...from Falkenblog (one of my favorite "economists", a word I hesitate to use since he clearly dips his ladle into many disclipllines). Von Mises could not have warned against false precision and the use of mathematics as a legitimizing tool (as opposed to a veracity tool) with any greater clarity.
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).