Among those concerned with mathematical logic and axiomatics, the impact of Gödel's 1931 theorems on first-order arithmetic was fairly rapid and positive, although a few notable figures were a little slow to recognize it. However, the reception in the mathematical community in general was much slower and more muted. Evidence for this impression is presented from the literature, especially from general books on mathematics. The slowness is itself evidence of the ambiguous attitude that mathematicians have usually shown toward logic.

Around 1930, developments in the foundations of mathematics were becoming dominated by the program of metamathematics led by David Hilbert (1862–1943) as well as by the reasonable expectation that all logical and mathematical theories that could be axiomatized would be shown to be complete and consistent. Gödel's own Dissertation at the University of Vienna was of this kind. In it, he showed that the first-order predicate calculus (as we now call it) with identity passed muster; a lightly revised version was published (Gödel, 1930). However, in that year, he unexpectedly changed the direction of research for ever. In a paper that was to be accepted as his Habilitation at the University of Vienna, he showed two properties of the system P of axiomatized first-order arithmetic (Gödel, 1931). In his first theorem, he proved that if P was ω-consistent, then it contained more truths than theorems. Some months later, he proved the second theorem: that any proof of the consistency of P would require a logically richer theory than that of P itself, not a poorer one.

INITIAL ENTRIES INTO FOUNDATIONAL STUDIES

Mathematics occupied a controversial place in Cambridge in the late nineteenth-century: the Tripos was being roundly criticised as a mere set of skills, and yet it must have helped the university to gain a high reputation in applied mathematics. Alfred North Whitehead (1861–1947) started off in this branch after graduating from Trinity in 1884, being quickly elected to a college Fellowship with a dissertation on Maxwell's theory of electromagnetism. Further work drew him to the algebraic methods of the German mathematician Hermann Grassmann, which he popularised in a large book called A Treatise on Universal Algebra, with Applications (1898). The title was a misnomer, in that no one algebra was presented but instead a range of them, including also George Boole's algebra of logic.

Pure mathematics at Cambridge was rather boring, with excessive emphasis laid upon linear algebras due to Professor Arthur Cayley, and rather routine treatments of the calculus and analysis. Bertrand Russell (1872–1970) took the Mathematics Tripos from 1890 to 1893 (with Whitehead as one of his tutors), but then abandoned the subject in disgust and moved over to philosophy. He united these two trainings in an attempt to find a foundations for mathematics, starting with a Trinity Fellowship dissertation in 1895 which he revised into the book An Essay on the Foundations of Geometry (1897). His philosophical training lay in the neo-Hegelian tradition then dominant, which he exercised with skill; but the results for mathematics were not satisfactory.

Introduction

Scenario

In the early nineteenth century, the principle of virtual work and other assumptions in analytical mechanics led to forays into equilibrium and optimization in economic contexts. Forays toward linear programming appeared in embryo forms at various times between the late eighteenth and the mid-nineteenth centuries, with a very clear formulation put forward by J. B. J. Fourier in the 1820s; but then they fell largely into desuetude. From 1900 to 1940, various studies concerning linear inequalities and/or convexity were carried out in many different branches of pure and applied mathematics; but they also did not launch linear programming, where progress remained slow until an extraordinarily rapid establishment after the Second World War. Similarly, some traces for nonlinear programming were laid, largely in connection with mechanics, but they were not seized when that topic advanced in the 1950s.

In another area, in 1829 two young French scientists, G. Lamé and B. Clapeyron, thought up all the basic ideas and applications of locational equilibrium; but their work made no impact, not even on the concerns of their own distinguished later careers. The topic saw only fitful and partial advances for the next century before establishment was effected.

Presentation

At the factual level this chapter is concerned with these three cases of mathematical economics, which saw their birth in France during this period and continued there and in other countries afterward. Each of them grew out of aspects of mechanics, which will be specified in the following section, along with an outline of the context of French science at that time.