Mathematical Knowledge and Divine Mystery: Augustine and his Contemporary Challengers

Christians have been active in philosophy of mathematics in recent years, but Steven D. Boyer and Walter B. Huddell III argue that the classical work of Augustine of Hippo in this field has been largely misunderstood or distorted even by its supposed advocates. This essay corrects that distortion and shows how the traditional Augustinian awareness of God’s incomprehensible mystery allows both a deeply Christian resolution to some perennial issues in mathematical ontology and also a surprisingly productive engagement with a contemporary anti-Christian alternative. Mr. Boyer is Professor of Theology and Mr. Huddell is Associate Professor of Mathematics at Eastern University.

It is surprising to some people that higher mathematics should be related to matters of passion, transcendence, mystery, and so on, or to anything beyond the immediate, prosaic practicalities of engineering or computation. It is not surprising to mathematicians. In 1995, when Princeton mathematician Andrew Wiles resolved the centuries-old enigma called “Fermat’s Last Theorem,” it was big news in the mathematics community, but it was even more than “news” for Wiles; it brought tears to his eyes.

In an interview for the “Nova” television series in 1997, Wiles sat at his office desk as he described, slowly and meditatively, one of the crucial moments of discovery. “At the beginning of September, I was sitting here at this desk, when suddenly…totally unexpectedly…I had this incredible revelation.” He paused, deeply moved by the memory, and obviously struggling to maintain his composure. In a moment he continued, in hesitant, sometimes broken sentences: “It was the most…the most important moment of my working life….” Again, he had to break off in mid-thought, as he attempted to contain emotions that threatened to overwhelm him. The pause grew longer, then longer still; he started to speak again, had to stop even before the first syllable emerged. At last: “Nothing I ever do again will—,” but once more Wiles had to stop, finally turning aside from the camera, barely able to squeeze out an apologetic “I’m sorry” as the beauty and power of the memory overcame him.¹

What is going on here? What is it about a mathematical theorem that would prompt one who puts it together to call it a “revelation,” to speak in hushed, emotion-laden tones, to treat it with what looks for all the world like the accouterments of the sacred? Is there some connection between mathematical truth at its highest level of abstraction and apparent impracticality, and the realm of sacred truth and beauty? What exactly is “mathematical truth” anyhow? And what is its relation to the sacred?

This set of questions has a long history, and one in which Christian voices have not been lacking, not least that of the great fifth century African bishop, Augustine of Hippo. But much of the standard Christian commentary on philosophy of mathematics in recent years has departed from the traditional Augustinian outlook in ways that have left it ill-equipped to address contemporary issues and challenges. In this essay, we would like to revisit the more traditional approach in order to see how it may provide for us not only a more coherent and satisfying Christian perspective on mathematics, but also a surprisingly productive response to contemporary anti-Christian alternatives.

Augustinian Mathematics, Then and Now

Many readers will already be familiar with the immense role of Augustine in the development of the Christian philosophy of mathematics. Prior to Augustine, the field was dominated by the work of Plato (early fourth century BCE), a work so lastingly significant that mathematical “realism” and “platonism” are often used as synonyms even today.² Plato held to the real existence of so-called “mathematical objects” (like numbers and sets) as part of his realm of transcendent Forms or Ideas, those supreme, immutable essences that constitute a reality that is of a higher order than the shifting, impermanent world of our ordinary experience. In Plato’s view, mathematical entities are eternal and necessary, and they exist in a manner completely independent of human knowledge. Their reality as changeless Forms governs and even gives existence to changing empirical things that are formed.

In the fifth century, Augustine famously incorporates Plato’s realism into the Christian tradition by positioning mathematical objects (and other Forms) as ideas in the mind of God. They are included somehow in the Logos, the eternal Word who, according to John’s gospel, was “in the beginning,” and was both “God” and also “with God” (1:1).³ Thus mathematical objects remain eternal, necessary, and mind-independent (at least as far as human minds are concerned), but they are now directly connected to God himself and can even be equated with the divine essence.⁴ Since God creates precisely by means of the Word (John 1:3), this approach maintains what we might call the “ontological flow” of the Platonic tradition, that is, the flow of creative energy from eternal, unchanging realities to the impermanent, mundane realities of our world.

Augustine’s Christianized reading of Plato becomes the standard mathematical ontology for most of Christian history—for most, but not quite for all. Plato’s great student Aristotle (mid-fourth century BCE) had provided a realist alternative to Plato by arguing that the Forms were not independently existing entities, but instead found their reality within the concretely existing substances of our experience. On this paradigm, the only way to consider the Forms in themselves was by means of an act of mental induction that gave rise to an “abstraction,” that is, to a mental notion that was “pulled off” (Latin, abs-tractus) of concrete reality. The mathematician observes particular instances of, for example, threeness and abstracts the universal “three” from the particulars. From a classical Platonic vantage point, this move is already a step in the direction of what would eventually lead to full-blown nominalism, for now the mathematical object does not exist in its own right, but is our name (Latin, nomen) for a mental construct that we derive from actual, concrete particulars. It is, we might say, merely an abstraction, and it owes whatever being it has to the particular, concrete reality of the empirical substance—a noteworthy reversal of ontological flow. As we shall see below, the most common contemporary accounts of mathematical realism already assume this Aristotelian outlook: they envision mathematical objects as one category of abstract objects, and these objects are therefore causally inert and impotent.

Indeed, this Aristotelian construal of mathematical realism comes to be associated even with Augustine, and in a fashion that will merit our extended attention here. In his carefully argued contribution to Russell W. Howell and W. James Bradley’s well-respected text Mathematics in a Postmodern Age: A Christian Perspective, philosopher Christopher Menzel of Texas A&M University provides a thorough, sophisticated defense of the classic, realist doctrine that mathematical objects (which he classifies as “abstract objects”) are ideas in God’s mind, and he describes his position as “simply an updated and refined version of Augustine’s doctrine of divine ideas.”⁵ Of course, as an “update” it does introduce some modifications of Augustine, most notably an affirmation that mathematical objects are created entities. By thinking of “creation” as God’s continuous ontological support of created things rather than as any kind of temporal origination, Menzel shows how mathematical truths can be eternal and necessary, yet still be “created in precisely the same sense in which concrete, contingent things are created.”⁶ Thus, we have a version of mathematical realism that can still allow the fundamental Christian belief in creation ex nihilo, and with it the utter ontological uniqueness of God, to be maintained with gusto.

The primary difficulty that Menzel encounters along the way, in typical Aristotelian fashion, is epistemological. How can we come to know necessary mathematical truths if they are concepts in someone else’s mind, namely, in the mind of God? This question he answers by appeal to the classical Christian notion of humanity as the image of God.⁷ Mathematical propositions are divine ideas to which a finite creature has access by virtue of being significantly “like” God as a knower. Just as any two finite knowers can be said to be thinking about “the same” concept or idea, so also a finite knower can be thinking about “the same” concept that is in the mind of the infinite God.⁸ In this way we do have epistemic access to mathematical propositions, and those propositions can have necessary validity by virtue of the precise character of God’s creative activity. Voilà—Augustine’s divine ideas, updated and vindicated.

Or not. The basic problem with this proposal is that Menzel’s update of Augustine is actually a substantial departure from Augustine, and a departure that loses sight of the very element in Augustine’s approach that made it so valuable: the mysterious character of mathematical objects as uncreated participants in the divine Logos. Menzel seems to think that his modification—construing mathematical and other abstract objects as created entities—is just a modest refinement of Augustine’s thought, or perhaps a noteworthy change but one still in keeping with Augustine’s overall theological project. But there is reason to think that Augustine himself would disagree.

In his most careful discussion of mathematical philosophy (in On the Free Choice of the Will⁹), Augustine goes out of his way to reject Aristotle’s view that “numbers are stamped on our mind not from some nature of theirs, but instead from the physical objects we come into contact with through bodily sense, as though they were some sort of ‘images’ of visible things.”¹⁰ Instead, like all Platonists, he insists that there does exist an “incorruptible numerical truth [that] is common to me and to any reasoning being.”¹¹ Then he makes the move for which he is so justly famous: having preferred Plato to Aristotle, he “Christianizes” Plato himself, in two short steps. First, he intentionally associates number with “wisdom”—that is, with that eternal, creative wisdom that all Christians had, by the fifth century, learned to identify with the Word who became flesh in Jesus Christ: “Wisdom and number are linked together in the most hidden and certain truth, [and] it is indisputable that they are one and the same thing.”¹² Second, he points out the supreme obscurity of all these matters, finally concluding that “we cannot be clear whether number is in wisdom or derives from wisdom, or whether wisdom itself derives from number or is in number, or whether each can be shown to be the name of a single thing.”¹³

In other words, Augustine’s steadfast conviction is that mathematical objects are (1) real in the full Platonic sense, (2) identical with the eternal divine Word, and (3) shrouded in mystery. We have already seen how Menzel’s suspicious talk of “abstract objects” belies 1; his doctrine that numbers are created entities clearly contradicts 2; we might even suspect that his clear, precise account of the relation between God and mathematical objects puts 3 at risk. It looks as if Augustine’s doctrine has not yet made it into the contemporary debate, or at least not very clearly or effectively.

To move in that direction, let us consider a bit more carefully what a fully Augustinian account of the philosophy of mathematics looks like, and how it differs from Menzel’s approach. We will suggest that at least two significant impulses must be at work in order to be faithfully Augustinian (and deeply Christian), and they point to two significant weaknesses in Menzel’s presentation and in much of the contemporary Christian discussion of mathematical ontology. Here is one way of summarizing these two crucial issues: we need an approach to mathematics that builds more faithfully on both the apophatic mystery and the trinitarian distinctiveness of God.

Mystery and its Mathematical Implications

Consider first the character of divine mystery. As everyone knows, both the Bible and the Christian tradition as a whole are heavy with the awareness that the living God is transcendent, incomprehensible, “past finding out” (Romans 11:33 KJV), and few Christians would disagree with Augustine’s own famous dictum, “If you have understood it, it is not God.”¹⁴ We can get at the significance of this teaching (and yet dodge some of its more technical details¹⁵) if we consider, as an analogy, the way a two-dimensional person (what Edwin Abbott used to call a “Flatlander”¹⁶) both could and could not grasp the geometry of a three-dimensional figure.

Think of the circularity of a closed cylinder, for instance. A two-dimensional observer could understand the roundness of a cylinder (how the cylinder looks when viewed from one end) with perfect mathematical clarity. Yet such a one would still have no grasp at all of how this perfectly round figure could be more than an ordinary disk—how, say, it could include a certain kind of rectangularity (picture the cylinder viewed from the side). There is real knowledge at work in our Flatlander’s grasp of this object, but it is combined with a limitation in perspective that makes the knowledge itself less transparent, less manageable, in a sense less trustworthy, than one would ordinarily expect it to be. Two-dimensional geometry can furnish a description of the cylinder that is entirely true, as far as it goes. But the two-dimensional observer cannot know “how far it goes,” and for just that reason can never be entirely confident that boundaries have not been crossed, that assumptions have not been introduced, that problematize the whole endeavor.

So also, say Christians, with the transcendent God. Creatures are made to know this God truly, and yet we inevitably know also as creatures, and therefore true knowledge is precisely knowledge that is constantly aware of an ontological supremacy that overflows every concept or category that creatures themselves can grasp—including the concepts and categories of mathematical logic. It is not exactly that these concepts cannot be applied to God.¹⁷ Logic, as a participation in the Logos who is God himself, certainly does apply to God: but we human reasoners may not know how to apply it. One can put the same point in more traditional terms: there is a real analogy between creaturely reasoning and the divine Reason, but it is an analogy in which unlikeness is at least as great as likeness,¹⁸ and an analogy, therefore, in which the contours even of the likeness are grasped in their fullness only by God himself. Hence, however far reason rightfully takes us in the knowledge of God, it is always accompanied by, or even powered by, an all-encompassing mystery that will not be tamed.

This abiding reality of mystery accounts for several features of historic Christian theology that will be relevant for us here, most obviously its strong experiential or mystical impulse and its frequent apophatic emphasis,¹⁹ but it also turns up in some places that may take us by surprise. Consider, for example, the classic doctrine of divine simplicity, the teaching that God is not “composite,” not made up of parts of any kind. Most of the contemporary philosophical literature regarding this doctrine, both pro and con, is devoted to investigation of simplicity as a positive attribute of God, that is, as a characteristic that tells us something affirmative about the divine nature—namely, that it is wholly uniform or homogenous, lacking all plurality, complexity, or internal distinction. But many scholars have argued convincingly that this is not what the historic doctrine of simplicity really aimed at.²⁰ Its purpose was not to give a substantial ontological description, but to provide a methodological guideline for an indirect account of the God who could not be grasped by creatures directly.²¹ In other words, the doctrine of simplicity expressed a constant, unyielding awareness that our knowledge of God is rooted in mystery, and that it therefore cannot be handled in the same light, breezy way that other, more straightforward instances of knowledge allow.

Now, with this first major piece of a more traditionally Augustinian ontology and epistemology in view, let us consider once again the account offered by Menzel of mathematical objects as “concepts in the mind of God.” Does it take seriously the sense in which the God whose mind we are discussing is an incomprehensible mystery? The answer seems to be No, for Menzel’s whole method is rooted in understanding “the mind of God” as ontologically parallel to human minds, which is exactly the rationalistic move that divine mystery forbids.

As we noted above, this parallelism is explicitly how Menzel deals with the problem of epistemological access that (he thinks) attends classical platonic realism. Since we cannot have knowledge of “abstract” objects such as numbers that are causally isolated from us, he relocates them into the divine mind, where they are accessible in the same way that concepts in other human minds are accessible, by virtue of the similarity between minds. Menzel explains that, if someone else seems to have “a concept with similar content that applies to the same things as our concept,” we say that it is the “same” concept. So also, since humanity is the image of God and is therefore significantly like God,

when we abstract a concept of pair or two-ness [from the concrete objects of our experience], it seems reasonable to say that God possesses the “same” concept, [and] hence we can be said to have knowledge of God’s number concepts, that is,…knowledge of the numbers per se.²²

Note well: this process begins with ordinary human ways of knowing ordinary human minds, and then it applies them univocally to the divine mind. There is every indication here that God is being thought of as proportionally analogous to ourselves—as if the Creator of all things were one entity (no doubt the biggest and strongest) alongside all the others in the created world.

Of course, Menzel knows that this parallel is only rough and inexact. Indeed, philosophically speaking, it has to be, lest too exact a parallel leave us with too “particular” an account of mathematical objects—with an account, in other words, that gives us no justification for the universality of mathematical truth. Hence, though he admits that “in one sense…concepts in the divine mind are just one more bunch of specific mental constructions,” he goes on to insist that identifying these constructions with universally binding mathematical truths “is far from arbitrary.” Why? Because “God’s is not just one more mind among many, but the fullest possible realization of everything that mind can be and do, and indeed, the very source of mind and consciousness.”²³ In other words, God’s concepts can have universal consequence precisely because God’s mind is fundamentally different from every other mind. But neither proposed difference (neither God’s mind as “fullest possible realization” nor as “very source”) is articulated by Menzel in anything like an apophatic way, and in any case, the coherence problem remains. God’s concepts must be different from ours in order to serve as universal mathematical principles, but Menzel’s system can allow the difference only at the expense of the likeness that made epistemic access possible in the first place. In other words, Menzel seems able to provide either universal mathematical truths or accessible mathematical truths, but not both. His attempt to secure the one renders the other unattainable, and the halfway house he commends is plausible for neither.

One course open to Menzel might grant both universality and accessibility, namely, a return to the historic doctrine of simplicity. This deeply Augustinian move would construe “God” and “God’s mind” and “concepts in God’s mind” as varying ways of pointing to the incomprehensible reality that God himself is, a reality that is therefore known in a manner that is sui generis. Unfortunately, Menzel rejects this option, apparently following the arguments of Thomas Morris, who in turn appeals to the widely influential case presented by Alvin Plantinga in his Does God Have a Nature?²⁴ But Plantinga’s critique of simplicity involves just the kind of logic chopping we noted above. In brief, for Plantinga, simplicity means there is no distinction between God and his properties, or between one property and another; hence, each property is identical to every other, and there is really only one divine property, and God himself is identical to that one property; hence, God is a property instead of the living, personal God, which is unacceptable to Christians. There have been many different responses, both positive and negative, to Plantinga’s argument.²⁵ But from the vantage point of a traditional, apophatic understanding of simplicity, it has all the earmarks of rationalistic obfuscation, for it assumes that simplicity is a positive, coherent element within God’s nature, hence one that we ought to be able to define rigorously and apply systematically. There is no hint of the sense in which simplicity functions to guard us against precisely this kind of rationalistic approach to God.

Menzel and Morris and Plantinga are not the only contributors to this discussion who dismiss simplicity and who thus treat God like just another object for our rational inspection. For example, William Lane Craig rejects simplicity on the grounds that any God who is identical with his properties cannot be a personal agent. Why not? Because, says Craig, “we clearly grasp some of the essential characteristics of properties and of abstract objects in general, so as to be able confidently to assert that anything that is a personal agent just is not a property.”²⁶ Really? Do we really have such a “clear grasp” of the nature of personal agency in the incomprehensible God?²⁷ Again (though not related to simplicity), Peter van Inwagen wonders how a “thought in anyone’s mind, God’s or anyone else’s, could possibly be identified with” something like a number or a set, and he concludes that it is impossible (or at least unhelpful).²⁸ In one sense, he is undoubtedly right: it certainly is difficult to see how an ordinary thought in an ordinary mind can be the same as a universal mathematical object. But is God’s simply an everyday mind that can be set alongside created minds in the way that the loaded phrase “God’s or anyone else’s” implies? Augustine would certainly say No.

We begin to suspect that this move—namely, treating God, God’s mind, God’s thoughts, God’s properties, God’s activities, in the same way we treat other entities, minds, thoughts, properties, and activities—is uncomfortably common in discourse about mathematical ontology and related subjects. Our complaint here is not simply that the move is unfaithful to Augustine (though it is), or even that it is unfaithful to the Christian tradition (though it is). Our complaint is that it is unfaithful to the very nature of God himself. To act as if we can understand God is to treat God as if he were a thing we can understand, as if he were a created thing. But to treat God like a created thing, or to treat a created thing like God, is one very simple definition of idolatry. This strikes us, therefore, as a move that a Christian approach to mathematical ontology ought steadfastly to avoid.

Trinitarian Relations and Mathematical Truth

But the loss of authentic mystery—the mistake of treating God as simply another mind, like ours but idealized in some fashion—is not the only failing of much current Christian discussion of mathematical ontology. We fear that the trinitarian distinctiveness of God is also at risk. This second issue is often closely related to the first, for it is quite common in the literature to hear, on the one hand, the doctrine of divine simplicity criticized precisely for standing in tension with robust trinitarian theology (as though simplicity calls for an unqualified, monistic unity in God rather than for an authentic, apophatic humility in us);²⁹ and to hear, on the other hand, that the doctrine of the Trinity provides some sort of straightforward evidence that the numbers one and three must be eternal, since God is eternal and God is both one and three (as though the divine Persons could be quantified in an ordinary, univocal fashion).³⁰ But there are also other trinitarian issues that come into play, issues connected especially to the Christian view of the relations between the divine Persons.

Let us consider those relations—or, in order to avoid unnecessary complexity, let us consider the particular relation between the Father and the Son, that is, between the unoriginate God and the only-begotten God, between the Person who is the source of deity in the Godhead and the Person who is the fully expressed “Word” or “Reason” or “Thought” of the Father (all of these terms being legitimate translations of the crucial Greek term Logos). Note that there is at work in this relation an odd but unambiguous interpenetration of ontological equality and ontological dependence.

It is unambiguous because the whole orthodox trinitarian tradition, from beginning to end, has affirmed both aspects of the relation, both equality and dependence.³¹ But it is also odd, for, historically speaking, it was precisely the ontological dependence of the Son on the Father that prompted fourth-century heretics to resist the full equality of the Son with the Father. After all, is not ontological dependence what we have in mind when we speak of God as “Creator” of all things, and of the world as “created”? Surely “created” means nothing else but dependent in the deepest, fullest sense—dependent for one’s very being on something outside oneself? If the Son is dependent, surely he must also be created? This line of thinking was found to be plausible enough in the early patristic era that orthodox Christians went out of their way to point out its inadequacy. The Son, they insisted, is dependent, is genuinely a Son; but the Son is not dependent in the way that ordinary things in the world are dependent, is not a created thing, however lofty or exalted. No, said the early Church in its first lasting statement on the question (the creed from the Council of Nicea in 325), the Son is “begotten, not made,” coming from the Father, but not created by the Father. The Son is from God, but he is God from God. Indeed, he is God (fully divine) precisely because he comes from God the Father not as a creature, but as God’s unique, eternal Son. Note how the equality and the dependence turn out not to conflict here, but to unite. This is historic Christian orthodoxy.

Now, as we did before, let us keep this vision of God in mind as we turn back to the mathematical ontology of Menzel and other recent Christian commentators. At least two significant points of tension arise, and both of them reveal the important connection between trinitarian theology and the philosophy of mathematics.

First, Menzel’s own handling of trinitarian theology leaves something to be desired. In a brief but pregnant paragraph,³² he argues that the Son’s (and the Holy Spirit’s) ontological dependence on God the Father provides grounds for believing that mathematical objects, too, can be eternal and necessary but still ontologically dependent and therefore created. Note well these last few words: dependent and therefore created. This phrase should stick in the throat of any orthodox trinitarian, since the divine Son (according to orthodoxy) most certainly is not created. Yet the logic of the argument requires that he must be. Menzel astutely recognizes the problem, and in response to it, he notes that one would need to introduce “qualifications” in order to dodge the heretical bullet. Unfortunately, such qualifications are just what Arians throughout the ages have always proposed, and so Menzel’s brief discussion leaves one with the uncomfortable impression that the ontologically dependent Son of God is really just a creature to whom God has communicated the divine nature.³³ Trinitarian orthodoxy demands more.

One need not conclude, however, that Menzel’s proposed analogy is sheerly wrong. On the contrary, it has much to commend it: the problem is that Menzel himself has misunderstood it. If the trinitarian dependence that we call “being begotten” is to give us analogical insight into how necessary mathematical truths are related to God, it makes more sense to draw conclusions just opposite those that Menzel draws. It is not that mathematical objects are dependent and therefore created things; it is instead that they are dependent as the Son of God is dependent and therefore that they are uncreated things—or rather, that they are not “things” at all, but are (incomprehensibly) exactly what the only-begotten Word is. Number and the eternal divine Wisdom are mysteriously the same, just as Augustine had maintained.

There is a second point of tension. When Menzel rejects this mysterious identity with all of its rich trinitarian contours, he inadvertently introduces other debilitating problems—most notably, the problem we have already mentioned regarding epistemic access to mathematical truths. It is true that, if we begin by assuming that mathematical objects arise from an Aristotelian process of “abstracting” so that they are best defined as “nonphysical, causally inert entities,”³⁴ we do have a problem. After all, how can we have direct knowledge of things that neither act upon us in any way nor are available to us empirically? This is the dilemma that drives Menzel to try to establish humanity’s “indirect access” to God’s concepts, since “it seems implausible in the extreme that, say, in first learning the number 2, a child is [directly] accessing the contents of the divine mind.”³⁵ Indeed.

But recall, by contrast, the “ontological flow” that is characteristic of Platonism, and that is studiously maintained by an Augustinian approach to mathematics. Here, the causal activity that leads to knowledge is not primarily a human activity (“abstracting”) exercised upon inert external objects: it is instead the creative energy of the higher worlds, an energy that, according to Christians, consists in nothing but the creative Word itself, the Word that brings all things into existence in Genesis 1, the Word who is equated with God himself in John 1. There is here no causal inertness, no epistemic isolation, to be overcome. On the contrary, mathematical objects are included in the primordial causal activity that stands behind the whole universe, and that is tacitly known any time knowledge of any kind whatsoever obtains.

Is it, then, so implausible that a child should know the divine mind? Only if the “divine mind” is construed as simply a bigger, fancier version of the human minds that we have (indirect) access to in our ordinary experience. It is true that we do not expect a child just beginning to learn what “2” means to grasp intuitively all of the ins and outs of mathematics in the much more complicated mind of a Euclid or a Gauss. But suppose the mind of God is not merely a souped-up version of a human mind. Suppose instead that, throughout eternity, the living God is never without his living Word, and that the “reasonings” of God are nothing else but this Word, this eternal Reason, this eternal Logos, which is itself “the true light that gives light to everyone” (John 1:9), even to young children just beginning to learn their numbers. Once this more rigorously trinitarian outlook is in mind, one can hardly help regarding Menzel’s charge of “implausibility” as an unwitting expression of the worst kind of academic hubris—as though the sophistication and erudition of the highly educated philosopher allows him an access to God’s mind unavailable to the ignorant child, as though the Logos himself had no better coin to trade in than philosophical sophistication.

As we saw before, Menzel is hardly the only thinker to make this move; in fact, talk about “causal impotence” or “inertness” or “isolation” is a staple of much contemporary mathematical ontology.³⁶ But those who have Augustine’s trinitarian vision of reality as part of their legacy should know better. There is simply no reason to view mathematical objects that participate in the eternal Word as causally and epistemically inaccessible, just as there is no reason to permit such a view to push us into an impoverished understanding of that Word in the first place.

When trinitarian concerns like these are combined with the apophatic issues noted earlier, a rather stark overall contrast emerges between the classical Christian approach to mathematical ontology and its contemporary Christian competitors (represented here, for the most part, by Menzel). We may summarize as follows. Menzel’s approach builds on some Christian concepts, but it rejects classical attributes like simplicity that the vast majority of Christians have found to be indispensable, it flirts with heresy in its view of the Trinity, and it can address the problem of epistemic access only by treating God as if he were simply like-us-but-bigger. By contrast, Augustine’s claim that number and wisdom are “the same,” that mathematical objects are part of the uncreated reality of the Logos, allows us to affirm exactly the incomprehensible transcendence of God that the whole Christian tradition has affirmed, it accepts trinitarian orthodoxy without hesitation, and it roots our knowledge of mathematical truths very plausibly in the eternal, creative activity of God himself. At every point, Augustine’s view is deeply coherent and faithfully Christian precisely because he rejects the blandishments of an idolatrous rationalism, remembering instead that any attempt to apply the standard strategies of discursive reasoning to the triune God, or to the eternal Word, or to the necessary truths that are “the same” as the Word, is doomed from the very outset. Better far, from a Christian perspective, to bow in worship before the incomprehensible Lord of all, and to allow our philosophy of mathematics to support that enterprise.

Toward a Wider Engagement

Now, up to this point we have concerned ourselves exclusively with a Christian philosophy of mathematics, and it is time to expand the discussion by bringing in a significantly different viewpoint. In particular, how might this Augustinian vision of a classically Christian mathematical realism help us to interact productively with something like the non-Christian, postmodern, anti-realist position of Paul Ernest, Emeritus Professor of the Philosophy of Mathematics Education at the University of Exeter in the United Kingdom? Ernest’s so-called “social constructivism” steadfastly opposes all of the supposed certainty, all of the absolutism, that he thinks characterizes the major contemporary options in mathematical epistemology and ontology.³⁷ We will understand why he takes this view when we grasp his basic convictions about the nature of mathematical statements, convictions that flow from two primary streams of philosophical reflection.

The first stream descends from the later work of well-known Austrian philosopher Ludwig Wittgenstein (1889-1951), who introduced into common parlance the notion of a “language game” to describe much human reasoning and interaction. The idea here is that human language (along with other cultural phenomena) is not a neutral medium for describing the world, but a binding set of rules for, in a sense, creating it. Just as the rules in a game of chess decisively shape the game by determining what moves are allowed and what are not, so also human language establishes conventions that decisively shape all human activity into particular “forms of life.” Of course, these are not “Forms” in any Platonic sense: the conventions of language do not depend upon correspondence to some higher, ideal reality any more than the rules of chess do. They are simply patterns of thought and activity that are accepted by those who speak that language, in order to provide coherence and order to the naturally chaotic jumble of human experience.

Ernest takes this hint from Wittgenstein and builds on it a comprehensive approach to mathematical ontology. Obviously it is an ontology that eschews the notion of eternal Forms, whether in the mind of God or elsewhere. Instead, Ernest argues that mathematical reasoning follows the rules inherent in mathematical language games and the forms of life practiced by the mathematical community. There are no eternal “laws of deductive logic”; there are human conventions for playing this “game” in the accepted way. Agreement that a mathematical player is playing according to the rules thus takes the place of logical necessity.

Hence, these rules are humanly defined or “constructed,” and this is why Ernest’s philosophy is known as “social constructivism.” Note well that this is not a constructivism of the kind represented in the early twentieth century by L. E. J. Brouwer (1881-1966), who looked in Kantian fashion to the natural, universal working of human minds in order to discover the rules that mathematical reasoning must inevitably follow. Instead, Ernest’s constructivism is rooted strictly in social convention, without reference to anything static and unchanging either inside or outside of us. Questions regarding the supposed “correspondence” between our understanding of a mathematical concept and what that concept “refers” to are simply misguided. Notions of “truth” and “proof” do not become purely arbitrary or subjective, but they do become intersubjective: that is, their public objectivity consists in having been formed socially, within the mathematical community.³⁸

This communal grounding is, for Ernest, precisely what allows the rules of the mathematical language game to be open to change and, in that sense, to be “fallible.” Here we come to the second stream of philosophical reflection that significantly informs Ernest’s view. This stream originates in the classical Hegelian notion of “dialectic,” but it comes to Ernest primarily through the work of twentieth-century Hungarian philosopher Imre Lakatos (1922-1974), who strongly emphasizes the role and character of historical development in mathematical knowledge.

As everyone knows, Hegel’s dialectical triad includes a basic assertion or proposition (a thesis), a second proposition that contradicts or refutes the first (an antithesis), and then a new, integrated proposition that sublates and unites the first two (a synthesis). Lakatos takes this basic framework and develops what he calls the “Logic of Mathematical Discovery,”³⁹ a sophisticated account of the gradual, historical development of mathematical reasoning, which Ernest happily appropriates for his own philosophy of mathematics. According to this model, mathematical understanding grows dialectically by means of the narrative of mathematical proofs. Of course, formal, written proofs do not look much like narratives or stories. They appear, at first blush, to be rooted not in dialogue (which relies on the give and take of dialectical logic), but in monologue. However, we must remember that, for the constructivist, proofs serve not to establish once-and-for-all certainty, but to convince a living audience. Ernest writes,

Ultimately a proof is a narrative for human consumption, a “procedure that is plain to view,” not a superhuman objective structure. For the primary function of a proof is to convince, and logical structure is a means to that end. Thus mathematical knowledge is founded on human persuasion and acceptance.⁴⁰

In this way, the formal proof is simply the beginning of the dialectical process—a process in which objection and counterexample and refinement play a crucial role. Every professional scholar will immediately recognize how these elements fit into the standard refereeing process for academic publication and advancement. Whether in mathematics or elsewhere, scholarly proposals—even those that will ultimately prove to be bold or revolutionary—are initially submitted for serious consideration in relatively small, communal contexts, perhaps with department colleagues in a casual bull session, or in a preliminary paper at an academic conference, or through submission to the editors of a scholarly journal. Each of these contexts invites or even requires active questioning, critical feedback, the raising of objections, the offering of counter-proposals.⁴¹ Even the most compelling proof does not descend from the platonic heavens, untouched by human hands. On the contrary, there is a concrete human element in the evolution of all mathematical knowledge, and it cannot be ignored.⁴²

Hence, Ernest’s approach, building on both Wittgenstein and Lakatos, gives us a philosophy of mathematics that takes seriously these universally acknowledged social and historical factors, yet without falling into a full-fledged, “anything goes” relativism. The explicitly social nature of the mathematical language game precludes the individualized arbitrariness that any strong form of relativism would presuppose, just as it also rules out the artificially “objective” arguments and conclusions that absolutism must demand. Avoiding these dubious errors on either side, Ernest proposes in their place a philosophy of mathematics that accounts for the actual practice of those who teach and learn mathematics, practice that inevitably includes an ongoing dialectical refinement that is deeply human at every point and that is therefore inevitably malleable and open.

A Preliminary Evaluation of Postmodern Mathematics

We earlier referred to Ernest’s philosophy of mathematics as “postmodern,” and by now it is not difficult to see why. Ernest is resistant to every expression of modern certainty, to anything that purports to be eternal or unchanging or absolute; he is unflinchingly attentive to the decisive role played by social and linguistic factors in all human knowing, including the dramatic limits these factors place on us as knowers. This is contemporary postmodernism in spades.

We are also setting Ernest’s view of mathematics over against the standard “Christian” view, and once again it is not hard to see why. Ernest has no interest whatsoever in God or God’s thoughts, or in any system of realism, which would posit some transcendent, independent existence for mathematical objects, including existence in the mind of God. His postmodern commitments might even serve to reinforce this “anti-Christian” character, insofar as many Christians will suspect that the denial of absolute truth inevitably signals the rejection of historic, biblical Christianity. If we follow this line, we will find ourselves drawn to an evaluation of Ernest that is quite negative, although a few points of agreement might be conceded.

Let us begin with the negative. Christian realists will want very strongly to point to at least three significant inadequacies in Ernest’s perspective. First of all, the history of mathematics suggests an ontology very different from Ernest’s. If social constructivism, with even its modestly relativistic emphases,⁴³ were correct, we would expect communities doing mathematics in independent spheres to arrive at significantly differing mathematical conclusions. But this does not seem to be the case. When we consider, for instance, Chinese mathematics prior to its encounter with the West, we find many quite astonishing parallels with developments in the West, including:

• Liu Hui’s extremely precise approximations of Pi;
• the development of a numerical method of solving algebraic equations that is virtually identical to what is known in the West as Horner’s method (named for British mathematician William Horner, who lived from 1786-1837);
• the development prior to the 1300s BCE of a Chinese version of the binomial theorem and Pascal’s triangle; and
• diagrams appearing in the ancient Chinese classic K’ui-ch’ang Suan-shu demonstrating a clear understanding of the Pythagorean relationships within right triangles.⁴⁴
  

These examples could easily be multiplied. At point after point, we find discoveries of various kinds (algebraic, analytic, arithmetic, geometric) that follow a methodology very different from that of Western, Hellenistic mathematics,⁴⁵ but with results indistinguishable from those in the West. These parallels point not to arbitrary cultural conventions, but to a mathematical reality that transcends culture.

Second (and related), it is hard to see how Ernest’s proposal can account for what physicist Eugene Wigner famously referred to as the “unreasonable effectiveness of mathematics” in describing the physical universe.⁴⁶ The assertion that mathematics is the language that the universe speaks, or the blueprint that the cosmos follows, has been made by thinkers from the Greeks, to Galileo, to Kurt Gödel, and the development of the modern natural sciences provides compelling evidence that this is so. According to mathematical physicist Paul Davies, the belief that nature appears mathematical only because we choose to describe it mathematically—that is, only because of cultural convention—is “altogether too glib” to be realistic, for

much of the mathematics that is so spectacularly effective in physical theory was worked out as an abstract exercise by pure mathematicians long before it was applied to the real world. The original investigations were entirely unconnected with their eventual application.⁴⁷

Such considerations suggest instead that there is some real relationship between mathematics and the Wisdom that is the “craftsman” at God’s side in creation (Proverbs 8). Indeed, if mathematics uncovers how the universe functions (as the sciences insist), and if the universe was created and is sustained by God’s eternal Wisdom or Word (as Christians have always believed), then there is every indication that the Word himself is mathematical.

Third, and from a more explicitly theological vantage point, Ernest’s proposal does not fit well with the overall ontological picture that almost any Christian philosophy of mathematics must presume. Not just for Augustine but for all Christians, the ontological flow of the act of creation moves unambiguously from a Creator God to the created order, with God’s eternal Word or Logos as the creative agent par excellence. Various Christians might want to quarrel with some of the ontological details of platonism, but that there exists at least some kind of eternal, creative reality that transcends human culture is non-negotiable. The divine Reason is not socially constructed, and insofar as social constructivism repudiates all such final, “absolute” claims, it must inevitably be found wanting.

On the other hand, despite these inadequacies in Ernest’s program, Christian realists need not jettison his approach altogether. There are at least two elements of constructivist thinking that may lend themselves to something akin to a “plundering of the Egyptians.” First, Ernest’s contention that mathematics is a fallible enterprise fits rather nicely with a Christian view of human limitations, especially in light of the transcendent otherness of God. At the very least, awareness of human finitude should allow Christians to applaud those developments in recent philosophy of mathematics that temper absolutist expectations: in particular, Gödel’s incompleteness theorems dealt a significant blow in the mid-twentieth century to the ambitious epistemologies of earlier mathematical philosophers, and Christians cannot help smiling at the thought.⁴⁸ When human fallenness, with its proneness to rationalization, misstep, and error, is also added into the equation, something not unlike Ernest’s fallibilism seems inevitable. Second, and for similar reasons, Christians can appreciate social constructivism’s emphasis on the value of the community in the discovery and warranting of new mathematical knowledge. Just as Christians have generally found that their understanding of Scripture and its application to doctrine should be checked against the reading of thoughtful Christians who have gone before,⁴⁹ so also the larger mathematical community rightly serves to correct the work of the individual mathematician at some times, and to inspire that work at others. One might even say that a kind of “dialectic” is operative in both spheres, and Ernest’s emphases can remind us of that fact.

A More Radical Appropriation

Now, this preliminary evaluation will strike many Christians as fair and balanced; indeed, it is not much different from the critical but hopeful evaluation of postmodernism proffered by Menzel as part of his discussion.⁵⁰ On the other hand, given the criticisms we have leveled against Menzel, this similarity may well be a bad sign—a sign that this evaluation of Ernest is guided too much by suspiciously “modern” presuppositions.⁵¹ We want to suggest now that a deeper engagement with Ernest’s approach is called for, an engagement whose results are a bit more positive, a lot more radical, and—for just that reason—significantly more Christian. Let us consider three different levels of productive appropriation of Ernest’s thought, each one reflecting an increasing appreciation of the significance of the apophatic incomprehensibility of the triune God for our approach to a Christian philosophy of mathematics.

First, we think Ernest is right to insist that mathematical knowledge is fallible, in a sense that goes quite a bit further than the bare acknowledgment of possible error that we grudgingly made a couple paragraphs back. There we said, in effect, “Of course mathematicians do make mistakes, and so it is important to be careful and to let others in the community verify our work.” This is true as far as it goes, but it acknowledges only the difficulty of obtaining logical certainty, not the impossibility of obtaining it. Yet impossibility is just what human finiteness and fallenness necessarily entail, in two different senses. On the one hand, certainty is impossible in the sense that, no matter how flawlessly consistent our mathematical system may be, we—even the entire mathematical community—can always make mistakes. This danger is illustrated by Alfred Kempe’s false “proof” in 1876 of the four-color theorem, a proof widely accepted for more than a decade before flaws were spotted by Percy Heawood in 1890.⁵² This decade of supposed certainty should provide a sobering lesson for other mathematicians who seek unqualified certitude. But on the other hand, certainty is also impossible in the sense that we cannot prove any mathematical system to be flawlessly consistent in the first place. As we noted above, this kind of demonstrated consistency was the lofty aim, in different ways, of more than one school of philosophy of mathematics in the twentieth century (especially Gottlob Frege’s logicism and David Hilbert’s formalism), but the work of Bertrand Russell (who discovered the problem in Frege’s system that became known as Russell’s Paradox) and Gödel laid such ambitions permanently to rest.⁵³ Here again, on a more theoretical level, absolute logical certainty proves to be out of reach.

Christian commentators on constructivism often concede this point in principle, but then seem to forget it as their work proceeds. For instance, David Klanderman admits that all human knowledge is fallible, but then he insists upon “the existence of absolute or universal truths even if our own humanly constructed truths may indeed be relative to the limitations of our humanity.”⁵⁴ One wonders just how seriously fallibility is being taken here: after all, what value is present in absolute or universal truths that are utterly inaccessible to us? Klanderman goes on to approve a quotation from Jim Jadrich: “Our imperfections (and imperfect constructions) do not impose a limitation on God or on his creation. It is our fallenness that holds us back from fully comprehending truth. Truth itself is not suspect.”⁵⁵ Fair enough. But what exactly is this “truth” that is not “suspect”? It cannot include any particular mathematical formula or theorem, since our very knowing of such a thing would make it a fallible construction rather than a pure, unadulterated “truth.”

In the end, it seems that, even if absolute mathematical truth really is “out there” in some sense, we ourselves can have no absolute knowledge of it, and in that sense final certainty simply is not available to us in the way we would prefer. Even our most direct and compelling proofs, worked out with ever-so-much care and cross-checked by ever-so-many brilliant colleagues and accepted for ever-so-long by the larger community, cannot “prove” in any full, unambiguous sense. Whatever we might say about ontology (more about that in a moment), epistemological limitations condition our mathematical knowing from top to bottom, just as Ernest suggests. And this is why, whatever may be the feeling of certainty and assured truth that mathematical reasoning engenders, it cannot give us unqualified logical certainty about our conclusions.

Now this is a hard pill to swallow—and not just for mathematicians. Can any of us seriously entertain the proposal that “2 + 2 = 4” could be wrong? The answer is probably No. But our inability here is a practical fact about human psychology, not a logical conclusion drawn from indubitable premises. From the premise “All human knowing is fallible,” the logical conclusion “The particular instance of human knowing expressed by ‘2 + 2 = 4’ could be wrong” is inevitable. To be sure, this whole line of thinking is also maddeningly paradoxical, since it destabilizes the very logic on which it is based. But this is just the flip side of the universal admission that we cannot logically prove that logical proofs work—or at least, we cannot do so without begging the question.⁵⁶ Hence, Ernest is right to note the inescapable fallibility of even our most cherished mathematical convictions. We are like Qoheleth (the “Teacher”) in the book of Ecclesiastes: so long as we are considering what is “under the sun”—and all humanly known mathematical truths are there—our ultimate conclusion will be, from one very important angle, “vanity.” Or, to put the point less pessimistically, we are like the apostle Paul in his eschatological hope: in our mathematics we walk by faith, not by sight (II Corinthians 5:7). As Anselm of Canterbury and many others have insisted, a faithful epistemology requires that we must first believe—without logical certainty of any kind—if we are ultimately to know or understand.⁵⁷

Second, we think Ernest is right to insist that mathematical knowledge is dialectical, indeed, that it is more dialectical than Ernest himself appreciates. Ernest is thinking primarily of classical Hegelian dialectic, which involves two parties on rather equal footing, engaging in personal dialogue as they attempt to move beyond their own limitations toward some higher (though still imperfect) grasp of the truth. We might refer to this as a “horizontal” dialectic, and we have already acknowledged that it is at work in mathematics, namely, in the mathematical community as proposals are entertained, clarified, refined, and formalized, and thus as our store of mathematical understanding grows and deepens.

But from a Christian vantage point, we should recognize that mathematical knowledge, like all knowledge, always and inevitably involves a “vertical” dialectic as well, as God empowers us to know and thus reveals himself to us. The Logos, as we have already had occasion to note, is the Word who actively enlightens everyone (John 1:9), and so knowledge is never simply a human achievement in the face of a neutral, passive, “objective” reality. The eternal Subject is always personally involved, as teacher, as leader, as Lord. And we, too, are personally involved, for God does not simply teach or reveal in some generalized, objective sense. God teaches us; God reveals to us. In this sense, God’s Word always presupposes a reader, and the whole aim of revelation (which means, by extension, the whole aim of knowledge as such) is that we should grow, gradually and with ever-so-many fits and starts, into full human persons who love and worship aright. Mathematics may be less speculative than other intellectual endeavors, but it is no less personal and therefore it is (so to speak) no less “subjective.” Each one of us is a subject, and God’s aim is to teach us at the deepest levels of our subjectivity not just to know but also to love—to love the truth we perceive and to love the God who is Truth. Andrew Wiles’ tearful recounting of his moment of “revelation” is, from a Christian perspective, wholly fitting: whether he realizes it or not, he has seen the face of God, by God’s own “dialectical” design.⁵⁸

An Unlikely Prescription

Finally, and most controversially, we think Ernest is right to insist that mathematical knowledge is not real, that mathematical objects do not exist, in the ordinary sense of these words. This will no doubt be a surprising move in light of our paper’s lengthy defense of a very definite Christian realism in the spirit of Augustine, and so we should explain as carefully as we can what we mean. It is true: we are realists. We have even argued that the realism of someone like Menzel does not go far enough, that Augustine gives us a realism that is somehow higher or more real because it is defined not by the reality of created things but by the more lofty, transcendent reality of the triune Creator himself. In this respect, we are prepared to agree with other Christian realists that Ernest’s explicit anti-realism is, in its most obvious sense, wrong.

On the other hand, one must be cautious about what is “obvious” when dealing with authentic transcendence. In the end, the language of Christian realism about the “reality” of God and the “existence” of God is no more exempt from ordinary creaturely connotations than any other language is, for it continues to be human language, colored inevitably by human associations. When we look around us, we notice all sorts of things that “exist”: this table, that wall, the tree out the window, the dog barking in the distance. If “existing” means being a thing like one of those (and what else can it mean?), then clearly the Lord of heaven and earth does not exist.⁵⁹ If being “real” means being what animals and plants and men and women and rocks and quasars are, if theirs is the kind of “reality” we have in mind (and what other kind do we know?), then the living God is not real.

To be sure, we still need to speak. And so it may be expedient to refer to God as some sort of greater “Existence,” some sort of higher “Reality”—note the capital letters, which attempt to point us beyond the usual limitations. But to be meaningful at all, these hyper-words must still involve some appeal to ordinary human experience, and at just that point the ordinary associations of human language threaten to creep in.⁶⁰ So what are we to do? If God is great in this language-devouring way—if God is literally unspeakably great—and if we as Christians still want to speak faithfully of him, then our very speaking must also include a certain kind of un-speaking, a paradoxical repudiation of every form of easy meaningfulness that would unwittingly domesticate the living God. We need an ever-present reminder of the permanent frailty of our language, even of language as apparently straightforward as words like “real” and “exists.”

Enter Paul Ernest. If we need something that bracingly renews our awareness that none of our language ever reaches fully to God, that all of our descriptions are false if taken literally or univocally, then one can hardly do better than Ernest’s ontological anti-realism, which can function as a kind of apophatic corrective to all of our accidentally idolatrous assumptions. Ernest never tires of reminding us that even those insights that strike us as utterly necessary, as free from all empirical taint, as “true” in the highest, purest fashion, are still fallible steps along a dialectical path whose end we cannot see. So when our successful arguments for Christian realism become too successful, when they put us in danger of forgetting human frailty and therefore of abandoning a clear, rich awareness of divine incomprehensibility, we might find that Ernest’s affirmation of a constructivist anti-realism is just what the doctor ordered.

Of course, we can hardly blame those who are suspicious of this rather unlikely prescription. Christian realists should regard anti-Christian anti-realism as a friend? Well, perhaps so—or at least as an unintentionally friendly amendment to a mathematical ontology that might otherwise be driven by modernist assumptions to rationalistic conclusions that belie authentic Christian faith. There certainly is precedent for this kind of inverted appropriation of postmodern, or even atheistic, thought for Christian purposes.⁶¹

In fact, there might even be precedent in the literature of Christian philosophy of mathematics, for not all mathematical anti-realism is anti-Christian. We referred earlier to the work of William Lane Craig, who criticizes divine simplicity for (we argue) insufficient reason: yet Craig seems to be attracted to an anti-realist mathematical ontology for what might be all of the right reasons. He discerns plausible elements in several different versions of anti-realist nominalism, but the driving force behind his attraction seems to be not the compelling power of nominalism per se, but the weakness that inevitably attends any realistic ontology construed in modern terms. In very brief form, here is his logic: he regards doctrines of divine aseity and creation ex nihilo as essential to any form of robust theism, but he sees that the existence of any non-divine, eternal, necessary realities (such as mathematical objects) would compromise these crucial doctrines. He considers various realistic proposals to see whether they can resolve this problem, but they cannot. The conclusion is inevitable: if mathematical objects exist, theism is doomed. Hence, to protect Christian theism, Craig argues that mathematical objects do not exist.⁶²

But what if each of the realistic proposals he considered was already beset with troublingly modern assumptions? What if one discovered a richly apophatic form of mathematical realism that no more threatened the non-negotiables of orthodox Christianity than anti-realism does—and for the same reason, namely, because this kind of realism did not posit the reality of any mathematical “things” alongside of or independent of God? We have argued that this is exactly what one finds in Augustine’s philosophy of mathematics. Here, there are no eternal “things” alongside of God: there is only the living God himself, the incomprehensible, multi-dimensional resplendence that Flatlanders understand in ways that are true, but always creaturely. A proposal like this would seem to allow Craig and those who follow him to hold fast to both their Christian theism and their anti-realism, precisely by following the higher realistic path of Augustine.⁶³

Mathematical Ontology and the Knowledge of God

We have seen, then, that Augustine’s classical ontology ingeniously draws together (1) mathematical realism and (2) mathematical anti-realism in a union that is grounded in (3) revealed Christian doctrine at its best. Consider each of these points in turn, as we conclude.

(1) On the one hand, Augustine’s model gives us a rock-solid platonic realism that helps us to account for the universal assurance that mathematical truths really are truths, hence, are not conventional fictions peculiar to human minds; and that mathematical truths really are known, hence, are not inert external objects isolated from human knowing. Menzel and other contemporary realists wrestle with these problems, but with only limited success. Whence come this immutable stability and this open accessibility? Augustine’s answer: number and the eternal Wisdom of God are the same. Mathematical truths are as solid and unyielding, and as knowable and accessible, as the God in whom they mysteriously participate.

(2) Yet, on the other hand, Augustine’s model pushes this realism to a height that explains the strange plausibility of anti-realism, too. The anti-realism of Ernest and his ilk is built on the elusiveness or frailty that we often encounter in our mathematical practice: our missteps as we learn, our hit-and-miss labors as we formulate proofs, our sudden perception of problems where yesterday we saw (or seemed to see) only solutions, our reliance on others to see what we ourselves are having trouble seeing, our expectation that truths we have no inkling of today will be developed with utter lucidity by future generations. But all of this makes perfect sense when we realize that even our highest, purest perceptions remain ours, and are therefore a long step removed from the lofty Reality that we perceive. We are mere Flatlanders, and final mastery of Truth itself is not ours to be had.

(3) Finally, Augustine draws these two contrary outlooks together precisely by rooting them in the deep Christian awareness that God himself is not an abstract principle for the intellect to grasp, but a supremely concrete tri-Personality who incomprehensibly overflows every created category of intelligibility. “There is none like you, O Lord; you are great, and your name is great in might” (Jeremiah 10:6, ESV). If perceiving mathematical truth is perceiving the Wisdom of God, if mathematical logic is a participation in the eternal Logos, if math itself is (so to speak) “begotten not made,” then our mathematical knowledge is authentic knowledge of a reality that transcends the world. Yet just for that reason, it is knowledge that tempts contingent creatures to forget their contingency, to construe their knowledge as independent, absolute, autonomous, and final. Fallen creatures are always tempted by idolatry, and this is exactly why it takes a healthy dose of anti-realism to keep a high, Augustinian realism from betraying itself. To borrow an image from Merold Westphal,⁶⁴ we might say that the writings of Paul Ernest make “good Lenten reading” for Christian mathematicians who want to celebrate the great truths that they know, but who also want to guard against the triumphalism that always threatens to taint their celebrations.

To be sure, this paradoxical union of realism and anti-realism is, from one angle, not particularly satisfying or comfortable. One might prefer a more straightforward answer, one that falls plainly on one side or the other of the ontological divide. Yet perhaps this discomfort, too, is something we should expect. The “vertical dialectic,” in which God himself is the personal tutor who draws us into real comprehension of what is incomprehensibly Real, may well turn out to be so extremely vertical that it causes—and ought to cause—a kind of intellectual vertigo. We are left off-balance and dazed by the profound awareness that we cannot fit the Lord of mathematics into any of our standard categories. This is as it should be, in our mathematical ontology as also in our theology.

Yet the ultimate aim of God’s dialectical instruction is not dazed discomfort, just as it is not simply “knowledge,” as that word is ordinarily understood. As we noted earlier, the aim of all knowledge, including mathematical knowledge, is love. And so, once again, Christians are not surprised to learn that those who see mathematical truth best tend, like Andrew Wiles, to be overwhelmed by the elegance and coherence and beauty of what they see. How could it be otherwise? What they see is, in a mystery, the eternal Word and Wisdom of God himself. It is seen with creaturely eyes, hence “through a glass, darkly” (I Corinthians 13:12, KJV). But when the eschatological day of real vision comes, when we have put on “the mind of Christ” fully and finally so that we can see Reality as it really is, we may discover that the perfect knowledge we enjoy then consists in the very same truth and beauty that mathematics affords us a frail glimpse of even now.

1. The program was originally aired on October 28, 1997. A brief clip of this moving scene at the beginning of the program has been posted on YouTube at http://www.youtube.com/watch?v=SccDUpIPXMO+index=1+list=PLVcrDOQFQd93h5ATTM_6fIadV8SQlmwX. For a transcript of the program in its entirety, see http://www.pbs.org/wgbh/nova/transcripts/2414proof.html.
2. Note the spelling of “platonism” with a lowercase “p,” which is often used to signal not the doctrine of the historical Plato himself but the more generalized, and often the more Christianized, mathematical realism of Augustine. See Stewart Shapiro, ed., Oxford Handbook of Philosophy of Math and Logic (Oxford: Oxford University Press, 2005), 6. We will adopt this convention in what follows.
3. As is often pointed out, John’s conception of the Logos also has important roots in the ancient Greek world, going back at least as far as Pythagoras and his followers (sixth century BCE), who held that the universe is governed by this fundamental rational principle that gives to all of reality a coherent order and structure that is deeply logical—and even deeply mathematical. Indeed, Pythagorean philosophy is often described as resting on the assumption that “whole number is the cause of the various qualities of man and matter” (Howard Eves, An Introduction to the History of Mathematics, 6th ed. [Fort Worth, TX: Harcourt Brace Jovanovich, 1990], 76). Here again we see that not only do numbers exist, but also that the whole material world exists only in a kind of ontological dependence upon them. In this respect (and in others), Plato was a convinced Pythagorean.
4. In light of this very positive vision of mathematical objects, it is worth noting, to avoid confusion, that Augustine sometimes refers in very negative terms to “mathematicians” (Latin, mathematici)—but only because he uses this term to refer to what we would today call “astrologers.” See, for instance, Confessions 7.6.
5. Christopher Menzel, “God and Mathematical Objects,” in Mathematics in a Postmodern Age: A Christian Perspective, eds. Russell W. Howell and W. James Bradley (Grand Rapids, MI: Eerdmans, 2001), 73. Note that though this chapter was originally penned by Menzel, the editors explain early on that they themselves and the other seven contributors to the larger volume have all “signed off” on the project, thus indicating their sympathy, if not unqualified agreement, with the approach presented by Menzel (vii). For a related perspective, see Howell and Bradley’s more recent volume, intended for a more popular audience, Mathematics Through the Eyes of Faith (New York: HarperOne, Council for Christian Colleges and Universities, 2011).
6. Menzel, “God and Mathematical Objects,” 71.
7. Ibid., 95.
8. Ibid.
9. See Book Two of Augustine, On the Free Choice of the Will, especially sections 8 and 11. All following page numbers refer to the recent translation by Peter King, reproduced in On the Free Choice of the Will, On Grace and Free Choice, and Other Writings (Cambridge: Cambridge University Press, 2010).
10. Ibid., 46-47.
11. Ibid., 47.
12. Ibid., 55.
13. Ibid., 56. It is worth noting that, throughout this discussion, Augustine uses the Latin term numerus, which the translator renders as “number,” but which in fact has a wider semantic range. The term also refers to rank, category, pattern, structure, and so on. Evidently Augustine has in mind the whole orderly arrangement of mathematical logic, not merely “numbers” in the narrow sense the contemporary English word connotes.
14. See, for example, Sermo. 117, 3, 5 (PL 38, 663), as well as Sermo. 52, 6, 16 (PL 38, 360).
15. For an exploration of many of these details, see the first three chapters of Steven D. Boyer and Christopher A. Hall, The Mystery of God: Theology for Knowing the Unknowable (Grand Rapids, MI: Baker Academic, 2012).
16. Edwin A. Abbott, Flatland: A Romance of Many Dimensions (London: Seeley & Co., 1884; repr. New York: Dover Publications, 1992).
17. Alvin Plantinga shows the problem with this self-refuting move in his landmark book, Does God Have a Nature? (Milwaukee, WI: Marquette University Press, 1980), especially pages 10-26. We will point out below at least one point at which we think Plantinga might be taking the opposite position—that logic can be applied to God—in an unhelpful direction.
18. “Between the Creator and the creature there cannot be a likeness so great that the unlike-ness is not greater” (Canons of the Fourth Lateran Council, Canon #2, available at http://www.fordham.edu/halsall/basis/lateran4.asp).
19. That is, its tendency to rely not on affirmations but on negations, to move not toward but “away” from “speaking” (Greek, apo-phanai) about God. A few examples: Gregory of Nazianzus declares, “To tell of God is not possible … but to know him is even less possible” (On God and Christ: The Five Theological Orations and Two Letters to Cledonius [Crestwood, NY: St. Vladimir’s Seminary Press, 2002], 39); Thomas Aquinas insists, “We cannot know what God is, but only what He is not” (Summa Theologica 1a.3, preface); Martin Luther often spoke of God as the deus absconditus, the “hidden God” (see Brian Gerrish, “‘To the Unknown God’: Luther and Calvin on the Hiddenness of God,” Journal of Religion 53.3 [July 1973]: 265-279).
20. See, for instance, William C. Placher, The Domestication of Transcendence: How Modern Thinking about God Went Wrong (Louisville: Westminster John Knox, 1996), 21-31; Christopher A. Franks, “The Simplicity of the Living God: Aquinas, Barth, and Some Philosophers,” Modern Theology 21.2 (April 2005): 275-300; and Andrew Radde-Gallwitz, Basil of Caesarea, Gregory of Nyssa, and the Transformation of Divine Simplicity (Oxford: Oxford University Press, 2009), especially chapters 5-7. See also the astute summary of the whole matter by William Alston in his lectures on the “divine mystery thesis”: “The concept of [divine] simplicity is a purely negative one. In its most careful formulations, such as the one by Aquinas, it consists in denying of God every possible form of composition, and that for Aquinas is the whole of it….The premise of divine simplicity, since it is itself a purely negative assertion, does not entail the denial of the divine mystery thesis. It does not mean that there is a positive concept, namely simplicity, [that is] applicable to God” (Divine Mystery and Our Knowledge of God (Nathaniel William Taylor Lectures, Yale Divinity School, 2005): Lecture I, “The Divine Mystery Thesis” (October 11, 2005), http://www.youtube.com/watch?v=Cz-Wt1HwEIE, at 24:49. Portions of this quotation may be a summary of Alston’s position by Nicholas Wolterstorff, who reads Alston’s lecture in the absence of the author himself. Yet it does seem to be an entirely accurate summary. In any case, much of Alston’s lecture, and additional material as well, is included in Alston’s essay on “Religious Language” in The Oxford Handbook of Philosophy of Religion, ed. William J. Wainwright (Oxford: Oxford University Press, 2005), 220-244.
21. The logic worked like this. All human knowing takes place by means of distinction and analysis: we break a thing up into its component parts, we understand the relation between the parts, and therefore we understand the thing. Perhaps we distinguish what a thing is made of from what it has been made into (in classical terms, this is the difference between “matter” and “form”); or we distinguish what a thing is doing from what it can do (“act” versus “potency”); or we distinguish what it truly is in itself from various characteristics it has (“essence” versus “attribute”); and so on. Each of these methods is both legitimate and useful, but each one knows a thing precisely by attending to a kind of “composition” in that thing—and this is exactly the method that cannot work with God, since (as we have noted) God is not a “thing” in the first place. God is not one of the ordinary objects that we can approach and know in the ordinary sense: God is instead an ontological plenitude that outstrips all creaturely knowledge, and the dogmatic insistence that God is not “composite,” that God is “simple,” is one way that we remind ourselves of that fact. Thus, simplicity serves as what might be called an “apophatic qualifier” of all of our other affirmations about God. (This handy phrase comes from Paul Gavrilyuk, The Suffering of the Impassible God: The Dialectics of Patristic Thought [Oxford: Oxford University Press, 2004], esp. chapter 2, although Gavrilyuk is talking about impassibility, not about simplicity.)
22. Menzel, “God and Mathematical Objects,” 95-96, italics original. Note the Aristotelian assumptions that undergird the simple phrase, “When we abstract a concept of….”
23. Ibid., 93.
24. For Menzel’s rejection, see Thomas V. Morris and Christopher Menzel, “Absolute Creation,” American Philosophical Quarterly 23.4 (October 1986): 359 and footnote 15. For Morris’s appeal, see his “On God and Mann: A View of Divine Simplicity,” Religious Studies 21.3 (September 1985): 300. For Plantinga’s own case, see his Does God Have a Nature?, 26-61.
25. See William F. Vallicella’s nice summary of responses and of the subsequent debate in “Divine Simplicity,” The Stanford Encyclopedia of Philosophy (Spring 2014 Edition), ed. Edward N. Zalta, <http://plato.stanford.edu/archives/spr2014/entries/divine-simplicity/>.
26. Paul Copan and William Lane Craig, Creation Out of Nothing: A Biblical, Philosophical, and Scientific Exploration (Grand Rapids, MI: Baker Academic, 2004), 178, italics added.
27. A few paragraphs further on, Craig criticizes one aspect of Thomas Aquinas’s doctrine of simplicity by analyzing what Craig thinks the doctrine implies about God’s “cognitive state” and then rejecting it as incoherent. The language of his actual rejection is ironic, for he states, “It is incomprehensible how the same cognitive state can be…” (179, italics added). Yes, incomprehensible is precisely what it is! But for Aquinas, incomprehensibility is not a justification for rejection; for Craig, it apparently is.
28. Peter van Inwagen, “God and Other Uncreated Things,” in Metaphysics and God: Essays in Honor of Eleanore Stump, ed. Kevin Timpe (New York: Routledge, 2009), 12.
29. See the literature cited (and briefly critiqued) in James E. Dolezal, “Trinity, Simplicity and the Status of God’s Personal Relations,” International Journal of Systematic Theology 16.1 (January 2014): 79-98.
30. See, for example, Menzel, “God and Mathematical Objects,” 69 (especially footnote 7), and Copan and Craig, Creation Out of Nothing, 170. Against such a move, recall Augustine’s famous exploration of God’s triune nature, in which it turns out that there is no “less” of God in one of the Persons than there is in the other two combined, or than there is in all three together (On the Trinity, viii.1.2). The application of standard mathematical categories to God is not something that Augustine would be likely to countenance.
31. To cite but one famous illustration, consider a well-known aphorism from Athanasius of Alexandria, the great fourth-century defender of orthodoxy: “The same things are said of the Son, which are said of the Father, except His being said to be Father” (Athanasius, Against the Arians III, xxiii, 4). On the one hand, the same things are said about the Father and the Son, for the Son is everything that the Father is: that is, the Son is divine in the fullest, most robust sense, fully equal to his Father as God (contra any form of Arian subordinationism). Yet on the other hand, the Son cannot be said to be the Father, for the Son is fully divine precisely as Son. The Father remains the fons divinitatis, the unique “font of deity” who is the source or cause of the divine nature in all the Godhead; the fully divine Son is fully divine by being the one and only Son of this fully divine Father, hence by being asymmetrically dependent on the Father (contra any form of Sabellianism or modalism). For a fuller discussion, see Steven D. Boyer, “Articulating Order: Trinitarian Discourse in an Egalitarian Age,” Pro Ecclesia 18.3 (Summer 2009): 255-272.
32. Menzel, “God and Mathematical Objects,” 72n10.
33. John Calvin was battling against a similar teaching in his exploration of the Trinity in the Institutes (i.13). See the careful discussion in Brannon Ellis, Calvin, Classical Trinitarianism, and the Aseity of the Son (Oxford: Oxford University Press, 2012).
34. Christopher Menzel, “God and Mathematical Objects,” 92.
35. Ibid.
36. For example, see Copan and Craig, 171, and van Inwagen, 7-8.
37. Paul Ernest, Social Constructivism as a Philosophy of Mathematics (Albany, NY: State University of New York Press, 1998), 24. No doubt Ernest would apply this summary critique to explicitly Christian proposals for epistemology and ontology, too, but he chiefly interacts with what are usually thought of as the three primary “schools” in contemporary philosophy of mathematics, namely, Logicism, Formalism, and Constructivism (or Intuitionism). For a lucid account of all three of these options in both their strengths and their weaknesses, see Alexander George and Daniel J. Velleman, Philosophies of Mathematics (Malden, MA: Blackwell, 2002).
38. Ernest, 145. This appeal to social intersubjectivity is how Ernest addresses criticisms like those of Michael Dummett, who regards any Wittgensteinian approach to mathematics as a “full blooded conventionalism” in which, when we have evaluated a proposed proof, “we could have rejected the proof without having done any more violence to our concepts than is done by accepting it; in rejecting it we could have remained equally faithful to the concepts with which we started out” (Wittgenstein’s Philosophy of Mathematics, eds. P. Benacerraf and H. Putnam [Englewood Cliffs, NJ: Prentice-Hall,1964], 495, 497). For Ernest’s explicit and biting response to Dummett, see Social Constructivism, 75. Still, not all are convinced: see, for instance, Bonnie Gold’s critical review of Ernest in The American Mathematical Monthly 106.4 (Apr 1999): 373-380.
39. See Imre Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery (Cambridge: Cambridge University Press, 1976). Ernest admits (Social Constructivism, 106-109) that, according to some editors of Lakatos, the Hungarian thinker rejected his fallibilist philosophy of mathematics late in life and purged all reference to Hegel in his later work. Ernest argues that, despite the rejection of explicit reference to Hegel, the actual influence of the notion of Hegelian synthesis on Lakatos’ work is clear, and also that the concrete evidence that Lakatos abandoned fallibilism is weak.
40. Ernest, 83. The phrase that Ernest uses in quotation marks is taken from Wittgenstein, Remarks on the Foundations of Mathematics, rev. ed. (Cambridge, MA: MIT Press, 1978), 173.
41. In his “Nova” interview, Andrew Wiles recounts this process in detail—including the moment when a decisive problem with his initial proof was pointed out, a problem to which he had to devote a full year of additional work before stumbling on a solution. See http://www.pbs.org/wgbh/nova/transcripts/2414proof.html, at approximately three-quarters of the way through the transcript.
42. For a similar account of how a “dialectical” process is at work in science education, and how this is relevant to Christian faith, see Lesslie Newbigin, The Gospel in a Pluralist Society (Grand Rapids, MI: Eerdmans, 1989), ch. 4 (“Authority, Autonomy, and Tradition”).
43. Even if an individual decision to accept the rules of the mathematical language game is not purely subjective or strongly relativistic, an ultimate arbitrariness at the larger social level still obtains. Recall Wittgenstein’s comment that “not only the axioms but the whole of syntax is arbitrary” (Ernest, 77, quoting Wittgenstein, Remarks on the Foundations of Mathematics, 104).
44. Eves, 213-216.
45. Much of Chinese mathematics appears to be inductive rather than deductive, moving from specific examples to general principles. The goal was to teach rather than to address skepticism, and proofs were often pictorial. See Howell and Bradley, 56-61; and Eves, 211-219, for more detail about China’s rich mathematical history. For a more extended account of the encounter between Western and non-Western mathematics, see Frank J. Swetz and T. I. Kao, Was Pythagoras Chinese?: An Examination of Right Triangle Theory in Ancient China (University Park, PA: Penn State University Press, 1977), and George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, 3rd ed. (Princeton, NJ: Princeton University Press, 2011).
46. Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications in Pure and Applied Mathematics 13.1 (February 1960): 1-14 (available online at http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html). Wigner even uses the theologically loaded word “sovereign” to describe the role of mathematics in the formulation of the laws of physics (6).
47. Paul Davies, The Mind of God: The Scientific Basis for a Rational World (New York: Simon & Schuster, 1992), 151.
48. Gödel’s theorems showed that, in any axiomatic system sufficiently powerful for the development of arithmetic, either there will be statements that cannot be proven within the system or the system will contain contradictions. For a non-technical exploration, see George and Velleman, 173-208; or Raymond M. Smullyan, Gödel’s Incompleteness Theorems (Oxford: Oxford University Press, 1992). For the technically inclined, see Kurt Gödel, On Formally Undecidable Propositions of Principia Mathematica and Related Systems, trans. B. Meltzer (Minneola, NY: Dover Publications, 1992; repr. of New York: Basic Books, 1962).
49. This methodological reliance on the community or on “tradition” has characterized not just Catholic and Orthodox Christians, but also (perhaps surprisingly) the Protestant Reformers. As Christopher Hall notes: “The ideal of the autonomous interpreter can more easily be laid at the steps of the Enlightenment than the Reformation. Rather, reformers such as Luther and Calvin wisely considered the history, councils, creeds and tradition of the church … as a rich resource ignored only by the foolish or arrogant” (Learning Theology with the Church Fathers [Downers Grove, IL: InterVarsity Press, 2002], 13-14).
50. Menzel, “God and Mathematical Objects,” 65-67.
51. Menzel freely acknowledges that his approach is “modern,” though he characterizes it, with a delightfully alliterative flair, as “a ‘moderate’ modernist model of the subject matter of mathematics” (67, italics added), and he concludes his chapter by noting that some observers may think he has gone too far in a postmodern direction (97). Our concern in this essay has been, of course, just the reverse.
52. This classic mathematical puzzle involved the attempt to show that only four colors were needed in a map to ensure that no two adjacent regions would be of the same color. Despite the embarrassing misstep in the 1870s and 1880s, a proof of the theorem was finally developed in 1979—note: a full century later!—and that proof was built on the same basic principles employed by Kempe. For the original “proof,” see A. B. Kempe, “On the Geographical Problem of the Four Colours,” American Journal of Mathematics 2.3 (September 1879): 193-200. The whole story is told in Robin Wilson, Four Colours Suffice: How the Map Problem Was Solved (Princeton, NJ: Princeton University Press, 2002).
53. For a concise account of this whole dramatic story, see George and Velleman, ch. 2 (“Logicism,” 14-43) and ch. 6 (“Finitism,” 147-172). For the highly technical details, see the relevant materials in Jean van Heijenoort, ed., From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (Cambridge, MA: Harvard University Press, 2002).
54. Howell and Bradley, 356.
55. Ibid., 356-357, quoting Jim Jadrich, “Constructivism in Science and Math: Is It Christian?,” Christian Educators Journal 38.1 (1998): 18. The words in parentheses are included in brackets in Klanderman’s quotation.
56. Of course, the question of whether there can be a proof for the validity of proofs is very different from the question of whether the acceptance of proofs is rationally justified or warranted. We happily grant that acceptance is warranted, given the nature and origin of reason as Christianity construes it. See the compelling, sophisticated account of these matters by Alvin Plantinga in Warranted Christian Belief (Oxford: Oxford University Press, 2000), especially chapter 7, “Sin and its Cognitive Consequences” (199-240). But Plantinga rightly realizes that his (intentionally Christian) argument for the reliability of human faculties cannot justify a naive assumption about the certainty of reason’s verdicts. Instead, he expresses sympathy with “such Christian thinkers as Pascal, Kierkegaard, and Kuyper,” who explicitly recognize that “there aren’t any certain foundations of the sort Descartes sought” (436). In a quite poignant paragraph, Plantinga describes this position as one that requires “a certain epistemic hardihood.” Why? Because “there is, indeed, such a thing as truth; the stakes are, indeed, very high (it matters greatly whether you believe the truth); but there is no way to be sure that you have the truth; there is no sure and certain method of attaining truth by starting from beliefs about which you can’t be mistaken and moving infallibly to the rest of your beliefs. … This is life under uncertainty, life under epistemic risk and fallibility. … I realize I can be seriously, dreadfully, fatally wrong, and wrong about what it is enormously important to be right [about]. That is simply the human condition: my response must be finally, ‘Here I stand; this is the way the world looks to me’” (436-437). Plantinga is not succumbing here to postmodern relativism (note his line “there is, indeed, such a thing as truth,” a line that would make Ernest cringe), but he is trying to take seriously the ambivalence of creaturely knowing.
57. For Anselm’s classic expression of this theological notion (Credo ut intelligam), see Proslogion 1. For the same idea in Augustine, see his Sermons on the Gospel of John 29.6, which is rooted in Jesus’s words in John 7:17, “Anyone who chooses to do the will of God will find out whether my teaching comes from God or whether I speak on my own” (NIV).
58. This “dialectical” reading of knowledge is, of course, a commonplace in the history of Christian thought. Witness, for instance, the standard Thomist account of reality as existing “between the absolutely creative, inventive knowledge of God and the imitating, ‘informed’ knowledge of us humans” (Josef Pieper, An Anthology [San Francisco: Ignatius Press, 1989], 96, originally published in Unaustrinkbares Licht [Munich: Kösel-Verlag, 1963]); or the classic Reformed distinction between “archetypal” knowledge, which is knowledge as God alone possesses it, and “ectypal” knowledge, the derivative knowledge available to creatures (see Willem J. van Asselt, “The Fundamental Meaning of Theology: Archetypal and Ectypal Theology in Seventeenth-Century Reformed Thought,” Westminster Theological Journal 64 [2002]: 319-335); or even the experiential “give and take” that characterizes God’s interaction with Israel in the Judeo-Christian prophetic tradition.
59. Attention to etymology may help us to see the point here. The verb “exist” comes from Latin and Greek roots that mean “to stand,” along with a prefix (ex-) meaning “out” or “forth.” Something that “ex-ists” is something that “stands out” from a larger, indeterminate background, something that “steps forth,” that “emerges” or “arises” or “appears” (all of these terms can be used to translate the Latin existere), something that displays a specific, definable character alongside of and over against the specific, definable character of other things. If we picture a vast space that is entirely empty, we might say that we are imagining a universe in which nothing at all “exists.” And if we go on to picture something in that universe, then lo and behold, it has become a universe in which something does “exist.” This is what “existence” involves: it involves being a particular thing that “stands out” from the empty blankness of non-being. But, as we are constantly repeating, God is not “a particular thing,” but the uncreated Creator of things, and therefore God cannot “ex-ist.” He does not “stand out” from some larger, all-inclusive background: if anything, God is, so to speak, the “background” against which all created things stand out.
60. Thinkers who have used this language with the greatest care throughout Christian history have usually been very aware of this danger. Take Thomas Aquinas as an example. It is clear that God’s “existence” and God’s “being” are very important for Aquinas’s Christian ontology, and it would be easy to think, therefore, that these terms have been defined precisely enough by Aquinas to eliminate the need for apophatic correctives. After all, does he not provide us with five classic ways to prove God’s “existence”? Does he not argue that God’s essence (God’s “being”) is his “existence”? Well, yes and no. Yes, Aquinas certainly does use some of this language, and in that sense he does offer a ringing affirmation of (so to speak) the “reality” of God. On the other hand, contemporary scholarship has repeatedly shown just how circumspect and deliberate this affirmation always is—as evident in the introductory warning with which Aquinas prefaces his whole discussion of God in the Summa: “We cannot know what God is, but only what He is not” (Summa Theologica 1a.3, preface [already quoted in footnote 19 above]). We might say that Aquinas’s ringing affirmations can ring so loudly precisely because they are so intentionally rooted in apophatic mystery. For a concise account of this aspect of Aquinas’s thought, see Placher, The Domestication of Transcendence, chapter 2 (“Aquinas on the Unknowable God”). For more detailed treatment, see Josef Pieper, The Silence of St. Thomas (South Bend, IN: St. Augustine’s Press, 1999); David B. Burrell, Aquinas: God and Action (Notre Dame, IN: University of Notre Dame Press, 1979); and Gregory P. Rocca, Speaking the Incomprehensible God: Thomas Aquinas on the Interplay of Positive and Negative Theology (Washington: The Catholic University of America Press, 2008).
61. See, among many others, James K. A. Smith, Who’s Afraid of Postmodernism?: Taking Derrida, Lyotard, and Foucault to Church (Grand Rapids, MI: Baker Academic, 2006); Merold Westphal, Suspicion and Faith: The Religious Uses of Modern Atheism (New York: Fordham University Press, 1998); Calvin O. Schrag, God as Otherwise Than Being: Toward a Semantics of the Gift (Evanston, IL: Northwestern University Press, 2002); John D. Caputo, What Would Jesus Deconstruct?: The Good News of Postmodernism for the Church (Grand Rapids, MI: Baker Academic, 2007); Donna Freitas and Jason King, Killing the Impostor God: Philip Pullman’s Spiritual Imagination in His Dark Materials (San Francisco: Jossey-Bass, 2007).
62. This overarching logic is pretty clear in the chapter on “abstract objects” in Copan and Craig, especially on pages 173 and 195. See also the summary of the issue on Craig’s website, including some comments regarding his own “panic” when he first encountered what appeared to him to be this “absolutely decisive refutation of theism” (http://www.reasonablefaith.org/current-work-on-god-and-abstract-objects).
63. One gets a hopeful glimmer of this kind of move in Craig’s own engagement with the work of Menzel in Copan and Craig, 173-177 and 191-195. Though usually quite critical, Craig sometimes seems to allow that Menzel’s thought is on the right track, especially when it can be framed as a species of “conceptualism,” the tag that Craig often uses to describe Augustine’s doctrine. Of course, we have already argued that Menzel’s scheme is not very faithful to Augustine, but that is beside the point here. Craig concedes that this “Augustinian” approach can boast many virtues (even a potential rapprochement with the doctrine of simplicity!), but he then quietly adds (195) that it can do so only because it “does not differ essentially” from the nominalism that he has commended just a few pages earlier. In other words, for Craig, a “conceptualist” version of realism turns out to be acceptable precisely insofar as it shades off into a kind of anti-realism. Our claim here is that this sword cuts in the opposite direction, too: an anti-realist ontology is acceptable precisely insofar as it shades off into an authentic Augustinian realism.
64. Merold Westphal, “Onto-theology, Metanarrative, Perspectivism, and the Gospel,” in Christianity and the Postmodern Turn: Six Views, ed. Myron B. Penner (Grand Rapids, MI: Brazos Press, 2005), 150.