The Copernicus Problem
In school, you learned that the Earth was the center of the universe. At least, you learned that people used to think that. Your science teacher then explained to you that actually, the Earth revolves around the sun instead. A man named Nicolaus Copernicus proved it during the Renaissance, and set off a scientific revolution.
If a teacher just wanted to teach you about orbits and the solar system, there would be no point in telling you that people used to be wrong about the sun and the Earth. There would be no point mentioning Copernicus, or Kepler, or Galileo, or the fact that for centuries many believed the sun and planets revolved around the Earth. Why bring it up, if it’s wrong?
It’s because the Copernicus story serves a different purpose. It teaches you that science is still a work in progress. It teaches you that scientists are still human, and humans can make errors. It teaches you that science can be wrong, and that the work of the scientist is to move closer towards truth.
I believe that we need a Copernicus story for mathematics. In school, math students rarely receive a reason why they have to endure the drudgery of times tables, why they have to confront the curlicue S’s of integral calculus, why they have to answer 30 worksheet problems about the quadratic formula, which to them is an abstract, unfeeling formula floating in mathland. In school we are led to believe that math is a pristine, completed work in an abstract utopia. This simply isn’t true.
When I was an undergrad, a TA changed the way I looked at math forever, but I don’t think he even realized it. The TA was a PhD student. I was in a discussion section for an abstract algebra class, and each week I would timidly spend the entire hour working up the courage to ask even a single question. One day, I asked the TA a question. I don’t even remember the exact question anymore (something about notation). I only remember the start of his response.
He thought about it for a second and said, “Well you have to understand that us mathematicians have no idea what we’re doing.” It sounds dumb, but this statement had a profound impact on me. The TA went on, explaining that algebraists wrote notation one way but then other people thought it should’ve been another way, and for years everyone had a hard time agreeing on which notation was best. It taught me what Copernicus taught science students; that math is developed by humans. Math is not a complete project. Math will always be a work in progress, and it is the job of the mathematician, like the scientist, to move closer towards truth.
I’m not sure what could serve as a good Copernicus story for math. Off the top of my head, I can’t think of a time where mathematicians thought one thing was true and then the opposite thing was true instead (but please feel free to let me know of any examples). Maybe a good story would be Hilbert and Godel arguing over incompleteness, but it could be too abstract for students. Most “incorrect proofs” that I can think of are proofs that reached the same conclusion as the final proof, but just had flaws originally. Two examples of that would be Kempe’s incorrect proof of the Four Color Theorem, and the first version of Wiles’s proof of Fermat’s Last Theorem before he corrected it.
I’ve written before that I wish math teachers spent more time on the personalities and the creative work of mathematics, kind of like we do with science. You’ve probably heard of Galileo, Einstein, Kepler, Salk, Curie, Darwin. Even if a layperson can’t understand their most complex papers, their accomplishments are so well-known that I don’t even have to type their full names for you to recognize them. But in the United States, can we say the same about Bernoulli, Euler, Gauss, Ramanujan, Fermat, or Poincaré? Maybe to math people yes, but the truth is it’s really hard to get students excited about a series of numbers floating in imaginary space, without any sort of rationale or personality behind it.
Instead, talk about how Pythagoras was low-key a cult leader who believed numbers had mystical properties, like the number “5” symbolizing marriage or “4” symbolizing justice. Talk about how Gottfried Leibniz, the guy who invented calculus independently of Newton, founded the philosophy of optimism and strongly believed that we lived in “the best of all possible worlds” (and then Voltaire wrote an entire novel arguing that he was wrong). And show that math is a work in progress, made by humans with lives and worries, struggles and triumphs, just like you and me.