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God Over All:
Divine Aseity and the Challenge of Platonism

by william lane craig
oxford, 280 pages, $88

Mathematics appears to describe a realm of entities with quasi-­divine attributes. The series of natural numbers is infinite. That one and one equal two and two and two equal four could not have been otherwise. Such mathematical truths never begin being true or cease being true; they hold eternally and immutably. The lines, planes, and figures studied by the geometer have a kind of perfection that the objects of our ­experience lack. Mathematical objects seem ­immaterial and known by pure reason rather than through the senses. Given the centrality of mathematics to scientific explanation, it seems in some way to be a cause of the natural world and its order.

How can the mathematical realm be so apparently godlike? The traditional answer, originating in Neoplatonic philosophy and Augustinian theology, is that our knowledge of the mathematical realm is precisely knowledge, albeit inchoate, of the divine mind. Mathematical truths exhibit infinity, necessity, eternity, immutability, perfection, and immateriality because they are God’s thoughts, and they have such explanatory power in scientific theorizing because they are part of the blueprint implemented by God in creating the world. For some thinkers in this tradition, mathematics thus provides the starting point for an argument for the existence of God qua supreme intellect.

There is also a very different answer, in which the mathematical realm is a rival to God rather than a path to him. According to this view, mathematical objects such as numbers and geometrical figures exist not only independently of the ­material world, but also independently of any mind, including the divine mind. They occupy a “third realm” of their own, the realm famously described in Plato’s Theory of Forms. God used this third realm as a blueprint when creating the physical world, but he did not create the realm itself and it exists outside of him. This position is usually called Platonism since it is commonly thought to have been ­Plato’s own view, as distinct from that of his Neoplatonic followers who relocated mathematical objects and other Forms into the divine mind. (I put to one side for present purposes the question of how historically accurate this standard narrative is.)

This view is the target of attack in William Lane Craig’s God Over All: Divine Aseity and the Challenge of Platonism. For a thing to exist a se, or “of itself,” is for it to be self-existent or to depend on nothing outside itself for its existence. The doctrine of divine aseity holds that God and God alone exists a se or in a self-existent way. Everything else exists only because it was created by him.

Platonism is at odds with the doctrine of divine aseity. For if numbers, geometrical figures, and other mathematical objects are eternal, immutable, necessary, and also exist in a realm of their own, apart from either the natural world or God, then they too would be self-existent. They would have no need for a creator, but would simply exist of their own nature, neither coming into being nor passing away. Indeed, since mathematical objects are infinite in number, it would follow that only a relatively small part of reality (namely, the material world) would have been created by God.

The issue is pressing because a Platonism that makes no reference to God has in recent decades come to renewed prominence in secular academic philosophy. The main ­motivation is known as the Indispensability Argument, associated with the late Harvard University philosophers W. V. Quine and Hilary Putnam. The argument holds that if we believe in the existence of the entities postulated by modern science and note that mathematics is indispensable to science, then we are bound to believe also in the entities postulated by mathematics. But mathematics, the argument says, makes reference to a realm of abstract objects (the expression “abstract objects” being preferred by contemporary philosophers over the more traditional Platonic talk of “Forms” or “Ideas”). Hence, acceptance of the results of science commits us to affirm the existence of such abstract objects.

Craig aims to refute this revived Platonism and thereby to rebut its implicit challenge to divine aseity. A prominent academic philosopher and Protestant theologian, he has made a specialty of defending the claims of traditional Christian theology against modern objections, and he is especially qualified to do so. A thinker of unusual breadth and depth, Craig has mastered and engaged with vast areas of modern thought, including the most technical reaches of contemporary analytic philosophy, the physics of relativity and Big Bang cosmology, and New Testament scholarship. In fact, Craig’s work played a key role in my own return to Christianity after a decade as an atheist.

Nonetheless, I have to disagree with him. Successfully answering the challenge that Platonism poses to divine aseity requires getting right the nature of God, the nature of mathematics, and the nature of the relationship between them. In my opinion, Craig’s position fails on all three counts.

In contemporary philosophy, the alternatives to Platonism that receive the most attention are variations on the medieval doctrine of nominalism, which treats numbers and other purported abstract objects as mere names or linguistic constructs rather than as really existing entities. An especially influential version of this idea is fictionalism, which holds that mathematics is like a made-up story, the elements of which are useful to science, but which is not literally true. Is it correct to say that Tony Stark is Iron Man and that Peter Parker is not? Yes, since that’s the way things are in the Marvel movies and comics. But of course, those stories are fictional; in reality, there is no Tony Stark, no Peter Parker, and no Iron Man. Similarly, it is correct to say that two and two make four rather than five. But this, too, reflects merely the way things are in a fictional ­story—the story of mathematics.

Craig discusses fictionalism in detail, along with variations like figuralism (which takes mathematical language to be figurative rather than literal) and pretense theory (which holds that we may speak of mathematics as if it were true without committing ourselves one way or the other about whether it is). These riffs on nominalism are all anti-realist insofar as they avoid Platonism by denying that mathematics certainly describes objective reality. And while Craig does not endorse any one of these specific variations to the exclusion of the others, he strongly favors the general anti-realist approach as the best way to rebut the Platonist challenge to aseity. In Craig’s view, the Indispensability Argument shows, at most, only that mathematics is a very useful fiction for scientific purposes, but not that it is anything more than that.

Now, Craig very skillfully exposes some technical problems with the specific way that thinkers like Quine develop the Indispensability Argument. But the problems with anti-realism are even greater. A non-Platonist realism is more defensible than either of these alternatives.

Anti-realists have trouble explaining how theories can work so well if they aren’t really true. For example, how could physics make such accurate predictions and yield such amazing technological applications if the fundamental particles and other unobservable entities it postulates don’t exist? As Hilary Putnam famously put it, anti-realism makes these successes seem like miracles. In the same way, how could mathematics contribute so much to the success of physics if it were not true? For the fictionalist, 2 + 2 = 4 is no truer than 2 + 2 = 5. The difference is just that science finds the first fiction useful and not the second. But why is it any more useful than the second is? If we based our system of criminal justice on the pretense that Iron Man and other superheroes exist, we would soon find this supposition to be far from useful. So how can mathematics be any more useful if it is merely a fiction on a par with the Marvel movies and comics?

Craig says little in response to this problem, other than to suggest that an anti-realist who is a theist can explain the effectiveness of mathematics by saying that God has simply caused the world to operate in a way that reflects the mathematics we know. But why did God opt for a world governed by the mathematics we know rather than, say, by some alternative mathematics in which two and two make five? Is it because there is no such alternative, and the mathematics we know is the only kind possible? If that is the case, then it seems we have abandoned anti-realism and admitted that there are, after all, objective mathematical truths that even God cannot alter—exactly the result Craig thinks it better for the defender of divine aseity to avoid.

The only alternative would seem to be some brand of voluntarism, according to which there is no rhyme or reason to what God wills, no objective standard of truth or rationality that guides his choices. On this view, the truths of mathematics are whatever God capriciously decides they will be. That two and two make four is a matter of sheer fiat or stipulation. Had God decided instead that two and two would make five—or that two and two would make zero, or 306, or anything else for that matter—then that is what would have been the case.

Craig briefly criticizes voluntarism early in his book, in the context of discussing the views of another philosopher and before he sets out his own views. But he never explains how his own position can avoid it. In fact, from William of Ockham onward, nominalism and voluntarism have tended to go together. And while Craig prefers the label “anti-realism” to that of “nominalism,” the substance of his views seems to threaten voluntarism for the same reasons nominalism does. If there is no objective fact of the matter about mathematical truth, but it nevertheless has its source in God, then divine whim is all that’s left to ground it.

One problem with voluntarism is that it violates what philosophers call the Principle of Sufficient Reason (or PSR), according to which, for everything that exists, every event that occurs, and every positive fact that obtains, there is a reason or explanation why it is that way rather than some other way (even if we don’t always know what the explanation is). For if mathematical truth were arbitrary, then there would be no reason why God gives the world the mathematical features that it has. It wouldn’t be that we merely can’t know what God’s rationale was, but that there just wasn’t any rationale at all. Caprice would lie at the very heart of creation.

But Craig has in other works vigorously defended PSR, and for good reason. Without PSR, it is difficult to articulate any argument for the existence of God as the ultimate explanation of why the world exists—and Craig is a prominent defender of such arguments. Inadvertently undermining the foundations of natural theology would surely be too high a price to pay in order to rebut Platonism. By embracing anti-realism as a way to save divine aseity—and thereby flirting with voluntarism and its ­irrationalist consequences—Craig is like the military officer who decided to destroy a village in order to save it.

The Aristotelian approach to mathematics offers a middle ground between Platonism and nominalism. For the ­Aristotelian, the Platonist is correct to regard mathematics as a description of objective reality rather than as mere linguistic convention. Mathematics works because it is true and the entities it refers to are real, just as the Indispensability Argument holds. But the Platonist is wrong to think that these entities exist in some exotic “third realm.” Rather, they exist only in the first two realms—in the natural world itself and in the minds that contemplate them. For example, the triangularity studied by geometers exists in actual particular triangles and in the minds that abstract this general pattern from these particular instances and consider its properties in isolation from them. Something similar can be said about other mathematical entities, such as numbers.

This approach has a distinguished pedigree in the history of philosophy, and in recent years it has been developed and defended by philosopher and mathematician James Franklin and others. Yet Craig says very little in response to it. In a footnote, he cites a common objection to the effect that it is mysterious what the Aristotelian means by saying that a pattern like triangularity is “in” particular things. But this usage is no more mysterious than other common usages of “in.” The way a person is in a club is very different from the way a spoon is in a drawer, and both are different still from the way a person might be in danger or the way World War II occurred in the twentieth century. Why is it any more mysterious to say that triangularity is in a billiards rack or a pyramid? As Aquinas would point out, the word “in” is one that is used analogically. There is something in the way a person is in a club or the way triangularity is in a pyramid that is analogous to the way a spoon is in a drawer, even if it is not exactly the same way. There is no reason to think that the spoon-in-a-drawer sort of case is the only one in which the word “in” has a legitimate use.

Another objection Craig raises in passing is that it is not clear how the Aristotelian approach can deal with mathematical entities that cannot plausibly be found anywhere in the physical world. For example, there are actual triangles in the physical world, but what about geometrical figures that are purely theoretical and have no physical instances? Or numbers that are larger than any collection that actually exists in nature?

This is where Aristotelianism can be combined with the Augustinian view I described earlier. For the Aristotelian, mathematical entities can be said to exist either in concrete particular objects in the physical world or as abstract patterns in the mind (ruling out only Plato’s “third realm”). Now, there are infinitely many numbers, infinitely many possible geometric shapes, and so on. No human mind or collection of human minds can contain them. But an infinite, divine mind could. Hence, to deal with the infinity of mathematical truths and entities that have no instances in the physical world, the Aristotelian approach to mathematics can be extended to include the mind of God. Indeed, the divine mind is where all mathematical truth can be said to have pre-existed, before it became embodied in the material world upon creation or abstracted from matter by finite human minds.

Note how this position does justice to the objective, necessary, eternal character of mathematical truth without threatening divine aseity, because it locates the mathematical realm precisely in God himself. One would think it the natural and obvious way to rebut the challenge posed by Platonism. Indeed, Craig is not only well aware of this view—which he labels divine conceptualism (since it identifies mathematical entities with concepts in the divine intellect)—but acknowledges that it has historically been the mainstream position in Christian theology, endorsed by the Church Fathers and the Scholastics. He even tells us that when embarking on his study of the challenge posed by Platonism to divine aseity, he expected that he would end up following the tradition and defending divine conceptualism. Yet he does not. Why not?

Craig raises the following objection. Divine conceptualism holds that eternal mathematical truths are to be identified with divine thoughts. Now, God’s thoughts are either caused or they are uncaused. But if we say that they are caused, then we have a problem. For God’s being omniscient involves, among other things, his having the thoughts he does. So, if God causes his thoughts, then he causes his omniscience—which, since being omniscient is just part of what it is to be God, entails that God causes himself. And that is incoherent. Yet if we say instead that God’s thoughts are uncaused, then we have another problem. Given divine conceptualism, that would entail that mathematical truths—which, again, are supposed to be eternal and identical with divine thoughts—are uncaused. And that would conflict with divine aseity, since it would mean that there are things distinct from God (namely, the thoughts in question) that exist in an eternal and uncreated way. The way out of this mess, in Craig’s view, is to abandon divine conceptualism in favor of anti-realism.

But divine conceptualism isn’t the source of this mess. The source of the mess is the assumption that divine thoughts are something distinct from God, which can intelligibly be said to exist apart from him in either a caused or uncaused way. And from the point of view of classical theism, this is simply wrong, for it conflicts with the doctrine of divine simplicity. According to that doctrine, there are no parts of any kind in God: no physical parts, and also no distinction in him between substance and attributes, between matter and form, between actuality and ­potentiality, between genus and differentia, or between his essence and his existence. He is as perfect a unity as can be conceived. Hence, there can be no distinction in God between him and his thoughts either.

Craig is well aware of the doctrine of divine simplicity but rejects it. Because it denies any distinction between God and his attributes, he believes it makes of God a kind of attribute. But this misses the point of the doctrine, which is that God transcends the distinction between substance and attributes. Indeed, the doctrine is no less central to the classical theist tradition than divine aseity itself is, being regarded as nonnegotiable by the Church ­Fathers, by Scholastics such as Anselm and Aquinas, by Protestant thinkers such as Luther and Calvin, and by the Catholic ecumenical councils Lateran IV and Vatican I.

Divine simplicity has traditionally been taken to be a necessary concomitant of divine aseity. Anything that is in any way composed of parts can exist only if something causes those parts to be combined. If God were not simple, he would need a cause—in which case, he would not be self-existent, contrary to divine aseity. If Craig thinks otherwise, he owes us more than a cursory treatment of the subject. For him to devote a book to defending divine aseity while giving such short shrift to divine simplicity is like writing a book on World War II that devotes only a few paragraphs to the European theater.

Having said that, we are in Craig’s debt for bringing back to our attention theological issues that are fundamental yet neglected, and for addressing them at the deepest philosophical level. We are sure to learn from a mind as fine as Craig’s, and to learn from his mistakes no less than from his many insights. 

Edward Feser is a professor of philosophy at Pasadena City College.