3; Tertullian Against Praxeas 5.13-15; Hippolytus Against Noetus 10.1; idem Refutation of All Heresies 10.28; Epiphanius Panarion 33.7.6; Athanasius Defense of the Nicene Definition 7: “On the Arian symbol ‘Agenetos’;” idem Discourses against the Arians 1.9.30-34; idem On the Councils of Ariminum and Seleucia 46-47; idem Statement of Faith 3.

  Justin Dialogue 5; Methodius On Free Will 5; Irenaeus Against heresies 4.38.3; Tertullian Against Praxeas 5.13-15; Hippolytus Against Noetus 10.1; idem Refutation of All Heresies 10.28; Epiphanius Panarion 33.7.6; Athanasius Defense of the Nicene Definition 7: “On the Arian symbol ‘Agenetos’;” idem Discourses against the Arians 1.9.30-34; idem On the Councils of Ariminum and Seleucia 46-47; idem Statement of Faith 3.

• [2]

  Athenagoras Plea for the Christians 15, 24; Tatian Address to the Greeks 4.10-14; Methodius Concerning Free Will;Hippolytus Refutation 6.16, 18, 19, 24, 43. Combining the Gospel of John’s Logos doctrine with Philo’s, the Greek Apologists grounded the intelligible realm in God rather than in some independent realm of self-subsisting entities like numbers or forms. According to Wolfson, every Church Father who addressed the issue rejected the view that the ideas were self-subsisting entities but instead located the intelligible world in the Logos and, hence, in the mind of God. For a discussion of texts taken from pseudo-Justin, Irenaeus, Tertullian, Clement of Alexandria, Origen, and Augustine, see (Wolfson 1970, chap. XIII).

  Athenagoras Plea for the Christians 15, 24; Tatian Address to the Greeks 4.10-14; Methodius Concerning Free Will;Hippolytus Refutation 6.16, 18, 19, 24, 43. Combining the Gospel of John’s Logos doctrine with Philo’s, the Greek Apologists grounded the intelligible realm in God rather than in some independent realm of self-subsisting entities like numbers or forms. According to Wolfson, every Church Father who addressed the issue rejected the view that the ideas were self-subsisting entities but instead located the intelligible world in the Logos and, hence, in the mind of God. For a discussion of texts taken from pseudo-Justin, Irenaeus, Tertullian, Clement of Alexandria, Origen, and Augustine, see (Wolfson 1970, chap. XIII).

• [3]

  I consider some suggested escape routes in (Copan and Craig 176-80) and find them unavailing.

  I consider some suggested escape routes in (Copan and Craig 176-80) and find them unavailing.