Foundational Anxieties

“This essay is adapted from Massimo Mazzotti’s 2023 book Reactionary Mathematics: A Genealogy of Purity”

“mathematics and politics as entangled”

“mathematical concepts and methods are anything but timeless or neutral; they define what “reason” is, and what it is not, and thus the concrete possibilities of political action”

“The technical and political are two sides of the same coin—and changes in notions like mathematical rigor, provability, and necessity simultaneously constitute changes in our political imagination”

“In 1806, the Kingdom of Naples was occupied by a French army and integrated into Napoleon’s imperial system”

“a distinctive mathematical culture that was hegemonic in that kingdom for several decades—from the late 1790s to the 1830s. Contemporaries called it the Neapolitan synthetic school. The name referred to synthetic (or pure) geometry, a geometry that does not use coordinates and algebraic formulas to study figures and solve problems”

“What the Neapolitans most adamantly did not trust was what they called, not without irony, the “very modern mathematics.” This body of knowledge, associated mainly with France, was characterized by the rapid advancements of an algebraized form of infinitesimal calculus and by its stunning and far-reaching practical applications”

“It had severed its connections with Euclidean geometry, and was referred to as “analysis”—a term that, in this context, meant a vast array of algebraic methods and algorithmic procedures that could be used to represent how things change, whether those things were, say, the trajectory of a cannonball or agricultural productivity”

“They often used metaphors of sight to make this point: synthetic geometry allowed practitioners to see with clarity, and this is why their results could be trusted; analysts were blind when they manipulated their formulas”

“We can see an early and radical manifestation of this anxiety in revolutionary Naples—in its bizarre and apparently backward attempt to return to a Greek-like pure geometry. The champion of this new old mathematics was Nicola Fergola (1753–1824), the charismatic and mystically inclined leader of a group of mathematicians and scientists who understood themselves as the last heirs of an ancient tradition”

“The tradition Fergola invoked was largely an invention—an imaginary mathematical lineage that ran through ancient Greece, late antiquity, and Christian Europe, all the way down to these self-proclaimed final paladins”

“Following the example of mathematicians like Condorcet, analysts were aiming to create a repertoire of finite and infinite algebraic methods that were abstract and general enough to apply to any kind of problem, be it in geometry, physics, economics, or even politics”

“The synthetics would say that it was legitimate only when they could see the geometry behind the formulas. But for complex problems this was not always possible, and in these instances algebra was blind; there was no way to reconstruct the geometrical meaning of the algebraic operations that led to the solution”

“For the analysts, this was irrelevant: algebra captured the essential relations expressed by the terms of the problem, which then served to guide the mathematician toward the solution. For the synthetics, by contrast, a solution to the original geometrical problem could only be geometrical in nature; and so, what the analysts were offering were not solutions but meaningless numbers”

“While the analysts strove for maximum generality, the synthetics argued for the specificity and locality of all mathematical methods”

“The synthetics’ world was, so to speak, epistemologically stratified. They recognized many kinds of truth, and thought it essential to keep them separated from one another. The truth of the geometer, they claimed, has nothing to do with the truths of the theologian, historian, or politician”

“For the synthetics, mathematical knowledge was the product of a process of recognition, the imperfect representation of metaphysical states of affairs that the gifted mathematician would be able to glimpse”

“For the analysts, mathematical reasoning was just a particular case of analytic reasoning—calculus, especially, was where analytic reason could be best seen in action. They saw themselves as the standard bearers of modernization and the promoters of rational action across both scientific and social life”

“Analysts enthusiastically compared their method to the clunky workings of a machine”

“way of arguing for the algorithmic nature and therefore accessibility of the method, as its standardized procedures could be easily learned, and deployed across different contexts”

“the knowledge of methods and their relative “strength” in getting useful results, including approximate ones, through the sheer power of calculation”

“Mathematicians in the 18th century had achieved stunning results in algebra and infinitesimal calculus, but to Cauchy’s eyes, they had been too casual in how they defined their concepts and devised and applied their methods”

“Cauchy was not interested in bringing back synthetic geometry. Rather, he aimed to reinterpret analysis within a new logical framework in which every concept and procedure would be logically justified”

“He set boundaries, in other words, within which certain techniques could be legitimately deployed. The modern mathematicians were those who, following Cauchy, could discipline themselves through a new kind of technical precision”

“Fergola and his students were initially marginal to Neapolitan scientific life. Their geometrical program was perceived as outdated, while the world of the salons scoffed at their baroque religious devotion and ascetic lifestyle. But this changed dramatically after the storming of the Bastille, when they quickly acquired an unprecedented cultural relevance”

“In 1794, the discovery of a Jacobin conspiracy to overthrow the monarchy sent the court into a state of panic. The Jacobins would succeed five years later, in 1799, when Naples became a republic. The leading revolutionaries were mathematicians. The chief conspirator of 1794, and the first president of the republic, Carlo Lauberg (1752–1834), was a teacher of chemistry and mathematics”

“It is no accident that almost every noteworthy figure in Neapolitan Jacobinism received some mathematical training: a basic understanding of analysis was an essential part of their worldview, as were republicanism, egalitarianism, and anticlericalism”

“The very structure of their secret society—a network of Jacobin clubs—was a working model of how analysis could be deployed in matters of social organization”

“they had become convinced that their vision of a just and equal society could be realized only through the universal implementation of analysis, which they understood as a revolutionary mathematics”

“This would detach politics from its metaphysical assumptions, turning it into a matter of rational and transparent administration”

“The analytic revolution could now be expected to transform society by making it possible to operationalize “the will of the people.””

“A programmatically impure mathematics, it was a universal language and reasoning style that could be applied across disciplinary boundaries to bring about immediate social change”

“The counterrevolutionaries reacted by turning these analytic features into the “Jacobin machine,” a deadly device for the control of public opinion, political life, and the state”

“In Naples, the Jacobin machine was viewed as foreign, disconnected from local political traditions”

“in France too, its effect was seen as one of contamination, this time from the inside. In both cases, the purity of the body politic had to be defended from a malignant mechanical-analytic threat”

“Many former revolutionaries, in France as in Naples, had turned the question of modernization into a technical problem, and had refashioned their personas and social function in terms of scientific neutrality and technocratic efficiency”

“Historian Ken Alder has aptly labeled them “techno-Jacobins.””

“In this normalized context, mathematics was a neutral tool, the distinctive expertise of technical elites who served the state. The direct connection between mathematics, egalitarianism, and republicanism, built through the notion of a universal analytic reason, had been severed, and with it vanished the very possibility of a revolutionary mathematics”

“Led by Fergola’s students, the synthetic school fought against the technical elites of the modern state, mostly civil engineers and statisticians, for scientific hegemony”

“The new technical experts had been charged with changing the kingdom’s physical and social landscape accordingly. Technical disciplines such as statistics or topography became key sites for negotiation, collaboration, and conflict between landed elites and the central government. On this technical terrain, the new experts would continuously clash with the synthetics”

“The technicians who supported the state’s modernizing action now argued for a mathematical reconciliation. What the two groups were defending, it was now believed, were simply two different ways of looking at mathematics, which should not be seen as opposed to each other but rather as complementary”

“The synthetics approach was useful for didactic purposes, while the analytic one was best suited for research and the discovery of new mathematical truths”

“This normalized reconstruction eliminated revolutionary and reactionary scientific aberrations, emphasized continuity in the history of mathematics, and aligned with the political life of Restoration-age Naples, which was hegemonized by new landed elites and their liberal and constitutional ambitions”

“The Jacobin’s analytic reason was universal, active, calculative, individual, a priori, and ahistorical; it was a completely autonomous reason that, when not obstructed, could truthfully describe and legitimately change the world—through revolutionary action, if necessary”

“The reason of the synthetic, by contrast, was local, passive, intuitive, collective, a posteriori, and eminently suited to historical thinking; it was a dependent reason, whose outcomes needed to be warranted by external sources of legitimation like tradition, custom, experience, religion, and metaphysical principles. It was, as such, a reactionary reason that envisioned the return to order as a return to hierarchy—order produced by subordination”

“It is only by contrast to an abhorred revolutionary reason, political theorist Corey Robin reminds us, that the invocation of ancient forms of wisdom can captivate the modern mind”

“When we craft logico-mathematical concepts and techniques, we design ways of ordering the natural and social world. These ways of ordering the world open up certain possibilities for action—including political action—while closing down others”

“Jacobin mathematics was deployed to critique and radically transform the existing social order, empowering traditionally subordinate social groups and bringing them into the space of politics as legitimate autonomous agents. The mathematics of the synthetics was designed to deny this possibility, to turn it, in fact, into a logical impossibility—hence it was, strictly speaking, a reactionary mathematics”