Calculation by Moonlight
I know nothing! I know everything! A midnight revelation
I’m not the sort of cringing milksop who likes to make confessions, and besides, when you’ve lead a shining, exemplary life like mine, there’s really not much to confess. (Hey you — indignant pipsqueak in the back row —stifle those gagging noises!) Normally I’d
rather puff up than own up,
but if the rest of this essay is going to make any sense, it’ll have to begin with a painful revelation. OK, my big confession is that … drumroll please … that is, at one time … I, uh, fl…arrgh grrzh mmph jeez this is hard! And right out in public, too! Uh, um, well … I flunked calculus in high school. There, I said it.
Now, I am not embarrassed about not knowing calculus. Most people don’t know it — in fact they couldn’t deal with any kind of math under any circumstances. They don’t walk around feeling awkward about their ignorance, so why should I?
Nope, my mortification is simply at flunking a class (that is, any class) in school, especially since I was an ‘A’ student starting as far back as kindergarten, if not earlier. Well, that’s not exactly true — they didn’t hand out letter grades in kindergarten, since we couldn’t read the letters yet. But I did get rewarded with little adhesive-backed stars, turtles, and smiley birds, plus other imaginative stickies like the used gum that Frankie the fledgling hood used to drop down the back of my shirt. (The school was in a tough neighborhood in Brooklyn, but that’s a story for another time.)
Are you impressed yet? I earned glowing grades like that right up through the summer of graduate school preceding my certification as an English teacher. I normally did well in all the subjects I cared about, such as Literature, History, Procrastination … and might have gotten an A in my drawing classes, too, if the separate art schools I attended gave out grades or certificates or anything. Other than the actual training, art school didn’t give me squat, but on the other hand it gave a buddy of mine the clap, so I guess I came out ahead.
A Litany of Excuses
As I said, I’m not of the confessorial breed, so now I have to balance my sordid admission of pulling an F-minus in calculus with a tedious explanation of the extenuating circumstances. You may tune out on the next part if you like; it’s just about sex and sports so you’d probably find it boring, plus its only purpose is to palliate my embarrassment at being a flunkee (not to be confused with flunky), and to mitigate my utter failure to master the cryptic craft of measuring the area under a curve, known to math nerds cognoscenti everywhere as calculus.
As a senior in high school, I had some choice in what courses to take. What possessed me to sign up for calculus is something of a mystery. I may have been under the impression that it was some kind of hygiene class, and therefore easy, having read somewhere that calculus is a yucky substance that builds up on your teeth if you don’t brush them. Eventually I discovered that it’s also a species of math, but by then it was too late.
Mathematics had always been my weakest subject. The sole exception was geometry, which I excelled in; I suspect this had something to do with my being, you know, an artist and stuff. My father, himself a painter and art teacher, later told me that he, too, had done well in geometry but poorly in other math.
Actually, there was one pleasant link between myself and the subject matter: in high school, one of the numeric numerous girls I fell in love with was a math devotee, and quite lovely. I had some interest in calculating the area under her curves — but my intuitive faculties sufficed, so that still doesn’t explain why I took the course.
Perhaps I thought my taking calculus would impress her, and by the time I flunked it, high school would be over and we’d both have gone off to separate colleges so who’d ever know? (Until now, that is.) It’s not like the girls I dated ever asked to see my report card or anything.
Well, the deeper meanings shall have to (as the pulp writers say) remain shrouded in the depths of time; we can’t unearth everything, after all, even in a big-time philosophical essay like this one.
By the way, my difficulties with the course were not simply that I had no interest in the subject, no aptitude for it, and no idea why anybody would want to calculate the area under a curve. There was an additional factor.
Calculus was my last class of the day, coming right after a required ordeal course, Physical Education. My strategy for avoiding the nerd-bullying gym jocks was to head straight out of the gym building to the athletic field, and run a few miles as fast as I could on the big oval running track. Then, still damp and utterly exhausted, I’d stagger into math class, hide in the last row, and promptly pass out.
What’s This All About, Anyway?
Okay, that was the steamy sex & sports part of this essay. I told you about all that to help underscore a single point, which I could have just stated outright, but I like to embellish these things. Anyway, pay attention now, because understanding this is critical to what comes later (drumroll please): despite my warming a seat throughout an entire term in that class, I have no more idea of how to solve a calculus problem than a cockroach does. In fact, I wouldn’t even recognize a calculus problem if it jumped out of an alley, barked at me, and peed on my shoe. Which I’m pretty sure has never happened.
Patient reader, by now you’ve probably come to the conclusion that this is an essay about flunking math, or perhaps just a former English teacher showing off his linguistic convolutions. Nope; this here essay is about epistemology. In case you flunked philosophy, epistemology is the study of how we know what we think we know. Also known as the theory of knowledge, it investigates how we can be confident that we really know something, as opposed to just having an unfounded opinion about it, or the delusion that we “know” it.
Hang with me; we’re almost up to The Big Dream itself, which is the epistemological heart of this opus. Just three more teeny-weenie prefatory points, and then, vexed by my own verbosity, we’ll be there:
One: the dream I’m about to recount is a true dream. By “true” I don’t mean that what transpired in it actually happened — it was just a dream, get it? I simply mean that I actually dreamt it, exactly as I’m going to retell it here. I know it may sound suspiciously pat, like a parable I concocted to make a point, but I didn’t. I shall set forth the dream precisely as dreamt, with no embellishment, exaggeration, extension, expansion, enhancement, or additional arbitrary alliteration.
Two: in the dream I was an adult, which is not surprising, because it transpired long after I got out of high school.
Three: during the dream I had no idea, no clue, that I was dreaming. I thought it was really happening.
Oh by the way, did I happen to mention that I lack any knowledge of how to solve a calculus problem? Nix, nil, naught, null, nothing, nada. Okay, I hope that’s clear, so here’s an actual dream I had. Just to confuse you, I’ve entitled it the same as this essay as a whole:
· · · Calculation by Moonlight · · ·
by Lawrence San
The huge, barn-like room seems featureless and almost empty. There are no windows, no furniture, nothing but one small card table in the middle, almost lost in the vast space. I’m sitting at the table on a folding chair, my skin slightly blue from the moonlight that seeps in between the wooden slats in the distant walls and illuminates me, the table, and the thin fog drifting up from the floor.
I’m holding a pencil and looking down at several sheets of paper, already partially covered with the equations I’ve been writing. Although my thoughts are fully inside the math, at the same time my mind hovers slightly overhead, peering down at myself seated there, witnessing what’s happening — this is calculus, and it’s just flowing out of me!
I feel exhilarated — I truly understand it, the logic is clear, the connections simple and obvious. It’s even fun! Why had I ever thought it difficult? I look at my equations one at a time, my mind flowing through them and examining each step. It all makes sense, and it’s easy to mark them down as I grasp all the relationships and look! I’ve done it!
But as I stare down at what I’ve done, doubt enters. Doubt slithers out from the fog under the table and sticks its ugly head up and nips at my joy, diminishes my certainty of achievement. Because how could I have written these equations, or even understand them? I flunked calculus in high school, remember? — slept through the class, never did the homework, never learned a thing. And I’m pretty sure that I’ve never studied calculus since then — at least I can’t remember having done so, and can think of no reason why I would have.
How in the name of all the gods of sloth and ignorance could I possibly have learned this? I look down at the papers again and see, in my own handwriting, a seamless web of mathematical logic that’s plain as day. “Yet how can it be — how can this really be calculus?” I say aloud.
“Well, would you like me to look at it?” asks the man, and I lift my head to see him standing at my side. He’s an older gentleman with a kindly expression, bushy white hair and mustache, rimless glasses, and a bow tie. In his shirt pocket is a pocket protector which holds one wooden pencil, one mechanical pencil, a ball- point pen, and a tiny steel ruler. He looks like a mathematician.
“I’m a mathematician,” he says, “and I’d be happy to look at your equations.”
I hand the papers to him and nervously await his verdict.
It only takes a minute or two for him to scrutinize them. “This seems like perfectly good calculus to me. You’ve solved the problems correctly — even elegantly — and I don’t see any errors.” He smiles and hands the papers back to me.
I feel a little better, but he must see in my face that I’m still somewhat dubious.
“You don’t trust me,” he says. There’s no anger in his voice, just a touch of sadness. “You don’t believe me — that I’m really a mathematician, that I’m perfectly qualified to judge a little simple calculus, there isn’t even anything complicated here — ”
“Oh, no,” I say, “please don’t take it personally. I’m sure you are what you say you are, and that you know what you’re doing. I just don’t see how it’s possible that I— ”
“Very well,” he sighs, “I can see that you need more proof. It’s a little ironic, and admirable in a way, because what you desire is proof of your own achievement; most people only expect proof of other peoples’ achievements. But never mind; you’re perfectly justified in your skepticism — the scientific approach is to never trust just one authority, to always expect replication of results by additional experts. That’s what we call peer review.”
I nod in agreement; always I have heard this. Scientific proof resides, to some extent, in numbers — I mean numbers of people. You expect, when possible, a large amount of consistent empirical data; or at least (since math isn’t exactly “empirical”) a large number of verifications from other experts who can reproduce and validate what the first expert claimed. In fact, large-scale agreement by outside authorities is an important part of what we call “proof.”
“Very well,” he says, “to satisfy your uncertainty, we shall have to ask for some, uh, outside authorities to validate your equations. How many mathematicians would you say is adequate?”
“I don’t know.” I’m embarrassed. He’s being gracious, but obviously the implication is that I’m questioning his competence, requiring a second opinion. “Whatever you think is sufficient —”
“How about thousands?” he asks.
“Huh?”
He strides over to the closed barn doors and, with a dramatic flourish, throws them wide open. Outside is only black sky, and the most intense full moon I’ve ever seen. He stands there rather formally, one arm stretched out the door, as the moon grows steadily brighter and more yellow until it’s the sun and daylight floods in, vanquishing the fog in the room. Only then does he make his announcement, gesturing grandly outdoors with one hand — his voice suddenly amplified, thunderous, completely filling the large space and echoing off the walls like the booming of drums and the blaring of many trumpets: “Behold! Ten thousand mathematicians!”
The echoes die down, and looking outside I see a pleasant landscape of gently rolling hills illuminated by bright sunshine. Starting just outside the door and winding around the path, stretching up and over the hills, is a long line of mathematicians, standing relaxed in their unkempt hair and tweedy jackets, smoking pipes and carrying little notebooks, chatting amiably to each other while they patiently await their turn.
Slowly they file into the room and amble up to the little folding table at which I sit. Each of them, one at a time, picks up the sheets of paper and scrutinizes my equations. “Yes, that’s calculus, it looks fine to me.” And he smiles reassuringly, puts the papers down, and saunters off to be replaced by another worthy expert. A couple of them even pat me on the shoulder. One after another they validate my equations, on and on and on, and I gradually grow confident that it must be true. After all, there are so many of them! And their opinion is unanimous!
Is that not the essence of the scientific method, after all — widespread peer review, numerous experts offering independent corroboration? And yet, a lingering doubt remains: how is it possible that I could know calculus? — but still, there are thousands of them, and not a single dissenting voice! Who am I to question the weight of all their combined authority? What more proof could anyone expect? If this is not the ultimate evidence of reality and truth, what is?
Then I woke up.
Dutifully reported by
your moƒt humble servant,
P.S. One of the things English teachers warn their students about is to never end a short story by having a character suddenly wake up and discover that it was all a dream. That’s the most unsophisticated, clichéd, cop-out ending imaginable. In my defense, I wish to remind you that this isn’t a short story, but a work of non-fiction, and true besides; and anyway, I really did wake up, as proven by the fact that I wrote this, and I don’t know how to type in my sleep. Q.E.D.
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