Writing the Rules of Reason: Notations in Mathematical Logic, 1847–1937

Abstract

This dissertation is a history of how logic became a mathematical science. Between the mid nineteenth century and the outbreak of World War II, scholars across Europe and North America replaced the venerable prose-based logic of Aristotle with a new and thoroughly mathematical enterprise. I approach this transformation on the level of practices, tracking how writers developed new symbolic systems for representing logic on paper—systems that eventually became not just tools but objects of scientific inquiry. Each new notation entailed a way of interacting with marks on paper, a manner of training students, and a vision for why people might need a science of logic.

I trace the development of writing practices during logic’s mathematical re-making by focusing on five major notations. In each case I show how the most abstract of sciences was rooted in local milieus around its transnational network of practitioners, entangled with their commitments from religious piety to nationalism and anti-Semitism. Chapter 1 explores the theologically-inflected calculus of English schoolmaster and self-taught mathematician George Boole. Chapter 2 considers the sprawling two-dimensional diagrams of German mathematician Gottlob Frege. Chapter 3 turns to American philosopher Charles Peirce and his students at Johns Hopkins University who, rather than sharing one notation, took individual variation as productive terrain to explore. Chapter 4 focuses on the internationalist symbolism of Italian mathematician Giuseppe Peano and the ends it served—quite different from his own—in the hands of English philosophers Alfred North Whitehead, Bertrand Russell, and later Susan Stebbing. Chapter 5 follows the concise unpunctuated strings of Polish logician Jan Łukasiewicz and his Warsaw School, showing how a notation shaped a local line of inquiry and became a symbol of national particularity.

Ultimately, students trained in this space of notational possibilities were socialized to accept a pluralism of inscriptive practices, disciplined by the cacophony of existing literature to see any given symbolic system as contingent. As the science of logic became mathematical, the diversity of writing practices through which that transformation took place made it a discipline that not only employed symbolic systems but took them as its fundamental concern.