Reactionary Mathematics: A Genealogy of Purity

A forgotten episode of mathematical resistance reveals the rise of modern mathematics and its cornerstone, mathematical purity, as political phenomena.   The nineteenth century opened with a major shift in European mathematics, and in the Kingdom of Naples, this occurred earlier than elsewhere. Between 1790 and 1830 its leading scientific institutions rejected as untrustworthy the “very modern mathematics” of French analysis and in its place consolidated, legitimated, and put to work a different mathematical culture. The Neapolitan mathematical resistance was a complete reorientation of mathematical practice. Over the unrestricted manipulation and application of algebraic algorithms, Neapolitan mathematicians called for a return to Greek-style geometry and the preeminence of pure mathematics.   For all their apparent backwardness, Massimo Mazzotti explains, they were arguing for what became crucial features of modern mathematics: its voluntary restriction through a new kind of rigor and discipline, and the complete disconnection of mathematical truth from the empirical world—in other words, its purity. The Neapolitans, Mazzotti argues, were reacting to the widespread use of mathematical analysis in social and political arguments: theirs was a reactionary mathematics that aimed to technically refute the revolutionary mathematics of the Jacobins. Reactionaries targeted the modern administrative monarchy and its technocratic ambitions, and their mathematical critique questioned the legitimacy of analysis as deployed by expert groups, such as engineers and statisticians. What Mazzotti’s penetrating history shows us in vivid detail is that producing mathematical knowledge was equally about producing certain forms of social, political, and economic order.  

Table of Contents:
Introduction: Mathematics as Social Order 1 Adventures of the Analytic Reason 2 Mathematics at the Barricades 3 Empire of Analysis 4 The Shape of the Kingdom Intermezzo: Algorithm or Intuition? 5 The Geometry of Reaction 6 A Scientific Counterrevolution 7 A Reactionary Reason 8 Mathematical Purity as Return to Order Notes Bibliography Index

About the Author :
Massimo Mazzotti is professor of history of science in the Department of History at the University of California, Berkeley. He is the author of The World of Maria Gaetana Agnesi, Mathematician of God and the coeditor of Algorithmic Modernity: Mechanizing Thought and Action, 1500–2000.

Review :
“The complex relationship between tradition and modernization is the pulsing heart of this engaging book. Beside a valuable historical analysis, Reactionary Mathematics offers an interesting and useful synthesis vision to help us understand, in these times of rapid and convulsive transformation, the mathematics of the present and, most importantly, the reasons for the mathematics that will come.” “Reactionary Mathematics is an ambitious book that is more than just a history of mathematics but an episode in the history of reason, furnished with a delightful display of different kinds of evidence, from archival documents to political satires to theological treatises to paintings to mathematics textbooks. . . . [It] is a deftly written and timely book brimming with empirical, conceptual and historiographical insights.” "For anyone interested in the "politics of mathematical modernity," this book shows how allegiances to particular types or styles of mathematics may indeed be related to Neapolitan academicians' personal responses to the urgent political pressures of their day." “One notable strength of Mazzotti’s book is its ability to transition seamlessly between different levels of analysis. It connects an in-depth historical exploration of a specific local context, such as Naples, with the social and political constraints unique to that site. Simultaneously, it addresses major upheavals and broad conceptual changes such as the evolution of purity, rigor, and abstraction and the very definition of 'modernity' in mathematics. In doing so, the book tackles a critical methodological challenge in the social history of mathematics, bridging the gap between the claim of universality associated with mathematical knowledge and the intricate study of the local contexts and social practices that underpin the production of such knowledge. Mazzotti’s thought-provoking narrative not only demonstrates . . . that mathematics is intimately connected to its cultural, social and political context, but it also prompts readers to consider new avenues of research.” “Mazzotti’s sustained treatment of the interlacing of mathematics and politics, which becomes evident once we recognize mathematics as historical, is fascinating and worth careful study. It is on the vanguard of a new and much-needed shift in perspective on mathematical practice.” “One of the most important books in the history of mathematics of the last few years. . . . Mazzotti’s book has many merits. The main one is the subtlety with which he depicts the political and religious culture of the Neapolitan mathematicians. . . . The story that Mazzotti narrates is made of blood, prayers and cilices, republican ideals and personal sacrifice, all the way up to the gallows. Thus, the history of mathematics acquires a profound human dimension.” “Reactionary Mathematics is an outstanding book that deserves to be read widely by modern historians, and not just by specialists in the history of science and mathematics. . . . It is a piercing study of how notions of intellectual purity have functioned to defend social hierarchy. It is an excavation, and a critique, of the epistemology of modern conservatism.” “This book is essential reading for anyone interested in the rise of modern mathematics.” “Mazzotti offers us a superbly crafted historical study of the interweaving of mathematics, politics, religion, social order, and even olive oil presses in the Kingdom of Naples around 1800. This gives him a distinctive, striking platform from which to address big questions: the relationship between science and politics, the connections between mathematics and modernity, and how we should understand mathematics’ past.” “Mazzotti has written a fascinating case study of ‘mathematical resistance’ in late eighteenth- and nineteenth-century Naples. On the most fundamental level, the book’s exploration of ‘mathematics as politics’ observes the reciprocal interactions between the mathematical imagination of historical actors and their sociopolitical circumstances. Mazzotti’s keen attention to the political actors themselves tells a very human story of mathematics, and of the events and changes that led to the development of this seemingly quixotic Neapolitan resistance to mathematical modernity.” “A landmark account of Neapolitan reactionary mathematics in context that contributes insightfully to the histories of Naples, reaction, and mathematics in their separate and interacting respects.”