The Way They Did Theory Then

Abstract

ABSTRACTIn this article, I take sixteenth-century mathematics as a test case for considering at its most general level what doing theory entails. Ideally, doing theory aims to change rather than merely describe reality, and in order to do so, it represents that reality differently. Such transformations, I argue, need special techniques of representation. In early modern mathematics, two techniques in particular enable new theoretical content: symbolic notation, and the real continuum. Together they create a language that can redescribe mathematical objects, for example, so that a single general formula can solve for all specific instances, and simple counting numbers can be reidentified in the real continuum as limits. I argue for the formative contribution of the imagination in such developments, that is, for an ability to reassemble givens of a problem in such a way as to solve it; and to concoct new objects or possibilities that do not exist in reality. In the sixteenth century, imaginary numbers (roots of negative values) most explicitly demonstrate the importance of speculating beyond what is real. As we consider what doing theory in the twenty-first century entails, it is worth remembering, in light of the article’s analysis, how form or technique itself has agency, and the ability to transform theoretical content.