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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 would become 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.

352 pages | 5 halftones, 2 line drawings | 6 x 9

History: European History

History of Science

Mathematics and Statistics


“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.”

Donald MacKenzie, University of Edinburgh

“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.”

Sean Cocco, Trinity College

“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.”

Michael Barany, University of Edinburgh

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

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