why I do math

I spent the past week on retreat with fellow faculty members. During part of this time, we each shared our “vocational journeys,” or the stories of how we have been led to this job and these fields of scholarship. I thought that part of my essay might have broader appeal, so I’m posting it here.

Why do I study and teach mathematics? My research is in the field of pure mathematics, for which it may seem harder to justify an investment of time than for its adjacent field of applied mathematics. Applied math at least tries to tie itself directly to the needs and concerns of our immediate physical world. Pure math is happy to oblige in improving how well we understand the world, but its primary concern is math for math’s sake. (The boundary between these two types of math is highly permeable, and even pure math almost always starts with inspiration from experience.) I’d like to address the question by comparing math with two other areas represented in academia: music and science.

First, math is like music. The aesthetic element in mathematics is essential, not peripheral. I’m not sure, but I think that in the minds of many people mathematics is reduced to a collection of more-or-less arbitrary facts, like the fact that the area of a circle equals pi times the square of its radius. Each of these facts, however, is like the final cadence of a symphony. It may be thrilling by itself, but it’s missing the indispensable context of “where did we start?” and “how did we get here?”

This is why mathematicians insist on proving things: the proof is a whole symphony, not a single chord. Mathematicians are lauded not for stating facts, but for demonstrating their necessity, the way composers and musicians are praised for the whole course of a piece or a performance, not just its ending. When executed well, a proof has rhythm. It has themes that are developed and interwoven. It has counterpoint. It sets up expectations that are satisfied or subverted. Economy of material is valued, but not exclusively; an argument that wanders into neighboring territory, like a modulation to a neighboring key, can provide fuller appreciation of the main theme.

Proofs have a variety of forms, some as common as sonatas and minuets: direct proof, proof by contradiction, proof by induction, proof by picture, proof by exhaustion. We have computer-generated musical compositions and computer-generated mathematical proofs, and in both communities there is healthy debate about whether these artificial creations are beautiful or desirable in such quintessentially human activities. We return over and over to the same pieces and theorems that have inspired us, whether they be simple or grand, and each performer gives her or his own interpretation and inflection to the presentation.

Second, math is like science. Often mathematics is categorized as a science, and that’s not entirely wrong. Science is built on careful observation, winnowing data from the chaff of noise. Science seeks explanation which can be turned into prediction. It invents new tools for collecting information and improves upon those that already exist. It creates models and theories that encompass and relate as many pieces of knowledge as possible.

Where science and math differ is that science deals with the world in which we live, while the world of math is imagined. Imagine that there are such things as points with no volume and perfectly straight lines that connect them. Imagine that numbers have enough solidity that we can move them around en masse by means of undetermined variables, the x, the y, the z. Imagine that once we start counting upwards 1, 2, 3, 4, 5, a thousand, a million, a trillion, a googol,…, we could never reach an end, not in any number of lifetimes in any number of universes. Or imagine that the filigree of a fractal truly exists at every scale, that we can examine it closer and closer and see the ever-increasing detail, that there is no quantum barrier to our exploration, beyond which sight and measurement cease to be meaningful.

When we imagine these things, we create the worlds in which we make our observations. The rules of these worlds are not completely arbitrary, at least not if we want to be able to know anything about them, but they are ours to choose. Each time we choose anew, we enter an undiscovered country. Once in this country, we must return to scientific methods of study. We look for patterns, try to explain them, and check that our explanations make accurate predictions. We must know when to trust the instruments we have—our minds, computer programs, results proved by other mathematicians—and when not to trust them. Like scientists, we have to winnow out the noise.

Mathematical truth persists across ages and cultures, and so it may seem timeless, but our experience of it certainly isn’t. The channels of logic through which a proof flows may be carved out once and for all in eternity or in the human mind (depending on your view of where mathematical truth lies), but like notes on a page they remain inert until they are brought to life by individual or communal study. Like the tree of life in biology or the standard model in physics, mathematical theories are crystallized around our experience and our perception of the world. As Bill Thurston wrote, “mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others.”

things you never thought you'd see #2112

Your fiancée’s father’s workplace shows up in a popular comic.

Girl Genius has high production values for a webcomic; in fact, it started as a graphic novel and then moved to the web. Right now they’re in between acts of the main storyline and so they’re running a retelling of Cinderella just for fun (the story starts here). There are of course lots of inside jokes based on the main strip, but just think “mad science” and “fictionalized late 19th century Europe” and you’ll get the gist.

The story of Cinderella has princes. Princes need a castle. The castle should look science-y, to fit with the theme. So the castle appears in the fourth panel of today’s strip. And what did they choose as a model? The high-rise office building of Fermilab, where Hannah’s dad works.

abide with me

Mammoth Cave preserves everything. The statistics were cited to us a dozen times in the last two days: 54° Fahrenheit, 87% humidity, all year round. Things stick around. We saw a log leaning against the wall that was dated at 2000 years old. Archaeologists working at the site told us that they had found shells of sunflower seeds from prehistoric times. And we heard about the mummified bodies that, in the early 19th century, cemented Mammoth Cave’s place as a tourist attraction.

The first one was found among a set of stone slabs that were set up as a rudimentary coffin. As more were found, some became traveling curiosities to advertise the cave, while others were set up in glass cases or on stands inside the caves for visitors to see. Our guide today told us about one mummified body that was found by a pair of explorers—it was a man that had been crushed under a rock. In the 1930s, while teenaged boys were working on laying out the trails through the cave that are still used today, they lifted the rock off the man, using a cable and tripod, and extracted the body. His was one that was shifted from room to room and put on display. Eventually, it was decided that showing off the corpse might be too disrespectful. Because it had no proper burial place, it was returned to the tunnel “near” where it had been found, so that it could be left undisturbed, with not even the guides knowing where it was.

I commented to Hannah at this point in the tour that humans sometimes have strange notions of respect for the dead. At times we put bodies on display, study them, or amuse and shock ourselves with them. Other times we treat them with the utmost reverence, hoping that giving rest to the bones will give rest to the spirit. One reads of Achilles dragging the body of Hector around the walls of Troy to dishonor his remains and dishearten the inhabitants. Why should these feelings be so powerful?

For one thing, memory is powerful. We remember the deeds of our ancestors, and we are grateful for their efforts that have led to our being here. Being good to their earthly remains seems to equate with being good to these people who are no longer with us in consciousness. For another, I think the hope of resurrection is embedded in us. It would be a strange superstition that the dead could walk again, because nothing natural suggests that it’s remotely possible. I suppose one, or a group of people, could find evidence that some personality is still at work even after their passing. Still, the desire to see certain people again, or the fear that we might see them again, is hope working beyond reason and stretching past the reality of loss. Respect for the dead is humane because we are connected to each other by more than physical interaction; the greater reality of our selves imbues dignity and meaning even to dry bones.

I’ll close tonight with a short passage from Psalm 73:

Whom have I in heaven but you?
And there is nothing on earth that I desire besides you.
My flesh and my heart may fail,
but God is the strength of my heart and my portion forever.

deep river

During today’s tour of Mammoth Cave, the discussion turned to the formation of the caves and where the water sits now. As Hannah and I learned on a trip to the Pink Palace Museum in Memphis a couple of months ago, there was a time when most of North America was covered in a shallow ocean (as I recall, the main exception was the Appalachian Mountains, because the Rockies hadn’t formed yet). The tour guide, Joe, said it’s believed that’s when the cave was formed. There are five (explored) levels to the cave, and our tour descended to the third. The water table sits about 360 feet below ground level; during floods it can rise about fifty feet about that. We got down over 200 feet below the ground, and someone asked if the cave ever floods. After explaining the levels and depths, Joe said, “We have a book we really respect, and it says that the rest of these caves aren’t ever going to flood again.” That was one of my favorite examples of the guides’ blend of humor and frankness. We also saw a cave cricket, and some amazing canyon-shaped passages, and the “flowstone” of “frozen Niagara”, which looked in some ways like a rank of organ pipes cascading to the ground and in others like a fleet of fins of some gorgeous aquatic creature. It was a stunning trip, if exhausting (over four miles of walking up and down, after already being somewhat tired from short nights of sleep). Tomorrow we’ll take the lantern-lit tour.

my lord is near me all the time

Inspired by my previous post on the moon, I’m going to use it as the focus of my Lenten meditation this evening, too. It appears pretty early in the Bible, by any reckoning:

And God said, “Let there be lights in the expanse of the heavens to separate the day from the night. And let them be for signs and for seasons, and for days and years, and let them be lights in the expanse of the heavens to give light upon the earth.” And it was so. And God made the two great lights—the greater light to rule the day and the lesser light to rule the night—and the stars. And God set them in the expanse of the heavens to give light on the earth, to rule over the day and over the night, and to separate the light from the darkness. And God saw that it was good. And there was evening and there was morning, the fourth day.

(That passage still reminds me of one of my favorite sections from Copland’s “In the Beginning”, which we sang in the Chorale last spring.) The Israelites were not, I think, very good astronomers. That phrase—“the lesser light to rule the night”—echoes through dozens of conversations I’ve had with people who don’t understand how the solar system works and think the moon is only/always out at night. Even in Guinea, which is the closest situation I’ve been in to ancient Israel, and where people’s schedules depend immensely on the phase of the moon, most people had no idea that when the moon isn’t out at night, it can be visible in the day. I don’t blame Moses, or whoever authored Genesis, for not saying something more precise about the nature of the moon; they had much more important things to discuss, like the fact that God is an amazing creator.

Once you do know something about the moon, though, it becomes a wonderful metaphor. It’s like the mirror Robert Fulghum wrote about in It Was On Fire When I Lay Down On It: not producing any light of its own, it takes the light of the greater light and shares it when the sun is inaccessible. It’s a physical manifestation of grace in the darkness.

This light is also genuinely the most peaceful place I’ve ever found to be. The expanse of stars is terrifying. But the moon is a neighbor, and it feels that way. While the Guineans would dance during the full moon, because it allowed them to stay up all night without fear of darkness, I would stand in awe. Moonlight feels like forgiveness. Debussy’s “Clair de lune” is gorgeous and almost captures it, but walking in moonlight gives an immense feeling of calm and starkness and security.

Tonight I just thank God for the moon, for all the dreams it has inspired and joys it has supported, for its beauty, and for the challenges it poses. May we treat it well, and may it always be a firm stepping stone to the rest of the cosmos.

will we find Alice there?

This evening, a friend asked if I saw the lunar eclipse last night. No, I said, because I didn’t know about it. This was actually an unheard of state of affairs for me; for a few years, I caught every lunar eclipse I could, some unexpectedly. Here’s an email I sent to a friend after the last one I saw (October 2004):

Well, although it seems somewhat unfair to me, I only got to watch about the last 20 minutes of eclipsing. To make up for it, I think I have to stay up past midnight to watch the emergence of the moon from Earth's shadow. I missed the first part because I was teaching. Ah, well.

This is, I think, the sixth total lunar eclipse I've seen since 2000, but right now I can't remember where I was for one of them:

January 2000: Minnesota. A group of friends (who were still attending college in a town a little over an hour from Minneapolis) organized a eclipse-watching bonfire. Which is a good thing to have around when one plans on spending two to three hours outside on a January night in Minnesota.

January 2001: Guinea. This one was completely unexpected. I was sitting around with my host family having dinner, when we noticed it was time for the moon to be out yet remained quite dark. A few minutes later, we realized the moon was rising and eclipsing as it rose. It was incredible. I was super exciting [sic—but not inaccurate as you’ll see], hopping around and shouting in English, French, and Malinke how excited I was. The Guineans were afraid: their story for an eclipse is that a cat is eating the moon, and you have to play drums and beg for the moon to come back. I got to explain the process to a few people, however. It was not only a great astronomical experience, but a great cultural exchange.

May 2003: Memphis. My dad and I went out to a field to watch. On the way there, just a few minutes into the partial eclipse, our tire blew out, and we were changing it as the moon grew darker. We arrived at the field about halfway through the process, and stayed basically until it was done. The other time I went out to that field with my dad was when the Leonids were especially bright, in November 2002.

November 2003: Ithaca. I talked my friend Matt into driving out to Varna, where Cornell has an off-campus observatory. We figured there would be someone there manning the observatory, but no one was. About twenty or thirty other people showed up with the same idea, so we just enjoyed being out there together. I played resident astronomy expert for some folks, which was easy since I was basically explaining rotation and revolution. Also, of course, I took the opportunity to point out my favorite constellations. :-)

May 2004: ? This is the one I don't remember, even though it's the most recent. I figure I must have made an attempt to see at least part of it, but I can't for the life of me remember now where I was.

Tonight: I guess this is the last one until sometime in 2007 or 2008. I'll still be here then.

“2007 or 2008” seemed so far away back then. And now I’ve missed the last three I had a chance to see (since I was in Europe last March I could have watched that one—although as I think back, I may have known about it but it was cloudy—and I didn’t get up last August; see the link above for the list of eclipses this decade). The next one for me will I guess be in 2010, unless I’m in Africa this August.

A bit perturbed at myself for being so ill-informed, and remembering that I used to know when these things were coming because I used to visit Space.com much more often, I hopped over to that site and signed up for their RSS feed. Here’s the first headline I came across: Private Race to the Moon Takes Off. It turns out, Google and X Prize (the latter of which awarded a $10 million prize in 2004 to the creators of a privately-funded spacecraft that successfully reached >100 kilometers altitude twice) are jointly offering $20 million to anyone who can send a privately-funded robot to the moon and complete a set of tasks. It’s called the Google Lunar X Prize, and—to express the obvious—it’s awesome. It’s another example of how Google really is working to make the world of technology better in all the ways we’ve dreamed about. The competition is in the Space.com headlines right now because nine teams have just joined the competition, to join the first team that entered back in December. I imagine there are a few Cornell students aching to make their way onto one of these teams.

There has been a spate of moon-related media lately. Last fall, Hannah and I went to see In the Shadow of the Moon, a documentary that interviews the astronauts who were on the moon, which at this point in history was certainly essential to accomplish. (This movie, inexplicably, is not nominated for an Academy Award. Seriously, this is a fantastic movie. In the metaphorical, not the literal, sense, because it’s not fantasy.) It made Hannah want to watch Apollo 13. Almost everything makes Hannah want to watch Apollo 13. In 2005, Tom Hanks released Magnificent Desolation, an IMAX quasi-documentary that, while it had some good moments, was not the best IMAX film I’ve seen, mainly because so much of it was CGI and sound stage stuff rather than actual footage. (We just saw it this past winter in Memphis.)

And just in the last couple of days, with the landing of Atlantis just in time for the U.S. government to shoot down a potentially dangerous satellite, space activity in general has gotten a lot of news. I personally am still thrilled that we have a permanent human presence in space with the International Space Station. (I along with many others was nervous back in 2003 that the Columbia disaster would abruptly end that promise of a permanent presence.) And who wouldn’t be inspired by the immense success of the Mars Rovers, Spirit and Opportunity, which are still chugging along (and providing images processed by—guess who—the Cornell Pancam Team).

I believe this work in space has to remain a priority. And I’m pleased Google has taken it on, as well. As usual, I don’t have any dreams myself to make it into space, but I will cheer and in fact support in any way I can.