The estimated reading time for this post is 7 minutes.

*This contribution is the third post in the four-part blog series on the history of mathematics in economics. For the first post on Philip Mirowski’s account of Irving Fisher, which also introduces the series, click here. For the second post on Marcel Boumans’s study of Jan Tinbergen, click here.*

**The previous contributions in the series have made you familiar with two stories in the history of mathematics in economics that reached radically different conclusions. For Philip Mirowski, the introduction of mathematics into 19 ^{th}-century economics was the result of physics-envy: economists without a proper understanding of physical theory, but with a fascination for its rigidity, borrowed from physics without noticing that these theories did not work in the economic domain. Marcel Boumans, on the other hand, showed how Jan Tinbergen used tools from physics to clarify economic problems by restricting this transfer to domains where the structure of the problem was the same for both fields – and, consequently, without presupposing a substantive analogy between the two disciplines, as was the case for Fisher. This third instalment will introduce the third and final author on the final economist in our series, with E. Roy Weintraub’s discussion of Gerard Debreu (1921-2004).**

In ‘How Economics Became a Mathematical Science’ (2002), Weintraub sketches the history of the mathematization of economics from the perspective of changing images in mathematics. In Weintraub’s account, Debreu explicitly rejected the use of physics and physical analogies in economics in favor of the pure Bourbakian mathematics.

**Constructing a new foundation for mathematics**

Before delving into Weintraub’s discussion of Debreu, it is necessary to provide some context about developments in the mathematics of the time. The mathematics of the second half of the 19^{th} century was thrown into disarray by the development of non-Euclidian mathematics. Until then, geometry was largely synonymous with Euclidian geometry. In the absence of an alternative, it was assumed that the postulates of Euclidian geometry were a necessary trait of space: as a result, the proposition that ‘the shortest distance between two points is a straight line’ was believed to be a definite truth about the world rather than a theory that could potentially be falsified. The idea behind non-Euclidian geometry was that it was possible to think of postulates that lead us to different conclusions than Euclidian postulates, without contradicting each other. A previously undoubted bastion of mathematical truth was thereby brought into question.

It was in this intellectual quicksand that David Hilbert aimed to restore some solid ground by what he called the ‘axiomatization of thinking’. Hilbert asserted that most sciences assume a particular theory or set of theories as its core, as the foundations upon which they build further (p. 87). Axiomatization implies that we try to underpin these theories with more fundamental axioms, which we in turn try to justify, all the way until we find it no longer reasonable to continue. We then ask whether the set of axioms we end up with are independent and consistent. Independent means that none of the axioms can be derived from any other; consistent means that no contradictions arise if we hold this set of axioms to be true simultaneously.

Hilbert’s views found a receptive audience, amongst others, in Bourbaki. Bourbaki was in fact a pseudonym for a group of French mathematicians, who shared an admiration for ‘pure mathematics’. To them, this pure mathematics was opposed to the applied mathematics of physics (and economics) – they upheld “the primacy of the pure over the applied, the rigorous over the intuitive, the essential over the frivolous” (Weintraub, p. 102). While Hilbert and Bourbaki shared their faith in axiomatic methods, the nature of their support for it was different (Corry, 1997). While Hilbert saw the axiomatic method as a useful way of generating knowledge in the sciences, he did not think the knowledge thus generated was infallible, nor did he see it as an independently sufficient guide to the organization of science. Bourbaki, on the other hand, believed that axiomatic methods would, if applied well, allow them to uncover the unchanging structure of mathematics – this was what they tried to do in their ‘Theory of Sets’. For Bourbaki, axiomatization meant more axiomatizing the assumptions of mathematical theories; it also meant relating these theories to each other, so that an underlying structure could be found. “Where the superficial observer sees only two, or several, quite distinct theories… the axiomatic method teaches us to look for the deep-lying reasons for such a discovery, to find the common ideas of these theories, buried under the accumulation of details properly belonging to teach of them, to bring these ideas forward and to put them in the proper light” (Bourbaki 1950, p. 223, cited in Weintraub 2002 p. 109). While the applied mathematics of physicists and economics tailored its tools to the domains to which it was applied, the pure mathematics of Bourbaki turned its gaze inward. The influence of the Bourbaki School was soon visible in French mathematics education and, as we shall soon see, was also exported beyond.

**The Truth about Economics is in the Mathematics**

This short introduction to Hilbert and Bourbaki has paved the way to the introduction of this contribution’s protagonist, Gérard Debreu. Debreu was a French economist, known amongst others for his 1954 article with Kenneth Arrow, in which they proved the existence of a general competitive Walrasian equilibrium, and his book on the ‘Theory of Value’. Trained by mathematicians in the tradition of Bourbaki, Debreu sympathized with their approach to axiomatization and their high esteem of mathematics and mathematical structures.

When Debreu joined the Cowles commission in 1950, it was in intellectual turmoil: firstly, empirical approaches to economics had lost credibility amongst Cowles commission economists, and secondly, the committee was left without a clear direction after it found itself in the line of fire of intellectual disputes between competing schools. The Bourbaki emphasis on theory and promise to look for the structure behind competing theories made it an appealing source of inspiration.

It in this context that Debreu’s Theory of Value made its entrance. This work was not innovative because it provided new theories or result – according to Weintraub, it does not do so at all – but because it gave economic theories a radically different form. Debreu tried to make economics axiomatic, in the sense of Bourbaki; according to Weintraub, the Theory of Value was meant to be analogous to Bourbaki’s Theory of Sets, right down to the title. Like the Theory of Sets, the Theory of Value was unsurpassed in its mathematical sophistication, and was only accessible to the mathematically trained.

Underpinning Debreu’s formalism was a new view on the relation between mathematical models and the economy that, again, seems to be inspired by Bourbaki. According to Weintraub, who takes the point from Ingrao & Israel (1990), Debreu’s mathematical model of the economy is no longer an analogy; it captures its essence. Like for Bourbaki, Debreu’s mathematics did not look outside towards its application, but inside towards justifications provided by the mathematics itself. Where the former saw the structure of mathematics as capturing something essential about the world, the latter saw his equilibrium model as the foundation that future economic research should take as its basis.

Next week’s contribution will compare the three accounts that have so far been introduced. In anticipation of that discussion, some contrasts between Debreu and the other two authors will already be clear to the reader. Debreu’s view that Walrasian equilibrium models captured the essence of economic systems implied a different view on the relation between physics and economics, when compared to Fisher and Tinbergen. Where the latter two explicitly drew on physics to dismantle economic problems, Debreu was much more pessimistic:

“[He] took this position even further by claiming that his Bourbakist program marked the definitive break with physical metaphors, since physics was dependent for its success upon bold conjectures and experimental refutations, but economics had nothing else to fall back upon but mathematical rigor. This is entirely consistent with the Bourbakist creed, which acknowledges that mathematical inspiration may originate in the special sciences, but that once the analytical structure is extracted the conditions of its genesis are irrelevant.” (p. 122)

On Weintraub’s account, Debreu decisively discarded physical analogies. His contribution to the mathematization of economics was based on pure mathematics, rather than applied mathematics as it was found in physics. He did not see his equilibrium models as tools, constructed in order to clarify a particular phenomenon or solve a particular problem, but as capturing something essential and unchanging about the economy.

The next contribution in the series will compare the three accounts that have so far been introduced. Specifically, it will focus on the different relations between physics and economics that these historians uncover, as well as implicit assertions about the nature of mathematics.

o-o-o

*I am grateful to Ivan Flis for helpful suggestions and corrections in producing this blog series. The paper from which this series is derived was written under the supervision of professor Geoffrey Hodgson (University of Hertfordshire), and has benefited greatly from his commentary.*

Manuel Buitenhuis is a graduate student at the University of Utrecht (MSc History and Philosophy of Science) and the Erasmus University Rotterdam (MA Philosophy and Economics). He holds a BA-degree from University College Utrecht, where he mainly studied economics and philosophy.

**Image credits**

http://www.freeimages.com/

**Bibliography**

Arrow, K., & Debreu, G. (1954). Existence of an Equilibrium for a Competitive Economy. *Econometrica* , 265-290.

Bourbaki, N. (1950). The Architecture of Mathematics. *American Mathematical Monthly* , 221-232.

Corry, L. (1997). The origins of eternal truth in modern mathematics: Hilbert to Bourbaki and beyond. *Science in Context* , 253-296.

Debreu, G. (1959). *Theory fo Value: An axiomatic analysis of economic equilibrium.* New Haven and London: Yale University Press.

Ingrao, B., & Israel, G. (1990). *The invisible hand: economic equilibrium in the history of science.* Cambridge, MA: MIT Press.

Weintraub, E. (2002). *How Economics Became a Mathematical Science.* Durham and London: Duke University Press.