Formalizing economic theory
from Lars Syll
What guarantee is there … that economic concepts can be mapped unambiguously into mathematical concepts? The belief in the power and necessity of formalizing economic theory mathematically has thus obliterated the distinction between cognitively perceiving and understanding concepts from different domains and mapping them into each other. Whether the age-old problem of the equality between supply and demand should be mathematically formalized as a system of inequalities or equalities is not something that should be decided by mathematical knowledge or convenience. Surely it would be considered absurd, bordering on the insane, if a surgical procedure was implemented because a tool for its implementation was devised by a medical doctor who knew and believed in topological fixed-point theorems? Yet, weighty propositions about policy are decided on the basis of formalizations based on ignorance and belief in the veracity of one kind of one-dimensional mathematics
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Real-World Economics Review Blog 
Of course, there is no guarantee that economic concepts can be mapped unambiguously into mathematical concepts. Finding a correspondence between the two classes of concepts is one of the most important parts of theoretical research.
The main part of Velupillai’s arguments in his Variations on the Theme of Conning in Mathematical Economics (Journal of Economic Surveys 21(3): 466-505, 2007) is that general equilibrium theory (GET) draws on the existence of an equilibrium, which draws on variants of fixed point theorems, and thus draws on the axiom of choice. He contends that the GET proved the existence of an equilibrium which is but uncomputable and non-constructive.
He is right in his contention. It does not imply that economics that uses mathematics is wrong by itself. What Veluppilai has proved is there is a defect in GET arguments. That does not mean that no mathematical economics is possible. Actually, Veluppilai himself noted almost at the same time that an alternative economics is possible citing Sraffa’s mathematical economics. See Velupillai, K. Vela (2007a) Sraffa’s Mathematical Economics: A Constructive Interpretation. Discussion Paper No.2, Department of Economics, University of Trento. Later published in Journal of Economic Methodology 15(4): 325-342, 2008.