Here’s what I’m asking you to forget: I want you to put away from your mind the pain of Mrs. Roger’s classroom, the slick palms and trembling twelve-year-old voice, as you searched the back of your classmate’s head for an answer, any plausible answer; the two trains were running towards each other on the same track, one at 60 miles per hour, and one at 80; when would they meet? And the absurdity of asking that question among all others. Would there be survivors? Would the explosion be heard for hundreds of miles around? Is it too late right now to board one of those trains, and be delivered from Mrs. Roger’s expectant stare?
It’s not true that everyone either loves math or hates it. I’ve met at least a handful of folks over the years who like math “okay,” but in my experience, this group seems far outweighed by the kids (and adults) who really, truly come to hate math, and often to fear it, through their K-12 experiences. There seems to be a vicious cycle, similar to my experience with piano lessons as a kid: when you dislike something, you don’t work as hard at it, which means you experience less success, which means you like it less and fear it more and feel even less motivated to work hard at it. Popular culture doesn’t help this situation, with its tiresome trope implying that you’re either “good” at math or you’re not. And, of course, the kids who are good at it are portrayed as the socially awkward, otherworldly genius-freaks. Research doesn’t support the idea of some kind of math gene that you either have or you don’t. With the exception of a handful of true math prodigies—the real headliners of the last few centuries—good mathematicians are made, not born. The kids who have the most success with math are the ones who believe that persistence and determination can lead to real improvement. In other cultures, that message is promoted clearly by teachers and parents, with very positive results.
All of which is to say: as we continue the tour, remember that you might not hate or fear math quite as much as you thought you did. It’s not only that math isn’t what you thought it was. You’re not what you thought you were, either. You’re actually not “bad” at math. You learned some math in school, you got stuck at some point, you got lost, or you lost faith in yourself, but perfect love casts our fear. I’m here today to share some of my love, and I hope a little rubs off on you.
Chapter 2: Real Analysis
So let’s say you’ve drunk the Kool-aid. You’re a true believer; you’re in love. You’re off to college to major in this beautiful subject we call Mathematics. Where would you start?
After Calculus, there are usually courses called Differential Equations and Linear Algebra, which are taken not just by math majors but also future engineers, doctors, scientists, and others who need a solid level of mathematical foundation in their chosen professions. But after this, the first course you might take, in which you find yourself surrounded only by fellow math majors, is called Real Analysis.
Some have said that Analysis is like Calculus on steroids. Maybe a better analogy would be that, if Calculus is a chocolate chip cookie recipe, Real Analysis is an exposition on the science of food, explaining why baking soda makes things rise and why eggs provide necessary binding. If Calculus answers the What of motion and measurement in our universe, Real Analysis give us the How and the Why. It’s the back story, running deeper and bringing the same satisfaction that The Magician’s Nephew gives to the Narnia series.
Most importantly, in Real Analysis, we grow up mathematically. Our work makes the foundational, irreversible shift from mere answers to solutions. In grade school and high school, solving a math problem meant performing a calculation and finding a number, or later on, solving an equation to find X. In higher math though, your solution to a problem isn’t a single number, but what we call a proof. A proof is actually a bit like a story; you start by introducing the characters (which we call defining variables), and then there’s some tricky conflict development before all the pieces fall into place and the conflict resolves. Just like writing a story is a personal, creative endeavor, it’s almost true that no two real analysis proofs will be alike. I say almost true because in a class of thirty creative writers, you should have thirty short stories as distinct as fingerprints, and that isn’t quite the case in real analysis. For many problems, the proofs will follow one out of two or three main structures.
But it’s also true that in some problems there are infinitely many paths from start to finish, meaning the proof you write—the style you use, the choices you make along the way—will bear your mark, just like a good poem or self-portrait. Unlike other art forms, though, a good proof bears a seal that sets it apart—a seal called objective truth. Another mathematician might criticize your technique or your style, but a proof is either (logically) absolutely true, or it’s not. And when you write a true proof, it’s as true as the proofs written by the greatest mathematicians of all time.
But maybe all this information about proofs still doesn’t quite fulfill my promise to tell you how math feels, on its most visceral level. I suppose Real Analysis feels a bit like picking up handfuls of dry, gritty sand at the beach, letting it run through your fingers, not as an adult but as a child, back when in the moment it was the most perfect and satisfying thing in the world. Real analysis deals powerfully with both the infinite and the infinitesimally small, and wrangling these notorious mathematical troublemakers successfully, and with finesse, gives the same sense of accomplishment you feel when mastering a tall mountain or getting a splinter out of your finger. With no frustration there is no satisfaction, and though the battle feels at times that it will kill us, there’s nothing to compare with the feeling of victory at the end of writing a good proof.
As your self-appointed tour guide, I feel a responsibility to mention that as I write, there’s a small, nagging voice in the back of my mind saying, “Are you sure that’s accurate? Is that really a universally true characterization?” I’m sure if there are any professional mathematicians (or amateurs more knowledgeable than I) in the audience, there are ideas in this travelogue that will make them cringe, that will strike them as distorted at best, or maybe flat-out wrong. It is not my intention to mislead, but to entice and captivate those visitors who would otherwise never dream of going near mathematics. My hope is that any ill effects from the shortcomings of my personal knowledge will be outweighed by the benefits of opening a new door in the mind of the reader—a threshold for a whole new land of discovery. After all, travel guides can only share their own stories of the land, not speak for a whole nation. Don’t take my views as the final word; think of them as a starting point for your own adventures.
Chapter 3: Abstract Algebra
Have you ever gotten a new album from your favorite musician, arrived at the song that made you say, “This is the best one,” only to be shocked when the following track was even better? This shock of joy—this whisper that you never knew something could be that good—this is Abstract Algebra.
Maybe I’m cheating by starting to tell you how it feels before I tell you what it is. Abstract algebra is the structure of the universe; it’s everything you see around you in the beautiful, messy, grand relationship of creation. Algebra—the real, grown-up stuff I’m writing about—is the result of taking your high school algebra experiences and zooming out, so to speak. It’s as if you realized you’ve been looking your whole life through a telescope at a two-foot patch of land on the opposite shore and missing the grandeur of the coastline altogether. In a way, you could argue that’s what the word “abstract” really means—to zoom out ideologically. It’s entirely analogous to what you do from arithmetic to middle school algebra, when you take specific calculations and formulate general principles from them (e.g., when you add zero to any real number, you always get that same number as the answer). When we go all the way to abstract algebra, we don’t just study real numbers and their relationships and properties anymore; we look at the relationships between all manner of mathematical objects—functions, shapes, and symmetries, and even imaginary numbers. More importantly, we look at how two sets of objects may seem different on the surface, while the relationships between corresponding objects in each set are similar or even identical.
As an example, suppose you lived in a world where there were only four hours in a day, so that an hour after 4:00 it was 1:00 again. You could make a chart showing the time it is now, and what time it will be a certain number of hours from now. For example, if it’s 2:00 now, in three hours it will be 1:00 again. On the other hand, think about the square root of -1, that mystical number we call i. We can make a chart showing the powers of i, and calculate that i2 times i3 is i. On a deep level, the structures in these two problems are identical, even though the relationship between a four-hour clock and the powers of i isn’t at all obvious. This is the tip of the iceberg—one of the simplest examples of what Abstract Algebra is all about.
Have you ever observed a pattern, and just when you thought you could predict what would come next, the field expanded and you saw an even bigger, more intricate, more complex pattern? That’s what it feels like to study algebra. Your intuition—your expectation of what will come next—grows and expands. It can seem like you’re always one step behind, but that makes every twist and turn in the road a new discovery, a new opportunity for wonder. C. S. Lewis wrote about this phenomenon in “The Great Dance” section of Perelandra: an unexpected joy, always tinged with the yearning that comes from encountering something just out of reach, far beyond oneself in every dimension.