**Formalists, Mathematics, and Pedagogy**

Mathematical economics has made it possible to state economic theory concisely and precisely and, by mathematical manipulations, to deduce the theoretical implications of a given set of assumptions. The formalists mathematically exposed inconsistencies and corrected logical errors in the literary reasoning that had been used to extend partial equilibrium analysis. Moreover, they showed that various aspects of the Marshallian model, such as demand theory and production theory, were simply specific applications of a generalized constrained maximization model. As they did, their mathematical techniques undermined the reason for using partial equilibrium analysis. Recognizing this, Samuelson went back to Walras to observe how he approached interconnected markets. Starting from Walras's analysis and applying algebra and calculus, Samuelson was able to determine the stability conditions necessary for equilibrium. This provided economic reasoning with a much more solid theoretical ground and an analytical core of multimarket equilibrium that would serve as the foundation of modern microeconomics.

But the introduction of formalism presented a pedagogical problem: the Walrasian general equilibrium approach is very difficult. In order to master it, one must learn a new language (mathematics) and be able to grasp highly abstract, noncontextual argumentation. But most economics undergraduates have no intention of becoming economists and hence have little incentive to acquire the considerable mathematical skill necessary to comprehend the complexities of general equilibrium interactions. This pedagogical problem has occasioned the current bifurcation of microeconomics, because the preferred graduate economics theory is too difficult for the typical undergraduate. Paul Samuelson responded to the special needs of undergraduate education by writing an elementary economics textbook that has sold several million copies and gone through many editions. Samuelson's text dominated the field for some thirty years from its first edition in 1947, and most other introductory texts have copied his format. This elementary text shaped modern undergraduate economics, just as his Foundations did graduate economics.

In his undergraduate text Samuelson graphically presented microeconomics as a logical extension of the interactions of rational individuals within a competitive market structure. Retaining the Marshallian tools but eliminating most of the platitudes and homey analogies of earlier economics texts, Samuelson constructed a largely noncontextual theory that is more consistent with general equilibrium analysis. It was in this manner that the current division of graduate and undergraduate economics developed. Undergraduate introductory texts kept the Marshallian approach, emphasizing two-dimensional graphical techniques rather than multivariate calculus, and graduate microeconomics moved on to the full mathematical approach that is far more consistent with Walras and Cournot than with Marshall.