by CJ Uberroth
As a high school math teacher, I have had my fair share of interesting conversations, ranging from my students’ favorite music to my addiction to American Spirit cigarettes. Sure, some may say that I shouldn’t speak with students about things like alcohol and nicotine but I can’t imagine a stupider objection. How else are they supposed to learn the real reasons one shouldn’t smoke? I can promise that you that the only conversations about “substances” that a teenager will listen to begin with the following sentence: “Well of course I smoked pot, but listen for a second and I’ll explain that the only reason I’m telling you not to is because you’re doing it stupidly.” But, I digress. The point simply is that I have interesting conversations with my teenage students. And the most interesting of all time may have occurred this past week. While I was writing the work for a related-rates problem on the board of my calculus class, I heard the most amazing word: “Why?”
This wasn’t some wasted breath. Nor was it some stray remark from Klark, one of my (brilliant) asshole seventeen year-olds, thinking he’d be funny by disrupting my class. This question came from one of the more thoughtful, bright seniors and was quickly followed up with a somewhat intense clarification:
“I mean, who came up with this? Why did they do it? Why was someone spending their time doing this?”
What a question! I felt like an only child walking into his living-room on Christmas morning. “Are all of these for me?” I was actually a bit stunned, but fearing that she might lose interest or Klark might strike without warning, I jumped into the sort of discussion I’ve only been able to have with a glass of Scotch and a cigarette and with people who already knew every idea I could possibly throw at them.
“Well,” I began, “it depends how far you want to consider this. As we all know – or at least should know – most theories in calculus were formed by Newton and Leibniz. The reasoning behind those theories will be tough to nail down. Each of them probably…”
And then she cut me off. Not because of boredom or regret for asking the question, but for even further clarification.
“I guess what I really meant was, how? How did this all come to be?”
At this point her friends were hanging on the conversation as well. Amazing. Here I had five or six high school seniors, in the middle of small-town Texas, asking about the philosophical theories behind mathematics. I figured I wouldn’t bother telling my colleagues, as I knew they wouldn’t believe it.
Then Plato hit me. For those of you who are unaware, philosophy of mathematics has arguably three branches of thought. There are the science-minded people who simply view mathematics as empirical and nothing more. This camp includes people from all flocks including empiricists and various other schools. Boring! I mean, sure, this is arguably the easiest concept and the most practical but it’s, well, boring. (Not to mention, probably wrong.)
Platonism describes those (including a younger version of myself) who believe that mathematics has its own independent existence and is waiting to be discovered and understood. What a wonderful idea! Imagine, the Pythagorean Theorem actually is an ancient Grecian monster, akin to Theseus’ minotaur, trained by Pythagoras in 500 BC to be available to all humankind for eternity. A tale worth telling, that! Metaphors aside, the idea is powerful, suggesting that mathematics exists as a thing of its own, with its own distinctive nature.
So, I jumped in head first, explaining that what seems like an absurd encyclopedia of random rules in calculus is really all tied together. Without the Intermediate Value Theorem, you have no Mean Value Theorem. Without the Fundamental Theorem of Calculus, you have no derivative. I was relentless. I even began to start showing the ties from integrals all the way down to Euclid’s Elements. Showing that everything was intertwined in these subtle ways really had their attention. But, it wasn’t meant to be.
Just as I was explaining the importance and acceptance of Euclid’s Elements, the fundamental axioms on which all of our basic mathematics are based, Klark struck. Who else but the smartest student in the class will point out the antithesis of this theory? If you’ve been following closely, you’ll notice that I left off the third of the three branches of mathematical philosophy. This was not by accident. I didn’t want to mention the final idea. But sure enough, Klark had thought of it.
Fictionalism, as the name suggests, is the view that mathematics is nothing more than a human invention, reflecting the human desire to find order in chaos. I hate it. I mean, why would someone strip away the beautiful idea that statements like “if the sum of two squares is equal to a third square, then the angle opposite the largest value in a triangle constructed from these values will be 90 degrees” describes a remarkable, delightful independent reality? But sure enough, Klark did it.
“That doesn’t make sense,” he sniveled. “How can mathematics ‘exist’ (he actually used air-quotes), if it all boils down to some set of rules that Euclid invented that have to be assumed to be true?”
I wanted to cry or scream or maybe smack him upside the head. I really can’t remember the exact feeling. But, I had to settle for a quiet nod of acknowledgment. He was right. I knew it, my students knew it, and, worst of all, he knew it. So after my twenty minutes of blowing my students’ minds with connected theories, sublime equalities, and incredible identities I had to come clean.
“Yes, Klark, I left one theory off my list.” I have long realized (grudgingly) that Fictionalism is the best philosophical account of mathematics. I’ve even developed a pseudo-proof with an old friend of mine. If I asked you what came after 10 in the sequence 4, 7, 10, … you would say “13.” But what if I told you it was 37? How would you argue with me? I never told you that the sequence was the progression of adding three each time. We could be following any rule you could imagine. Hell, I never even told you that the sequence had to continue, which means the answer could have been “nothing.” But we want it to be 13 don’t we? To me, that is enough to prove Fictionalism drips with the truth (along with the blood of my Platonic youth).
I couldn’t believe it. Thwarted by the student who sings “All Star” by Smash Mouth in the back of my class just to piss me off. Yes, I know Platonism is false, but hear me out. I want to help educate a group of people that will go out into the world and become scientists and mathematicians, whether by trade or by hobby. Imagine Dan Kaufman and I are standing in front of you. Beside each of us is a boulder that we ask you to roll up a hill. Dan offers you ten days and nights at his private resort on the Miami Beach, with all expenses paid, and I tell you that I’m gonna’ kick it back down the hill the minute you’re done. Who will you do it for? How devastating to a young learner is Fictionalism? Realizing that even if you master everything that mathematics has to offer, it is ultimately an exercise in futility, as it all could have been entirely different; or could be rewritten next week; that none of it describes anything that’s really there.
Having explained all three theories to my students, now, and feeling quite deflated, I wrapped up with a snarky quip at Klark. It was over. My students would continue in mathematics just until it was done paying off for their future careers in medicine or engineering or whatever. Ugh, it hurt, that feeling of failure. To think I was so close to convincing a group of young people that mathematics was beautiful and profound. Something more than a mere tool. The bell rang to signal us to class shortly after that. As everyone packed up, the student who had asked the question that started this whole thing stopped at my desk.
“You know, Uberroth,” she said, “that, Platonism thing…?”
“Yeah?” I replied.
“It’s way more of a reason to do math than what Klark said. Thanks!” And then she left.
I leaned back in my chair, with a satisfied sigh. I had made a believer out of at least one student that day. Then, I contemplated what practical joke I should play on Klark during the next day’s class. I swore to myself that it would be brutal.