“Unless a grain of wheat falls into the earth and dies, it remains alone; but if it dies, it bears much fruit.” – John 12:24
“The last enemy to be destroyed is death...Death is swallowed up in victory.” – 1 Corinthians 15:26, 54
"The greater the sin, the greater the mercy, the deeper the death and the brighter the rebirth.” - C. S. Lewis
"This story...has the very taste of primary truth." - J. R. R. Tolkien

Tuesday, May 13, 2008

A Theology of Mathematics

For my Physics Seminar class, I had the opportunity to write a paper on the nature of mathematics, a topic that has interested me for some time. I argue for a "realist" perspective of mathematics and for the view that number is inherent to the nature of God.

A Theology of Mathematics

1. Introduction

What are numbers? This may seem like a simple question, and perhaps it is, but is a very significant question. After all, what is the “number” 2? What is 3, or e, or 239? A simple and accurate definition is elusive, and perhaps it is impossible to convey verbally the full idea of numbers. In any case, we perceive in numbers something seemingly universal and essential – an underlying structure to reality. Various thinkers throughout history have perceived something powerfully real in mathematics, even describing a sublime beauty to the structure of numbers. In this paper I will discuss the nature of mathematics and the idea of mathematical beauty from a distinctly Christian perspective. I will argue that numbers are real entities, independent of physical reality, and that our perception of things beyond the physical world poses problems for a naturalistic worldview. Furthermore, our perception of mathematical beauty points to the necessary existence of a deeper Reality that is the ontological foundation for the physical world. This Reality has been viewed by many of the greatest thinkers as an omnipotent and transcendent Being, and this view provides the most plausible account of reality and of the existence of mathematics in particular.

In speaking of abstract mathematical truths or realities or entities, I wish to refer to the field of mathematics in general, and to numbers in particular, leaving open the possibility that more abstract concepts like sets are not real abstract entities but merely useful concepts. Professor of mathematics Reuben Hersh points out that mathematicians have invented various different formulations, theories, or definitions in order to more accurately describe reality or to solve certain problems (Hersh 74). In this paper, though, I am concerned primarily with numbers, which have always been understood intuitively rather than having been formulated or invented as useful concepts.

2. Epistemology and Ontology

In considering the nature of mathematics, it is essential that we understand the nature of human knowledge and how it relates to actual reality. First, as Ben Carter states in “The Limits of Mathematics in Assessing Causality,” “the world we perceive is distinct from the world as it is” (Carter 283). What is true and real is a separate issue from what we think or perceive or believe to be true or real. However, in the words of John Polkinghorne, “epistemology models ontology.” That is, although we cannot have exhaustive knowledge of reality, what we do perceive should be taken as a generally reliable guide to truth. Our basic reaction to our perceptions and/or intuitions, whether sensory or otherwise, should be one of trust rather than suspicion. Of course, we cannot know with 100% certainty that we perceive correctly in every case (for example, there is a finite chance that the physical world which we perceive to exist does not exist), and each incidence of perception or intuition should be considered probabilistically. That is, how confident am I that what I perceive or think to be true is true? The answer may vary, and we must judge carefully and try to identify biases in our thought or feeling, recognizing that we have “no grounds for supposing that, by using our virtual world as a standard, we can model the actual world in any genuinely exhaustive way” (Carter 280, emphasis added). In general, though, we can do have some knowledge of what is out there and can increase our level of knowledge by further consideration of reality – epistemology models ontology.

3. Do Numbers Exist?

The basic question in philosophy of mathematics is whether purely mathematical entities, and numbers in particular, are in some sense real or are only useful inventions of man. That is, do numbers, as abstract entities, really exist, or are they merely tools that are helpful in describing the physical world (for example, in saying that there three cows)? Is mathematics discovered or invented? It should be noted that the written equations themselves are merely representations of the mathematical concepts. Other representations are possible, but we are concerned with the mathematical idea to which the invented representations refer.

3.1. Philosophies of Mathematics

In his book Philosophy of Mathematics, professor of philosophy Stephen Barker lays out three main interpretations of abstract objects, and of numbers in particular: nominalism, which states that numbers have no ontological status independent of physical reality, conceptualism, which holds that numbers are merely inventions of the mind, and realism, which maintains that numbers exist independent of physical reality or human thought (Barker 69). The mainstream interpretation, taken by many of the greatest philosophers and mathematicians throughout history (Descartes, Kant, Russell, Gödel, etc.) has been the realist interpretation; “mathematics is superhuman – abstract, ideal, infallible, eternal” (Hersh 92). This view of mathematical truth as real, eternal, absolute, and almost divine has its roots in Pythagoras and Plato, and it was embraced by Augustine, perhaps the most influential Christian thinker, who stated that “this incorruptible truth of number is common to me and to any reasoning person whatsoever” (cited in Hersh 104). Despite the rise of naturalism and materialism in the last few centuries, most mathematicians continue to see abstract numbers as real (Howell and Bradley 3). The non-realist interpretations of mathematics are much younger in their origins and have been less popular throughout the history of mathematics.

3.2. Non-realist Approaches

Nominalism and conceptualism share a good deal of common ground with one another, subjecting number to dependence on mind and/or matter, whereas the mainstream approach of realism is strikingly different. Conceptualism is less far from realism, though, in granting the reality of abstract universal concepts within the mind (Holmes 4). Both perspectives, however, fail to provide an adequate account of reality, and for similar reasons. In general, the restriction of mathematics to dependence on either the mind or on matter invalidates too much of what we actually perceive in mathematics. Certainly the mathematician would say that 3410000 is no less valid a number than, say, 13; indeed, there are infinitely many valid natural numbers. But since the human mind is finite, it is at least difficult to imagine how it could “create” an infinity of numbers.1 One might object that “It’s not the infinite that our minds/brains generate, but notions of the infinite” (Hersh 75). But Barker argues that number theory implies an actually infinite number of successive natural numbers (Barker 70) – if our minds do not generate this infinity, but only notions of it, then “number theory cannot be given any thoroughly nominalistic interpretation under which it will come out literally true” (Barker 72). That number theory is false is hard to swallow, though, given (at the very least) the extensive usefulness of mathematics in describing the physical world. One might respond that we create a finite number of axioms, which in turn imply an infinity of numbers, but this answer still implies that our minds generate, at least indirectly, an infinity of numbers, and it is hard to fathom infinity coming from something finite. In distinguishing between actual infinity and notions of the infinite, Hersh makes a valid point which demonstrates the subtlety and complexity of the issue and lessens the tension for nominalism and conceptualism somewhat. Still, there is a definite tension here which reckons nominalism and conceptualism less plausible.

Furthermore, in claiming that number is real only as a creation of the mind, conceptualism fails to give a realistic (no pun intended) account of the practice of mathematics. First, it struggles to explain the universality of mathematics, as understood by humans, across time and space. Mathematicians in different cultures and throughout history have arrived at the same mathematical ideas and worked from the same intuitive understanding of numbers. For example, mathematicians in pre-modern China, ancient Greece, and medieval Islam arrived at many of the same theorems or equations despite the fact that their mathematical formulations or methods differed; there is a “universality of content” (Howell and Bradley 62) throughout different cultures and worldviews. Can this universality be explained if mathematics is a mere invention of the mind? If number is created in the mind, then why do mathematicians feel constrained to “create” only certain intuitively sensible axioms or definitions? One might reply that a “number sense” is built into humans genetically. While this may explain the cultural universality of mathematics to an extent, it is not a philosophical answer and offers no solution to the numerous other problems faced by non-realist interpretations. Furthermore, there still remains the fact of our strong intuitive understanding of mathematical truth, which is no less reliable if there are biological descriptions of how it exists.2 Mathematicians have always done their work and arrived at conclusions as if they are trying to discover an absolute reality that is already there (for example, in the finding of larger and larger prime numbers), rather than merely finding an efficient way to describe the world.

Intuitionism is a more recent interpretation, which Barker describes as a variation on conceptualism that states that mathematical truth is no more than what we intuitively sense to be true in our minds. In other words, the intuitionist restricts mathematical truth to the intuitively obvious. Possibilities such as Fermat’s last theorem and Goldbach’s conjecture, which are unproven but may well be true, Barker argues, are rejected by the intuitionist (Barker 76-77). This kind of approach, I would argue, flies in the face of logic and is incompatible with the philosophical distinction between epistemology and ontology made at the beginning of this paper. It is an essential logical premise that a given proposition is either true or false whether we know so or not.

1 Similarly, the physical universe is finite, but it is difficult to imagine the infinite range of numbers depending on a finite physical reality.
2 Our thoughts cannot be dismissed as unreliable because our brains may be affected by the physical world – this would cause us to doubt our perception of the physical world, which is the very premise of the train of thought. Evolutionary biologist J.B.S. Haldane wrote “If my mental processes are determined wholly by the motions of atoms in my brain, I have no reason to suppose my beliefs are true...and hence I have no reason for supposing my brain to be composed of atoms” (Byl 6).

3.3. Logicism

There are several other philosophical interpretations of mathematics which vary in the extent to which they are realist or non-realist. Here I will only briefly touch on one of them. Logicism is closer to realism, but is not necessarily in contradiction with non-realism. Formed primarily by Gottlob Frege and expanded by Bertrand Russell and A. N. Whitehead, all of whom were mathematical realists, logicism is the idea that mathematical concepts can be reduced to logic and deduced from the basic laws of logic. According to Barker, this idea was formulated be realists in an attempt to ground the reality of mathematics in the laws of logic (Barker 81). Although it is a more plausible theory than nominalism or conceptualism, logicism faces several problems. According to professor of mathematics Paul Zwier, “the additional axioms created by Russell and Whitehead…are widely held not to be purely logical in character” (Howell and Bradley 25). Even in Zermelo-Fraenkel Set Theory, the favored system for presenting mathematics, “there are axioms…generally acknowledged as not purely logical” (Howell and Bradley 29). Furthermore, Kurt Gödel’s incompleteness theorems demonstrated that it is impossible to formulate a set of axioms that are both consistent and give a complete description of mathematics. In other words, “mathematics ultimately rests on intuitions that cannot be proved” (Carter 280). Thus, it is highly doubtful that mathematical truth could be proven from logic alone, at least in any complete sense. Our intuition of mathematical truth then seems to be that of something irreducible and fundamental – related to logic yet not logically verifiable. It is all the more remarkable that we have such a strong “sense” of numbers.

3.4. Yes, Numbers Exist

Even if it requires further axioms beyond logic, there are numerous reasons to think of mathematical objects, and numbers in particular, as real abstract objects. One of the strongest reasons for realism is the fact of our strong intuitive perception of mathematical truth. When we consider the idea “2+3=5” we realize intuitively that it is a true statement. Similarly, we understand that “2+3=7” is a false statement, and this intuitive perception is of number in a general and abstract sense. If epistemology models ontology, as we have assumed, then it would seem our knowledge of pure numbers and of a distinction between true and false statements describing numbers indicates some sort reality for pure numbers.

In summary, realists are free to look at the infinite range of numbers in mathematics as a reality that exists independent of the restrictions of the finite physical world or finite human mind. The realist perspective also offers the best account of the culturally universal content of mathematical conclusions, the practical approach of mathematicians which aims to discover rather than invent, and of our intuitive understanding of mathematical truth. The most significant piece of evidence for the abstract reality of mathematics, though, is perhaps our perception of mathematical beauty, and it is this perception that we will consider next.

4. Mathematical Beauty

Further support for a realist interpretation of mathematics can be found in the “beauty” and “depth” perceived in mathematical laws and equations. Anyone who has studied mathematics in some depth perceives, at least to an extent, the elegance of mathematics and of the mathematical description of physical reality. For example, Bertrand Russell wrote that “mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere” (Russell 47). In stating “God made the integers; all else is the work of man” (Bell 477), Leopold Kronecker expressed a similar sense of awe towards numbers. What is it about simple numbers that brings about such striking statements? This perception of beauty and depth is quite remarkable (as is our perception of beauty and goodness more generally).

The natural numbers, and the prime numbers in particular, have been thought to be foundational and central to mathematical truth and beauty. Mathematicians today are still hard at work to discover a deeper structure behind the numerous mysterious patterns formed by prime numbers. In fractals we see how beautiful visual structures and order can be constructed, as it were, out of chaos. The “divine proportion” φ=(√5+1)/2 is the ratio for which the ratio of two quantities equals the ratio of the sum of the two to the larger of the two, but it is also the ratio approached by adjacent terms in the Fibonnaci sequence. This proportion has been found to be the built into the structure of many things in the physical world. These and other instances of mathematical beauty point to an underlying simplicity and interweaving of what appears on the surface to be complexity or randomness.

The so-called “imaginary” number i adds a whole new dimension to mathematics, revealing deeper connections and an elegance that suggests it may not be so imaginary after all. Perhaps the most “beautiful” and “deep” equation in mathematics is Euler’s identity, eiπ+1=0. If any equation touches upon the core of mathematical truth, this is it. In the words of Stanford mathematics professor Keith Devlin, Euler’s equation “reaches down into the very depths of existence” (Nahin 1). Nineteenth century mathematician Benjamin Pierce described it as “absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth” (Kasner and Newman 103–104). According to Charles Edward White, professor of Christian thought and history at Spring Arbor University, “a mathematics professor at MIT, an atheist, once wrote this formula on the blackboard, saying, ‘There is no God, but if there were, this formula would be proof of his existence’” (White 4). Although the professor apparently demands God’s existence as a condition for the proof thereof and thus implicitly uses circular reasoning, he nevertheless offers a striking statement. White goes on to explain that Euler’s equation brings together what are probably the five most important numbers in mathematics with the central operations of addition, multiplication, and exponentiation, and goes on to say that:

The idea that these two irrational numbers should combine with an imaginary one to yield so utilitarian a result is breathtaking…That these three strange numbers with such diverse origins should work together to produce a result so basic to mathematics argues that there is a profound elegance or beauty built into the system…It seemed to argue that there was a plan where no plan should be (White 4, emphasis added).
This unity and simplicity of the amazingly diverse and complex areas of mathematics calls to mind Colossians 1:17, “in him all things hold together.”

This perception of mathematical beauty is not limited to pure mathematics, but extends into the physical world, which is saturated with things that can be described mathematically. In fact, “Sophisticated theories, such as relativity or quantum mechanics, can be aptly summarized in just a few small mathematical equations and their logical implications” (Byl 4). Simple mathematical equations such as the Gaussian distribution arise again and again in descriptions of vastly different physical phenomena. John Polkinghorne writes “It is a technique of proven fruitfulness in fundamental physics to seek those theories whose expression is in terms of beautiful equations” (Polkinghorne 72). Indeed, the elegance of string theory is acknowledged as evidence in favor of it, regardless of what experimental evidence there may be. Because of these numerous discoveries of mathematical simplicity and elegance, physicists have “at least unconsciously…abandoned a raw naturalism in favor of a theory formation method that has principles of beauty embedded in its core” (Howell 502). Some mathematicians and physicists may be naturalists, but in giving so much weight to abstract principles like beauty, they run the risk of being philosophically inconsistent. In his celebrated paper, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Eugene Wigner describes and demonstrates the miracle of mathematics in the world, arguing that “the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language” (Wigner). Wigner connects this to man’s search for a deeper, “ultimate” truth.

Having looked at several specific instances of the “diverse unity” of mathematics and of the perception of this beauty among both mathematicians and physicists, let us consider what this may imply for our understanding of the nature of mathematics.

5. Beauty and Epistemology

When we examine the human reactions towards numbers and the intricate yet simple patterns and connections that they form, it becomes clear that these thinkers have indeed perceived something powerfully real and true. In considering the implications of this remarkable perception, and of knowledge in general, it is helpful to conceive of different probabilistic levels of epistemology. Obviously we are more certain of some proposed truths than of others. For example, we perceive a real physical world, yet we must acknowledge that there is a small but finite possibility that our sensory perceptions are incorrect, and there is no world. However, when we carefully consider seemingly more distant realities such as moral values or the beauty of music, intuition tells us that these values are absolute, powerfully real, and foundational to existence in a way that physical perceptions are not. We are more certain of their reality. In the same way, when we perceive the “beauty” and “depth” of mathematics, it is intuitively clear that we are seeing something deeply and absolutely real. That is, the beauty indicates reality in a more powerful way. When we consider mathematics, then, the magnitude of the beauty of number shows us that number is real (after all, only real things are beautiful). One might object that all this relies too much on emotion at the expense of rational thought. However, there is no reason to think that emotion does not indicate something important about reality, or that emotion must necessarily be at odds with rational thought. In the words of Blaise Pascal, “the heart has its reasons of which reason knows nothing.”

This perception is really quite remarkable; it is, in a sense, an observation of some component of reality beyond the physical world. We perceive these “very puzzling and mysterious” abstract realities (Barker 69) which cannot be located in space or time – and not only numbers, but also similarly abstract realities such as beauty and moral values. The human mind seems to touch on a realm of reality beyond the physical.

In summary, unless numbers are real, they cannot be described as elegant. There is, however, no good reason to doubt the reality of this perceived beauty and order any more than there is reason to doubt the reality of the world we perceive. If we hold to the premise that “epistemology models ontology,” which we all assume to be true in our everyday lives (that is, we live as if the world is real), then it would seem that the mainstream realist interpretation of mathematics is correct.

6. Can Reality Stand Without God?

Having come to a point of reasonable certainty that numbers are real, let us briefly consider the nature of reality in general. The fact that abstract numbers exist is truly remarkable when we think about it, and it causes us to wonder why this is the case. In fact, the existence of mathematics is a good reminder of the existence of anything at all. The fact that anything exists is, quite simply, shocking! The phenomenon of existence should make us step back in awe. Not only is there a reality, but it is intelligible to us! It is essential that we not take these things for granted, and that we strive to maintain a sense of awe at reality simply because it exists.

In his paper on “Theism and Mathematical Realism,” professor of mathematics John Byl describes the three ontological realms of abstract platonic truth, physical reality, and mind. Each “emerges” from the other in a way – the physical world is fundamentally mathematical, mind is tied to the material brain, and numbers and other abstract objects emerge as mental concepts. Byl argues that from a naturalistic framework, it is very difficult to find an ontologically satisfying account of the interactions of these realms of reality (Byl 5-6). An ontological foundation that accounts for all of these realms simply cannot be satisfactorily found within these realms, so these components of reality, in and of themselves, fail to give us a reason for reality to be here, and more specifically, for this specific, contingent reality.

We began by asking “are numbers real?” and eventually concluded that they are. Now the question is “why are numbers real?” More generally, why is there something rather than nothing? Here is not the place to consider in depth so significant a question, but let us briefly examine a theistic response to it. This question essentially reduces to asking, “what could be so extraordinarily and immeasurably great – so ontologically supreme and original – that it exists of necessity and exists simply (sustaining its own being)?” From what we have seen so far, logic or numbers might seem to stand as this ontological foundation for reality, but despite their beauty and depth numbers are merely abstract realities. Mathematics may or may not take us close to the rock-solid ontological foundation of reality, but it leaves us unsatisfied. Is that really it? Is there no more at the bottom of it all than real but abstract things like numbers or beauty? Such a foundation would be too weak and nebulous – to build reality on mere abstractions would be like building a house on water. The ontological foundation we are searching for must be of greater magnitude than this. Indeed, this quality of greatness can only be true of conscious being. That conscious being is at the foundation of reality also provides a satisfactory explanation for the existence of consciousness in humans. Following this line of thinking, it becomes clear that by far the most plausible candidate for an eternal, necessarily existent, and self-sustaining thing is God.

7. God in the History of Mathematics

But where does this leave us in our quest to understand the nature of mathematics? If it is not all the way at the bottom of reality, then where exactly is it in the grand scheme of things? Before considering this question in greater detail, let us look briefly at the role of spirituality and religious sentiments in the history of science and mathematics.

Throughout the history of mathematics, many of the greatest philosophers have understood mathematics to be at the foundation of existence and deeply connected with the spiritual and the divine. This view was central to the philosophy of the Pythagoreans and Platonists. Professor of philosophy and psychology Richard Tarnas writes that for the Pythagoreans,

[T]he mathematical patterns discoverable in the natural world secreted, as it were, a deeper meaning that led the philosopher beyond the material level of reality. To uncover the regulative mathematical forms in nature was to reveal the divine intelligence itself, governing its creation with transcendent perfection and order…there exists a deeper, timeless order of absolutes (cited in Hersh 93, 96).
This connection between mathematics and the divine has continued for millennia as a mainstream concept in philosophy of mathematics. Augustine commented on similar lines that “Even if it cannot be clear to us whether number is in wisdom or from wisdom, or wisdom itself is in number and from number, or whether it can be shown that they are names for one thing; it is certainly manifest that both are true and true immutably” (cited in Hersh 105). In fact, “Many contemporary philosophers of science perceive theism to be the basis for classical mathematics and mathematical realism, both of which are therefore found to be objectionable” (Byl 1).

Even in recent decades, as naturalism and atheism have become popular in the academic circle, various thinkers such as Stephen Hawking and Albert Einstein (despite their lack of belief in an omnipotent personal being) have described the physical world with religious language and personified the laws of physics and/or mathematics as God. Quarks are named “truth” and “beauty,” and the laws of physics and mathematics are thought of as divine in a pantheistic way. Even atheists such as Richard Dawkins express wonder at the natural world and stress the importance of “truth.” Clearly, the trend of connecting science and mathematics to the idea of God or to some deeper, mysteriously spiritual reality has been a prominent theme in scientific and mathematical thought throughout history.

8. God and Mathematics

In light of this religious view of mathematics or mathematical laws, how is the Christian idea of a personal God to be brought together with mathematical realism? There are two main questions to consider here. First, does mathematics as it is exist of necessity, or could it have been otherwise? Second, did God create mathematics as an external expression of his internal being and framework for reality, or is it inherent to his being? Here I will assume a very important connection between these two questions: there is no necessary reality apart from God. Thus, anything that exists of necessity must be inherent to God’s nature. Furthermore, there is nothing in the being of God that is not a necessary reality. Thus, contingent realities, although they are ordained by the will of God, are separate from his divine nature. It is also important to approach the nature of mathematics with the theological mindset that there is nothing external to God which exists independent of God; all things exist only because God upholds their very being.

8.1. God and Logic

In approaching this issue, it will be helpful to consider the nature of logic. Orthodox Christianity has held that God is rational, acting and existing in accordance with the laws of logic. This is really the only option. In all our thinking and reasoning we must assume logic to be an accurate description of reality. Like mathematics, it is foundational and universal. To say that God created the laws of logic as entities separate from his nature or chose to structure reality to work logically implies that these laws are not true of necessity (that is, it is not essentially and necessarily true that two contradictory realities cannot both be true). This, however, goes against the most basic intuitions of human thought. The only alternative is that God himself is logical (truth and falsehood are inherent to his being), and that the laws of logic exist of necessity because God, in all his diverse yet united perfections, exists of necessity – and God exists of necessity because of who he is. In the words of philosopher and theologian John Frame, “Does God, then, observe the law of non-contradiction? Not in the sense that this law is somehow higher than God himself. Rather, God is himself non-contradictory and is therefore himself the criterion of logical consistency and implication. Logic is an attribute of God, as are justice, mercy, wisdom, knowledge” (cited in Byl 7).

8.2. Numbers Are Divine

Could the same be said of mathematical truth? That is, might mathematics exist of necessity and thus be intrinsically linked to God? Without bringing God into the picture, there is good reason to believe numbers exist of necessity. One can hardly fathom a reality where the basic numbers 1, 2, 3, etc., are somehow different or nonexistent. Our considerations of mathematical beauty provide further support for this conclusion. Still, when number is considered in and of itself, there remains some uncertainty as to whether it exists of necessity.

When the Triune God is considered alongside mathematics, however, we see a very interesting connection. If God is one being in one sense yet also three persons in another sense, how can mathematics be said to be a creation of God? The reality of numbers must be inherent in the being of the Triune God. Otherwise, how can God eternal and self-existent be said to be triune? It seems, then, that number is part of who God is just as, for example, goodness is who God is. Goodness is not a thing unto itself independent of God. Rather, goodness is defined in the being of God: God is good. In the same way, mathematics has no ontological status independent of God. Rather, mathematics is defined in the nature and being of God: God is “mathematical.” In summarizing Aquinas’s theology, John Polkinghorne says “no distinctions are to be made within God’s nature, no separation of God from God’s goodness or from God’s reason, as if they were independent constituents of the divine nature” (Polkinghorne 67). This idea of numbers or mathematical truth as part of the divine logos is very similar to the mainstream interpretation of mathematics given by theistic or Pythagorean-minded philosophers. “Augustine argued that mathematics implied the existence of an eternal, necessary, infinite Mind in which all necessary truths exist” (emphasis added, Byl 2). Nearly two thousand years ago, Nicomachus of Gerasa, who followed in the Pythagorean tradition of connecting mathematics and the divine, expresses a very similar idea:

[A]rithmetic…existed before all the others…in the mind of the creating God like some universal and exemplary plan, relying upon which as a design and archetypal example the creator of the universe sets in order his material creations…the pattern was fixed, like a preliminary sketch, by the domination of number preexistent in the mind of the world-creating God, number conceptual only and immaterial in every way, but at the same time the true and eternal essence (cited in Hersh, 94).
More recently, Alvin Plantinga has espoused the similar view that mathematical truth is linked to God’s nature and eternally present in his mind (Byl 8). Plantinga, however, seems to separate necessary truths from God, saying that they are not in his control (Zwier 4). This idea of necessary truth outside of God is problematic, though, because “if abstract objects also exist a se [of necessity], then God’s uniqueness in the universe appears now to be compromised” (Howell and Bradley 70). However, if all necessary truth is intrinsic to God’s nature, as we have assumed, then God’s uniqueness and sovereignty as the ontological source of all reality is preserved. Furthermore, if something is eternally existent in the mind of God, it would seem that it is part of God’s nature to have that thing eternally present in his thought. That is, there is a natural progression from eternality in the mind of God to ontological reality in the being of God and aseity as part of the divine essence.

Professor of philosophy Christopher Menzel agrees that “the abstract objects that exist at any given moment, as products of God’s mental life, exist because God is thinking them,” (Howell and Bradley 73) and he agrees that abstract objects like numbers exist of necessity. If we accept that the Trinity exists of necessity, then number must exist of necessity because the Trinity implies number. However, Menzel argues that objects that exist necessarily need not be uncreated or ontologically independent: “God necessarily thinks them; that is, God creates them necessarily” (Howell and Bradley 73). While necessary existence may not strictly imply uncreated, ontologically independent status, things that exist of necessity must in a sense remain grounded ontologically in the uncreated and necessarily existent being of God, as part of his nature, and if they are part of God’s nature, then they are in some sense uncreated. This idea of the necessary creation of abstract objects like numbers is, in my mind, a possible but less likely description of reality; it is more natural to place such foundational, necessary objects fully within the divine nature than it is to separate them from God while at the same time holding that they are necessarily in God’s mind and necessarily existent.

The reality of numbers and mathematical relationships is best understood in this light – as part of the ontologically essential and foundational (yet incomprehensible) unity of the diverse perfections of the Trinity. Furthermore, the beauty of mathematics is best understood in this light, that is, as the beauty of God himself. Our God is such a God that in him all the diverse tapestry of mathematics is brought together into a coherent whole with a deep beauty and sublime majesty that takes our breath away. Not only so, but there is also a unity among all the attributes of God. Mathematical truth is intrinsically tied to each facet of God and to the whole, and there is a unity in the being of God that transcends the diversity of his nature. The beauty of unity amidst diversity in mathematics is part of a greater oneness of the vast and endless perfections of God, and of the unity of three distinct persons in one being.

Incidentally, it would seem that if number and logic are both essential to the nature of God, and there is unity in this nature, then logic and number are closely tied. Thus, although we may not be able to epistemologically reduce math to logic, there may be a deeper connection beyond our knowledge, and the logicists may have been on the right track in seeing a tie between number and logic.

8.3 Two Objections

One might object that if mathematics and logic are necessarily true or real, God is somehow constrained or limited. In response to this, it must first be noted that without its identity being ontologically grounded in the being of God, mathematical and logical truth would lose its source for existence along with all else. God is the great ontological rock at the foundation of reality; without his supreme ontological greatness, reality would have no source of being and would fall out of existence. Thus, mathematical realism loses it ground without theism. Hersh seems to realize this and writes that “The trouble with today’s Platonism is that it gives up God, but wants to keep mathematics a thought in the mind of God” (Hersh 135). Byl points out that the decline in theism in the 20th century led mathematicians to seek a foundation for mathematics in “self-evident axioms” (Byl 2), thus moving the search for a foundation for mathematics from ontology to epistemology, which, I would argue, is a fruitless endeavor. Numbers cannot exist as independent free-floating entities anymore than the justice of God exists as an abstract reality independent of God. Thus, mathematical truth cannot be separated from God. Each facet of the divine being of God only exists of necessity because the united whole of the being of God exists of necessity. Because neither mathematics nor logic itself have any reality external to God or independent of God, God is not bound or constrained in any way by logic or mathematical truth.

One might also object that sliding mathematics into the divine nature would put us on a slippery slope towards pantheism. However, other entities of beauty, such as the laws of physics, need not follow. It is only because of the absolutely universal and fundamental truth of mathematics that one would expect it to be true of necessity. Again, a Triune God cannot be conceived of without the number three. Indeed one could hardly conceive of it not existing of necessity (that is, numbers being created entities). Furthermore, we are comfortable believing that similarly abstract entities of beauty, such as absolute moral values, love, and goodness, are foundationally divine and find their grounds for existence in God. Numbers are comparable to these truths in beauty and necessity of existence and, therefore, are probably divine as well.

9. Epistemology and Revelation

If this is the nature of the being of God, what implications might there be for theological revelation and our knowledge of God? Barker asks of abstract objects, “If they are intangible, immaterial entities, how then can we have knowledge of them, and how can they be so important in our thinking?” (Barker 69). Perhaps we will never fully understand how abstract objects independent of mind and matter are made known to minds, which are linked to matter and yet are something more. However, we can, from a Christian perspective, say something of why God gave us this knowledge.

As Jonathan Edwards wrote in his great work The End for Which God Created the World, creation, in general, should be viewed in part as an overflowing of the fullness of the perfections of God, somewhat like a fountain (cited in Piper 165). Thus, the knowledge of mathematical truth and beauty in the minds of those made in the image of God is not only revelation of the divine nature to us, but also an overflowing of the nature of God into lesser creatures made in his image. Similarly, the mathematical structure of the physical world is the result of the action of a Creator who imbues all things with his likeness, in some way or another. Furthermore, in making the laws of physics and the patterns of this world so thoroughly mathematical, God is making himself known to his creatures and displaying his glory. He is saying to us, “look! – this is a small part of what I, your Maker, am like.” This revelatory understanding of epistemology and of creation as an expression of God’s character is confirmed in Scripture. In the words of the apostle Paul, “For since the creation of the world God’s invisible qualities, his eternal power and divine nature, have been clearly seen, being understood from what has been made” (Romans 1:20). Jonathan Edwards writes of the glory of God on a similar note:

The thing signified by the that name, the glory of God, when spoken of as the supreme and ultimate end of all God’s works, is the emanation and true external expression of God’s internal glory and fullness…The emanation or communication of the divine fullness…has relation to God as its fountain, as the thing communicated is something of its internal fullness…The refulgence shines upon and into the creature, and is reflected back to the luminary. The beams of glory come from God, are something of God, and are refunded back again to their original. So that the whole is of God, and in God, and to God; and he is the beginning, and the middle, and the end (cited in Piper 242, 247).

10. Conclusion

To sum up, we have examined the various philosophies of mathematics, concluding that realism offers, in general, the best account of observation and intuition. The perception of mathematical beauty provides strong support for this interpretation. Remarkably, we perceive real things beyond the physical world. Mathematical realism, however, is difficult to uphold without theism; more generally, the idea of God is by far the most plausible candidate for an ontological foundation to reality and an explanation of why anything exists. Indeed, mathematical realists throughout history have leaned towards theism and have thought of mathematical truth as almost divine. The foundational nature of mathematics, perception of mathematical beauty, and the triune nature of God suggest that mathematical truth exists of necessity in the being of God, as part of the diverse yet united perfections God. Thus, theism in turn supports realism by providing an ontological foundation. Finally, if numbers are divine rather than created, then our interest in mathematics and in the physical world as revelation of God should increase all the more. The theme of beauty and perfection in unity amidst diversity (and perhaps also of simplicity beneath complexity) is a key idea, both with respect to mathematics and with respect to God. As Ravi Zacharias has said, God is able to take the sublime and make it simple, and he is able to take the simple and make it sublime. I close with two excerpts from Scripture expressing the beauty and majesty of the revealed perfections of the divine nature.

And these are but the outer fringe of his works; how faint the whisper we hear of him! Who then can understand the thunder of his power? – Job 26:14

Oh, the depths of the riches of the wisdom and knowledge of God! How unsearchable his judgments, and his paths beyond tracing out!...For from him and through him and to him are all things. To him be the glory forever! Amen. – Romans 11:33, 36


Barker, Stephen. Philosophy of Mathematics. Prentice-Hall: Englewood Cliffs NJ, 1964.

Bell, Eric Temple. Men of Mathematics. Simon and Schuster: New York, 1986.

Byl, John. “Theism and Mathematical Realism.” Journal of the Association of Christians in the Mathematical Sciences (2001). <http://acmsonline.org/Byl-realism.pdf>.

Carter, Benjamin M. “The Limitations of Mathematics in Assessing Causality.” Perspectives on Science and Christian Faith 57.4 (2005): 279-283.

“Euler's Identity.” Wikipedia. 23 Apr. 2008. <http://en.wikipedia.org/wiki/Euler>.

Hersh, Reuben. What Is Mathematics, Really? Oxford University Press: New York, 1997.

Holmes, Arthur F. “Wanted: Christians Perspectives in the Philosophy of Mathematics.” Journal of the Association of Christians in the Mathematical Sciences (1997). <http://acmsonline.org/Holmes77.pdf>.

Howell, Russell W. “Does Mathematical Beauty Pose Problems for Naturalism?” Christian Scholar’s Review 35.4 (2006): 493-504.

Howell, Russell W., and W J. Bradley, eds. Mathematics in a Postmodern Age: A Christian Perspective. Grand Rapids: William B. Eerdmans Company, 2001.

Kasner, E., and Newman, J. Mathematics and the Imagination. Bell and Sons, 1949.

Piper, John. God’s Passion for His Glory. Crossway Books: Wheaton IL, 1998.

Polkinghorne, John. Science and Theology: An Introduction. Minneapolis: Fortress Press, 1998.

Nahin, Paul J., Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills. Princeton University Press: Princeton, 2006.

Russell, Bertrand. Mysticism and Logic. Dover Publications, 2004.

White, Charles E. “God by the Numbers: Coincidence and Random Mutation are Not the Most Likely Explanations for Some Things.” Christianity Today Mar. 2006. 13 Apr. 2008. <http://www.christianitytoday.com/ct/2006/march/26.44.html>.

Wigner, Eugene. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Communications in Pure and Applied Mathematics 13.1 (1960). <http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html>.

Zwier, Paul. “A Comparative Study of Christian Mathematical Realism and its Humanistic Alternatives.” Journal of the Association of Christians in the Mathematical Sciences (1997). <http://acmsonline.org/Zwier83.pdf>.


  1. Wow, I loved reading this! It really helped me formulate my thoughts as well since I have a presentation on truth in mathematics and theology for a course. Thanks for sharing!

  2. Mathematics is a human artefact, a human conception, in which there is no truth … there is no absolute unity; no absolute space and no absolute time, there is no mathematics.
    Gerrit Mannoury; (1867-1956); Quoted in Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer; Dirk van Dalen; 1999; p121

    Modern astronomers might agree with Kepler that the heavens declare the glory of God and the firmament showeth His handiwork; however, they now recognize that the mathematical interpretations of the works of God are their own creations …
    Morris Kline; (1908-1992); Mathematics and the Search for Knowledge; 1985; p85

    The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter.
    Carl G. Hempel; (1905-1997); On the Nature of Mathematical Truth; 1945


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