Destroy or be destroyed—such were the stakes when it came to infinitesimals, according to Tacquet. Strong words indeed, but to the Fleming’s contemporaries, they were not particularly surprising. Tacquet was, after all, a Jesuit, and the Jesuits were then engaged in a sustained and uncompromising campaign to accomplish precisely what Tacquet was advocating: to eliminate the doctrine that the continuum is composed of indivisibles from the face of the earth. Should indivisibles prevail, they feared, the casualty would be not just mathematics, but the ideal that animated the entire Jesuit enterprise.
When Jesuits spoke of mathematics, they meant Euclidean geometry. For, as Father Clavius had taught, Euclidean geometry was the embodiment of order. Its demonstrations begin with universal self-evident assumptions, and then proceed step by logical step to describe fixed and necessary relations between geometrical objects: the sum of the angles in a triangle is always equal to two right angles; the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of the long side; and so on. These relations are absolute, and cannot be denied by any rational being.
And so, beginning with Clavius and for the next two hundred years, geometry formed the core of Jesuit mathematical practice. Even in the eighteenth century, when the direction of higher mathematics turned decisively away from geometry and toward the newer fields of algebra and analysis, Jesuit mathematicians held firm to their geometrical practice. It was the unmistakable hallmark of the Jesuit mathematical school. If only theology and other fields of knowledge could replicate the certainty of Euclidean geometry, they believed, then surely all strife would be at an end. The Reformation and all the chaos and subversion that flowed from it would never have taken root in such a world.
This vision of eternal order was, to the Jesuits, the only reason mathematics should be studied at all. Indeed, as Clavius never tired of arguing to his skeptical colleagues, mathematics embodied the Society’s highest ideals, and thanks to his efforts the doors were opened at Jesuit institutions for the study and cultivation of the field. By the late sixteenth century, mathematics had become one of the most prestigious fields of study at the Collegio Romano and other Jesuit schools.
Just as Euclidean geometry was, for the Jesuits, the highest and best of what mathematics could be, so the new “method of indivisibles” advocated by Galileo and his circle was its exact opposite. Where geometry began with unassailable universal principles, the new approach began with an unreliable intuition of base matter. Where geometry proceeded step by irrevocable step from general principles to their particular manifestations in the world, the new methods of the infinitely small went the opposite way: they began with an intuition of what the physical world was like and proceeded to generalize from there, reaching for general mathematical principles. In other words, if geometry was top-down mathematics, the method of indivisibles was bottom-up mathematics. Most damaging of all, whereas Euclidean geometry was rigorous, pure, and unassailably true, the new methods were riddled with paradoxes and contradictions, and as likely to lead one to error as to truth.
If infinitesimals were to prevail, it seemed to the Jesuits, the eternal and unchallengeable edifice of Euclidean geometry would be replaced by a veritable tower of Babel, a place of strife and discord built on teetering foundations, likely to topple at any moment. If Euclidean geometry was, for Clavius, the foundation of universal hierarchy and order, then the new mathematics was the exact opposite, undermining the very possibility of universal order, leading to subversion and strife. Tacquet was not exaggerating when he wrote that in the struggle between geometry and indivisibles, one must destroy the other or “must itself be destroyed.” And so the Jesuits proceeded to do just that.
THE CENSORS, PART I
The issue of the structure of the continuum could hardly have been further removed from the minds of the early Jesuit fathers as they faced off against Martin Luther and his followers in a battle for the soul of Europe. The first Jesuit to take any notice of the issue was none other than Clavius’s old nemesis at the Collegio Romano, Benito Pereira. In 1576, at the height of his struggle with Clavius over the proper place of mathematics in the Jesuit curriculum, Pereira published a book on natural philosophy intended to establish the proper principles that should be adopted by the Jesuits. Following the guidelines established by the Society’s founders, Pereira adhered closely to the teachings of Aristotle, and so he also addressed that ancient philosopher’s teachings on the subject of the continuum. In the best tradition of medieval scholasticism, Pereira first posed the thesis that a line is composed of separate points, and presented all the arguments offered in support of the thesis by ancient and medieval masters. He then demolished the arguments one by one, until he was left to conclude, along with Aristotle, that the continuum is infinitely divisible, and not composed of indivisibles. Pereira, it is clear, was not concerned about mathematical innovations or their subversive implications: writing decades before Galileo and his disciples developed their radical mathematical techniques, he had no reason to be. And since he saw no value for the Jesuits in the study of any kind of mathematics, he was unlikely to concern himself with determining the right “kind” of mathematics that should be taught. For him the question of the continuum was merely one more topic to be addressed in a discussion of Aristotle’s natural philosophy.
It was a full two decades before another Jesuit took up the question of the continuum, and this time it was a much more authoritative one: Father Francisco Suárez, the leading theologian of the Society of Jesus. In 1597, Suárez devoted thirteen folios to the question of the composition of the continuum in his Disputation on Metaphysics, but like Pereira, he addressed the matter as part of a broader discussion of Aristotelian physics. Unlike Pereira, however, the great theologian did not peremptorily reject the notion that the continuum is composed of indivisibles; admitting that the question is difficult, he gives up any hope of certainty, and seeks only an answer that “appears to be true.” He cites the doctrine that the continuum is composed of indivisibles, and then the complete denial of indivisibles, arguing that both are “extreme” positions. He then proposes some intermediate positions that he thinks more likely, while conceding that a definite solution is beyond reach. For Suárez as for Pereira, the entire question was technical, or what we would call “academic.” Neither one thought that there was much at stake here, except the correct interpretation of Aristotelian physics.
But as the troubled century of Charles V, Luther, and Ignatius was drawing to a close, an unmistakable sense of urgency entered into Jesuit discussions of infinitesimals. At the time, the general superior, Father Claudio Acquaviva, was increasingly concerned with the growing diversity of opinions within the Society. This was undoubtedly the price of success, as the rapid expansion of the Society in those years, in the form of hundreds of colleges and missions across the known world, brought many new peoples into its orbit. But for General Acquaviva this was no excuse for the soldiers of Christ to deviate from the correct teachings of the Church. As far as the Jesuit hierarchy was concerned, the increase in the Society’s numbers and influence was all the more reason it should speak with a single and clear voice. “Unless minds are contained within certain limits,” warned Father Leone Santi, prefect of studies at the Collegio Romano some years later, “their excursions into exotic and new doctrines will be infinite,” leading to “great confusion and perturbation to the Church.” To prevent this, the general superior in 1601 instituted a college of five Revisors General at the Collegio Romano, with the power to censor anything that was taught in the Society’s schools anywhere in the world or published under the Society’s aegis. With the Revisors’ oversight, Acquaviva hoped, only correct doctrine would be taught in Jesuit schools, and books published by Jesuit fathers would speak with a single authoritative voice, approved by the authors’ superiors. It did not take long for the Revisors to begin issuing prohibitions on the teaching and promotion of infinitesimals.
The first decree
by the Revisors General on the composition of the continuum dates from 1606, when the office was only five years old. Responding to a proposition sent in from the Society’s schools in Belgium that “the continuum is composed of a finite number of indivisibles,” the Revisors, quickly and without comment, ruled that the proposition was an “error in philosophy.” Only two years later another missive from Belgium brought the very same doctrine before the Revisors. This time they were somewhat more expansive, though just as firm: “everyone agrees that this must not be taught, since it is improbable and also certainly false and erroneous in philosophy, and against Aristotle.” Only a decade before, Suárez had merely expressed some concern about whether the notion that the continuum was composed of indivisibles was philosophically viable, and offered some alternatives. The Revisors, in contrast, banned it outright as “false and erroneous.”
What had changed? The Revisors themselves offer no clue, and the summaries they left provide no details on the sources of propositions brought before them, except for their country of origin. But we do know that those early years of the seventeenth century saw a significant uptick in interest in the infinitely small among mathematicians. In 1604, Luca Valerio of the Sapienza University in Rome published a book on calculating the centers of gravity of geometrical figures in which he employed rudimentary infinitesimal methods. Valerio was well known to the Jesuits, having studied under Clavius for many years, and even receiving doctoral degrees in philosophy and theology from the Collegio Romano. His work could not have gone unnoticed by the Jesuit fathers, who likely felt they needed to better define their position on this new approach. We also know that in 1604, Galileo, then at the University of Padua, was experimenting with indivisibles in formulating his law of falling bodies. Galileo thought very highly of Valerio: years later he nominated him for membership in the Lincean Academy, and in his Discourses of 1638 he refers to Valerio as “the Archimedes of our age.” Whether the two drew upon each other or developed their ideas independently, their work marked a significant change in the status of infinitesimals: instead of an ancient doctrine definitively discussed by Aristotle and his later commentators, infinitesimals now seemed to be entering the arena of contemporary mathematics.
For the Jesuits, this was a critical change. Clavius had only recently won his battle to establish mathematics as a core discipline in the Jesuit curriculum, and the order’s mathematicians were beginning to be recognized as leaders in the field. When, in the early seventeenth century, infinitesimals began seeping into mathematical practice, the Jesuits felt compelled to take a stand on the new methods. Are they compatible with the Euclidean approach so central to the Society’s mathematical practice? The Revisors’ answer was a resounding no. Yet despite the stern pronouncements, the problem never seemed to go away. Mathematically trained Jesuits across Europe were closely following developments at the forefront of mathematical research, and were well aware of the growing interest in infinitesimals. Conscious of the sensitivity of the subject, they kept appealing to the Revisors with different versions of the doctrine, each one deviating slightly from those that had already been banned. Consequently, when the Revisors turned their attention once more to the infinitely small, the catalyst once again was developments in the field of mathematics.
Johannes Kepler (1571–1630) is remembered today as the man who first plotted the correct elliptical paths of the planets through the heavens. Nor did Kepler go unappreciated in his own day. In the early seventeenth century he was the only scientist in the world whose fame rivaled Galileo’s, and though a Protestant, he held the most coveted mathematical position in the world: court astronomer to the Holy Roman Emperor in Prague. In 1609 Kepler published his masterpiece Astronomia nova (The New Astronomy), in which he demonstrates that the planets move in ellipses and not perfect circles, and codifies his observations in two laws of planetary motion. (Kepler’s third law was published later, in his Harmonices mundi of 1619.) To calculate the precise motion of the planets at varying speeds along their orbit, Kepler made rough use of infinitesimals, assuming that the arc of their elliptical path was composed of an infinite number of points. Six years later Kepler further developed his mathematical theory in a work dedicated to calculating the exact volume of wine casks, where he calculated a whole range of areas and volumes of geometrical figures using infinitesimal methods. To calculate the area of a circle, for example, he assumed that it was a polygon with an infinite number of sides; a sphere was composed of an infinite number of cones, each with its tip at the center and its base on the surface of the sphere, and so on. Titled Nova stereometria doliorum vinariorum (“A New Stereometry of Wine Caskets”), it was a mathematical tour-de-force that hinted at the power of the approach that Cavalieri would later systematize and name. Once again the Jesuits felt compelled to respond, and once again the job fell to the Revisors General in Rome. In 1613 they denounced the proposition that the continuum was composed of either physical “minims” or mathematical indivisibles. In 1615 they reiterated their condemnation, rejecting, first, the opinion that “the continuum is composed of indivisibles” and, several months later, the opinion that “the continuum is composed of a finite number of indivisibles.” This doctrine, they opined, “is also not permitted in our schools … if the indivisibles are infinite in number.”
Once the Revisors had issued their decision, a well-oiled machinery of enforcement sprang into action. The numerous Jesuit provinces across the globe were informed of the censors’ verdict, and they then passed it on to lower and then lower jurisdictions. At the end of this chain of transmission were the individual colleges and their teachers, who were instructed on the new rules on what was permissible and what was not. Once a decision by the Revisors in Rome descended the Jesuit hierarchy and reached an individual professor, he was now responsible for carrying it out to the fullness of his ability and of his own free will, regardless of his previous views on the subject. It was a system based on hierarchy, training, and trust—or, as an unfriendly observer might suggest, on indoctrination. Either way, there was no doubt that it was remarkably effective: the Revisors’ pronouncements became law in the many hundreds of Jesuit colleges worldwide.
THE FALL OF LUCA VALERIO
The Revisors’ decree of 1615 against infinite indivisibles may have been directed against the admirers of Kepler. But whatever the intent, it was the Jesuits’ former associate Luca Valerio who fell victim to the Society’s new and harsher stance. Three years had passed since Galileo proposed Valerio for membership in the prestigious Lincean Academy, which served as the institutional center of the Galileans in Rome. The academy was an exclusive club made up of a select group of leading scientists and their aristocratic patrons, but Valerio seemed like a perfect fit: not only was he a mathematician renowned for bold ideas and a professor at the ancient Sapienza University, but he was also an aristocrat and a personal friend of the late pope Clement VIII (1592–1605), who had been his student. He brought with him sparkling social prestige, as well as personal creativity and institutional respectability, and the Linceans promptly elected him on June 7, 1612. From the moment of his election, Valerio became a leader among the Linceans, given overall editorial responsibility for all the academy’s publications.
Valerio, who had studied for years under Clavius, remained on good terms with his former mentors and colleagues at the Collegio Romano, and that, too, made him valuable to the Linceans. At a time of increasing tensions between the Galileans and the Jesuits of the Collegio Romano, Valerio served as a means of communication and possible compromise between the two camps. Indeed, there was nothing Valerio wanted more than to heal the rift that had opened between his two groups of friends. It was not to be. Showing no concern for Jesuit sensitivities, Galileo published his Discourse on Floating Bodies—which attacked the principles of Aristotelian physics—debated the nature of sunspots, and circulated his views on the proper interpretation of Scripture. For the Jesuits of the Collegio Romano, this intrusion into theology was the l
ast straw. They determined to strike back against the man they had once honored with a full day of ceremonies but whom they now viewed as a bitter enemy.
The Jesuits had learned from their past mistakes. Time and again they had been outmatched by Galileo’s brilliant polemics, coming off as rigid and didactic pedants standing in the way of scientific progress. So, instead of engaging in public debate, they turned to the arena in which their power was unchallengeable: the hierarchy and authority of the Church. In 1615 cardinal Bellarmine issued his opinion against Copernicanism, which soon became official Church doctrine. He followed this up with a personal warning to Galileo to desist forever from holding or advocating the forbidden doctrine. It was an impressive demonstration of the Jesuits’ ability to harness the Church apparatus to their cause, and a stinging defeat to the Galileans. As regards infinitesimals, nothing so public as the decree against Copernicanism took place. But it is probably not a coincidence that the Revisors’ ruling against indivisibles in April 1615 coincided precisely with Bellarmine’s issuing his opinion against Galileo.
Valerio felt besieged. The two great intellectual schools that he had hoped to reconcile were now openly at war. The middle ground on which he stood was quickly melting away, and he was being pulled in opposite directions. The Revisors’ decree of April 1615 on the composition of the continuum was a reminder that, as a mathematician identified with infinitesimal methods, he could not long remain above the fray. When the Revisors repeated their decree in November, this time adding that it applied even “if the indivisibles are infinite in number,” he may well have concluded that he himself was their target. We do not know what he was told in private by either the Jesuits or the Linceans, but the pressure must have been unbearable. Finally, in early 1616, with the tide turning decisively against the Galileans, Valerio made his decision: he tendered his resignation to the Lincean Academy, siding openly with the Jesuits.
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Page 14