I have argued elsewhere that it is meaningless to criticize mathematics per se. As Steve Keen puts it, “while it is undeniable that an inordinate love of mathematical formalism has contributed to some of the intellectual excesses in economics, generally this reaction is as erroneous as blaming the piano for the discordant notes of a bad piano player. If anything should be shot, it is the pianist, not the piano”. His book combines sound theory (Marx – Schumpeter – Minsky – Keynes) and chaos theory (non-linear differential equations) to analyze the inherent instability of a monetary economy.
He argues that poor scholarship leads to bad theory, both regarding the lack of knowledge in the history of economic thought as well as the lack of proper mathematical training. It is interesting to note that neoclassical economists are so vain about their unlimited mathematical skills and reject their critics on the basis that they do not understand math and therefore aren’t in a position to criticize their work. This can be traced back to Walras:
“As for those economists who do not know any mathematics […] let them go their way repeating that ‘human liberty will never allow itself to be cast into equations’ or that ‘mathematics ignores frictions which are everything in social sciences’ and other equally forceful and flowery phrases”
Ironically, Kalecki who was a great mathematician (see Libânio) was left at the backwaters of economics, which leads to the conclusion that the neoclassical claim to be scientific does not hold up to scrutiny. Since he had a Marxist background where classes were the basic unit of analysis and did not use the conventional tools – equilibrium, utility and production functions, optimization – he was ignored, as anyone who doesn’t follow the dogmas will. To put it short, mathematics on itself won’t take you to the ivory tower of neoclassical economics.
Tony Lawson, trained as a mathematician, supports Keen’s point that you don’t need to be a bad mathematician to be critical of economics. Eric Weinstein, in the same vein, proposes Gauge Theory to deal with economic issues. Evolutionary economics and complexity theory, as well, provide alternatives based on evolutionary game theory and computer simulation that are far from nonmathematical and take the social and historical much more seriously than standard economics.
For every flaw of standard theory there are ancillary assumptions to save the day, however, as Lakatos argues, this is a sign of a degenerative scientific research program, one in which the hard core is constantly being incapable of explaining new phenomena and most effort is applied at adjusting the ancillary assumptions.
Hence, whether you are enthusiastic of Keen, Kalecki, Weinstein, Evolutionary Economics, Complexity Theory (or what have you) you are entitled to say: “don’t shoot me, I’m only the piano”. However, be aware you won’t be invited to play in the lousy gigs where most of the audience goes. But, who said the best parties are the most crowded?