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Response To Bridges And van Inwagen

William Lane Craig


Bridges’ so-called “moderate realism” is really a misnomer, since Aquinas’ view was that mathematical objects and universals are mere entia rationis, having no existence in the world.  Van Inwagen takes for granted the criterion of ontological commitment expressed in the Indispensability Argument.  The metaphysical idleness of properties on van Inwagen’s view ought to motivate re-examination of that criterion.  As for paraphrastic strategies, one can meet van Inwagen’s challenge to provide a nominalistically acceptable paraphrase of Eudclid’s proof of exactly five Platonic solids.  Concerning fictionalism, van Inwagen should allow the anti-Platonist the same privilege that he invokes for treating supposed composite, inanimate objects.  Finally, van Inwagen is too quick in dismissing the absolute creationist view that abstracta can be effects, if not causes.

I take it that the first 5/6 of van Inwagen’s paper targets Bridges’ so-called “moderate realism,” while the final sixth is directed at my theological critique of Platonic realism.  Since I think that van Inwagen has still not seriously engaged the biblical and patristic texts undergirding that critique, we can dispense quickly with his remarks on that head and spend most of our time on his defense of Platonism.

Theological Acceptability of Platonism

Basically, van Inwagen just reiterates his earlier claim[1] that there is no inconsistency in affirming both that “God has created all things” and “There are things uncreated by God” because these quantified statements assume different domains.  This situation is illustrated by the compatibility of Jesus’ statement “God can do everything” and the statement “There are things God cannot do,” statements whose quantifiers plausibly range over different domains.  While I am tempted to comment on van Inwagen’s provocative illustration, to do so would be a distraction.  For at most it is just an illustration of how a universal affirmative and an opposing existential negative statement can both be true if they have different domains, which we already knew.  This goes no distance at all toward showing that the domains assumed by the biblical and patristic writers in the texts I cited were restricted.  Indeed, it was partially van Inwagen’s original claim in this regard that motivated me to undertake a serious exegetical study of the biblical and patristic texts and their respective backgrounds in order to determine whether these authors did intend the domain of their quantifiers to be unrestricted, once God is exempted.  The would-be Christian Platonist now needs to get serious about these texts.

One final correction before we move on:  I emphatically do not claim that “when we say that God is the creator of all things, . . . we mean—at any rate that we have to mean—that he is the creator of abstract objects.”  For, first, as I explained, the biblical writers may have had no conception of abstract objects, in which case their statement “God is the creator of all things” did not mean, even if it implied, that God is the creator of abstract objects.  Second, more importantly, to say that “God is the creator of abstract objects” is to endorse absolute creationism, and I certainly do not think that we must do that!  Rather, when we affirm with the biblical writers and ante-Nicene Church Fathers that God is the creator of all things, we mean that simply nothing exists, apart from God, which is uncreated by God.

Argument for Platonism

Let us turn now to van Inwagen’s argument that there are shapes which nothing has.  I take it that the target of this argument is a sort of immanentism, usually associated with Aristotle, which affirms the reality of abstract universals but denies that unexemplified universals exist.  Universals are not, as in Platonism, transcendent, non-spatiotemporal realities but exist immanently in the things which exemplify them.  For the immanentist, there are no uninstantiated shapes.  I suspect that van Inwagen believed that Bridges advocates such a realism and so goes after it.

Bridges’ Thomism

Unfortunately, Bridges greatly misleads his reader by characterizing Thomism as a form of “moderate realism.”  For “moderate realism” is already employed in the literature to designate immanentism regarding abstract universals.[2]  But Thomas rejects moderate realism, as Bridges himself explains.  Once one frees oneself of the misimpression that Bridges affirms that abstract universals exist immanently in things, it becomes quite clear that for Bridges universals “exist only in the mind” and “the only things that exist outside the mind are concrete particulars.”  Thomas Aquinas is thus what moderns would call a nominalist.[3]  Ed Feser bristles at the suggestion:  “This is not nominalism, for it holds that universals exist.”  But in what sense?  Only in the mind!  What does that mean?  Is this a sort of quasi-existence, a sort of Meinongian watered-down existence?[4]  Being quasi-existent makes about as much sense as being quasi-pregnant; either one is or one is not.   “Existing only the mind” seems to be just another way of saying that something does not exist even though one can have thoughts of such a thing.  Every anti-realist agrees that properties, numbers, and other abstracta exist in that sense.[5]  Bridges is therefore correct to surmise that I would find his Thomist perspective theologically unobjectionable, for it just is another form of anti-realism that one might add to my Fig. 1.

Before we return to van Inwagen’s argument, it is worth noting that due to his denial of immanent universals, Bridges cannot claim for his view any advantage based upon a supposed common nature in Adam, Christ, and ourselves.  For on Thomism natures are multiplied by particulars:  you and I do not have the identical human nature, even though we have two qualitatively similar natures.  Neither can similarity be explained in terms of things’ having an identical metaphysical constituent, for there is no such thing.  By abandoning immanentism, Thomism gives up the supposed advantages of historic moderate realism.  No great loss, however; for to explain, for example, the similarity in color of an apple and a tomato, one can explain why the apple is red in scientific terms, for example, its absorbing and reflecting certain light spectra, and then simply repeat the account when it comes to the tomato.

Ontological Commitment and Neutralism

I am inclined to agree with van Inwagen that “if there are shapes at all, there must be shapes nothing has.”  But why think that there are shapes at all?  Sure, there are things that are (approximately) cubical and spherical and icosahedral, but why think that in addition to these objects there also exist (in a metaphysically heavy sense) things like the cube and the sphere?

It is evident that van Inwagen takes for granted some instance of Balaguer’s Indispensability Argument schema.  In particular, he assumes the criterion of ontological commitment expressed in premiss (I).  Although van Inwagen interacts with a number of anti-Platonist perspectives in his paper, it is noteworthy that he never thinks to discuss neutralism’s challenge to that criterion, without which the whole rationale for realism collapses.  If neutralism is right, as I think it is, then we may affirm mathematical truth without commitment to mathematical objects.

Van Inwagen would do well to reconsider his adherence to premise (I).  He has elsewhere characterized Quine’s metaontological theses as a set of rules or a strategy for settling ontological disputes.  So why should we think that those theses are true rather than merely useful or effective in settling disputes?  Van Inwagen rejects a constituent ontology which takes properties to be metaphysical constituents of things.  He also rejects any attempt to appeal to exemplification of properties as an explanation of why a particular is as it is or of why particulars resemble one another.  It is no wonder, then, that van Inwagen would rather not be a Platonist!  For his abstract objects are not doing any metaphysical work.  They are just idle components of his worldview which he is stuck with because of his adherence to Quine’s metaontological theses, particularly the criterion of ontological commitment.  In light of this fact, why not, with the neutralist, reject one or more of those theses and rid oneself of these unwanted freeloaders?

Paraphrastic Strategies

Alas, instead, van Inwagen considers paraphrastic strategies to rid oneself of shapes.  Unfortunately, he does not consider, for example, the paraphrases which Charles Chihara’s constructibilism or Geoffrey Hellman’s modal structuralism might offer for Euclid’s proof of exactly five Platonic solids.  With his customary chutzpah, van Inwagen just asserts that “no one knows of any other proof of his conclusion every step of which can be given a nominalistically acceptable paraphrase.” 

Well, how about this?

1.  It is not possible to construct a solid angle with two triangles.

2.  It is possible to construct the angle of the cube with three squares.

3.  It is possible to construct the angle of the pyramid with three triangles, the angle of the octahedron with four, and the angle of the icosahedron with five.

4.  It is possible to construct the angle of the dodecahedron with three pentagons.

5.  It is not possible to construct a solid angle with six or more plane angles, for the six angles would be equal to four right angles, which is impossible, since any solid angle is comprised of angles less than four right angles.

6.  Therefore, “it is possible for there to be five distinct physical objects each of which is platonic and each of which is shaped differently from each of the others; it is not possible for there to be six distinct physical objects each of which is platonic and each of which is shaped differently from each of the others.” Q.E.D.

Van Inwagen cannot protest that this proof commits us to the existence of possible geometrical objects, since the modal operator forms an intensional context into which one cannot quantify.

Oddly enough, however, van Inwagen appears to do precisely this in his proof that every shape exists necessarily, or that there are no contingent shapes.  The key line in his argument is:  “‘the cube’ is a possible shape. And it can’t be even a possible shape if it’s not there at all.”  Van Inwagen apparently infers from “Possibly, there is some x such that x = the  cube”  to “There is some x such that, possibly, x = the cube.”  He seems to be assuming the Barcan Formula ◇∃ Φx → ∃x◇ Φx. But then he will find himself ontologically committed to possibilia, which he earlier seemed to reject.[6]  Absent the Barcan Formula, he has no justification for thinking that because it is possible that something is the cube, there is something that is possibly the cube.  The cube could be a contingent shape.


Van Inwagen also rejects fictionalism, a view which accepts Quine’s criterion of ontological commitment but which holds that sentences quantifying over mathematical objects are false.  He does so because he thinks that a proof of a nominalistically acceptable conclusion requires the truth of all the argument’s premises.  But a fictionalist may hold that some of the premises are to be treated merely instrumentally but can be relied upon to yield nominalistically acceptable conclusions.  Van Inwagen regards such a position as a non-starter because it leaves unanswered the question:  why is mathematics reliable?[7]  This is the question of mathematics’ applicability, which, it seems to me, the theist, be he realist or anti-realist, is best positioned to answer.[8]  God has imbued the physical world with a certain structure such that the standard model of arithmetic can be counted on to deliver reliable conclusions. 

Van Inwagen is himself a fictionalist when it comes to the existence of chairs and other inanimate composite objects.  If so bold a thesis is acceptable, why not similarly allow fictionalism about abstract objects?  When van Inwagen denies that chairs exist, he has said that he is not speaking ordinary English, but Tarskian, the language of the metaphysics seminar.  So why not accord the same privilege to the fictionalist anti-realist?  He, too, can allow that when we say, “The number of Martian moons is two,” we make a true statement in ordinary language, even if this sentence is false in Tarskian. 

Absolute Creationism

Van Inwagen is intolerant not only of anti-realism.  He is equally unsympathetic to non-Platonic realism.  He criticizes absolute creationism on the grounds that (1) it is hard to make sense of God’s deciding to create a shape, and (2) creation is a causal relation, and shapes cannot enter into causal relations.  With respect to (1), absolute creationists (unless they are radical theistic activists) do not typically root mathematical necessities, including geometrical truths, in God’s will but in God’s nature.[9]  Hence, though shapes depend ontologically on God, they are not the result of His deciding.  As for (2), while abstracta have no causal powers and so cannot be causes, why can they not be effects?  Indeed, most of those philosophers of aesthetics who are, like van Inwagen, realists about fictional characters take them to be created by their authors.[10]  I carry no brief for absolute creationism, but van Inwagen’s cursory dismissals do not do this viewpoint justice.


We have noted van Inwagen’s claim that one should not want to be a Platonist unless there is some rationally compelling argument for Platonism.  Does he really think, then, that his argument for abstract objects is rationally compelling?  Does he really think that all those anti-realist philosophers like Jody Azzouni and Charles Chihara and Geoffrey Hellman and Mark Balaguer and Penelope Maddy and Stephen Yablo are irrational in rejecting that argument?  His objections to absolute creationism, paraphrastic strategies, and fictionalism, as well as his silence concerning neutralism, make it hard to believe that that bar has been reached.

  • [1] Peter van Inwagen, “God and Other Uncreated Things,” in Metaphysics and God, ed. Kevin Timpe (London:  Routledge, 2009), p. 19.

  • [2] See Marilyn McCord Adams, “Universals in the Early Fourteenth Century,” in The Cambridge History of Later Medieval Philosophy, ed. Norman Kretzmann, Anthony Kenny, and Jan Pinborg (Cambridge:  Cambridge University Press, 1982), pp. 411-39, esp. 422-34; Marilyn McCord Adams, William Ockham, 2 vols., Publications in Medieval Studies 26 (Notre Dame, Ind.: University of Notre Dame Press, 1987), 1: 30.  She lists three features of the moderate realism of a theologian like Walter Burleigh: (1) a universal is a thing that exists in reality as a metaphysical constituent of particulars; (2) the universal exists in many particulars simultaneously without being numerically multiplied itself; (3) the universal is really distinct from particulars and other universals.

  • [3] Jeffrey E. Brouwer, “Aquinas on the Problem of Universals,”  Philosophy and Phenomenological Research (2015): preprint, in agreement with Frederick Copleston and Brian Leftow.  Brouwer holds that Aquinas is not a nominalist in the medieval sense because he holds that distinct things belong to the same kind in virtue of their having non-identical, individual natures that are intrinsically similar but individuated by matter.  Nonetheless, Brouwer recognizes that Aquinas denies the existence of universals and so counts as a nominalist in the modern sense.  Armand Maurer explains that Aquinas treated mathematical objects just as he treated universals:  they are entia rationis devised by the mind as a consequence of our knowledge of the world.  “Nothing like them exists in the external world” (Armand Maurer, “Thomists and Thomas Aquinas on the Foundation of Mathematics,” Review of Metaphysics 47 [1993]:  59).  I’m grateful to J. T. Bridges for sharing these articles with me. 

  • [4]  Meinong notoriously resorted to a sort of depotenzierte Existenz in order to explain why objects which exist in the understanding like the existing x do not actually exist (Alexius Meinong, Über Möglichkeit und Wahrscheinlichkeit: Beiträge zur Gegenstandstheorie und Erkenntnistheorie [Leipzig:  Verlag von Johann Ambrosius Barth, 1915], p. 282).  For a comparison of scholastics’ entia rationis with Meinong’s non-existent objects, see Daniel D. Novotny, “Scholastic Debates about Beings of Reason and Contemporary Analytical Metaphysics,” in Metaphysics: Aristotelian, Scholastic, Analytic, ed. Lukas Novak et al. (Frankfurt: Ontos Verlag, 2012), pp. 11-24.  According to Novotny “ens rationis” was for the scholastics a term for a non-existent, merely intentional object.

  • [5] See, e.g., Mark Balaguer, “Realism and Anti-Realism in Mathematics,” in Philosophy of Mathematics, ed. Andrew D. Irvine, Handbook of the Philosophy of Science (Amsterdam:  North Holland, 2009), p. 82.

  • [6] The Barcan Formula and the specter of necessitism (the view that nothing exists contingently) provide additional motivation for a neutralist view of first-order quantification.  For discussion see Takashi Yagisawa, critical notice of Modal Logic as Metaphysics, by Timothy Williamson, Notre Dame Philosophical Reviews (October 15, 2013),  Necessitism illustrates the sort of metaphysical debacle which the assumption of Quine’s criterion of ontological commitment helps to generate.

  • [7] Peter van Inwagen, “Fictionalist Nominalism and Applied Mathematics,” Monist 97 (2014):  486; cf. pp. 495-6.

  • [8] On the one hand, the theistic realist can argue that God has fashioned the world on the structure of the mathematical objects.  On the other hand, the theistic anti-realist can claim that God has created the world according to a certain blueprint which He had in mind.  Thus, the theist—whether he be a realist or an anti-realist about mathematical objects—has the explanatory resources to account for the mathematical structure of the physical world and, hence, for the otherwise unreasonable effectiveness of mathematics—resources which the naturalist lacks.  

  • [9] See Thomas V. Morris and Christopher Menzel, “Absolute Creation,” American Philosophical Quarterly 23 (1986):  355-6, 360.  Cf. Brian Leftow, God and Necessity (Oxford: Oxford University Press, 2012). Among theistic modal theories, Leftow distinguishes between theories which ground modality in God’s nature and theories which ground modality in God’s activity. Leftow endorses a partial activist view but even so grounds necessary truths of logic and mathematics in God’s nature, not His activity.

  • [10] Christy Mag Uidhir, “Introduction: Art, Metaphysics, and the Paradox of Standards,” in Art and Abstract Objects, ed. Christy Mag Uidhir (Oxford: Oxford University Press, 2012), p. 7.

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