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.
Part II
Leviathan and the Infinitesimal
The calculus is to mathematics no more than what experiment is to physics.
—BERNARD LE BOVIER DE FONTENELLE
6
The Coming of Leviathan
DIGGERS
On Sunday, April 1, 1649, a group of poor men gathered with their families on St. Georges Hill, near the town of Kingston in Surrey, England. The hill was barren and seemed an unpromising locale for a new settlement. But the newcomers had come to stay: they had brought their belongings with them, and quickly set about building huts to shelter them from the elements. Then they began to dig. Day after day they continued digging, carving out trenches and planting vegetables on the rocky hill, while calling on others in the nearby towns to join them. “They invite all to come in and help them,�
� noted one observer, “and promise them meat, drink, and clothes.” They confidently predicted that “they will be four or five thousand within ten days,” and while this proved overly optimistic, the community did attract newcomers, their numbers soon reaching several dozen families. And yet they went on digging.
As the community slowly grew, suspicion of the “Diggers” in the surrounding towns and villages grew along with it. “It is feared they have some design in hand,” noted the same observer, and he was not mistaken. Digging trenches on a barren hill may seem like an innocent act to us, but things were different in seventeenth-century England. With their actions, the Diggers were asserting ownership and their right to cultivate enclosed lands that were owned or controlled by the local grandees. It was a calculated and open assault on the ownership rights of the propertied classes, and if their intentions were not sufficiently plain from their actions, the Diggers soon followed up with a pamphlet they distributed far and wide. “The work we are going about is this,” they explained: “To dig up Georges Hill and the waste Ground thereabouts … that we may work in righteousness, and lay the Foundation of making the Earth a Common Treasury for All, both Rich and Poor … not one Lording over another, but all looking upon each other, as equals in the Creation.”
Such a bold denial of the rights of private ownership would have been enough to send chills down a landowner’s spine, then and now. But there was more: “that this Civil Propriety is the Curse, is manifest thus, Those that Buy and Sell Land, and are landlords, have got it either by Oppression, or Murther, or Theft.” All private property was, according to this logic, stolen, and should by all rights be returned to its rightful owner: the people. True, the Diggers professed pacifism and made a point of disavowing the use of force to reclaim the land. But since several of their members were veterans of the English Civil War and its ravages, the “better sort” of people in Weyburn and surroundings were far from reassured. Having been labeled thieves and murderers, and their property rights denied, they were understandably alarmed. Fearing for their land and possessions, not to mention their lives and safety, they struck back.
As established members of society, they first turned to the authorities: Sir Thomas Fairfax, commander of the New Model Army, was stationed nearby, and the landowners appealed to him to remove the squatters. Fairfax was probably the most powerful man in England at the time, having led the Parliamentary forces to decisive victories over the Royalist armies of Charles I. A gentleman and a knight, Fairfax had little sympathy for the revolutionary demands of the Diggers, and the landowners expected him to take their side. Fairfax, however, equivocated: He arrived at St. Georges Hill with his troops and engaged in several discussions with the Diggers’ leader, Gerrard Winstanley. Beyond this, however, he took no action. If the landowners had an issue with Winstanley’s band, Fairfax informed them, they needed to take it up with the courts.
Though disappointed at Fairfax’s response, the landowners did precisely that—and more. They charged the Diggers with sexual licentiousness, and prevailed on the courts to bar them from speaking in their own defense. Meanwhile, Francis Drake, lord of the nearby manor of Cobham, organized raids on the Diggers’ settlement, ultimately succeeding in burning down one of their communal houses. Faced with a concerted legal and physical assault, the Diggers gave way. By August they had been forced to leave St. Georges Hill and move to a new location some miles away. When this new refuge also came under attack, they abandoned the land and largely dispersed. The landowners had won.
The drama at St. Georges Hill is one of the best documented attempts to subvert the established social order in early modern England. But it was not an isolated incident. Other digging colonies sprang up during the period, and other forms of protest, subversion, and even insurrection abounded. For, in the middle decades of the seventeenth century, from 1640 to 1660, England was a land in turmoil, and traditional institutions were in flux, if they had not disappeared altogether. Less than forty years after the death of the brilliant “Virgin Queen,” Elizabeth I (1558–1603), her successor once removed, Charles I, was driven from London by Parliament, his armies defeated in the field and he himself imprisoned and ultimately executed. The Church of England, created by Elizabeth and her father, Henry VIII (1509–47), was effectively dissolved, its bishops forced into exile and its great cathedrals seized by rival Protestant churches. A Scottish army had invaded and, for a time, occupied the northern counties; while in Ireland, a Catholic uprising had laid waste to the lands of English lords and settlers, massacring many of them and forcing others to flee. Amid this national crisis, with the state decapitated, the official Church suppressed, the law of the land ignored, and print censorship removed, a multitude of groups emerged from the shadows, dedicated to turning the old world upside down. The Diggers of St. Georges Hill were but one of them.
THE LAND WITHOUT A KING
The causes of what is variously known as the English Revolution, the English Civil War, or simply the Interregnum, are debated by historians to this day. Political, religious, social, and economic causes are all cited, and indeed there is no doubt that all contributed in some way to the collapse of the English government in 1640. This much, however, is clear: ever since 1603, when James I (1603–25) of the House of Stuart succeeded Elizabeth I on the English throne, England’s kings were increasingly at odds with Parliament, a body that represented large swaths of the English propertied classes. In part, this was a straightforward power struggle. Parliament, whose roots went back to the thirteenth century, had by the time of Elizabeth’s reign acquired the exclusive right to levy taxes. Since raising and maintaining an army and a navy were by far the most costly undertakings of an early modern state, and could be financed only through taxes, this meant that the king could not pursue a foreign policy without Parliament’s approval. Due to its control over state revenues, Parliament had the power to veto policies it did not like, and it did not hesitate to use it. As long as royal policies were acceptable to Parliament, there was little trouble. Such was the case with Elizabeth’s long, expensive, and inconclusive war with Spain, which nevertheless retained broad popular support. But when James I made peace with Spain, and when Charles I decided to aid King Louis XIII of France in crushing the Protestant Huguenots, things changed. Parliament refused to authorize taxes to fund what it viewed as “godless” and “tyrannical” actions, making it impossible for the English king to effectively carry out his policies.
The Stuart monarchs found this situation intolerable. Only the king, they insisted, had the power to set policy and levy taxes, and Parliament’s stranglehold over taxation was an illegal usurpation of royal power. They looked enviously at the French kings, who had humbled their own assembly, the Estates-General, and were successfully concentrating all power in their own hands. James I, perhaps the most scholarly of English kings, even penned a treatise entitled The Trew Law of Free Monarchies, in which he argued that kings ruled by divine right, and under no circumstances could the people legitimately resist royal decrees.
With Parliament growing ever more assertive and the Stuart kings ever more furious, a confrontation was inevitable: in 1629, Charles I dissolved Parliament and refused to call a new one. For the next eleven years he ruled alone, while the treasury was slowly drained and his freedom of action became increasingly restricted. Finally, in 1640, following a disastrous attempt to reform the Scottish church that led the two nations to the brink of war, Charles could hold out no longer and recalled Parliament. His intent was only to approve funds for the Scottish war and then quickly dissolve the unruly body. But Parliamentary leaders struck first: to prevent a repeat of Charles’s “tyrannical” rule, they immediately passed a resolution that Parliament stay in session until it dissolved itself. It remained formally convened for the decade, and is known to history as the Long Parliament.
The constitutional crisis of 1640 was a clash of two fundamentally opposing views of the proper political order. The Stuart kings struggled mightily to est
ablish an absolutist monarchy on the French model, in which all authority resided with the divinely sanctioned king. Parliament, meanwhile, stood for a constitutional monarchy (though the term had not yet been coined). Even the king, in its view, could not trample on the ancient rights of freeborn Englishmen. Royal power must be tempered and, when necessary, resisted by “the people,” as represented in Parliament. Needless to say, parliamentary leaders never dreamed of including the lower classes and the poor among “the people” of England. Only property owners were represented in Parliament, so only they had the right to share power with the king. Even so, the Parliamentary party stood for a vast expansion of the political class in England, which was precisely what the Royalists were determined to prevent.
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Page 21