What is learning?

Part of learning a subject is assimilating the facts. For instance, if you were to take a complex analysis class, you would learn that $e^{i \pi} = -1$ (Euler’s Identity). But what makes some facts more important than others? Why is Euler’s Identity any more important or interesting than the fact that the largest mammal is a blue whale?

Learning is more than facts. An introductory course in complex analysis is a story. The facts comprise the plot, each successive fact allowing it to advance a little more. There is a climax to the story: the Fundamental Theorem of Algebra. (Of course, we leave plenty of room for sequels.) And though it may not be obvious, there is a moral to the story.

Let me actually give you another example where the moral is clearer: consider the basic category theory relevant to a first course in abstract/commutative algebra. You learn that a concept such as “product” or “free group” is not just some object; it is really the object equipped with the relevant morphisms.

1. For example, the product of groups $G$ and $H$ is not just the group $G \times H$; it is also the projections $\pi_G : G \times H -> G$ and $\pi_H : G \times H -> H$.
2. Likewise, the free group of $S$ is not just the group $\text{Free}(S)$. It is also the relevant “inclusion map” $S \hookrightarrow \text{Free}(S)$.

Furthermore, such objects are usually characterized by some universal property (i.e. it is the initial or terminal object satisfying some description). And that is the moral of category theory, that the right way to define something in relation to other objects is as such an arrow.

Blue whales are interesting too

I doubt there are many marine biologists on this website. Odds are most people reading this essay have gladly nodded their heads along, agreeing that of course Euler’s Identity is more important. In mathematics? Sure.

But if you really start to think about blue whales, you might wonder, “what allows it to be so large?” And you start looking at similarities between all the largest mammals, and they turn out to all be sea creatures living relatively near the surface of the ocean. Why?

Land mammals have to support their entire weight, and the square-cube law means the pressure exerted on their feet by gravity grow as they do. But blue whales don’t have to support their entire weight; the buoyant force of water is doing a lot of it for them! So they can grow larger. And they live near the surface of the ocean because otherwise, water pressure would counteract this effect — and then some. (Deep sea creatures are freakish!)

Even something mundane like a blue whale fact can be part of an interesting story: sea creatures can grow larger because they are less constrained by gravity. Your initial reaction to such a fact might be, “okay, that’s interesting, I guess”. I know mine was. But there really is a story behind every fact, and when you string facts together in the right way, that’s when you actually learn.

Little kids understand this concept. That’s why they keep asking “why?” They want to know the story behind everything. You and I are older than that, but it would be unwise to forget this wisdom that we had as children.

Why do we have stories?

Excessive wealth causes people to detach from others and the world, leaving them unable to form meaningful relationships.

I doubt you would disagree with the moral above. But do you really believe these platitudes?

But if you have read the Great Gatsby, seen how Daisy destroys lives with no real consequence because of her wealth, and seen how Gatsby ultimately dies alone despite the extravagant parties he throws, you would believe it. And I’m sure you can think of contemporary examples of very wealthy people who have no real connections.

Yes, such a platitude rarely holds absolute. But the potential for excessive wealth to diminish empathy is a powerful force in the world, and the only way to further understand and internalize this idea is through examples. These examples are stories.

To put it another way, the existence of gravity is such an obvious truth to you not because you know the formula for the gravitational force between two objects, but because you see objects fall to the ground on the daily. And not only that, you see them fall at the same rate (air resistance notwithstanding).

Going back to math: I could tell you that universal properties are the right way to define certain constructions like the product. But you must work through quite a few facts before you see how this definition can save you a lot of trouble and unlock useful new perspectives. I can tell you the moral of the story, but for you to actually apply it effectively, you must truly believe it. That is why every textbook works through examples of any definition or theorem.

But it is important to always keep in mind that there is some moral to be learned. And it helps even more to be aware of it on some level as you learn a subject. When you know such a moral exists, it is easier to relate the facts you learn to it.

Case study: linear algebra

My first study of linear algebra was not very enlightening. The course I took in community college had a very matrix-first perspective. They were just blocks of numbers, and all we learned through the semester was how to manipulate them.

Of course, you and I know that linear algebra really involves the study of linear maps. But is it just that?

No, not really. Linear algebra was essential for pure mathematics; for starters, the derivative is really a linear approximation of a function at a point. In single-variable calculus we just treat it as a number, but it really is a linear map! If I perturb the input by $v$ in a certain direction, then the output is perturbed by approximately $D f (v)$, where the linear map $D f$ is the derivative of some function $f$. And approximating functions linearly is very nice because linear maps are easily understood. This justifies the study of linear algebra as the study of linear maps.

But linear algebra is useful for applied math as well. A long time ago, it could be used to solve systems of equations. But I want to talk about a more recent development: machine learning. Machine learning fundamentally relies on gradient descent, the idea that iterating the best local improvements (which we approximate via the derivative) will converge to a good global solution. When computers became commonplace, we finally had a computational way to train models via gradient descent.

So the question becomes how we represent linear maps computationally, and how we can manipulate these representations to support faster computations. Certain matrix operations such as SVD, PCA, etc. are not so fundamental to linear maps. They are truly important for matrices in particular. And we care about these techniques largely for computational reasons. This justifies the study of linear algebra as the study of matrices.

But even in pure mathematics, matrices are a useful computational tool. Take the proof of the structure theorem for finitely generated modules over a PID. It appeals to the Smith Normal Form which is constructed via a matrix algorithm.

Linear algebra is not just the study of matrices.

Nor is it just the study of linear maps.

It is the study of matrices and linear maps, and the correspondence between the two.

Of course, you cannot learn all of linear algebra at once. If you are going through a first study of linear algebra (especially if you have had brief exposure to linear algebra as matrix manipulations), it is better to tell you that it really is about linear maps instead. But it is useful to keep the entire picture in mind, because you will hit a wall when reasoning about everything as a linear map, and the only way to surmount it will be to appeal to blocks of numbers.

Here is the moral: if you have been told a moral like “linear algebra is just about linear maps”, and then you encounter some truly necessary matrix facts that do not fit this moral, then you need to consider how your story fits into the bigger story.

Stories are (distortions of) the truth

Historically, category theory was invented to discuss natural transformations. My introduction instead treats category theory as the natural extension of “mathematical objects are their properties”. Historically speaking, this is a serious distortion of the truth. Why do I do it?

It is because the easiest way to convince someone category theory is useful is not necessarily the most natural way to come up with it. Yes, if you study anything requiring category theory, natural transformations will inevitably come to be a large part of the story. But I only am trying to entice you to read it for now.

Mathematics is not really as simple as a story. While some universes like Star Wars have sprawling, expansive lore, it does not compare to the natural beauty of mathematics. But stories are the best way for us to interface with mathematics. Even if a story is not the entire picture, the morals it reveals still hold.