Providence and freedom

I wanted to develop a thought from yesterday on providence and free will. But where to start...

Imagine a world governed entirely by God's arbitrary will, that is, with no discernible regularities in it. Everything is an immediate invention of God's mind and need not have any connection with anything before or after. One is tempted to say that we would have no free will in that case, as an immediate corollary. But then, a compatibilist view of free will, in which human agency does not contradict determinism (which latter could be the result of scientific or theological suppositions), might still hold in such a universe. God could give each of us the sensation of having free will, producing in us the experience of making decisions and having some perceptible impact on our environment. If that doesn't sound like genuine free will to you, then I guess you're not a compatibilist. (I'm not either.)

Anyway the real problem isn't whether we might have the sensation of a will of our own. The real problem is that we couldn't possibly learn or understand anything. Learning is always a matter of seeing connections between things. We find causal or logical relationships between events and concepts, we perceive similarities and differences allowing us to categorize things and experiences, and in so doing we build a web of knowledge. Now if God were always there to possibly interrupt all such connections, we would have no choice but to view life as fundamentally absurd. Learning would be not just pointless, but impossible. Scientific knowledge (in the modern sense) would be impossible, since repeated experiments would not necessarily yield the same results. Personal relationships would be just as impossible, since the bond of trust could never exist in a world without predictable patterns.

Only if God's will is bound by definite patterns that are in some sense universal can human beings come to understand the world as such. This fact need not infringe on God's supremacy. Those patterns could have their origin in God's own will or character. The point is not to say what God must be like in any conceivable universe, but rather to say what He must be like given that we exist.

By contrast, a world completely governed by unchanging laws is, ironically, the most fertile ground for free will. In this case the mind can discern order in the universe, because it is actually there. We acquire knowledge through experience and memory. Memory doesn't fail us because the present really is connected to the past, and experience doesn't fail us because there is a true link (causal, rational) between our actions and their consequences. One potential irony is that we ourselves are governed by physical laws, and so arguably we aren't genuinely free; all our decisions are perhaps the inevitable result of evolution given certain initial conditions. Or maybe not. It is entirely conceivable that in a world governed by universal physical laws, there may exist rational beings with the power to choose between several physical possibilities; by what logical principle must a physical law allow at most one outcome for each initial condition? The universe still remains intelligible if possibilities are limited in a consistent way.

What impresses me the most is the analogy that exists between this seemingly abstract consideration and real world politics. In free societies, laws are consistently applied, so that individual members can be assured that if they abide by a certain base line code of conduct, they will be free to pursue their own ends. In tyrannical societies, it is often observed, a general atmosphere of suspicion falls on the entire people. The concept of truth itself becomes degraded. When the actions of the state are totally unpredictable, people resign themselves to a life with no intelligible order. Only might makes right.

Is God a god whose might makes right? In the world we actually live in, certainly not. We live in a universe so well ordered by laws that those who understand them may question the existence of any exceptions whatsoever. In other words, in a world where scientists tend not to believe in miracles, you can be assured that God is not a tyrant. On the contrary, you might criticize Him for being too libertarian. But for the Christian who believes that everything in the universe is under God's control, these universal laws become windows into His character. There is, presumably, a reason why He adheres to these laws in particular. Reflecting on those reasons can be particularly painful when the consequences of said laws seems so devastating. It hardly seems comforting to think that God is in control of tsunamis that wipe out hundreds of thousands of mostly innocent people, or that He is in control of diseases that kill millions of people every year. Why can't He break these laws for our sake, that is, for the sake of justice?

I think part of the resolution of such questions lies in God's ultimate destiny for human beings, and part lies in the inherent goodness of the entire created order (why should the wind and waves stop because we want them to leave us alone?). But another important factor is, I wager, our capacity for freedom. God knows that the only way for rational minds to exist in the created world is for the order of creation to be consistent. We would not be able to discern His character if He erratically imposed His will on us. Instead, He acts toward us according to a coherent set of principles, allowing us to accumulate knowledge of Him and His creation. In so gaining understanding, it's worth restating that we also become more free to pursue our own ends. It would appear that God values our freedom, the kind that comes from learning and growing in wisdom.

I make no attempt to actually address the classical problem of "free will" here. I am more interested in the nature of God's providence as it relates to our freedom. As best I can tell, it matters for our freedom what kind of control God exercises over the world, and it seems to me that the kind of control He in fact does exercise permits the growth of free, rational minds. This says a great deal about the purpose of creation. It says also a great deal about our purpose as rational beings.

Free will and theodicy

Reading through Origen's On First Principles, I discovered that he used free will as a way of justifying God, to the point where he quite apparently says that all souls are born into different stations in life because of the good or bad they did in previous lives. Aside from the fact that this sounds more like a Hindu doctrine than a Christian one, I was struck by how much it differs from the Augustinian interpretation of grace. Calvin took Augustine to the extreme. For Calvin, there is no reason to justify God in the first place. God foreordains everything according to His good purpose, and if it makes no sense to us, it is only because we are so limited and/or corrupted.

Now being acquainted with these two extremes in Christian thought, I'm tempted to go the moderate route. It seems perfectly reasonable that God would create the world with creatures of various stations, not because of any question of merit, but rather simply because diversity is a good and beautiful thing (contra Origen). On the other hand, it also seems reasonable to think that many of these creatures have free will, and will be judged on the basis of what they do with their limited capacities, in proportion to the extent of those capacities. To suggest, as Calvin seems to do, that we can't use our common sense to understand what is right and wrong for God to do puts us in a pretty bad position when it comes to theology--how do we know that God is not actually a demon?

But the moderate route isn't entirely satisfying. If God isn't truly in control of everything that happens, how can we call Him Lord of all creation? There are many passages of Scripture that seem to attribute to God absolute power over human actions, even the human mind. So Isaiah 63:17--"Why, O Lord, do you make us stray from your ways and harden our heart, so that we do not fear you?" Or Proverbs 16:9--"The human mind plans the way, but the Lord directs the steps." Or Jeremiah 31:33--"I will put my law within them, and I will write it on their hearts." There are plenty of other verses which affirm human responsibility, but these are hard to ignore.

The difficulty in trying to justify God is trying to determine just what kind of universe He should have made. I suppose Calvin saw this more than most of us. It seems impossible to imagine a universe for which God could not receive some reproach. The fact that anything exists at all is a miracle and, at the same time, a condemnation--if I exist, I am doomed to be whatever it is I am and not something else. One could imagine that this horrible tension is the root of all sin. Adam and Eve had a paradise to live in, yet they still found themselves tempted by what it was not.

At the same time, to jettison all sense of justice by which to evaluate God's action is to destroy prayer as it is found in the Bible. There are, of course, the psalms, which make very direct complaints to God, calling on Him to remember His steadfast love and His promises. But the paragons of prayer are Abraham and Moses. Abraham negotiated with God over the destruction of Sodom: "Shall not the Judge of all the earth do what is just?" And Moses interceded for his people on the basis of reason: "Why should the Egytians say, 'It was with evil intent that he brought them out to kill them in the mountains, and to consume them from the face of the earth'?" If these men had been willing merely to accept God's plans, convincing themselves by some theological argument that whatever God did was right, they would never have been heroes of the faith.

I don't see any way out of the tension between what is and what is not. It seems to me the essence of life itself. We desire to reach a point of perfection, where we can finally stop and say, "It is enough." Yet we know that if we stopped completely, that would be an end to life itself. All life is motion and change. A perfect, finished worked of art is lifeless. It is only in the contemplation of that art, in the active appreciation, that we derive any pleasure from it.

Behold the central paradox of Christian theology. Christ did not consider equality with God as something to be grasped, but emptied himself, taking the form of a human being, and become obedient even to the point of death on a cross. He who knew no sin became sin. God took on imperfection in order to attain perfection.

What does this have to do with free will? Only that in Christian thought, free will relates to theodicy, the justification of God. Some will justify God's wrath by reminding us of our responsibility. Others will justify it by asserting God's supremacy. The argument goes back and forth eternally. The disagreement is honest. It is a living response; any resolution to the argument would mean we are no more than a lifeless work of art.

Prayer, more than debate, is a living response. Abraham and Moses argued their case before God. Jesus, and then Stephen the martyr, prayed, "Father, forgive them, for they know not what they do." Ultimately, the resolution of God's wrath is on the cross. Christ died for us while we were still sinners. We are told he even went and proclaimed the good news to the dead (1 Peter 3:19). For the Christian, then, there is limitless hope for the salvation of the world, of every soul. There is no reason to resign ourselves to believing in the eternal damnation of some for the sake of God's eternal purpose (contra Augustine and Calvin). God's eternal purpose is the cross, that impossible event which changes everything.

Jesus told his disciples to take up their own cross. This is the ultimate living response. Instead of justifying God by our theories, we ought to vindicate Him by imitating Him. Whether or not this is a free choice, I still don't know, but it is a choice: to take up our cross daily, to embrace the contradiction in our own lives between our desire to be and our desire to be something else.

Florensky on evolutionism -- "What is the differential equation?"

From At the Crossroads of Science and Mysticism, Lecture Four:

Renaissance culture also gave attention to the other principal form of being--time. The examination of the concept of time occurred later, and the fragmentation of being in time was performed later than the fragmentation in space: evolutionism came after mechanism.

Evolutionism does have a healthy seed--gentism, according to which an essence does not unfold in some single moment, and the spiritual meaning of an object is not exhausted by some single state, but exists in the totality of its states. But evolutionism errs when it states that this genesis is made up of infinitely small additions, so small that each separately can be considered not a creative act. ...

For example, let us consider the question of the origin of man. Evolutionism denies that there is a qualitative and fundamental difference between man and animals. But if this process occurs discontinuously, if man is descended from the apes, this theory loses its anti-religious character, since a qualitative change suddenly occurs. And if man was created from the dust of the earth by a special creative act, why not then allow in principle--only in principle--that man was created by a momentary addition of spiritual qualities to the ape?

What I am pretty sure Florensky is saying is that one need not take the creation story in the Bible literally to have a Christian worldview, but on the other hand he does insist on a "discontinuous" development of spiritual qualities. On the one hand, I suppose that science can't now and perhaps never will be able to address the important qualitative differences between man and the apes. For example, there will never be a fossil record of spoken language, so we will never know definitively when or how speech developed. On the other hand, Florensky (writing in 1920, mind you) could be taken as throwing his lot in with what might be called the "intelligent design" crowd. How else to interpret this statement, particularly with its wording about "creative acts"?

There's a very thought-provoking passage on how the differential calculus was absorbed into modern science:

For the immediate consciousness rest is opposite to motion, while the essence of science is the fact that we can study motion only by separating it into states as if of rest, with the result being the differential equation. The latter becomes the universal instrument of mathematics, and then of all science and the whole Renaissance epoch, since, to quote Kant, every science is a science insofar as it incorporates mathematics.

 He goes on:

What is the differential equation? Some sort of process is occurring, and we stop it and break it up into a series of instants and see it as if in sectional view. The differential equation is a general formula suitable for the sectioning and study of any process. With this method we are not concerned with the past, with what occurred earlier in time, with whether these are people or whether they are statues who suddenly started moving again after we had stopped them. Only the present is important for us here, not the past; but for many phenomena it is precisely the past that is important.

Writing in 1920, I suppose Florensky could simply not have known about all the modern developments in differential equations. Now it is commonplace to deal with integro-differential equations which were developed to take into account nonlocal interactions across both time and space. I wonder what he would say if he could have lived to see the era in which most differential equations do not involve continuous functions and Taylor series, but rather wild, mysterious functions and generalizations thereof.

Even today, one might think at first glance that Florensky is basically right in his description of differential equations, in that they are meant to break down a process into infinitesimal moments, each of which may be viewed as "not a creative act" but rather a blind obedience to some universal law. But even creative acts or subjective experiences can be modeled quantitatively. The whole use of optimal control theory to predict social behavior comes from the idea not that human beings follow mechanical laws but rather that they exhibit optimizing behavior, i.e. genuine acts of will. Florensky simply could not have seen this, since its development came only after his death.

Still, Florensky challenges me like no other thinker, perhaps because so few theologians know or write anything about differential equations! It would be naive to assume that mathematical objects appeared in a philosophical vaccuum. I should pay closer attention to how today's concepts are used (or could be used) to shape our world view.

To me, there will always be one fundamental discontinuity that cannot be ignored by any scientific theory, and that is the mere fact of existence. Between existence and non-existence there is no continuum. There can be no "probability" that the universe would come into being. Rather, its existence is pure impenetrable mystery. For me, this is basically the same thing as to say that God spoke the universe into being.

Anthropocentric religion, part 2

A while back I wrote on this blog against the complaint that Christianity is essentially anthropocentric. Yes, the universe is vast, and humans are very insignificant in relation to the whole sum of existence. That is hardly an argument against a faith which, historically, strongly affirms these claims.

But then again, experience often contradicts this fact and reinforces the idea that Christianity is a religion focused on satisfying some human psychological need. I was reminded of this a week ago during Sunday worship.

Perhaps the vast majority of self-identified evangelical churches would summarize the gospel as follows: we are great sinners, but Jesus is a great Savior, who through his sacrifice on the cross atones for our sins and loves us unconditionally, so that through faith in him we may approach God with confidence and find eternal life.

All the main elements of worship at the church where I worshiped were clearly designed to reinforced this message. From the opening remarks (in which the associate pastor confessed that he, too, is a sinner, awkwardly eliciting faux surprise from the congregation) to the songs (including a rendition of "It Is Well with My Soul" in which the instruments made sure to put a musical climax around the verse, "My sin, O the bliss of this glorious thought... is nailed to the cross, and I bear it no more...") to the sermon (an interpretation of Psalm 139 which, though quite sophisticated, nevertheless boiled down to, "God loves you and thinks about you even more than you do"), everything pointed one's thoughts toward our individual need for love and redemption and the good news that God satisfies these needs. (Ironically enough for an evangelical church, it was perhaps the Lord's table which made an exception to this rule, the focus being not merely Christ's atonement on our behalf, but the spiritual life he gives us by feeding us. But I digress.)

The closing hymn was none other than "Just a Closer Walk with Thee," which has the following lyrics:

I am weak, but Thou art strong,
Jesus, keep me from all wrong,
I’ll be satisfied as long
As I walk, let me walk close to Thee.

Just a closer walk with Thee,
Grant it, Jesus, is my plea,
Daily walking close to Thee,
Let it be, dear Lord, let it be.

Through this world of toil and snares,
If I falter, Lord, who cares?
Who with me my burden shares?
None but Thee, dear Lord, none but Thee.

When my feeble life is o’er,
Time for me will be no more,
Guide me gently, safely o’er
To Thy kingdom's shore, to Thy shore.

Now, far be it from me to say this song is without merit as a hymn of praise. It is not wrong to celebrate one's personal walk with Jesus Christ. And yet, an outside observer could be forgiven for concluding from these words that Christianity is fundamentally about worshiping an imaginary friend who comforts individuals through hard times and assures them that one day this sad, physical, temporal existence will give way to an eternal, motionless bliss.

To be clear, I don't think the evangelical summary of the gospel is a lie. I merely think it is not a summary of the whole gospel. Where is the cosmic significance of Jesus Christ? Where is the story of God's creation? Where is the hope for the redemption of the universe? These things are also part of the gospel, and they beckon us toward something much larger than the affairs of human beings.

Consider a different hymn, "Of the Father's Love Begotten":

Of the Father’s love begotten,
Ere the worlds began to be,
He is Alpha and Omega,
He the source, the ending He,
Of the things that are, that have been,
And that future years shall see,
Evermore and evermore!

At His Word the worlds were framèd;
He commanded; it was done:
Heaven and earth and depths of ocean
In their threefold order one;
All that grows beneath the shining
Of the moon and burning sun,
Evermore and evermore!

He is found in human fashion,
Death and sorrow here to know,
That the race of Adam’s children
Doomed by law to endless woe,
May not henceforth die and perish
In the dreadful gulf below,
Evermore and evermore!

O that birth forever blessèd,
When the virgin, full of grace,
By the Holy Ghost conceiving,
Bore the Saviour of our race;
And the Babe, the world’s Redeemer,
First revealed His sacred face,
evermore and evermore!

O ye heights of heaven adore Him;
Angel hosts, His praises sing;
Powers, dominions, bow before Him,
and extol our God and King!
Let no tongue on earth be silent,
Every voice in concert sing,
Evermore and evermore!

This is He Whom seers in old time
Chanted of with one accord;
Whom the voices of the prophets
Promised in their faithful word;
Now He shines, the long expected,
Let creation praise its Lord,
Evermore and evermore!

Righteous Judge of souls departed,
Righteous King of them that live,
On the Father’s throne exalted
None in might with Thee may strive;
Who at last in vengeance coming
Sinners from Thy face shalt drive,
Evermore and evermore!

Thee let old men, Thee let young men,
Thee let boys in chorus sing;
Matrons, virgins, little maidens,
With glad voices answering:
Let their guileless songs re-echo,
And the heart its music bring,
Evermore and evermore!

Christ, to Thee with God the Father,
And, O Holy Ghost, to Thee,
Hymn and chant with high thanksgiving,
And unwearied praises be:
Honour, glory, and dominion,
And eternal victory,
Evermore and evermore!

I have no idea why this is considered a "Christmas carol." It is a song which literally summarizes the entire gospel--from Christ's divine identity, to his creation, to his incarnation, to his redemption of humankind, to his final judgment. It calls in the entire universe to praise him, reminding the singer that human beings are not alone in receiving this good news.

This isn't just a long complaint about how the church needs "better music." Every element of worship is a choice affecting what members of a congregation will fix their minds on and remember. Worship is a reflection of our beliefs about God, and attending regularly will tend to shape and reinforce our beliefs about God. Just as one needs to be concerned with eating a balanced diet, so also, it seems to me, we ought to be concerned as Christians, whether our worship gives a balanced view of the gospel.

It is rather trivial to observe that most people focus on themselves most of the time. I'm not actually sure how true this is, exactly, but I know that any pastor can easily proclaim it as an unquestionable axiom of human existence. What is perhaps less trivial is how focused we are on our need for acceptance from others. Human beings have always been social creatures, dependent on family and tribe for survival, and this is only more true in a civilization in which most of our professions depend not at all on nature and the elements but wholly on ideas, technology, and our relationships with others. Beginning as children raised in schools, and continuing on as adults whose survival depends on successful performance reviews, we are obsessed with assessment. Perhaps it is no wonder that we present the gospel in these terms--God's view of our performance.

While it is certainly comforting to know that God is not like our boss--that he is loving, compassionate, patient, and forgiving--I think it would be even more eye-opening to have our gaze turned toward something other than God's assessment of us. When God spoke to Job in response to Job's complaints about suffering, he gave Job a tour of the whole creation. There is something about being reminded of our place in this universe which allows us to flourish in ways we had not understood before.

If Christian worship does not direct our minds and hearts toward the majesty of all creation, what will? Modern life doesn't reinforce the splendor of creation. We are obsessed with human creations--democracy, economy, technology, popular culture. I think it is our Christian duty to direct the eyes of human beings back toward the heavens, so that we might say with the psalmist, "The heavens are telling the glory of God," and, "What are mortals that you are mindful of them?"

Why proofs of God's existence don't thrill me

The other day one of my Catholic apologist friends (I have many online) posted a link to "An Aristotelian Proof of the Existence of God" given by Edward Feser. My friend claimed that atheists shouldn't start with the Bible, but rather with reason and logic, in order to arrive at belief in God's existence.

Now, I don't think atheists should start with the Bible, either. Most people don't when they come to faith--at least, not the whole Bible. If you try to read the good book cover to cover in search of some sort of proof of God's existence, you will only find it by a miracle (which I would not rule out, but neither would I hold my breath). Ordinarily, such a straightforward reading of Scripture will lead you to many scandalous passages and incomprehensible claims about God, and to stories whose meaning is rather obscure without some additional cultural context. Certainly the story of Jesus himself is provocative enough to sell itself, but even in the gospels one is as often perplexed as not.

But I'm not sure the classical "proofs" of God's existence hold much water. Professor Feser is certainly a brilliant man, and his exposition of the argument is solid, but I've just never been impressed with Aristotelian/Thomist arguments that The Unmoved Mover/Uncaused Causer/Purely Actual Being/whatever exists. The same holds for Anselm's Ontological Argument, and any similar methods of proof.

It's not that I wouldn't want to have such a proof. It's not even that I don't like entertaining the arguments. On the contrary, such abstract lines of reasoning are fun for me (which is why I chose mathematics as a profession).

The problem is, I just don't think abstract reasoning reveals God. As much as I love my profession, I don't think its tools are appropriate for theology. In fact, I think of mathematics as the simplest and humblest of all the sciences, whereas theology is the loftiest and most difficult. But I'll get to that.

The word "abstract" literally means "to draw away." It is well known that logical arguments in philosophy or mathematics lead us farther away from our direct experience. This makes them considerably less attractive as aids to knowing God, whose presence is everywhere and is therefore constantly part of our everyday experience.

Not that things which are omnipresent can't be the subject of abstract arguments. Atoms are everywhere, but we need theoretical physics to understand and know them. Yet the problem with atoms is that they are small, not that they are difficult to understand. In many ways, atoms are like numbers. They are highly predictable, following precise patterns expounded by the laws of physics.

Anyone reading this might find remarkable that I say an atom is not hard to understand. But I stand by that claim, and I say that math is even simpler. I appeal to a wonderful quote from the great John von Neumann: "If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." Mathematics and physics have made such incredible progress by concentrating on things which are simple, meaning predictable, utterly bound by rules. Thanks to this form of simplicity, our abstract arguments are applicable no matter how "drawn away" they become from direct experience.

Neumann hit the nail on the head: how complicated life is! How amazingly different it is from proving a theorem! Living things are not utterly rule bound. Human beings make choices, act spontaneously, and often actively resist the rules which are supposed to apply. How, then, do we seem so comfortable with other humans, and how do we know them so well?

The question would really be a mystery if abstract reasoning were our primal way of knowing, but that is not the case. There is nothing more natural in the world than getting to know other human beings, and it is not because we know what they will do at any given moment. It is rather because we have a built-in desire to be with others. More is gained through the direct experience of their company than through being able to predict their behavior. Only marketers are interested in understanding human beings the way physicists understand atoms. The rest of us just want to be loved.

Which brings me to proofs of the existence of God. No son has ever asked himself, "How do I know my mother exists? What are the properties she should have? From what starting principles might I argue toward the existence of such a being?" A son knows his mother because she is there--to feed him, clothe him, clean him, comfort him--and she always has been. Before a boy can do any abstract reasoning whatsoever, he must learn words, most likely from his parents. The primal form of knowledge is the direct experience of someone. Their presence alone is the start of every other kind of knowledge we could possibly have.

It is true that a son whose mother has long been absent must reason abstractly, saying, "All humans are born to a mother and father, therefore I too must have a mother whom I don't know." This is, at best, the type of theological knowledge which Aristotelian/Thomist/Anselmian/etc. arguments can give us. It is a theology of orphans. It is only useful or necessary if God is absent, and has been for a long time.

"Blessed are the pure in heart, for they will see God." If I understand the Christian faith correctly, I would say that the only thing separating us from God is our sin. God's constant presence ought to be perfectly clear to us, but we are blind. Christ does not offer theorems but rather a way to see again.

I catch glimpses of God from time to time. If I really began to fully repent, perhaps I would see him all the time in everything. But "surely there is no one on earth so righteous as to do good without ever sinning." (Ecc. 7:20) I only know it is that direct experience which sustains me.

I don't want to say that "existence proofs" are a total waste of time. God does often feel very absent, and perhaps we need these arguments to give us a hint of his presence. But I think it's very telling that in Scripture, to know God is to do justice, more than anything else. Christianity's great intellectual tradition notwithstanding, I say it is more a religion of doing than of reasoning.

And I think our theology ought to be done accordingly...

Edible words

And he said to me, “Son of man, feed your belly with this scroll that I give you and fill your stomach with it.” Then I ate it, and it was in my mouth as sweet as honey. (Ezekiel 3:3)

And he humbled you and let you hunger and fed you with manna, which you did not know, nor did your fathers know, that he might make you know that man does not live by bread alone, but man lives by every word that comes from the mouth of the Lord. (Deuteronomy 8:3)

I have sometimes wondered whether the Bible's prohibition against idolatry was really consistent. After all, aren't words themselves symbolic windows into God's existence, a sort of image for the ears rather than the eyes? One could argue that using words to describe God's ineffable existence is just as dangerous as trying to depict it through images.

Yet recently I was once again struck by the Bible's description of words as food. Funny thing about food: whatever you eat was once a living thing, and then it died.

And isn't that exactly how words function in reality? A single word is so short-lived. It has hardly left your mouth before it is dead, yielding totally to the words or silence which come after it. Even the written word functions this way. You cannot read by staring at a single word; you must let it die, letting your eyes advance continually in order to gain the whole idea of phrases and paragraphs.

Images do just they opposite. They impose upon you. If an image is before you, even if you want, you cannot simply let it pass away. It demands that you at least allow it a place in the background of your mind. It can change your mood without you even realizing it. It insists on immovability, a sort of immortality, if you will.

But Jesus said, "Heaven and earth will pass away, but my words will never pass away." The most permanent fixtures in all of ordinary human experience will die, but these words--which die as soon as they are born--will live forever.

It is the great paradox of Christianity once again. "Whoever would save his life will lose it, but whoever loses his life for my sake will find it." Images insist on their own immortality, but they are in fact lifeless. Words exist only to die, and thus they feed us, and so they live forever.

I don't mean this meditation to be polemical, but I will confess that it confirms my iconoclasm, which is standard for Protestants. Roman Catholics and Eastern Orthodox have good arguments in favor of icons. They argue that the incarnation of Jesus, the Son of God, changes and sanctifies physical reality in such a way that to prohibit icons would be to deny that incarnation. But I think these arguments nevertheless miss the most radical part of Christ's incarnation--he came to die. It is only because he died that he can feed us with himself. And let's not forget that when he rose again, he rose with a new body that was barely recognizable to his disciples. If there is an image of Christ to be used in worship, it is of course the bread and the wine; but even these are meant to be consumed.

Thoughts on Augustine's City of God

A while back I promised myself to read City of God. It was a bit like reading through Calvin's Institutes, in that it is a massive work written at a pivotal time in Christian history, and also in that Augustine and Calvin really do sound a lot alike (as much as two men separated by more than a thousand years can sound alike). I have not blogged through my entire reading of the book, as I did for Calvin, and I did not space my reading evenly over a year, but I think I still retain enough of an idea to summarize some of my reactions here.

Augustine was writing at a time just after Rome had been sacked. The world had changed. In light of that change, a defense of Christianity had to be given. I confess that I'm so underinformed about history that I still have trouble making sense of this--hadn't Christianity already become the religion of the empire? But perhaps this book alone sheds some light on how naïve it is to think that Christianity was ever really the religion of Rome. The Roman pagans gave what would be, I suppose, the natural response to Rome's fall: it's the Christians fault. And so Augustine launches himself into a critique of paganism in all its forms, followed by a philosophy of history based on the Christian Bible, in order to put historical events into a larger narrative.

I'm not an expert in how this book shaped history following that moment, but the very fact that we're here (Christians in the West, that is) means that it probably had a massive influence. After Rome fell, Christianity not only held on but continued to spread, eventually shaping all of what we now think of as "Western culture." (As an aside, I always find it terribly confusing when people think of Christianity as merely the "local religion" of something called "Western culture." As if there would be anything called "Western civilization" had it not been for the spread of Christianity which eventually united all of these European barbarians!)

Now to the text itself. The first ten books are devoted to refuting paganism, starting with gods and goddesses, then moving on to various pagan philosophies, and finishing with the Platonists, who Augustine says are the closest to the truth of Christianity. If you've ever engaged in modern debate on religion and theism, you've probably heard someone ask something like this:

"Why should we believe in the Christian God and not, say, Zeus or Poseidon? What evidence is there for one which there isn't for the other?"

It's true that in the modern world we don't feel the pressing need to refute the existence of Zeus. Augustine did. So if you'd like to know why Christians reject the existence of Zeus and other gods, you can read about 300 pages of painstaking refutations. Fair warning: you will, as a modern person, feel a bit silly reading them. Yet Augustine took the religions of his day seriously, as I think modern Christians take (or ought to take) Hinduism, Buddhism, Islam, and of course Judaism very seriously. (Note that Augustine never "refutes" Judaism, which was also an important competitor to Christianity. The implication seems to be that the only dividing line between Judaism and Christianity is the resurrection of Jesus Christ, and I suppose that despite the divergent evolution of the two religions, that's still ultimately true today.)

What gives me occasion to reflect very deeply about my faith is the fact that Augustine's refutations of paganism rest largely on moral grounds. The gods were either immoral, or were worshipped in immoral ways, or were celebrated in tales of immorality. Is not the same charge leveled today against the Christian God? The argument worked very well for Augustine, because in fact Greek and Roman philosophers really did find the popular religion quite embarrassing (I recall that Plato wanted to ban poetry from his ideal Republic, because it gave people false impressions of the gods). In some sense that argument works today on Christians, does it not? In my experience, when you talk to more sophisticated Christian theologians, they tend to finesse difficult biblical passages until they are allegorized or spiritualized into oblivion, or else they flat out reject those passages! On some occasions you will find a very stern theologian willing to bite the bullet on every biblical passage, but these find it more and more difficult to get a hearing in today's world, even within the Church. And I think this is not without reason. It is just plain difficult to see how the same God who would lovingly die on the cross for our sins, refusing to fight evil with evil, would also order the wholesale slaughter of women and children or even of anyone at all, as we read in the Old Testament. To me this question remains perpetually unresolved, which is either part of the beautiful mystery of Christianity or else a grave weakness which will ultimately be its undoing. Yet this is not a new problem by any means--it has been there since the beginning of Christianity--and here we are after 2000 years.

So much, I suppose, for the first half of the book. The second half is Augustine's attempt to explain all of human history. I read in the preface that many scholars think of this is the first "philosophy of history" ever written, and I suppose that's one way to think of it. It's hard to imagine you'd ever consider a "philosophy of history" something worth writing unless you already had one given to you by Scripture or something else. And that is indeed the case for Augustine. He looks at the Bible as a whole, and from its contents deduces an outline to all of human history based on the presence of two cities: the city of man and the city of God.

I won't outline the whole argument. I will instead tell you to simply go read the Bible, cover to cover, with Augustine's basic framework in mind. The city of man is all the children of Adam after being expelled from Eden, and the city of God is all those who by faith in God lived and now live in the hope of new life to come in Jesus Christ. Indeed, one way to think of Augustine's work is as a "philosophy of history," but perhaps a more direct way to think of it is as a synthetic exposition of all of Scripture. In that it really is akin to Calvin's Institutes.

There are other ways in which Augustine's and Calvin's works are similar, and that is because they appear to have similar theological bents. First, they both emphasize regularly the power and foreknowledge of God, the weakness of the human will after the fall, and the sheer unmerited grace which allows us to attain to the hope of new life in Christ. Second, they both base their arguments on the sheer authority of Scripture as an unquestionable guide in matters of doctrine. I think it's good to put this second point after the first, though some might believe it should go first--after all, it's from the Scriptures that we know about God's power and foreknowledge, isn't it? Yet it pretty clearly acts as an interpretive framework over Scripture, so that any time God is said to change his mind or make a decision, this is interpreted as merely a way to describe God in human terms, and not literally true. Everything else in the Bible might be literally true, but not those parts! This comment is not meant to prove that Augustine and Calvin are wrong, but only to show which ideas they truly put first. In my view, there simply is no interpreting Scripture without some first principle(s) guiding you, which must come from outside of Scripture itself.

Now, the reason I am emphasizing similarities between Augustine and Calvin is mostly because of my own experience reading through Calvin, and not to do any favors to Calvinists, who will take such comparisons as a compliment. Personally, I find that the things Augustine and Calvin hold in common are what make them both difficult to read. Consider their view of Scripture. Now, their justification of the authority of Scripture is slightly different: Calvin emphasized the direct confirmation of the Holy Spirit, while Augustine uses a somewhat different argument. Isn't it amazing, he says, how the whole world has come to accept the authority of the Bible? And this, he says, is also a fulfillment of biblical prophecy, which only bolsters that accepted authority. The argument isn't circular in either case, but it is rather frustrating, because it seems completely inaccessible to reason.

Let me stay with Augustine's argument for a moment. It interests me, because its plausibility in Augustine's mind must have been related to the historical context. Christianity had just seen, over the course of a few hundred years, a breathtaking rate of growth throughout a pagan empire, despite (or indeed because of) persecutions, all by the transmission of these words of Scripture. It is commonplace today to insist that "many other religions" have had similar experiences, but how many are there, really? Buddhism counts, I suppose, and perhaps certain sects within both the Christian and Islamic traditions. And of course there is Judaism with its own rather unique story, but it bears repeating that the line between Christianity and Judaism has always been, frustratingly, very thin. I honestly can't think of many others which have spread and flourished in quite the same way that Christianity did in its early years. Given all of that, I think we should be sympathetic to Augustine's argument. The words of Scripture changed people in a way that suggests real power.

Yet of course it's quite a logical leap to conclude that the Bible is therefore completely accurate in everything that it says. This is an aspect of Augustine's thought that disappointed me. I was set up for this by modern pastors and theologians of a certain intellectual type who like to bang on about how we don't read the Bible today the way ancient Christians did. They bring this up in the context of modern debates about faith and science, claiming that the early Church Fathers never did take Genesis 1 literally, and so why should we do any different? We should relearn to give sophisticated, spiritual readings of biblical texts, not focusing on their historicity but on how they point to Christ. I find this approach attractive.

It's true that from what I've read of Origen, that man truly did not take Scripture literally. But Augustine, let's face it--he's a fundamentalist. Now, that does not make him less intelligent, nor does it mean that his philosophical reflections are any less sophisticated. Some of the most profound reflections on joy and suffering in this world can be found in Augustine. And of course, Augustine loves allegorical interpretations of the Bible--as everyone does! I find it absolutely laughable when modern Catholic and Orthodox Christians assert or imply that Protestants hate allegorical interpretations, insisting only on the literal. They have clearly never listened to a Bible-thumping Baptist sermon, full of allusions and comparisons between Scripture and present-day events. It is simply impossible to read the Bible and not draw analogies between the texts and our own world, or between different texts themselves. No one has ever seriously denied this. But Augustine, like traditional Protestants (and all traditional Christians, I suppose, depending on what you mean by "traditional"), believes that the Bible should also be taken historically. He leaves no doubt that he thinks Adam was the first human being, that the world is only a few thousand years old, and that every word of the Bible is to be taken as telling us how human history happened exactly. To be fair, he is not so simplistic as not to notice all the problems of translation, all the mysterious numbers that appear in certain texts, and all the competing historical claims from nonchristian sources. Yet to a modern reader, it is quite difficult, if not impossible, to share Augustine's confidence in the veracity of Scriptural history.

All of this causes me more and more problems as the book draws to a close, where Augustine finally deals with judgment and salvation. I will have to save that for another post, because there is a lot to unpack.

Let me conclude this summary post by saying what I learned about myself by reading City of God. I learned that I am truly (sadly?) unfamiliar with the Greco-Roman world, but perhaps more at home than I thought I would be in the ancient Christian world. When Augustine talks about Homer or Plato, I struggle to recall the few basic lessons I learned in school, and my reading is slow. When Augustine quotes Scripture, I find myself reading quickly. It is not just what he says, but even the objections he responds to, which I find amazingly familiar. Truly, there are few theological controversies which divide Christians today which did not already exist in the early Church. We may have new language in which to express them, and perhaps new reasons to find them important, but we are not the first people to think of such problems.

Finally, I have to mention that City of God is a very different reading from, say, Confessions. I learned to love Augustine from reading the latter when I was much younger, and so I assumed (perhaps naïvely) that reading City of God would only reinforce that love. That is not quite true. I find now that I both love and hate Augustine, much in the same way that I both love and hate Calvin. That is perhaps an inevitable reaction to anyone with enough gumption to lay out a definitive history of the human race. You will find incredible beauty in this book, but you will not be able to look away from all the things which perplex and terrify you.

If I find time, I will come back to some particular passages of this book, because they deserve some meditation, but I think I've managed here to touch on all my general reactions. It's a work that certainly makes an impression, and I'm glad I took the time to read it.

Historical and Theological Adam

Peter Leithart's argument on historical Adam confuses me. On the one hand, he subscribes to Pope John Paul II's distinction between "historical" and "theological" humanity. In order to understand Christian ethics, it is necessary to go back to the beginning:

Historical humanity is the humanity enslaved by sin, turned here and there by mixed motives, the humanity of domination and control, of objectification and shame. But the Bible begins with an account of a different form of human life, which John Paul labels the “theological,” according to which humanity is made in the image of God to be God’s covenant partner.

...And this theological form of humanity is still normative. Christians can’t stop at the boundary. We can’t concede that there is only “historical man” in all his evil and shame and abuse. We can’t really understand the career of historical man, John Paul says, without knowing that it is a deviation from the original state of man.

On the other hand, having accepted this distinction between "historical" and "theological," Leithart goes on to assert that,

Unfortunately, Evangelicals appear eager to replicate the liberal error. Today, many, perhaps most, Evangelicals still believe that Adam was an actual man, but there is a growing body of opinion asserting the contrary.

What does "actual" mean in this context? Does it mean ... historical?

Leithart gives theological arguments which suggest that if we give up on the historical reality of the creation of Adam, we will lose our theological and ethical foundation. This is problematic because it simply forbids us from asking any questions about the material reality of Adam. If we buy his argument, we accept the existence of Adam as recounted in the Bible because of the theological implications that come with it. Forget whether or not archaeology, biology, geology, or physics might have anything to say about the possible physical existence of such a human being, or the age of the earth, etc.

While I have never much appreciated the notion that theology and science are wholly separated spheres, the other extreme seems equally problematic. N. T. Wright's description of biblical cosmology has always stuck with me: it's the intersection of two worlds. Just as in C. S. Lewis's Narnia series there are two distinct worlds which intersect, so heaven and earth in the Bible appear to have such a relationship.

So what of historical Adam? Is it necessary that we accept this belief on theological grounds? Would it not be more appropriate to accept the existence of theological Adam?

The common objection from evangelical circles is that this is really a way to say Adam never existed in any real way. My response: this is a limited view of "real." If heaven is real, indeed if God is real, but is not perceptible by the methods of historical humanity--that is to say by the sciences and reason--I don't see why the same can't be true of Adam.

It's difficult to capture the distinction, here, but I'll try to do so with the Resurrection, the fundamental turning point for Christian theology. Christians historically believe that the Resurrection really happened--that is, Jesus physically, literally, really rose from the dead. But what we don't believe is that there is any scientific explanation for this. How could there be? Science insists on explaining things in terms of things we can control or observe repeatedly. We can't repeat the Resurrection. If it happened, it happened once and for all, and we hope it will happen to the rest of us some day. What can we say about it, then? Is there an explanation of the Resurrection? The most we can say is that the power to raise the dead came from heaven. This is inaccessible to us, but we believe it is there, and we believe we will have access to it again one day.

So with Adam, what analogy can one draw? Adam is inaccessible to the sciences. We can't see him with archaeology, we can't verify the historical records and find he is there. But we believe he was there, in the sense that we believe humanity was in a real sense created in the image of God. We don't have access to this. We believe that one day we will, but for now, all we have is the historical tools of historical humanity, which don't have access to theological Adam.

I am not suggesting the Bible is describing another world which has no relation to ours. But then I am suggesting that the Bible is not describing solely our world. If there really are two world which intersect (and sometimes collide?) then it should not be surprising of sometimes the Bible describes things which are simply inaccessible to any rational tools we have.

Whatever the appropriate response may be, I've found conservative evangelicals consistently failing in addressing this issue, because they always give me theological reasons to believe something historical, rather than theological reasons to believe something theological (or historical reasons to believe something historical). As someone who believes both science and theology are very important, I believe the distinction is worth making and then exploring.

Correcting assumptions about religion and science

From the Browser, which is always a great source for reading up on the latest in modern thought, I found an article called "Christianity and the rise of western science" by Peter Harrison. Excerpt:

In spite of this widespread view on the historical relations between science and religion, historians of science have long known that religious factors played a significantly positive role in the emergence and persistence of modern science in the West. Not only were many of the key figures in the rise of science individuals with sincere religious commitments, but the new approaches to nature that they pioneered were underpinned in various ways by religious assumptions.

Could modern science have arisen outside the theological matrix of Western Christendom? It is difficult to say. What can be said for certain is that it did arise in that environment, and that theological ideas underpinned some of its central assumptions. Those who argue for the incompatibility of science and religion will draw little comfort from history.

Those who have magnified more recent controversies about the relations of science and religion, and who have projected them back into historical time, simply perpetuate a historical myth. The myth of a perennial conflict between science and religion is one to which no historian of science would subscribe.

This part was really on the money:

What historical record also suggests is that insofar as modern science posits natural laws and presupposes the constancy of nature, it invokes an implicit theology. Most important of all, perhaps, religious considerations provided vital sanctions for the pursuit of scientific knowledge and, arguably, it is these that account for the positive attitudes to science which have led to the high status of science in the modern West.

And this part actually surprised me a bit:

The adoption of more literal approaches to the interpretation of the bible, usually assumed to have been an impediment to science, also had an important, in indirect, role in these deveolopments, promoting a non-symbolic and utilitarian understanding of the natural world which was conducive to the scientific approach.

The whole thing is a good read. Very little of it was stuff I didn't already know, but I realize that sometimes it's not so much how new your ideas are, it's how often they get repeated. One could take the article as a healthy antidote to misleading "New Atheist" rhetoric. On the other hand, you could also take it as an exhortation to Christians in the west to earn once more the healthy relationship with science that we once had.

Folks on all sides need to remember that religion is extremely diverse, and as a result it's hard to pin down ideas and label them correctly. Christian ideas played a role in bringing about the rise of modern science, but not all Christian ideas played the same role. All I would ask of atheists is that they admit that science comes with a certain theology--not necessarily a Christian or otherwise traditional theology, but a theology nonetheless. Likewise, I would ask Christians not to label something as outside the realm of Christian thought simply because it doesn't align with their own traditional beliefs.

What the article addresses most thoroughly is the Christian conception of nature and the relation between that conception and western science. Where I think it gets interesting is in more recent years, when, starting with Darwinian biology and continuing on with psychology and neuroscience, science begins to penetrate into questions about ourselves. No longer is science about this vast sphere of reality known as nature, with human beings attempting to be objective observers. Now human beings are studying themselves, trying to determine where we came from and by what laws we behave.

Small wonder that this creates such a great deal of tension with religion. Christianity, like many religions, is largely a story about who we are and where we came from. As modern science begins to challenge that story, it creates conflict where there was none before. It's one thing to say that time started with the big bang; that doesn't necessarily change the story about ourselves. It's quite another to say human beings have a common ancestor with other species of living things. That definitely changes the story.

Then again, it's all in how you tell it, isn't it? The Genesis account has human beings made both in the image of God and from the dust of the earth. One suggests great power and dignity, the other suggests fragility and corruptibility. Does the story modern science tells really change so much?

I don't know the answer to that, I'm just offering some space for better theological discussion. At the very least, we all need to realize that at the heart of everything we do is some sort of theology, good or bad. The false assumption that we can get by without thinking theologically at all is helped along by these great historical myths about the inherent conflict between science and religion. We'd be much better off having real theological discussions, instead of arguments in which we either assume our theology is basically correct or else pretend we don't need any.

On faith and mathematics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The essence of mathematics lies in its freedom.

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

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

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

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

...

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

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

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

A theological defense of freedom

I was going to write my own essay on this topic, but I found as I was reading The Brothers Karamazov that the chapter entitled "The Grand Inquisitor" says all I need to say and more. Ivan tells the story of The Grand Inquisitor, a priest during the Inquisition who finds that Jesus Christ (or one we may take to be Jesus) has returned and is healing people. Upon this discovery, the priest has Jesus arrested, and goes to address his prisoner:

"'Is it Thou, Thou?' but receiving no answer, he adds at once, 'Don't answer, be silent. What canst Thou say, indeed? I know too well what Thou wouldst say. And Thou hast no right to add anything to what Thou hadst said of old. Why, then, art Thou come to hinder us? For Thou has come to hinder us, and Thou knowest that. But dost Thou know what will be tomorrow? I know not who Thou art and care not to know whether it is Thou or only a semblance of Him, but tomorrow I shall condemn Thee and burn Thee and the stake as the worst of heretics. And the very people who have today kissed Thy feet, tomorrow at the faintest sign from me will rush to heap up the embers of Thy fire.

...'Didst Thou not often say then, "I will make you free"? But now Thou hast seen these "free" men,' the old man adds suddenly, with a pensive smile. 'Yes, we've paid dearly for it,' he goes on, looking sternly at Him, 'but at last we have completed that work in Thy name. For fifteen centuries we have been wrestling with Thy freedom, but now it is ended and over for good. Dost Thou not believe that it's over for good? Thou lookest meekly at me and deignest not even to be wroth with me. But let me tell Thee that now, today people are more persuaded than ever that they have perfect freedom, yet they have brought their freedom to us and laid it humbly at our feet. But that has been our doing. Was this what Thou didst? Was this Thy freedom?'"

The Inquisitor goes on to explain to Jesus how during his trial in the wilderness, the three questions asked him by the devil were really the three questions which define mankind: the need for bread, the need for someone to worship, and the need for a figure of authority.

"They will find us and cry to us, 'Feed us, for those who have promised us fire from heaven haven't given it!' ...And we alone shall feed them in Thy name, declaring falsely that it is in Thy name. Oh, never, never can they feed themselves without us! No science will give them bread so long as they remain free. In the end they will lay their freedom at our feet, and say to us, 'Make us your slaves, but feed us.' They will understand themselves, at last, that freedom and bread enough for all are inconceivable together, for never, never will they be able to share between them!

..."But only one who can appease their conscience can take over their freedom. In bread there was offered Thee an invincible banner; give bread, and man will worship thee, for nothing is more certain than bread. But if some one else gains possession of his conscience--oh! then he will cast away Thy bread and follow after him who has ensnared his conscience. In that Thou wast right. For the secret of man's being is not only to live but to have something to live for. Without a stable conception of the object of life, man would not consent to go on living, and would rather destroy himself than remain on earth, though he had bread in abundance. That is true. But what happened? Instead of taking men's freedom fro them, Thou didst make it greater than ever! Didst Thou forget that man prefers peace, and even death, to freedom of choice in the knowledge of good and evil? Nothing is more seductive for man than his freedom of conscience, but nothing is a greater cause of suffering. And behold, instead of giving a firm foundation for setting the conscience of man at rest for ever, Thou didst choose all that is exceptional, vague and enigmatic; Thou didst choose what was utterly beyond the strength of men, acting as though Thou didst not love them at all--Thou who didst come to give Thy life for them! Instead of taking possession of men's freedom, Thou didst increase it, and burdened the spiritual kingdom of mankind with its sufferings for ever. Thou didst desire man's free love, that he should follow Thee freely, enticed and taken captive by Thee. In place of the rigid ancient law, man must hereafter with free heart decide for himself what is good and what is evil, having only Thy image before him as his guide."

The Inquisitor goes on to explain to Jesus how the Church will correct his work, for they will take away their freedom, and they will gladly surrender it to them in exchange for happiness:

"Freedom, free thought and science, will lead them into such straits and will bring them face to face with such marvels and insoluble mysteries, that some of them, the fierce and rebellious, will destroy themselves, others, rebellious but weak, will destroy one another, while the rest, weak and unhappy, will crawl fawning to our feet and whine to us: 'Yes, you were right, you alone possess His mystery, and we come back to you, save us from ourselves!'

The whole story is very much worth the read.

It's worth noting that Ivan has some sympathy for the priest in this story. What if it is not sheer corruption that leads to the desire for more control, more power? What if it really is a desire to save humanity, because God cannot? Making human beings free often seems the worst punishment of all. If only we were instead taken care of!

Perhaps I'll still write my own theological defense of liberty, but Dostoevsky does what I could never do: he forces you to face the grand contradictions which make life what it is.