Noah Smith is encountering the frustration that most graduate students in economics experience when their professors put them through a math wringer that the students correctly suspect is largely useless. (In the more than 20 years I have worked on economic policy, as a consultant in several countries, in the U.S. Congress, and at the U.S. Treasury, I do not recall having used anything beyond basic algebra and some elementary statistics that you can run in Excel, such as a regression line in a scatterplot diagram. Only a few people I know who work on economic policy use more math than that.) Bryan Caplan at EconLog maintains that "economath fails the cost-benefit test."
I made a few comments on models in a post months ago. Now I add that what graduate school professors call, often disparagingly, "intuition" is in fact the real economics. When mathematicians say they are giving an intuitive explanation, they mean they are skipping important steps for the sake of making the presentation more understandable, or in some cases, they mean they suspect they know the answer but are missing some steps to prove it. When economists say they are giving an intuitive explanation, they mean they are explaining how people think and act. But people's thoughts and actions are the building blocks of economics! Any satisfactory account of economic phenomena has to be able to show how they arise from those building blocks. The building blocks are also what we use to navigate our way through the human world. The main reason that graduate school economath is so useless later in most careers is that it leaves out the thinking, acting people who make the economy hum. Instead of stories about people, it offers much less satisfying stories having to do with aggregates, propensities, and the like.
"Intuition" is not the right word for the explanations in terms of thoughts and actions that good economics offers. A more precise term is fortunately at hand: Verstehen, the German for "understanding." As many Austrian economists and German-language or German-influenced writers in other social sciences have used it, it means our understanding of other people's intentions and actions. More English-speaking need to become familiar with it.