It can't be true!

Yesterday I had the joy of sharing Cantor's famous diagonal argument, proving that the real numbers form an uncountable set. This semester, instead of teaching my own class, I get to be a TA for an honors Calculus III course, which goes through many advanced topics to prepare students for higher level mathematics. As a result, I get to share moments like these, in which students go from being calculators to being real mathematicians.

Mathematics finally means something to a student when he encounters a proof of something that he can't believe. I had at least one student express his inability to accept the diagonal argument after I gave it. Of course, the proof is irrefutable--it is logically sound, and despite the large number of attempts made every year by crank mathematicians, it can never be overturned.

That's part of the sheer beauty of mathematics. It shows how pure logic can yield surprising results. Why are we surprised by things that are logically irrefutable? This is not a mathematical question, but a question about the human spirit; yet in some respects it is one which only the mathematician can encounter. No one else can experience quite the same thrill, or frustration, at coming face to face with those results of pure logic that seem to break down every sense of intuition you ever had.

Why does Cantor's proof frustrate students so?

The result itself frustrates people in general, at least in my experience. When you explain that the real numbers can never be counted, even if you counted them for all eternity, even if you go all the way to infinity, they do not believe you. Or they don't understand what you even mean. The young student of mathematics, on the other hand, can see and accept each line of the proof; yet much like any other person, he is intuitively troubled by the result itself. This manifests itself in the form of complaints about the proof; yet what it always seems to come down to is that the result is incomprehensible on any intuitive level.

Why is this? I don't really think that it is because the logical steps are hard to follow. Indeed, I think that even a person relatively untrained in mathematics can understand each step in the proof. What I think people can't really and truly deal with are the starting assumptions themselves. To embrace those assumptions takes imagination.

What is infinity, in the first place? To the extent that it touches upon our every day experience, infinity can actually be something quite small. The national debt, for instance, is for all practical purposes, infinite. No single human being can fathom having control of that much money. If I had $13 trillion in my bank account, I would simply never run out of money. I could spend $5000 every second for the next 80 years and still not run out of money.

Infinity, then, is a guarantee: there's always something left. The truth is, though, 13 trillion can be a very small number of some things. Avogadro's constant is somewhere on the order of a trillion trillions; yet this ridiculously large number is equal to the number of molecules in a measly 32 grams of oxygen. Numbers in this universe get so ridiculously large that our minds stop distinguishing between them all. (This has unfortunate political consequences, as people have yet to truly comprehend how astronomical their own governments' spending really is.)

So then, one who is willing to stretch out with the imagination can make this assumption: for every number n, there is always a number n + 1. You can always go, as Christopher Guest on Spinal Tap might put it, "one louder." We have formalized the guarantee of infinity. We have assumed the existence of an infinite set of numbers. One, two, three, four, five, ... ten, ..., twenty, ... thirty, ... one hundred, ... one thousand, ..., one million, ..., one trillion, ... why should it ever end?

But that isn't the only kind of guarantee one seeks in this world of computation. We also seek the guarantee of being really close to something. For instance, there is this ugly business of finding the circumference of a circle. A circle, of all things! A beautiful, simple shape--nothing could be simpler, really. And yet, the ratio between its circumference and its diameter is such a monster of a number that we are forced to give it a Greek name--"pi"--and leave it at that.

Or are we? We all heard in grade school that pi = 3.14... People sometimes wonder if I have all the digits of pi memorized, although some more sophisticated folks know that you can't memorize all of them but still wonder how many I know. Well I know up to about eight, I think: 3.1415926... And I've heard school children will sometimes have contests to see how many digits of pi they can memorize. Yet no matter how many digits they memorize, they will never have actually gotten pi. Never. Not in a million billion trillion years--not ever.

But they are getting closer and closer. How much closer is rather easy to quantify. If I guess pi = 3.14, then I am within one-hundredth of pi--that is, pi is between 3.14 and 3.15. If I guess pi = 3.1415926, then I am within one 10 millionth of pi. That's pretty close. Never exactly right, but closer and closer. So that's my guarantee. I can always get closer.

There are plenty of other common ratios in the real world which have no exact finite decimal representation--in fact, no repeating pattern ever emerges in their decimal expansions. The square root of 2 is one of them--this is simply the length across the diagonal of a square. The square roots of most numbers are the same way, and these can all be represented using common, everyday shapes. In each case, we do have a guarantee, as we did with pi. We can get closer and closer by taking more and more decimal places. That is, we can take the ratio between two ordinary natural numbers and be as close as we want to be to a weird number like pi.

Now what if I imagine that there is a number at the end of every conceivable decimal expansion? I say to myself, "Look, I can just start typing numbers after a decimal point, and there's no stopping me." In fact, just like at me as I cough up numbers right now:

.011923985461928384093745601092983562390523984691348612340213901098234908123984...

Try it! It's kind of therapeutic, actually.

Now I stretch out my imagination and put my faith in a new assumption: that this process of picking new digits can continue on forever and ever, and that no matter how the digits are picked at each step, the result can rightfully be called a number. I don't know what that number is, but I know that I'm getting closer and closer to it with every arbitrary choice of a new digit. Just as it took imagination to take on the assumption that there is always one more number, so it also takes imagination to embrace this new assumption. It means formalizing a guarantee.

It is, in a sense, an act of faith. Implicitly it means trusting that guarantee to have some sort of meaning. Otherwise, what would be the point of studying the logical results of that guarantee?

If you've gone with me this far, if you have enough faith to believe that after every number n there's always n+1 and that any decimal that can be continued on forever should be considered a number, then congratulations! You have, more or less, just embraced what mathematicians call the real numbers. If not, don't worry. Chances are your world can do without such big numbers. Even the national debt is probably higher than you'll ever need to imagine.

Young students of mathematics have generally been unwittingly indoctrinated into having faith in the real numbers. Duh, there's always a bigger number, and of course every decimal is a number. Why would we have to "imagine" that? Why, indeed! The ancient Greeks didn't even believe in zero.

Our educational system teaches these principles from a very young age. Enter pi in on your calculator. See? A decimal comes up! And those digits can keep going and going... Thus the youth are catechized into the traditions of their elders, unaware that without imagination, none of the structure in their mathematical universe could ever have arisen.

No wonder young mathematicians are so shaken by Cantor's proof! Indoctrinated into the assumptions which the proof begins with, these young minds are totally unprepared to handle the consequences of those assumptions. For as soon as the mind willingly submits to the two assumptions I have fleshed out here, then simple logic reveals a truth that is so staggering that it has drawn downright hostility from philosophers and logicians ever since Cantor first made his argument.

The claim is simple. Take all of those counting numbers: 1, 2, 3, 4, 5, 6, 7, ...
And I mean all of them, all infinitely many of them, because you know there is always one more! And now to each one of those counting numbers, assign some decimal, like this:

1: 0.123846130865861902034109826394384...
2: 0.988349102039136358923403948190234...
3: 0.238463985658654865435864238653428...
4: 0.843565643854843643874398340954893...
5: 0.458430938230483240893291293048239...
6: 0.789234923902309238942394823094872...
7: 0.097138428094217089340873078320683...
...

and so on. Except, don't just do it how I just did it--do it however you want! Be completely arbitrary!

Here's what happens: you'll never get all of the decimals!

Never! Never ever! No matter how cleverly you chose which decimals to match to each of your counting numbers! Even though you always have one more counting number--that is, even though there are infinitely many counting numbers--you still don't have enough. The decimals are to the counting numbers what the national debt is to your savings account.

And the argument is quite simple: just read down the diagonal of that list you just made. Take the first digit of the first number, change it to another digit, and write it down. Take the second digit of the second number, change it, and it write it down next to the first one you just wrote. Do the same with the third, and the fourth, and so on. You'll get a new decimal. Mine would start to look like this:

0.2996455...

All I did was add 1 to each of the "diagonal elements" of my list. You can't add 1 to a 9, but you can just change 9 to 8, and the same idea holds.

Now, is this new number in your list? No! It can't be. It's not the same as your first number, because the first digit is different. It's not the same as your second number, because the second digit is different. It's not the same as your third number, because the third digit is different. And so on, even for every single counting number.

That's Cantor's proof. As shocking as the result is, the proof is nothing more than simple logic. Yet logic has to build on certain assumptions, and it's really those assumptions that set up this amazing result. It must have been that Cantor was the first person to fully buy into all of those assumptions. He was a mathematician of true faith, and true imagination.

And also true bravery. When I say that Cantor fully embraced those assumptions about numbers, I mean that he was even willing to embrace the logical consequences of those assumptions. That is faith. And without it, there can be no progress.

I'm allowed to say such things, because it's my blog. But I seriously wonder, how many assumptions do we take for granted, yet without being willing to accept their logical consequences? It is often only when someone shows you what the logical consequences are that you're able to see what the assumptions actually mean.

This, to me, is what's so liberating about mathematics. Whatever faith you have in your assumptions will be thrown to the fire to be tested. You must seek out the logical consequences of whatever you start with. And if you are able to overcome your initial fear, you might just find that the universe is a much grander, more majestic, and more mysterious place than you had ever imagined.

Beauty for Truth's Sake - a review

If I could pick my top five topics to think about on a regular basis, three of them would be faith and reason, education, and, of course, mathematics. Thus Beauty for Truth's Sake: On the Re-enchantment of Education seemed like a perfect book for me to read and contemplate in my spare time. In it Stratford Caldecott offers something of a "manifesto" summarizing a view of education which sees the classical Western Liberal Arts tradition as a means of integrating faith and reason, art and science.

There is much to appreciate about the book. The beginning is quite strong, displaying a clear sense of what has gone wrong in modern education on the philosophical level. The opening lines read,

"In the modern world, thanks to the rise of modern science and the decline of religious cosmology, the arts and sciences have been separated and divorced. Faith and reason often appear to be opposed, and we have lost any clear sense of who we are and where we are going."

Why is this a problem? I think Caldecott sums it up nicely when he says,

"The purpose of an education is not merely to communicate information, let alone current scientific opinion, nor to train future workers and managers. It is to teach the ability to think, discriminate, speak, and write, and, along with this, the ability to perceive the inner, connecting principles, the intrinsic relations, the logoi, of creation...

As the title of the book suggests, one of the key integration points between these seemingly disparate areas of knowledge is beauty. "Everything," Caldecott says, "is true, good, and beautiful in some degree or in some respect.... Beauty is the radiance of the true and the good, and it is what attracts us to both." In particular--and this is part of what I found intriguing about the book--mathematics is a key to understanding how the classical tradition sought to "perceive the inner, connecting principles" of the universe.

"Theology, therefore, has an important place in the integration of the arts and sciences. Equally important, however, is a symbolic approach to number and shape--that is, the awareness that mathematics has a qualitative, as distinct from a purely quantitative, dimension."

Caldecott spends a couple of chapters in the middle of the book illustrating that qualitative dimension of mathematics, using examples from the ancient Pythagoreans, numerology from the Bible, and various other musings on the relationships between numbers, shapes, and the world around us. It is a rather delightful survey, ranging from the Tetractys to the five Platonic solids to the golden ratio. It really is a shame, in my opinion, that modern mathematical education leaves very little room for this kind of appreciation of mathematical objects.

Let me briefly outline the structure of the book, to show how these ideas are expressed as a whole. The Introduction states the problem as I have, and offers three guiding principles. The first is, "The way we educate is the way we pass on or transform our culture.... The fragmentation of education... is a denial of ultimate meaning. Contemporary education therefore tends to the elimination of meaning...." The second is, "The "re-enchantment" of education would open our eyes to the meaning and beauty of the cosmos." The third is, "The cosmos is liturgical by its very nature." In this way Caldecott makes it clear from the start that education can never find true integration without a religious foundation. This raises interesting practical questions, but this book doesn't deal with them.

Chapter 1 sets about calling us to return to an idea of education as a means of becoming "truly free, fully human." Caldecott wants us to see that education is not just about what is useful. It is, in the great Socratic tradition, about gaining knowledge of "the forms, or the highest causes," which one can only attain "through the systematic ordering of the soul." In Chapter 2, he shows us that this path requires awakening the "poetic imagination," the ability to find "within the self something that corresponds to the object, thus leaping over the barrier between self and other." Thus symbolism comes to play a key role throughout the remainder of the book. Chapters 3, 4, and 5 essentially serve to illustrate the "poetic imagination" at work throughout the Western tradition. I've already mentioned the mathematical objects; there are also a number of excursions into theology, as well as music, architecture, ecology, and astronomy. Throughout these chapters one gets the sense that while Caldecott may not be advocating a return to a medieval understanding of the cosmos, he certainly seems to have an affinity for it. Chapter 6 argues quite strikingly for that third principle mentioned in the introduction, that "the cosmos is liturgical by its very nature." He thus argues that any real education must involve elements that are at least implicitly religious.

The conclusion is perhaps the most explicitly religious, in fact explicitly Catholic, part of the whole "manifesto." He says, "As we have seen, the Liberal Arts were intended to conduce to freedom of mind, and they were developed and nourished by the Catholic Church." He then explains that the modern conception of freedom is deprived of a certain fullness that is granted by the Christian (Catholic) view.

"The best way to put this might be that the Christian conception of freedom is larger and fuller than the modern conception, for it includes both vertical and horizontal dimensions. The horizontal dimension encompasses the world we see directly, and the vertical allows for degrees of being and value, invisible realms, formal causality, and so on. ...

In the traditional "three-dimensional" world, the self was encouraged to collect itself together in a point, in order to attach itself to a vertical axis, a spiritual "path." ... Modernity, on the other hand, rejects the existence of the vertical altogether, or the very possibility of thinking in terms of up and down. ...

In a flatter universe, freedom had to be reconceived as entirely a matter of movement within the horizontal plane. I am assumed to be "freer" the more places I can go to, the more things I can choose on the supermarket shelf, the more people I can have relationships with. And that is why the Church claims today to be in the business of liberating human freedom, by making known the beauty of truth in its fullness.

Now that I've summarized the book, let me get into my complaints. I agree with Caldecott that the fragmentation of knowledge is symptomatic of some deep problems. I also agree with the basic idea of seeking beauty in the universe, and that the pursuit of knowledge is, at its core, about love. But I would challenge some of the assumptions of his "Christian Platonism."

Before I do that, though, let me make some slightly more superficial comments. I have to say, and I think many readers would agree with me, there were many times during the reading of this book when I thought "Re-enchantment of Education" simply meant redecorating the universe with medieval superstitions. Certainly Caldecott was aware of this as he was writing, which is why he made the occasional remark that he is not trying to undo the Enlightenment or go back to the Middle Ages. Yet these remarks have little force behind them; he doesn't seem to have anything good to say about the Enlightenment in any meaningful sense.

On a related note, Caldecott relies so heavily on his own Catholic tradition that it is difficult for readers outside that tradition to understand the appeal of his illustrations. For instance, he mentions more than once how cosmically significant it was for Christian that there are seven days in a week and seven sacraments. Well, suppose there aren't seven sacraments... Does that mean the universe is less enchanted with meaning? In one part of Chapter 4 he wanders off into a discussion about the filioque controversy, a subject which is thoroughly uninteresting to many of us. Even more importantly, such controversies aren't settled by geometric arguments, and Caldecott's references to the relationship between mathematics and theology are more likely to offend believers of different theological persuasions than they are to enlighten anybody.

Chapter 4 ends with a paragraph that begins, "Speculations like those I have mentioned in this chapter will appear forced to many." Believe me, they did. I found circles and lines to be wholly inappropriate for trying to visually represent the Trinity. I found the comparison between Jesus and the line perpendicularly connecting a point on a circle to a given diameter also rather "forced." And the ratio between this line in the "golden circle" and the circumference, why, it's miraculously just over 7! The amount in excess of 7 which we find in this ratio is, of all things, what Caldecott thinks might correspond to that "tiny and indispensable human contribution needed if heaven is truly to descend to earth." In other words, heaven divided by earth = 7 (God's number) + some tiny human contribution = pi * ((2 * phi) - 1) = 7.02481473...

It is worth noting here that the number of man's symbolic contribution in this calculation is irrational. All this to say, one has to be very careful before going off to find the logoi of creation. This search can easily degenerate into such absurd arguments as "there are seven sacraments because there are seven days in a week." It might be postmodern of me, but different cultural perspectives really are worth keeping in mind as we examine the connections underlying things in the world. For instance, the octave interval in music might very well have a special relationship with the number "eight" in the West, where eight might have special theological meaning (on the eighth day Christ rose again) or other kinds of meaning. But, lest we forget, this association is based on Western musical scales. Other cultures have more varied intervals, and therefore the association doesn't work. That's often how it goes with Platonism. You think you've found the form of which all the world is a reflection, but then you realize it's just your own perspective being forced on the world around you.

Now I am starting to get into my deeper qualms with Caldecott's assumptions. Consider this third vertical dimension of human experience, which he proposes in his conclusion. It is as if he says we ascend to heaven through the illumination of education (the right kind of education, anyway). This is indeed a very Platonic way of viewing things, but not a very Christian one. In the gospels, wisdom is given to ordinary people. It is all about grace, not enlightenment through systematic human effort. As I have already suggested, there is no guarantee that such systematic efforts would lead one closer to heaven. Human beings have difficulty seeing the difference between the beauty inherent in the universe and their own prejudices based on cultural conditioning.

Another point I would make is that this Platonic notion that Ideas are ultimate reality (the "thoughts of God," for a Christian Platonist), and all else is reflective of these pure Ideas, is in some sense to deny the goodness of creation. Creation has its own reality that is not a mere shadow of something else. I'm sure this point has been made plenty of times by people much more theologically astute than I, but from my own perspective it is important to recognize that every single thing you see and feel has its own existence. Yes, it is all made of the same basic stuff--i.e. matter and energy governed by universal laws of physics. But to have the underlying principles is not to have the thing itself. The world is not translucent. I don't agree with this idea of looking at the world as if the only thing that makes it good is being a channel through which to see something else.

In terms of application I think this point can be rather significant. As in I don't think it's necessary or fruitful to link every scientific discovery to some theological precept. I don't think mathematical objects have some inherent mystical meaning. In terms of mathematics education, I do think it would be helpful for people to be taught that mathematics is more about inner relationships than it is about formal operations, which can have nothing to do with reality. Far from vindicating Platonism, however, I think this just points for the need for human beings to be connected to things. Caldecott gets it right when he talks about the "poetic imagination," at least insofar as he describes human beings as inherently connected to creation. But I think he gets it wrong when he posits a realm of pure Ideas which have some higher reality than the world of tangible experience.

The last thing I'll point out is that Caldecott seems, in spite of himself, to miss the strong distinction between Platonic idealism and Christian realism. Related to Platonism is, I think, a tendency to try to escape realism. If the most important thing is to ascend to the Forms, then it becomes less important to actually deal with the gritty details of the real. I submit that this is a theological weakness in Caldecott's understanding. He spends some time in first chapter talking about "Beauty on the Cross." There is, paradoxically, a great deal of beauty on the Cross of Christ, but the other side of that paradox must always be remembered. The crucifixion was a gruesome, grotesque thing. The Incarnation itself was an "emptying" of Christ to the point of humble obedience.

It is important that we as Christians remember this, so that we, like Christ, can enter into the world as it actually is, and not as a mere reflection of perfect Ideas. True education demands a certain realism. We cannot gloss over the details. We must be willing to face the world as a complicated, often frustrating place. We're not going to be able to transcend uncertainty and confusion. Even Christ himself prayed in the Garden of Gethsemane, "Father, if it is your will, save me from this hour." And on the cross he cried out, "My God, why have you forsaken me?" How much less will we be able to transcend the uncertainty of this life? Caldecott criticizes the postmoderns for doubting the human ability to obtain truth, but to a certain extent the postmoderns are right. It is not necessary to believe that truth is "relative" to have a healthy skepticism about the human ability to possess truth. We have to be realistic about ourselves, and about the world we live in. And instead of trying to transcend this world, we ought to follow Christ's example; he stepped down from his position of transcendence, that he might enter into this world--not, I believe, to show us the way out of this world, but to begin to transform it.

Caldecott is right about the most important thing: it all comes down to love. It is difficult to know what direction love should take us. While I don't agree with the direction Caldecott proposes, I think he is right to give us an alternative to our current approach to education. We do live in a beautiful universe, and it would be a waste to treat it in the purely utilitarian way that students are encouraged to now. Perhaps there is a better approach, one that relies not so much on enchantment as on the love of the universe for what it really is.

Beautiful Mathematics

Peter Kuchment is here helping to teach us about Radon transforms and medical imaging. He described the way in which Johann Radon originally discovered the Radon transform: it was just a side project, for fun, without any "useful" purpose. We now know that the mathematics Radon did as a side project in 1917 are foundational for computerized tomography, for which Allan Cormack and Godfrey Hounsfield won Nobel Prizes in physiology in 1979.

Kuchment suggested that Radon probably worked on the Radon transform out of sheer curiosity, simply because the idea was beautiful. He said, "If it's beautiful, it must be important." Mathematicians work on problems for their own sake; but the great mathematicians, according to Kuchment, always seem to find problems beautiful which also have a long term impact on the sciences. The lesson to be learned from this is that one ought to pursue mathematics for the sake of its beauty, and that "if it's not beautiful, it must not be that important, anyway."

There seems to be an implicit theology at work here. Truth, beauty, and goodness are unified in a transcendent way. The pursuer of truth needs not only logical skill but also a heightened sense of beauty. He who discovers what is truly beautiful also discovers both what is true and what the world needs. That the universe should really be this way is a pretty remarkable presupposition. I find this presupposition both deeply theological and deeply compelling.

It might take a bit of sheltering to preserve this kind of faith; I don't know for sure. After all, this world often seems fragmented. Rampant consumerism, promising to yield all that the world needs, has often dulled our sense of what is beautiful. The cultivation of "high art" has often been disconnected from anything true or meaningful. And the rigorous pursuit of scientific "facts" has often been done without regard for anything actually worth valuing, right down to scientific progress being used to destroy human life rather than preserve it. Truth, beauty, and the good seem miles apart from one another, and most people actually seem to stop believing in one or more of them. Thus cynics on the one hand accept the hard facts of life while rejecting its inherent beauty, while optimists on the other hand reject facts for the sake of beauty and goodness.

But then, mathematics is a subject that demands patience. It is a pursuit of the beautiful, but it doesn't always appear so. If we want to find beauty, we cannot listen simply to our hearts; we have to take the hard path of contemplation and rigorous logic. And if we seek what is truly beneficial to society, there again we must be patient. Without taking the long journey of pursuing what is beautiful and abstract, we can never lay a foundation for more powerful applications. So in some sense the belief in the unity of truth, beauty, and goodness is hard earned. It doesn't come naturally but must be slowly cultivated over time.

In this sense, I find mathematics one of the most deeply spiritual things we do as human beings.

Logic and Faithfulness

One question I'm fascinated by is whether or not the rules of logic are necessarily right. That is, can we imagine a universe where the rules of logic are different? I realize this is an ill-posed problem. It is very easy to point out that the word necessarily is rather ill-defined, and therefore one can go around in circles all day arguing both sides of the question to no avail. But the question still intrigues me, especially since it has to do with the very foundations of my work as a mathematician.

Where do the laws of logic come from? Do we simply take them as axiomatic and then move on? Why do we attempt to constrain our use of language to a set of precise rules that can never be broken? I am speaking, of course, only of subjects such as mathematics where logical precision is a defining characteristic. Venture into poetry, speculative philosophy, or mystical theology and I'll grant that logic does not need to constrain our use of language.

As I grow accustomed to reading mathematical papers, and especially now that I have been introduced to the task of reviewing a paper (which involves the daunting task of checking it for correctness) I have noticed just how essential it is that words are, in some sense, faithful. For one thing, every proof depends on the fact that symbols don't change their meaning from page to page. For another, proofs of theorems constantly rely on lemmas that must in turn be based on earlier knowledge. This, too, demands some sort of faithfulness on the part of words and their corresponding concepts--I need to know that every time a certain set of hypotheses is satisfied, I get a desired conclusion.

When a mathematician defines a term, he is really establishing certain relationships between words. He is building a certain kind of trust. Whenever you see these symbols, he says, you can count on these other symbols being connected in this way. Theorems are maps of those relationships that take you much longer distances than the bare definitions. Think of a family tree. Each child is the child of two parents--this corresponds to a mere definition. But every child can be connected to myriads of other people by tracing his ancestry back to common ancestors and then down through some other line of descent. This is kind of what theorems are. Definitions are immediate connections that we create; theorems are broad connections that establish some sort of common bond that may have been unexpected.

How could there be any logic in the universe if there were no faithful relationships? If we had never encountered the possibility of trusting in someone or something, it's hard to imagine developing logical reasoning. If our minds never retained connections, never established relationships between things, then all hope of doing mathematics in particular and logic in general would be lost.

Whatever codified rules of logic we may have inherited from the Greeks on down, it seems to me that logic is really the simple manifestation of the faithfulness of the Creator. In some sense, logic is faithfulness. Language becomes illogical when words are used in an untrustworthy manner. You say one thing but you mean another; you use the same word to mean multiple things; you follow one sentence with another sentence without remaining faithful to the former.

There is a moral element to logic. It isn't just some abstract, impersonal idea. We hear this constantly if we listen to scholars talk about their interpretations of historical documents, particularly in biblical studies. They say, "I have tried to remain faithful to the text." There must always be a moral commitment in order for there to be any logic. Hence the Proverb, "The fear of the Lord is the beginning of knowledge."

So do other things such as poetry deviate from this faithfulness? I don't think so. As long as the reader understands that words may not always mean exactly one thing, and that words may even contradict themselves to create something beautiful, then no faithfulness is lost. Like I said, our codified laws of logic are no substitute for the real heart of the matter: logic is faithfulness in words. Poetry and music have their own logic, for they have their own inner faithfulness. And when that faithfulness is not present, a good listener will notice. It is no different from a trained mathematician being able to spot a mistake in a mathematical research article.

Thus whenever we hear of a modern society based on reason and logic as opposed to faith in God, we must ask ourselves, on what are reason and logic based? They are always based on a relationship of some kind. Throw out any God save ourselves, and what are we left with? To what, exactly, are we to be faithful to?

Imagining God

I've been thinking for a little while about this "Name-Worshiping" idea that I've blogged about recently. It reminds me of something I've thought for a long time.

A lot of times you hear more or less an accusation that religion invents God for whatever reason--we need something to justify our existence on this planet, we need a crutch to get through tough times, we need a way to control the masses, etc.

So no matter what kind of experience one has had with the divine, perhaps even supernatural encounters with miracles or visions, and no matter what evidence you see for the divine, there will be those who say, well, that's just your imagination.

My response to this would be, I don't think it's just my imagination... but I do think imagination has a lot to do with it.

In the modern world people often try to pick sides between a cold, rational way of looking at things and a romantic view that incorporates mystery and wonder. Often things we say seem like attempts to convert each other from one point of view to the other: "Oh, that's just your imagination," or "Where's your sense of wonder?"

If you pick one side or another, though, you're just cutting yourself in half. Reason and imagination go together, always, and when they don't you're bound to go wrong somewhere.

This is especially clear to a mathematician, actually, since imagination is essential to discovering new mathematics, but reason and logic are the tools by which we tie these new discoveries to old ones.

On a more basic level, imagination and reason are both involved when we ask the question, how do I know this is real? How do I know the world around me is real? How do I know God is real?

How can I really know another person? Oh sure, my capacity to reason plays a role. I have to learn patterns in a person's speech and behavior. Social interaction can be just as much a game as anything else, with rules and logic that a master can figure out and manipulate.

But that's not all there is to really knowing a person. My ability to know a person has everything to do with my ability to imagine her. Imagine what it really means for her to be a person. Imagine her as more than a face in the crowd, an individual with an imagination just like mine. She has stories, hopes, and dreams, and she wants to imagine someone else imagining her.

A lot of philosophy is done with the intention of musing on a thought for a while because it's interesting to think about. I could never do philosophy that way, and I'm not doing that now. To me philosophy begins with pain.

There really is nothing quite like the pain of knowing a person deeply, only to be faced with the reality that it really was just your imagination.

It's tempting for someone who has experienced this to retreat and say, it was my imagination that did this to me. If I simply put on my guard, hide behind the cold shell of logic and reason, then I won't experience this pain anymore.

But only those who lose their life find it. It is only by risking yourself enough to imagine someone who might, after all, not even be there, that you can ever truly love someone.

(Am I speaking metaphorically? Sometimes it's hard to tell. Perhaps metaphor just expresses reality so much better than facts.)

Reason grasps for answers; imagination creates possibilities. They need one another. I would not diminish the importance of anyone's attempt to provide reasons for God's existence; but I emphatically believe that imagination is essential to the spiritual life.

Very often our reason flourishes after a rush of imagination. That's how it always works in mathematics, anyway, and I suspect it works that way in theology. It's no wonder the prophets were always telling people to tear down their idols. Idols limit the imagination; they teach us to think of God as smaller than He really is.

Christian traditions have doctrines that they've built up to direct people toward God. I've noticed that when pastors preach about doctrine, they tend to focus on the weakness of ideas outside their own tradition.

What if, instead, pastors focused on the weaknesses of their own tradition? What if each pastor could act as a prophet for his own congregation, urging them to tear down their own idols, and not self-righteously look down on the idols of others?

Would this not liberate the imagination from the chains of "pure doctrine" that we Christians so quickly wrap around ourselves out of sheer vanity? But we are too afraid of a God who speaks out of a whirlwind, a God who meets our questions with more questions, rather than answers.

There are many Christians these days who say we need more apologists, more people explaining why the Bible is fully reliable, showing that evolution is false and creation is true, proving that Jesus rose from the dead, or giving five reasons why there must be a God.

But my honest answer to anyone who asks me why I believe in God is that faith is as risky as love. My faith looks forward in hope to something I don't even understand, but I try to imagine it every day. I try to imagine a world without evil, and I think, God, that's what I pray for.

Sometimes I catch just a glimpse when I see the sun rise on a clear autumn day, or when I hear a voice that I've been waiting all day to hear, or maybe when I just stop and let myself be alone with my own thoughts. Just a glimpse is all. Isn't that all any of us ever really see of those we love?

I don't suppose that will convince anyone of anything, nor do I even think it should. But it's honest, and maybe the more I think about it the more fruit will come of it. That's the cool thing about reason--eventually it can catch up to imagination, before imagination takes another turn.

Until my reason does catch up, though, this little blog of mine will have to act as a bookmark so I can come back to these thoughts. It's hard to shake the feeling sometimes that there's something just around the corner that I'm not seeing.

Life is like that, though, isn't it?

Math Art

This picture represents a good deal of what I accomplished my senior year of college. I didn't make this picture; that credit belongs to my former Professor, Jacob Siehler, who is quite amazing at using computers to represent mathematical data.

Basically this picture represents a result of my senior thesis. Here's more or less what I did:

Take a positive integer m. Make a table with m columns and infinitely many rows. Label the columns 0, 1, ... , up through m - 1. Now place a ball in the 0 column of some row, say row x. Then define a sequence of rotation remainders as follows:

• When the ball is in row x, you move it x spaces to the right, and if there aren't that many spaces left in the row, just wrap around to the next row (just like reading a page left to right, top to bottom)
• After each move, write down the column position the ball is in

This sequence of column positions is called the sequence of rotation remainders, which is a term I made up, and it's what I titled my thesis. Surely I could have picked a more exciting and/or cute name for it, but what can you do...

So you might ask how the above picture represents this sequence at all. In fact, each row of pixels in the picture represents an integer (1, 2, 3, 4, 5, ...) and the pixels in a given row represent the numbers in the rotation remainder sequence corresponding to x. In this picture, m = 63.

Isn't it pretty? I promise there's a lot more complicated stuff behind it, too. I'm sure you'd be very impressed.

New Creation

The Tree of Life, pictured on the right, is a sculpture made entirely of decommissioned weapons. It is on display in the British Museum.

I first heard about the tree when I was listening to N. T. Wright's speech from InterVarsity's Following Christ Conference. Wright was talking, as he so often does, about Christians acting in ways that exhibit God's creative power in the world.

I finally got around to looking up the picture, and I was truly amazed. Both the title and the sculpture are ingenious. Rather than deny that evil exists or avoid thinking about it, the artists have redeemed wicked and ugly things by making them into something beautiful.

I can't think of a more beautiful image of what human striving is all about. When we long passionately for justice, isn't our longing most fulfilled when that which is evil is stripped of its evil nature, and made into something new?

Without that hope, that one day all things will be renewed in this way, I honestly don't know what else would drive me. Words fail to describe how powerful this image is for me.