Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (28 page)

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 philosophize (“
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.

Taking place at the dawn of the modern age, the struggle over the infinitely small was a contest between opposite visions of what modernity would be. On the one side were the Jesuits, one of the first modern institutions the world had seen. With rational organization and unity of purpose, they were working to shape the early modern world in their image. Theirs was a totalitarian dream of seamless unity and purpose that left no room for doubt or debate, a vision that has appeared time and again in different guises throughout modern history. On the other side were their opponents, which in Italy were the friends and followers of Galileo. They believed that a new age of peace and harmony would be brought about not by the imposition of absolute truths, but through the slow, systematic, and imperfect accumulation of shared knowledge and shared truths. It was a vision that allowed for doubt and debate, freely acknowledging that some mysteries remained unsolved, but insisting that much could nevertheless be discovered through investigation. It opened the way for scientific progress, but also for political and religious pluralism and limited (as against totalitarian) government. This group, too, has had many different incarnations in the modern world, but their views are still recognizable in the ideals of liberal democracy.

In seventeenth-century Italy, the enemies of the infinitely small prevailed. The principles of hierarchy, authority, and the absolute unity of truth were affirmed, and the principles of freedom of investigation, pragmatism, and pluralism were defeated. The consequences for Italy were profound.

THE WELL-ORDERED LAND

For nearly two centuries, Italy had been home to perhaps the liveliest mathematical community in Europe. It was a tradition that stretched back to the counting houses of Italy’s commercial hubs, and later encompassed professional and university mathematicians, forebears of today’s academics. In the early sixteenth century, Cardano, Tartaglia, and their fellow “cossists” (as they were known) wagered money and possessions on their ability to solve cubic and quartic equations. Some decades later, classicists such as Federico Commandino and Guidobaldo del Monte idolized the ancients and produced new editions and translations of their work. And most recently, champions of the infinitely small (Galileo, Cavalieri, and Torricelli) pioneered new techniques that would transform the very foundations of mathematical inquiry and practice.

But when the Jesuits triumphed over the advocates of the infinitely small, this brilliant tradition died a quick death. With Angeli silenced, and Viviani and Ricci keeping their mathematical views to themselves, there was no mathematician left in Italy to carry the torch. The Jesuits, now in charge, insisted on adhering close to the methods of antiquity, so the leadership in mathematical innovation now shifted decisively, moving beyond the Alps, to Germany, France, England, and Switzerland. It was in those northern lands that Cavalieri’s and Torricelli’s “method of indivisibles” would be developed first into the “infinitesimal calculus” and then into the broad mathematical field known as “analysis.” Italy, where it all began, became a mathematical backwater, a land in which there was no future for those seeking to pursue a mathematical career. In the 1760s, when the young mathematical prodigy Giuseppe Luigi Lagrangia of Turin sought to make a name for himself among the “great geometers” of the day, he was obliged to leave his homeland and travel first to Berlin and then to Paris. He succeeded, but his Italian roots were soon forgotten. To future generations he was and remains a Frenchman: Joseph-Louis Lagrange, one of the greatest mathematicians in human history.

The extinction of the Italian mathematical tradition was the most immediate result of the suppression of the infinitely small, but the Jesuit triumph had far deeper and more wide-ranging effects. Going back to the High Middle Ages, Italy had led all Europe in innovation—political, economic, artistic, and scientific. As early as the eleventh and twelfth centuries, it was home to the first flourishing cities to emerge from the Dark Ages. These cities played a vital role in reviving the long-dormant commercial economy, and were also sites of lively political experimentation in different forms of government, from autocratic to republican. In the thirteenth century, Italian merchants became Europe’s first and wealthiest bankers, and beginning in the middle of the fourteenth century, Italy led the way in an artistic and cultural revival that transformed Europe. Humanists from Petrarch to Pico della Mirandola, painters from Giotto to Botticelli, sculptors from Donatello to Michelangelo, and architects from Brunelleschi to Bernini, made the Italian Renaissance a turning point in human history. In the sciences, Italians from Alberti to Leonardo to Galileo made crucial contributions to human knowledge and opened up new vistas of investigation. As a land of creativity and innovation, it is fair to say, Italy had no peer.

All this, however, came to an end around the close of the seventeenth century. The dynamic land of creativity and innovation became a land of stagnation and decay. The thriving commercial hubs of the Renaissance became marginal outposts in the European economy, unable to match the rapid expansion of their northern rivals. Religiously, the Italian peninsula came under the sway of a conservative Catholicism, in which no dissent from papal edicts was permitted and no other sect or belief was allowed a foothold. Politically, Italy was an amalgam of petty principalities ruled by kings, dukes, and archdukes, and by the Pope himself. With few exceptions, all were reactionary and oppressive, and all forcefully stifled any hint of political opposition. In the sciences, a few brilliant men, such as Spallanzani, Galvani, and Volta, worked at the forefront of their disciplines and were admired by colleagues throughout Europe. But these few exceptions only emphasized the overall impoverishment of Italian science, which in the eighteenth century was but an appendage to the flourishing Parisian science. By 1750 there was little trace of the bold spirit of innovation that had characterized Italian life for so long.

It would be an exaggeration to attribute all these developments to the defeat of the infinitely small in Italy in the late seventeenth century. There were many causes for Italian decline—political, economic, intellectual, and religious—but it is undeniable that the struggle over the infinitely small played an important role among them. It was a key site in which the path of Italian modernity was fought over and decided, and the victory of one side and defeat of the other helped shape Italy’s trajectory for centuries to come.

It did not have to be so: the struggle was a close one, and if the Galileans had won and the Jesuits lost, it is easy to imagine a quite different way forward for Italy. The land of Galileo would likely have remained at the forefront of mathematics and science, and may well have led the way in the scientific triumphs of the eighteenth and nineteenth centuries. Italy might have been a center of Enlightenment philosophy and culture, and the ideals of freedom and democracy could have resonated from the piazzas of Florence, Milan, and Rome, rather than the places of Paris and the squares of London. It is easy to imagine Italy’s petty dynasts giving way to more representative forms of government, and the great cities of Italy as thriving hubs of commerce and industry, fully the equal of their northern counterparts. But it was not to be: by the late seventeenth century the infinitely small had been suppressed. In Italy, the stage was set for centuries of backwardness and stagnation.