Imagine that you come across a new game, and want to learn how to play it. How would you go about it? Chances are that it would involve reading the rules, watching the game being played, or some combination of the two. These two approaches look vastly different in practice, but they both serve the same purpose, namely gaining a better understanding of the game, and each approach provides insight into the other. Obviously, learning the rules of a game allows a person to be able to play the game, and similarly, watching a game being played gives some insight into the game’s rules.
We can even imagine that, in theory, having a perfect understanding of one approach gives one a perfect understanding of the other. Take the game of chess for example. Certainly, a person who understands the rules of chess could, given enough time, systematically produce every possible game of chess that can be played. And similarly, someone who has seen every possible game of chess could theoretically deduce the exact rules of the game (how the pieces move, how the game ends, etc.).
As you’ve probably guessed, this correspondence applies to mathematics as well. A couple of posts ago, I talked about the idea of a mathematical language, which consists of some collection of things (usually numbers), and some operations that acted on those things to create “words” in that language. In that post, I went on to try to stretch the language analogy further, and in retrospect, I believe that I pushed it a bit too far, so for today, “language” will just mean a collection of things, some operations, and the words that can be built from these.
Given some language of this kind, we can define some rules about what words are equal to each other. Here are a few examples that go quite nicely together.
First, assuming that we have a language with a binary operation called “*”, we could require that (x*y)*z=x*(y*z) holds for any three elements x, y and z from our collection.
Next, let’s require that our collection contain an element called 1 with a special property, namely that x*1=x=1*x for any element x from our collection.
Finally, we’ll say that given any x from our collection, there exists some other element y from our collection with the special property that x*y=1=y*x. The intuition for this rule might not be as obvious, but the point here is that we want to think of y as being x-1.
It turns out that these three rules hold true for a wide variety of things that are of interest to mathematicians, to the point where we have a name for any structure in which these rules hold: a group. We would call the set of three rules listed above the theory of groups, and any example of a group could equivalently be called a model of the theory of groups.
Rotations and groups often go well together. For example, the manipulations of a Rubik’s cube (the ways in which each face can be rotated) form a group, if we consider combining two manipulations to be doing one after the other.
Notice just how similar this situation is to what we said about games at the beginning. Taking the chess example again and updating it with our new terminology, the set of rules of chess would be the theory of chess, and any individual game of chess would be a model of that theory. The correspondence that we observed, then, is between the theory of chess and the collection of all models of that theory.
You might have observed that this interplay between theory (the rules defining a thing) and collections of models (instances of that thing) can be extended far beyond just games and abstract math. Admittedly, I’m no philosopher, but I would argue that we arrive at our understanding of a great many concepts via some version of this idea.
Remarkably, this idea of the correspondence between theories and models can be encoded into the language of categories, giving us, for example, a category that represents the theory of groups. Those types of categories, which are called Lawvere theories, are why I decided to do a Ph.D. in category theory in the first place, and I am very excited to (finally) talk about them in my next post.
Photo by leogg via Open Clipart
