Although the method of Lagrange multipliers has been embedded in contemporary mathematics (notably in multivariable calculus and constrained optimization) and is nowadays considered as a genuine (i.e., justified entirely within mathematics) piece of mathematical knowledge, in his Mécanique analytique (1780, 2nd ed. 1811) Joseph Louis Lagrange introduced it in the context of statics, in order to describe the equilibrium of a system of points subject to some constraints.
In the Seminar, I offer a reconstruction of how Lagrange developed the method of multipliers. In fact, accounting for the success of this well known method requires an understanding of how Lagrange introduced it, but once such a reconstruction is made, it is easy to see what makes the multipliers’ method successful in its application to static problems.
Moreover, the method applies with success also to other scientific settings that (seem to) have very little to share with the mechanical scenarios for which it was developed by Lagrange. For instance, it applies with success in dynamics, when we find the maximum range of the cannonball launch, or in economics, when it is necessary to find a maximum production level for a manufacturer’s production that is modeled by a particular function, or even in chemistry, to determine the equilibrium composition of large chemical systems subject to generalized linear constraints. Should we consider this extended success a miracle? What do the problems of static equilibrium studied by Lagrange have in common with such problems? How can we account for the success of mathematics in these scenarios? The interesting point in the context of the “applicability issue” concerns the analysis of how Lagrange presented the mathematical method, in tandem with the tracking of the application-setting.I argue that the cultural significance of this case-study analysis is twofold. In the context of the applicability debate, my historically-driven considerations pull towards the reasonable effectiveness of mathematics in science. The effectiveness of mathematics in Lagrange’s treatment of bodies at equilibrium is not unreasonable, as some philosophers would be tempted to say (without taking account an historical reconstruction). Secondly, the analysis of the multipliers' case has the merit of disclosing interesting and potentially fruitful interconnections that exist between history of mathematics, philosophy of mathematics and mathematics education in the context of the applicability issue.
Il seminario avrà luogo presso il Dipartimento di Matematica e Fisica
Largo San Leonardo Murialdo,1 - Pal.C - Stanza 311