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God and Abstract Objects

William Lane Craig

Used by permission of Philosophia Christi 17 (2015):  269-76.


Central to classical theism is the conception of God as the sole ultimate reality, the creator of all things apart from Himself.  Such a doctrine is rooted in Hebrew-Christian Scripture and unfolded by the Ante-Nicene Church Fathers.  Platonism, which postulates the existence of uncreated abstract objects, is therefore theologically objectionable.  I thus find myself in agreement with Peter van Inwagen, though for different reasons, that the Christian philosopher, at least, “should not believe in abstract objects unless [he] feels rationally compelled by some weighty consideration or argument.”  But there is no such argument.  The principal argument offered on behalf of Platonism comes in the various incarnations of the Quine-Putnam Indispensability Argument.  But there are a plethora of anti-Platonist responses, both realist and anti-realist, to that argument.  In order to overcome the presumption which anti-Platonism enjoys theologically, the Platonist would have to show that all of these other positions are rationally untenable.  No one has even attempted so audacious a project, nor is there any reasonable expectation that it could be carried out.

Central to classical theism is the conception of God as the sole ultimate reality, the creator of all things apart from Himself.  Such a doctrine is rooted in Hebrew-Christian Scripture.  To select but one text, in the prologue of the Gospel of John, we read:  “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).[1]  The evangelist gives us to understand that 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 (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 by God. 

The strongest challenge to the traditional doctrine of divine aseity comes from the philosophy of Platonism.  Although contemporary Platonism differs vastly from classical Platonism in various respects,[2] both views are united in holding that there exist uncreated entities--for example, mathematical objects--other than God.  Would-be Christian Platonists must maintain that John’s domain of quantification is restricted in such a way that abstract objects escape his universally quantified statements.

Now it’s important that we understand clearly the question before us, since it is so often misunderstood.  The question is not:  did John have abstract objects in mind when he said “all things came into being through him”?  Probably not!  But by the same token, neither did he have in mind quarks, galaxies, and black holes; yet he would doubtless take such things and countless other things, were he informed about them, to have been created by God and to be in the class of things he is talking about.

The question is not what John thought lay in the domain of his quantifiers; rather the question is whether John intends his domain of quantification, once God is exempted, to be unrestricted.  Does he think that apart from God everything else that exists is created by God?  It is more than probable that he did.  For God’s status as the only eternal, uncreated being is an earmark of first century Judaism.  In his influential work on the character of ancient Jewish monotheism, Richard Bauckham identifies two characteristics that uniquely mark off Israel’s God from all others, namely that “he is Creator of all things and sovereign Ruler of all things.”[3] There is in the Judaism of John’s day a bright dividing line which separates God ontologically from everything else, a bifurcation which Bauckham attempts to capture by the term “transcendent uniqueness.” God’s status as the sole ultimate reality comes to practical expression in the Jewish restriction of worship as properly directed toward God alone.  According to Bauckham this restriction “most clearly signaled the distinction between God and all other reality.”[4]

The crucial point here is that the unrestrictedness of the domain of quantification is based, not in what kinds of objects were thought to lie in the domain, but rather in the Jewish doctrine of God as the only being which exists eternally and a se.  It is who or what God is that requires that the domain of quantification be unrestricted, whatever beings might be discovered to lie in the domain.

John himself identifies the Word (Logos) alone as existing with God and being God in the beginning.  The creation of everything else through the divine Logos then follows.  Bauckham calls such a view “Christological monotheism:”  the divine Logos is on God’s side of the dividing line between God and the rest of reality.  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, like Philo, thought that the intelligible realm (kosmos noetos) of what we would today call abstract objects was contained in the divine Logos, so that creation comprised only concrete objects.[5] 

Be that as it may, there is no reason to doubt that John believed that every existing thing apart from God had come into being through the Logos. To postulate an infinite plentitude of abstract objects as real as planets existing independently of God, so that the realm of concrete objects brought into being by God is literally infinitesimal by comparison, would betray Jewish monotheism and trivialize the doctrine of creation.   So I think it clear that John did intend his domain of quantification to include everything apart from God, whatever idea he may have had concerning what objects lay in the domain.

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 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.[6]

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.  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:  agenetos/genetos and agennetos/gennetos.[7]  The word pair agenetos/genetos derives from the verb “ginomai,” which means to become or to come into being.  “Agenetos” means unoriginated or uncreated, in contrast to “genetos,” that which is created or originated.  The second word pair agennetos/gennetos derives from the verb “ginnao,” which means to beget.  That which is agennetos is unbegotten, while that which is gennetos is begotten.  These distinctions allowed the Fathers to hold that while both God the Father and God the Son are agenetos, only the Father is agennetos.

The ante-Nicene and Nicene Church Fathers, like the Arian heretics,  rejected any suggestion that there might exist ageneta apart from God alone.[8]  According to patristic scholar Harry Austryn Wolfson,[9] the Church Fathers all accepted the following three principles:

1.  God alone is uncreated.

2.  Nothing is co-eternal with God.

3.  Eternality implies deity.

Each of these principles implies that there are no ageneta apart from God. 

But 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.[10]  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 unique aseity 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.

I have belabored this point because the grounds of my rejection of Platonism are not philosophical but theological.  I press no philosophical objections against Platonism; rather, rejecting Quine’s epistemological naturalism, I offer theological grounds for thinking Platonism false.  I thus find myself in agreement with Prof. van Inwagen, though for different reasons, that the Christian philosopher, at least, “should not believe in abstract objects unless [he] feels rationally compelled by some weighty consideration or argument. . . . a philosopher should wish not to be a platonist if it’s rationally possible for the informed philosopher not to be a platonist.”[11]  Only if anti-Platonism is rationally impossible to hold, only if there is a rationally compelling argument for Platonism, should the Christian philosopher feel torn to abandon his theological commitment to God’s being the sole ultimate reality.

It hardly needs to be said that there is no such argument.  The principal argument offered on behalf of Platonism comes in the various incarnations of the Quine-Putnam Indispensability Argument.  Mark Balaguer succinctly formulates the Indispensability Argument as follows:[12]

(I) If a simple sentence (i.e., a sentence of the form ‘a is F’) is literally true, then the objects that its singular terms denote exist. (Likewise, if an existential sentence (e.g., ‘There is an F’) is literally true, then there exist objects of the relevant kinds.

(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.

(I) is a metaontological thesis expressing a criterion of ontological commitment. (II) is the affirmation that abstract discourse is be construed as literally true.  How might the Christian philosopher respond to the Indispensability Argument?  Taking mathematical objects as a case in point, Figure 1 displays some of our many options.

Flowchart of responses to indispensability arguments concerning the existence of mathematical objects.

Fig. 1:  Some responses to indispensability arguments concerning the existence of mathematical objects.

The various options can be classed as realist (mathematical objects exist);  anti-realist (mathematical objects do not exist); or arealist (there is no fact of the matter concerning the existence of mathematical objects).  Look first at the realist branch.  As Fig. 1 illustrates, 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, absolute creationism is a sort of modified Platonism, holding that mathematical objects have, like concrete objects, been created by God, thus safeguarding divine aseity.  Concretist versions of realism can 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 Philo and the Church Fathers, according to which there are no mathematical objects independent of God.

Moving left to right, we next come to 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.[13]  No philosopher today would defend Carnap’s verificationism; but his conventionalism does find an echo today in what we might call ontological pluralism.[14]  According to thinkers of this persuasion, certain ontological questions, though meaningful, do not have objective answers.  Some non-realists, notably the philosophers of mathematics Mark Balaguer and Penelope Maddy, would deny that the question “Do mathematical objects exist?” has an answer that is objectively true or false.[15]  Now at first blush arealism might seem a quick and easy solution to the challenge posed by Platonism to divine aseity.  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.

When we turn to anti-realist responses to the Indispensability Argument, we find a cornucopia of different views.   Neutralism rejects the criterion of ontological commitment expressed in Premiss (I), taking the use of singular terms and existential quantification to be neutral with respect to ontological commitments.[16]  Fictionalism accepts the Platonist’s criterion of ontological commitment but denies that mathematical statements are true.[17]  Figuralism holds that mathematical discourse is true but denies that it must be taken literally.[18]  Neo-Meinongianism holds that there are objects referred to by abstract singular terms but takes these objects to be nonexistent.[19]   Pretense theory considers mathematical discourse to be a species of make-believe, so that mathematical objects are akin to fictional characters.[20]   Paraphrastic strategies like Charles Chihara’s constructibilism or Geoffrey Hellman’s modal structuralism hold that we can offer paraphrases of mathematical statements which will preserve their truth value without ontological commitment to abstract objects.[21]  And so on!

In order to overcome the presumption which anti-Platonism enjoys theologically, the Platonist would have to show that all of these other positions are rationally impossible to hold, that all the informed philosophers espousing such positions are irrational in so doing.  No one has even attempted so audacious a project, nor is there any reasonable expectation that it could be carried out.

Had time permitted, I should like to have shared which anti-realist view (or combination thereof) I find most persuasive.  For that you’ll have to read the book!  But the more important task here today was to explain why I think you should not be a Platonist and then to open up some options on how you might avoid it.

  • [1] Ἐν ἀρχῇ ἦν ὁ λόγος, καὶ ὁ λόγος ἦν πρὸς τὸν θεόν, καὶ θεὸς ἦν ὁ λόγος.  οὗτος ἦν ἐν ἀρχῇ πρὸς τὸν θεόν.  πάντα δι᾽ αὐτοῦ ἐγένετο, καὶ χωρὶς αὐτοῦ ἐγένετο οὐδὲ ἕν.

  • [2] Principally in taking abstract objects to be causally unrelated to the concrete world; neither do contemporary Platonists consider abstract objects to be more real than concrete objects.  Nor do they think that concrete objects participate in some way in abstract entities, as Plato thought physical objects participate in ideal objects.

  • [3] Richard Bauckam, “God Crucified,” in Jesus and the God of Israel (Grand Rapids, Mich.: William B. Eerdmans, 2008), p. 8.

  • [4] Ibid., p. 11.

  • [5] The doctrine of the divine, creative Logos was widespread in Middle Platonism, and 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 the first chapter of Genesis (Jutta Leonhardt-Balzer, “Der Logos und die Schöpfung:  Streiflichter bei Philo (Op 20-25) und im Johannesprolog (Joh 1, 1-18), in Kontexte des Johannesevangelium, ed. Jörg Frey and Udo Schnelle, WUNT 175 [Tübingen:  Mohr-Siebeck, 2004], pp. 309-10, cf. pp. 318-19).  A hallmark of Middle Platonism was Plato’s bifurcation between the realm of static being (τί τὸ ὄν ἀεί) and the realm of temporal becoming (τί τὸ γιγνόμενον μὲν ἀεί) (Timaeus  27d5-28a4).  The realm of becoming was comprised primarily of physical objects, though it would also include immaterial objects like souls, while the static realm of being was comprised of what we would today call abstract objects.  The former realm is perceived by the senses, whereas the latter is grasped by the intellect.  For Middle Platonists, as for Plato, the intelligible world served as a model for the creation of the sensible world.  But for a Jewish monotheist like Philo, the realm of Ideas does not exist independently of God but as the contents of His mind (On the Creation of the World [De opificio mundi]16-25).  For Philo the intelligible world (κόσμος νοητός) may be thought of as either formed by the divine Logos or, more reductively, as the divine Logos itself as God is engaged in creating.  On Philo’s doctrine, then, there is no realm of independently existing abstract objects.  In Runia’s words, while not part of the created realm, “the κόσμος νοητός, though eternal and unchanging, must be considered dependent for its existence on God” (D. T. Runia, Philo of Alexandria and the “Timaeus” of Plato [Amsterdam:  Free University of Amsterdam, 1983], p. 138).  John does not tarry to reflect on the role of the divine Logos causally prior to creation, but given the provenance of his doctrine it is not at all implausible that he, too, thought of the Logos as the seat of the intelligible realm of what we would call abstract objects.

  • [6] Πιστεύω είς ενα Θεόν, Πατέρα, παντοκράτορα, ποιητήν ουρανού καί γής, ορατών τε πάντων καί αοράτων.

    Καί είς ενα Κύριον, Ίησούν Χριστόν, τόν Υιόν του Θεού τόν μονογενή, τόν εκ του Πατρός γεννηθέντα πρό πάντων τών αιώνων. φώς εκ φωτός, Θεόν αληθινόν εκ Θεού αληθινού γεννηθέντα, ού ποιηθέντα, ὁμοούσιον τώ Πατρί, δι’ ού τά πάντα εγένετο.

  • [7] For a survey of texts see the nice discussion by George L. Prestige, God in Patristic Thought (London:  SPCK, 1964), pp. 37-55, which I follow here.  See also the references in J. B. Lightfoot’s “Excursus on the Words gennethenta ou poiethenta” reprinted in Nicene and Post-Nicene Fathers, 2d series, vol. 14:  Seven Ecumenical Councils, ed. Philip Schaff and Henry Wace (rep. ed.: Peabody, Mass.:  Hendrickson, 1994), pp. 4-7.

  • [8] 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.  All references to the Church Fathers in this note and note 10 are from Ante-Nicene Fathers, ed. Alexander Roberts and James Donaldson et al. (rep.ed.:  Peabody, Mass.:  Hendrickson, 1994) and Nicene and Post-Nicene Fathers, 2d series, ed. Philip Schaff and Henry Wace et al. (rep. ed.: Peabody, Mass.:  Hendrickson, 1994).

  • [9] Harry A. Wolfson, “Plato’s Pre-existent Matter in Patristic Philosophy,” in The Classical Tradition, ed. Luitpold Wallach (Ithaca, NY:  Cornell University Press, 1966), p. 414.

  • [10] 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 presentation of Christ as the pre-existent Logos who in the beginning was with God and was God and through whom all things came into being (Jn 1.1-3) with Philo of Alexandria’s conception of the Logos as the mind of God in which the Platonic realm of Ideas subsists (On the Creation of the World 16-25), 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 Harry Austryn Wolfson, The Philosophy of  the Church Fathers, vol.: I:  Faith, Trinity, and Incarnation, 3rd ed . rev. (Cambridge, Mass.:  Harvard University Press, 1970), chap. XIII:  “The Logos and the Platonic Ideas.”

  • [11] Peter van Inwagen, “A Theory of Properties,” in Oxford Studies in Metaphysics, vol. 1, ed. Dean Zimmerman (Oxford: Clarendon Press, 2004), p. 107.

  • [12] Stanford Encyclopedia of Philosophy,  s.v.  “Platonism in Metaphysics,” by Mark Balaguer, April 7, 2009,

  • [13] Rudolf Carnap, “Meaning and Necessity:” A Study in Semantics and Modal Logic (Chicago:  University of Chicago Press, 1956), pp. 206-17.

  • [14] See David J. Chalmers, “Ontological Anti-Realism,” in Metametaphysics:  New Essays on the Foundations of Ontology, ed. David Chalmers, David Manley, and Ryan Wasserman (Oxford:  Clarendon, 2009), pp. 77-129.  I prefer the nomenclature “ontological pluralism” to “ontological anti-realism.”  Chalmers’ terminology is misleading, since anti-realism on the level of ontology would involve the denial of the existence of mathematical objects.  It is only on the metaontological level that anti-realism is the denial that the ontological question has an objective answer. 

  • [15] See Mark Balaguer, Platonism and Anti-Platonism in Mathematics (New York:  Oxford University Press, 1998), pp. 151-79; Penelope Maddy, Defending the Axioms:  On the Philosophical Foundations of Set Theory (Oxford:  Oxford University Press, 2011), p. 98.  Maddy offers various characterizations of arealism.

  • [16] See Jody Azzouni, “On ‘On What There Is’,” Pacific Philosophical Quarterly 79 (1998): ***1-18; Jody Azzouni, Deflating Existential Consequence:  A Case for Nominalism (Oxford:  Oxford University Press, 2004); Jody Azzouni, “Ontological Commitment in the Vernacular,” Noûs 41 (2007):  204-26; Jody Azzouni, “Ontology and the Word ‘Exist’:  Uneasy Relations,” Philosophia Mathematica 18 (2010): ***74-101.

  • [17] See Hartry Field, Science without Numbers:  A Defence of Nominalism (Princeton, N.J.:  Princeton University Press, 1980); Hartry Field, Realism, Mathematics, and Modality (Oxford:  Basil Blackwell, 1989); Mark Balaguer, Platonism and Anti-Platonism in Mathematics (New York:  Oxford University Press, 1998); Stanford Encyclopedia of Philosophy, s.v. “Fictionalism in the Philosophy of Mathematics,” by Mark Balaguer, April 22, 2008,


  • [18] See Stephen Yablo, “Does Ontology Rest on a Mistake?”  Proceedings of the Aristotelian Society (Supplement) 72 (1998): ***229-261; Stephen Yablo, “A Paradox of Existence,” in Empty Names, Fiction, and the Puzzles of Non-Existence, ed. Anthony Everett and Thomas Hofweber (Stanford:  Center for the Study of Language and Information, 2000), pp. 275-312; Stephen Yablo, “Go Figure:  A Path through Fictionalism,” in Figurative Language, ed. Peter A. French and Howard K. Wettstein, Midwest Studies in Philosophy 25 (Oxford:  Blackwell, 2001), pp. ***72-102.

  • [19] See Alexius Meinong, “The Theory of Objects” [“Über Gegenstandstheorie” (1904)], trans. Isaac Levi, D. B. Tenele, and Roderick M. Chisholm, in Realism and the Background of Phenomenology, ed. Roderick M. Chisholm (Atascadero, Calif.:  Ridgeview, 1960), pp. 76-117; Richard Routley, Exploring Meinong’s Jungle and Beyond:  An Investigation of Noneism and the Theory of Items (Canberra:  Australian National University Research School of Social Sciences, 1979); Kenneth J. Perszyck, Nonexistent Objects:  Meinong and Contemporary Philosophy, Nijhoff International Philosophy Series 49 (Dordrecht:  Kluwer Academic Publishers, 1993).

  • [20] See Kendall L. Walton, Mimesis as Make-Believe:  On the Foundations of the Representational Arts (Cambridge, Mass.:  Harvard University Press, 1990); Kendall Walton, “Existence as Metaphor,” in Empty Names, Fiction, and the Puzzles of Non-Existence, ed. Anthony Everett and Thomas Hofweber (Stanford:  Center for the Study of Language and Information, 2000); Mary Leng, Mathematics and Reality (Oxford University Press, 2010).

  • [21] See Charles S. Chihara, Constructibility and Mathematical Existence (Oxford:  Clarendon Press, 1990); Charles S. Chihara, The Worlds of Possibility:  Modal Realism and the Semantics of Modal Logic (Oxford:  Clarendon Press, 1998); Charles S. Chihara, “Nominalism,” in The Oxford Handbook of Philosophy of Mathematics and Logic, ed. Stewart Shapiro (Oxford:  Oxford University Press, 2005), pp. ***483-514; Geoffrey Hellman, Mathematics without Numbers (Oxford:  Clarendon Press, 1989); Encyclopedia of Philosophy, 2d ed., ed. Donald M. Borchert (New York: Thomson Gale, 2006), s.v. “Structuralism, Mathematical,” by Geoffrey Hellman.

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