This is a wonderfully lucid (and somewhat surprising) discussion of the role of abstraction in research. Chomsky reminds us that mathematics isn’t literally a language. The sentence I’ve used as my click-baiting title comes in his very clever analogy between (I presume) long jumping, chicken flight, and the flight of eagles. He points out that people can jump about 30 feet (the record is in fact under 25 feet) and chickens can only manage about an order of magnitude (about 300 feet) more than that. Eagles, meanwhile, can stay in the air for hours.
It’s true that humans fly more or less like chickens and neither are like eagles. But that’s not the way it works. Chickens fly like eagles and humans don’t fly at all.
That is, humans “fly” (metaphorically) almost as well as chickens, but eagles (literally) fly much better than chickens. Similarly, we might compare pidgin languages, i.e., “chickens” (not “pigeons”) with Eagle … sorry, English. One is more of a language than the other but both are, as Chomsky points out, natural phenomena. Mathematics is a deliberate human invention. So, it’s true that mathematics is a language more less like pidgin and neither is like English. But that’s not how it works. A pidgin is language like English and mathematics isn’t a language at all.
(I’m happy discuss my stretching of this metaphor and the quality of my puns in the comments.)
You may have been struck by Chomsky’s matter-of-fact denial of a meaningful concept of the “physical” as something distinct from the “mental”, saying there just is what there is and some of the things there are are thoughts. I like that way of approaching it. It’s an approach to a problem I took up a while back about the facts and “propositional states”. It’s sort of liberating to just talk about facts and things how they are related and not worry about thoughts and concepts as some weird other kind of “stuff”.
We must recognize that all our thinking about things, and certainly all methodical inquiry, is really about “abstract objects”, not things as such, separate from our mental life. The things are simply part of our mental life. Idealization, as Chomsky points out, thereby gets us closer to reality by making us more precise. It lets us understand specific aspects of the world more clearly. By the time we can apply a mathematical formula, we’ve made things sufficiently simple, very precise, that’s all. I think there’s much to learn from this.