I feel like I might be being a little coy stating this, but I feel like “heterogeneous preferences” may not be as inadequate as it seems. At least, if you allow that those heterogeneous preferences are not only innate like taste preference for apples over oranges.
If I have a comparative advantage in making apples, I’m going to have a lot of apples, and value the marginal apple less than the marginal orange. I don’t think this is a different kind of “preference” than liking the taste of oranges better: Both base out in me preferring an orange to an apple. And so we engage in trade for exactly that reason. In fact, I predict we stop trading once I value the marginal apple more than the marginal orange (or you, vice versa), regardless of the state of my comparative advantage. (That is, in a causal sense, the comparative advantage is covered by my marginal value assignments. My comparative advantage may inform my value assignments, but once you know those, you don’t need to know if I still have a comparative advantage or not for the question “will I trade apples for oranges?”)
Comparative advantage is why trade is useful, but I don’t know if it’s really accurate to say that heterogeneous preferences are an “inadequate answer to why we trade.”
I doubt my ability to be entertaining, but perhaps I can be informative. The need for mathematical formulation is because, due to Goodhart’s law, imperfect proxies break down. Mathematics is a tool which is rigorous enough to get us from “that sounds like a pretty good definition” (like “zero correlation” in the radio signals example), to “I’ve proven this is the definition” (like “zero mutual information”).
The proof can get you from “I really hope this works” to “As long as this system satisfies the proof’s assumptions, this will work”, because the proof states it’s assumptions clearly, while “this has worked previously” could, and likely does, rely on a great number of unspecified commonalities previous instances had.
It gets precise and pedantic because it turns out that the things we often want to define for this endeavor are based on other things. “Mutual information” isn’t a useful formulation without a formulation for “information”. Similarly, in trying to define morality, it’s difficult to define what an agent should do in the world (or even what it means for an agent to do things in the world), without ideas of agency and doing, and the world. Every undefined term you use brings you further from a formulation you could actually use to create a proof.
In all, mathematical formulation isn’t the goal, it’s the prerequisite. “Zero correlation” was mathematically formalized, but that was not sufficient.