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Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

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by Amir Alexander

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Pulsing with drama and excitement, Infinitesimal celebrates the spirit of discovery, innovation, and intellectual achievement-and it will forever change the way you look at a simple line.

On August 10, 1632, five men in flowing black robes convened in a somber Roman palazzo to pass judgment on a deceptively simple proposition: that a


Pulsing with drama and excitement, Infinitesimal celebrates the spirit of discovery, innovation, and intellectual achievement-and it will forever change the way you look at a simple line.

On August 10, 1632, five men in flowing black robes convened in a somber Roman palazzo to pass judgment on a deceptively simple proposition: that a continuous line is composed of distinct and infinitely tiny parts. With the stroke of a pen the Jesuit fathers banned the doctrine of infinitesimals, announcing that it could never be taught or even mentioned. The concept was deemed dangerous and subversive, a threat to the belief that the world was an orderly place, governed by a strict and unchanging set of rules. If infinitesimals were ever accepted, the Jesuits feared, the entire world would be plunged into chaos.

In Infinitesimal, the award-winning historian Amir Alexander exposes the deep-seated reasons behind the rulings of the Jesuits and shows how the doctrine persisted, becoming the foundation of calculus and much of modern mathematics and technology. Indeed, not everyone agreed with the Jesuits. Philosophers, scientists, and mathematicians across Europe embraced infinitesimals as the key to scientific progress, freedom of thought, and a more tolerant society. As Alexander reveals, it wasn't long before the two camps set off on a war that pitted Europe's forces of hierarchy and order against those of pluralism and change.

The story takes us from the bloody battlefields of Europe's religious wars and the English Civil War and into the lives of the greatest mathematicians and philosophers of the day, including Galileo and Isaac Newton, Cardinal Bellarmine and Thomas Hobbes, and Christopher Clavius and John Wallis. In Italy, the defeat of the infinitely small signaled an end to that land's reign as the cultural heart of Europe, and in England, the triumph of infinitesimals helped launch the island nation on a course that would make it the world's first modern state.

From the imperial cities of Germany to the green hills of Surrey, from the papal palace in Rome to the halls of the Royal Society of London, Alexander demonstrates how a disagreement over a mathematical concept became a contest over the heavens and the earth. The legitimacy of popes and kings, as well as our beliefs in human liberty and progressive science, were at stake-the soul of the modern world hinged on the infinitesimal.

Editorial Reviews

The New York Times Book Review - Jennifer Ouellette
Packed with vivid detail and founded on solid scholarship, this book is both a rich history and a gripping page turner.
Publishers Weekly
UCLA historian and mathematician Alexander (Geometrical Landscapes) gives readers insight into a real-world Da Vinci Code–like intrigue with this look at the history of a simple, yet pivotal, mathematical concept. According to classic geometry, a line is made of a string of points, or “indivisibles,” which cannot be broken down into anything smaller. But if that’s so, how many indivisibles are in a line, and how big are they? And what happens when you divide the line into smaller segments? It seemed that indivisibles weren’t really indivisible at all, a “deeply troubling” idea to the medieval Church and its adherents, who demanded a rigidly unchanging cosmos with no surprises. Churchmen and respected thinkers like Descartes railed against infinitesimals, while Galileo, Newton, and others insisted the concept defined the real world. The argument became an intellectual and philosophical battleground, in a Church already threatened by doctrinal schisms and social upheaval. Focusing on the Jesuits, beginning with the German Jesuit mathematician Christopher Clavius, Alexander explores this war of ideas in the context of a world seething with political and social unrest. This in-depth history offers a unique view into the mathematical idea that became the foundation of our open, modern world. Agent: the Garamond Agency. (Apr.)
From the Publisher

“You probably don't think of the development of calculus as ripe material for a political thriller, but Amir Alexander has given us just that in Infinitesimal.” —Jordan Ellenberg, The Wall Street Journal

“Packed with vivid detail and founded on solid scholarship, [Infinitesimal] is both a rich history and a gripping page turner.” —Jennifer Ouellette, The New York Times Book Review

“[A] finely detailed, dramatic story.” —John Allen Paulos, The New York Times

“Alexander pulls off the impressive feat of putting a subtle mathematical concept centre stage in a ripping historical narrative . . . this is a complex story told with skill and verve, and overall Alexander does an excellent job . . . There is much in this fascinating book.” —Times Higher Education

“A triumph.” —Nature

“Every page of this book displays Alexander's passionate love of the history of mathematics. He helps readers refigure problems from over the centuries with him, creating pleasurable excursions through Euclid, Archimedes, Galileo, Cavalieri, Torricelli, Hobbes, and Wallis while explaining how seemingly timeless and abstract problems were deeply rooted in different worldviews. Infinitesimal captures beautifully a world on the cusp of inventing calculus but not quite there, struggling with what might be lost in the process of rendering mathematics less certain and familiar.” —Paula E. Findlen, The Chronicle of Higher Education

“With a sure hand, Mr. Alexander links mathematical principles to seminal events in Western cultural history, and has produced a vibrant account of a disputatious era of human thought, propelled in no small part by the smallest part there is.” —Alan Hirshfeld, The Wall Street Journal

Infinitesimal is a gripping and thorough history of the ultimate triumph of [a] mathematical tool . . . If you are fascinated by numbers, Infinitesimal will inspire you to dig deeper into the implications of the philosophy of mathematics and of knowledge.” —New Scientist

“Brilliantly documented . . . Alexander shines . . . the story of the infinitesimals is fascinating.” —Owen Gingerich, The American Scholar

“Back in the 17th century, the unorthodox idea [of infinitesimals], which dared to suggest the universe was an imperfect place full of mathematical paradoxes, was considered dangerous and even heretical . . . Alexander puts readers in the middle of European intellectuals' public and widespread battles over the theory, filling the book's pages with both formulas and juicy character development.” —Bill Andrews, Discover

“In Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, Amir Alexander successfully weaves a gripping narrative of the historical struggle over the seemingly innocuous topic of infinitesimals. He does an excellent job exploring the links between the contrasting religious and political motivations that lead to acceptance or refusal of the mathematical theory, skillfully breathing life into a potentially dry subject. Infinitesimal will certainly leave its readers with a newfound appreciation for the simple line, occasion for such controversy in the emergence of modern Europe.” —Emilie Robert Wong, The Harvard Book Review

“Fluent and richly informative” —Jonathan Rée, Literary Review (UK)

“Alexander tells this story of intellectual strife with the high drama and thrilling tension it deserves, weaving a history of mathematics through the social and religious upheavals that marked much of the era . . .The author navigates even the most abstract mathematical concepts as deftly as he does the layered social history, and the result is a book about math that is actually fun to read. A fast-paced history of the singular idea that shaped a multitude of modern achievements.” —Kirkus (starred review)

“[Infinitesimal] gives readers insight into a real-world Da Vinci Code–like intrigue with this look at the history of a simple, yet pivotal, mathematical concept . . . Alexander explores [a] war of ideas in the context of a world seething with political and social unrest. This in-depth history offers a unique view into the mathematical idea that became the foundation of our open, modern world.” —Publishers Weekly

“A bracing reminder of the human drama behind mathematical formulas.” —Bryce Christensen, Booklist

“A gripping account of the power of a mathematical idea to change the world. Amir Alexander writes with elegance and verve about how passion, politics, and the pursuit of knowledge collided in the arena of mathematics to shape the face of modernity. A page-turner full of fascinating stories about remarkable individuals and ideas, Infinitesimal will help you understand the world at a deeper level.” —Edward Frenkel, Professor of Mathematics, University of California, Berkeley, and author of Love and Math

“In this fascinating book, Amir Alexander vividly re-creates a wonderfully strange chapter of scientific history, when fine-grained arguments about the foundations of mathematical analysis were literally matters of life and death, and fanatical Jesuits and English philosophers battled over the nature of geometry, with the fate of their societies hanging in the balance. You will never look at calculus the same way again.” —Jordan Ellenberg, Professor of Mathematics, University of Wisconsin–Madison, and author of How Not to Be Wrong

“You may find it hard to believe that illustrious mathematicians, philosophers, and religious thinkers would engage in a bitter dispute over infinitely small quantities. Yet this is precisely what happened in the seventeenth century. In Infinitesimal, Amir Alexander puts this fascinating battle in historical and intellectual context.” —Mario Livio, astrophysicist, Space Telescope Science Institute, and author of Brilliant Blunders

“With considerable wit and unusual energy, Amir Alexander charts the great debate about whether mathematics could be reduced to a rigorous pattern of logical and orderly deductions or whether, instead, it could be an open-ended and exciting endeavor to explore the world's mysteries. Infinitesimal shows why the lessons of mathematics count so much in the modern world.” —Simon Schaffer, Professor of the History of Science, University of Cambridge

“In Infinitesimal, Amir Alexander offers a new reading of the beginning of the modern period in which mathematics plays a starring role. He brings to life the protagonists of the battle over infinitesimals as if they were our contemporaries, while preserving historical authenticity. The result is a seamless synthesis of cultural history and storytelling in which mathematical concepts and personalities emerge in parallel. The history of mathematics has rarely been so readable.” —Michael Harris, Professor of Mathematics, Columbia University and Université Paris Diderot

“We thought we knew the whole story: Copernicus, Galileo, the sun in the center, the Church rushing to condemn. Now this remarkable book puts the deeply subversive doctrine of atomism and its accompanying mathematics at the heart of modern science.” —Margaret C. Jacob, Distinguished Professor of History, University of California, Los Angeles

Kirkus Reviews
★ 2014-02-16
In the mid-17th century, debate raged over a mathematical concept of the infinitely small—and nothing less than modernity as we know it was at stake. At its core, the public argument over the infinitesimal—the idea that a line is composed of an endless number of immeasurably small component parts—is rooted in the ideological scope of post-Reformation Europe. The church, struggling to maintain autonomy over an increasingly disparate populace, fought to bar the infinitesimal from mathematical doctrine due to its implication that nature itself is not orderly, logical and completely subject to deductive reasoning. At the same time, leading intellectuals like Thomas Hobbes and John Wallis insisted that embracing the idea of the infinite in mathematics would open up a remarkable new opportunity to experimentally explore the world around us. Alexander (History/UCLA; Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics, 2010, etc.) tells this story of intellectual strife with the high drama and thrilling tension it deserves, weaving a history of mathematics through the social and religious upheavals that marked much of the era. For the people of Europe, more than just academic success was on the line: The struggle for civil liberties and rebellion against the rigid doctrines of the establishment were entrenched in the conceptual war over the infinitesimal. The fact that progressive mathematics prevailed was unquestionably momentous, as the addition of the concept of the infinitesimal eventually led to calculus, physics and many of the technological advances that are the bedrock of modern science and society. The author navigates even the most abstract mathematical concepts as deftly as he does the layered social history, and the result is a book about math that is actually fun to read. A fast-paced history of the singular idea that shaped a multitude of modern achievements.

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How a Dangerous Mathematical Theory Shaped the Modern World

By Amir Alexander

Farrar, Straus and Giroux

Copyright © 2014 Amir Alexander
All rights reserved.
ISBN: 978-1-4299-5392-4


The Children of Ignatius


On August 10, 1632, five men in flowing black robes came together in a somber Roman palazzo on the left bank of the Tiber River. Their dress marked them as members of the Society of Jesus, the leading religious order of the day, as did their place of meeting—the Collegio Romano, headquarters of the Jesuits' far-flung empire of learning. The leader of the five was the elderly German father Jacob Bidermann, who had made a name for himself as the producer of elaborate theatrical performances on religious themes. The others are unknown to us, but their names—Rodriguez, Rosco, Alvarado, and (possibly) Fordinus—mark them as Spaniards and Italians, like many of the men who filled the ranks of the Society. In their day these men were nearly as anonymous as they are today, but their high office was not: they were the "Revisors General" of the Society of Jesus, appointed by the general of the order from among the faculty of the Collegio. Their mission: to pass judgment upon the latest scientific and philosophical ideas of the age.

The task was a challenging one. First appointed at the turn of the seventeenth century by General Claudio Acquaviva, the Revisors arrived on the scene just in time to confront the intellectual turmoil that we know as the scientific revolution. It had been over half a century since Nicolaus Copernicus published his treatise proclaiming the novel theory that the Earth revolved around the sun, and the debate on the structure of the heavens had raged ever since. Could it be possible that, contrary to our daily experience, common sense, and established opinion, the Earth was moving? Nor were things simpler in other fields, where new ideas seemed to be cropping up daily—on the structure of matter, on the nature of magnetism, on transforming base metals into gold, on the circulation of the blood. From across the Catholic world, wherever there was a Jesuit school, mission, or residence, a steady stream of questions came flowing to the Revisors General in Rome: Are these new ideas scientifically sound? Can they be squared with what we know of the world, and with the teachings of the great philosophers of antiquity? And most crucially, do they conflict with the sacred doctrines of the Catholic Church? The Revisors took in these questions, considered them in light of the accepted doctrines of the Church and the Society, and pronounced their judgment. Some ideas were found acceptable, but others were rejected, banned, and could no longer be held or taught by any member of the Jesuit order.

In fact, the impact of the Revisors' decisions was far greater. Given the Society's prestige as the intellectual leader of the Catholic world, the views held by Jesuits and the doctrines taught in the Society's institutions carried great weight far beyond the confines of the order. The pronouncements coming from the Society were widely viewed as authoritative, and few Catholic scholars would have dared champion an idea condemned by the Revisors General. As a result, Father Bidermann and his associates could effectively determine the ultimate fate of the novel proposals brought before them. With the stroke of a pen, they could decide which ideas would thrive and be taught in the four corners of the world and which would be consigned to oblivion, forgotten as if they had never been proposed. It was a heavy responsibility, requiring both great learning and sound judgment. Little wonder that only the most experienced and trusted teachers at the Collegio Romano were deemed worthy to serve as Revisors.

But the issue that was brought before the Revisors General that summer day in 1632 appeared far from the great questions that were shaking the intellectual foundations of Europe. While a few short miles away Galileo was being denounced (and would later be condemned) for advocating the motion of the Earth, Father Bidermann and his colleagues were concerning themselves with a technical, even petty question. They had been asked to pronounce on a doctrine, proposed by an unnamed "Professor of Philosophy," on the subject of "the composition of the continuum by indivisibles."

Like all the doctrinal proposals presented to the Revisors, the proposition was cast in the obscure philosophical language of the age. But at its core, it was very simple: any continuous magnitude, it stated, whether a line, a surface, or a length of time, was composed of distinct infinitely small atoms. If the doctrine is true, then what appears to us as a smooth line is in fact made up of a very large number of separate and absolutely indivisible points, ranged together side by side like beads on a string. Similarly, a surface is made up of indivisibly thin lines placed next to each other, a time period is made up of minuscule instants that follow each other in succession, and so on.

This simple notion is far from implausible. In fact, it seems commonsensical, and fits very well with our daily experience of the world: Aren't all objects made up of smaller parts? Is not a piece of wood made of fibers; a cloth, of threads; an hour, of minutes? In much the same way, we might expect that a line will be composed of points; a surface, of lines; and even time itself, of separate instants. Nevertheless, the judgment of the black-robed fathers who met at the Collegio Romano that day was swift and decisive: "We consider this proposition to be not only repugnant to the common doctrine of Aristotle, but that it is by itself improbable, and ... is disapproved and forbidden in our Society."

So ruled the holy fathers, and in the vast network of Jesuit colleges, their word became law: the doctrine that the continuum is composed of infinitely small atoms was ruled out, and could not be pursued or taught. With this, the holy fathers had every reason to believe, the matter was closed. The doctrine of the infinitely small was now forbidden to all Jesuits, and other intellectual centers would no doubt follow the order's example. Advocates of the banned doctrine would be excluded and marginalized, crushed by the authority and prestige of the Jesuits. Such had been the case with numerous other pronouncements coming out of the Collegio, and Father Bidermann and his colleagues had no reason to think that this time would be any different. As far as they were concerned, the question of the composition of the continuum had been settled.

Looking back from the vantage point of the twenty-first century, one cannot help but be struck, and perhaps a bit startled, by the Jesuit fathers' swift and unequivocal condemnation of "the doctrine of indivisibles." What, after all, is so wrong with the plausible notion that continuous magnitudes, like all smooth objects, are made of tiny atomic particles? And even supposing that the doctrine is in some way incorrect, why would the learned professors of the Collegio Romano go out of their way to condemn it? At a time when the struggle over Copernicus's theory raged most fiercely; when the fate of Galileo, Copernicus's ardent advocate and the most famous scientist in Europe, hung in the balance; when novel theories on the heaven and the earth seemed to pop up regularly, didn't the illustrious Revisors General of the Society of Jesus have greater concerns than whether a line was composed of separate points? To put it bluntly, didn't they have more important things to worry about?

Apparently not. For, strange as it might seem to us, the condemnation of indivisibles in 1632 was not an isolated incident in the chronicles of the Jesuit Revisors, but merely a single volley in an ongoing campaign. In fact, the records of the meetings of the Revisors, which are kept to this day in the Society's archives in the Vatican, reveal that the structure of the continuum was one of the main and most persistent of this body's concerns. The matter had first come up in 1606, just a few years after General Acquaviva created the office, when an early generation of Revisors was asked to weigh in on the question of whether "the continuum is composed of a finite number of indivisibles." The same question, with slight variations, was proposed again two years later, and then again in 1613 and 1615. Each and every time, the Revisors rejected the doctrine unequivocally, declaring it to be "false and erroneous in philosophy ... which all agree must not be taught."

Yet the problem would not go away. In an effort to keep abreast of the most recent developments in mathematics, teachers from all corners of the Jesuit educational system kept proposing different variations on the doctrine in the hope that one would be tolerated: Perhaps a division into an infinite number of atoms was allowable, even if a finite number was not? Maybe it was permitted to teach the doctrine not as truth but as an unlikely hypothesis? And if fixed indivisibles were banned, what about indivisibles that expanded and contracted as needed? The Revisors rejected all of these. In the summer of 1632, as we have seen, they once again ruled against indivisibles, and Father Bidermann's successors (including Father Rodriguez), when called to pass judgment on it in January 1641, again declared the doctrine "repugnant." In a sign that these decrees had no more lasting effect than their predecessors, the Revisors felt the need to denounce indivisibles again in 1643 and 1649. By 1651 they had had enough: determined to put an end to unauthorized opinions in their ranks, the leaders of the Society produced a permanent list of banned doctrines that could never be taught or advocated by members of the order. Among the forbidden teachings, featured repeatedly in various guises, was the doctrine of indivisibles.

What was it about the indivisibles that was so abhorrent to the Jesuit Revisors in the seventeenth century? The Jesuits, after all, were a religious order—the greatest one of the day—whose purpose was saving souls, not resolving abstract, technical philosophical questions. Why, then, would they bother to proclaim their opinion on so inconsequential a matter, pursue it and its advocates decade after decade, and with the sanction of the highest authorities of the order, make every effort to stamp it out? Clearly the Black Robes, as the Jesuits were popularly known, saw something in this apparently innocuous thesis that is completely invisible to the modern reader—something dangerous, perhaps even subversive, that could threaten an article of faith or core belief the Society held dear. To understand what this was, and why the largest and most powerful religious order in Europe took it upon itself to eradicate the doctrine of indivisibles, we need to go back a century, to the founding days of the order in the early sixteenth century. It was during that time that the seeds of the Jesuit "war on indivisibles" were sown.


In the year 1521 the young emperor Charles V convened a meeting of the estates of the Holy Roman Empire in the west German city of Worms. Only two years past his election to his high office, Charles was titular head of the Holy Roman Empire, commanding the allegiance of its princes and vast populace. In fact, he was both less and more than that: less because the so-called "empire" was in reality a patchwork of dozens of principalities and cities, each fiercely protective of its independence and as likely to oppose as to aid its imperial lord in time of need; and more because Charles was no ordinary prince; he was a Habsburg, a member of the greatest noble family the West has ever known, with possessions extending from the coast of Castile to the plains of Hungary. Consequently, Charles was not only the elected emperor of Germany, but also, by birthright, the king of Spain and the duke of portions of Austria, Italy, and the Low Countries. Moreover, in those very years, Castile was fast acquiring new territories in the Americas and the Far East, making Charles, in a phrase from the time, "the emperor in whose lands the Sun never sets." And though Francis I of France and Henry VIII of England might have bridled at the suggestion, to his contemporaries as well as to himself, Charles V was the leader of Western Christendom.

In the winter of 1521, however, it was his fractured German empire, not his vast overseas possessions, that were chiefly on the emperor's mind. It had been three and a half years since Martin Luther, an unknown Augustinian monk and professor of theology, nailed a copy of his Ninety-Five Theses to the door of the Castle Church in Wittenberg. The theses themselves were narrowly focused, confronting what Luther saw as an unconscionable abuse practiced by the Church: the sale of "indulgences," which were guarantees of divine grace, absolving the purchasers of their sins and sparing them the torments of purgatory. Luther was far from alone in denouncing the sale of indulgences, which was one among many Church practices that were routinely condemned as abuses by both clerics and laypeople. Nevertheless, Luther's open challenge to Church authorities struck a nerve with both scholars and the common people like nothing before it. Over the following months, with the aid of the newly invented printing press, the theses were disseminated all across the Holy Roman Empire, and were enthusiastically received nearly everywhere.

If this had been where things ended, then the affair would have been of no concern to Charles V. Like many in his day, Charles, too, was distressed by the more egregious practices of the Church, and he may even have felt some sympathy toward the audacious monk. But events soon acquired a momentum of their own. Alarmed by Luther's success, his Augustinian superiors called him to account at a meeting in Heidelberg, but by the time he left, he had converted many of them to his position. When he was then summoned to Rome, he sheltered under the protection of his prince, the elector Frederick the Wise of Saxony, who arranged for a hearing for him in Germany instead. In an effort to discredit this irksome critic, Church authorities sent the Dominican professor Johann Eck of Ingolstadt, a professional debater and theologian, to confront Luther. The two met for a public debate in 1519, in which Eck skillfully maneuvered his opponent into admitting to clear heresies: that divine grace is granted to believers through faith alone, not through the sacraments of the Church; that the Church is a purely human construct and holds no special power to mediate between men and God; and that its supreme head, the Pope, is fundamentally an impostor. Luther made no apologies for his beliefs; Eck denounced him as a heretic.

Unfortunately for Church leaders, this designation did nothing to slow down the zealous Luther. In 1520 he published three treatises that outlined his basic doctrines in deliberate defiance of established teachings. No longer a critic, he was now a rebel, openly calling for the overthrow of the Church hierarchy and institutions. His influence continued to spread, first in Wittenberg, then in Saxony, and soon clear across Germany and beyond. Everywhere, it seemed, Luther was acquiring followers in all classes and stations of life—men and women, nobles and peasants, country people and city dwellers—all of whom saw him as a leader of a religious awakening that would displace the ossified and corrupt Church of Rome. At long last growing alarmed at the fast-deteriorating situation, Pope Leo X excommunicated Luther, but by this time the drastic action had little effect. Luther's teachings were spreading like wildfire throughout German lands.


Excerpted from Infinitesimal by Amir Alexander. Copyright © 2014 Amir Alexander. Excerpted by permission of Farrar, Straus and Giroux.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Meet the Author

Amir Alexander teaches history at UCLA. He is the author of Geometrical Landscapes and Duel at Dawn. His work has been featured in Nature, The Guardian, and other publications. He lives in Los Angeles, California.
Amir Alexander teaches history at UCLA. He is the author of Geometrical Landscapes and Duel at Dawn. His work has been featured in Nature, The Guardian, and other publications. He lives in Los Angeles, California.

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Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World 5 out of 5 based on 0 ratings. 1 reviews.
David_A_Bassett More than 1 year ago
The best summer read this year. Prof Alexander illuminates a little explored segment of intellectual history in an engaging and approachable narrative. The conventional story of tension between religion and science is shown to be a multidimensional and nuanced difference in world-view and process.