Discussions at Commonwealth have a sneaky habit of wandering out of classrooms. Whether it's debating Romeo’s decision-making skills, unpacking resistance movements in colonial Central America, or finding problems that don’t conform to the Church-Turing thesis, these spirited chats spill out into the hall, follow friends to lunch, and stick in your head long after the day is done. Here, you get to listen in on one of those conversations, as senior Anto ’24 and veteran math teacher Al Letarte dive into wildly and wonderfully abstract math. 

Anto: Here’s the thing about linear algebra: you don't actually need to have an application for it. I mean, I’m sure it has its applications, and I just haven’t seen them yet. But these concepts can exist in the void. And they are just mind blowing. 

Mr. Letarte: Correct me if I'm wrong, but it sounds like you're drawn to that aspect of math—the math that doesn’t necessarily have a connection to reality.

Anto: Yeah, I’m more skilled in the completely abstract areas of math, and I feel like they are a new frontier in a way that math grounded in reality is not. There's just so much more to do. Like, for my sophomore-year Project Week, I focused on the way paper folds. I suppose, technically, that’s grounded in reality, but at a certain point, it gets so abstract. Then it isn't about paper or about folding; it's about matrices that get bigger and bigger. And I think that's beautiful.

Mr. Letarte: A person after my own heart. Next year, when you take Abstract Algebra, you'll spend a lot of time studying group theory, which came up in your paper-folding project, and I think will capture your interest. If you like what you’ve been doing in Theo Calc, you will discover that you're in a world where one needs linear algebra all the time to understand what's going on.

Anto: There were lots of moments in Theoretical Calculus referencing the axiomatic set theory that underlies it all. Everything was just defined in terms of sets. And it completely blew my mind in Geometry when I learned it doesn't matter that this is a line and this is a point. They are just two objects that obey this set of rules. That's it!

Mr. Letarte: This is what mathematical logic does: it tries to step back and build a world of objects that obey certain rules of inference that do not necessarily have a connection to reality. I wish we were offering our Logic class this year, because you could have taken it. Shoot! 

Anto: Logic’s the best! It's so interesting. It was quite possibly my favorite part of Geometry Advanced. Going into that class, I had no idea how to write a proof. I just felt completely lost. Logic-wise, it seemed like an art instead of a science. There were axiomatic systems that were completely abstracted from any semblance of reality. But I learned to just say, Okay, it's not about what I know on an intuitive, instinctual level; it's about what I can prove.

Mr. Letarte: So, your Geometry Advanced book uses a system of axioms that's based quite closely on Hilbert's axioms. He was this German mathematician around at a time, in the late 1800s, when all of the great geometry problems were being solved using abstract algebra, things like finding an algorithm for trisecting an angle with a compass and unmarked straightedge. Hilbert was involved at the sort of inauguration of mathematical logic, and he was the one talking about this great new field that was being used to solve problems all over math: linear algebra. 

Anto: Linear algebra is fairly new, right? 

Mr. Letarte: It is. It's surprisingly new. 

Anto: It's just fascinating how new a lot of math is. 

Logic’s the best! It's so interesting. It was quite possibly my favorite part of Geometry Advanced. Going into that class, I had no idea how to write a proof. I just felt completely lost. Logic-wise, it seemed like an art instead of a science. There were axiomatic systems that were completely abstracted from any semblance of reality. But I learned to just say, Okay, it's not about what I know on an intuitive, instinctual level; it's about what I can prove.

Mr. Letarte: Yeah, I think the greatest example of that is the invention of coordinate geometry. Have you heard me talk about that? I actually talked about it in Dive In this morning. We started plotting points, and I asked them, Have you seen this in school? They all said yes, which was great. But they had no idea it came from Descartes, who invented it in the early 1600s, or just how important his discovery was. They had just been told how to do it. So I made a graph on the board with the amount of math that was known in the world from 500 BC to 1600 AD—and the graph just takes off. The subject just exploded. You had Newton, pulling together the ideas of calculus, right after the invention of coordinate geometry. People often don't know that math is dynamic in that way, but it really is. And then you had all these fields like abstract algebra and linear algebra. Even newer than linear algebra is topology. Even in the 1940s, mathematicians were not in agreement about what a topological space should be. 

Anto: Oh, wow. So what do you think is the next big fundamental field that we are just only starting to discover in math?

Mr. Letarte: The category theorists would have you believe it theirs! It’s used in theoretical computer science.

Anto: What is category theory? I might be familiar with some of it from computer science.

Mr. Letarte: So you think of functions as arrows between sets, and those arrows can be composed under the right circumstances. And composition of functions is associative. There are identity functions, which leave things unchanged, and if you compose them with other functions, it doesn't change those functions. Like if I apply a function to take set A to set B, and then I apply the identity mapping on B, it doesn't change it. So category theory is what you get if you take the sets and you just pour concrete in them so that you can no longer look inside of them. Now they're just nodes in a graph. And there are arrows between those—

Anto: Oh, wait! This is like all of Computer Science 3, right?

Mr. Letarte: Could be. Wouldn't surprise me. I believe you will do some graph theory, and a category is a type of graph. But there are rules that govern the behavior of the arrows. So the composition operation has to be there, and it has to be associative, and there have to be identity arrows. Then the sets don’t have to be sets anymore—they can be objects. But here's the thing, it turns out, you can study properties that you thought required looking under the hood, and looking at the sets, without actually looking under the hood. For instance, you can tell whether a function is injective, or 1:1, without actually looking in the sets. You just look at the way it behaves with respect to other arrows. 

Anto: What?! 

Mr. Letarte: It's weird, right? Basically, category theory is an example of what we do often in math, which is to throw away properties that are not central to what it is we're trying to study. You did that in linear algebra: the classical vector spaces actually aren't part of what mathematicians later decided should be the axioms of a vector space, like, say, the magnitude of a vector. We don't need that notion to study vector spaces profitably. And there's a tremendous amount that we can say about the behavior of functions without ever actually looking at the sets. You can show that in the category of sets, which is the classic category, that a function is injective. A function F is injective. If and only if, whenever F is composed with G is equal to F composed with H, G and H are the same arrow. So that’s a category-theoretic definition of one to oneness. You might say, “Wait, how do you know this is one to one? You didn't look at the set!” But you don't have to. So when you do that, you can accomplish a lot. Category theorists believe that category theory can be the new foundations of mathematics; they think set theory is not needed.

Anto: Oh, that's wild. 
 
Mr. Letarte: But the trouble is, they have to talk about the sets of arrows that make up the category, and there are categories in which the set of arrows is too big to be a set. And to understand how to deal with that you have to study axiomatic set theory. So category theorists haven't found a way to break free of classical mathematical logic. Yet. 

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