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The Mathematization of Economic Theory Gerard Debreu The American Economic Review, Vol. 81, No. 1. (Mar., 1991), pp. 1-7. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28199103%2981%3A1%3C1%3ATMOET%3E2.0.CO%3B2-N The American Economic Review is currently published by American Economic Association. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/aea.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Thu Aug 23 23:26:40 2007 Number 92 of a series of photographs of past presidents of the Association The Mathematization of Economic Theoryt Morgenstern published the Theory of Games and Economic Behavior. As the Second World War was drawing While the professional journals in the field near its resolution, economic theory entered of mathematical economics grew at an un- a phase of intensive mathematization that sustainably rapid rate, the American Eco- profoundly transformed our profession. In nomic Review underwent a radical change several of its main features that phase had in identity. In 1940, less than 3 percent of no precedent, and it will have no successor. the refereed pages of its 30th volume ven- Assessing it requires a multidimensional tured to include rudimentary mathematical analysis acknowledging the contributions to expressions. Fifty years later, nearly 40 per- economics that were made, as well as the cent of the refereed pages of the 80th vol- tensions among economists that were ume display mathematics of a more elabo- heightened. rate type. The development of mathematical eco- At the same time, the mathematization of nomics during the past half-century can be economists proceeded at an even faster pace read in the total number of pages published in the 13 American de~artmentsof eco- each year by the leading periodicals in the nomics labeled by a recent assessment of field, an index that I will follow at first. research-doctorate programs in the United From 1933, the date when they both started States (Lyle V. Jones et al., 1982) as "dis- publication, to 1959, those periodicals were tinguished" or "strong" according to the Econometrica and the Review of Economic scholarly quality of their faculties. Every Studies, and the index tells of the decline year the Fellows of the Econometric Society from a high point, above 700 pages in 1935 (ES) certify new members by election into to the lowest point, below 400 pages in their international guild, which increased in 1943-1944. But 1944 marked the beginning size from 46 in 1940 to 422 in 1990. For of a period of explosive growth in which those 13 departments together, the propor- Econometrica and the Review of Economic tion of ES Fellows among professors was Studies were joined in 1960 by the Interna- less than 1 percent in 1940; it is now close tional Economic Review, in 1969 by the to 50 percent. It equals or exceeds 50 per- Journal of Economic Theory, and in 1974 by cent for six of them, which were among the Journal of Mathematical Economics. In those assessed as the eight strongest. So 1977, these five periodicals together pub- mathematized a faculty expects its students lished over 5,000 pages. During the period to have what it considers to be minimal 1944-1977, the index more than doubled mathematical proficiency, and knowledge of every nine years. By that measure, 1944 was calculus and linear algebra is required, or a sharp turning point in the history of math- forcefully recommended, for admission to ematical economics. It was also the year in all 13 graduate programs. which John von Neumann and Oskar Several scholarly recognitions lay addi- tional emphasis on the role that mathemati- cal culture is now playing in our profession. Of the 152 members of the economics sec- 'presidential address delivered at the one-hundred tion of the American Academy of Arts and third meeting of the American Economic Association, December 29, 1990, Washington, DC. Sciences, 87 are Fellows of the Econometric *Department of Economics, University of California Society; and of the 40 members of the eco- at Berkeley, Berkeley, CA 94720. nomics section of the National Academy of 2 THE AMERICAN ECONOMIC REWEW MARCH 1991 Sciences of the United States, 34 are ES been an inaccessible ideal toward which Fellows. From 1969 to 1990, 30 economics economic theory sometimes strove. During Nobel awards were made, and 25 of the that period, this striving became a powerful laureates are, or were, ES Fellows. Since it stimulus in the mathematization of eco- was first presented to Paul Samuelson in nomic theory. 1947, the John Bates Clark medal of the The great theories of physics cover an American Economic Association has been immense range of phenomena with a given to 21 economists, of whom 20 are ES supreme economy of expression. Of this, Fellows; and of the 26 living past presidents James Clerk Maxwell (1865) had given a of our Association, 13 are ES Fellows. notable example, as he described the elec- One may wish that those counts had not tromagnetic field by means of eight equa- been made. One may argue about points of tions at the time when mathematical eco- their interpretation. But they belong in our nomics was born and came of age in the common knowledge, and their thrust is un- middle of the 19th century. This extreme equivocal. They indicate how extensive the conciseness is made possible by the privi- mathematization of economics and how leged relationship that developed over sev- deep the accompanying change of our field eral centuries between physics and mathe- were over the past five decades. matics. In turn, the former presented the The perception of the depth of that latter with open problems, or found to ques- change is reinforced by a comparison of the tions raised by physical theory ready-made levels of mathematics required in 1940 and answers discovered by mathematicians in in 1990 to follow the development of eco- their abstract universe. Sometimes the nomic theory in every direction it was tak- causal linkage of research done in each one ing. Fifty years ago, basic undergraduate of the two fields could not easily be unrav- preparation in mathematics was almost al- eled; and, on occasion, the same scientist ways sufficient. Today, graduate training in made inextricably intertwined contributions mathematics is necessary. If, instead of be- to both disciplines. ing a follower, one wishes to be an active The benefits of that special relationship participant in that development along its were large for both fields; but physics did most technical avenues, a high degree of not completely surrender to the embrace of mathematical professionalism is called for. mathematics and to its inherent compulsion Several faculty members of the 13 depart- toward logical rigor. The experimental re- ments of economics mentioned previously sults and the factual observations that are at were actually identified as mathematicians the basis of physics, and which provide a by their doctorates; four of them served as constant check on its theoretical construc- chairmen of those departments during the tions, occasionally led its bold reasonings to past 25 years. If still sharper focus brings violate knowingly the canons of mathemati- out the intellectual leaders of that develop- cal deduction. ment, prominent among them is John von In these directions, economic theory could Neumann, one of the foremost mathemati- not follow the role model offered by physi- cians of his generation. cal theory. Next to the most sumptuous In that development process, mathemati- scientific tool of physics, the Superconduct- cal economics was continuously redefined as ing Super Collider whose construction cost new territories were included within its out- is estimated to be on the order of $10"' ward-moving frontier and as topics that were (David P. Hamilton, 1990; see also Science, once at that frontier became standard parts 5 October 19901, the experiments of eco- of the graduate, if not of the undergradu- nomics look excessively frugal. Being denied ate, economic-theory curriculum. a sufficiently secure experimental base, eco- nomic theory has to adhere to the rules of logical discourse and must renounce the facility of internal inconsistency. A deduc- Before the contemporary period of the tive structure that tolerates a contradiction past five decades, theoretical physics had does so under the penalty of being useless, VOL. 81 NO. I DEBREU: MA THEM TIZATION OF ECONOMIC THEORY 3 since any statement can be derived flawlessly commodity (opposite signs differentiating and immediately from that contradiction. inputs from outputs). That list can be treated In its mathematical form, economic the- as the list of the coordinates of a point in ory is open to an efficient scrutiny for logi- the linear commodity space.

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