Starting from zero

I tend to think of the origin of all things as a tension between One and Zero, that is, between Something and Nothing, or between Existence and Nonexistence.

In a sense, I start counting from zero.

To get to one from zero is an infinite leap, an unprecedented creative act by which existence comes into being. To reach one is to declare that something exists, that counting is possible.

Zero is a place where one can naturally start and remain for all eternity. It is complete in itself--add zero to zero, multiply by zero, and you're right back where you started. To reach one is, in the greatest of ironies, to reject uniformity, to create a chasm between the utterly self-sufficient zero and the utterly insufficient unit.

Indeed, in creating such a chasm, it is clear that what has been created is not one but two--a choice, between something and nothing. But two is not nearly so remarkable at one. The chasm between zero and one is infinite. The chasm between one and two is one itself.

One is insufficient. If we add one again, we get two, then three, then four... We tumble off into infinity, and we find that really this one generates something that cannot be contained. The leap was indeed infinite, and we see an infinite set emerge because of that.

But something really strange happens when we try to close this system. If we can go out from zero by adding one, we also want to be able to go back toward zero, by subtraction. Suppose we reach zero and decide to continue subtracting. Then we create a mirror image of the natural numbers. Perhaps there's nothing strange about this yet, until we consider that now we have an infinite set extending symmetrically in two directions. As a result, zero becomes arbitrary. There is no reason to think of zero is the "true middle" of this infinite set, precisely because it is infinite. One can simply move to a different "origin" and the two infinite branches proceeding from it are still equal in length.

This is the nature of Euclidean space: it has no center. In other words, a center (very suggestively called "origin" in mathematics) can be chosen arbitrarily. By convention we call this point "zero," but by a change of coordinates any point can be zero.

In geometry, then, the meaning of zero becomes distorted. It is merely a reference point, having no particular identity for the purposes of ontology. Indeed, no point in a geometric space possesses any such identity. Geometry is about relationships, without any center. Everything is relative.

Yet the existence of space is absolute. Either a thing exists or it doesn't. There is an infinite chasm between existence and nonexistence. Geometry has no infinite chasms. In geometry, everything is ultimately rather close to everything else, in an absolute sense; only in a relative sense can something be close or far away.

Kronecker is supposed to have said that God made the integers, and that all else was the work of man. I suppose the intuition for this comes from the fundamental difference between the study of number and the study of measure. For the latter, we take existence for granted. The construction of the real number line is a matter of "filling in the gaps" between points which are already imagined to have some spatiotemporal existence. But the construction of the set of natural numbers is something else entirely. It bursts into existence from nothingness. To measure the difference between existence and nonexistence is meaningless.

As moderns we laugh at the quaintness of a geocentric view of the universe. But I think we should try to be aware of what we might have lost in shedding the innocence of that view. Our universe no longer has any center, or rather we can choose one arbitrarily. Once upon a time space was every bit as real as number. Now it is wholly relativized. We might as well measure everything only in relation to ourselves.

But there is a point of reference far more absolute than we realize. When we envision our universe as nothing more than a space-time continuum, talking of a (geometric) origin becomes meaningless. It is only when we reflect upon its existence or nonexistence that we realize the true center. The true "origin," to which we must compare everything, is nonexistence.

How is it that the universe bursts into existence? How is this infinite chasm bridged? This is the fundamental question. The distance that everything around us traveled to get where it is now is a pitifully small question compared to the fundamental one.

The center of the universe does not lie geometrically in the center of our world, underneath the ground below us. Rather, it lies ontologically in the fires of hell--that is, in nonexistence. The Bible describes judgment as a fire that is never quenched. That is because fire obliterates flesh, and eternal destruction is the return to the center of the universe--to utter nothingness. There can be no greater torture than this. The sheer contemplation of ceasing to exist terrifies me more than words can express.

To exist is always to be away from this center. There is an infinite chasm between heaven and hell. Heaven means eternal existence, where one continually marvels at the fact of being, where there is infinite joy because there are infinite possibilities. In hell there are no possibilities.

What will the redemption of all things look like? Will it mean an end to the story, the end of time? Yet to imagine an end is to cut off all these infinite possibilities distinguish existence from nonexistence. It is zero that stays fixed forever; one, by contrast, can't help but generate infinite sets beyond itself. Heaven cannot be a place of eternal inactivity. It is not a place where all stories end.

God creates out of nothing. Even if space has no center, even if time itself has no real beginning, nevertheless the creation is the most fundamental fact of the universe. If we lose sight of this, we become disoriented. When the universe stops becoming a gift and is rather a meaningless background on top of which our lives are arbitrarily thrown, it is because we have lost the center. The center of all existence is nonexistence. Christ descended into hell, so that all might be raised to heaven.

Aristotelian realism

An article I read in Aeon Magazine by James Franklin gives me a good springboard for some of my own thoughts about the philosophy of mathematics. The author (who has a book on the subject) essentially opposes two extreme positions, the one nominalist and the other Platonist. The nominalist seems to say that mathematics doesn't study any real objects; it is merely a language, a series of tautologies that has great instrumental value but has no content on its own. The Platonist says that, on the contrary, mathematical objects exist in their own realm, and that the human mind has access to that realm through contemplation and logical reasoning.

The problem with the first view is that to any mathematician, it seems fairly straightforward to assert that we actually discover something--not just logical relationships between symbols, but actual content. The problem with the second view is that the world of mathematical concepts seems remote; how can we physical beings have access to it?

The alternative is Aristotelian realism, which asserts that mathematical objects inhere in nature. Our minds have access to them initially through observation, then through abstraction and logical reasoning.

This alternative is very attractive for at least two reasons. One reason is that it makes sense of applications far more easily than either Platonism or nominalism. Why should mathematical models be so good at describing real world phenomena? Under the Platonist view, there's not much reason even to wonder about it, since mathematical objects are eternal and inherently separate from the contingent world we live in. Under the nominalist view, the puzzle is why a mere language would be so effective in discovering things about the universe before they are even observed (think about the mathematical development of general relativity). Realism has a simple explanation: we draw mathematical concepts out of the real world, so it's natural that we should use them to explain how it works.

Another reason is that it's satisfying from the point of view of a practicing mathematician. Platonism also has that trait, in that it elevates the objects of mathematical study themselves. But Aristotelian realism allows us to assert that mathematics has real content without divorcing it from common experience. I find this accords well with my own practice of mathematics, both in research and teaching. I always emphasize to my students that common sense should be the starting point for thinking about any mathematical problem. Of course we have to take a long journey out from that starting point, but ultimately each step is grounded in reasoning that any flesh and blood human being can understand.

For me there's a third, more theological reason to appreciate Aristotelian realism. Franklin alludes to theological import himself:

Aristotelian realism stands in a difficult relationship with naturalism, the project of showing that all of the world and human knowledge can be explained in terms of physics, biology and neuroscience. If mathematical properties are realised in the physical world and capable of being perceived, then mathematics can seem no more inexplicable than colour perception, which surely can be explained in naturalist terms. On the other hand, Aristotelians agree with Platonists that the mathematical grasp of necessities is mysterious. What is necessary is true in all possible worlds, but how can perception see into other possible worlds? The scholastics, the Aristotelian Catholic philosophers of the Middle Ages, were so impressed with the mind’s grasp of necessary truths as to conclude that the intellect was immaterial and immortal. If today’s naturalists do not wish to agree with that, there is a challenge for them. ‘Don’t tell me, show me’: build an artificial intelligence system that imitates genuine mathematical insight. There seem to be no promising plans on the drawing board.

This paragraph is delightfully provocative. I suspect many proponents of artifical intelligence believe they are not so far off as Franklin believes, but I can neither confirm nor deny such claims. In any case, artificial intelligence is not what interests me most. Instead, I tend to fixate on this question, "What is necessary is true in all possible worlds, but how can perception see into other possible worlds?"

To me the advantage Aristotelian realism has over Platonism is that it lets us see the eternal, even the sacred, in all things. Whereas the Platonist sees objects in the world as mere shadows on the wall, as it were, the Aristotelian sees them as sources of truth in themselves. For this reason I think Aristotelianism can affirm creation in a way that Platonism can't.

It is common for applied mathematicians to point out that their models are only approximations of reality, and that real life, unlike beautiful mathematical theories, is "messy." And I think that both for the nominalist and the Platonist, there is a sense in which one must choose between the beautiful realm of theory and the messy realm of facts. I reject this dualism by taking the radical position that eternal, necessary truths are inherent in real objects. I do not thereby deny the contingency of the universe; of course it could have been different from the way it is. Yet every object reveals necessary truths; paradoxically, we find the infinite and the eternal in the finite and temporary.

To put it in starkly theological terms, I would compare Platonism to gnosticism and nominalism to idolatry. The one would have discovery be a way of escaping the created order; the other would have discovery be entirely about finite, contingent reality. Instead, I think discovery involves an interlocking of the temporal and the eternal. From real world objects we discover eternal, necessary truths; in return, we can use these eternal truths to understand--and also care for--the world we inhabit.

Indeed, is it not the mystery of whether physical laws are truly necessary that drives so much of theoretical physics? One encounters mathematical relations between objects with fundamental constants which can be measured empirically, and it is natural to wonder whether such constants could actually be deduced from some deeper principle. Or whether the laws of physics themselves are actually corollaries of some more fundamental Law. Could the universe have "come into being" through some means other than what we call the "big bang"? Such questions magnify the interlocking of the eternal and the temporal, the necessary and the contingent. God's glory shines in all things, to such an extent that it is difficult to see where his invisible glory ends and the more visible nature of things begins.

As a corollary, I see mathematics not so much as a way of escaping into abstract truths in a higher realm, nor as a mere tool of the sciences, but rather as a humble servant of empirical investigation. We study mathematics not only to understand what the world is like but also how it must be, and in that sense it gives some of the deepest insight of any science. Yet the inspiration for its progress is not so much a desire to ascend toward heaven as to see the heavenly on earth. Whose heart can be so cold as to resist finding the beauty in Euler's formula? Yet if we never saw such things as oscillations in common experience, I'm sure we never would have seen such a beautiful equation.

Creation story

In the beginning, there was One.

All was One, and One was all, One alone.

No one was with One, and One was with No one.

No One loved One, because One was alone. So No One was lonely, for All was One. Yet One was with No One, and One loved No One.

One was No One, for if All is One then there is only

1
one
1
iness.

And No One was One.

One beheld No One, as in a mirror, and No One beheld One. Their love was impossible, yet real, all at once. Because One was, One alone, One was not, that is, No One was. One begot No One.

There was One and No One, Two. And One and Two made Three.

Thus all the numbers were born.

For in the very ambiguity between existence and non-existence proceeds a relationship, that is, "two-ness" or duality. But the duality is actually a third, that is, a bond between two. Once this bond proceeds from the first two, which are really One, then all other relationships extend outward to infinity.

One and One are Two, and One and Two are Three. One and Three make Four, and then Five, Six, and Seven.

And so it goes, forever and ever.

A humble science

Mathematics is the lowest and humblest of all the sciences. It makes slow but steady progress, and it says little to nothing about the questions dearest to the hearts of human beings, but what it says is absolutely certain. What gives it such certainty? It is content to study objects which are wholly abstract, so that the relations between them are absolute, eternal, and necessary. The word abstract implies that they are drawn away from reality. In this way they are lifeless. Rather than exploring strange new worlds with their own vibrant existence, the mathematician is content to study lifeless forms which are absolutely transparent to the mind. Let the brave adventurers go off to study living creatures, the cosmos, and that most mysterious object of all, human pyschology. The mathematician will humbly work at understanding that which is already closer to home than anything can be--that is, close to the mind.

It would be a fatal mistake, then, to view mathematics as a lofty venture, towering over all the other sciences, for that is the opposite of the truth. Mathematics, like anyone who wishes to be great in God's kingdom, must be the servant of all. We happily live in a world which follows certain patterns, often not at all obvious, and the objects we encounter, though they have in some sense a life of their own, can be seen to follow abstract rules which can be studied at the level of theory. In this way the applications of mathematics are abundant, starting with the simplest of all objects (physics) and working our way up even to the most complex and spontaneous (biology, economics).

But it is a slippery slope from application to assimilation. The problem with much modern thought is the way it treats all objects as abstract, lifeless forms. What mathematics so useful and reliable is that it studies objects which have no independent existence and no context. To study living things--and particularly the human mind--in such a way would be (and often is) disastrous. Psychologists have noticed, for example, that searching for universal principles governing human thought by observing only Western, educated, industrialized, rich, democratic people is a flawed idea. Yet in observing that our methods do not lead to what we are searching for, rarely do we ask whether the search itself is misguided.

Why does modern thought push relentlessly toward that which is universal? I suspect the reason is as much moral as scientific. On the one hand, science seeks that which is universal because it gives us a deeper understanding of life as a whole, and that allows us to do the greatest amount of good. On the other hand, we would also find it unfair if our particular context really mattered. The universe should be, above all, fair, even if that implies it is meaningless. History must be random, because otherwise that would imply something special about the way things happen to be. We have concluded, after thoroughly deconstructing the moral pretentions of past generations, that there can be nothing special about our heritage, culture, or anything else passed down to us.

As an aside, I highly suspect this attitude explains why physicists have come up with the idea of a multiverse. If there really is one universe, with exactly one history, that means a practically infinite number of possibilities are shut off forever from all reality. That would mean all of reality is dependent on a particular context--it is no longer held captive by abstract concepts. Such a conclusion is intolerable in our intellectual climate.

The modern reaction against the Judeo-Christian tradition can be explained in these terms. If there is a God, its existence should be explicable in rational, abstract terms that do not depend on context--that is, mathematically. There is a long tradition of such proofs in Western tradition. But that is not what we find in the Bible. What we find there is a God who, though he is supposed to be the creator of all things, has attached himself to a particular people in the Middle East. "I Am Who I Am," God says, affirming his utterly transcendent identity, and then adds soon after, "the God of Abraham, Isaac, and Jacob." How can all of reality find its source in a God so particular? How could one petty group of people out in the middle of nowhere happened to have stumbled onto the source of life and hope for all humanity, and indeed all the universe?

As much as it offends our sensibilities, we ought to be able to understand this. Life is a serious of decisions and commitments, each of which cuts off others which were at one time possible. Once you say a word, it will be forever true that you said it. Once you are married, it will be forever true that you decided to marry. Once you go to your grave, whatever you have done with your life is all that you have done. There is no going back. Indeed, we might perhaps better understand this than our ancestors, since because of the Internet, nearly everything we say in public will be forever recorded somewhere in this vast ocean of data.

Is it really so impossible to believe that God himself would make such commitments, and that those commitments would be the basis of all reality? In fact I myself find it very hard. If at one time in human development it was natural to anthropomorphize God, today it seems difficult to think of God as anything other than an abstract concept.

Theology isn't the only thing at stake. It is not just the living God whom we try to kill with our abstract thinking. It is anything living. We moderns are increasingly detached from our own history, living in the dream that we can transcend history, which was random and arbitrary up to exactly our generation, and then build the future on abstract principles from here on. Naturally, we refuse to believe that this dream came from anywhere other than our own reason.

To be sure, looking at the abstract principles behind living realities is a good thing. It helps us to simplify problems and find solutions agreeable to everyone. It can even help us know better what we observe, so that we can appreciate it all the more. (I find this especially true of music, and I suspect it is true of art. For me, music theory makes great works come to life even more than they already do.)

The problem comes when we confuse abstract principles with true or ultimate reality. That is the path to self-destructive rationalism--it empties the world of all meaning, it jettisons history as a source of knowledge, and it risks degrading civilization itself, which is built on living traditions. No, reality is not a set of equations. It is a living, spontaneous, external universe which imposes its particularity on us. I say this not to denigrate my own field, but merely to put it in its rightful place as a loving servant of the real.

Aubin, Bayen, and Saint-Pierre on the science of complexity

I'm trying to teach myself viability theory using Aubin, Bayen, and Saint-Pierre's tome bearing that title. In the  part of the introduction outlining applications and motivations of the theory, the reader stumbles upon a rather remarkable philosophical challenge: to radically change the current mathematical paradigm in order to better understand complex systems. The relationship between mathematics and the physical sciences is well established. We have developed mathematical tools to study the simple pieces of nature, particularly waves and particles in motion. But what about living systems or, more complex still, societies? "Simplifying complexity," say the authors (emphasis in the original), should be the purpose of an emerging science of complexity, if such a science will emerge beyond its present fashionable status."

That's an intriguing challenge. What do they have in mind? The following paragraph needs a full quote.

Quoting from Section 1.1, p. 8:

So physics, which could be defined as the part of the cultural and physical environment which is understandable by mathematical metaphors, has not yet, in our opinion, encapsulated the mathematical metaphors of living systems, from organic molecules to social systems, made of human brains controlling social activities. The reason seems to be that the adequate mathematical tongue does not yet exist. And the challenge is that before creating it, the present one has to be forgotten, de-constructed. This is quite impossible because mathematicians have been educated in the same way all over the world, depriving mathematics from the Darwinian evolution which has operated on languages. This uniformity is the strength and the weakeness of present day mathematics: its universality is partial. The only possibility to mathematically perceive living systems would remain a dream: to gather in secluded convents young children with good mathematical capability, but little training in the present mathematics, under the supervision or guidance of economists or biologists without mathematical training. They possibly could come up with new mathematical languages unknown to us providing the long expected unreasonable effectiveness of mathematics in the social and biological sciences.

What's awfully strange about this, aside from the desire to hide children away in convents, is the implicit belief that somehow mathematics must be capable of describing social and biological phenomena in a direct way as in physics. The reader is left with the sense that he has just read not so much a discussion of the state of the art in mathematics as an eschatological vision in which mathematics will conquer all of the corners of reality. It seems rather utopian.

Indeed, the next paragraph reveals, in my opinion, why this utopianism ultimately fails. "Even the concept of natural number is oversimplifying," say the authors, "by putting a same equivalence class so [sic] several different sets, erasing thier qualitative properties or hiding them behind their quantitative ones." This is an issue I've mused on myself: is there a way to "break away" from the discrete number system which gave birth to our mathematics? There are subtle issues to be dealt with, here, but I think the way the authors have phrased it leads quickly to absurdity: if what we care about is the qualitative and not the quantitative, it seems safe to say we're no longer doing mathematics. We might as well call it--ahem--biology and economics.

Though my initial reaction is critical, I don't mean to be overly negative. On the contrary, I think the vision expressed here is important. Mathematics should remain in the tradition of seeking to clarify and extend our understanding of reality. Yet I humbly suggest that this will not require children raised by economists. It will require, as the authors suggest, a great deal of trial and error--Darwinian evolution, if you like. Ultimately, however, I think mathematics is the science of quantity, and to the extent that it describes qualitative properties, it does so by mapping them to numbers. I am not saying that the concept of number is self-evident and unchanging. History proves that it isn't. But we need not rely on some forced revolution in mathematical training in order to reimagine (and sometimes redefine) the concept of number.

Progress in the direction of mathematical life/social sciences already seems underway. The very text I am citing here is evidence of that. Another example I would point to is mean field game theory, with which the authors are surely familiar. The insight that the behavior of a multitude of rational beings can be understood quantitatively in the aggregate appears to be a profound boost to the project of mathematical social sciences, in particular. Based on my general sense of the field (from a young perspective), I would like to believe that the current interest in biological, economic, and social questions in mathematics is more than a passing fad. Over time, we may think of mathematics as intrinsically tied to these subjects just as it is to physics.

But all things in their place. Mathematics may be helpful in addressing a plethora of questions, but I very much doubt it is without limits (no pun intended).

Edward Rothstein on musical expression

Here's a provocative passage from Rothstein's Emblems of Mind, Chapter III, challenging certain contemporary presuppositions about the purpose of music:

But does music really represent feelings in this way, either by reflecting our own or by expressing the composer's? Is it emotion we feel when we listen--anger or sadness or envy or desire? When music brings tears to our eyes, is it because it makes us sad? Some music unquestionably does stir or inspire us; that is the purpose, after all, of national anthems and masses and even some folk songs. Some music also prompts unexpected emotion and thought. But this view of music's purpose is far too limited. The Indian raga serves the function of neither pleasure nor expression, nor does most of the great music of our Western tradition--even the Romantic music that claims to be fundamentally self-expressive. Only products of the pop culture industry unambiguously aim to inspire identification with musical "expression," and seek the avid consumption of such expression through purchase or use.

What Rothstein is trying to do for music here mirrors what he has attempted to do (in Chapter 2) for mathematics. He has previously used examples from modern mathematics to problematize the notion that mathematicians mechanically search for objective, universal, "external" truths. Here he is going the other direction: he is trying to dispel the notion that music is merely a subjective experience tied solely to one's personal feelings.

The strength of Rothstein's writing is the way in which he veils the "big ideas" he's getting at, hinting that they are too mysterious to be fully encapsulated in concrete definitions of terms. What does a mathematician really explore when he does mathematics? What does our mind experience when we listen to music? Somehow the two are linked, but it takes patience to really see what the link is, or what the answer to either question might be.

I haven't made it all the way to the end, but one gets the feeling that there's a big payoff waiting.

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.

The Unreasonable Effectiveness of Zen and the Art of Motorcycle Maintenance

So today, like the total nerd that I am, I actually read a couple of the articles which came out in the latest Notices of American Mathematical Society. One article that caught my attention was entitled, "A Perspective on Wigner's 'Unreasonable Effectiveness of Mathematics,'" which refers to a famous essay by Eugene Wigner on the mysterious way in which mathematics actually seems to tell us true things about the physical world. (As opposed to merely the world of abstract ideas. It would be unsurprising of mathematics told us something about that.) In the article, Jason Nicholson appeals to the philosophy of one Robert Pirsig, who wrote what is claimed to be one of the most widely read books of philosophy ever written, Zen and the Art of Motorcycle Maintenance. Nicholson seeks to bring Pirsig's "Metaphysics of Quality" into academic discussion, particularly among mathematicians interested in mathematical metaphysics.

Since I have never read Pirsig, let me just give a few excerpts from Nicholson's summary so that you get the main idea:

The Metaphysics of Quality is, in some limited sense, as follows. He had in his first book realized (and made the case) that Quality is an undefinable entity that is the precursor of subjects and objects; everyone knows what it is but no one can define it. He proceeds to understand that subjects and objects are only one dual pair of defined things into which the undefined Quality event gets split as it is “realized”—that is, made real through a necessarily incomplete attempt to define it. ...

In his second book, however, he is led to a different split into what he calls “static” and “dynamic” aspects of reality as the best split possible, the most useful. He actually terms them static quality (or value) and Dynamic Quality, and with them he builds his Metaphysics of Quality, a metaphysical framework that provides a different, and, he demonstrates, better way of understanding the world we live in. Dynamic Quality is the undefined Quality that was described in his first book, but now he introduces static patterns of quality alongside it to reflect the “realization” of that undefined Quality which makes up our world. They act like a ratchet: the Dynamic Quality is the constant stimulus to move to something “better”, to ratchet up, but the static quality is the latch of the ratchet itself, the making tangible of the motion up into something concrete which will prevent falling down into something “worse”. Dynamic Quality is the creative urge, whereas static quality, or patterns of static quality, is what is created in response.

In building his Metaphysics of Quality, Pirsig classifies patterns of static quality into four discrete yet interrelated levels: Inorganic, Biological, Social, and Intellectual. He describes the relationship between these levels as being analogous to the relationship of computer hardware to computer software—the software is run on the hardware, but has nothing, really, to do with it. The program that you run on your computer and write your article with has nothing to do with the computer hardware itself. Furthermore, the content of your article has nothing to do with the program you write it in. In this way the levels of static quality are related to each other: Biological is built on Inorganic, Social is built on Biological, and Intellectual is built on Social, but each level is independent of the other.

Using this idea, Pirsig makes the case that Darwinian evolution is just Dynamic Quality at work by understanding “survival of the fittest” as meaning the movement of static quality (survival) towards Dynamic Quality (fittest). Then the four levels of static quality are levels of evolution.

Nicholson then goes on to apply this metaphysics to the problem of "the unreasonable effectiveness of mathematics."

The key word in Wigner’s thesis is “unreasonable”; he actually hit on the solution to the problem in the title of his article. Since Dynamic Quality cannot be defined, it is by definition (so to speak) unreasonable. But that is the reason that any explanation of Wigner’s observation requires an expanded metaphysics. In our tacitly assumed subject-object metaphysics, as Pirsig makes clear, anything “unreasonable” is discarded, and so the effectiveness of mathematics in describing the natural world is an insoluble quandary. Once an “unreasonable” entity, Quality is seen as the root or precursor to all subjects and objects, the quandary fades. ...

The Metaphysics of Quality also easily solves another long-standing dilemma among mathematicians regarding the nature of their subject: the “is mathematics invented or discovered?” debate. The solution to this debate is reminiscent of the Metaphysics of Quality’s resolution of the “free will versus determinism” debate referred to above. Mathematics is invented insofar as it is a process of following Dynamic Quality—that is, insofar as it is “free”. It is discovered insofar as it is a process of fleshing out previously unknown consequences within the static patterns of quality that are mathematics as it stands. Most Ph.D. theses and much published mathematics are more of this latter type—original work, that is, new consequences of existing static patterns, but not in the sense of following only Dynamic Quality. In fact, one might say that any new development comes as a mixture of both types of originality; it lies on a continuum between purely static quality at one end and purely Dynamic Quality at the other. The most “creative” and “original” mathematics obviously sits toward the Dynamic Quality end of the spectrum.

As I understand this, it feels like simply an evolved, 20th century version of Platonism, with a dash of eastern religion added for flavor. And I'm entirely comfortable with that. The only thing I want to point out is how easily this Metaphysics of Quality fits into the idea of participation in the divine. Whereas the Platonist might say that concrete assertions in mathematics are reflections or shadows of a higher reality, Nicholson, drawing on Pirsig, is saying that mathematical ideas are "static patterns" emerging in response to Dynamic Quality. And whereas the Platonist might imagine this higher reality as unchanging while changeable things are merely shadows, Pirsig's philosophy flips that around and says that what is more real is changing, so that all of reality is cast in a Darwinian light.

Which leads me to the question: can we really escape the moral question about God's goodness? Whether God is the creator or simply the selector, whether he is the personal God of the Old Testament or the impersonal God of physics, it does seem like the process of discovery is tied up with this grand question: is it all worth it? In other words, if my intelligence is the result of a selection process that ultimately has no moral worth, why not rebel against it? Perhaps the answers we've been getting from science really are wrong on some fundamental level--not wrong in the sense that they have predictive power, but wrong in the sense that the universe is not worthy of our comprehension. Much like a young Christian who suddenly discovers he can't find it in himself to worship a God who banishes people to eternal torment in hell, maybe one day humanity will rebel against the very idea of knowledge, on the grounds that the universe is just too cruel or depressing to be worthy of our careful study.

Theodicy, it seems to me, is not merely a question for theists.

The mathematics of causality

The other day some of my fellow grad students got an e-mail forwarded from our professor. The e-mail was from a student in electrical engineering who wanted help with a system of nonlinear differential equations. In particular, the student wanted "the solution." Keep in mind that most differential equations don't work that way. When you take a class in differential equations, you always start with those basic examples that you can solve explicitly. Once you get out into the "real world," you quickly realize how futile it is to dream of finding an exact, closed-form solution. Our professor had to respond to this student by explaining something about the general theory of differential equations, to which the student replied with apparent dissatisfaction. She was only interested in "the answer." Of course, our professor is not planning on wasting any more time with this e-mail.

The incident highlights an important fact: very few people actually understand what I do. (Most days I don't even understand what I do, which is why we call it "research.") To me this is rather unfortunate, because I actually think what I do has a tremendous philosophical contribution to make to the sciences. Too often the contribution of mathematics is seen in purely utilitarian terms: we can model a "real-world" process in mathematical terms, thereby understanding it in rigorous quantitative terms. I think there's a lot more to it than that.

Concerning "Axioms and Inferences"

Here's the video I'm referring to:

Axioms and Inferences: A Mathematician Thinks About Faith from The Veritas Forum on Vimeo.

Here are my comments:

On the whole, I'm pretty underwhelmed by John Lennox's argument. Firstly, the vast majority of the argument could be summed up as "anything but atheism," which is hardly an argument for Christianity. I admit that some arguments concerning first principles can be interesting, but when it comes to religion I find them less and less so over time. When your entire argument for Christianity seems to hinge on the meaning of the word "faith" in the English language, there appears to be something missing. True, mathematicians don't primarily deal in empirical matters, but rather in matters of logic. Yet is it too much to ask that a mathematician who identifies as a Christian also be held responsible for the pressing empirical questions on which the whole of Christianity is based?

Second, I find this attack on atheism using "simple logic" very glib, and quite probably uncharitable. Consider the quote from Bertrand Russell near the beginning of the talk: "What science cannot tell us, mankind cannot know." Lennox quickly dismantles this statement by saying, well, this isn't a statement of science, so you cannot know it. Many responses could be given to this refutation, but Lennox passes them over as if basic logic can easily refute atheism.

He is also conflating atheism with scientism, and this leads me to a third point. I find this "two competing worldviews" narrative rather unhelpful and even deceptive (perhaps unintentionally so). If it is true that atheists don't fully appreciate the diversity within Christian thought, it is still more true that Christians apparently don't have a clue when it comes to the diversity of secular thought. I have grown quite tired of Christians trying to claim that atheism is a "worldview" which, like Christianity, must stand on its own. That is false. It is true that atheists must have some sort of worldview, but among the competing possibilities, we find atheist representatives in all of them. Some atheists are collectivists, and others are individualists; some are modernists, and others are postmodernists; some hold to the myth of progress, others are nihilists; some put their faith in science, others put their faith in power, and others put their faith in personal (even mystical) experience. And I really haven't begun to list all the real alternatives. So the idea that there is an "atheist worldview" is nonsense, as most atheists will be quick to tell you. Christians really should be a little less blind to this. If you're merely trying to refute Sam Harris, Daniel Dennett, and Richard Dawkins, just say so. Don't bring everyone else into the picture without acknowledging how big and complicated the world is.

Mathematics through the Eyes of Faith: a review

James Bradley and Russell Howell have put together this exploration of the relationship between mathematics and Christian faith. The book is intended primarily for students, and could even be used as curriculum supplement for a Christian educational setting. Each chapter is completed with exercises for the student, some of which are legitimate mathematical exercises that would engage even quite advanced undergraduate mathematics students. Even without the exercises, it is a good read for anyone who wants an introduction to the philosophy of mathematics from a Christian perspective.

The main questions addressed in this book are philosophical in nature; concepts from mathematics are used primarily as instruments to stimulate thinking about the "big questions." Chapters 1 and 2 give an introduction to these big questions and an historical background in order to set the stage for the next eight chapters, which are entitled, Infinity, Dimension, Chance, Proof and Truth, Beauty, Effectiveness, Epistemology, and Ontology, respectively. These titles all allude to the "big questions" introduced in Chapter 1. The final chapter is "An Apology" for mathematics, encouraging students who might take an interest to pursue mathematics as part of a greater search for truth, and as a particular way of serving God in the world.

The authors have given a very balanced and sophisticated treatment of each of the subjects they have introduced. They have refrained from picking a side on any issue for which there appears to be room for more than one consistently Christian view (which is virtually every issue). Consider, for instance, the issue of "Chance" in the universe. It is common for believers to contrast chance with God's sovereignty; however, this can hardly be the whole story, for reasons both scientific and theological. Howell and Bradley have laid out in Chapter 5 a case for theistic determinism and a case for theistic nondeterminism. At the heart of the debate is the notion of ontological uncertainty, the state of being actually governed by chance, so that no additional knowledge could possibly remove uncertainty. Theists naturally divide on this issue as much as non-theists do; the determinist may argue that God's sovereignty and omniscience excludes ontological uncertainty, whereas the nondeterminist may argue that God has created this universe with a freedom of its own.

Perhaps that is the great puzzle of this book: is there any particular way of seeing mathematics "through the eyes of faith"? It is not so much in the answers as in the questions that Howell and Bradley demonstrate the relationship between mathematics and faith. There is no one Christian position on the ontology of numbers, or on the nature of proof; rather, there are distinctively Christian questions that we may ask. For instance, if mathematical certainty implies genuine, sure knowledge about reality, does that mean we can "know the mind of God" through mathematics? Or is mathematics merely a creaturely activity, which, just like all human thought, is in an important way eternally distinct from God's thought?

Mathematical style

This semester I have the privilege/responsibility of teaching a class whose students, all very intelligent and highly motivated, are trying to get a jump start on higher level mathematics. The course is advertised as Multivariable Calculus "honors" course. I hesitate to even call it a course in multivariable calculus, as we won't cover any of that material until the second half of the course. The first half of the course will be laying a very theoretical groundwork for what students in other classes are already learning. Our class will have the advantage of seeing these concepts the "right" way--understanding the derivative in terms of linear functions, understanding functions in a rigorous way, and understanding vectors and vector spaces abstractly.

This week I am focusing on function theory and set theory. That means proofs--and lots of them. One of my students in office hours was complaining as much. (My office hours, by the way, had to be held in the common room in the math building, because apparently my office must now be under construction until an unspecified time. There was no notice of this occurrence before it started, and those of us who occupy said office still have no idea when it will end. Also, today I went in my office to see if it was finished, and I noticed that someone had messed with my Rubick's cube! All of the stickers had been taken off! Who messed with my stuff, anyway? But I digress...)

So, proofs... Yes, entering the world of mathematical rigor is not easy for most students. I have to be honest, it was pretty easy for me when I first started seeing proofs in college. Part of that was that I had a proof-based course in high school through the EPGY. Another part of it is that I've always been a naturally skeptical person, and very particular about the meaning of statements. For instance, once in fourth grade I actually battled my entire class, including my teacher, over the statement, "A square is a rectangle, but a rectangle is not a square." Obviously, the correct statement would be, "Not all rectangles are squares," but the subtlety was lost on my peers, and, yes, even my teacher, as far as I can tell to this day.

Proof writing is a beautiful and delicate art. The truly great mathematical writers are the ones who get beyond just the brute calculations and really explain what's going on. Often people ask me how one actually does research in mathematics. The answer is the same as in any other field; you come up with a new idea and you present it. This is done on a scale from acceptable to great. Acceptable mathematics presents true statements and sufficient details to prove those statements. Great mathematics presents not only true statements, but powerful ones; and it doesn't just prove the results, it also expounds the key ideas that make the results true. In other words, great mathematics is great writing, just as in any other field.

What does it mean to be infinite?

Taking a break from my intensely political posts to write about mathematics again. I'm going to teach an honors class this fall which will deal with some advanced topics like set theory and cardinality, which deal with basic questions like, what is the size of a set?

I've been thinking about such questions ever since I ran into the possibility some years ago that math could have evolved without counting. You see, sets in mathematics are inherently discrete constructions. They consist of elements, each element being perfectly distinct from all the other elements. The size of a set can only be measured, then, by counting the number of distinct elements. You can tell if sets have the same size by matching elements together; if one set is bigger than another, you'll always have some elements left over in that set. Just think of boys and girls at a dance; if every boy has found a girl partner and there are still girls left over, obviously there are more girls than boys.

But counting isn't the only way we think of size. You can tell one object is "more" than another in several ways. Visually something could just be bigger than something else. One cup could have more water in it than another cup--just compare the height of the water in matching containers. One object could be heavier than another--just use a balance. None of these comparisons require actually coming up with numbers. "More" and "less" still have meaning even if there's nothing distinct about a given quantity.

I have written before about how the mere ability to distinguish two things can conceivably lead to an idea of discrete elements, thereby leading to counting, thereby leading to all of classical arithmetic, etc. But that seems pretty unnatural.

What I thought about doing instead was a thought experiment, built on the premise that mathematics develops with commerce. For this thought experiment we will want to assume that very different resources are being traded than what we are accustomed to. We count things because we have so many resources which can be conveniently counted--you know, cows, chickens, ears of corn, that sort of thing.

Musical Mathematics

Tell me you don't find this beautiful.



One could object that this is just artificially imposing an artistic interpretation on an otherwise abstract sequence of numbers with no inherent aesthetic value. But that is a biased way of putting it. It is not so much an imposition as it is a response; art is possible only because the universe beckons for our interpretation.

That was Pi; here's 2Pi:

From mathematics to theology

[T]heology is, for all its modesty, in an exemplary way a free science. This means it is a science which joyfully respects the mystery of the freedom of its object and which, in turn, is again and again freed by its object from any dependence on subordinate presuppositions.

--Karl Barth, Evangelical Theology: An Introduction

In modern usage, science is the study of a particular object or range of objects. Physics takes as its objects the most basic or "fundamental" objects of the empirical world, e.g. particles, energy, forces. Biology takes as its objects all living things, though it is not at all times clear what this means. Economics may take as its object particular markets, e.g. the housing market or the automobile industry, or it may take that grand object which Hayek liked to call the "catallaxy." In all cases, the amount of knowledge we can obtain about something is often inversely proportional to the level of complexity with which we are dealing. This is how it is usually described: physics studies the most simple objects, chemistry studies one level of complexity higher (i.e. compounds of the most basic materials), biology studies another level, psychology and neuroscience perhaps another level higher, environmental science and ecology another level still, and finally the social sciences: sociology, economics, and political science.

If we can classify mathematics as a science, it precedes even physics on this staircase of complexity. As John von Neumann said, "If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." Mathematicians study the very simplest objects of all: those which can be defined precisely and axiomatically, and which therefore can be understood with absolute certainty.

Prerequisites for calculus

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Florensky on learning mathematics

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.

(From Avril Pyman's biography, p. 160.)

Mental liberty

Talking to my grad student friends about teaching reveals a lot of agreement. For instance, one of the things that strikes all of us who teach calculus is the way our students tend to ask this particular question:

Am I allowed to do that?

Are you allowed to do that? I think to myself. Who ever suggested to you that you were somehow under someone's authority? Notice what the student doesn't ask. She doesn't ask, "Is this true?" She doesn't ask, "Is this the correct meaning of the symbol I'm using?" She doesn't even ask, "Will this get me the right answer?" Just a simple, submissive question: Am I allowed to do this?

Semantics, you say. All of those questions are really just different ways of asking the same thing. Nonsense! It is no accident that students ask this wretched question. From the time they are little children, they are trained to think that truth is a matter of authority. They are trained to look in the back of the book for answers to their math problems, they are trained to solve problems using the exact step by step process spoon fed to them by their teachers, and they are trained out of the creativity they once possessed as children. Because they also happen to be imbibed with an American sense of anti-authoritarianism, they simply become relativists, not having any sense of truth as an objective reality. American individualism notwithstanding, this view of truth degrades human freedom and is the source of all political evil.

I try to tell my students, There are no rules! You are constrained solely by what is true! Only a free person understands this; indeed, it is the definition of freedom. You are never free of all constraints. Objective reality does not give in to your whims. But you are free to poke it and prod it as you wish. What students fundamentally misunderstand about their education is the relationship between themselves and reality. They believe that in some aspects they can stand in authority over reality, whereas in others they must bow in submission. Neither is correct; the human mind never transcends the world, but neither is it simply drifting in the wind.

In short, no one can be free from God; but to be free of every will that is not God's is true freedom. There is a reason we call it liberal education--it is the education of free people. It is not my students' ability to reason quantitatively that I most worry about. What I worry about is their desire for truth, which is fundamentally linked to their creativity and above all their ability to exist as free people. Truly, no one can enter the Kingdom of Heaven except as a child. It is when we lose our child-like tendency to ask questions out of sheer curiosity, to poke and prod and play with, to seek out boundaries and see how far we can push them--then we lose our freedom, and with it our thirst for truth, and with that the things that make us most fully human.

Cantor's Miraculous Function

Secular historians often define a miracle (implicitly or explicitly) as the least likely thing to happen. Thus the historian must never use a supernatural explanation for any event. This implies that any secular conception of history must assume an historical continuum: it must be possible to measure the likelihood of a particular event by comparing it to other events we are used to.

What if there were a simple mathematical illustration showing how those events which have negligible probability, i.e. those events which are by definition the least likely to happen, can have a dramatic effect? In fact, there is, and it has been well-known for quite a long time. It's called the Cantor function and it looks a little something like this:

What's so special about the Cantor function? Here are a few of it's properties. First, it increases continuously from 0 to 1 as x ranges from 0 to 1. Second, it has a derivative defined almost everywhere with respect to the uniform probability distribution (or Lebesgue measure) on the unit interval. Third, this derivative which is defined almost everywhere is zero.

To put it in laymen's terms, pick a random point x in the interval from 0 to 1. The probability is 100% that the Cantor function will be staying perfectly still at that point, i.e. the instantaneous rate of change of the function at the point is zero. And yet! The function increases continuously--that is, without any jumps--from 0 up to 1. (In fact, multiplying by any constant changes none of these properties; the Cantor function could increase as high we like!) This absolutely mystifying property has earned the Cantor function the title "devil's staircase"
by some. I think a more fitting title is this: the Cantor function is a miracle.

Why do I say miracle? Because the event which enables the Cantor function to increase is, in the language of probability, an event which almost never happens. The set on which the Cantor function's derivative is undefined is called the Cantor set. This set meets the historian's criterion of being an event which, by definition, is the least likely event to happen, ever. Yet it is precisely this set which connects this "devil's staircase" from top to bottom. Hence I call this ascension from 0 to 1 a miracle. The historian's explanation will not explain the Cantor function. If one looks at the event most likely to happen, one sees that the Cantor function is not moving. Yet the function moves. Not only does it move--it bridges the entire gap from start to finish, never missing any point between.

The universe is full of such miracles. The continuum is teeming with undetected and undetectable events, whose impact cannot be readily assessed by scientific measurement. Part of the joy of mathematics is discovering such events and bringing them to light! Can there be a more spiritual endeavor?

Discontinuity

I read somewhere that the theory of functions had a profound impact on Florensky's philosophy. The assumption of continuity, for him, meant that the philosophy of the modern era was doomed to see everything in purely deterministic terms. The discovery of discontinuity, of functions that do surprising and almost incomprehensible things, signaled for him a turning point. The world was full of wonder again. Learning could be spontaneous, and the mind could leap forward in faith into the unknown.

There has to be something to this. When I teach calculus, I can't help but notice that students have the hardest time processing the definition of continuity--probably because they can't imagine discontinuity. Oh, sure, maybe a function jumps a few times here and there. Or maybe a function like f(x) = 1/x has a vertical asymptote. Big deal. For the most part in my beginning calculus class we deal with functions that are continuous everywhere they're defined, aside from maybe some removable discontinuities.

The definition of continuity is simply that the local behavior of a function f(x) near x = a is the same as the value f(a). The surprising thing to students is that this might not be the case. Take a function like f(x) = x^2, and obviously finding the limit is just a matter of "plugging in" the appropriate value of x. Discontinuity is jarring. It means the function doesn't behave the way one would expect. I find it necessary to hammer this point home for students, because otherwise they'll have a hard time perceiving that continuity is really an issue at all.

There is an intimate relationship between discontinuity and freedom of the will. To construct a continuous function one simply needs a general rule, e.g. f(x) = x^5 - 5. To construct a discontinuous function, one must dissect the continuum, picking out individual points of interest and declaring that a function behave differently at those points. It is this declaration by which a mind exercises its freedom of will. Even if it is silly or strange to do so, I may simply declare that f(5) = 0, even though for every other real number x I say f(x) = 2x. It is an act of will, a bold separation of chosen points from the rest of the continuum, an assertion that my word alone defines what a function is. Its local behavior may be perfectly continuous, and it may have a limit as x approaches 5. I am by no means bound by such trends.

One who is truly skilled at dissecting the continuum can do much more terrible things than making a function with a single point of discontinuity; indeed, it is not hard to define a function which is continuous nowhere, even though it is defined everywhere. I've given my students that problem as an extra credit assignment; I'm asking them to look it up and see if they can explain it to me. They won't be able to produce the example themselves, which is not a reflection on their intelligence so much as on the nature of mathematics. There are certain things one simply cannot imagine unless one has seen.

This is why mathematics is essential to a truly liberal education, and why I find it incredible how many educated people consider math a mere specialization totally closed off even to the average intellectual. There is nothing so liberating to the mind as being able to enter a world of pure thought, a world which is simultaneously the work of our own minds and beyond the mind's control. Every field in the humanities sounds to me to be obsessed with paradigms and paradigm shifts. Mathematics has no such discussions, because there is only one paradigm: discovery through creation. Mathematics is a creative act of will in the purest sense. "Let x be," and it is. What holds most people back from the world of mathematics is their unwillingness to take that leap of faith, to make that bold assumption of creativity, to step out of continuity with the world of the concrete and "practical" (concepts which any intellectual ought to address with suspicion). More than all the scientific applications, more than all the quantitative reasoning and critical thinking skills, mathematics has this one thing to offer the world: intellectual courage.

That is, of course, if one understands it the way, for instance, Florensky did.