"Nothing is more likely than that the verbal expression of the result of a mathematical proof is calculated to delude us with a myth." -- Ludwig Wittgenstein, Remarks on the Foundations of Mathematics (1978) Part III., 26.
"Justin Fox sums up the overwhelming majority of economics papers in one sentence:The problem isn't that one has no good basis for thinking this assumption true. Rather, the problem is that lots of evidence indicates this assumption is false.'The basic form of an academic economics paper is a couple of comprehensible paragraphs at the beginning and a couple of comprehensible paragraphs at the end, with a bunch of really-hard-to-follow math or statistical analysis in the middle.'What he doesn't (need to) mention is the way that journalists, myself included, read economics papers: we generally have no ability or inclination to try to understand the details of the formulae and regression analyses, so we confine ourselves to reading the stuff in English, and work on the general assumption that the mathematics is reasonably solid." -- Felix Salmon
For example, macroeconomists will claim to be presenting models with microfoundations, and thus invulnerable to the Lucas critique. Typical assumptions in such models included the existence of only one good and of a representative agent. The one good serves as both a means of production and a consumption good. The agent decides how much of the good in each period to allocate to consumption and to the accumulation of capital. The Cambridge Capital Controversy and the Sonnenschein-Mantel-Debreu results show this approach lacks microfoundations.
Some economists might say this is old news. For example, they might point to models supposedly with heterogeneous capital goods. But some of these lauded models use the quantities of the capital goods, as measured in numeraire units, as arguments in production functions. Economists who model "heterogeneous" capital in this way typically seem to be ignorant of the impact of price Wicksell effects.
Further, Philip Mirowski has shown that economists fail to investigate the conservation laws built into their equations.
As with economists themselves, journalists covering developments in academic economics vary in their understanding of the theoretical issues embodied in the math. Chris Hayes made quite a splash with his Nation article a couple years ago on heterodox economics. John Cassidy examines, for example, the impact of the Sonnenschein-Mantel-Debreu results, on general equilibrium theory, in his book, How Markets Fail: The Logic of Economic Calamities. (I haven't read Justin Fox's book.)