what's the ideal analogy

I spent a decent fraction of today trying to explain to Scott Sumner that an analogy is like a functor, or really, a functor is the idealization of an analogy. Math in general is the idealization of these things. You take a thing the rest of the world does badly, you find the property that’s actually doing the work, you demand that property, and you throw the rest out. That’s what mathematicians did to “is like” when they coined functor. The property is composition. A functor preserves composition, that’s the whole point. If A is like B and B is like C, then A had better be like C, and the path A → B → C had better land in the same place as A → C. 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 is that it’s supposed to let you carry a fact from one place to another. And that move only carries if the analogy survives composition. Most analogies don’t survive…