From Schwartz Space to Structural Aging (Towards a small history of Old Age mathematics)

Abstract

Starting with some historical considerations, we examine the various attempts to model old age that fall within the realm of mathematical physics. For example, we study the mathematical theory of declining functions, which, since Laurent Schwartz's work on distributions, are generally defined on what is called a "Schwartz space." We explain how this essentially analytic formalization applies to the representation of the decline of vitality (or of a number of vital functions). We then show that there exist many other possible formalizations that utilize the resources of algebra, in particular the theory of "inclines" of Cao, Kim, and Roush and that of relational structures, on which an algebraic notion of "age," in Cameron's sense, can be defined. But we maintain, in conclusion, that a more positive representation of old age, consistent, moreover, with what a number of cultural sources from Antiquity to the 20th century tell us, should mobilize another type of formalization, capable of selecting, within the overall debate between the living being and its environment, persistent structures that do not decline amidst the decline of others.