I think discussions about science and faith are important. I think it's necessary to ask whether the latest scholarship and research contradicts or affirms traditional beliefs. I like to be involved in those debates, because I think sometimes skeptics can be convinced that Christianity is not all baloney and sometimes Christians can be convinced that not every traditional doctrine is Truth with a capital 'T'.
But there's a deeper, more fundamental question that surprisingly few Christian apologists ever bother to answer (and perhaps few Christians ever think to ask themselves). Is Christian faith consistent with a vibrant intellectual life at all?
Perhaps the question seems so blatantly offensive that it doesn't seem worth considering. I can certainly understand why. So many of the greatest minds, not only in science but in all areas of philosophy and the arts, have been faithful believers. Why should we doubt that one can have both sincere devotion to God and at the same time vigorously pursue intellectual questions?
Yet it seems to me there is naturally a great deal of tension between faith and reason for the intellectual. By "intellectual" I mean someone who seeks relentlessly to know what is true, who is indeed so committed to the pursuit of understanding that they will not allow any tradition, force of habit, feeling, prior commitment or anything else to stand in the way of rational inquiry. Thus intellectuals necessarily leave themselves open to changing every opinion, even those they hold most dear.
Christian religion, on the other hand, most certainly demands that we believe something. We are called to believe and warned not to fall away. We are expected to be convinced, and once convinced we are warned never to doubt. Faith is indeed a form of loyalty. It means devotion to a person--Jesus Christ--and to his mission, and to all the other people who have also made themselves loyal.
Can the intellectual truly hold such loyalty? I suppose the same question applies to any sort of loyalty. Can one be a committed member of a political party and be an intellectual? But changing political parties is certainly not unheard of. Perhaps a more dire question would be, can one honestly be loyal to one's country and be an intellectual? Is it not the case the one's country might get in the way of the truth, in which case such loyalty must be abandoned?
Less dire, more personal: can one be an intellectual and faithfully devoted to a family? This question seems to me far less hypothetical than the others. Families fall apart all the time in our day. That seems to be in large part because we are quite committed to discovering ourselves as we go, which means old commitments might sometimes have to give way to new self-discoveries. Some people seem to think it's worth it; others of us aren't so sure.
Maybe one could legitimately ask whether it's right or good to truly be an intellectual. After all, is it not self-defeating? To be committed to the pursuit of truth at the cost of any and all loyalties is itself a kind of loyalty of the most demanding kind. Yet that kind of loyalty is exactly the kind Jesus himself demanded: "Whoever loves father or mother more than me is not worthy of me, and whoever loves son or daughter more than me is not worthy of me, and whoever does not take up the cross and follow me is not worthy of me." Indeed, Jesus declared that he is the truth.
If we are committed to the truth at the cost of all other loyalties, there is no internal inconsistency, and moreover we are found merely to be doing what Jesus himself demands of us. The only question is, what gives Jesus the right to call himself the truth? Do there truly exist such overwhelmingly compelling arguments in support of such a claim?
Some apologists begin with five (or so) arguments for the existence of an omnipotent, omniscient, and omnibenevolent creator. Others begin with historical evidence for the resurrection of Jesus Christ. Others begin with an appeal to our sense of justice and innate belief in moral objectivity. Still others appeal to the universal human thirst for spiritual meaning.
When we add up such arguments together, do we yet get anywhere close to where we need to be, in order to convince the intellectual that Jesus himself is indeed the truth? Can intellectuals' loyalty to God ever be higher than their devotion to the truth? Must their devotion, if it is sincere, lead to faith? (Such a demand seems to defy common experience.) Or is the ultimate discovery rather that their devotion to the truth is a sort of faith in God, albeit unbeknownst to them?
Showing posts with label faith and reason. Show all posts
Showing posts with label faith and reason. Show all posts
Saturday, September 10, 2016
Friday, July 15, 2016
Between two worlds
One of the fruits of my reflections on Stephen Weinberg's The First Three Minutes is a renewed shock at the way modern people, especially scientists, can walk around knowing how pointless and inhospitable the universe is and simultaneously lead happy and productive lives. It is not so much that I think it surprising that human beings should be so self-absorbed as to ignore anything outside their own minuscule sphere of existence. Rather, I wonder how anyone who has gazed straight at the grandeur of this vast, pitiless universe can afterward return to diligently fulfill his daily responsibilities. Is there a sort of intentional amnesia that happens? Do scientists like Weinberg simply choose to forget how pointless their lives really are? Or do they instead have doubts about the validity of their own assertions? Perhaps there is some source of hope that they missed while they were looking through telescopes and performing calculations.
Christian faith trains the imagination to hold on to two seemingly opposite realities at the same time. Front and center is Jesus Christ, who is said to be both fully human and fully divine. In the same way he is both absent--seated at the right hand of God--and fully present, for the church is his body. It is a theme woven throughout the Bible. God is too big for even heaven and earth to contain him, yet he chose Jerusalem as a dwelling place. When Moses asks God what his name is, he responds, "I Am Who I Am," but then he adds that he is in fact the God of Abraham, the God of Isaac, and the God of Jacob. God is always making a bridge between the utterly unknowable and the known, the transcendent and the imminent, the universal and the particular.
Even if someone were to prove to me that Christianity is not true--if someone could identify the remains of Jesus, or find some other way to wholly discredit the New Testament's witness--I would still find myself compelled to search for this bridge between two worlds, like the God of the Bible. Somehow we humans are caught between the two worlds, scrambling to find ground beneath our feet, hoping against hope that we might live comfortable with one foot on either side. We live to satisfy temporal desires, for good food and entertainment, for success and prosperity, for meaningful accomplishments within our brief lifetimes. But we also gaze up at the stars, and down through the microscope, and wonder about the big picture. We try to measure the age of the universe and guess its ultimate destiny. We try to understand the laws of physics and how to master them. We try to answer questions about life's ultimate purpose, about what is really beautiful, and about what truly lasts forever.
In my daily life I find myself regularly called away from my ordinary tasks to contemplate just how little a difference it makes whether I complete them or not. True, as concerns my own life and the lives of those around me, it can make a tremendous difference. But on a larger scale, it makes practically none. One human life does not change the ultimate fate of humanity, and even if all humanity were to pass away, the earth would keep on turning, and even if the earth itself were destroyed, the sun would continue burning, and even if the sun itself died, the galaxy would keep on spinning, and the universe would go on as it always has...
Yet it is in these very tasks which I perform daily that I become witness to this grand spectacle. I perform calculations, and I write articles for journals of mathematics. Every theorem correctly proven is a small bit of insight into an eternal truth that will never be taken away. There is ground underneath my feet. The human mind is not adrift. There is a bridge, somewhere.
I wish scientists talked about this more, but I suppose that would involve matters of faith rather than rigorous empirical evidence.
Christian faith trains the imagination to hold on to two seemingly opposite realities at the same time. Front and center is Jesus Christ, who is said to be both fully human and fully divine. In the same way he is both absent--seated at the right hand of God--and fully present, for the church is his body. It is a theme woven throughout the Bible. God is too big for even heaven and earth to contain him, yet he chose Jerusalem as a dwelling place. When Moses asks God what his name is, he responds, "I Am Who I Am," but then he adds that he is in fact the God of Abraham, the God of Isaac, and the God of Jacob. God is always making a bridge between the utterly unknowable and the known, the transcendent and the imminent, the universal and the particular.
Even if someone were to prove to me that Christianity is not true--if someone could identify the remains of Jesus, or find some other way to wholly discredit the New Testament's witness--I would still find myself compelled to search for this bridge between two worlds, like the God of the Bible. Somehow we humans are caught between the two worlds, scrambling to find ground beneath our feet, hoping against hope that we might live comfortable with one foot on either side. We live to satisfy temporal desires, for good food and entertainment, for success and prosperity, for meaningful accomplishments within our brief lifetimes. But we also gaze up at the stars, and down through the microscope, and wonder about the big picture. We try to measure the age of the universe and guess its ultimate destiny. We try to understand the laws of physics and how to master them. We try to answer questions about life's ultimate purpose, about what is really beautiful, and about what truly lasts forever.
In my daily life I find myself regularly called away from my ordinary tasks to contemplate just how little a difference it makes whether I complete them or not. True, as concerns my own life and the lives of those around me, it can make a tremendous difference. But on a larger scale, it makes practically none. One human life does not change the ultimate fate of humanity, and even if all humanity were to pass away, the earth would keep on turning, and even if the earth itself were destroyed, the sun would continue burning, and even if the sun itself died, the galaxy would keep on spinning, and the universe would go on as it always has...
Yet it is in these very tasks which I perform daily that I become witness to this grand spectacle. I perform calculations, and I write articles for journals of mathematics. Every theorem correctly proven is a small bit of insight into an eternal truth that will never be taken away. There is ground underneath my feet. The human mind is not adrift. There is a bridge, somewhere.
I wish scientists talked about this more, but I suppose that would involve matters of faith rather than rigorous empirical evidence.
Monday, May 4, 2015
Evolution, tradition, reason, faith
To me the Bible is the greatest book never written. Its contents were collected and edited over centuries until finally becoming the self-contained cornerstone of Judeo-Christian tradition that it is today. Like any great institution, the Bible was grown, not designed.
As a result of its history, the Bible carries around it now a sort of magical fence which, Catholic-Protestant debates notwithstanding, prevents any serious changes to be made to its contents. There is a marvelous double effect of the Bible on the community of Christian believers: on the one hand, conservative Bible believers are forced to confront a wealth of confusing, frustrating, and downright bizarre stories and passages which, by their own standard, cannot be erased; and on the other hand, liberals are forced to confront the reality that faith is not the result of pure reason, that rationalistic belief can only be something other than Christianity, and that it is in ancient tradition rather than current that the mind continues to receive its greatest stimulation and challenge.
It's a delicious irony. Conservatives, who hate evolution because they love the idea of God the designer of all things, are in reality relying on an evolved tradition, while liberals, who love evolution because they love the process of reason which discovered evolutionary theory, have in reality found the very principle which destroys their own rationalism.
If you really want to be a rationalist, the logical belief is not that the creation story was too short (thousands of years vs. billions of years) but rather too long, as Origen pointed out. Everyone knows God the great architect really created everything at once. Those seven days are all just metaphors.
Indeed, it's hard to accept that God might just like watching things grow. Evolution requires both patience and spontaneity; that is, one must go into it knowing it will take a very long time but not at all knowing the final outcome. And the really strange thing is that the conservative hates this because he is too impatient, while the liberal hates it because he isn't spontaneous--when it should really be the other way around.
As a result of its history, the Bible carries around it now a sort of magical fence which, Catholic-Protestant debates notwithstanding, prevents any serious changes to be made to its contents. There is a marvelous double effect of the Bible on the community of Christian believers: on the one hand, conservative Bible believers are forced to confront a wealth of confusing, frustrating, and downright bizarre stories and passages which, by their own standard, cannot be erased; and on the other hand, liberals are forced to confront the reality that faith is not the result of pure reason, that rationalistic belief can only be something other than Christianity, and that it is in ancient tradition rather than current that the mind continues to receive its greatest stimulation and challenge.
It's a delicious irony. Conservatives, who hate evolution because they love the idea of God the designer of all things, are in reality relying on an evolved tradition, while liberals, who love evolution because they love the process of reason which discovered evolutionary theory, have in reality found the very principle which destroys their own rationalism.
If you really want to be a rationalist, the logical belief is not that the creation story was too short (thousands of years vs. billions of years) but rather too long, as Origen pointed out. Everyone knows God the great architect really created everything at once. Those seven days are all just metaphors.
Indeed, it's hard to accept that God might just like watching things grow. Evolution requires both patience and spontaneity; that is, one must go into it knowing it will take a very long time but not at all knowing the final outcome. And the really strange thing is that the conservative hates this because he is too impatient, while the liberal hates it because he isn't spontaneous--when it should really be the other way around.
Thursday, February 9, 2012
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:
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:
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:
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:
(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,
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,
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:
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.
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:
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.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.
(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.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?
...
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."
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.
Thursday, September 8, 2011
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?
Wednesday, August 31, 2011
A little fragmentation
At the request of my uncle, who helps run the Splintered Light Bookstore, I'm reading through a book entitled, Mathematics through the Eyes of Faith, which is part of "through the eyes of faith" series. I intend to write a review fairly soon, once I finish, but at the moment I just had a general comment to make about the "through eyes of faith" series. The concept of such a series seems both familiar and strange to me. Many Christians I interact with regularly are of the kind to ask the "big questions" about how faith and life connect, about how to view everything in light of the gospel, about how to bring all things under the authority of Jesus Christ. From a Christian point of view, this seems like the right mission, even if it is an enormous undertaking. Thus, in particular, many Christians are wondering how the various academic disciplines relate to the faith; hence, this series.
On the other hand, something feels a little strange about the kinds of ideas which result from this grand endeavor. Some will argue (not the book I'm reading) the rather extreme position that mathematics makes no sense outside of a Christian worldview. That would be a powerful apologetic if it made any sense to anyone other than those making the argument. Others take the more moderate approach of just trying to draw vague but highly stimulating connections between a particular discipline and Christian faith. For instance, what does Cantor's theory of infinity say about God? Does belief in a personal infinite being open up avenues of inquiry to us which may not have otherwise existed? (Such might be suggested by the very interesting book, Naming Infinity.)
But whatever ideas you might generate by looking "through the eyes of faith," the simple fact is that all disciplines, from mathematics to history to philosophy, continue onward without adhering permanently to one guiding framework. Paradigms come and go. The notion that we need to be constantly aware of how all knowledge fits into a bigger picture can be quite stifling, or at best irritating. No, I'm really not thinking about God every time I recall a result from functional analysis in order to prove a theorem about a system of partial differential equations. Asking me to explain how this new truth I've discovered relates to the gospel is, from my point of view, silly. Frankly, I don't expect mathematics to look noticeably different "through the eyes of faith" than through any other set of eyes.
I think a case can be made for a little fragmentation in our lives. This is precisely what many Christians seem to think is wrong with the secular world: we live in many different parts of reality without being integrated into a whole. But I'm not convinced this is so much of a problem. In fact, it might be worth being more intentional about this. For instance, I highly recommend that everyone make a point of fragmenting political issues from one another. Gay marriage, abortion, prayer in schools, immigration, medicare, and terrorism are all completely unrelated issues. By saying "completely unrelated" I may have made the above statement false, but if you let yourself believe that it is true for a least a little while, you might end up thinking about each issue more rationally, without allowing your views to be predetermined by the one big picture that now informs all of your beliefs. The more we are in the habit of seeing everything through a single lens, the harder it becomes to resolve our differences without entering into conflict.
We may not be willing to say that some beliefs and ideas are unrelated to faith, but perhaps we can at least say that we do not know what the relationship is. Maybe we can never know. In our vain life, sometimes we must simply be content to solve the problems which are solvable, and to leave to God the things that are mysterious. A holistic worldview is perhaps but a chasing after wind.
On the other hand, something feels a little strange about the kinds of ideas which result from this grand endeavor. Some will argue (not the book I'm reading) the rather extreme position that mathematics makes no sense outside of a Christian worldview. That would be a powerful apologetic if it made any sense to anyone other than those making the argument. Others take the more moderate approach of just trying to draw vague but highly stimulating connections between a particular discipline and Christian faith. For instance, what does Cantor's theory of infinity say about God? Does belief in a personal infinite being open up avenues of inquiry to us which may not have otherwise existed? (Such might be suggested by the very interesting book, Naming Infinity.)
But whatever ideas you might generate by looking "through the eyes of faith," the simple fact is that all disciplines, from mathematics to history to philosophy, continue onward without adhering permanently to one guiding framework. Paradigms come and go. The notion that we need to be constantly aware of how all knowledge fits into a bigger picture can be quite stifling, or at best irritating. No, I'm really not thinking about God every time I recall a result from functional analysis in order to prove a theorem about a system of partial differential equations. Asking me to explain how this new truth I've discovered relates to the gospel is, from my point of view, silly. Frankly, I don't expect mathematics to look noticeably different "through the eyes of faith" than through any other set of eyes.
I think a case can be made for a little fragmentation in our lives. This is precisely what many Christians seem to think is wrong with the secular world: we live in many different parts of reality without being integrated into a whole. But I'm not convinced this is so much of a problem. In fact, it might be worth being more intentional about this. For instance, I highly recommend that everyone make a point of fragmenting political issues from one another. Gay marriage, abortion, prayer in schools, immigration, medicare, and terrorism are all completely unrelated issues. By saying "completely unrelated" I may have made the above statement false, but if you let yourself believe that it is true for a least a little while, you might end up thinking about each issue more rationally, without allowing your views to be predetermined by the one big picture that now informs all of your beliefs. The more we are in the habit of seeing everything through a single lens, the harder it becomes to resolve our differences without entering into conflict.
We may not be willing to say that some beliefs and ideas are unrelated to faith, but perhaps we can at least say that we do not know what the relationship is. Maybe we can never know. In our vain life, sometimes we must simply be content to solve the problems which are solvable, and to leave to God the things that are mysterious. A holistic worldview is perhaps but a chasing after wind.
Tuesday, June 7, 2011
OK, so I just have to post another one...
This guy is great.
This would be a great video for discussion among Christian intellectuals. Can stories be dangerous? Should we be suspicious of stories? Are we comfortable with being "agnostic"? Are we OK with messes? All these questions seems to get right at the heart of the relationship between faith and reason.
It's good to pay attention to economists.
Saturday, August 28, 2010
Beauty for Truth's Sake - a review
If I could pick my top five topics to think about on a regular basis, three of them would be faith and reason, education, and, of course, mathematics. Thus Beauty for Truth's Sake: On the Re-enchantment of Education seemed like a perfect book for me to read and contemplate in my spare time. In it Stratford Caldecott offers something of a "manifesto" summarizing a view of education which sees the classical Western Liberal Arts tradition as a means of integrating faith and reason, art and science.
There is much to appreciate about the book. The beginning is quite strong, displaying a clear sense of what has gone wrong in modern education on the philosophical level. The opening lines read,
As the title of the book suggests, one of the key integration points between these seemingly disparate areas of knowledge is beauty. "Everything," Caldecott says, "is true, good, and beautiful in some degree or in some respect.... Beauty is the radiance of the true and the good, and it is what attracts us to both." In particular--and this is part of what I found intriguing about the book--mathematics is a key to understanding how the classical tradition sought to "perceive the inner, connecting principles" of the universe.
Let me briefly outline the structure of the book, to show how these ideas are expressed as a whole. The Introduction states the problem as I have, and offers three guiding principles. The first is, "The way we educate is the way we pass on or transform our culture.... The fragmentation of education... is a denial of ultimate meaning. Contemporary education therefore tends to the elimination of meaning...." The second is, "The "re-enchantment" of education would open our eyes to the meaning and beauty of the cosmos." The third is, "The cosmos is liturgical by its very nature." In this way Caldecott makes it clear from the start that education can never find true integration without a religious foundation. This raises interesting practical questions, but this book doesn't deal with them.
Chapter 1 sets about calling us to return to an idea of education as a means of becoming "truly free, fully human." Caldecott wants us to see that education is not just about what is useful. It is, in the great Socratic tradition, about gaining knowledge of "the forms, or the highest causes," which one can only attain "through the systematic ordering of the soul." In Chapter 2, he shows us that this path requires awakening the "poetic imagination," the ability to find "within the self something that corresponds to the object, thus leaping over the barrier between self and other." Thus symbolism comes to play a key role throughout the remainder of the book. Chapters 3, 4, and 5 essentially serve to illustrate the "poetic imagination" at work throughout the Western tradition. I've already mentioned the mathematical objects; there are also a number of excursions into theology, as well as music, architecture, ecology, and astronomy. Throughout these chapters one gets the sense that while Caldecott may not be advocating a return to a medieval understanding of the cosmos, he certainly seems to have an affinity for it. Chapter 6 argues quite strikingly for that third principle mentioned in the introduction, that "the cosmos is liturgical by its very nature." He thus argues that any real education must involve elements that are at least implicitly religious.
The conclusion is perhaps the most explicitly religious, in fact explicitly Catholic, part of the whole "manifesto." He says, "As we have seen, the Liberal Arts were intended to conduce to freedom of mind, and they were developed and nourished by the Catholic Church." He then explains that the modern conception of freedom is deprived of a certain fullness that is granted by the Christian (Catholic) view.
Now that I've summarized the book, let me get into my complaints. I agree with Caldecott that the fragmentation of knowledge is symptomatic of some deep problems. I also agree with the basic idea of seeking beauty in the universe, and that the pursuit of knowledge is, at its core, about love. But I would challenge some of the assumptions of his "Christian Platonism."
Before I do that, though, let me make some slightly more superficial comments. I have to say, and I think many readers would agree with me, there were many times during the reading of this book when I thought "Re-enchantment of Education" simply meant redecorating the universe with medieval superstitions. Certainly Caldecott was aware of this as he was writing, which is why he made the occasional remark that he is not trying to undo the Enlightenment or go back to the Middle Ages. Yet these remarks have little force behind them; he doesn't seem to have anything good to say about the Enlightenment in any meaningful sense.
On a related note, Caldecott relies so heavily on his own Catholic tradition that it is difficult for readers outside that tradition to understand the appeal of his illustrations. For instance, he mentions more than once how cosmically significant it was for Christian that there are seven days in a week and seven sacraments. Well, suppose there aren't seven sacraments... Does that mean the universe is less enchanted with meaning? In one part of Chapter 4 he wanders off into a discussion about the filioque controversy, a subject which is thoroughly uninteresting to many of us. Even more importantly, such controversies aren't settled by geometric arguments, and Caldecott's references to the relationship between mathematics and theology are more likely to offend believers of different theological persuasions than they are to enlighten anybody.
Chapter 4 ends with a paragraph that begins, "Speculations like those I have mentioned in this chapter will appear forced to many." Believe me, they did. I found circles and lines to be wholly inappropriate for trying to visually represent the Trinity. I found the comparison between Jesus and the line perpendicularly connecting a point on a circle to a given diameter also rather "forced." And the ratio between this line in the "golden circle" and the circumference, why, it's miraculously just over 7! The amount in excess of 7 which we find in this ratio is, of all things, what Caldecott thinks might correspond to that "tiny and indispensable human contribution needed if heaven is truly to descend to earth." In other words, heaven divided by earth = 7 (God's number) + some tiny human contribution = pi * ((2 * phi) - 1) = 7.02481473...
It is worth noting here that the number of man's symbolic contribution in this calculation is irrational. All this to say, one has to be very careful before going off to find the logoi of creation. This search can easily degenerate into such absurd arguments as "there are seven sacraments because there are seven days in a week." It might be postmodern of me, but different cultural perspectives really are worth keeping in mind as we examine the connections underlying things in the world. For instance, the octave interval in music might very well have a special relationship with the number "eight" in the West, where eight might have special theological meaning (on the eighth day Christ rose again) or other kinds of meaning. But, lest we forget, this association is based on Western musical scales. Other cultures have more varied intervals, and therefore the association doesn't work. That's often how it goes with Platonism. You think you've found the form of which all the world is a reflection, but then you realize it's just your own perspective being forced on the world around you.
Now I am starting to get into my deeper qualms with Caldecott's assumptions. Consider this third vertical dimension of human experience, which he proposes in his conclusion. It is as if he says we ascend to heaven through the illumination of education (the right kind of education, anyway). This is indeed a very Platonic way of viewing things, but not a very Christian one. In the gospels, wisdom is given to ordinary people. It is all about grace, not enlightenment through systematic human effort. As I have already suggested, there is no guarantee that such systematic efforts would lead one closer to heaven. Human beings have difficulty seeing the difference between the beauty inherent in the universe and their own prejudices based on cultural conditioning.
Another point I would make is that this Platonic notion that Ideas are ultimate reality (the "thoughts of God," for a Christian Platonist), and all else is reflective of these pure Ideas, is in some sense to deny the goodness of creation. Creation has its own reality that is not a mere shadow of something else. I'm sure this point has been made plenty of times by people much more theologically astute than I, but from my own perspective it is important to recognize that every single thing you see and feel has its own existence. Yes, it is all made of the same basic stuff--i.e. matter and energy governed by universal laws of physics. But to have the underlying principles is not to have the thing itself. The world is not translucent. I don't agree with this idea of looking at the world as if the only thing that makes it good is being a channel through which to see something else.
In terms of application I think this point can be rather significant. As in I don't think it's necessary or fruitful to link every scientific discovery to some theological precept. I don't think mathematical objects have some inherent mystical meaning. In terms of mathematics education, I do think it would be helpful for people to be taught that mathematics is more about inner relationships than it is about formal operations, which can have nothing to do with reality. Far from vindicating Platonism, however, I think this just points for the need for human beings to be connected to things. Caldecott gets it right when he talks about the "poetic imagination," at least insofar as he describes human beings as inherently connected to creation. But I think he gets it wrong when he posits a realm of pure Ideas which have some higher reality than the world of tangible experience.
The last thing I'll point out is that Caldecott seems, in spite of himself, to miss the strong distinction between Platonic idealism and Christian realism. Related to Platonism is, I think, a tendency to try to escape realism. If the most important thing is to ascend to the Forms, then it becomes less important to actually deal with the gritty details of the real. I submit that this is a theological weakness in Caldecott's understanding. He spends some time in first chapter talking about "Beauty on the Cross." There is, paradoxically, a great deal of beauty on the Cross of Christ, but the other side of that paradox must always be remembered. The crucifixion was a gruesome, grotesque thing. The Incarnation itself was an "emptying" of Christ to the point of humble obedience.
It is important that we as Christians remember this, so that we, like Christ, can enter into the world as it actually is, and not as a mere reflection of perfect Ideas. True education demands a certain realism. We cannot gloss over the details. We must be willing to face the world as a complicated, often frustrating place. We're not going to be able to transcend uncertainty and confusion. Even Christ himself prayed in the Garden of Gethsemane, "Father, if it is your will, save me from this hour." And on the cross he cried out, "My God, why have you forsaken me?" How much less will we be able to transcend the uncertainty of this life? Caldecott criticizes the postmoderns for doubting the human ability to obtain truth, but to a certain extent the postmoderns are right. It is not necessary to believe that truth is "relative" to have a healthy skepticism about the human ability to possess truth. We have to be realistic about ourselves, and about the world we live in. And instead of trying to transcend this world, we ought to follow Christ's example; he stepped down from his position of transcendence, that he might enter into this world--not, I believe, to show us the way out of this world, but to begin to transform it.
Caldecott is right about the most important thing: it all comes down to love. It is difficult to know what direction love should take us. While I don't agree with the direction Caldecott proposes, I think he is right to give us an alternative to our current approach to education. We do live in a beautiful universe, and it would be a waste to treat it in the purely utilitarian way that students are encouraged to now. Perhaps there is a better approach, one that relies not so much on enchantment as on the love of the universe for what it really is.
There is much to appreciate about the book. The beginning is quite strong, displaying a clear sense of what has gone wrong in modern education on the philosophical level. The opening lines read,
"In the modern world, thanks to the rise of modern science and the decline of religious cosmology, the arts and sciences have been separated and divorced. Faith and reason often appear to be opposed, and we have lost any clear sense of who we are and where we are going."Why is this a problem? I think Caldecott sums it up nicely when he says,
"The purpose of an education is not merely to communicate information, let alone current scientific opinion, nor to train future workers and managers. It is to teach the ability to think, discriminate, speak, and write, and, along with this, the ability to perceive the inner, connecting principles, the intrinsic relations, the logoi, of creation...
As the title of the book suggests, one of the key integration points between these seemingly disparate areas of knowledge is beauty. "Everything," Caldecott says, "is true, good, and beautiful in some degree or in some respect.... Beauty is the radiance of the true and the good, and it is what attracts us to both." In particular--and this is part of what I found intriguing about the book--mathematics is a key to understanding how the classical tradition sought to "perceive the inner, connecting principles" of the universe.
"Theology, therefore, has an important place in the integration of the arts and sciences. Equally important, however, is a symbolic approach to number and shape--that is, the awareness that mathematics has a qualitative, as distinct from a purely quantitative, dimension."Caldecott spends a couple of chapters in the middle of the book illustrating that qualitative dimension of mathematics, using examples from the ancient Pythagoreans, numerology from the Bible, and various other musings on the relationships between numbers, shapes, and the world around us. It is a rather delightful survey, ranging from the Tetractys to the five Platonic solids to the golden ratio. It really is a shame, in my opinion, that modern mathematical education leaves very little room for this kind of appreciation of mathematical objects.
Let me briefly outline the structure of the book, to show how these ideas are expressed as a whole. The Introduction states the problem as I have, and offers three guiding principles. The first is, "The way we educate is the way we pass on or transform our culture.... The fragmentation of education... is a denial of ultimate meaning. Contemporary education therefore tends to the elimination of meaning...." The second is, "The "re-enchantment" of education would open our eyes to the meaning and beauty of the cosmos." The third is, "The cosmos is liturgical by its very nature." In this way Caldecott makes it clear from the start that education can never find true integration without a religious foundation. This raises interesting practical questions, but this book doesn't deal with them.
Chapter 1 sets about calling us to return to an idea of education as a means of becoming "truly free, fully human." Caldecott wants us to see that education is not just about what is useful. It is, in the great Socratic tradition, about gaining knowledge of "the forms, or the highest causes," which one can only attain "through the systematic ordering of the soul." In Chapter 2, he shows us that this path requires awakening the "poetic imagination," the ability to find "within the self something that corresponds to the object, thus leaping over the barrier between self and other." Thus symbolism comes to play a key role throughout the remainder of the book. Chapters 3, 4, and 5 essentially serve to illustrate the "poetic imagination" at work throughout the Western tradition. I've already mentioned the mathematical objects; there are also a number of excursions into theology, as well as music, architecture, ecology, and astronomy. Throughout these chapters one gets the sense that while Caldecott may not be advocating a return to a medieval understanding of the cosmos, he certainly seems to have an affinity for it. Chapter 6 argues quite strikingly for that third principle mentioned in the introduction, that "the cosmos is liturgical by its very nature." He thus argues that any real education must involve elements that are at least implicitly religious.
The conclusion is perhaps the most explicitly religious, in fact explicitly Catholic, part of the whole "manifesto." He says, "As we have seen, the Liberal Arts were intended to conduce to freedom of mind, and they were developed and nourished by the Catholic Church." He then explains that the modern conception of freedom is deprived of a certain fullness that is granted by the Christian (Catholic) view.
"The best way to put this might be that the Christian conception of freedom is larger and fuller than the modern conception, for it includes both vertical and horizontal dimensions. The horizontal dimension encompasses the world we see directly, and the vertical allows for degrees of being and value, invisible realms, formal causality, and so on. ...
In the traditional "three-dimensional" world, the self was encouraged to collect itself together in a point, in order to attach itself to a vertical axis, a spiritual "path." ... Modernity, on the other hand, rejects the existence of the vertical altogether, or the very possibility of thinking in terms of up and down. ...
In a flatter universe, freedom had to be reconceived as entirely a matter of movement within the horizontal plane. I am assumed to be "freer" the more places I can go to, the more things I can choose on the supermarket shelf, the more people I can have relationships with. And that is why the Church claims today to be in the business of liberating human freedom, by making known the beauty of truth in its fullness.
Now that I've summarized the book, let me get into my complaints. I agree with Caldecott that the fragmentation of knowledge is symptomatic of some deep problems. I also agree with the basic idea of seeking beauty in the universe, and that the pursuit of knowledge is, at its core, about love. But I would challenge some of the assumptions of his "Christian Platonism."
Before I do that, though, let me make some slightly more superficial comments. I have to say, and I think many readers would agree with me, there were many times during the reading of this book when I thought "Re-enchantment of Education" simply meant redecorating the universe with medieval superstitions. Certainly Caldecott was aware of this as he was writing, which is why he made the occasional remark that he is not trying to undo the Enlightenment or go back to the Middle Ages. Yet these remarks have little force behind them; he doesn't seem to have anything good to say about the Enlightenment in any meaningful sense.
On a related note, Caldecott relies so heavily on his own Catholic tradition that it is difficult for readers outside that tradition to understand the appeal of his illustrations. For instance, he mentions more than once how cosmically significant it was for Christian that there are seven days in a week and seven sacraments. Well, suppose there aren't seven sacraments... Does that mean the universe is less enchanted with meaning? In one part of Chapter 4 he wanders off into a discussion about the filioque controversy, a subject which is thoroughly uninteresting to many of us. Even more importantly, such controversies aren't settled by geometric arguments, and Caldecott's references to the relationship between mathematics and theology are more likely to offend believers of different theological persuasions than they are to enlighten anybody.
Chapter 4 ends with a paragraph that begins, "Speculations like those I have mentioned in this chapter will appear forced to many." Believe me, they did. I found circles and lines to be wholly inappropriate for trying to visually represent the Trinity. I found the comparison between Jesus and the line perpendicularly connecting a point on a circle to a given diameter also rather "forced." And the ratio between this line in the "golden circle" and the circumference, why, it's miraculously just over 7! The amount in excess of 7 which we find in this ratio is, of all things, what Caldecott thinks might correspond to that "tiny and indispensable human contribution needed if heaven is truly to descend to earth." In other words, heaven divided by earth = 7 (God's number) + some tiny human contribution = pi * ((2 * phi) - 1) = 7.02481473...
It is worth noting here that the number of man's symbolic contribution in this calculation is irrational. All this to say, one has to be very careful before going off to find the logoi of creation. This search can easily degenerate into such absurd arguments as "there are seven sacraments because there are seven days in a week." It might be postmodern of me, but different cultural perspectives really are worth keeping in mind as we examine the connections underlying things in the world. For instance, the octave interval in music might very well have a special relationship with the number "eight" in the West, where eight might have special theological meaning (on the eighth day Christ rose again) or other kinds of meaning. But, lest we forget, this association is based on Western musical scales. Other cultures have more varied intervals, and therefore the association doesn't work. That's often how it goes with Platonism. You think you've found the form of which all the world is a reflection, but then you realize it's just your own perspective being forced on the world around you.
Now I am starting to get into my deeper qualms with Caldecott's assumptions. Consider this third vertical dimension of human experience, which he proposes in his conclusion. It is as if he says we ascend to heaven through the illumination of education (the right kind of education, anyway). This is indeed a very Platonic way of viewing things, but not a very Christian one. In the gospels, wisdom is given to ordinary people. It is all about grace, not enlightenment through systematic human effort. As I have already suggested, there is no guarantee that such systematic efforts would lead one closer to heaven. Human beings have difficulty seeing the difference between the beauty inherent in the universe and their own prejudices based on cultural conditioning.
Another point I would make is that this Platonic notion that Ideas are ultimate reality (the "thoughts of God," for a Christian Platonist), and all else is reflective of these pure Ideas, is in some sense to deny the goodness of creation. Creation has its own reality that is not a mere shadow of something else. I'm sure this point has been made plenty of times by people much more theologically astute than I, but from my own perspective it is important to recognize that every single thing you see and feel has its own existence. Yes, it is all made of the same basic stuff--i.e. matter and energy governed by universal laws of physics. But to have the underlying principles is not to have the thing itself. The world is not translucent. I don't agree with this idea of looking at the world as if the only thing that makes it good is being a channel through which to see something else.
In terms of application I think this point can be rather significant. As in I don't think it's necessary or fruitful to link every scientific discovery to some theological precept. I don't think mathematical objects have some inherent mystical meaning. In terms of mathematics education, I do think it would be helpful for people to be taught that mathematics is more about inner relationships than it is about formal operations, which can have nothing to do with reality. Far from vindicating Platonism, however, I think this just points for the need for human beings to be connected to things. Caldecott gets it right when he talks about the "poetic imagination," at least insofar as he describes human beings as inherently connected to creation. But I think he gets it wrong when he posits a realm of pure Ideas which have some higher reality than the world of tangible experience.
The last thing I'll point out is that Caldecott seems, in spite of himself, to miss the strong distinction between Platonic idealism and Christian realism. Related to Platonism is, I think, a tendency to try to escape realism. If the most important thing is to ascend to the Forms, then it becomes less important to actually deal with the gritty details of the real. I submit that this is a theological weakness in Caldecott's understanding. He spends some time in first chapter talking about "Beauty on the Cross." There is, paradoxically, a great deal of beauty on the Cross of Christ, but the other side of that paradox must always be remembered. The crucifixion was a gruesome, grotesque thing. The Incarnation itself was an "emptying" of Christ to the point of humble obedience.
It is important that we as Christians remember this, so that we, like Christ, can enter into the world as it actually is, and not as a mere reflection of perfect Ideas. True education demands a certain realism. We cannot gloss over the details. We must be willing to face the world as a complicated, often frustrating place. We're not going to be able to transcend uncertainty and confusion. Even Christ himself prayed in the Garden of Gethsemane, "Father, if it is your will, save me from this hour." And on the cross he cried out, "My God, why have you forsaken me?" How much less will we be able to transcend the uncertainty of this life? Caldecott criticizes the postmoderns for doubting the human ability to obtain truth, but to a certain extent the postmoderns are right. It is not necessary to believe that truth is "relative" to have a healthy skepticism about the human ability to possess truth. We have to be realistic about ourselves, and about the world we live in. And instead of trying to transcend this world, we ought to follow Christ's example; he stepped down from his position of transcendence, that he might enter into this world--not, I believe, to show us the way out of this world, but to begin to transform it.
Caldecott is right about the most important thing: it all comes down to love. It is difficult to know what direction love should take us. While I don't agree with the direction Caldecott proposes, I think he is right to give us an alternative to our current approach to education. We do live in a beautiful universe, and it would be a waste to treat it in the purely utilitarian way that students are encouraged to now. Perhaps there is a better approach, one that relies not so much on enchantment as on the love of the universe for what it really is.
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