Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  It is not difficult to see why aristocrats and wealthy commoners across Catholic Europe clamored for the establishment of Jesuit colleges in their towns. Traditional parochial schools were of dubious quality, and student life at the great universities was reputedly dissolute and immoral, and little concerned with actual studies. The Jesuits offered something else altogether: a rigorous and demanding curriculum taught by highly qualified teachers and regularly updated by the luminaries of the Collegio Romano. And whereas university students were free to indulge in a life of drunken debauchery, the students in the Jesuit colleges were closely supervised and filled their days with study and prayer. An aristocrat or merchant who sent his son to a Jesuit school was confident that the boy would be immeasurably bettered, both intellectually and morally.

  The Collegio Romano, designed by Bartolommeo Ammannati, as it appears today. The building currently houses a public high school. (Alinari / Art Resource, NY)

  The long list of distinguished alumni of Jesuit colleges fully bears out this assessment. In addition to the leading Jesuits themselves, graduates include royalty such as Emperor Ferdinand II (1620–37), statesmen such as Cardinal Richelieu, humanists such as Justus Lipsius, and philosophers and scientists such as René Descartes and Marin Mersenne. Jesuit education, as even enemies of the Society acknowledged, was simply the best available in all Christendom. Even Francis Bacon, Lord Chancellor of England, and no friend of the Jesuits, ruefully remarked, “Talis quus sis, utinam noster esses” (“you are so good, would that you were ours”).

  Bacon had good reason to rue the Jesuits’ educational excellence. For, of all the services the Society of Jesus offered the Papacy in its struggle against Protestantism, none proved more powerful or more effective than the colleges. Wherever one was established, it became a center of Catholic life and a living demonstration of what the Roman Church could accomplish. Rare was the Lutheran or Calvinist school that could match the Jesuits for sheer educational quality, or compete with them in attracting the sons of the lay elite. Once they had them in their care, the Jesuits spent years imparting Catholic teachings to their charges, complete with learned and authoritative refutations of Protestant doctrines. Inevitably the students became imbued with the Jesuit devotion to the Papacy, and with the Jesuit spirit of dedication and sacrifice for the cause of the Church and its hierarchy. With hundreds of such colleges across Europe, and with hundreds and sometimes thousands of students enrolled in each one, the Jesuit educational system turned out generations of well-educated and devoted Catholics who would ultimately take up leadership positions in their communities. In effect, as the chief educators of the Catholic elite, the Jesuits ensured the survival, as well as the revival, of the Roman Church in large parts of Europe.

  The impact of the Jesuit colleges was unmistakable. The first Jesuit college in the Holy Roman Empire was founded in Cologne in 1556, at a time when the empire appeared on the verge of succumbing to the Lutheran surge. But with the college in place, Cologne became a Catholic stronghold, and a base for future expansion of Jesuit activities. In the following decades, with strong support from the ruling Wittelsbach and Habsburg families, the Jesuits founded dozens of colleges in Bavaria and Austria, and took over the administration of existing universities. They even went so far as to found a special school in Rome dedicated to training the promising young Germans for positions as high Church officials. Upon completing their studies, the graduates of this “Collegium Germanicum” returned home, where they became bishops and archbishops, and the backbone of the Catholic revival in Germany. In the Low Countries, too, the Jesuits were exceedingly active: when the northern provinces turned to Protestantism and took up arms against their Habsburg sovereign, the Jesuits helped make the southern provinces into a Catholic bastion. Thanks in great part to their efforts, the region was saved for the Catholic Church, acquired its own separate identity, and ultimately gained independence as the modern state of Belgium.

  Much like Germany, sixteenth-century Poland seemed well on its way to accepting one form or another of Protestantism when Catholic noblemen invited the Jesuits to open their colleges there in the 1560s. They soon gained the trust and support of the Polish royal family, which helped the Jesuits expand from five colleges in 1576 to thirty-two colleges by 1648. The Jesuits became the educators of the Polish ruling class, both the rural aristocracy and the urban elite, while in Rome they educated a devoted cadre of priests who returned to Poland to take up the leadership of the Church. So close were the Jesuits to the Polish monarchs, that King Sigismund III (1587–1632) was known as the “Jesuit King” and his son Jan II Kazimierz (1648–68) was a member of the order and a cardinal before assuming the throne. Poland was transformed: a nation that had previously prided itself on religious tolerance, and had opened its churches and parishes to the reformers, became the devout Catholic land we still recognize today. In Poland as elsewhere, the Jesuit intervention proved decisive.

  The upright disciples of Ignatius accomplished what the worldly Renaissance popes could not: they arrested the seemingly unstoppable progress of Protestantism across Europe and revived the power and prestige of the Roman Church. Wherever the Society raised its standard and opened its colleges, a new energy of spiritual devotion and purposefulness of action infused the old Church and inspired its followers to make a stand against the heretics. A grateful Pope Gregory XIII acknowledged as much when he addressed the general congregation of the Society in 1581:

  Your holy order … is spread throughout the entire world. Anywhere you look you have colleges and houses. You direct kingdoms, provinces, indeed, the whole world. In short, there is this day no single instrument raised up by God against heretics greater than your holy order. It came to the world at the very moment when new errors began to be spread abroad. It is all important therefore … that this order increase and prosper from day to day.

  ORDER OUT OF CHAOS

  The Miracles of St. Ignatius, a massive painting originally intended to grace the altar of Antwerp Cathedral, hangs today at the Kunsthistorisches Museum in Vienna. It is the work of the Flemish painter Peter Paul Rubens (1577–1640), whose modern-day reputation rests largely on his erotic depictions of ample women that challenge our ideal of female beauty. But Rubens was a devout Catholic who attended Mass every morning, and was on intimate terms with the Jesuits in his home city of Antwerp. In 1605, when the Jesuits were campaigning to have their founder canonized, Rubens contributed eighty engravings to a Jesuit hagiography, The Life of Ignatius. Four years later, when Ignatius was beatified, placing him one short step from canonization, Rubens was commissioned by the Society to produce several large portraits of the future saint for the Gesù, the Jesuits’ home church in Rome, and for the Antwerp Cathedral. In the imposing Miracles of St. Ignatius, he accomplished what was probably his greatest masterwork for the order.

  The painting brings us to a scene of high drama taking place within a large hall, most likely a church, which is depicted from its vaulted ceiling to its stony floor. At the top, near a brightly lit cupola, floats a band of playful angels and cherubs, who seem to pay no heed to the human chaos beneath them. Indeed, the floor of the church is a scene of pain, fear, and confusion, where a large group of men, women, and several children are caught up in an agonizing frenzy. One man flails about on his back as if in a seizure, while another man, with bloody streaks on his back, tends to him. A disheveled woman, her fists clenched, her face twisted wildly, and her eyes glazed, struggles to get away as two men try to support her. A gray-haired man, only his head visible, gazes up in desperation, his face twisted in a mask of horror. The rest, those who have not been overcome themselves, look upward in a tortured mixture of supplication and hope: can they be saved from that which torments them?

  The figure that is the object of their gaze is Ignatius himself, standing upright, resplendent in his priestly robes. On his dais, Ignatius is only a few steps above the floor, but he inhabits a completely different realm. Calm and commanding, his right
hand raised in benediction, he is performing an exorcism, expelling the evil spirits from the people, bringing peace and order to those afflicted with torment and chaos. Evil demons, on the left side of the painting, have emerged from the people and are fleeing before Ignatius’s holiness as one of the angels in heaven bids them an ironic good-bye. Ignatius, though the unchallenged focus of the painting, is not alone: behind him on the raised platform are his followers, a long line of black-robed Jesuits stretching into the distance. Like him, they are calm and somber, surveying the suffering before them. They are Ignatius’s army, there to learn from their master, follow his directions, and ultimately take over his mission of turning chaos into order and bringing peace to the afflicted.

  For that was indeed the “miracle” of St. Ignatius and his followers. Like no one else, they managed to restore peace and order in a land torn apart by the challenge of the Reformation. In place of heresy and confusion they brought unity and orthodoxy; where the rule of the holy Church was subverted and priests and bishops disowned, they rebuilt that grand old edifice and reestablished the sway of its hierarchy; where confusion reigned, they restored an unyielding certainty in the truth and rightness of the Roman Church. Their success in doing all this was indeed nothing less than miraculous. The keys to this miracle, as the Jesuits saw it, were simple: truth, hierarchy, and order.

  The Jesuits did not believe in plurality of opinion: the truth was absolute. They did not believe in pluralism of power and authority: once the truth is known, all power must flow from those who know and recognize it, and be imposed upon those who do not yet accept it. And they certainly did not believe in democracy, which allows the expression of different and opposing views and thrives on lively debate and competition for power. The truth has no room for such dissent or challenges. Only absolute authority of God’s emissaries and the divine truth they carry, they believed, would allow for peace and harmony to prevail. Such was the worldview of the Jesuits, and they worked hard to implement it within their order, within the Church as a whole, and in the world at large. Structured in a clear hierarchy, The Miracles of St. Ignatius puts this entire narrative into visual form. At the top is the realm of divine light and truth; at the bottom are the tormented and confused people. In between are Ignatius and his men: disciplined, calm, and commanding, they expel the demons of strife and confer the light of truth upon the people. Thanks to the Jesuits, peace will prevail.

  Peter Paul Rubens, The Miracles of St. Ignatius, 1617. (Erich Lessing / Art Resource, NY)

  2

  Mathematical Order

  TEACHING ORDER

  Ignatius of Loyola, founding father of the Society of Jesus, was not enamored of mathematics. As an aristocratic courtier and dashing cavalier in his early life, he learned to despise the pedantries of scholars and mathematicians. The ecstatic revelations of his later years led him, if anything, even farther away from the cold, logical world of numbers and figures, and his university studies in Barcelona, Alcalá, Salamanca, and Paris did not apparently include any mathematics. By 1553, when under his leadership the Jesuits were in the process of launching a worldwide network of colleges, Ignatius came to see the value of some mathematical education, writing that the colleges should teach “the parts of mathematics that a theologian should know.” And that, it should be admitted, was not much.

  The low standing of mathematics in the early days of the Jesuit educational system is not, in truth, surprising. The Jesuit colleges, after all, had a very specific and urgent goal, very different from the aims of their modern successors: to stop the spread of Protestantism and reestablish the prestige and authority of the Catholic Church. As Ignatius’s lieutenant, Juan de Polanco, explained in a 1655 letter, in the Society’s colleges “men of those nations” where the true faith is threatened “are taught with example and sound doctrine … to keep what remains, and restore what was lost, of the Christian religion.” A remote and abstract subject such as mathematics had little to contribute to this mission.

  The goal of reversing the progress of the Reformation, however, did not mean that the Jesuit colleges focused their curriculum exclusively on religious teachings. Ignatius firmly believed that proper religious instruction must be grounded in broader teachings in philosophy, grammar, classical languages, and other humanistic fields, and it was also essential that the colleges live up to their promise of providing broad and up-to-date education. Otherwise, the local elites would turn elsewhere for the education of their sons, which would spell disaster for the order’s spiritual mission. As Jerónimo Nadal put it in 1567, “For us lessons and scholarly exercises are a sort of hook with which we fish for souls.”

  The “hook” that Ignatius recommended included the languages that might be required in order to read the ancient masters: Latin, Greek, and Hebrew, but also, in some colleges, Chaldee, Arabic, and Hindi. In philosophy, he ruled that the colleges would follow the teachings of the ancient Greek philosopher Aristotle, by far the most influential philosopher in the West ever since his writings had been translated into Latin in the twelfth century. His corpus of writings, covering fields as diverse as logic, biology, ethics, politics, physics, and astronomy, was the most comprehensive then known, and was accepted as authoritative by the majority of European scholars. It was therefore easy for Ignatius, who had studied Aristotle at the universities, to rely on him in setting the curriculum for the Society’s colleges. In theology, Ignatius decreed, the Society would follow St. Thomas Aquinas, the thirteenth-century Dominican who reconciled the teachings of Aristotle and the Church. “The Angelic Doctor,” as Aquinas was known, became after his death the most authoritative theologian in the West, and Ignatius regarded him as well-nigh infallible. Since Thomism (as Aquinas’s theology was known) relied heavily on Aristotelian philosophy, it was essential that students in the Jesuit colleges be immersed in Aristotle before they fully engaged in religious study.

  But if the curriculum of the Jesuit colleges was diverse and wide-ranging, it was also rigorous, clearly ordered, and hierarchical. The relative value of the different disciplines was never in doubt: At the top was theology, comprised of the infallible teachings of the Catholic Church. Below it was philosophy, both moral and natural, which taught truths about the natural and human world and might be required in order to understand religious teachings. And below philosophy were the ancillary fields such as languages and mathematics, which did not deal with truth themselves but could prove useful in understanding the higher disciplines. Here, as elsewhere in the Jesuit world, order prevailed. Each field had its place in the grand scheme of the disciplines. The truths of theology were the highest, and no philosophical doctrine, even if supported by the authority of Aristotle himself, could ever contradict a theological truth. Mathematical sciences ranked lower still, and their results did not even qualify as truth, but only as hypotheses. It was a seamless hierarchy of knowledge in which Thomist theology reigned supreme.

  The clear order of the disciplines at the Jesuit colleges contrasted well with the offerings at the universities of the time, where studies were often haphazard and students typically attended unrelated lectures. Many students lost their way in this unstructured maze. The Jesuits, in contrast, offered a clear sequence of learning, beginning with languages and the many branches of Aristotelian philosophy, then moving on to theology. Along with the regulated and orderly life of the colleges and the upstanding moral example of the Jesuit instructors, this rigid progression kept the students on track and away from the temptations that afflicted their peers.

  But the hierarchy of truth was, for the Jesuits, more than a pedagogical device. It reflected their unyielding faith that a clear and undisputed hierarchy was essential for reconstituting the godly order lost in the Reformation. It governed society itself, and it governed the Church, from the Pope to the lay congregation. Hierarchy, the Jesuits believed, must prevail in the world if heresy were to be defeated and if truth were to triumph over error. After all, was not the scourge of the Reformation itsel
f the result of a breakdown in the proper order of knowledge? Did not Luther, a mere monk, dare to challenge the authority of the Pope himself? Did not Luther and, later, Zwingli, Calvin, and others posit their own novel theologies in opposition to the authoritative teachings of the Church? And what was the result? Chaos and confusion, in which the single authoritative voice of the Roman Church was drowned by a cacophony of competing voices. It seemed obvious to the Jesuits that the collapse of the ancient unity of Christendom, and the chaos that followed, were the direct result of the collapse in the proper order of knowledge. Only by preserving this strict hierarchy of knowledge would truth prevail and heresy be defeated.

  Since truth, for the Jesuits, was unchanging, and eternal, and founded on the authority of the Church, then novelty and innovation posed an unacceptable risk, and must be fervently resisted. “One should not be drawn to new opinions, that is, those that one has discovered,” warned theologian Benito Pereira of the Collegio Romano in 1564. Instead, one must “adhere to the old and generally accepted opinions … and follow the true and sound doctrine.” Two decades later, General Acquaviva exhorted his cohorts to avoid not only innovation, but also having “anyone suspect us of trying to create something new.” Innovation, so prized today, was regarded with deep suspicion by the Jesuits.