How Economics Became a Mathematical Science by Roy Weintraub

In the book How Economics Became a Mathematical Science, Weintraub spent a whole chapter 4 discussing Gerald Debreu of the Arrow Debreu fame. The title of that chapter is Bourbaki and Debreu. The following are some quotes from that chapter.

Murray Gell-Mann has written, 'The apparent divergence of pure mathematics from science was partly an illusion produced by the obscurantist, ultra-rigorous language used by mathematicians, especially those of a Bourbakist persuasion, and by their reluctance to write up non-trivial examples in explicit detail. . . . Pure mathematics and science are finally being reunited and, mercifully, the Bourbaki plague is dying out" (1992, 7). (103)

Comments: The Bourbaki plague is still dominating economics, through Arrow Debreu model.

The serious issues were intellectual strategy, in mathematics and beyond, and raw political power. An obvious manifestation of intellectual strategy concerns "taste." For Bourbaki, the fields to encourage were few in number, and the fields to discourage or suppress were many. They went so far as to exclude (in fact, though perhaps not in law) most of hard classical analysis. Also unworthy was most of sloppy science, including nearly everything of future relevance to chaos and to fractals. (Mandelbrot 1989,10-11) (103)

For many scientists, Bourbaki became the watchword for the chasm that had opened up between mathematics and its applications, between the rigor of axiomatization and rigor in the older sense (see chapter 2) of basing argumentation on the physical problem situation. In such a world, would it not appear that a Bourbakist-inspired discipline of "applied mathematics" would be an oxymoron? It is our thesis that such a thing did occur in economics, and indeed, it took root and flourished in the postwar American environment. The transoceanic gemmule was Gerard Debreu; (104)

the "founders" seemed to believe, "… Bourbaki... is … a very well arranged cemetery with a beautiful array of tombstones. . . . There was something which oppressed us all: everything we wrote would be useless for teaching" (107)

"Bourbaki did not adopt formalism with full philosophical commitment, but rather as a facade to avoid philosophical difficulties." Others now concur in this assessment (Mathias 1992). Bourbaki gave the impression of elevating his choices in mathematics above all dispute: but that was all it was—just an impression. (112)

These details concerning Bourbaki's history and Corry's reading of it, seemingly so far removed from economics, are instead absolutely central to understanding its postwar evolution. The reason is that very nearly everything said about Bourbaki will apply with equal force to Gerard Debreu. (113)

In retrospect, it is hard to read Theory of Value as anything else, since it also provides no "new" theorems or results; it is Chevalley's "very well arranged cemetery with a beautiful array of tombstones" (Guedj 1985, 20).(123)

Some others quotes from the book.

Klein ends his lecture with some observations on the educational implications of these connections between mathematics and applied sciences. "I am led to these remarks by the consciousness of the growing danger in the higher educational system of Germany,—the danger of a separation between abstract mathematical science and its scientific and technical applications. Such separation could only deplored; for it would necessarily be followed by shallowness on the side of the applied sciences, and by isolation on the part of pure mathematics" (p. 27)

The distinction that Volterra makes in this passage between the two approaches to doing physics, the distinction between grounding explanations on the physical characteristics of the problem, or grounding explanations on mathematico-logico reasoning chains, mirrors the distinction between nonformalist and formalist responses within the mathematics community to the crisis of the foundations of mathematics, the paradoxes of set theory, of almost the exact same period. In the case of both physics and set theory mathematicians could, with the formalist response, ground the unknown upon the known. For mathematics, the grounding was to be an axiomatization of the settled parts of mathematics, logic, set theory, and arithmetic, as a basis of both more "advanced" mathematical theory and the sciences built upon the axiomatized mathematical structures so created. For Volterra, this formalist response was not rigorous: scientific reasoning chains had to be based not on the free play of ideas, or axioms, or abstract structures. Rather, scientific models had to be based directly and specifically on the underlying physical reality, a reality directly apprehended through experimentation and observation and thus interpersonally confirmable. (48)

This point is important, and bears repeating because the present-day identification of rigor with axiomatics obscures the way the terms were being used at the turn of the twentieth century.5 Today we tend to identify the abstract reasoning chains of formal mathematical work with the notion of rigor, and to set rigor off against informal reasoning chains. Unrigorous signifies, today, intellectual informality. This distinction was not alive in Volterra's nineteenth century world, however. For Volterra, to be rigorous in one's modeling of a phenomenon was to base the modeling directly and unambiguously on the experimental substrate of concrete results. The opposite of "rigorous" was not "informal" but rather "unconstrained/' To provide a nonrigorous explanation or model in biology, or economics, or physics, or chemistry was to provide a model unconstrained by experimental data or by interpersonally confirmable observations. (49)