How Economics Became a Mathematical Science. (Book Reviews).

By E. Roy Weintraub 

Durham, NC: Duke University Press. 2002. Pp. xiii, 313. $18.95 (paperback).

It is a contemporary truism that if you hope to make a contribution to the field of economics, or even to study it, you had better first get a solid background in mathematics. It should also be evident that this has not always been the case, as anyone who has read Adam Smith or Karl Marx, or for that matter Frank Knight, J. M. Keynes, or F. A. Hayek, well knows. Sometime in the twentieth century economics changed, and changed profoundly. In How Economics Became a Mathematical Science, Roy Weintraub attempts to make some sense of the transformation.

I was not prepared to enjoy this book. I knew Weintraub to be a lively writer, but his topic was daunting. I was half expecting a forthrightly pedantic and in the end (at least, given my own tastes) ploddingly dull monograph that followed the usual formulaic listing of "seminal contributions" on the road to full mathematization. In glum anticipation I imagined the questions to be covered: Shall we start with Cournot? With Bentham and the felicific calculus? Was Walras, as Schumpeter wrote, the greatest economist who ever lived, at least that is until his twentieth century successors, Arrow and Debreu and McKenzie, surpassed him? There would of course be the obligatory chapter on Samuelson's Foundations, another on the introduction in economics of fixed point theorems, still another on the fine points of proof strategies. At least for me, reading such a book would be a real challenge. It would be a duty, and a grim one.

To my great good fortune, Roy Weintraub's How Economics Became a Mathematical Science is neither pedantic nor dull. It is, in fact, an altogether extraordinary book. Organized not in terms of a grand narrative, it is instead a series of snapshots. The snapshots are not necessarily of seminal moments, but of representative ones, and they are well chosen. Like all good history, the stories Weintraub recounts are multilayered, complex, even messy, a word he uses (p. 207). It would be unfair to expect that such a collection of vignettes should hang together. Incredibly, though, they do. As a final bonus, there are real surprises to be found in nearly every chapter.

This, then, is an enjoyable book to read. But it is also an important one. I do not think it is an exaggeration to say that How Economics Became a Mathematical Science itself promises to change the way that people view the relationship between mathematics and economics. Weintraub says as much at the start of one of his chapters: "Modern controversies over formalism in economics rest on misunderstandings about the history of mathematics, the history of economics, and the history of the relationship between mathematics and economics" (p. 72). I usually hate such bald statements. It turns out, though, that he's right.

Weintraub starts from a startlingly simple premise--both economics and mathematics have changed over the course of the past 100 plus years, so it makes sense to look at them both. Looking at how they have changed, and their interaction, might reveal some things that would be lost if one followed the all-too-usual Whig history route, the latter a form of backward induction in which one uses the present state of economics as a guide for picking out which episodes in the past are worthy of attention.

The surprises begin in the first chapter. When most economists think of mathematics, they think of a stable discipline consisting of a set of related subject areas (algebra, geometry, calculus, topology) that yield tools for economists to use in constructing models of varying levels of generality. The tools are out there, as it were, sitting in books on the shelf, only to be learned and applied. (1) This "bookshelf view of mathematical knowledge" is challenged in the very first chapter, one that carries the intriguing title, "Burn the Mathematics (Tripos)." Weintraub takes us back to Alfred Marshall's Cambridge at the turn of the century. It was Marshall who famously wrote in a letter to Arthur Bowley that after one had used math as a shorthand language to figure out a problem, one should rewrite the problem in English and "bum the mathematics." Marshall had a reputation as a competent mathematician, so his letter has long been considered a bit of a historical puzzle. Weintraub offers a plausible solution to the mystery, and changes in math are at the heart of his explanation.

Mathematics had been a part of the Tripos, or the central examination that all Cambridge undergraduates had to take, for much of the nineteenth century. The math Tripos was required of all students because mathematics was thought to be a tool for the discovery of truth, a discipline whose mastery would help students to construct sound arguments. The mathematics taught was applied rational mechanics, and as the century progressed the examination questions became ever more elaborate, meant to see how well a candidate could handle a "set of tricks and details, based on Newton, which were linked to applied physics and mechanics, and which could be tested in a time limited fashion" (p. 14). Weintraub provides sample test questions, some of which are very bizarre indeed.

By the end of the century, new ways of conceptualizing the nature of mathematics were beginning to emerge. The development of non-Euclidean geometries had undermined the notion that mathematics was an engine for the discovery of timeless truth. One program looked to the sciences to provide models for the discovery of new approaches in mathematics, which reversed the usual way of thinking. Another hoped that economics could be modeled on physics, which was taken to mean that better measurements were needed. The latter program, quite popular for a while, was itself soon to be undermined by developments in physics that reduced its own claims to exact measurement. Still another program advocated that mathematicians employ methods that resemble the more formalist mathematics of today.

In sum, there was plenty of contention around the turn of the century about what counted as a mathematically rigorous approach. Mathematics was changing, and the debates that people had about the uses of mathematics within disciplines like economics must be seen against that background. To return to poor Alfred Marshall, here is how Weintraub summarizes his dilemma: "Marshall was caught. His image of mathematics was formed by the early Victorian Mathematics Tripos of simple geometry, the drawing of cord segments and conic sections, simple statics, dynamics and the like. His conception of mathematics was incompatible with either the late-nineteenth-century mathematics of physical-model-based analysis, or that which was to supplant it in turn, the early-twentieth-century move to axiomatics and mathematical-model-based analysis." (p. 24). Marshall was caught, and his ambivalence helps explain his remarks in his letter to Bowley. His approach to mathematics was soon to be retired. In the next chapter, we meet ano ther short-lived interpretation, one that Weintraub associates with Griffith Conrad Evans. Why have we never heard of Evans? Although he was famous enough in his own time, he has been neglected because his approach, based on the physical-model-based approach to analysis, also did not prevail.

Let us assume that after two chapters the reader is convinced that understanding changes in mathematics will help us better to understand specific positions held by economists in the past. The implication seems simple enough; we should learn more about the history of mathematics. But again, there are more surprises. Trying to figure out the history of mathematics brings us to more contested territory. Weintraub illustrates the problem in his chapter "Which Hilbert?" Here he shows that the Hilbert program in mathematics, one that helped initiate the turn toward a more austere mathematical formalism (and as such, a key development), is itself the center of interpretative debates among historians of mathematics. This is a somewhat difficult chapter to follow, chiefly because Weintraub is responding to other historical interpretations, which he must summarize at the same time that he is trying to develop his own reading. But the message is clear enough: just as there is no bookshelf of mathematics to borrow from , there is no bookshelf of the history of mathematics to borrow from either.

No book on the mathematicization of economics would be complete without a discussion of general equilibrium theory (GET), which for a number of decades during the past century held the dubious distinction of being both the standard exemplar of mathematically rigorous theoretical economics and the standard target for all those critical of the aridity and empirical emptiness of high theory. We encounter GET in two chapters, both of which underline the importance of historical contingency in the development of our discipline.

In Chapter 4, titled "Bourbaki and Debreu," Weintraub shows how the formalist movement in mathematics arose in part as a result of the efforts of the Bourbaki school in France, whose goal in a series of volumes was to elucidate the basic structures that formed the foundations for all of mathematics, so that as their "immense project took shape in print over the decades, mathematics was presented as self-contained in the sense that it grew out of itself, from the basic structures to those more derivative, from the 'mother-structures' to those of the specific areas of mathematics" (p. 108). Although the Bourbaki School never got much beyond programmatic statements, its influence in French mathematics was substantial, and it entered into economics in part through the work of Gerard Debreu. As Weintraub notes, Debreu's 1959 book, The Theory of Value, "wore its Bourbakist credentials on its sleeve," and that although "more than one member of the profession might have thought this species of economist had dropped f rom Mars, in fact he had merely migrated from France" (pp. 114-5). Debreu managed to appear at the Cowles Commission in 1949 at just the right moment, and for a variety of reasons within a year Bourbakism "became the house doctrine" (p. 119), and from there went on to influence the discipline as a whole. If one combines Weintraub's stories with Philip Mirowski's (2002) recent account of developments at the Cowles Commission, it is easier to see just how economics became the way it is today. At the same time, it is also evident that the changes that took place were certainly neither preordained nor constitutive of any objective scientific progress, but rather were the results of the sort of contingent events that populate the histories of the development of all disciplines.

At this point in the book, the reader is aware that mathematics is different from what we thought, that its own history has been contentious, that it has gone through phases and enthusiasms just as our own discipline has, and that its relationship to economics has been colored by all of this. But surely, the relationship between the two disciplines is not wholly idiosyncratic and contingent. Can we not at least still accept the basic idea that a mathematician can tell us when, say, a particular derivation is right or wrong? Alas, the answer here may also be no.

In two chapters that feature Ted Gayer as a coauthor, Weintraub further debunks the standard view of the economics--mathematics relationship. In one of these we meet Professor Cecil Phipps, a mathematician at the University of Florida who took it upon himself to check up on the applied work of economists. He described his role in a letter to Don Patinkin as follows: "I examine [applied work] for soundness of the mathematics in them. If it is faulty, the article is worthless until the defect is corrected" (p. 164). Phipps' own self-image, then, corresponded to the standard one. But as Weintraub and Gayer show, things are not so simple. Their examination of an extended (and, as time went on, an increasingly exasperated) correspondence between Phipps and Patinkin demonstrates that mathematicians sometimes have very different concerns from those of economists. They give different answers to questions of what constitutes an adequate proof, and may not see why an economist, trying to highlight a specific economic r elationship, might want to set up a problem in a particular way. In short, Weintraub and Gayer believe that translation across the two disciplines is typically difficult and in some instances may be impossible, or as they put it, "mathematics is a separate and distinct set of discursive practices and arguments, in which Kuhn's incommensurability problem occurs in spades, and translation fails necessarily" (p. 171).

It is within this context that GET comes up again. It turns out that Professor Phipps was one of the referees of the original 1954 Arrow-Debreu paper, "Existence of an Equilibrium for a Competitive Economy." As was true for many other economics papers that Phipps refereed, he urged rejection. The editors had a real problem on their hands, for only a few economists had the mathematical skills to even read the article, much less assess it. In the end, the editors found a handful of prominent economists, and all supported publication, so it was published. How did these prominent economists argue for what would become overnight one of the most famous papers in their discipline? Most of them didn't. They just asserted that they trusted the judgment of the authors over that of someone like Phipps! Ain't science wonderful?

As noted at the outset of this review, there was a time during the twentieth century when economists like Knight, Hayek, and Keynes could make contributions to economic theory without making recourse to mathematics. When did that change? Weintraub answers that final question with yet another biographical sketch, this one of his economist father, Sidney Weintraub. The younger Weintraub presents us with an abbreviated and remarkably honest family history, one that at times is difficult to read and must have been excruciating (though doubtless cathartic) to write. Sidney Weintraub was a figure on the cusp. He had aspirations to make theoretical contributions, and he recognized the importance of mathematics. We read over and over again in his wartime letters to his bride of his efforts to teach himself advanced calculus. But he never really succeeded, and it limited what he was able to do as an economist. Like Marshall, he was caught. Roy Weintraub completes his family history with another very honest portrait, t his last one autobiographical.

The final chapter is probably of interest principally to other historians of thought. Over the years, Roy Weintraub has couched his historical studies within a variety of larger frameworks. In recent times, his frameworks were less absolutist, less like frameworks, but still, they were there. This book, like Keynes' (1936) General Theory, marks his final escape from the "myth of the framework" that plagued his earlier work. It is an escape that I applaud.

How Economics Became a Mathematical Science is a gem, the work of a mature historian of thought at the peak of his powers. It is also a book that virtually any economist will enjoy. Whether you call it a late holiday present or anticipated beach reading, treat yourself to this book.

(1.) I owe the "bookshelf of knowledge" metaphor to Hands 2001.

References

Arrow, Kenneth, and Gerard Debreu. 1954. Existence of an equilibrium for a competitive economy. Econometrica 22: 265-90.

Debreu, Gerard. 1959. The theory of value: An axiomatic analysis of economic equilibrium. New York: John Wiley & Sons.

Hands, D. Wade. 2001. Reflection without rules: Economic methodology and contemporary science theory. Cambridge, UK: Cambridge University Press.

Keynes, John Maynard. 1936. The general theory of employment, interest and money. New York: Harcourt, Brace.

Mirowski, Philip. 2002. Machine dreams: Economics becomes a cyborg science. New York: Cambridge University Press.