I spent a decent fraction of today trying to explain to Scott Sumner that an analogy is like a functor, or like a functor is like the idealization of an analogy. What makes an analogy useful isn’t saying A is like B because, you know, his jacket is like that plant. The jacket has a kind of leafy texture, the plant has a kind of leafy texture, you’ve made a sentence, you haven’t done anything yet.
What makes the analogy useful is in the functoriality. The functor preserves the composition property. That’s the move math always pulls. You take something everyone gets wrong, you find the one property that’s actually doing the work, you demand that property and throw the rest out. That’s what mathematicians did to “is like” when they coined functor.
The property is composition.
So to make an analogy useful, if A is like B and B is like C, A should be like C. And when transferred to a domain under some functor F, F of A, F of B, F of C should all chain together too. That’s transitivity. It’s chainability. It’s modus ponens if you tilt your head: A implies B, B implies C, A implies C. The whole reason “is like” is supposed to be useful in the first place is that it lets you carry a fact from one place to another, and that move only carries if the analogy survives composition.
Most analogies don’t, because most analogies only act on objects.
His jacket is like that plant. They put two things next to each other and say look. The trouble with object-only analogies is that if all you do is point at objects then X can be like any Y. Like anything resembles anything in some respect, color, shape, mood, the way it looked at noon, so “is like” restricted to objects contains everything and therefore says nothing. The only way to cut “is like” down to a relation that actually has content is to demand it preserve the maps. The arrows. The morphisms.
That’s what a functor does. A functor sends objects to objects, sure, but the part nobody pays attention to is that it also sends arrows to arrows, and the rule is the F of a composition is the composition of the F’s. It doesn’t fall apart when you chain things. That’s the property. That’s the only property that actually separates a real analogy from a poster.
Posters are fine. They just don’t carry information. They look at a thing.
This generalizes upward. Functors themselves can be related, and the way you relate functors is via natural transformations, which are exactly the analogies of analogies. So what makes better analogies is then the higher tower, like natural transformations, and the squares that commute, and the towers above that.
In practice, when somebody reaches for a metaphor, the move isn’t to ask is it vivid, it’s to ask does it compose. Run it three steps. If you still come out at something true, you’ve got an analogy. If not, you’ve got a sentence. You’ve got his jacket and a plant, and the plant is just sitting there.
The ideal analogy is the one that survives chaining. Everything else is X is like Y where X and Y are anything you want them to be.