The Mathematical Economists
Economics faces six fundamental questions: (i) will the society organized on the principles of exchange (the Market Economy) stay composed or will it fall apart (the question of existence of equilibrium)?, (ii) will such an equilibrium be unique (a multiplicity of equilibria poses difficult and embarrassing questions)?, (iii) will such an equilibrium be robust (the question of stability of equilibrium)?, (iv) will such an economy (society) be efficient?, (v) will it grow or expand forever?, and (vi) will it be just? The classical economists, Adam Smith in particular, answered all these questions affirmatively using a characteristic methodology. However, Karl Marx challenged the entire structure of faith in the merits of the exchange economy and shattered all optimism regarding the said order. The neoclassicists, mostly using their own new (mathematical, marginalist, rationalistic, atomistic, hedonistic, etc) methodology set out to prove that answers to all those six questions were in affirmative. They restructured the faith in the said order. In so doing, they had to distance themselves from the reality and they did not mind doing so. This endeavour made neoclassical economics dogmatic and religious in nature. Leijonhufvud (1973) characterized neoclassical economics in the most sarcastic manner.
In his A History of Western Philosophy Bertrand Russsell wrote: "The origin ... is in the achievements of mathematicians who set to work to purge their subject of fallacies and slipshod reasoning. ... It gradually became clear that a great part of philosophy can be reduced to something that may be called "syntax". ... when errors in syntax are avoided, a philosophical problem is thereby either solved or shown to be insoluble. ... there can be no doubt that the utility of philosophical syntax in relation to traditional problems is very great." (pp. 783-785).
The economists who were guided by the philosophy hinted at by Russell made serious attempts to mathematically formulate (often the Neo-Classical) economic theories in the mathematical syntex and purge the economic philosophy (the system of theories) of the prevailing fallacies and slipshod reasoning. Those economists are the Mathematical Economists.
Mathematical economics is, unfortunately, often thought to be as any and every application of mathematics to represent economic theories and analyze problems posed in economics. If one argues in the same vein, any and every use of mathematics in physics will amount to doing mathematical physics. Now since no branch of physics is such where mathematics is not used; ergo, the entire body of physics is mathematical physics. But that will not do. Mathematical physics is a term used in a very special sense. It denotes research aimed at (or the body of knowledge at hand) studying and solving problems inspired by physics within a mathematically rigorous framework. Mathematical physics in this sense covers a very broad area of topics with the common feature that they blend pure mathematics and physics. Although related to theoretical physics, 'mathematical' physics in this sense emphasizes the mathematical rigour of the same type as found in mathematics. In the same manner, mathematical economics denotes research aimed at (or the body of knowledge at hand) studying and solving problems inspired by economics within a mathematically rigorous framework (such as the one we have in Arrow's impossibility theorem). It refers to the axiomatic development of mutually consistent theories in economics making a coherent system. It is not the representation of an 'indifference curve' by, say, a rectangular hyperbola and its tangency with a downwards sloping price line. It is concerned with the implications of the said mathematical specification on the theory of consumer's behaviour immediately and the theory of general (competitive) equilibrium finally. It is also concerned with representing a cardinally conceived price line tangent to an ordinally conceived indifference curve (innocently specified as a rectangular hyperbola) in the same plane.
There is one more important point to note as indicated by Nicholas Georgescu-Roegen: "There are endeavors that now pass for the most desirable kind of economic contributions although they are just plain mathematical exercises, not only without any economic substance but also without any mathematical value". Georgescu-Roegen cites as an example a paper that assumes more traders in existence than there are points in the set of real numbers.
Dynamic General Equilibrium Economics is attracting many new authors. Most of their works may be free downloaded.