The sentence structure of mathematics

by countedblessings2 min read7th Oct 201915 comments


Logic & Mathematics Philosophy of LanguageCategory Theory

"Alice pushes Bob."

"Cat drinks milk."

"Comment hurts feelings."

These are all different sentences that describe wildly different things. People are very different from cats, and cats are very different from comments. Bob, milk, and feelings don't have much to do with each other. Pushing, drinking, and (emotionally) hurting are also really different things.

But I bet these sentences all feel really similar to you.

They should feel similar. They all have the same structure. Specifically, that structure is

Because these sentences all share the same fundamental underlying structure, they all feel quite similar even though they are very different on the surface. (The mathematical term for "fundamentally the same but different on the surface" is isomorphic.)

When you studied sentence structure back in grammar school (it wasn't just me, right?) you learned to break down sentences into their parts of speech. You learn that nouns are persons, places, or things, and verbs are the activities that nouns do. Adjectives describe nouns, and adverbs describe pretty much anything. Prepositions tell you where nouns go. Etc.

Parts of speech are really abstract and really general. When you look at the surface, the sentence

the ant crawls on the ground

and the sentence

the spaceship flies through space

could not possibly be more different. But when you look at the sentence structure, they're nearly identical.

The concept of "parts of speech" emerge when we notice certain general patterns arising in the way we speak. We notice that whether we're talking about ants or spaceships, we're always talking about things. And whether we're talking about crawling or flying, we're always talking about actions.

And so on for adjectives, adverbs, conjunctions, etc., which always seem to relate back to nouns and verbs—adjectives modify nouns, for example.

Next we simply give things and actions, descriptors and relational terms some confusing names to make sure the peons can't catch on—nouns and verbs, adjectives and prepositions—and we have a way of breaking down any English sentence into its fundamental parts.

That is to say, if you know the abstract rules governing sentence structure—the types of pieces and their connections—you can come up with structures that any English sentence is but a particular example of.

Like how "Alice pushes Bob" is but a particular example of "Noun verb noun."

At the most basic level, category theory breaks down mathematics into its parts of speech. It turns out that mathematics is pretty much just nouns and verbs at its simplest—just like how, if you read between the lines a bit, any English sentence can be boiled down to its nouns and verbs. Those are the "main players" which everything else just modifies in some fashion.

In mathematics, a noun is called an object.

A verb is called a morphism or arrow. We'll explore the terminology of morphism a bit more next time. As to why they can also be called arrows, that's because verbs appear to have directions: One noun does the verb, and another noun (potentially the same noun, like pinching yourself) receives the verb. So you could draw that as an arrow like so:

This is exactly how we diagram objects and morphisms in category theory, with one difference: we typically use single letters in place of full names. (I'd explain the value of concision here, but it seems hypocritical.) So if Alice and Bob are objects in our category, and Alice's push of Bob is the morphism, then we might write it this way:

Equally legitimate is to highlight the morphism up front. (We'll see they're the real stars of the show):

So now you understand objects and morphisms, the basic pieces of any category, just like how nouns and verbs are the basic pieces of any sentence.

Of course, making a sentence isn't as simple as mashing nouns and verbs together. We need to make sure that the sentence makes sense. To paraphrase Harrison Ford, you can write "colorless green ideas sleep furiously", but you sure can't think it.

We'll explore the rules that define a category in the next post.