Mathematical Economics
A year later, in the spring of 1937, Samuelson audited Mathematical Economics, the other course Wilson taught for the Economics Department. Schumpeter also attended this course and appears to have had more success in attending than the previous year, for at the end of it he wrote to Wilson to express his gratitude both for what he had learned from it and for the support Wilson had given “to a line of advance which at Harvard still has to fight for its existence.”23 His letter made it clear that Wilson had not been teaching the course in a conventional way.
The approach Wilson was taking was, he claimed, “indispensable if economics and economic policy is ever to emerge from the stage of phraseology on the one hand, and of pedestrian fact-eliciting on the other.” Schumpeter then went on to explain his view of the mathematics that young economists needed to learn, in a way that strongly implies he was echoing views Wilson had expressed in his course, for he urged Wilson to expand the first part of the course into its main content.Schumpeter believed that it was important for economists to learn from physics how to build up an exact argument. This would involve the use of concepts that were intermediate between pure and applied mathematics: they were neither pure mathematics (dealing with abstract concepts) nor applied mathematics (relating to specific problems).
[W]e can learn [from physics] to understand the relation of mathematics to the reality to which it is applied. Most important of all is the consideration that there are obviously a set of concepts and procedures which, although belonging not to the field of pure mathematics but to the field of more or less applied mathematics, are of so general a character as to be applicable to an indefinite number of different fields.24
Given Wilson's belief that concepts such as friction and inertia had to be defined in relation to economics before mechanical and physical analogies could be used, this may have been the result of Wilson, in his lectures, criticizing what he saw as Schumpeter's overemphasis on theory.25 When faced with difficult factual problems, it was important to learn to pay close attention to how existing mathematical tools could be applied, and to develop new ones, rather than simply copying techniques from one discipline to another.
This is entirely consistent with Samuelson's view of what he learned from Wilson about the relation between economics and physics: that there were similarities between the structures of economic problems and certain problems in physics. He could thus make use of these similarities to solve economic problems without implying that there was any deeper relation between the physical and the economic concepts.The main example of this was the Le Chatelier Principle, the idea for which he attributed specifically to Wilson's lectures: “In particular, I was struck by his statement that the fact that an increase in pressure is accompanied by a decrease in volume is not so much a theorem about a thermodynamic equilibrium system as it is a mathematical theorem about surfaces that are concave from below or about negative definite quadratic forms. Armed with this clue, I set out to make sense of the Le Chatelier principle.”26 He was to make the Le Chatelier Principle central to Foundations (1947a).c It should be noted that though accounts of it are often couched in terms of differential calculus, the principle is much more general and applies, as Samuelson argued, equally to systems in which discrete choices mean there is no smooth substitution between variables.
During the course Wilson asked Samuelson to present two lectures, afterward telling him that he had done “a beautiful job both with respect to the selection of his material which was mostly his own and with respect to the presentation of it.”27 Notes taken by Lloyd Metzler give an idea of how Wilson may have approached some of the material, but because they date from 1938 or 1939, care has to be taken not to attribute to Wilson material that he had learned from Samuelson in the intervening years.28 The notes began with a discussion of consumer theory, assuming continuous, differentiable functions before proceeding in the next class to the discontinuous case.
Here it is notable that an equation very similar to one that Samuelson was to use to sum up all that was known about consumer theory was described as the “J. Willard Gibbs condition.”dA week later, Wilson brought in thermodynamics, describing it as “a problem of constrained equil. since the system must always be closed—is similar to econ. in this respect.”29 Wilson then wrote down the maximization problems for thermodynamics and the consumer to show that, though there were similarities, they were not exactly the same. He even claimed that had Pareto been familiar with “the notion of physical equilibrium which was then prevalent in scientific work,” he would have worked out equilibrium conditions using finite differences rather than with derivatives, for it was more general.30 The notes went on to discuss integrability (deriving a utility function from demand functions) and the notion of a utility index.
c. See chapter 22 this volume.
d. Wilson's equation is ∆ξ∆x > 0, where ξ denotes marginal utility and x quantity (it is possible that Metzler transcribed the inequality sign incorrectly). The equation to which Samuelson attached importance was ∆p∆x < 0, where p is price.