Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (50 page)

But the story does not end there, for the implications of the struggle over the infinitely small ranged far beyond the mathematical world, or even that of science and technology. The fight was over the face of modernity—and indeed, in the two lands in which the struggle raged fiercest, modernity took starkly different paths. Italy became very much what the Jesuits had made it. Profoundly Catholic, it was imbued with the eternal, unchanging truths of Catholic dogma and dominated by the absolute spiritual authority of Pope and Church hierarchy. This spiritual order supported a secular order that shared many of the same characteristics. While papal supremacy did not allow for the development of a strong state, the petty princelings who ruled Italy as tyrannical autocrats owed their positions to ancient dynastic rights and claimed absolute authority over their domains. Religious dissent was unthinkable, political opposition fiercely repressed, intellectual innovation frowned upon, and social mobility nearly nonexistent. As nations to the north became hotbeds of intellectual debate, technological innovation, political experimentation, and economic progress, Italy remained frozen in time. The land that for centuries had led all Europe in the arts and sciences, whose famous city-states were at one point wealthier and more prosperous than many a great kingdom, became stagnant, backward, and poor.

In the same years that Italy was falling behind, England was becoming the most dynamic, forward-looking, and fastest-growing land in Europe. Long considered a wild and semibarbarous country at the northern edge of European civilization, it became a leading center of European culture and science and a model for political pluralism and economic success. Here was a vision of modernity opposite in every way to the one that dominated Italy: Instead of dogmatic uniformity, England exhibited a marked and ever-increasing openness to dissent and pluralism. Politically, religiously, and economically, England became a land of many voices, where rival views and interests competed openly, relatively free from state repression. And it was in this relative freedom that England discovered its path to wealth and power.

The efforts of the Stuart kings to establish an absolute monarchy were defeated by fierce resistance from Parliament, first during the civil war and finally in the Glorious Revolution of 1688. When the last of the Stuarts, James II, was driven into exile and replaced by the constitutionally minded William of Orange, Parliament became the supreme governing body of the state. To be sure, the English Parliament of the seventeenth and eighteenth centuries was very far from the democratic institution we know today. It was a conservative body that represented the landed and propertied classes, and feared social unrest by the landless far more than it feared royal domination. Nevertheless, it was a deliberative body that allowed for an unprecedented degree of dissent, debate, and the free expression of ideas. Over time, the openness inherent in a parliamentary system triumphed over the class and social loyalties of its members. Throughout the eighteenth and nineteenth centuries the franchise was slowly but irreversibly expanded, and Parliament’s membership expanded to include ever broader segments of the population. The process was not completed until 1928, when all women were given the right to vote.

Political pluralism in England was matched by an unprecedented degree of religious tolerance. The Puritans of the early seventeenth century were not a tolerant lot. They viewed themselves as God’s elect, and their zeal to impose their beliefs and morals on the broader population played no small part in launching the crisis of 1640 and the civil war that followed. But the tragedy of the civil war, broadly attributed to the clash of competing dogmas, gave rise to a more forgiving attitude toward religious truth. Many of the Anglican bishops in the aftermath of the Interregnum, and nearly all the leaders of the newly founded Royal Society, advocated a latitudinarian, rather than dogmatic, approach to religion. Instead of insisting on a strict religious doctrine and excluding anyone who did not fully subscribe to it, they advocated broad latitude in doctrinal matters, acknowledging that ultimate truth was something to search for, not a given. A range of different beliefs was welcomed within the Anglican fold, as long as those beliefs agreed on certain fundamental tenets, such as the Trinity and the supremacy of the king (rather than the Pope).

First advocated for the Church of England itself, religious pluralism was soon extended beyond the Church’s confines. The 1689 Act of Toleration, coming fast on the heels of the Glorious Revolution, granted freedom from persecution to nonconforming Protestant sects such as Presbyterians, Quakers, and Unitarians. Though restricted (until 1828) from many spheres of public life, and from the universities of Oxford and Cambridge, the dissenters were nevertheless secure from state interference and prospered both economically and intellectually. They formed their own churches and their own academies, which were often far more advanced in their teachings than the plodding Anglican universities. Catholicism was not as easily tolerated, however. Both detested and feared, it was associated with the danger of foreign intervention and papal claims of supremacy, and with the claims of the deposed Stuarts. But even though they suffered systematic discrimination, English Catholics were largely left in peace until their official emancipation in 1829.

Political and religious pluralism went hand in hand with scientific, intellectual, and economic openness. The Royal Society, along with the French Academy of Sciences, soon became the leading scientific academy in Europe, and English science set the standard for all Europe. In the realm of letters, England became a locus of public philosophical and political debates in which luminaries such as John Locke, Jonathan Swift, and Edmund Burke took opposing, brilliantly argued, sides. Political liberalization also made possible economic liberalization and private entrepreneurship on an unprecedented scale. The accumulation of capital and the growing size of workshops made it profitable to invest in new technologies, particularly the steam engine. As a result, by the late eighteenth century, England became the first industrialized country in the world, pulling far ahead of its continental rivals and leaving them scrambling to keep up.

Whether the continuum is made up of infinitesimals seems like the quaintest of questions, and it is hard for us to fathom the passions it unleashed. But when the struggle raged in the seventeenth century, the combatants on both sides believed that the answer could shape every facet of life in the modern world that was then coming into being. They were right: when the dust cleared, the champions of infinitesimals had won, their enemies defeated. And the world was never the same again.

DRAMATIS PERSONAE

T
HE “
I
NFINITESIMALISTS”

Luca Valerio (1553–1618):
A mathematician and friend of Galileo who made important contributions to infinitesimal methods. However, when Galileo clashed with the Jesuits in 1616 Valerio sided against him, and was fiercely denounced by his former friends. He died in disgrace shortly thereafter.

Galileo Galilei (1564–1642):
The most celebrated scientist of his day. He was persecuted by the Jesuits for his advocacy of Copernicanism, which led to his trial and downfall. Galileo made use of infinitesimals in his work, and supported and encouraged a generation of young mathematicians to develop the concept. Even after his condemnation, he was still the undisputed leader of the Italian infinitesimalists.

Gregory St. Vincent (1584–1667):
A Jesuit mathematician who developed a novel method for calculating the volumes of geometrical figures that involved infinite division. His Jesuit superiors considered the method too close to infinitesimals, and forbade him from publishing his work.

Bonaventura Cavalieri (1598–1647):
Galileo’s disciple, later professor of mathematics at the University of Bologna, and a member of the order of the Jesuats. His books
Geometria indivisibilibus
(1635) and
Exercitationes geometricae sex
(1647) became the standard works on the new mathematics, which he called “the method of indivisibles.”

Evangelista Torricelli (1608–47):
Galileo’s disciple and ultimately his successor in Florence. An ardent infinitesimalist, and far less concerned with technical rigor than Cavalieri, he was famous for his powerful and creative techniques, which involved calculating the “width” and “thickness” of infinitesimals. His
Opera geometrica
of 1644 was widely read by mathematicians across Europe, and Wallis in particular modeled his
Arithmetica infinitorum
on Torricelli’s work. Among Torricelli’s most surprising results was his success in calculating the volume of a solid of infinite length.

John Wallis (1616–1703):
An ardent Parliamentarian and Puritan divine in the early years of the Interregnum, Wallis served as secretary to the Westminster Assembly of Divines. From the mid-1640s he was a regular participant in the private meetings that would later lead to the establishment of the Royal Society of London, and in 1649 he was appointed Savilian Professor of Geometry at the University of Oxford. Wallis made his name in mathematics as a leading infinitesimalist, and in politics as a relative pragmatist and moderate, in line with his fellows at the Royal Society. He was engaged in a decades-long war with Hobbes over his mathematics and his authoritarian politics.

Stefano degli Angeli (1623–97):
A friend and disciple of Cavalieri, professor of mathematics at the University of Padua, and a member of the order of the Jesuats. In the 1650s and ’60s he was the last public voice in Italy defending infinitesimals and openly denouncing the Jesuits. But when the Jesuats were unceremoniously dissolved by the Pope in 1668, Angeli finally desisted, and never published on infinitesimals again.

T
HE “
A
NTI-
I
NFINITESIMALISTS”

Christopher
Clavius (1538–1612):
Professor of mathematics at the Jesuit Collegio Romano and founder of the Jesuit mathematical tradition. Clavius insisted on a geometrical approach, which he valued for its orderliness, rigorous deductive method, and absolutely true results. He hoped to apply this methodology to all fields of knowledge, and was not interested in mathematical innovation. Clavius did not address infinitesimals directly, since they were hardly used by mathematicians during most of his career, but he was the author of the core principles of Jesuit mathematics, which led directly to the war on infinitesimals.

Paul Guldin (1577–1643):
Leading Jesuit mathematician, charged with discrediting infinitesimals. He attacked Cavalieri’s method in his
De centro gravitatis
of 1641.

Mario Bettini (1584–1657):
Mathematician who became the leading Jesuit critic of the infinitely small after Guldin’s death. He ridiculed infinitesimals in his collections of mathematical curiosities, the
Apiaria universae philosophiae
of 1642 and the
Aerarium philosophiae mathematicae
of 1648.

Thomas Hobbes (1588–1679):
The author of
Leviathan
and advocate of an absolutist authoritarian state, Hobbes considered himself a mathematician as well. He believed that his philosophy was founded on mathematical principles, and that it was therefore as certain as a geometrical demonstration. The decrees of the Leviathan, he believed, will be as incontestable as geometrical proofs.

André Tacquet (1612–60):
Leading Jesuit mathematician, also charged with discrediting infinitesimals. He denounced infinitesimal mathematics in
Cylindricorum et annularium
, published in 1651, but accepted their limited use as heuristic devices. He was subsequently directed by his superiors to refrain from publishing original work, and to focus exclusively on writing textbooks. He did so.

J
ESUITS

Ignatius of Loyola (1491–1556):
A Spanish nobleman and soldier from the Basque region, who experienced a religious awakening after being wounded in the Battle of Pamplona in 1521. With ten devoted followers he founded the Society of Jesus, which was officially recognized by Pope Paul III in 1540. Under his leadership the Jesuits became the most dynamic order of the Church and the most effective in battling the Reformation. By the time of Ignatius’s death, the Society had grown to one thousand members and several dozen schools and colleges, and was still expanding rapidly.

Benito Pereira (1536–1610):
Clavius’s nemesis at the Collegio Romano, who insisted that mathematics did not qualify as a science. He was also the first Jesuit to directly condemn infinitesimals, although the context was not mathematics, but rather a commentary on Aristotle.

Claudio Acquaviva (1543–1615):
Superior general of the Jesuits from 1581 to 1615, Acquaviva established the office of the Revisors General, and supported the early campaign against infinitesimals.

Mutio Vitelleschi (1563–1645):
Superior general of the Jesuits from 1615 to 1645, during their period of eclipse (1623–31) and their return to power in Rome. He presided over the launch of the final campaign against infinitesimals, and wrote to the provinces to forbid the doctrine.

Jacob Bidermann (1578–1639):
Leader of the Revisors General in 1632, when the Jesuits renewed their assault on infinitesimals.

Vincenzo Carafa (1585–1649):
Superior general of the Jesuits from 1646 to 1649. Enforced the ban on infinitesimals, and humiliated Pallavicino by compelling him to retract his views. He wrote to his underlings to remain vigilant against infinitesimals, and began the process to add infinitesimals to a list of permanently banned doctrines.

Rodrigo de Arriaga (1592–1667):
Leading Jesuit philosopher. In 1632 he published his
Cursus philosophicus
, which surprisingly concludes that infinitesimals are plausible. By the time the book was published, however, the Jesuits were back in power in Rome and determined to quash the support for infinitesimals. Superior General Carafa declared that there would be no more Arriagas.