This is a preliminary version of an entry for the new edition of The New Palgrave, a Dictionary of Economics

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1 This is a preliminary version of an entry for the new edition of The New Palgrave, a Dictionary of Economics. Its publisher, Macmillan, will be the copyright holder of record. Please do not quote from or cite this paper without permission. Mathematics and Economics Abstract The interconnection of mathematics and economics reflects changes in both the mathematics and economics communities over time. The respective histories of these disciplines are intertwined, so that both changes in mathematical knowledge and changing ideas about the nature of mathematical knowledge have effected changes in the methods and concerns of economists. Mathematics and Economics Understanding the nature and role of mathematical economics is not the same as understanding the connection between mathematics and economics. Mathematical economics, as Debreu in this volume (mathematical economics) argues, is the employment of mathematics in economics itself. Explaining or justifying mathematical economics often involves essentialist arguments concerning the true nature of economic objects, and the true nature of the economy, as well as arguments suggesting that employing mathematics is appropriate since the underlying economy is quantitative in nature. Consequently an historical discussion of mathematical economics will be a narrative of increased sophistication over time in economics as mathematical tools, techniques, and methods move into economic discourse and enrich economic analysis. Alternatively, one can discuss the relation between mathematics and economics in terms of separate intellectual activities performed in separate intellectual communities, and in that case one will wish to look over time at the interpenetration of the ideas and practices of each community across their highly permeable boundaries. The history of mathematics concerns the changing body of mathematical knowledge such as new theorems proved, new research areas opened, and new techniques developed. But the history also involves changing images of mathematical knowledge: changing perspectives and understandings, for example, about what is the nature of mathematical objects, what constitutes a proof, what constitutes rigor, what constitutes useful versus not useful mathematics, etc.? (see Corry 1996: 3). Similarly the history of economics involves a history of not only the development of economic knowledge, but the development and changes in images of economic knowledge: what constitutes the economy, what constitutes a good explanation in economics, what constitutes serious empirical work in economics, what is a good model, etc? Consequently a discussion of the interconnection of mathematics in economics requires attention to the interconnection not just of the bodies of knowledge, as is reflected in the historical discussion of mathematical economics, but a historical

2 discussion of the interconnection of their respective images of knowledge. Put another way, a discussion of the connection of mathematics in economics must reflect economists changing conceptions of the image of mathematical knowledge and not just their changing understandings of the body of mathematical knowledge. This distinction between the body of knowledge and the images of knowledge provides a different perspective on the relation between mathematics and economics. The central point for economists to understand is that there were three distinct shifts in the image of mathematics from the beginning of the nineteenth century to the end the twentieth century. As a starting point, consider the conditions and perspectives under which mathematics was produced early in the 19 th century. Looking closely, we see, particularly in England, the importance of both Euclid s Elements and Newton s Principia. That is, from relatively early in the 19 th century, through the modifications of the Cambridge Tripos in 1849, and on through the middle third of the 19 th century at Cambridge, mathematics was understood as flowing, in its purpose and nature, from both Euclid and Newton. From Euclid one understood that geometry was the paradigm of mathematics, and that it was a path to truth. Theorems were derived from assumptions called axioms, where the truth of those assumptions was self-evident from our understanding of the physical world. To learn geometry was to understand how rigorous arguments could lead to truth. One studied mathematics, specifically geometry, as an exemplar of how one deduced truths about the world, and thus mathematics was the paradigm of deductive thought and logical ratiocination. Parallel to this view of how deductive reasoning from true premises could lead to true conclusions, Newton s Principia (his mathematical proofs of course were all based on Euclidian geometry -- even the calculus derivations were geometrical), suggested how this kind of mathematics could also open up an understanding of the physical world. Students were required to study mathematics because it provided a way of achieving truth. This image of mathematics is at the root of Ricardo s arithmetical models, and is present in Whewell s papers on economics using mathematics, for Whewell himself was central in reconstructing the Cambridge Tripos around Euclidean geometry and Newton s Principia at midcentury. Economics was to employ a particular kind of mathematics, Euclidean geometry, to demonstrate its propositions. Just as Newton employed geometrical proofs of his propositions, so too did Marshall. It is an interesting exercise to open Alfred Marshall s Principles next to the Newton s Principia and see the physical similarity of the proofs or demonstrations of the propositions in each book. Marshall, as Second Wrangler in the Mathematical Tripos of January 1865, had had to master both Euclid and Newton. The first change in the image of mathematics was developed from a new conception of what mathematical truth might mean. It occurred over the second third of the nineteenth century and was then well-incorporated in the Continental tradition in mathematics. That is, outside England there was a change in the image of mathematics between the time of Whewell s defense of mathematics in the educational process, a defense based in the notion that mathematics (vide Euclid, Newton) was the paradigm of certain and secure knowledge (the time of Marshall s student days), and Marshall s later time as Professor of Political Economy. The emergence of non-euclidean geometries had made Whewell s argument about axiomatics, and inevitable truth,

3 ring hollow long before the turn of the twentieth century. In the time of the new geometries, the difficulty of linking mathematical truth to a particular (Euclidean) geometry produced a real crisis of confidence for Victorian educational practice (Richards, 1988). This first crisis prepared the late Victorian mind for the new idea that mathematical rigor had to be associated with physical argumentation. And it was this new image of mathematics in science that helps us to understand the concerns of individuals like Edgeworth and Pareto. An emergent set of themes in mathematics developed from the increased awareness of alternatives to Euclidean geometry, and the recognition that no one set of axioms could be selected for demonstrating the truth of all mathematical propositions. Thus the success of the new rational mechanics (Lagrange s program of applying techniques of advanced calculus to the study of motions of solids and liquids) in making sense of the world of physical systems encouraged a refinement of the truth-producing view of mathematics. That is, in the last third of the nineteenth century, in England as well as Italy, France, and Germany, a rigorous mathematical argument began to be seen as one based on a substrate of physical reasoning. For an argument to be rigorous, and thus believable, the mathematical structure had to be founded generally on the most successful of applied mathematical practices, namely rational mechanics. A valid and good and useful mathematical model was a model that had physical interpretations. The marginal revolution in economics was precisely this new understanding. One sees this very clearly in Marshall who was at the cusp of this changed image of mathematics, for his derivations were offered using Euclidian geometry, but whose mathematical arguments about equilibrium and stability are instantiations of mechanical devices like an egg in a bowl, or a pair of scissors. Put another way, through much of the 19 th century in British mathematics, and thus to a degree among insular British economists for whom British mathematics was mathematics, rigor in argument was associated with geometric proofs based on assumptions, called axioms, that could be linked to constrained optimization processes associated with particular physical systems. Rational mechanics was taken as a paradigm for what economists came to call the marginal revolution, which though was hardly revolutionary but rather the migration of rational mechanical ideas into economic discourse (Mirowski 1989). Thus by the last decades of the nineteenth century one finds economists employing specific mechanical models of economic behavior. Walras, Pareto, Marshall, Edgeworth, and Fisher were producing rigorous mathematical models of economic processes, where rigor was associated with a mathematics tied to physical processes. But by 1900 the images of, and styles of doing, mathematics were beginning to change again in response to new challenges in mathematics and physics. In mathematics, there were problems associated with the foundations of mathematics. There were apparent inconsistencies in set theory associated with Georg Cantor s new ideas on infinity (i.e. transfinite cardinals, and the continuum of real numbers), and apparent inconsistencies in the foundations of arithmetic and logic, associated with work by Frege and Peano. Similarly troubling was the failure of physics, particularly rational mechanics, to solve the new problems associated with black-body radiation, quanta, and relativity. If the deterministic mechanical mode of physical argumentation was to be replaced by an alternative physical theory, what constituted a rigorous mathematical argument had to be re-described. In any event, some established areas of mathematics were no longer connected to a canonical physical model (Weintraub, 2002).

4 Consequently around the end of the 19th century, just as economists had begun to understand that constructing a mathematical science required basing argumentation of the physical reasoning of rational mechanics, and the measurement of quantities to further ground those reasoning chains, the image of mathematical knowledge was again changing. Modeling the concerns of the new physics appeared to require a new mathematics, a mathematics less based on deterministic dynamical systems and more on statistical argumentation, algebra, and new beliefs about appropriate axioms for logic and arithmetic. Just as the objects of the physical world appeared changed -- gone were billiard balls, newly present were quanta -- the recognition that the paradoxes of set theory and logic were intertwined led mathematicians to seek new foundations for their subject. Analysis of those foundations of set theory, logic, and arithmetic, and thus the foundations of sciences based on mathematics, were now to be based on axiomatic thinking. A rigorous argument was to be one built on strong foundations, and axiomatizing the structure of theories, in both physics and mathematics, was a path to the development of those theories (Hilbert, 1918). Thus following a late 19 th century period in which mathematical rigor was to be established by basing the mathematics on physical reasoning, around as understanding of the physical world became less secure -- mathematical truth was to be established not relative to physical reasoning but relative to other mathematical theories and objects. From a physical reductionism mathematics moved to a mathematical reductionism, in the guise of one or another set of ideas about formalism: problems and paradoxes and confusions of the turn of the century mathematics were to be resolved by a reconceptualization of the nature of the fundamental objects of mathematics. The images of mathematical knowledge, ideas of rigor, truth, formalization, and proof all changed over this period. It took a number of decades for this new image of mathematics to become securely established in the mathematical community. From Hilbert s 1918 call for axiomatization as the road to knowledge in mathematics and science, through the interwar years, mathematicians were slow to reframe their working concerns. So too did economists use of mathematics in the interwar period reflect the earlier perspectives of modeling economic problems as constrained optimization demonstrations imitating 19 th century mechanics. Beginning in the 1930s however, a group of French mathematicians, collectively called Nicholas Bourbaki, began rewriting mathematics from the foundationalist perspective (Weintraub and Mirowski, 1994). Mathematics was conceived, in their project, to grow organically from very basic ideas about sets, which led inexorably to the identification of a small number of mother structures (algebraic, order, and topological) from which other structures, other branches of mathematics, could be derived. Rigorous mathematics was not grounded in physical models, but rather in mathematics itself. Mathematics was to concern itself with analyses of mathematical structures. Over the next few decades pure mathematics, or mathematics uncontaminated by applications and disengaged from the world of applications, gained sway in the mathematics community. It was in this period that the eminent mathematician Paul Halmos (1981) famously titled an article Applied Mathematics is Bad Mathematics. In economics, this concatenation of ideas moved into mainstream theory with the work of Gerard Debreu, Kenneth Arrow, and Tjalling Koopmans. The Cowles Commission, in the 1940s at the University of Chicago, became the site for production of this kind of work in mathematical economic theory, particularly general equilibrium theory.

5 Yet even as a pure mathematics was taking hold in economics, the exigencies of World War II, and economists involvement with scientists, engineers, and other social scientists, moved mathematical economists concerns back from axiomatization and into what would become operations research. This of course was not pure at all, but based on concrete problems of real systems. As the historian of mathematics Amy Dahan Dalmedico noted: The second World War initiated what I shall call image war or representation war concerning what mathematics was about, what it dealt with, and how. Over the course of the 1950s and 1960s, this war was progressively developed until the balance of power began to shift perceptibly at the end of the 1970s and during the 1980s. This war was focused mainly on the cleavage between pure and applied mathematics, and on the tacit hierarchy of concepts as much as of values informing these categories of pure and applied. (Dalmedico, 2001, 224) Thus Bourbakist images of mathematics were becoming dominant in economics at the same time that the major challenge to those ideas was forming outside pure theory. The image of mathematics as a discipline concerned with understanding the structures of mathematical objects was indeed dominant in the 1950s and 60s not only in the United States, but in a number of other countries. Yet from World War II on through the Cold War, applied mathematics was taking root in disparately profound ways, and was attracting more and more support in form of grants and contracts and students. New fields of statistics, computer science, and operations research flourished. Consequently, economists ideas about mathematics began to undergo changes, as usual with some time lag, mirroring the changing images of mathematics that were reshaping interests and methods in the mathematics community itself. While structure was the emblematic term of the 1960s, model has now taken its place. In the physical sciences, climatology, engineering science, economics, and the social sciences, the practice of model-building has gradually dominated the terrain. It is today absolutely massive and intrinsically bound up with numerical experimentation and simulation. (Dalmedico, 2001, 249 If the important lesson from mathematics in the first third of the nineteenth century was that economics needed to become a deductive science (as geometry was), in the late nineteenth century the lesson from mathematics was that economics needed to model itself on rational mechanics. Over the first two thirds of the twentieth century the lesson was that economics was to become scientific by grounding its models and theories on a modest set of axioms concerning pure economic agents preferences and choices. But beginning nearly at mid-century, mathematics was re-imagining itself as a discipline that historically had developed by solving real problems presented to it from other sciences. And in a similar fashion, and partially in response to that changing image of mathematical knowledge, the notion of a serious economic science, connected to data-based reasoning, was reshaping the idea of rigorous argumentation in economics. Econometrics and applied microeconomics were to form the reconstructed core of economic science much as work in algorithmics and applied mathematics were re-commanding attention in the mathematics community. At the Berlin International Congress of Mathematicians in August 1998, the old opposition between the pure and the applied still widely shared in the community has been formulated in quite different terms: mathematicians

6 who build models versus those who prove theorems. [Mumford, 1998]. But the respect enjoyed by the former is now definitely as high as that of the latter. (Dalmedico, 2001, 249) So too in economics, as the prestige accorded good work in applied economics now rivals that accorded to work in pure theory. References Corry, L. (1996). Modern Algebra and the Rise of Mathematical Structures. Boston, Birkhäuser. Dalmedico, A. D. (2001). An Image Conflict in Mathematics After Changing Images in Mathematics: From the French Revolution to the New Millenium. U. Bottazzini and A. D. Dalmedico. London and New York, Routledge: Halmos, P. R. (1981). Applied Mathematics is Bad Mathematics. Mathematics Tomorrow. L. A. Steen. New York and Heidelberg, Springer-Verlag: vi Hilbert, D. (1918). "Axiomatisches Denken." Mathematische Annalen 78: Mirowski, P. (1989). More Heat Than Light. New York and Cambridge, Cambridge University Press. Mumford, D. (1998). "Trends in the Profession of Mathematics." Berlin Intelligencer, International Congress of Mathematicians Berlin. Richards, J. L. (1988). Mathematical Visions: the Pursuit of Geometry in Victorian England. San Diego, Academic Press. Weintraub, E. R. (2002). How Economics Became a Mathematical Science. Durham, NC, Duke University Press. Weintraub, E. R. and P. Mirowski (1994). "The Pure and the Applied: Bourbakism Comes to Mathematical Economics." Science in Context 7(2): E. Roy Weintraub Cross reference: mathematical economics