Unity and identity of opposites in calculus and physics (Q1923796)

Unity and Identity (UI) in a category in the title of this paper means a map equipped with two sections, or equivalently, a common retraction for two subobjects whose underlying objects are identical. The particularly important case of UI is UIAO (unity and identity of adjointly opposites), known also as an ``essential localization''. The author argues that a significant fraction of notoriously disputable dialectical philosophy, developed by Hegel, Marx and others, can be grasped firmly with the above mathematical apparatus of UI and UIAO. The examples discussed in the paper include van der Waals' formula for the unity of the liquid and gaseous states, the foundations of differential calculus, and \textit{W. Noll}'s [Arch. Ration. Mech. Anal. 122, No. 3, 197-212 (1993; Zbl 0787.73006)] generalization of the usual concept of continuous body.