BSHM Meeting News

Abstract

s from past meetings History of Decision Mathematics Saturday 15 May 2021 Online from Birkbeck College, London Tinne Hoff Kjeldsen (University of Copenhagen) The emergence of nonlinear programming: Duality and WWII The significance of internal and external driving forces in the history of mathematics. In this talk we will discuss the emergence of nonlinear programming as a research field in mathematics in the 1950s. We will especially focus on various kinds of driving forces both from inside and outside of mathematics, and discuss the significance of their influence on its development. The term ‘nonlinear programming’ entered into mathematics when the two Princeton mathematicians Albert W Tucker and Harold W Kuhn at a conference in 1950 proved what became known as the Kuhn-Tucker theorem. Later it turned out that a similar result had been proved earlier, even twice: in 1939 and 1948, but nothing came of it. Kuhn and Tucker’s workon nonlinear programming grew out their work on duality in linear programming, which in itself originated from investigations of a mathematical model of a logistic problem in the US Air Force from the Second World War. This short outline prompts several questions: Why could the result of the Kuhn-Tucker theorem all of a sudden launch a new research field in mathematics in 1950? How did ideas of duality emerge in linear programming, and what role did they play for the development of nonlinear programming? How did the Air Force logistic problem cross the boundary to academic research in mathematics? What role did the military play and what influence did it have for the emergence of mathematical programming as a research area in mathematics in academia? The talk will be governed by these questions, and the answers will show that both internal and external factors influenced the mathematicians’ work in crucial ways, illustrating the interplay between developments of mathematics and the historical conditions of its development. Norman Biggs (London School of Economics) Linear Programming from Fibonacci to Farkas Linear Programming is a 20th-century invention, but its roots can be traced back to the tenth century, when the Islamic mathematician Abu Kamil wrote about ‘The Problem of the Birds’ This was one of several problems on ‘mixtures’ that appeared in Fibonacci’s 1202 manual of commercial arithmetic, the Liber Abbaci — in a chapter on ‘The Alloying of Monies’. His work was repeated in the early printed books of arithmetic, many of which contained chapters on Alligation, as the subject became known. Around 1600 the introduction of modern notation clarified the link with the study of linear inequalities and Diophantine problems. The next step was Fourier’s work on Statics, which led him to suggest a procedure for handling linear inequalities based on a combination of logic and algebra. He also introduced the idea of describing the set of feasible solutions geometrically. In 1898, inspired by Fourier’s work, Gyula Farkas proved what we now regard as the fundamental theorem about systems of linear inequalities. This topic eventually found many 88 British Journal for the History of Mathematics