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 growi
ng 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:
25. The succession continuum and the intensity of qualities are composed of sole indivisibles.
26. Inflatable points are given, from which the continuum is composed.
30. Infinity in multitude and magnitude can be enclosed between two unities or two points.
31. Tiny vacuums are interspersed in the continuum, few or many, large or small, depending on its rarity or density.
Thesis 25 is the broadest of the four, referring to any possible continuum and its composition. The question of the “intensity of qualities” refers to medieval debates in which the intensity of qualities such as “hot” or “cold” existed along a gradient, raising the question of whether there was a finite or infinite number of grades of these qualities forming the continuum. Thesis 26 was a response to widespread speculation in the seventeenth century about what caused a material’s change in density, a question that was considered one of the toughest challenges to an atomic theory of matter. The thesis claimed that matter was composed of innumerable inflatable points, whose size at any given moment determined the degree of density or rarity, but to the Jesuits this was no more acceptable than the simpler doctrine that the continuum was composed of indivisibles. Thesis 30 is the most explicitly mathematical of the four, referring directly to the method of indivisibles practiced by Cavalieri and Torricelli, which relied on the division of finite lines or figures into an infinite number of indivisible parts. Thesis 31 appears to address Galileo’s theory of the continuum as expounded in the Discourses of 1638. Relying on analogies with matter and the paradox of Aristotle’s wheel, Galileo concluded that the continuum is interspersed with an infinity of tiny vacuums. Between them, these four propositions encompass the different variations of the method of indivisibles under dispute in the middle of the seventeenth century. All were unequivocally banned.
The Ordinatio of 1651 was a turning point in the Jesuit battle against the infinitely small. The prohibition on the doctrine was now permanent, and was backed by the highest authority in the order, the General Congregation. Printed, published, and widely disseminated, the Ordinatio was brought to the attention of every Jesuit father teaching at every one of the Society’s institutions across the world. The repeated blasts by the Revisors, who every few years issued their censures of the doctrine, were now at an end. No further condemnations were needed: the prohibition was now permanent and compulsory, and every member of the order knew it well. And so it stood for the next century, as the Ordinatio remained the fundamental guide to Jesuit teachings. In fact, the document did much more: it set the tone for intellectual life in the lands where the Jesuits dominated. For the few and lonely mathematicians still defending infinitesimals in Italy, the consequences were devastating.
5
The Battle of the Mathematicians
GULDIN VERSUS CAVALIERI
“The reasoning by indivisibles convinces all the famous geometers brought up here,” wrote Stefano degli Angeli (1623–97), professor of mathematics at the University of Padua, in 1659. Angeli, as was his way, expressed himself with great bravado, but the truth was very different. By the time he was writing, Angeli was likely the last mathematician left in Italy to adopt the method of indivisibles, and one of the even fewer to publish their work in the field. Most of the individuals named in his own list of the “famous” adherents of the method resided north of the Alps, many in France and England. The Italians on the list belonged to an old and dying generation of Galileans, who had stopped publishing on the method decades before. When he wrote those words, Angeli was not in fact reporting on a favorable state of affairs, but rallying the troops for a desperate rearguard action on behalf of infinitesimals, which were in danger of being extinguished in the land where they had once flourished. Who his enemies were, he had no doubt: “the three Jesuits, Guldin, Bettini, and Tacquet” were, he claimed, the only ones who remained unconvinced by the method of indivisibles. “By what spirit they are moved,” he continued, with evident frustration, “I do not know.”
Paul Guldin, Mario Bettini (1584–1657), and André Tacquet were among the most notable mathematicians of the Society of Jesus in the mid-seventeenth century. Tacquet, whom we have already met, was the most original and creative of the three, but Guldin, too, was a widely respected mathematician. Bettini was perhaps less so, known mostly as a prolific author of rambling collections of mathematical results and curiosities, and less as a creative thinker in his own right. But he, too, was known in the Society and beyond as a man of broad learning and considerable authority on things mathematical. Together, Guldin, Bettini, and Tacquet were a formidable trio, exemplifying the intellectual prowess and the cultural and political cachet of the Jesuit mathematical school. And in the 1640s and ’50s, all three were engaged in the same mission: to discredit and undermine the method of indivisibles using sound and incontrovertible mathematical arguments. Theirs was a dimension of the Jesuit war on infinitesimals that was just as critical as the repeated condemnations of the Revisors and the Ordinatio of 1651. For if the use of infinitesimals in mathematics was to be permanently abolished, it was not enough to declare them philosophically, theologically, or even morally wrong, and legally banish them. It was also crucial to prove them mathematically wrong.
Guldin, the oldest of the three, was the first to take the field. Born Habakkuk Guldin to Protestant parents of Jewish descent in St. Gall, Switzerland, he may be the first in a long and illustrious line of Jewish (and converted Jewish) mathematicians that continues to this day. Guldin was not raised to be a scholar but an artisan, and he was working as a goldsmith when he began having doubts about his Protestant faith. At age twenty he converted to Catholicism and joined the Jesuits, changing his name in the process from the Old Testament prophet Habakkuk to Paul, the most famous Jewish convert, who preached the Christian faith to the Gentiles. Guldin’s was an eclectic and unusual background for a Jesuit, encompassing as it did many of the religious and ethnic fault lines of the early modern world, but this did not prevent his full acceptance in the Society. Indeed, it is one of the most admirable characteristics of the early Jesuits that despite pressure from the Iberian kingdoms, which placed the highest value on limpieza de sangre (p
urity of blood), the Society remained one of the most welcoming Catholic institutions for converts of all kinds.
The Society of Jesus was also, to a large degree, a meritocracy, and although high-born noblemen such as Marchese Pallavicino enjoyed enviable advantages, there was also a path forward for men of humble origins such as Guldin. As a bright young man with a talent for mathematics, he rose steadily through the ranks of the order, and was ultimately sent to the Collegio Romano to study with Clavius. Guldin spent fewer than three years under Clavius’s tutelage before the old master passed away in 1612. Five years later he was sent to teach mathematics in the Habsburg Austrian lands, and spent the rest of his life at the Jesuit college in Graz and at the University of Vienna. Nevertheless, it is clear from Guldin’s subsequent career that his years with Clavius shaped his mathematical outlook for a lifetime. Guldin was Clavius’s follower in every way: he adhered to the old Jesuit’s view that mathematics lies halfway between physics and metaphysics, he believed in the primacy of geometry among mathematical disciplines, and he insisted on following the classical Euclidean standards of deductive proof. All these positions made him an ideal choice as a critic of the method of indivisibles.
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Page 17