EconSoc Editorial: "We Need a Sociology of Mathematical Economics"

By E. Roy Weintraub (Duke University)

A couple of years ago there was an exchange among some economics graduate students on an internet newsgroup for economics (sci.economics.res): Student number one asked:

In this exchange, we have a central conundrum for any study of the nature of modern economics: What is the connection the community of economists, and the community of mathematicians? That the communities are interrelated cannot now be questioned in fact, no matter what one’s likes or dislikes about this fact may be. But how might we attempt to understand this interrelationship?

At the beginning of the new millennium, many historians affirm: 1) knowledge is always associated with a larger or smaller group of individuals who share some beliefs which join them in community; 2) the transformation of beliefs into knowledge serves the community’s purposes in a local and contingent manner; 3) beliefs are transformed into knowledge in response to arguments that members of the community make to one another; and 4) the methods and arguments used within the community to transform beliefs into knowledge are mutable and indeed have not been stable over time. Thus to address the interrelationship of the two communities, we need to have histories and sociologies of those communities, and their conjoin, available to draw upon. That we do not have.

As the sociologist Steve Shapin wrote:

"We traditionally and formally warrant scientific truth by pointing to individual empirical foundations, yet nothing recognizable as scientific knowledge would be possible were that knowledge actually to be individually sought and held. Nor would the paradox be resolved if we were to conceive of scientific knowledge as the aggregate of what individuals hold in their heads. To the aggregate of individuals we need to add the morally textured relations between them, notions like authority and trust and the socially situated norms which identify who is to be trusted, and at what price trust is to be withheld. The epistemological paradox can be removed only by removing solitary knowers from the center of knowledge-making scenes and by replacing them with a moral economy." (Shapin 1994, 27)

How can we approach this issue for the community of mathematical economists? In the book (Mehrtens 1981) which reported the papers given at a conference in West Berlin in 1979, the historian of mathematics Herbert Mehrtens attempted to provide an overview of the issues that are involved in writing any social history of mathematics:

"We have to construe mathematics as both a body of knowledge and a field of social practice at the same time. These are not halves of a circular area embedded in the larger area equally divided into science and society. While the social practice of mathematics is determined by the nature of mathematics as a special type of knowledge, the historical process of extension and change of mathematical knowledge is a social process inseparably embedded in the societal environment. An individual new idea in mathematics is brought forward as a knowledge claim.' This is an act of communications subject to specific social regulations. The evaluation of such a knowledge claim within the community of mathematicians again is a process of social interaction. . . . The inclusion of an interaction into the dogmatized body of taught mathematics, its dissemination into areas of application and other mathematical or scientific sub-disciplines are social processes as well as subject to regulations imposed by norms and institutions." (265)

We are beginning to have some such social histories of mathematical economics, although the three I know best are all "to appear". Philip Mirowski’s Machine Dreams, Robert Leonard’s From Red Vienna to Santa Monica, and my own In the Image of the Queen all interrogate the social context of the interconnected mathematics and economics communities. Thus were I Research Director uber alles, I would encourage serious study of the sociology of mathematical economics, and the sociology of mathematical economics knowledge, which is not of course the same thing. And that sociology would need to be joined with these emerging social histories of mathematical economics, different from the intellectual histories which currently dominate the literatures.

That the communities themselves, and not Nobel-size individuals, must be the exploratory object should be clear. Indeed in some of my own writings I have borrowed from Stanley Fish, who in a 1980 book argued that:

"Interpretive communities are made up of those who share interpretive strategies not for reading but for writing texts, for constituting their properties. In other words the strategies exist prior to the active reading and therefore determine the shape of what is read rather than, as is usually assumed, the other way around. . . [But] an interpretive community is not objective because as a bundle of interests, of particular purposes and goals, its perspective is interested rather than neutral; but by the same reasoning the meanings and texts produced by an interpretive community are not subjective because they do not proceed from an isolated individuals but from a public and conventional point of view. . . [M]embers of the same community will necessarily agree because they will see (and by seeing, make) everything in relation to that community's assumed purposes and goals; and conversely, members of different communities will disagree because from their respective positions the other `simply’ cannot see what is obviously and inescapably there." (Fish 1980, 14-16)

Fish’s argument mirrors that first developed in science by Ludwig Fleck with his idea of "thought collectives", which itself was further shaped and refined by individuals like Imre Lakatos, with his idea that identifiable groups of scientists were linked by interests which could be termed a `scientific research program’. The idea is likewise related to the social groupings identified by Thomas Kuhn in his book The Structure of Scientific Revolutions. In any case a community of researchers, whether they define a subdiscipline, or a "school" in one sense or another, or an "invisible college", or a "network", shares strategies for understanding and constituting belief into knowledge. Indeed, it is because they share a number of understandings that the move from belief to knowledge is relatively easier for members within the group than it is with respect to exchanges between members of the group and individuals outside the groups. The innocuous observation that scientific knowledge is constructed within particular scientific communities has the corollary that different communities may understand differently.

Mathematicians and economists are different people. Of course they are communities of people who speak ordinary languages, have social roles, religious and philosophical beliefs, loves, desires, hopes, wishes, beliefs, and fears. We do not deny that they may separately have membership in the community of Roman Catholics say, or Chicago Clubs fans. Yet to be socialized as an economist is different from being socialized as a mathematician. In the sense that words have specialized meanings, problems have specialized histories, beliefs have specialized justifications, and the networks in which all of these activities take place remain separate and distinct, mathematicians and economists live in different worlds.

But there really are very few materials in the sociology of mathematics. There were some early attempts to look at the products of mathematics as socially determined, as Marxist and Marxist influenced sociologists argued that this or that development in mathematics was culturally forced by the particular forms of economic organization (Struik 1942). But leaving such arguments aside, we do not really have a very good understanding of what makes a good mathematician as opposed to what makes a good economist for example. Who gets trained as a mathematician, and are those people systematically different from the kinds of people who seek training as economists? The folklore of the subject suggests that the mathematician is a misfit, at least in American culture, developing from teenage oddballs who develop a passion for mathematics entirely unreinforced by the larger culture. The belief in departments of mathematics that a mathematician, if not smarter than the everyone else in the University, is at least able with a short period of study to do any other scholarly work that would appear in the University is a belief that differentiates the mathematician from say the sociologists who would never suggest, even in the sociology lounge, that the sociologists could do a better job teaching mathematics with a month's preparation than a mathematician could do.

Hubris, o'erweening pride, characterizes the mathematician's view of his (and until the past decade almost never her) learned profession. An economist can look back to Adam Smith perhaps, and feel a glow and a connection, but the mathematician can claim Euclid, and Archimedes, and Greeks, and Arabs, and Newton, and Galois, and Gauss, and Hilbert and von Neumann, and all such geniuses of the past. Compared to these, economists have a past of error and advice to monarchs and claims about a transient social order to their credit -- hardly anything, even a pin factory, compares with the prime number theorem. And the Arrow Impossibility Theorem is not the Riemann Hypothesis in depth and complexity.

This recognition of genius, and the longing for mathematical immortality connected to having a theorem, or lemma, or inequality named for oneself produces a competitiveness quite beyond what is typically known among academics.

What kind of vocabulary is available to us to construct a sociology or social history of a mathematical economics? In his 1973 paper "Some social characteristics of mathematicians and their work" C. S. Fisher tells us that:

"The first relevant aspect of the training of mathematicians is a learned feeling of independence. This contributes to the isolation of mathematicians. Mathematicians like to claim that they could carry on their activities independently of the rest of the people in the world. They feel that, having reached a certain stage of maturity, their discipline is so internalized that they could, given the fundamentals of a theory, develop all of its conclusions while living on a desert island. This assertion is usually accompanied by examples of the few famous mathematicians who, isolated in their youths and unaided, were able to make great progress in mathematics. These exotic claims do correspond to some everyday mathematical realities. For instance, mathematicians often work alone for years with a feeling of assurance that their output is mathematically correct… The second aspect of training is the common course of study which mathematicians of the same generation and locale is experience. This promotes the feeling that the discipline is unified and vitiates against the forces toward diffuseness to be mentioned next. Students of mathematics follow a common course of study until the last years of their graduate education. . . The third factor lies in the theoretical structure of mathematics. It contributes to the diffuseness of the discipline. . . a mathematician could cultivate either of these specialties (algebra, geometry, and analysis) for long periods of his scientific life and remain completely unaware of the activities of men and the other. . ." (232-234)

Aside from the above mentioned Colandar-Klamer work, we have nothing like this kind of discussion moving into the history and sociology of economics, let alone mathematical economics. In calling thus for a sociology of mathematical economics, and a sociology of the knowledge products of that initially hybrid, now mainstream community, I would call for a moratorium on publishing new contributions to the literature of complaint which drives contemporary discussions of mathematics in economics, and a redirection of energies away from "This shouldn’t be so" toward "What is this and how did it happen?". The researchable topics are vast, and cross a number of different fields and methodologies and historiographic sensibilities. Failing this kind of effort, I see little hope of constructing even partway interesting histories of twentieth century economics, let alone twenty-first century economics.

References

Barnes, B., D. Bloor, et al. (1996). Scientific Knowledge: A Sociological Analysis. Chicago, University of Chicago Press.

Bloor, D. (1991). Knowledge and Social Imagery. Chicago, University of Chicago Press.

Fish, S. (1980). Is There a Text In This Class? Cambridge, Harvard University Press

Fisher, C. S. (1984 [1972/3]). Some Social Characteristics of Mathematicians and Their Work. Mathematics: People, Problems, Results, Volume III. D. M. Campbell and J. C. Higgens. Belmont, CA, Wadsworth: 230-247.

Klamer, A. and D. Colander (1990). The Making of an Economist. Boulder, Colorado, Westview Press.

Mehrtens, H. (1981). Social History of Mathematics. Social History of Nineteenth Century Mathematics. H. Mehrtens, H. Bos and I. Schneider. Boston and Basel, Birkhauser: 257-280.

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Shapin, S. (1994). A Social History of Truth. Chicago, University of Chicago Press.

Struik, D. J. (1942). "The Sociology of Mathematics." Science and Society VI(1): 58-70.