Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (23 page)

Vitelleschi moved quickly. Only six months after the Revisors issued their edict, he found himself writing to Father Ignace Cappon at the Jesuit College in Dôle, in eastern France, to complain that his repeated instructions were not being followed. “As regards the opinion on quantity made up of indivisibles,” he wrote impatiently, “I have already written to the provinces many times that it is in no way approved by me and up to now I have allowed nobody to propose it or defend it.” Indeed, he had made every effort to suppress the doctrine: “If it has ever been explained or defended, it was done without my knowledge. Rather, I demonstrated clearly to Cardinal Giovanni de Lugo himself that I did not wish our members to treat or disseminate that opinion.” The fight to wipe out any vestiges of the offending doctrine from the Society was one the general would undertake himself.

The lines were now firmly drawn. On the one side were the Jesuits, determined to wipe out the doctrine of the infinitely small. On the other side was a small band of mathematicians who still saw Galileo as their undisputed leader, despite his public humiliation. And the struggle raged. In 1635, Cavalieri published his
Geometria indivisibilibus
, giving the most systematic exposition of the method of indivisibles. Three years later, Galileo published his
Discourses
in Holland, which included his treatment of the paradox of Aristotle’s wheel and his discussion of infinitesimals. Galileo’s towering fame ensured that his views on the continuum would be taken very seriously by scholars across Europe, and his praise of Cavalieri established the Jesuat monk as the leading authority on the subject of indivisibles. In response, the Jesuits struck back with repeated condemnations of the infinitely small. From this time onward a steady stream of denunciations of the offending doctrine began issuing from the pens of the Revisors General in Rome.

On February 3, 1640, for example, the Revisors were asked to comment on the proposition that “The successive continuum … is composed of separate indivisibles,” and ruled that “the doctrine is prohibited in the Society.” Less than a year later, in January 1641, they were once again confronted with ideas “that were invented or innovated by certain moderns,” including the proposition “that the continuum is made of indivisibles,” and a variation that claimed “that the continuum is made of indivisibles that expand and contract.” Both propositions, the Revisors ruled, were “contrary to the common and solid opinion.” On May 12, 1643, they came down hard on an author who allegedly preferred the opinions of Zeno to those of Aristotle: “We do not approve or concede of these,” they wrote, “as it is against the Society’s constitution and rules, as well as the decrees of the General Congregation.”

A pattern set in. In Rome the Jesuits were strong enough to quash any talk of the forbidden doctrine; but farther away, some mathematicians could still defend and promote it. Torricelli, for example, was in Rome in the 1630s, toiling in private on the new mathematics but publishing not a word. But it took only two years after his installment as Galileo’s successor at the Medici court in Florence before he published his
Opera geometrica
, which contains the fruits of his silent labor in Rome. Even from the safety of his court position, however, Torricelli tried to avoid open conflict with his powerful critics. Unlike Galileo, he did not directly engage with them in his book, argue over the merits of his method, or ridicule their motives or reasoning. He let his powerful results speak for themselves, and that they undoubtedly did: the
Opera geometrica
was admired and emulated by mathematicians from Germany to England. In Italy the long shadow cast by the Society of Jesus was sufficient to ensure that no mathematician there expressed similar enthusiasm for Torricelli’s work. But the Jesuits nonetheless took note of his success, and prepared to strike back.

Three years later, Cavalieri fired his last salvo in the fight over the infinitely small. As professor of mathematics at the University of Bologna, he enjoyed the protection of his own order, the Jesuats of St. Jerome, as well as of the Bolognese Senate, to which he dedicated several of his books. Like Torricelli, Cavalieri, too, was far from Rome, and in a position in which he could afford to risk the wrath of the domineering Jesuits. In 1647 he was in any case mortally ill, and did not need to concern himself with any future retaliation by his enemies. Shortly before his death, he managed to see through the publication of his second, and final, book on indivisibles, the
Exercitationes geometricae sex
. As regards to mathematical method, the
Exercitationes
contained little that was new to anyone familiar with Cavalieri’s monumental
Geometria indivisibilibus
. As it happened, no one except for the author saw any fundamental difference between the two. But the book did play a significant role in the ongoing struggle over infinitesimals: in a wholly new section, Cavalieri directly attacked the Jesuit mathematician Paul Guldin, who had harshly criticized Cavalieri’s approach. It was the last statement on indivisibles from the man who was widely recognized as the greatest authority on the method, but it could not stem the tide of Jesuit denunciations. In 1649 the Revisors ruled on two more variations of the doctrine of infinitesimals, proposed by the Jesuits of Milan. As usual, they decreed that the doctrine was forbidden and could not be taught in the Society’s schools.

THE HUMBLING OF THE MARQUIS

Arriaga’s views on the continuum were unequivocally condemned by Father Bidermann and his Revisors in 1632, but the
Cursus philosophicus
prospered nonetheless. Not only did the book retain its popularity, but Arriaga managed to secure permission from his superiors to publish new editions every few years, the last one appearing two years after his death in 1667. The reason for this leniency was not, as Arriaga suggests in his introduction to the posthumous edition, that he had published his opinions in good faith and that they “do not pertain to matters of faith,” as such considerations did not stop the Jesuit hierarchy from taking an extremely grim view of any advocacy of indivisibles. It was surely Arriaga’s enormous popularity as a teacher and his standing as one of the leading intellectuals in Europe that allowed his unorthodox book to remain in print. Arriaga brought intellectual stature and prestige to an order eager to assert its intellectual leadership all across Europe, and the Jesuit superior generals therefore thought it best to leave him in peace. But once Arriaga died, there was no more need to accommodate such dissent. The 1669 edition, approved before Arriaga’s death and published shortly after it, was to be the last of the
Cursus philosophicus
.

At least one other high-ranking Jesuit tried to emulate Arriaga’s example: the Marchese Pietro Sforza Pallavicino, who as a young maverick in 1620s Rome dared to challenge the Jesuits in a public oration in their own halls. When the political tide in Rome turned against the Galileans, Pallavicino paid a price for his cockiness. In 1632 he was exiled from the papal court and sent to govern the provincial towns of Jesi, Orvieto, and Camerino. But by 1637 the marchese had tired of country living and was ready to see the light: in a reversal that left Roman society dumbfounded, Pallavicino took the monastic oaths and entered the Society of Jesus as a novice. For the Jesuits it was a stunning coup. Not only was Pallavicino a high-ranking nobleman and a renowned poet and scholar, but he was also famous for his open criticism of the Society. Nothing demonstrated the Jesuits’ triumph as clearly as the defection of the brilliant marchese from their enemies’ camp, to enter their own ranks as a humble novice.

Even so, Pallavicino was no ordinary novice. It took Clavius, who had come from humble stock, two decades to climb from novice to professor at the Collegio Romano. The noble Pallavicino completed the same journey in two years, before being appointed professor of philosophy—an honor Clavius never attained. It seems likely that General Vitelleschi had struck a deal with the young marchese, promising him a shortened apprenticeship and a prestigious appointment in Rome as the price of his entry into the Society. Be that as it may, by 1639, Pallavicino was teaching philosophy at the Collegio Romano, and several years later he was also appointed professor of theology, the highest academic post at the college. In 1649 he published a comprehensive defense of the Jesuits, entitled
Vindicationes Societatis Iesu
, a particularly appropriate task for one who some years before had been among the Society’s most public critics. At the personal request of the Pope, he then authored a history of the Council of Trent, intended as an official rebuttal to the controversial (and, to the Papacy, defamatory) history that Venetian Paolo Sarpi published back in 1619. Pallavicino’s volumes came out in 1656 and 1657 as
Istoria del Concilio di Trento
, and earned the marchese the crowning honor of his life: the cardinal’s purple.

Pallavicino’s career among the Jesuits went from triumph to triumph, but in the 1640s he did suffer some embarrassing setbacks. Despite throwing in his lot with Galileo’s enemies, the marchese still considered himself a progressive thinker and an admirer of the Florentine master. Such residual allegiance to the Jesuits’ vanquished enemy was bound to be viewed with suspicion by the Society’s hierarchy, and indeed, Pallavicino frequently came under the Revisors’ scrutiny for what they considered his “novel doctrines.” Nevertheless, encouraged no doubt by the example of Arriaga and believing that his stature would protect him from censure, Pallavicino forged ahead, lecturing on his unorthodox views to the Collegio’s students. But the marchese had miscalculated. He himself hints as much in the
Vindicationes Societatis Iesu
, when he recalls having “to face a fight several years ago” when he wished to express himself on a matter he considered “common or well known.” Pallavicino, it seems, was criticized and perhaps chastised for his position, but he was not one to concede that he was in the wrong. To the contrary, he insists, although the propositions he mentioned may be false, in a Society so devoted to the well-being of its students, “a certain freedom to speak of positions less accepted should, up to a point, not be eliminated, but promoted.”

Pallavicino tried to put the best face on the incident, but a much clearer picture of the events emerges in an angry letter from Superior General Vincenzo Carafa, who had succeeded Vitelleschi, to Nithard Biberus, Jesuit provincial for Lower Germany. “When I came to know there are some in the Society who follow Zeno, who pronounced in a philosophy course that a quantity is composed of mere points, I let it be known that it is not approved by me,” the superior general wrote irately on March 3, 1649. “And since in Rome Father Sforza Pallavicino taught this, he was ordered to retract it in the very same course.” It was a stinging rebuke, and undoubtedly a humbling experience for the marchese to be forced to retract his own words before his own students. The bitterness of the experience still echoes in the
Vindicationes
, but as a member of a society that valued obedience above all, refusing a direct order from the superior general was out of the question. So Pallavicino swallowed his pride, retracted his teachings on the infinitely small, and having learned his lesson, quietly resumed his climb up the ladder of the Jesuit hierarchy.

Carafa’s letter to Biberus makes clear that the superior general could not allow even the marchese to get away with teaching the forbidden doctrine. When he recently wrote to censure a professor in Germany for teaching it, Carafa continued, the professor answered, “by way of excuse, that Arriaga and a certain Portuguese of ours have expounded these views in print.” The general, however, would have none of it: “I then wrote again that these [two] works being given, there will be no third who will imitate them.” Arriaga (as well as the unnamed Portuguese) was a special case, grandfathered in from laxer times. But no one, not even the Marchese Pallavicino, should consider it a precedent. The doctrine of the infinitely small was banned to all Jesuits, and anyone who dared promote it would suffer the consequences.

A PERMANENT SOLUTION

But even as the superior general was personally admonishing his subordinates and publicly humbling an excessively proud Jesuit, pressure was growing for a more permanent solution to the problem of dissent within the Society. Already in 1648, Carafa had instructed the Revisors to search their records and come up with a provisional list of theses that should be permanently banned from the order. When, following Carafa’s death, a General Congregation convened in December 1649, it instructed the newly elected superior general, Francesco Piccolomini, to follow up on his predecessor’s initiative. Over the next year and a half, a Jesuit committee met and devised an authoritative list of prohibited doctrines. The results of their labors were published in 1651 as part of the
Ordinatio pro studiis superioribus
(“Regulations for Higher Studies”), designed to preserve the Society’s “solidity and uniformity of doctrine.” Henceforth, any Jesuit anywhere in the world would have access to an authoritative list detailing which doctrines were anathema to his order and must never be held or taught.

The sixty-five forbidden “philosophical” theses cited in the
Ordinatio
(there are also twenty-five “theological” ones) make for an eclectic list. Some banned propositions infringed on accepted interpretations of Aristotelian physics, such as “primal matter can naturally be without form” (number 8), or “heaviness and lightness do not differ in species, but only in regard to more or less” (number 41). Some offending propositions were tinged with materialism, such as “the elements are not composed of matter and form, but only from atoms” (number 18). Other theses were banned for challenging divine omnipotence, such as “a creation so perfect is possible that God is incapable of creating a more perfect one” (number 29); and still others were banned for teaching of the diurnal motion of the Earth (number 35) or promoting the magical curing of wounds at a distance (number 65). But no fewer than four of the prohibited propositions directly addressed the question of the composition of the continuum from indivisible parts: