Full Circle

With Twitter collapsing, my social media circles have fragmented. I am not really hopeful for reunification, either; social media seems to revolve around anchors, and the anchors for different communities I love have scattered to different platforms. But I miss writing to everybody all at once. So here I am.

I’ve tried to get into blogging before. I couldn’t keep it up last time. I got too obsessed with everything being perfect, so writing posts felt like a huge chore. “I am going to do my best to not obsess over that here,” I type into the text editor with a totally straight face as I obsessively spend the next ten minutes thinking of what header text to use for the next section. Godspeed.

Math

I’m a computer science professor, and all of my work these days is on tools that make it easier to write proofs that computers can check. Techniques I use span all the way from dependent type theory to machine learning, which already makes me kind of a weirdo. But everything centers machine-checkable proofs. Mostly about programs. But sometimes about languages. Sometimes about computer architectures. Sometimes about math. The “math” thing is becoming more important these days than it used to be, and that’s giving me all sorts of warm and fuzzy feelings.

Math and I have a long, complicated, beautiful, and rocky history all at once. I think usually when people who like math a lot talk about math, it’s something they’ve always loved and excelled at. I definitely loved and excelled at math as a kid, but I felt really discouraged late in high school, and by the time I reached my senior year of high school, I felt super worried that maybe, since I wasn’t even good at math anymore, I just wasn’t good at anything. I was wrong about that, but I really did think that.

The Wall

A prominent “IQ” realist recently said some annoying things about hitting The Wall in math on the internet, so I felt this extreme irrational compulsion to respond. I Tweeted about how I’d once hit The Wall and then gotten over it. Because The Wall was pedagogically induced, as it often is.

But I Tweeted this all in a narrative way, invoking a two hour coffee conversation I’d had with Terry Tao the day before, so it went wildly viral and I think people got distracted by the whole Terry Tao thing. Honestly I feel kind of bad about that. Not the distracting part, but the talking about details of our conversation without realizing it’d go viral part.

OK so for the rest of this post, if you’ve read that Tweet thread, I want you to forget about coffee with Terry Tao or whatever. I instead want to elaborate on my experiences with The Wall.

But first, let’s go back to before I ever hit it.

Smanyonyo

I think the first thing I remember about math is Smanyonyo. I know you’re thinking, “Talia that definitely is a made-up word,” and like, yes. Absolutely. My childhood best friend Danielle and I made it up when we were six. We did that to win arguments with our mutual friend Steven.

Smanyonyo meant “more than whatever you say.” It was a clever catch-all that won many arguments until Steven came along and invented Smanpeko, which meant “more than Smanyonyo.” I tried to convince Steven that this was nonsensical because, by definition, if one invokes Smanpeko after Smanyonyo, Smanyonyo means more than Smanpeko, and it can’t be true that Smanyonyo is more than Smanpeko which is more than Smanyonyo. (No, I didn’t use those particular words. I was six.)

I didn’t know this was math, but it was important because I wanted to win arguments with Steven.

Invention

I didn’t just want to win arguments. I also wanted to be more like my dad. My dad is a really incredible mechanical engineer. So as a kid I’d try to invent things out of toys or whatever. I was not any good at this. I also was not any good at physics, later.

But math gave me that feeling of invention I’d been craving. In the fourth grade, I remember learning something about triangles, and then declaring to the teacher that I wanted to “invent new math.” I just started writing down everything I knew about triangles and trying to figure out new things about triangles. Probably it was nonsense, but it was fun.

I think it’s really cool that this is basically my job these days, just with computer math. It’s still fun. It’s no longer nonsense though!

Cake

I was also motivated by food. And this is something that Shlomo, an electrical engineering professor and my friend’s dad, knew well. Shlomo would drive us to piano lessons sometimes, and then he’d have to sit there and entertain two hyperactive kids while we waited. So he’d give us math puzzles and bribe us with cake.

One day, he drew a house-shaped object with an “x” through it, and said:

I’ll give you a slice of cake if you can draw this without lifting up your pencil, and without retracing any lines, and you can only touch these points.

We got this too quickly. There was cake on the line. So then:

I’ll give you a huge cake if you can do the same thing for this.

He erased the roof of the house-shaped object, leaving just a square with an “x” through it. We tried, we looked for technicalities, and we failed. After the piano lesson, I kept trying. I kept failing. I wasn’t sure why. But I was obsessed.

I forgot about this for a while. But in college, I learned about graph theory and Eulerian paths. And so I finally figured out why I had failed: such a graph cannot have an Eulerian path since it has more than two vertices with odd degree. I emailed him. He bought me a cake.

Candy

My sixth grade math teacher’s name was Mrs. Fox. She bribed us not with cake, but with candy.

Mrs. Fox’s class was not a traditional lecture. Instead, we all sat at tables and worked on problems together, in between explanations. But the most exciting part of all of this was the Problem of the Day.

The Problem of the Day came from a calendar of math problems. They were like puzzles. Some were “guess a number that satisfies these properties,” some were about shapes, some used physical objects we could manipulate. Some involved telling time on other planets, tricking us into learning modular arithmetic.

At the start of each class, every table would work on the Problem of the Day. We would work together and try to get the answer first. If we did, we’d get candy.

I loved (and still love) all of candy, puzzles, teamwork, and friendly competition. So I got super into all of those problems. I worked with my table to win most of those. And I learned so much.

Math was my favorite subject.

Divergence

In the years that followed, math seemed get further and further away from that. It seemed to diverge.

Some of my teachers after that were still thoughtful and kind and did similar things. Like Ms. Hickey in high school would give us homework passes if we won class-wide games of Set. I hated homework, and I loved the feeling of playing Set, which kind of reminded me of Zoombinis from when I was a kid. So I got really good at Set and won a lot of homework passes.

But most teachers were not like this. And all were limited by the curriculum.

Limits

Gone were the fun abstract puzzles with cake and candy. In their place? Memorizing sines and cosines. Meticulously calculating derivatives and integrals that quickly spun out of control if you made a single mistake early on. And at the foundation of everything, this thing called a “limit.”

What is a limit?

It looks like this.

A graph. A curve. A dotted line.

I didn’t get it. Nor did I get integrals, which were explained partly in terms of these mythical limits, and partly by drawing more curves and sketching the area under them and saying:

It looks like this.

This probably worked for most people. It just didn’t work for me. So without really knowing what was going on, I tried to memorize and calculate. But I’d forget things, and I’d mess up details.

By the time I got to my second year of calculus, I was convinced I’d hit my limit. I think I got a C- in that class the first time around (I took it a second time in college). I remember crying about this. I’d chosen to major in math because I’d remembered liking it once upon a time, and anyways I didn’t like anything else except technical theater, and my mom wouldn’t let me major in that. But maybe I wasn’t even good at math. Maybe I was bad at everything.

Five Things

It turns out I was wrong. There were five things about myself I didn’t know back then that I’ve learned over the years:

1. I have ADHD. I wasn’t diagnosed until sophomore year of college. This made remembering boring things and getting details right really hard. Coping mechanisms and medication made a huge difference, as did learning more about ADHD and what it meant for me and how I think. These days I view it as more of a blessing than a curse. ADHD brains are beautiful in the right environment.
2. I need rigorous foundations. It bothers me to leave questions unanswered. The first time I really felt like I understood calculus was when I took Real Analysis I and II. I actually really loved Real Analysis, even though it was supposed to be the math weed-out course. Because by building everything up from scratch, I really learned what limits and derivatives and integrals actually are. Only then did I feel like I could do anything with them.
3. I do not find visual learning natural. I can learn visually, but I basically have to install a formal semantics in my head first. The first time I figured out how to learn visually was when I took a crash course in category theory during graduate school, because someone explained to me that categorical diagrams actually have a semantics, so I learned how to interpret the diagrams. Now I do not need to try to interpret them; it comes naturally. But it didn’t naturally come naturally; I had to learn.
4. I find more abstract things easier. When I took Calculus III, and we finally moved into higher dimensions, I felt relieved and annoyed, like “this isn’t bad, but why didn’t we start here to begin with?” Restricting things arbitrarily to one or two dimensions made things harder for me to understand and relate to. This is the same reason I find dependent type theory easier to understand than, say, System F. So the further along I got in my undergraduate math (and computer science, though that was an accident of history) degree, the easier things felt. It felt like my brain was upside-down; I wish I could’ve started at the end.
5. I’m an algebraist at heart. I like structures. Abstractions. Relations. When I took abstract algebra, I really fell in love. (Honestly, I also felt annoyed that we had spent so much time learning about integers and so on when we could have just learned about the general structures that acted like them from the start, but yeah, I’m weird.) I have strong intuition, strong structure sense, and I find it easy to see how two apparently very different things are actually the same if you look at them under the right light. I think a lot of type theorists are algebraists at heart.

I had to learn about myself in order to learn to learn. I’m glad I did.