Foundational Anxieties, Modern Mathematics, and the Political Imagination

This essay is adapted from Massimo Mazzotti’s 2023 book Reactionary Mathematics: A Genealogy of Purity, available now from the University of Chicago Press.

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A FORGOTTEN EPISODE in French-occupied Naples in the years around 1800—just after the French Revolution—illustrates why it makes sense to see mathematics and politics as entangled. The protagonists of this story were gravely concerned about how mainstream mathematical methods were transforming their world—somewhat akin to our current-day concerns about how digital algorithms are transforming ours. But a key difference was their straightforward moral and political reading of those mathematical methods. By contrast, in our own era we seem to think that mathematics offers entirely neutral tools for ordering and reordering the world—we have, in other words, forgotten something that was obvious to them.

In this essay, I’ll use the case of revolutionary Naples to argue that the rise of a new and allegedly neutral mathematics—characterized by rigor and voluntary restriction—was a mathematical response to pressing political problems. Specifically, it was a response to the question of how to stabilize social order after the turbulence of the French Revolution. Mathematics, I argue, provided the logical infrastructure for the return to order. This episode, then, shows how and why 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.

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In 1806, the Kingdom of Naples was occupied by a French army and integrated into Napoleon’s imperial system. The French and their local supporters had a clear agenda: they wanted to transform the semifeudal society into a centralized administrative monarchy with a liberal economy. This ambitious plan, however, soon ran up against obdurate realities like muddy roads, brigandage, popular insurgencies, and the thinly disguised hostility of powerful local elites. There was another problem, too. Open a Neapolitan university textbook of the time and you will see that the French had to fight their battles in a land where their mathematics was wrong.

While armed and cultural resistance against the French invaders’ imperial ambitions happened in other parts of Europe, the Neapolitan case is particularly interesting because it includes a mathematical resistance. This resistance took the form of 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. Leading Neapolitan mathematicians embraced it as the veritable foundation of all mathematics. Only its methods and assumptions, they believed, could be trusted.

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.

Driven by eminently practical goals, analysis had become highly abstract: a versatile tool that could be applied to describe and control natural and social phenomena. The Neapolitans were quite sure it was morally suspect—a degenerate form of knowledge, and dangerous for the stability of society. Its proliferation across Europe and globally was, to them, an unmitigated disaster. While these allegations sound extravagant to our ears, some of their concerns resonate with ones in this century—e.g., about how even programmers who fashion certain complex algorithms do not understand their inner workings or why they come to the conclusions they do. Synthetics, for instance, pointed out that the analysts prioritized practical success over understanding: they aimed to model phenomena using algebraic tools that they could not fully justify, neither through some form of intellectual intuition nor through logic. By contrast, synthetic geometers clarified and grounded every single step of their procedures. 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. Using another set of metaphors: Analysts followed the fast flights of their feverish and uncontrolled imagination, while synthetics kept their feet on the ground. Their procedures were slow but safe.

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The mathematical 19th century was suffused with a distinctive foundational anxiety. 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, a tradition that was now under attack and needed to be defended.

Skeptics, however, understood the synthetic school’s mathematical resistance as backwardness, the rearguard action of a group of isolated practitioners. And yet, Fergola’s puzzling quest for purity was something more. For one thing, there was no established tradition of synthetic geometry worth defending in Naples. 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. In fact, Fergola, who was well aware of recent mathematical developments, breathed new life into forgotten mathematical techniques. Neapolitan synthetic mathematics, in other words, was not a remnant of the past, but a new way of understanding mathematics, characterized by new canons of rigor and founded on a core of “pure” mathematics. In the name of a mythical tradition, it imposed a new discipline on its practitioners by emphasizing self-restraint as the key epistemic virtue. Fergola, his admirers reported, was a champion of self-control: he controlled his body, passions, and imagination, and knew well when to stop trusting his own mathematical techniques.

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The Neapolitans did not reject modern analysis simply because they considered it French. What triggered their anxiety were its technical features—the way it worked. 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. This zealous quest for universal problem-solving algorithms is precisely what made the synthetics uneasy. Interestingly, the analysts themselves knew well that the manipulation of these algorithms—the way they produced results—was not grounded in either geometric intuition or logical arguments. What warranted their use, they believed, was that algebraic procedures mirrored the fundamental workings of the human mind. If, as the analysts believed, the mind was an analytic machine, then analysis was the quintessential expression of human reason—and, as such, isomorphic to nature: analysis was effective because it mirrored the deep structures of reality. This was a mathematics essentially interwoven with the world of experience—one that, as d’Alembert had written, “gives us the most perfect examples of the manner in which one should use the art of reasoning.”

When asked to solve a geometric problem, the analysts would find an appropriate system of coordinates that would allow them to turn figures into algebraic formulas, then would manipulate these formulas to obtain the “solving equation,” as they called it. They would interpret the solutions as solutions to the original geometric problem. Their operations had thus shifted from geometry to algebra. Was this a legitimate move? 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. It followed that the nature of the problem had changed. 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. They saw this as a question of jurisdiction: there are many different ways of reasoning and many different methods, and they all have their legitimate function and scope. It would be illusory—and deceitful—to try to solve a geometrical problem using purely algebraic methods, or a political problem using the methods of geometry. Even more misleading would be to believe that there is a single universal method that can be applied to all kinds of problems. 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.

These two mathematical cultures differed sharply in the way they conceived mathematical reasoning. 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. Their teaching reflected this view: a close-knit school with an inner circle of students who worked with their maestro, engaging in an endless reflection on geometrical problems received from antiquity. 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. For them, mathematical training was a matter of learning to frame problems analytically and then solving them following a set of standard procedures. It was not a matter of intellectual intuition, gift, and genius. On the contrary, anyone, with proper training, could become an effective problem-solver.

Analysts enthusiastically compared their method to the clunky workings of a machine: to them, the machine was an emblem of rational thinking. It was also a 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. At the core of their science was not the “pure and simple knowledge of truths,” they argued, but the knowledge of methods and their relative “strength” in getting useful results, including approximate ones, through the sheer power of calculation. The synthetics countered that results needed always to be precise and perfectly interpretable. The analysts were teaching their students blind methods that deformed their young minds; they were turning them into automata—soulless, machinelike number-crunchers—who ignored at their peril the meaning of the formulas they were manipulating. Well before the dawn of digital computing, the questions of the meaning of formal procedures and of the social implications of their extensive use were at the core of a debate that, using the language of morals, addressed basic questions of social order.

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Why did some mathematicians perceive logical gaps in analysis in the years around 1800, and why did they consider them unbearable? It turns out that anxieties about the foundations of analysis were growing even among its supporters. By the 1820s, these anxieties were spreading across Europe. Consider Augustin-Louis Cauchy (1789–1857), who played a key role in the creation of modern mathematics as we know it. His much-celebrated revolution had less to do with his specific contributions and more with his overall transformation of mathematics as a discipline.

He brought order to the world of mathematics. He modernized it by imposing rigor. 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. This blithe attitude needed to end, he declared, and a new Euclidean spirit—a spirit of rigor—needed to replace it. 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. Cauchy’s program of rigorization redefined the meaning of mathematical techniques, providing precise definitions and limits for the application of each method. 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.

Cauchy’s rigorous analysis seems distant from Fergola’s Greek-like geometry. But it was shaped by the same foundational anxiety and urgency to restore order to a mathematical world that—in the views of both men—had gone badly astray. What we learn from the comparison is that the fundamental opposition was not between geometry and algebra but between mathematics as a pure, rigorous, self-contained, and reliable body of knowledge and mathematics as a set of highly general and universally applicable algorithmic procedures expressing an all-encompassing analytic rationality. The fact that Fergola tried literally to reinstate geometry at the core of mathematics while Cauchy injected a rigorous Euclidean spirit within analysis was more about their local conditions than anything else. What really mattered is that both programs promised a return to mathematical order.

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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. The Neapolitan government would take a decisive anti-French stance, reorient its entire cultural politics accordingly, and align itself with the Catholic Church. Within the Church itself, enlightened reformism gave way to a Jesuitic religiosity that reactivated baroque devotions and mobilized popular piety to support the alliance of throne and altar.

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.

Neapolitan Jacobins aimed to find universal methods to address pressing social and political problems—above all, the problems of political representation and wealth redistribution. By the mid-1790s, 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. Analysis was already being applied to the natural sciences and now, they said, it was time to apply it to the science of society as well, and to political matters. The analytic method would turn the art of politics into a science, replacing tradition, prejudice, and private interests with rational decision-making. To apply analysis to politics meant to reduce it to its elementary components, study their relations, and use algebraic procedures to intervene. 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.”

Neapolitan Jacobins thus took the tools and basic assumptions of analysis and turned them into a militant mathematics—the veritable “backbone of society.” 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. In fact, mathematics and politics merged seamlessly in the lived experience of these Jacobins, who saw themselves as agents of change, able to escape the logic of reform and the apparent fatalities of history.

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. The metaphor emphasized discipline, organization, and the capacity for control, but also gestured toward the extraneous and polluting character of what was described as a set of manipulation techniques. In Naples, the Jacobin machine was viewed as foreign, disconnected from local political traditions. But 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.

The breathtaking adventure of the Jacobin Republic, characterized by sweeping plans for popular education and redistribution of wealth, ended abruptly five months after it started, in June 1799, when British, Russian, and Turkish forces joined a local counterrevolutionary army and stormed the walls of Naples. In its aftermath, about 120 prominent Jacobins were put on show trials and executed. Fergola had moved to the countryside during these months. A biographer reported that he could not stand “the noise” of the city. When he returned, he and his students were asked to reorganize scientific life in the university and in the entire system of public education. Education, which had been “infected” by the Jacobins, a royal dispatch read, now needed to be brought back to its ancient order.

Fergola, a celibate vegetarian who found the presence of women extremely unpleasant, was a tormented man. He was not someone who could easily fit into the salons of Enlightenment Naples. His religiosity was deliberately untimely and baroque. He chose the most anti-modern and anti-rationalist religiosity, the popular piety of the Neapolitan crowds, when this religiosity was under attack. A spate of crying and bleeding Madonnas signaled the crisis of a subaltern agrarian world that would soon explode in massive counterrevolutionary insurgencies. Embracing popular religion in the 1790s meant embracing it as resistance. In general terms, it was a resistance against the modern state’s secularized and rational principles of organization. In this sense, Fergola’s public display of his rosary and bloodied scourge was a politicized act of resistance. And in this sense, his mathematics, too, was a politicized act of resistance.

Fergola suffered numerous and often inexplicable “organic” and “moral” ailments that progressively hampered his activity. Yet, we are told, he gazed serenely at his sore body as if that flesh was not his own: his entire life could be recounted as a triumph of spirit over matter. Exhausted by mysterious convulsions and prostrated by horrific demonic visions, Fergola felt that his faith was constantly put to the test. But even as his own health failed, his school prospered. To the end, he railed about the degeneration of learning and the “sacrilegious horde,” which included Freemasons, Jacobins, liberals, and even some of his legitimist colleagues.

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With the French occupation of 1806, the synthetic school had to face, once again, the threat of analysis. The world, however, had changed. Most surviving Jacobins had come to terms with the Napoleonic normalization. The synthetics controlled the university, but outside of it, analysis thrived, constituting the core training in the new schools of engineering fostered by the French. The continuity, however, was mostly apparent. 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. But reframing modernization as a technical rather than a political problem meant detaching analysis’s formal tools from their original source of legitimation. Intriguingly, the political choice of reframing modernization in exclusively technical terms had produced a profound and pervasive mathematical problem.

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 old regime had long been imploding in Naples, but a new order struggled to consolidate itself. The French had arrived in Naples with a promise of order through modernization, and had found receptive interlocutors in the landed elites who could most benefit from the abolition of the feudal-communal system. 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.

It is not a coincidence that the synthetic school faded into oblivion when the reactionary position lost ground as a viable political option. From the 1820s onward, what was really at stake was the form of the new relations between the centralized administrative state, the landed elites, and the largely dispossessed peasant masses. Analysis had morphed into a set of allegedly neutral administrative tools, and the controversy between analytics and synthetics, which had long defined Neapolitan academic life, became increasingly meaningless. 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 compromise was an elegant way of disposing of what, at that point, was an embarrassing anomaly for Neapolitan science. 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. Emblematic of this cultural climate was the success of philosophical positions grounded on consciousness, which insisted that, within certain limits, individual reason was autonomous and legislative, and that fighting for liberty of conscience coincided with fighting for political and economic liberty.

The mathematical controversy between synthetics and analytics had been a controversy about the nature of reason all along. At stake was reason’s nature and limits. 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. This reactionary reason was a militant reason, its arguments forged in battle.

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. The image of reason that emerged with the consolidation of the modern liberal state valued individual reason while acknowledging its clear limits. The principles of a liberal economy and the new relationships of subordination between landed elites and peasant masses were not to be questioned. The autonomy of individual reason was celebrated against prerevolutionary obscurantism and absolutism, but it was only a relative autonomy, to be exercised within the boundaries of postfeudal order. The newly rigorous mathematics and the constitutional project set those boundaries with precision. If absolutism was a thing of the past, so should be revolutionary anarchy.

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Logical inference, Wittgenstein quipped, is how we refer to what we do not intend to question. The case of the Neapolitan mathematical resistance should not be seen as one in which mathematics was temporarily distorted by politics. 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. They discriminate between what is visible, plausible, and logical, and what is none of those things. 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.

Modern mathematics, as it took shape with Cauchy and those who continued his program, was constitutive of the postrevolutionary political normalization. It was the logico-mathematical infrastructure of the new moderate liberal discourse. It retained analysis’s operative orientation while embracing the synthetics’ quest for a foundational core of mathematical knowledge. The image of reason it embodied was bounded and self-disciplined, and while it legitimated a neutral, instrumental technical dimension, it confined the truth of mathematics to the ethereal and otherworldly realm of pure mathematics. Fergola’s mathematics, it turns out, was modern.

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Massimo Mazzotti is a professor at UC Berkeley, where he holds the Thomas M. Siebel Presidential Chair in the History of Science. He is the author of Reactionary Mathematics: A Genealogy of Purity (2023).