Consider me an ambassador: an emissary from a foreign land, a place whose name I dare not mention for risk of causing you to turn away before the end of this very sentence. Do you truly have faith in that age-old hope of finding beauty where you least expect it? Can you let go of what you thought you knew, what you thought you feared, what you remember feeling when you were young? Can you forgive?
Imagine a land of exquisite, other-worldly splendor. Some would call it ‘exotic,’ though when you look more closely you see but an amplified, glorified version of the types of landscape features that are familiar to you. The hidden connections and relationships between the myriad pieces of our own natural world have their incarnation here; what is invisible is here proclaimed, and the complexity will quickly overwhelm you if you aren’t careful to take in the scenery in small bits. Natives of this land continually experience the bittersweet sense of knowing and loving a place of profound beauty, with few people around to share it. The rest of the world believes only what they’ve heard, the misconceptions about the place that lead them to believe it uninhabitable. Those misconceptions fall into three main categories.
First: people believe that this land is a boring, monotonous, uniformly dull terrain of tedium and gloom. This impression is falsely formed by the experiences of early childhood, in which our well-meaning educational systems show students a small lot of land on the shore, a particularly uninspiring, rather barren patch of dirt, weeds, and small rocks. As this is the only glimpse of the country that children are presented with, they naturally feel the whole thing must be this way and lose any interest in spending any time voluntarily in such a place.
Second: when the children are a bit older– the educational system compounds the problem by leading children to believe that the land is also frustrating. When children graduate from the patch of dirt, they move on to a steep path up a hill with increasingly larger rocks to climb over, and they’re told that the view from the top is spectacular. The only problem is that along the path there’s not much reward; there’s a troublesome, intermittent fog that comes and goes, bothering some students enough to make them give up long before reaching ‘the top,’ and making some students unable to see much of anything even if they do finally arrive.
And finally, the worst of the three: children learn that this land can be frightening. This comes in connection with the rocks and the fog, for children learn they can stumble, they can get lost; they can get hurt. Children are not provided with what they need to persist and to triumph in this land. They need caring, wise guides. They need good lanterns and proper climbing gear. They need encouragement and the honest truth– that there is beauty, unmatched in all the world, ready to be discovered, and that they can and will find it if their hearts are open and they are willing to make the journey, even when the journey is hard.
In case you haven’t guessed, I am talking about the land of (deep breath, please; hold it a moment, and breathe out slowly) Mathematics, yes, that subject you learned to hate, and fear, and avoid at all costs, somewhere in your youth. And if any of those descriptions above (boring, frustrating, frightening) describe your view of my homeland, I’m here to tell you today– you’ve been deceived! Robbed even! You’ve seen only the smallest, most limited facets of the place, and you must know, before you die– you must know there’s a whole world out there beyond the bit that you saw, and it’s like nothing you can imagine! Mathematics is surprising, creative, incredibly diverse, intricate, mind–blowing, satisfying, and– I know you won’t believe this one yet– it’s fun. It’s downright fun.
I’m here to give you the tour– the tour you should have gotten when you were young. The real tour.
Before we embark– you should know a bit about your tour guide.
I remember each week in my third grade Sunday school class we ‘interviewed’ a student, recording pets’ names and favorite colors and the like. Every other girl listed ‘math’ as her favorite subject in school, while mine was Bible. Math obsession didn’t infect me until later in life, when things got more interesting– when we started algebra most of my friends jumped overboard, while I fell in love. Even more fell away at the fork in the road labeled ‘Calculus,’ where my rapture only grew. I was a girl with a goal in high school: get as many credits as possible to get things like history and literature out of the way, so I could focus on My One True Calling in college.
But a funny thing happened when I reached my Christian liberal arts school– I rediscovered the rest of the world. Philosophy and art and poetry and music called my name, and though my focus on math didn’t waver, there were new threads weaving into the tapestry. As a senior in my dorm one evening, the urge to try my hand at a poem took over, and I sat down and poured out a poem that began, “I would give both my hands to be a writer.”
It shocked me when I wrote it, but it really shouldn’t have; the impulse was there all along, growing steadily under the surface of my math-obsessed life. I’ve struggled all these years to reconcile those two disparate impulses; to find out what it means to write about math, or more specifically, to find out what I am meant to write about math. It’s always been a balancing act– much like a tightrope walker’s need for and struggle with the counterweight of a balance pole. Math is hard, and writing provides such a welcome outlet when the gears of my problem-solving brain just can’t anymore. But writing is hard, too, in a completely different way, and after struggling in vain I always think (rather guiltily– like a cheating spouse?) “thank goodness I have math to go back to.”
Enough about me. It’s time we embark on this fantastic ‘tour’ I’ve promised you!
Interestingly enough, in recent decades there’s been an explosion of a new genre of books on the shelves of your local Barnes and Noble– these are works I would call ‘popular math’ books. They vary in quality of course– many are quite good! – but what they all have in common is a goal to, in some way, explain real mathematics to non–math people. These are not math textbooks; they don’t go through the long journey of hard, rigorous development to really arrive at a destination; they sort of ‘fast forward’ through the really tricky parts (skipping several hundred pages of post-calculus math that you never learned at all), to give you a taste of the really cool stuff that actually motivates mathematicians to put in the effort. To use (abuse?) my extended metaphor above, it’s like showing you gorgeous photos of the really beautiful, amazing parts of Mathematics, the parts you never dreamed existed in your experiences in school.
But I’ve always wanted to do something different. Photos in a book are wonderful and enjoyable, of course, but we all know it’s just not the same as really visiting a place, in person. What does the breeze in your hair feel like? Is it a warm, gentle breeze, or the cool blasts of late autumn? What does the gravel sound like under your feet? What does the sun feel like? You can’t truly capture those things with a photo.
So dear reader, my goal today isn’t to explain any mathematics to you. Rather, I want to try to capture, just a tiny bit– what does mathematics, real mathematics, feel like while you are doing it? Not a photograph– more like some kind of advanced, high–tech virtual reality simulator. Because in all honesty, that’s what keeps research mathematicians going… it’s a feeling like none other.
Part 1: Calculus
What does calculus feel like? It feels like perfection– the most objectively real perfection we’ll find this side of Heaven. Calculus is a sharp break from the years of slow, steady progress that comes before it. After trudging up a hill for a long time, there are some difficult boulders to get over and THEN, all of a sudden, you see it– there’s suddenly a natural system with the potency to illuminate and give us dominion over the inordinate intricacy of the real world (as opposed to the simplified fake-world of your algebra book.)
Draw a circle. Now look at that circle with the wonder it deserves: before we had algebra and decimals and calculators and pi, your local shepherdess wanted to put her pet lamb on a rope tied to a stake, and wondered, how long should the rope be to ensure the circle of available grass would be sufficient?
Or to put it more plainly– if you didn’t have a formula handed to you, no ‘pi’ button on a calculator or a suggestion to use 3.14… how would YOU find the area of a circle?
Well, one thing you could do would be what the curious (and desperate) have done for centuries– you could get an estimate of the area of the circle by putting a shape inside it that you do know how to measure– for example, a square:
… and then you could observe that you could find better and better estimates for the area by using more and more sides for the shape on the inside:
So for a long time, people were happy with this, because, let’s face it– how precisely do you need to find the amount of grass in your lamb’s reach? If you’re off by a few blades, no big deal, right?
But God created us as humans with a collective desire to know more– to know more perfectly. And the human community persisted in this, and a dozen other problems like it, until we discovered the gem in the mind of our Creator: the concept called a limit. In the context of the circle problem, this means we keep increasing the number of sides of the shape, and getting more and more accurate estimates, until something magic happens: we zoom out of the real numbers altogether, into something called infinity.
And it’s there– when the number of sides of the shape ‘goes to infinity’– that the polygon transforms into a circle and the area becomes exactly pi times the radius squared.
Where else can we see a limit? I remember the row of street lights, as a child, when I rode in the car, and at first I saw the row of twenty of them, all distinct, and as the car moved, they would fall more and more into line until at a single instance, a single fleeting moment, I did not see twenty street lights, but one: they were perfectly aligned in my line of sight. That was a limit. Or think about your speedometer on that same trip in the car: algebra tells you how to find out how far you’ve traveled if you know how long you’ve been going exactly 60 miles per hour. But with the limit method of calculus, you can find the truth: your speedometer was moving continually during that time period– you slowed down for the cop, sped up to pass the slow RV– so your ‘speed’ isn’t a fixed number, but a constantly fluctuating number, and yet calculus still provides a way to find your exact distance. Or more simply: God is working on our hearts to draw us continually closer to Him; and somehow we will never BE Him, but at the same time we will never stop getting closer.
This desire to transcend what is fixed and simple and finite, and pass into what is perfect and infinite– surely it lies quiet in our hearts until it finds a way out, a place to be incarnate and visible. We see only glimpses here, earthside, of the transformation we long for, the unspoken yearning of our souls to be united with our Creator. Calculus is one of many ways to find such a glimpse.
Editor's Note: Nancy Elizabeth Wentzel is a college math instructor, wife of a physics professor, and mom to four adorable up-and-coming nerds. The Wentzels make their home (and their stories) in Johnson City, Tenn.