Claims that economics is close in spirit and principle to mathematics have been reiterated many times; a well-known one is William S. Jevons’s “our science must be mathematical, simply because it deals with quantities” (1871 [1888]: 1.5, original emphasis).
The use of mathematics is now widely recognized as an essential ingredient of both research and teaching in economics, and attempts to build mathematical accounts of economic phenomena are as old as the discipline itself (Theocharis 1961 [1983]).
The history of mathematical modelling in economics has not been a linear one. The profession privileged the verbal form for long, and even when the formal approach gradually gained ground, it often had to defend itself against hefty criticisms. Today’s widespread consensus around formalization does not prevent doubts from occasionally resurfacing.
Formalization and mathematical modelling in economics need to be distinguished from quantification, or the systematic effort to measure reality for purposes of administrative intervention, for instance, by counting the population in a census or by devising accounting rules to estimate national income. Quantification dates back to the very origins of government and received substantial impulse in the seventeenth century, with the seminal contributions of William Petty and Charles Davenant; it remains a necessary support of economic policy today. More deeply intertwined with the development of economic thought, formalization is an approach to theory-building which, starting from the definition of theoretical constructs and of linkages between them, allows deriving logical conclusions from given premises. It does not even require its objects to be measurable or quantifiable, in so far as they can be expressed as variables or functions of variables, and they can be compared (larger/smaller, more/less, positive/negative, and so on). Advocating this approach, Jevons wrote that “The data are almost wholly deficient for the complete solution of any one problem”, yet “we have mathematical theory without the data requisite for precise calculation” (Jevons 1871 [1888]: 1.9).
Because formalization purports to develop hypotheses and explanatory arguments independently of, or prior to, confrontation with empirical data, it is not to be confused with a third and more recent use of mathematics, namely, empirical validation and verification, testing (formal) theory against (quantitative) data according to some version of the scientific method. It is with the rise of econometrics (see this entry) in the twentieth century that the latter approach has blossomed.
The history of formalization is a succession of trials and errors, of failures and successes, and of difficulties that needed to be surmounted for mathematical models to provide relevant economic insight. At the beginning, only loose analogies were established between mathematical symbols or diagrams and economic concepts, so that the former could at most vaguely illustrate or summarize the latter, but could not guide reasoning. It took a long time to understand how symbols and formulas could be made to embody economic content in such a way that any conclusion reached mathematically - for example, by solving a system of equations - allows substantive economic interpretation.
The history of formalization is also a history of inter-disciplinary exchange and influence: far from being a set of ready-made tools for the economist to apply, mathematics is an independent discipline with its own constructs and modes of reasoning, whose use may contribute to forging economic ideas. It matters which mathematics is used: for example, do the assumptions of continuity and derivability required for calculus excessively constrain the choice of appropriate forms for the production and/or utility function? Is the assumption of linearity a reasonable approximation, or a deceptive misrepresentation of a more complex reality? Notice, also, that the choice of a particular branch of mathematics at a given point in time also depends on the state of mathematics at that time, so that its implications for economic theory may change with the further progress of mathematics (see also Weintraub 2002).
For example, the continuity assumption was introduced in economics to authorize the use of calculus, before the discovery that it is a necessary but not a sufficient condition for derivability; it remained widespread afterwards, supporting use of other mathematical techniques. Another example is the notion of limit, which was formed after the first applications of calculus to economics and required reinterpretation of existing results.A related issue is the relationship between economics and physics, the first example of a successful mathematical science. To name just two, Jevons himself and another founding father of mathematical economics, Leon Walras, made systematic use of mechanical analogies in justifying their theories and methodological choices. Even leaving aside the empirical dimension to focus on formalization alone, the questions arise of how physics may have influenced economics through mathematics and of the extent to which economics can meaningfully imitate physics in its choice of mathematical methods and procedures. Economists’ use of calculus is a case in point, as this area of mathematics was originally developed in close relation with Newtonian mechanics.
The next sections outline the history of formalization and mathematical modelling in economics, in two phases (up to 1925 and since 1925); a third part on recent developments has been added to this basic scheme. There is insufficient room here to cover all aspects of the intellectual reflection in mathematical economics over such a long time span. For this reason, the presentation is limited to a sketch of what each period contributed to the formal study of two foundational issues that, more than others, have lent themselves to mathematical treatment - namely, the theories of individual economic behaviour and of the market mechanism as a coordinating device.