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.
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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
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 19th 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 19th century, through the modifications of the Cambridge Tripos in 1849, and on through the
middle third of the 19th 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,
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 19th 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).
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 19th century period in which mathematical rigor was to be established by basing the
mathematics on physical reasoning, around 1900 -- 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 19th 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.
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
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 1945. Changing Images in
Mathematics: From the French Revolution to the New Millenium. U. Bottazzini and A. D.
Dalmedico. London and New York, Routledge: 223-253.
Halmos, P. R. (1981). Applied Mathematics is Bad Mathematics. Mathematics Tomorrow. L. A.
Steen. New York and Heidelberg, Springer-Verlag: vi + 250.
Hilbert, D. (1918). "Axiomatisches Denken." Mathematische Annalen 78: 405-415.
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): 245-272.
E. Roy Weintraub
Cross reference: mathematical economics