M. H. I. Dore
Economics purports to be a science about reality, about real entities such as production processes, firms, prices, consumers and so forth, all of which are real and finite entities. Mathematical theorems have no intrinsic empirical content; yet the growing use of mathematical theorems to portray and represent economic entities and their interrelations requires a greater attention to method. John von Neumann’s growth model (1937), translated in 1945, was the first major attempt to use mathematical structures and their related theorems to develop an economic analysis. To use mathematical results in this way, it is necessary to establish a strict one-to-one correspondence between mathematical object and entities. Without such correspondence, the economic interpretation lacks credibility.
The plan of this chapter is to examine the aspects of mathematical philosophy and method that are relevant to the use of mathematics in economics. Section II deals with a specific mathematical result and its economic interpretation: this mathematical result is von Neumann’s minimax theorem of rectangular games, and the growth model is its economic interpretation. A well-known feature of the growth model is the duality of prices and outputs; however, when seen in terms of the minimax theorem, this duality is in fact the bilinearity of prices and outputs. It will be argued that this bilinearity is very special, and was rejected by Sraffa in 1926.
The linear methods of von Neumann have been adopted by both the neoclassical school and the reformulated classical school based on Ricardo and Marx. Because linear methods are dynamically unstable they are appropriate only for “snapshots” of the economy, as carried out, for example, by Sraffa.
The instability of linear methods suggests the need for nonlinear methods if mathematics is to be useful for dynamic economic analysis. But unfortunately nonlinear methods require the use of even more complicated mathematical structures and theorems, with the added danger of a growing distance between the mathematical structures and the economic interpretation, von Neumann’s growth model is reexamined from the above perspective, and his method is contrasted with that of Sraffa.