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Response To Van Inwagen And Welty

William Lane Craig

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In response to my critics, I argue that Peter van Inwagen, despite his protestations, is an advocate of an indispensability argument for Platonism. What remains to be shown by van Inwagen is that his version of the argument overcomes his own presumption against Platonism and survives defeat by besting every anti-Platonist alternative. While acknowledging Greg Welty’s helpful responses to my worries about divine conceptualism as a realist alternative to Platonism, I express ongoing reservations about some of those responses.

It’s an honor to have God over All: Divine Aseity and the Challenge of Platonism[1] selected as the subject of an “author meets critics” session at the 2019 American Philosophical Association Eastern Division meeting and a special privilege to have two so prominent critics as Peter van Inwagen, today’s most important Platonist metaphysician, and Greg Welty, today’s most prominent divine conceptualist. Their criticisms are bracing and serve to advance the discussion.

God over All is a semi-popular distillation of my larger book, God and Abstract Objects, published in 2017 by Springer Verlag.[2] Its expanded discussion treats additional antirealist responses to Platonism such as Charles Chihara’s and Geoffrey Hellman’s paraphrastic strategies.[3]



Since most of you will doubtless not have read God over All, let me take a step back and give you a panoramic view of my project. My aim is to de- fend the doctrine of divine aseity, that is to say, the view that God is, in Brian Leftow’s helpful phrase, “the sole ultimate reality,” against what I take to be the most serious challenge to that doctrine, namely, Platonism, the view that there are uncreated abstract objects. I make it clear that I am talking about metaphysically heavyweight Platonism, which takes abstract objects to be as real as the fundamental particles that make up the physical world, not a light- weight Platonism which takes abstract objects to be merely semantic objects, not objects in the ordinary sense.

In the book I offer a taxonomy of positions on the existence of mathematical objects (see fig. 1). I choose mathematical objects as my paradigm example rather than abstract objects in general because the taxonomy then makes clear, as Welty remarks, that there are realist as well as antirealist versions of anti-Platonism and because the center of gravity in the debate over abstract objects lies in the philosophy of mathematics. I take it that in our session today we have two realist views of mathematical objects represented.


Figure 1.


It’s true, as Welty reminds us, that he has not actually defended conceptualism for any other allegedly abstract objects than propositions and pos- sible worlds, but given the conceptualist’s desire to preserve divine aseity, it is natural that he will also seek to identify mathematical objects like sets with God’s mental activities, as Alvin Plantinga has proposed.[4] So I place Welty on the left-hand side of the diagram as someone who affirms the reality of mathematical objects, not as abstract but as concrete objects, not as physical objects as James Franklin has argued, but as mental objects, not in some human mind, a view mercilessly criticized by Gottlob Frege, but in the divine mind.[5] I also place van Inwagen on the left-hand side, for he has defended the reality of mathematical objects like shapes, not as concrete but as abstract objects. Although van Inwagen has rejected absolute creationism, I argue that his objections to that position are weak and that, therefore, given his commit- ment to classical theism, he ought to switch to absolute creationism. Indeed, his favored, anticonstituent ontology is uniquely suited, I think, to avoid the dreaded bootstrapping objection that attends the absolute creationism of Thomas Morris and Christopher Menzel.[6]

For the rest, the taxonomy reveals a panoply of non-Platonist alternatives, such as Penelope Maddy’s arealism, Jody Azzouni’s neutralism, Hartry Field’s fictionalism, Richard Routley’s neo-Meinongianism, Mary Leng’s pre- tence theory, and so on and so forth.[7] I find that Christian philosophers are generally unfamiliar with these non-Platonist alternatives, so that these views tend to be overlooked. For example, in his massive book, God and Necessity, Brian Leftow never mentions these antirealist alternatives to Platonism until the book’s final paragraph.[8]

It is my goal in the book, not to foreclose, but to open up various alternatives to Platonism. I want realist alternatives like Absolute Creationism and Conceptualism to succeed. So Welty is quite right in seeing me as an ally in this venture. I have not offered knockdown arguments against Conceptualism but expressed “worries” that will, I hope, motivate Christian philosophers to look more seriously at antirealist alternatives, which I, at least, find much more plausible. Indeed, I can imagine some future philosopher remarking, “William Lane Craig never met an antirealism that he didn’t like!”


Presumption of Anti-Platonism

It’s important to understand that my anti-Platonism is theologically motivated. I do not suggest that Platonism is philosophically untenable. I think that the general consensus among philosophers of mathematics is that all of these views are controversial and that therefore no one can justifiably claim that his is the correct view. Rather my conviction is that Platonism is theo- logically untenable.

I have taken to heart Alvin Plantinga’s advice to Christian philosophers to think more integratively, to argue unapologetically from an explicitly Christian point of view.[9] That is why I undertook at the start a serious exegesis of key New Testament passages, an effort which yielded rich rewards. I was intrigued by Philo of Alexandria’s conceptualism and surprised to learn of the influence of Middle Platonism upon the New Testament.[10] My investigation of the church fathers’ views was actually motivated by van Inwagen’s remark that when he recites the Nicene Creed’s affirmation that God is the maker “of all things, visible and invisible,” he tacitly restricts the domain of quantification to objects standing in causal relations. Is that justified, I wondered. How did the ante-Nicene fathers understand the domain of quantification? I was surprised to find that that question could be decisively answered, that they unanimously held that there is a sole agenētos or uncreated reality which is God Himself and explicitly rejected the views of Plato and Pythagoras that posited a plurality of agenēta like numbers or forms.[11]

I thus come to the discussion with the theologically based conviction that Platonism is untenable. The question, then, is how best to defeat any arguments for Platonism. It intrigued me that van Inwagen also comes to the table with a deep presumption against Platonism, though in his case philosophically, not theologically, motivated. He takes it that abstract and concrete objects are so utterly dissimilar that it is preferable to assume that one of these categories is empty. Since we know that concrete objects exist, we should presume that abstract objects do not exist. Now, as I say, for van Inwagen, thispresumption is very deep. He says, “one should not believe in abstract objects unless one 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.”[12] Now in my mind and, I’m sure, in the minds of most philosophers, the debate ends right there. For there is no rationally compelling argument for Platonism, or any of the alternatives we have mentioned. One argues at best for the rational tenability of one’s favorite view. Indeed, van Inwagen has himself expressed such an understanding of doing metaphysics. “No one,” he says, “is in a position to be confident about the answers to these questions.”[13]


Van Inwagen’s Indispensability Argument

But perhaps van Inwagen exaggerates the strength of the anti-Platonist presumption. What overriding argument for Platonism is there? The only serious argument for Platonism is the Indispensability Argument. Now the original Quine-Putnam version of the Indispensability Argument is dead.   It is that version of the argument that van Inwagen has explicitly rejected. But that is not the contemporary version with which I’m concerned. Mark Balaguer provides the following stripped down version of the argument:

  1. If a simple sentence (i.e., a sentence of the form ‘a is F’, or ‘a is R- related 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.
  2. 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.
  3. Therefore, abstract objects exist.[14]

Now it seems to me that van Inwagen affirms this argument or something very like it. (I) expresses his criterion of ontological commitment. We can leave aside the use of singular terms if desired and focus on existential quantification, as he has done today. (II) expresses his conviction that no para- phrases are available that get rid of the quantification over abstract objects. Thus, one is stuck with abstract objects.

In the book I try to show that both of the premises can be undercut, if not refuted, and thus the conclusion is defeated. Now van Inwagen declines to respond to my attempts to undercut (I), not to mention (II), because, he says, he understands “very little” of what I say.[15] Moreover, he says that I have grossly misunderstood him. In short, he doesn’t understand me, and I don’t understand him! This is a fine kettle of fish!

Well, on the basis of his explanations today, let me try once again to understand him. Reading what he said today, the thought occurred to me that maybe he isn’t really offering a criterion of ontological commitment after all. Maybe he’s not saying that you are ontologically committed to the values of the variables bound by the first-order existential quantifier in sentences you take to be true. Maybe he’s  saying merely that one should not make contradictory claims like “~∃x x is an abstract object & ∃x x is a sentence-type.”

Even if one has an ontologically lightweight interpretation of the existential quantifier, still one shouldn’t make both claims.

But that can’t be the whole story. Of course, philosophers such as neutralists and neo-Meinongians, who deny that the quantifier expresses heavy- weight existence claims, will generally agree that one should not make even lightweight contradictory claims. But they will explain that there is no con- tradiction because these quantificational claims are neutral and so can be used in a lightweight or a heavyweight sense. As van Inwagen would put it, they can be used to express different propositions.

So reconsider A. J. Ayer’s claim that “it makes sense to say, in a case where someone is believing, that there is something that he believes. But it does not follow from this that . . . something must exist to be believed, in the way that something must exist to be eaten or to be struck.”[16] Van Inwagen’s claim that Ayer’s words involve a glaring apparent contradiction is purchased only at the expense of ignoring Ayer’s full claim, which is not even apparently contradictory. Again, what Ayer said was that “it makes sense to say, in a case where someone is believing, that there is something that he believes. But it does not follow from this that . . . something must exist to be believed, in the way that something must exist to be eaten or to be struck.” The first sentence is a lightweight quantificational claim; the second is a heavyweight existence assertion. You get a contradiction only by assuming that the first- order existential quantifier carries heavy ontological commitments, which is exactly what van Inwagen has elsewhere affirmed but what neutralists and neo-Meinongians deny. So van Inwagen is not just saying that we shouldn’t contradict ourselves; he is saying that the first-order quantifier is a device of ontological commitment.

But this is a controversial claim. I agree with Jody Azzouni that the informal quantifiers of ordinary language are neutral in their ontological com- mitments. We say things like “There’s a lack of compassion in the world,”  or “There’s a better way to do that,” without thinking that we are making ontological commitments or need to offer a paraphrase. Van Inwagen affirms that the first-order existential quantifier is no more than an abbreviation of these ordinary language claims. Accordingly, it ought to be ontologically neutral. Only by assuming that Ayer’s quantificational claim involves ontologi- cal commitment does the neo-Quinean generate a contradiction where there was none. This was one of my complaints about the neo-Quinean criterion: far from helping us to notice contradictions, it actually creates them.

Van Inwagen goes on to affirm the necessity of offering an acceptable paraphrase of the claim which commits the would-be nominalist to abstract objects, and he expresses his pessimism that this can be done.

His procedure is analogous to his treatment of inanimate composite objects. He holds that existential quantification over composite objects in sentences we take to be true will commit us to the existence of such objects unless we can offer acceptable paraphrases of such claims. In the case of inanimate composite objects, van Inwagen is confident that such paraphrases can be found and so ontological commitment avoided. But in the case of abstract objects he is pessimistic about such paraphrases and so finds himself unwillingly committed to the reality of abstract objects.

So it seems to me that van Inwagen is, indeed, defending an indispensability argument for Platonism. The following seems a fair formulation of van Inwagen’s argument presented today:

  1. If a true simple sentence expresses a proposition involving quantification over certain objects, then we are ontologically committed to those objects.
  2. There are true simple sentences which express propositions involving quantification over objects that could only be abstract objects.
  3. Therefore, we are ontologically committed to abstract objects.

What remains to be shown by van Inwagen is that this argument overcomes the presumption against Platonism and survives defeat by besting every anti- Platonist alternative.


Conceptualism as an Anti-Platonist Alternative

One of those alternatives is divine conceptualism. I have misgivings about conceptualism, which Welty tries to allay. This is exactly the sort of response I hoped to elicit from committed conceptualists.

I don’t claim, but I do worry, that Welty’s version of conceptualism violates divine aseity because he thinks that God’s thoughts exist as uncreated objects which are not identical with God. One issue that needs clarification is the status of divine thoughts. Conceptualism is a form of realism that affirms that divine thoughts are actually existing objects. I myself prefer van Inwagen’s more austere view that we don’t really need thoughts as items in our ontology, so we might as well get rid of them. Certainly a thinking God exists, but we needn’t reify His thoughts into existing things. But on conceptualism God’s thoughts do exist and are not identical to God (God is not a thought). What kind of objects, then, are thoughts? Well, they seem to be mental states or mental events of some sort. In that case, it seems to me almost obvi- ous that God is the cause of His thoughts. He is engaged in thinking and the result is thoughts, just as I produce various thoughts by thinking. If this is right, then there is no threat to divine aseity, since thoughts are dependent beings. However, such a version of conceptualism will affect Welty’s handling of the bootstrapping objection, since his response depends on God’s not being the cause of His thoughts.

But consider Welty’s view that God’s thoughts are uncreated objects. He suggests two ways to reconcile such a view with divine aseity. First, we take God’s thoughts to be necessarily existing parts of God. This would solve the problem; but my reservation about this response is that God’s thoughts seem to be neither parts of God, as the Trinitarian persons might be, nor necessarily existing parts. God’s thoughts seem to be contingent, in that God is plausibly not thinking about everything He knows. That was the point of another of Graham Oppy’s objections to conceptualism.[17] God knows that for any real number r, r is not identical to the Taj Mahal. But is God actually thinking about each such truth? That doesn’t seem a plausible account of God’s thought life. So I like Welty’s second response better: There is just “God engaged in thinking,” and there’s an end on it.[18] Yes! That seems right; but that, I maintain, is antirealism, not realism.

Next, pace Welty, I do think that conceptualists can handle identifying false propositions as divine thoughts, and, as I explained, Morris and Menzel show us how. We distinguish between God’s thoughts and God’s beliefs. God’s thoughts are characterized by a range of doxastic attitudes toward what He conceives, of which belief is only one. No problem!

Next, I do continue to worry about conceptualism’s ascribing to God inappropriate thoughts. Denying that God has such thoughts would be a prob- lem for omniscience only if we think that everything God knows He is thinking about. But I know innumerable things that I am not now thinking about; I have no such thoughts at this time. Indeed, in the case of bawdy thoughts we who are Christians try to put such thoughts out of mind when they arise. Surely, Welty can see the problem of having God constantly thinking aboutsuch things! When I was speaking on this topic at Rutgers University, the late Marilyn Adams reminded us that the medievals held that God’s thoughts are directed only toward Himself, that if He thought of anything else He would get “dirty,” so to speak. He knows creatures only insofar as He knows Himself as their cause. Since I think that God is temporal since the moment of creation, I have no problem with God’s having a stream of consciousness, thoughts which are constantly changing as new events become present.

A really significant worry about conceptualism is its privatization of certain propositions. I don’t think that Welty has fully grasped the problem. As a personal being, God surely has first-person indexical thoughts. But on conceptualism, if God has any first-person indexical thoughts, then those thoughts are propositions. Since we cannot access God’s first-person thoughts, it immediately follows that there are purely private propositions, a most unwelcome conclusion. As Welty says, we want to say that when God has the thought I am the God of Israel and we have the thought Yahweh is the God of Israel, we grasp the same proposition from different perspectives. The thoughts are different but the propositional content is the same. But on conceptualism we cannot make such a distinction, since God’s thoughts just are propositions.

It seems to me that the only way out of this problem for the conceptualist is to hold that while all propositions are divine thoughts, not all divine thoughts are propositions. Welty reminds us that he has said this very thing: “it is no part of the conceptualist thesis to say that just any divine thought counts as a proposition. The conceptualist view is more modest than this: ‘a particular range of the uncreated divine thoughts function as abstract objects.’”[19] Good point! So on this view, first-person indexical divine thoughts are not propositions, despite their alethicity (truth-aptness) and doxasticity (being the objects of doxastic attitudes). But that looks terribly ad hoc to me. So the conceptualist has some more work to do on this score.

The next problem is similar: since we cannot distinguish on conceptualism between God’s thoughts and the propositional content of those thoughts, the thoughts just being propositions, God’s ontological commitments seem to become unclear. In particular the neo-Quinean procedure of paraphrasing in order to reveal propositional content of a thought won’t work in God’s case. We don’t want to say that all of the ontological commitments of divine beliefs should be accepted because that would saddle us with all the excesses of ordinary language that Quine wanted to get rid of by his paraphrastic pro- cedure.

Nevertheless, I think Welty  is right that this worry is not so serious as  I imagined. Suppose God has the thought There’s a crack in the Liberty Bell and also the thought The Liberty Bell is cracked. The latter cannot express the propositional content of the former, since both thoughts just are propositions. But that doesn’t commit God to an inflated ontology that includes cracks. For suppose that cracks don’t really exist. Then the thought There’s a crack in the Liberty Bell is false, and so God doesn’t believe it and thus isn’t committed to cracks. Since God’s beliefs infallibly track reality, His ontological commitments cannot err. What’s odd is that whereas the availability of a paraphrase can annul our apparent ontological commitments, it can’t annul God’s.

I’m out of space. But I thank my esteemed commentators for their stimulating remarks!


[1] William Lane Craig, God over All: Divine Aseity and the Challenge of Platonism (New York: Oxford University Press, 2016).

[2] William Lane Craig, God and Abstract Objects: The Coherence of Theism (Cham, Switzerland: Springer, 2017).

[3] Charles S. Chihara, Constructibility and Mathematical Existence (Oxford: Clarendon, 1990); Charles S. Chihara, “Nominalism,” in The Oxford Handbook of Philosophy of Mathematics and Logic, ed. Stewart Shapiro (New York: Oxford University Press, 2005), 483–514; Geoffrey Hellman, Mathematics without Numbers: Towards Modal-Structural Interpretation (New York: Oxford University Press, 1989); Geoffrey Hellman, “On Nominalism,” Philosophy and Phenomenological Research 62 (2001): 691–705.

[4] See, e.g., Alvin Plantinga, “Theism and Mathematics,” Theology and Science 9 (2011): 27–33.

[5] James Franklin, An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure (Hampshire, UK: Palgrave Macmillan, 2014); Gottlob Frege, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, trans. J. L. Austin, 2nd rev. ed. (Evanston, IL: Northwestern University Press, 1968), §I.7 (pp. 8–11); §II.26–7 (pp. 34–8).

[6] Thomas V. Morris and Christopher Menzel, “Absolute Creation,” American Philosophical Quarterly 23 (1986): 353–62. On the bootstrapping objection, see Michael Bergmann and Jeffrey Brower, “A Theistic Argument against Platonism (and in Support of Truthmakers and Divine Simplicity),” in Oxford Studies in Metaphysics, vol. 2, ed. Dean Zimmerman (New York: Oxford University Press, 2006), 357–86.

[7] See, e.g., Penelope Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory (New York: Oxford University Press, 2011); Jody Azzouni, Deflating Existential Conse- quence: A Case for Nominalism (New York: Oxford University Press, 2004); Hartry Field, Realism, Mathematics, and Modality (Oxford: Basil Blackwell, 1989); Richard Routley [Sylvan], Exploring Meinong’s Jungle and Beyond: An Investigation of Noneism and the Theory of Items (Canberra: Research School of Social Sciences, Australian National University, 1979); Mary Leng, Mathematics and Reality (New York: Oxford University Press, 2010).

[8] Brian Leftow, God and Necessity (New York: Oxford University Press, 2012), 551.

[9] Alvin Plantinga, “Advice to Christian Philosophers,” Faith and Philosophy 1 (1984): 253–71.

[10] Helpful works include Philo, On the Creation of the Cosmos according to Moses, trans. David T. Runia (Leiden: Brill, 2002); D. T. Runia, Philo of Alexandria and the ‘Timaeus’ of Plato (Amsterdam: Free University of Amsterdam, 1983); Jutta Leonhardt-Balzer, “Der Logos und die Schöpfung: Streiflichter bei Philo (Op 20–25) und im Johannesprolog (Joh 1, 1–18),” in Kontexte des Johannesevangeliums, ed. Jörg Frey and Udo Schnelle (Tübingen: Mohr Siebeck, 2004), 295– 320; Gregory Sterling, “‘Day One’: Platonizing Exegetical Traditions of Genesis 1:1–5 in John and Jewish Authors,” paper presented at the Philo section of the Society of Biblical Literature, San Antonio, TX, November 20, 2004; Audrey N. M. Rich, “The Platonic Ideas as the Thoughts of God,” Mnemosyne, 4th ser., 7, fasc. 2 (1954): 123–33.

[11] See George L. Prestige, God in Patristic Thought (London: SPCK, 1964); Harry Austryn Wolfson, “The Logos and the Platonic Ideas,” in The Philosophy of the Church Fathers, vol. 1, Faith, Trinity, and Incarnation, ed. Harry Austryn Wolfson, 3rd rev. ed. (Cambridge, MA: Harvard University Press, 1970), chap. 7.

[12] Peter van Inwagen, “A Theory of Properties,” in Oxford Studies in Metaphysics, vol. 1, ed. Dean Zimmerman (New York: Oxford University Press, 2004), 107.

[13] Peter van Inwagen, Metaphysics, 3rd ed. (Boulder, CO: Westview, 2009), 304.

[14] Mark Balaguer, “Platonism in Metaphysics,” in Stanford Encyclopedia of Philosophy, April 7, 2009,

[15] Peter van Inwagen, “Response to William Lane Craig’s God over All,” Philosophia Christi

[16] >A. J. Ayer, Thinking and Meaning (London: H. K. Lewis, 1947).

[17] Graham Oppy, “Response to Welty,” in Beyond the Control of God? Six Views on the Problem of God and Abstract Objects, ed. Paul M. Gould (London: Bloomsbury, 2014), 104–6.

[18] Greg Welty, “Do Divine Conceptualist Accounts Fail? A Response to Chapter 5 of God over All,” Philosophia Christi 21 (2019): 260.

[19] Ibid., 265.