"Lyrics just don't hold up without the music," says Billy Collins, professor and former poet laureate. When his students argue that the lines by their favorite rock stars should be assessed as literature, he demurs: "I assure them that Jim Morrison is not a poet in any sense of the word."
When I was young, I knew a lot of kids that thought lyrics were really deep. I think it only seems so when it accompanies good music, the mood swings of youth, depression, or drugs. That is, in combination with other emotional drivers, lyrics seem much deeper than they are. By themselves many tunes that sound emotionally powerful when sung, are almost parodies of themselves when read silently (MacArthur Park? Muskrat Love? Careless Whisper?). I watched a documentary of the making of Bohemian Rhapsody, a great road tune, and they went over the lyrics, and it was clear that many words and images were picked merely because they rhymed, or the words or phrases out of context matched the mood of the song in that measure. As a poem it fails.
Alas, something similar is true for economics, where silly ideas, when combined with rigorous mathematics, don't seem so silly. Take an argument like the idea of infant industries, where if you protect an industry from overseas competition, it may increase a country's growth because only after infancy it can thrive; in infancy it may die from competition. The idea is predicated on increasing returns to scale, such that there's an inflexion point at which firm rapidly move down a cost curve relative to smaller producers. 200 years of empirical evidence, not logic, demonstrate this infant industry argument is wrong, that infant industries invariably become coddled, spoiled infants that never grow up. As per increasing returns, it is hardly a rigorous point: Microsoft and Google have both benefits (via increasing returns to scale) and costs (via decreasing returns to scale) peculiar to their large size. Yet if you dress the argument up in mathematics, where the idea is derived given algebraic assumptions for utility, production, and market clearing conditions, you get Paul Krugman's Nobel Prize. The argument isn't any more compelling with mathematics when you try to explain it in words via Krugman's model, and that should tell you something.
Krugman might say in his defense that his argument was meant to explain the stylized fact that trade between two large nations can hardly be explained by mere natural resources, and so how does one explain persistent German excellence in cutlery, or why Sweden exports and imports autos, except via some Guthian-inflation that occurs in certain firms. Fine, I get that, but Krugman's algebraic model is not really useful in isolating some idea in that argument. Without the faux-rigorous math model, there's no there there. Economics as a 'science' might be poorer without math, but as an understanding of our world it would have been the same.
Math helps ideas when it isolates the relationships or implications with greater parsimony or precision. In practice, economic models are abstruse arguments with mere potential for great scope, but the history of economics suggests great skepticism for this potential. What irreducible-to-words model, other than derivatives models in finance or statistical models in econometrics, have generalized to create a greater consensus on anything of economic importance? My best guess would be Ricardo's theory of comparative advantage, but note 1) that was made over 150 years ago and 2) it hasn't really convinced anyone that free trade is a good idea (it's easy to add some assumption that invalidates it).
Paul A. Samuelson's Foundations of Economic Analysis, in 1947 argued that rigorous modeling based on optimizing behavior of agents and stability of equilibrium was the essence of scientific economics; that is, modeling. It was one of the greatest intellectual errors of the 20th century. After all, for all of Samuelson's noted brilliance, what economic idea remains, other that methodology? I would say the Law of Iterated Expectations, which basically says that rational expectations implies a martingale, which is like Brownian motion. Nice, but hardly proportionate to his outsized reputation while alive.
Having a sense of the rhythms and emotions of music helps poetry, and so to does the logic of mathematics help economics. But good economics, like good poetry, almost always stands by itself without the adjunct. Just as we should beware of poetry that only sounds good when there's music playing, we should distrust economic analysis that only seems profound within mathematical models. There's the hope that if we use some rigorous method of reasoning economics will avoid the stagnation of ideas we see in politics or journalism. Would that it were true.