On faith and mathematics

The occasion for writing this piece is a talk I will give at Graduate Christian Fellowship (GCF) at the Center for Christian Study this Friday evening.

To put this in context, we will be having the Oxford mathematician John Lennox come to speak at UVA on February 20 about the relation between faith and science. As it turns out, there are several in the group at GCF who have surprisingly strong opinions about why mathematics is not one of the sciences. Be that as it may, the relation between math and science is undeniable, not only historically but sociologically. In particular, mathematics is as much a part of the modernist agenda of the so-called "New Atheists" as any of the sciences. The reason should not be surprising: both mathematicians and scientists are professional rationalists, basing all of their propositions on reason and facts. It behooves us, then, to think about mathematics in this context, and to think about it as Christians.

Many of the remarks I will make could just as easily apply to faith and reason more broadly. However, mathematics is in many ways unique, and I make no apologies about speaking about it very specifically. If nothing else, it will be good to raise awareness about a field of human endeavor which many people, even educated people, seem to know so painfully little about.

Since the subject of this talk is the relation between faith and mathematics, I suppose I'd better start by addressing the question that is probably on everyone's mind: is there a relation between faith and mathematics at all? My sense is that many of us, include us Christians, do not see any relationship between that actual doing of mathematics and the actual believing in God or Jesus Christ. And I will concede, indeed I will insist, right from the outset, that there is a great deal of truth in this, more than some Christian intellectuals might like to admit. When mathematicians get together to decide whether a theorem has really been proved or not, matters of faith simply do not enter into the equation, so to speak. Any mathematician from any cultural background or religion will be forced by the same universal principles of deductive inference to acknowledge certain facts as true, certain other claims as refuted, and other claims as conjectures whose truth value is yet to be determined. There is simply no difference between a Christian and a non-christian at a mathematics conference. This much, as far as I am concerned, is indisputable fact.

And it is, indeed, quite a different story with many other fields. The assumptions about reality made by a psychologist might have a profound influence on how people in our society come to treat the human mind. Or the worldview of a sociologist might have a great deal to do with the conclusions he draws from so-called "data." It may indeed matter whether an economist believes in God, not just to his own spiritual existence, but in fact also to his work, for all economics boils down to certain assumptions about human nature. But it simply does not matter to a mathematician as a mathematician. He can prove his theorems and perform his calculations free from any theological or metaphysical controversies. "Interpreting the data" is simply not an issue in mathematics. When it becomes an issue, we are dealing not with mathematics per se, but with some other science.

If we were to end there, we would miss out on the extremely rich and beautiful relation between mathematics and the divine. It is worth mentioning right at the outset that mathematics has always had a rather privileged place in philosophy, not only because of its practical value but also because of its connection to transcendent truth. As Plato put it in The Republic:

Then this is knowledge of the kind for which we are seeking, having a double use, military and philosophical; for the soldier must learn the art of number or he will not know how to organise his army, and the philosopher also, because he has to rise out of the transient world and grasp reality, and therefore he must be able to calculate.

Today, of course, we have sadly too much of organizing armies, and too little of grasping reality. Perhaps this is not due to a lack of mathematics but to a lack of real mathematical education, in which students are taught to behold something beyond the abstract formalism in their computations.

This leads me to the main thesis of my talk. I do not want to say that mathematics leads us to God or that the real meaning of mathematics is only perceived in light of God. Rather, I merely wish to suggest two things: first, that mathematics can tell us something about God if we are willing to listen; and, second, that mathematics really is a worthy enterprise, not just for the specialist, but for anyone who is willing to stretch his mind toward the heavenly.

We've had so far this semester two talks on the attributes of God. Not that I feel any pressure to fit into that mold in this talk, but I did think about this theme as I was writing. We live in an age of what I would call "Christian sentimentalism," in which the main attributes of God are related to how caring he is, and how, in spite of life's turmoil, we always have a friend in Jesus. I don't wish to denigrate those truths about God, but only to point out that it is clearly not the whole story. The attribute of God I want to focus on for the moment is that of beauty--not, of course, of a sentimental kind, but of the kind to which mathematics is a window. Bertrand Russell, who was of course an atheist, had this to say about mathematics:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

If you have not had the pleasure of proving the Pythagorean theorem or the Fundamental Theorem of Calculus, then perhaps such a statement sounds astonishing. But there is indeed a supreme beauty in seeing the bare structure of the universe unveiled before you. Moreover I think this beauty in mathematics points to certain underplayed attributes of God: his wisdom, his immovability, and his often stern impartiality.

The idea that we touch on divine truth when we do mathematics is a very old one. (I recall reading some time ago in one of Augustine's works, I believe it was On Free Choice of the Will, that the most certain truth is the truth of number. After some searching, I was unable to verify this, but given Augustine's Platonist influence, it hardly seems implausible.) From Plato we inherit a tradition which posits that all real-world objects are but reflections of transcendent forms, to which we have direct access when we do mathematics. Generations of mathematicians, even if they weren't "Platonists" in a broad sense, have been what we will call "mathematical platonists," in that they agreed with Plato that mathematics possesses a certain ontological reality.

For many reasons, I would rather avoid the subject of ontology, but it is inevitable that it should come up in a discussion about mathematics and the Christian faith. There is no doubt that the biblical vision of the world includes the idea that all things participate in Christ and in a heavenly reality, and for much of the Christian tradition this has struck a Platonist chord. Let me say to begin with that there is a wide consensus on at least the practical matter of doing mathematics: all working mathematicians are practical platonists. To actually solve a problem working with abstract concepts, you have to believe implicitly that they really exist. Undoubtedly the worst mathematics students are those for whom the symbols on the page remain merely symbols on the page. Manipulating symbols according to the correct procedure can only be so enlightening. At some point a student really has to get it, by somehow gaining "direct access" to the needed concepts.

It is difficult for me to list example without delving into rather advanced subjects, but I will list them anyway. One does not prove anything about convex functions without first intuiting something about the canonical example f(x) = x^2, and immediately thinking about the parabola, that Platonic form of which all other convex functions seem merely shadows. In the theory of differential equations there is really only "one" linear differential equation, namely x' = Ax + f, whose properties are derived precisely from those of A and f, seen as objects in a highly abstract space. The entire field of topology is really a matter of classifying all sorts of spaces into several categories; there are, for instance, only so many smooth manifolds in one, two, and even three dimensions. One could make similar comments about group and ring structures in algebra, and so on. The point of this digression is to say that mathematicians are always striving for those true "forms," of which everything else is but a reflection.

The practical experience of mathematicians seems to make mathematical platonism so attractive that, when a mathematician is put into a debate on the subject with a philosopher, the mathematician will more likely side with platonism than the philosopher! (See the discussion on this topic in Mathematics through the Eyes of Faith.) But does that prove that platonism is true? That is, is there anything "real" about any of these beautifully abstract (and austere) concepts? Do we reach the heavenly realm through mathematics?

It's difficult to say yes. Let me explain by telling some history. Once upon a time Euclid's Elements was absolutely these seminal textbook in geometry. Anyone who wanted to be an educated person had to read it. Euclid's geometry was based on a set of axioms and postulates on which any reasonable person would surely agree. They were so "self-evident" as to negate the need for any proof. But one postulate seemed especially cumbersome, the so-called "parallel line" postulate. The simplest way to explain this postulate is this: two lines are defined to be "parallel" if ever transversal (that is, any third line passing through them) intersects the two lines at equal angles. (It helps if you draw a picture, but unfortunately I can't.) The parallel line postulate is what most of us learned in grade school to think of as the definition of parallel lines: they do not intersect. This fact was taken as so obvious that Euclid wrote it down as a postulate, but many mathematicians for centuries following him were not satisfied. They felt sure that one should be able to prove the parallel line postulate from the other axioms in Euclid's book.

However, that just wasn't true. It turns out that the parallel line postulate is what we call an independent axiom, meaning that both the postulate and the negation of the postulate are logically compatible with the other axioms in Euclid's geometry. The example which demonstrates this is both embarrassingly obvious and, at the same time, dramatically brilliant: just try to do geometry on a sphere. It turns out that parallel lines always intersect on a sphere, because of the way angles work. Just think of lines of longitude on a globe; these are all parallel, but they intersect at the north and south poles. If you insist on parallel lines being lines that do not intersect, then what you will get is that transversals no longer cut through parallel lines at equal angles. You can't have both in spherical geometries.

Sadly, this discovery meant that Euclid's timeless geometrical truths became relegated to a mere branch of geometry, namely "Euclidean geometry." (Euclid has the last laugh, of course, because the theory of manifolds is based precisely on the notion that locally every space should be Euclidean; this is the definition of a manifold. So perhaps "Euclidean space" really is the ultimately Platonic form after all. Or perhaps our brains are just natural more attuned to measurements using rectangular pieces.)

After such discoveries as non-euclidean geometry, it appeared that what we had always taken to be truly transcendent principles were really just inventions of our own minds, abstractions which had no necessary relation to the real world. Physical discoveries such as general relativity only confirmed this idea; the universe, it seems, is not necessarily a big three-dimensional space, but a four-dimensional manifold (if string theory has any merit, perhaps there is even more to the story). As a result of such discoveries as well as certain philosophical trends in the 19th century, by the turn of the 20th century many mathematicians, such as David Hilbert, were proposing an entirely non-platonic justification for the existence of mathematics. "Hilbert's program" was to justify mathematics entirely on the basis of its self-consistency. The symbols didn't have to mean anything, they just had to be used consistently according to certain rules. This formalism gave way to instrumentalism, meaning that mathematics, rather than pointing to any transcendent reality, could be used merely as a handmaiden of the sciences, describing natural phenomena in a rigorously quantitative way.

There was one problem with Hilbert's program, which was later to be discovered by Kurt Godel. The famous incompleteness theorem shows that no complete self-consistent axiomatic system could possibly exist (or at least none that contained enough axioms to make arithmetic possible). Thus in order for Hilbert's idea of a self-consistent system of symbols to work, mathematics would always have to contain propositions which could neither be proved true nor false. Without any outside reference point, there could be no way to decide whether such propositions should be true. Godel himself, as I understand it, was a mathematical platonist, but whether or not his theorem really necessitates platonism is a matter of considerable dispute.

There are a number of other reasons to be suspicious of even a moderate form of mathematical platonism. It has been noticed that many mathematical developments seem culturally relative. The Greeks did not seem to think 0 worthy of being called a number, but they were fascinated by prime numbers. Words like "irrational" and "imaginary" betray an obvious bias in our thinking about the meaning of numbers. Even while mathematics has seemed ultimately to transcend cultural assumptions, its development also seems to be tied to very human assumptions, which could have gone another way. Consider the following thought experiment from Sir Michael Atiyah:

"[L]et us imagine that intelligence had resided, not in mankind, but in some vast solitary and isolated jelly-fish, buried deep in the depths of the Pacific Ocean. It would have no experience of individual objects, only with the surrounding water. Motion, temperature and pressure would provide its basic sensory data. In such a pure continuum the discrete would not arise and there would be nothing to count." (from Is God a Mathematician?)

Could a jellyfish-like creature ever do mathematics? It's something I've mused on before. What I do know for sure is that many of our seemingly timeless abstractions appear, upon inspection, to be rather tied to our neural circuitry, rather than to the heavenly realm of pure thought. This is something we must take seriously as we explore the relationship between mathematics and ultimate truth.

But despite all of these reasons to doubt, the mathematical platonist has a good deal of evidence to support his position. I'll discuss here two major themes. One is the mathematical encounter with the infinite; the other is what is famously discussed as the "unreasonable effectiveness of mathematics."

For the first theme, I highly recommend the book Naming Infinity, which tells the story of three Russian mathematicians from the early 20th century, whose faith led them not only toward deep mathematical discoveries but also to political persecution and martyrdom. (At this point I'll also mention Avril Pyman's Pavel Florensky: A Quiet Genius, a wonderful biography of this extraordinary man. Let me also recommend Everything and More: A Compact History of Infinity by David Foster Wallace.) The story of the infinite goes back as long as we have recorded mathematics. One can think of Zeno's paradox as motivation: how is it that the arrow ever actually reaches its target? The seemingly infinite divisibility of nature creates all sorts of puzzles.

And so does the infinite countability of things. I've heard that recent studies show children are amazingly receptive to the concept of infinity. Many of them from an early age recognize the principle that there is no biggest number; if I think I have such a number, I can always add one more and get an even bigger number. This principle is probably our earliest encounter with infinity. There are, of course, many metaphorical ways of understanding it, such as imagining a hallway that continues forever, or imagining looking down an infinite staircase. But really the concept of infinity is a statement about what we can't experience. We can't name a highest number; we can never count to infinity. That seems to be a good old-fashioned Aristotelian account of the infinite.

Enter Georg Cantor, a German mathematician of the nineteenth century who introduced an entirely new level of infinity. The best explanation I can think of is as follows. Instead of thinking about how many numbers there are, let's think about how many ways there are to count up toward infinity. We could go the very obvious and traditional route: 1,2,3,4,5,6,7,... Or we could go by even numbers: 2,4,6,8,10,12,... Or we could go by powers of 2: 1,2,4,8,16,32,64,... There are easily infinitely many ways we could do this. But here's the really jarring thing: there are more ways to do this than we could ever "count," even if we used all infinitely many counting numbers. This can be rigorously proved using set theory, pioneered by Cantor using axioms which would at first appear quite innocuous. However, his conclusions were not popular, and at first many mathematicians did not accept the notion of treating infinitely many things as a unified "set," especially given the absurd conclusion one must then draw about infinite degrees of infinity!

Indeed, set theory has its limitations. The famous Russell Paradox demonstrates that making up sets willy-nilly doesn't work, particularly if you allow infinite sets. The paradox goes like this: let S be the set containing all sets that don't contain themselves. Does S contain itself? If it does, then it doesn't, and if it doesn't, then it does. Clearly, the definition of S is meaningless. It doesn't even get the dignity of being an empty set. It just doesn't exist. For similar reasons, but more difficult to explain, there is no "set of all sets." This is also a corollary of Cantor's theorems about sets.

As Naming Infinity recounts, the French rationalists had sufficient trouble with the puzzles and paradoxes of set theory that it actually caused progress to stagnate. Not so with the Russians: their connection with Eastern Orthodox mysticism ("name-worshiping" comes up more than once) inspired in them a belief in the ontological reality of the mathematical objects they studied. P.A. Nekrasov wrote of the Moscow Mathematical Society, contrasting it with the French and Petersburg schools:

While they ascribe great importance to facts, experiment, and the experimental sciences, the founders of the Mathematical Society are opponents of the slavish worship of facts by certain scholars. They were among the first to protest this enslavement of modern scientific thought and clearly explained the value of imagination and will equipped with the prerequisite objective and subjective (authoritative and nonauthoritative) world-views and the more or less exact theories that consciousness, living by its own pure process and internal experience, combines with the phenomena of external facts in motivating actions to be taken.

The authors of Naming Infinity explain that due to the best of their historical accounting, they can only conclude that this philosophical bent actually enabled the Russian school to resolve mathematical problems that remained a mystery among mathematicians schooled in the Western rationalist tradition.

(I will warn the readers of Naming Infinity that the authors seem to misunderstand the religious tradition of Florensky, Egorov, and Lusin. If one wishes to glimpse Florensky's theology, I recommend the biography of Pyman as well as Florensky's own works, particularly The Pillar and Ground of the Truth.)

There is far more to say about this episode in history than could ever be said here. I only wish to point out how profound the experience of mathematics is in connection with the divine. While I think it is more than a bit dangerous to ascribe mathematical descriptions to God (such as the "set of all sets" or some such nonsense), I also think it is fitting to ascribe to God that kind of infinity which is simply inaccessible to the human mind. There exists, as Cantor proved, an infinite hierarchy of infinities--and God surpasses all of this. What we touch upon through the exploration of the infinite is but a taste of that truly sublime attribute of God, which is his supreme impassibility.

Now there is a second theme which also inspires the mathematical platonist, namely the "unreasonable effectiveness of mathematics." This is based on a famous essay by Eugene Wigner, about the remarkable way in which mathematics tells us something about the physical world. Now, it is not totally surprising that the universe can be described quantitatively. The real reason the effectiveness of mathematics is "unreasonable" is that we seem to get so much more out of it than we put in. This really is the profound discovery of the scientific revolution: a simple mathematical rule can describe not merely what is happening, but why. The fact that the rule is simple means that we can explain the world in terms of concise laws, from which we can deduce facts about nature which we can then verify through observation.

As Einstein said, the most incomprehensible thing about the universe is that it is comprehensible.

Remarkable as it is, I've heard scientists get up and say how silly Einstein was for saying this. The most common argument that I hear is evolutionary: the suggestion is that it is not surprising for us to be in tune with symmetries of the universe because we are natural products of those symmetries. This argument, in my opinion, misses the point. It is not remarkable that we are parts of this universe, such as it is. What is remarkable is that the universe is the way it is. I can't help but feel that it is a bit perverse not to have what Einstein called that "cosmic religious feeling," the overwhelming sense of awe one feels at the breathtaking spectacle of order. The world is not chaos. If you look closely, everywhere you find the same law universally obeyed. This law is recovered through mathematics, and from it we may deduce the behavior of everything from the stars down to the smallest atom, if we are clever enough. Even if we grant that our mathematics will never precisely describe the order of the universe as it actually is, the fact that such a project is successful at all is a testament to an intrinsic structure--indeed, an austere beauty--in creation.

I admit, however, that this creates a bit of a conundrum for the Christian. If this divine law draws us to worship God, what, then of miraculous intervention? That is a real question, one I don't believe I'll be able to answer any time soon. You see the problem: the very same principle that fills a person with awe also seems to deny any possibility of anything like the Christian God. If all is ordered according to one universal law--and I would not be the only person to suggest there might indeed be one law--a so-called "unified theory of everything"--then what are we to make of the radical working of God's grace? Perhaps we are simply to leave it at that: his grace is radical. It is beyond the natural order of the universe, which is itself good. Perhaps, as Florensky suggested, we downplay the discontinuity of God's relationship with the universe to our own peril. Continuity and symmetry are beautiful, but perhaps they do not tell the whole story. I leave it to the listener, and indeed the reader, to decide.

Whatever the philosophical answers to these riddles may be, there is no denying the power of these experiences--beholding the infinite in the mind's eye, and beholding the intrinsic order of the universe--to evoke a sense of the divine, and to inspire worship in the believer's heart. It is enough that we acknowledge this power, without taking a firm platonist position on the ontological question. For my part, I will admit that I am no mathematical platonist. Mathematics seems to be a construct of human minds that have learned to follow certain patterns of thought, evolving much the same way language does. Its symbols do not point to heavenly realities, although they may indeed illuminate physical realities. That is not to say there is no real truth in mathematics--far from it. Mathematical theorems are irrefutable precisely because mathematical language must be spoken only with strict adherence to a certain pattern of thought, and this pattern necessitates certain conclusions, just as a piece of music necessitates a certain style of play from a musician. As Florensky said in a letter to his daughter from prison,

In mathematics try not just to memorise what to do and how but take it in gradually, bit by bit, as though it were a new piece of music. Mathematics should not be a burden laid on you from without, but a habit of thought.

As I see it, mathematics is a very human activity, perhaps one of the most human of all. We humans love to play games and enact rituals. You would not think that we would enjoy submitting ourselves to contrived rules of behavior, but in fact that is exactly what we do all the time. A mathematician is the ultimate example of this. He cannot always claim a perfect correspondence between his habit of thinking and the way the world really is, but such habits of thought as his have been so wildly successful in aiding human beings in our understanding of the world that the tradition surely will not die any time soon. Now because mathematics is a deeply human activity, and humans are made in the image of God, I believe in that way it does bring us closer to understanding God himself--not by direct access, but only through reflection. Is God a Mathematician? I don't think God needs the help, to be quite honest.

Here are some ways I think mathematics does not point to God. (Unfortunately, you can find these examples in two books which I would otherwise recommend, namely Beauty for Truth's Sake and Mathematics through the Eyes of Faith.) I don't put much stock in delightful constants such as the golden mean or the number 10. I don't put much stock in brilliant equations such as Euler's identity--though I will qualify that by saying it really should warm your heart, that is not the kind of pure, austere beauty that I ultimately see in mathematics. I certainly don't put any stock in mathematical explanations of Christian doctrine--they usually end up being heresies. I had a brief exchange with Peter Leithart about "mathematical modalism" once. Rest assured, mathematics is no way to explain the Trinity. (See, however, Florensky's exposition in The Pillar and Ground of the Truth.) If we can just avoid these pitfalls, then I think we still have a powerful argument that mathematics helps us to witness a small piece of the glory of God.

So much for the first part of my thesis. It would take me ages, I think, to really fully explain what mathematics can tell us about God and the world we live in, but I hope even this cursory explanation has been valuable. I will now move on to briefly talk about the second part of my thesis, which is that mathematics is a worthy enterprise for any human being, because it has a profound way of shaping the mind and the soul. It does this in two ways, I think. First, mathematics makes us more attuned to the truly universal, i.e. to the theoretical principles that bind together all the particulars. Second, mathematics makes us more skeptical, training us in a certain level of rigor that will not accept flimsy arguments. In some respects these two ways reinforce one another, while in others they are actually in tension. But whether through consonance or dissonance both of these influences have a way of making us truly free creatures. As Georg Cantor said,

The essence of mathematics lies in its freedom.

Perhaps nothing needs to be said here about the way in which mathematics directs us toward the universal. But let me say a few words about skepticism. It is not surprising to me that most mathematicians are atheists (and all the evidence I've seen suggests they are). Skeptics in our culture tend to be atheists, for many reasons. However, if we are concerned about the souls of skeptics, it will do no good to morally oppose skepticism. After all, skepticism is to some degree the marker of an advance civilization. It is a sign of amazing wealth and opportunity that we can afford the time and resources it takes to rigorously analyze the world around us with logic and scientific experimentation.

Moreover, skepticism can be pointed inwardly as much as outwardly, and in this way I firmly believe it becomes one of the highest moral virtues. One of the greatest contrasts between a mathematics class and a class in other disciplines is that you'll find far less "discussion" in a mathematics class. Our modern prejudice seems to be in favor of hearing out students' opinions in the hopes that discussion will become enlightening. Frankly, I rather admire the way in which mathematics (and many of the sciences) has a way of politely yet firmly assuring students that their opinion really doesn't matter. They must conform to the truth through hard work and self-discipline. As my advisor in fact put it once, "We must learn through suffering." Mathematics is submission, a form of dying to self. Only thoughts that pass the absolutely rigorous test of deductive logic are allowed to survive.

And finally, I believe that we need Christians in mathematics like Pavel Florensky, who are willing to challenge the philosophical presuppositions of the modern age. This passage from Naming Infinity says volumes about his character:

"Florensky was convinced that intellectually the nineteenth century, just ending, had been a disaster, and he wanted to identify and discredit what he saw as the 'governing principle' of its calamitous effects. He saw that principle in the concept of 'continuity,' the belief that one could not make the transition from one point to another without passing through all the intermediate points.

...

Florensky faulted his own field, mathematics, for creating this unfortunate monolith. Because of the strength of differential calculus, with its many practical applications, he maintained that mathematicians and philosophers tended to ignore those problems that could not be analyzed in this way--the essentially discontinuous phenomena. Only continuous functions were differentiable, so only those kinds of functions attracted attention... Differentiable functions were 'deterministic,' and emphasis on them led to what Florensky saw as an unhealthy determinism throughout political and philosophical thought in general, most clearly in Marxism."

It takes a certain kind of skepticism, combined with a habit of seeing universal principles underlying all things, to offer a powerful critique of cultural assumptions. Florensky's critique has indeed been vindicated by discoveries in twentieth century mathematics and physics. What else might new generations of Christian intellectuals have to say by gaining a broad view of their own disciplines and their connections to others?

As a mathematician and a Christian with many questions about life, I cannot pretend any of the answers I have given in this talk are really answers. I think the more important point is which questions we are open to asking. If I could leave my friends with one thought, it would be that mathematics might just have something to teach us about things that matter. This is not simply a matter of mathematics having "applications." It is a matter of mathematics being part of a broader vision of the universe, in which order and beauty actually matter, and in which we ought to glorify God with all our minds. I can only hope that my small contribution is a genuine step in the right direction.