Integration by Parts: Wordplay, Abuses of Language, and Modern Mathematical Theory on the Move

This paper accounts for the intercontinental elaboration of French mathematician Laurent Schwartz’s theory of distributions in the years immediately following World War II by tracing how mathematicians explained the theory to each other, advanced new interpretations, and reconciled existing ones. Situating distributions in mathematicians’ changing contexts of funding, travel, and publication, especially in connection with the postwar reconstruction of international science, I argue that wordplay and suggestive comparisons—often termed “abuses of language”—helped tie communities of scholars together across disparate geographies and fields of study. Material limits and linguistic ambiguity, here, offered important resources for asserting relevance and unity in a fragmented and heterogeneous discipline. I show in particular how reinterpretations and puns of the calculus technique of integration by parts helped advocates of Schwartz’s theory create a far-reaching community of students and researchers that was itself partially integrated—with distributions’ scholars believing themselves to be using a common theory while understanding and using that theory in considerably different (if sometimes mutually recognizable) ways. If exponents of modern mathematical research and pedagogy tend to emphasize settled theories and stabilized innovations, the history of these activities demands a converse emphasis on the variable and ongoing labor required to reconcile techniques and concepts—a labor that often hinges on theories’ instability, pliability, and susceptibility to play.