The sentence structure of mathematics

“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.