The Role of Mathematical Reasoning in Economics

Entering the study of economics, Marshall set himself the task of enhancing its scientific status to the standard of the natural sciences. Economics, as he found it in the late 1860s, was in a mess.

Mathematics helped Marshall to build up the analytical machinery of book V of Principles, often considered to be the only part of his system that survived the test of time. To better understand Marshall’s views on the use of mathematics in economics, we need to look at his mathematical training. In his time, Cambridge mathematics was outdated when compared with France, where Lagrange and Laplace had established “the supremacy of analytic over synthetic-geometric mathematics” (Becher 1980: 1). The “impure” character of Cambridge nineteenth-century mathematics laid stress on the cal­culations needed for mathematical physics to the detriment of pure mathematics. In 1848, the reform of the tripos restrained the diffusion of analysis, “undermining the analysts’ emphasis on pure mathematics” and “preserving the tradition conducive to the develop­ment of physics” (ibid.: 46).

Given this background, Marshall almost naturally came to think that the fundamen­tal ideas of classical economics could be clarified by the use of functions that correlated variable quantities:

The most powerful engines for such a purpose are supplied by the various branches of the mathematical calculus. But diagrams are of great service, wherever they are applicable, in inter­preting to the eye the processes by which the methods of mathematical analysis obtain their results Diagrams present simultaneously to the eye the chief forces which are at work, laid

out, as it were, in a map. (Whitaker 1975, II: 133)

The mechanical model of the balance of forces, that originally belonged “to the older science, physics” (Marshall 1898: 43), was transferred, by analogy, to the “balance between efforts and the satisfactions resulting from them” (Marshall 1920: 353).

The theories of value and distribution of the early manuscripts bear witness to Marshall’s excitement at the discovery of how far mathematics could go to improve the consistency and perspicuity of economic doctrines. “On value” presents a whole set of equilibrium models, “depending mainly on the length of the period of time to which the investigation applies” (Whitaker 1975, I: 134).

At first sight, Marshall’s initial enthusiasm for the applications of mathematics to eco­nomics looks to be in sharp contrast with his later critical remarks on the “mathematical toys”, which make Arthur Bowley lose touch with the real world (Whitaker 1996, II: 301). Similar charges are levelled against Edgeworth, who “has crushed his instincts between the cog wheels of his mathematical machinery” (Whitaker 1996, II: 307; see also III: 130). This leads to thinking of a life parabola, owing to the progressive loss of mathematical competencies that Marshall himself avows (Whitaker 1996, III: 91). Harshness aside, however, there is nothing new in these criticisms. The same concern about excessive trust in mathematical analysis permeates the early review of Edgeworth’s Mathematical Psychics (Whitaker 1975, II: 267), and the late warnings against pure mathematical exercises remind us of the “mist of symbols” that worried Whewell (Becher 1980: 17), and was censured in the student’s guide to the tripos.

When Marshall thought of mathematical training as a privileged way of accessing the study of economics, and successfully strove to attract students from the Mathematical Tripos, he looked for students who were trained in applied rather than pure mathemat­ics, and had “mastered some branch of the physical sciences” (Guillebaud 1961, II: 173).