Angeli goes on to claim that there is nothing particularly troubling in Tacquet’s criticism of indivisibles. His arguments are old, he writes, and were already raised by Guldin and satisfactorily answered by Cavalieri years ago. But Tacquet did provide Angeli with an opportunity to proclaim how influential the method of indivisibles had become by the late 1650s. “Who does this reasoning convince?” Tacquet asks rhetorically, pointing to what he considered the inherent implausibility of the method of indivisibles. “Whom does it convince?” Angeli repeats incredulously. Everyone, he responds, except the Jesuits.
Angeli here is trying to turn the tables on the Jesuits: rather than the indivisiblists being a lonely and diminishing band under attack from powerful enemies, it is the Jesuits who are lone holdouts against a method that is being universally accepted. Indeed, at first reading, the list cited by Angeli is impressive, and appears to support his case. But a closer inspection tells a very different story: Yes, Beaugrand, Boulliau, White, and van Schooten did indeed adopt Cavalieri’s method, but they resided in faraway lands, north of the Alps. Of the three Italians whom Angeli cites—Torricelli, Rocca, and Raffaello Magiotti—only Torricelli had in fact published on indivisibles, whereas Rocca and Magiotti had remained mum; and in any case, by 1659 all three were dead. Despite his protestations to the contrary, Angeli was, in his own land, alone.
Satisfied with his rhetorical salvos, Angeli then confronts Tacquet’s dark warning that unless it were destroyed first, the notion that the continuum is composed of indivisibles would destroy geometry. Cavalieri insisted that the question of the composition of the continuum was irrelevant to the method of indivisibles, and Angeli here follows his teacher—but only up to a point. Like Cavalieri, he, too, argues that Tacquet was wrong, and that “even if the continuum is not composed of indivisibles, the method of indivisibles nevertheless remains unshaken.” But he adds a twist: “if in order to approve the method of indivisibles, the composition of the continuum from indivisibles is necessarily required, then certainly this doctrine is only strengthened in our eyes.” In other words, unlike his cautious teacher, Angeli is perfectly willing to accept that the continuum is indeed composed of indivisibles. The power and effectiveness of the method of indivisibles is proof enough of its correctness, and if it leads to the conclusion that the continuum is composed of indivisibles, then that, too, must be correct. The fact that this doctrine leads to contradictions and paradoxes bothers him not at all.
Angeli, the flamboyant Jesuat, took on the Jesuits like no one had dared since the days of Galileo himself. He called them names, ridiculed their exorcism practices, and pretended never to have heard of the most illustrious mathematician among them. But nothing demonstrated the clash between Jesuit and Jesuat like their contradictory approaches to the question of the composition of the continuum. For the Jesuits, the notion that the continuum is composed of indivisibles led to paradoxes, and on that account alone it must be banned from mathematics. A method based on it, even if effective and fruitful, was unacceptable because it undermined the very reason for which mathematics was studied: for its pure logical structure. Angeli’s view was the exact opposite: because the method of indivisibles was effective, he reasoned, its underlying assumptions must be true, and if they involved paradoxes, then we would just have to live with them. One approach emphasized the purity of mathematics, the other emphasized practical results; one approach insisted on absolute perfect order, the other was willing to coexist with ambiguities and uncertainties. And never the twain shall meet.
THE FALL OF THE JESUATS
Thanks to the protection of his order and the Jesuit-unfriendly Venetian Senate, it appeared that Angeli would get away with his open defiance of the Society of Jesus. He kept up his work, and over the next eight years he published an additional six mathematical books, in all of which he used and advocated the method of indivisibles. His greatest triumph came in 1662, when he was appointed to the chair of mathematics at the University of Padua, a position once held by Galileo. The Jesuits, so powerful elsewhere in Italy, could only fume as the upstart Jesuat was raised to one of the most prestigious mathematical posts in all Europe. They never responded to his taunts, or denounced him openly, but quietly, patiently, bided their time.
The Jesuits were in a tight spot. As long as Angeli continued with his insolence, there was always a danger the forbidden doctrine might be revived in Italy and their decades-long campaign would come to naught. But what could they do? Angeli was safe in Venice, and if they ever thought they could persuade the authorities there to silence him, then surely his appointment to the Padua professorship showed them this was unlikely. So the Jesuits changed tactics: in Venice they might be of little account, but in Rome they were still ascendant. And so, in order to silence the last Italian voice advocating the infinitely small, they turned to the papal Curia.
The evidence for what happened next is circumstantial, the documents relating to the events buried to this day deep in the Vatican archives. But what we do know is this: On December 6, 1668, Pope Clement IX issued a brief suppressing three Italian religious orders: one was a community of Canons Regular, residing on the island of San Giorgio in Alga, in the Venetian lagoon; the second were the Hieronymites of Fiesole, a popular order that, at its height, had forty houses across Italy; the third were the Jesuats of St. Jerome. As the brief put it, “no advantage or utility to the Christian people was to be anticipated from their survival.”
The Canons Regular was a tiny community, and confined to a single Venetian island, and it is plausible that the bureaucrats at the Vatican indeed came to the conclusion that it served no real purpose. The Hieronymites of Fiesole was a much larger order, but in their case the term suppression is misleading. For while it is true that they ceased to exist independently, the order was not in fact dissolved, but rather merged with its sister order of the Hieronymites of Pisa; the houses themselves continued to exist much as before. But for the Jesuats of St. Jerome the suppression was a death sentence: from one day to the next, the order simply ceased to exist, its houses dissolved and its brothers dispersed. It was a stunningly violent and unexpected end to an old and venerable order. Founded by the Blessed John Colombini in 1361 to tend for the poor and the sick, it had survived for exactly three centuries and seven years.
The official reason given, and the one quoted in all public sources today, is that “abuses had crept into the order.” But this explanation is no more helpful than the claim that the order’s survival served no purpose. Some scholars note that the Jesuats were often referred to as the “Aquavitae Brothers,” a designation that may suggest lax morals and loose living. But this was far from the case: the nickname was given to them because of their dedication to treating victims of the plague, which they did by administering an alcoholic elixir produced in their houses. There is no evidence that the Church hierarchy objected to the Jesuats’ medical practices or tried to put an end to them.
In fact, by all indications the Jesuats were a flourishing order. Pope Gregory XIII (1572–85), who patronized the Jesuits, also supported the Jesuats, and entered their founder, the Blessed John Colombini, in the official Church calendar, fixing July 31 as his feast day. The order expanded rapidly in the sixteenth and seventeenth centuries, and established dozens of houses across Italy. They must have been popular with the upper classes in Italian cities, since both the Cavalieris of Milan and the Angelis of Venice saw fit to send their gifted sons to be educated in the order. The fact that two members of the order occupied academic chairs in Bologna and Padua, among the most prestigious universities in Europe, added intellectual luster to the Jesuats that few orders could match. Although it is not easy to learn what life was like inside the Jesuat houses, nothing that we know suggests moral decrepitude. Cavalieri’s letter to Galileo in 1620 about life in the Jesuat establishment in Milan, in which he complained about being besieged by old men who expected him to study theology, does not give one a sense that this was in any way a party house. Nor does
one get that feeling about the house in Bologna, where Cavalieri resided for the last eighteen years of his life, ill with gout, and where he engaged in mathematical discussions with the young Angeli. The inescapable impression is that these were establishments with a serious focus on academic learning and religious ministry, and the rapid advancement of both Cavalieri and Angeli to positions of authority in the order indicate that intellectual achievement was highly prized. Before 1668 the Vatican generally saw no reason to intervene in Jesuat affairs, except in 1606, when, for the first time, it allowed clergymen to enter the order—a change that suggests a rise rather than a diminishment in the order’s standing. There is nothing in all this that would explain why this old and venerable brotherhood was singled out for sudden annihilation.
But the Jesuats did stand out in one way: they counted among their members the most prominent Italian mathematicians promoting the doctrine of infinitesimals. First Cavalieri and then Angeli, each in turn, was the leading advocate for indivisibles in his generation, and both received the full backing of their order. Not only were they promoted rapidly through the ranks, but many of their books were personally approved by the general of the Jesuats. Inevitably, when Angeli and Cavalieri entered into a bitter conflict with the Jesuits over the infinitely small, the fight became not just their own, but that of their order as well. Whether by design or happenstance, the Jesuats of St. Jerome became the chief obstacle for the Jesuits in their drive to eradicate the infinitely small.
It is possible that if the Jesuits had found a way to silence Angeli while leaving his brothers in peace, they would have done so. But it is just as likely that they were eager to make an example of the smaller order, a warning to all those within the Church who would dare challenge the Society of Jesus. In the end, the result was the same. Unable to persuade the Venetian authorities to discipline the insolent professor, they turned instead to the papal Curia in Rome, where their influence was decisive. They could not punish Angeli directly, so they let their fury rain on the order that sheltered him and his late teacher. When faced with the wrath of the mighty Society of Jesus, the Jesuats never stood a chance. The order that had survived three hundred years of political and religious upheaval, whose brothers administered the waters of life to victims of the plague, and two of whose members rose to the heights of mathematical distinction, simply evaporated at the stroke of a papal pen.
Surreally, the man at the eye of the storm remained unmoved—at least geographically. Although the brotherhood that had been his home since his youth had suddenly dissolved around him, Angeli was still professor of mathematics at the University of Padua, and still protected by the Venetian senate. He remained in Padua for the next twenty-nine years, until his death in 1697. But even though he still professed himself an admirer of Galileo, and although he had previously published no fewer than nine books promoting and using the method of indivisibles, Angeli did not publish a single word on the topic ever again. The Jesuits had won.
TWO DREAMS OF MODERNITY
By the 1670s the war over the infinitely small had ended. With Angeli at long last silenced, and all the Jesuits’ rivals driven underground or melting away, Italy was a land cleansed of infinitesimals, and the Jesuits reigned supreme. It was, for the Society, a great triumph, and it came at the end of a difficult campaign that had claimed many victims along the way. Some of them were famous in their day, such as Luca Valerio and Stefano degli Angeli, but many more will remain forever anonymous. The cold hand of the Society of Jesus drew a curtain over this lost generation of Italian mathematicians and left them in the dark.
The Jesuits did not fight this battle out of pettiness or spite, or merely to flex their muscle and humiliate their opponents. They fought it because they believed that their most cherished principles, and ultimately the fate of Christendom, were at stake. The Jesuits were forged in the crucible of the Reformation struggle, which saw the social and religious fabric of the Christian West tearing at the seams. Competing revelations, theologies, political ideologies, and class loyalties were all vying for the minds and souls of the people of western Europe, leading to chaos, hunger, pestilence, and decades of warfare. The one and only Truth of the ancient Church, which had united Christians and given purpose to their years, had suddenly disappeared amid the clamor of rival creeds. Reversing this catastrophe, and ensuring that it never recurred, was the overriding purpose of the Society of Jesus from the day of its founding by Ignatius of Loyola.
The Jesuits pursued this goal in many ways, but always with energy, skill, and determination. They became expert theologians dedicated to formulating a single religious truth, and expert philosophers to support their theology. And they founded the largest educational system the world had ever seen, in order to disseminate knowledge of these truths far and wide. They were the engine of the Catholic revival in the second half of the sixteenth century and played a key role in halting the spread of the Reformation and reversing some of its gains.
But the Jesuits were confronted with a pesky problem: different opinions were everywhere, and every religious or philosophical doctrine was seemingly in contention between different authorities. Except, that is, in mathematics. This at least was the opinion of Christopher Clavius, who began advocating for the field at the Collegio Romano in the 1560s and ’70s. In mathematics, and especially in Euclidean geometry, there was never any doubt, Clavius argued, and he ultimately made mathematics a pillar of the Jesuit worldview.
It is because of their deep investment in mathematics, and their conviction that its truths guaranteed stability, that the Jesuits reacted with such fury to the rise of infinitesimal methods. For the mathematics of the infinitely small was everything that Euclidean geometry was not. Where geometry began with clear universal principles, the new methods began with a vague and unreliable intuition that objects were made of a multitude of minuscule parts. Most devastatingly, whereas the truths of geometry were incontestable, the results of the method of indivisibles were anything but. The method could lead to error as often as to truth, and it was riddled with contradictions. If the method was allowed to stand, the Jesuits believed, it would be a disaster for mathematics and its claim to be a fount of incontestable knowledge. The broader implications were even worse: If even mathematics was shown to be riddled with error, what hope was there for other, less rigorous disciplines? If truth was unattainable in mathematics, then quite possibly it wasn’t attainable anywhere, and the world would once again be plunged into despair.
It was to avoid this catastrophic outcome that the Jesuits pursued their campaign against infinitesimals. But were the mathematicians in Italy who championed the method of indivisibles truly dangerous individuals who would revel in overthrowing authority? This hardly seems likely. Galileo and Cavalieri, Torricelli and Angeli were, after all, academics and professors, hardly a breed of men inclined to overthrow civilization. Galileo may have been a flamboyant individualist, but he was no opponent of order, as he made clear when he chose to leave republican Venice to take a post at the court of the Grand Duke of Tuscany. Cavalieri was a sedate cleric and professor who left the city of Bologna only once in the last eighteen years of his life, and Torricelli, after settling in Florence, did his best to avoid conflict with his critics. Angeli undoubtedly showed a great deal of spirit in taking his last stand for indivisibles, but it would be exceedingly hard to describe him as a subversive. He was, after all, a cleric and a professor who relied on the protection of his ancient order and the Venetian Senate to keep his enemies at bay. It would indeed be difficult to find anyone among the proponents of the infinitely small who justified the Jesuits’ ferocious reaction to the doctrine, or their fears of its implications.
So were the Jesuits simply wrong to fear the proponents of the infinitely small? Not exactly. For although it is true that the Galileans were not social subversives, it is also true that they stood for a degree of freedom that was unacceptable to the Jesuits. Galileo was a brilliant public advocate for the freedom to philosophi
ze (“libertas philosophandi”), by which he and his associates meant the right to pursue their investigations wherever they led. He openly mocked the Jesuits and their reverence for authority, writing that “in the sciences the authority of thousands of opinions is not worth as much as one tiny spark of reason in an individual man.” Not only did Galileo argue that when Scripture and scientific fact collide, it is the interpretation of Scripture that must be adjusted, but he publicly transgressed the authority of professional theologians. Not surprisingly, the Jesuits were furious. It was precisely the kind of transgression they believed could lead to chaos.
Galileo was the chief public spokesman of his group, but his fellow Linceans, his students, and his followers shared his views. All of them believed in the principle of libertas philosophandi and saw in the trial and condemnation of their leader a monstrous crime against the freedoms they cherished. For them, the Jesuit quest for a single, authorized, and universally accepted truth crushed any possibility of philosophizing freely. By championing the mathematics of the infinitely small, they were taking a stand against the Jesuits’ totalitarian demand that truth be officially sanctioned.
The core conflict between the Jesuits and the Galileans was on the questions of authority and certainty. The Jesuits insisted that truth must be one, and in Euclidean geometry they believed they had found the perfect demonstration of the power of such a system to mold the world and prevent dissent. The Galileans also sought truth, but their approach was the reverse of that of the Jesuits: instead of imposing a unified order upon the world, they attempted to study the world as given, and to find the order within. And whereas the Jesuits sought to eliminate mysteries and ambiguities in order to arrive at a crystal-clear, unified truth, the Galileans were willing to accept a certain level of ambiguity and even paradox, as long as it led to a deeper understanding of the question at hand. One approach insisted on a truth imposed from above through reason and authority; the other pragmatically accepted the possibility of ambiguity and even contradiction, and sought to derive knowledge from the ground up. One approach insisted that the infinitely small must be banned, because it introduced paradox and error into the perfect, rational structure of mathematics; the other was willing to live with the paradoxes of the infinitely small as long as they served a powerful and fruitful method and led to deeper mathematical understanding.
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Page 20