# Mathematical economics

 Mathematical Economics(Harvard University, 1934-1938) A depiction of American polymath Edwin Wilson's steam engine and physical chemistry based course on "mathematical economics" taught at Harvard.
In hmolscience, mathematical economics, in regards to physical science based economics, is a term that tends to refer to the early 20th-century extension of the Lausanne school of physical economics, particularly the work of Vilfredo Pareto (1897-1912), into the American economic school, with the slight incorporation, assimilation, and upgrade of the chemical thermodynamics work of Willard Gibbs, particularly via Gibbs' student American polymath Edwin Wilson, who taught a physical chemistry based “mathematical economics” at Harvard for some years, and, in turn, his student economist Paul Samuelson, in the form of what can be classified as mathematical thermodynamics applied to economic theory, or "mathematical isomorphisms" as Samuelson seemed to refer to his subject.

Early history
The earliest efforts to outline a history of “mathematical economics” dates to Stanley Jevons’ 1879 “List of Mathematico-Economic Books, Memoirs, and Other Published Writings”. This was followed by Irving Fisher, who updated and revised Jevon’s list, publishing it in the appendix of his 1892 doctoral dissertation, entitled “Bibliography of Mathematical Economics”. (Ѻ)

Wilson | Physicochemical economics
In 1912, American polymath Edwin Wilson, the last protégé of Willard Gibbs, was dissecting the rational mechanics based socio-economics work of Vilfredo Pareto, supposedly with a Willard Gibbs’s chemical thermodynamics like approach in mind. [1] Sometime thereafter, he began teaching some type of steam engine/thermodynamics based course on economics, wherein equilibrium was defined to economics students the way it was being defined to physical chemistry students.

In 1932, Wilson participated in Lawrence Henderson's Gibbs-Pareto based Pareto seminar (1932-38), i.e. Harvard Pareto circle.

In 1934, Wilson discussed the teaching of “mathematical economics” a thermodynamics/steam engine based course in economics with Harvard economics chairman Harold Burbank, about his past teaching outlines and or upgraded teaching proposals, per influence of the Pareto seminar and Henderson. [3]

In 1935-38, Wilson was one of the guest lecturers in Henderson's Gibbs-Pareto based Sociology 23 course.

In 1938, e.g. Wilson communicated the following to Burbank in letter:

Schumpeter has suggested that it would be particularly well for me to give as I gave last time a general theory of equilibrium such as this is understood by physical chemists including the phase [see: social phase] systems of Willard Gibbs. Most of our equilibrium theory in economics really has for its background the notions of equilibrium which arise in mechanics. Although Pareto was certainly quite familiar with the types of equilibrium which arise in physical chemistry and are necessary in fact for the study of the steam engine he doesn’t use this line of thought in economics.”

This so-called "physicochemical economics" course, as is classified in modern terms, e.g. compare: physicochemical sociology, or “mathematical economics”, as Wilson seemed to refer to it, according to Roy Weintraub (1991), was taken by American economist Paul Samuelson. [3]

In 1938, to exemplify, Wilson wrote the following to Samuelson, in commentary on one of Samuelson's papers:

“Moreover, general as the treatment is I think that there is the possibility that it is not so general in some respects as Willard Gibbs would have desired. [In] discussing equilibrium and displacements from one position of equilibrium to another position [Gibbs] laid great stress on the fact that one had to remain within the limits of stability. Now if one wishes to postulate the derivatives including the second derivatives in an absolutely definite quadratic form one doesn’t need to talk about the limits of stability because the definiteness of the quadratic form means that one has stability. I wonder whether you can’t make it clearer or can’t come nearer following the general line of ideas [that] Gibbs has given in his Equilibrium of Heterogeneous Substances, equation 133.”

The very impressive mention of "equation 133", from Gibbs' subsection "Internal Stability of Homogeneous Fluids as indicated by Fundamental Equations", is the following:

$U - TS + PV - M_1 m_1 - M_2 m_2 ... - M_n m_n \,$

Wilson, in other words, is suggesting, as it seems to be, to Samuelson that he use the Gibbs fundamental equation to formulate a theory of economic stability.

Nine years later, in 1947, Samuelson, taking Wilson's advice, used some of this logic, in outline (e.g. Le Chatelier's principle), to pen his magnum opus Foundations of Economic Analysis, which invariably put economics into a new form of a more rigorous, semi-physical science conceptualized or analogized, mathematics-based science; see: Harvard Pareto circle (section: Wilson | Schumpeter | Samuelson) for more on this.

Quotes
The following are related quotes:

“... struck by a remark made by an old teacher of mine at Harvard, Edwin Wilson. Wilson was the last student of J. Willard Gibbs at Yale and had worked creatively in ... had become interested early in the work of Pareto and gave lectures in mathematical economics at Harvard …”
Paul Samuelson (1970), “Maximum Principles in Analytical Economics” [3]

References
1. Wilson, Edwin B. (1912). “Mathematical Economics: a Review of Manuel d’Econmie Politique by V. Pareto” (Ѻ), Bulletin of the American Mathematical Society, pgs. 462-74, Jun.
2. Weintraub, E. Roy. (1991). Stabilizing Dynamics: Constructing Economic Knowledge (Wilson letters, pg. 60; also 63-65). Cambridge University Press.
3. Samuelson, Paul. (1970). “Maximum Principles in Analytical Economics” (pdf), Nobel Memorial Lecture, Dec 11; in: The American Economic Review (abs), 62(3):249-62, 1972.