This book is not only an excellent contribution to the history of economics, clarifying how economics became mathematicized in the twentieth century, it also provides valuable insights into the epistemological roles mathematics plays in the special sciences. To account for this history, Weintraub employs a distinction, developed by the historian of mathematics Leo Corry (Reference Corry1989), between the “body” of mathematical knowledge, and the “image” of that mathematical knowledge. Corry's argument is that to speak of change in mathematics is to speak not only of change in mathematical knowledge, in the sense of new theorems proved, new definitions created, and new mathematical objects described. Change in mathematics also involves changes in the image of mathematics, in changed standards for accepting proofs, changed ideas about rigor, and changed ideas about the nature of the mathematical enterprise. In each chapter Weintraub illuminates how economics has been shaped by economists' ideas about the image of mathematics. Chapters 1 and 2 discuss the changing ideas about rigor, chapters 3 and 4 deal with the difference between formalism and axiomatics, and chapters 5 and 6 treat the problems of mathematics considered as a language. Chapters 7 and 8 localize these issues in two microhistories of the experiences of the author's father in the 1940s and 1950s and those of the author himself in the 1960s. The last chapter considers alternative historiographical standpoints to deal with these issues. To avoid the danger of approaching history from philosophy, namely that the history of economics becomes a history of winners and losers with respect to particular theories of good and bad economic science, Weintraub chooses to write from a naturalistic perspective, based not on asking of science how it should be done, but rather how it was and is done. These close investigations of practice can only but enrich any philosophy dealing with the role of mathematics in science.

To clarify how changing ideas about rigor influenced the way economists assessed the nature and role of mathematics, Weintraub emphasizes that one should not identify “rigor” with “axiomatics” by referring to Giorgio Israel's (Reference Israel1981) argument that late-nineteenth-century mathematics considered “rigor” and “axiomatization” antithetical. Rigor was then understood to mean “based on a substrate of physical reasoning.” The opposite of “rigorous” was not “informal” but rather “unconstrained,” as with a mathematical argument unconstrained by instantiation in a natural science model. However, this view of science and scientific explanation, which entailed rigor in modeling in the sense of developing economic explanations from mechanical ones, was increasingly unsatisfactory as a solution to the crisis of those days in the natural sciences. The crisis was resolved by the formalist position on explanation whereby mathematical analogy replaced mechanical analogy, and mathematical models were cut loose from their physical underpinnings in mechanics. The result was that in the first decades of the twentieth century a rigorous argument was reconceptualized as a logically consistent argument instead of as an argument that connected the problematic phenomenon to a physical phenomenon by use of empirical data. The image of mathematics, with respect to rigor, had changed. However, in practice, the distinction between rigor as materialist-reductionist quantification and rigor as formal derivation, a distinction contested at the end of the nineteenth century but which disappeared as formalism took hold in mathematics, reestablished itself in the distinction between econometrics and mathematical economics, between applied economics and economic theory.

“Rigor” should not be confused with “axiomatics,” ditto Weintraub argues, “axiomatics” should not be identified with “formalism.” To clarify this latter distinction he unravels two intertwined lines in David Hilbert's research program: one called for a change in mathematical knowledge, a quest for certainty in mathematics by seeking a proof of the consistency of arithmetic or logic or set theory. The other called for a change in the image of mathematics and can be understood as the quest to develop axiomatic formulations of not only mathematical theories but also scientific theories more generally. Following Corry's suggestion, Weintraub labels the first as Hilbert's “Finitistic Program for the Foundations of Arithmetic” (FPFA) and the second as Hilbert's “Axiomatic Approach” (AA). Although Gödel's impossibility theorem has changed forever the optimistic hope of the Hilbert FPFA, it did not dash the optimism of the Hilbert AA, nor did it touch Hilbert's arguments for the introduction of formal rigor wherever possible in scientific work. Gödel's theorem thus did not touch the change in the mathematician's image of the activity of making mathematical knowledge more secure and how to pursue new scientific knowledge in an organized and rigorous manner. On the contrary, the theorem opened up an alternative approach to certainty: for any system, truth as consistency was to be relative to the structure in which that system was embedded. Truth and consistency were thus intertwined. In contrast to the image of mathematics using mechanical reductionism to make scientific arguments rigorous, the emerging view of mathematical truth, Hilbert's AA, appeared to require a different conceptualization. This new image of mathematics shaped an emergent mathematical economics. To preserve the relationship between rigor and truth, economists began to associate rigor with axiomatic development of economic theories, since axiomatization was seen as the path to discovery of new scientific truths.

Hilbert's AA found its way into mathematical economics by the Bourbakist interpretation of it, upheld by Gerard Debreu. Bourbaki became the watchword for the chasm that had opened up between mathematics and its applications, between the rigor of axiomatization and rigor of basing argumentation on the physical problem situation. It is striking that the Bourbaki orientation entered economics at the Cowles Commission, that up until that moment had been primarily known as a center for the development of econometrics and as such the treasurer of the idea that rigor must develop from observations, so that that the truth of a model may be tested or confirmed by reality. However, disappointments in their program of structural estimation were causing the commission to contemplate a turn away from empiricism to a program of mathematical formalization, and Debreu was invited to work in this new direction.

Mathematical economists often claim that one can translate between mathematics and economics. An implied corollary of this position is that any disagreement between an economist and a mathematician on the nature of a mathematical proof is due to a misunderstanding of the assumptions or the logical reasoning of the proof. And any disagreement on the economic implications of the mathematical proof is due to mistranslation or a lack of understanding of the underlying economic reality. Consequently there is an implicit “right way” to understand how economists and mathematicians can negotiate the more or less rigid boundary which separates their disciplines. However, two chapters, both written with Ted Gayer, discussing the validity of mathematical proofs in economics, document communication failures between mathematicians and economists. Both chapters shed light on the complexity of achieving a common understanding about the image of mathematics (role of assumptions, the nature of proof, and the meaning of mathematical modeling). The communication failures arise because economists and mathematicians are trained in different discursive practices, each containing their own set of persuasive techniques and rules of assessing evidence.

According to Weintraub, this Kuhnian incommensurability problem challenges not only the metaphor of translation from economics to mathematics and vice versa (about which he is right), but also the idea that a mathematical model can be used as a tool to describe an autonomous economic reality. He therefore uses the image of a model as a map of a region, a representation of the region different from that region, a picture as a representation of an object different from the object. He criticizes this image because it would imply a homomorphical mapping in the sense that the map is “relevantly like” the region, which is not possible to assess because it presupposes knowledge about structures loose from these representations. This of course comes close to Ronald Giere's (Reference Giere, Magnani, Nersessian and Thagard1999) idea of models, though he emphatically avoids using the concept of homomorphy—for the same reasons. Mathematical models are distinct bodies of knowledge that can be and in fact are negotiated across boundaries, like hammers can be used for different purposes in different contexts. However, the criteria for assessing which model is most accurate, or most true, belong to the image of knowledge. Corry's framework that Weintraub so convincingly uses throughout his book is regrettably forgotten for a moment when it comes to discussing the role of models in the process of mathematization. Maybe it isn't that odd to recall (and paraphrase) Lakatos's (Reference Lakatos1976) view on the relationship between history and philosophy: the history of economics, lacking the guidance of (not necessarily normative) philosophy, is in danger of becoming blind, while the philosophy of science, turning its back on the most intriguing phenomena in the history of science, is in danger of becoming empty.