On August 10, 1632, the Revisors General of the Society of Jesus, better known as the Jesuit Order, forcefully condemned the doctrine that the continuum is composed of infinitely small atoms and forbade it from being pursued, taught, or even entertained. In a blog that reads like a thriller, historian of mathematics Amir Alexander (UCLA) delves into the question what, in the eyes of the Jesuits, was wrong with the notion that continuous magnitudes are made of tiny atoms. At a time when the struggle over Copernicus’ theory raged most fiercely and Galileo’s fate hung in the balance, didn’t the Revisors General have greater concerns than whether a line is composed of separate points?
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 Biderman, but in their day these men were nearly as anonymous as they are today. Their high office was not: They were the “Revisors General” of the Society of Jesus, appointed by the General of the order 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 17th century by General Claudio Aquaviva, the Revisors arrived on the scene just in time to confront the intellectual turmoil that we know as the Scientific Revolution. From across the Catholic world, wherever there was a Jesuit school, mission, or residence, a steady stream of questions came flowing in to the Revisors General in Rome: Were 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 teachings 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, however, 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 its confines. The order’s pronouncements were widely viewed as authoritative, and few Catholic scholars would openly champion an idea condemned by the Revisors General. With the stroke of a pen Father Biderman and his colleagues could decide which ideas would thrive and be taught in the four corners of the world, and which would be consigned to oblivion.
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 later condemned, for advocating the motion of the Earth, Father Biderman and his colleagues concerned 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”:
“The permanent continuum can be constituted solely of physical indivisibles or atomic corpuscles having mathematical parts identified with them. Therefore the said corpuscles can be actually distinguished from each other.”[i]
Shorn of its technical clothing, the proposal is very simple: any continuous magnitude, it states, whether a line, a surface, or a length of time, is 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 seems commonsensical, and fits very well with our daily experience of the world: Aren’t all objects made up of smaller parts? Nevertheless, the judgment of the fathers 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. 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 21st century, one cannot help but be struck by the Jesuit fathers’ swift and unequivocal condemnation of the “the doctrine of indivisibles.” What, after all, is so wrong with the plausible notion that continuous magnitudes 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’ theory raged most fiercely, when the fate of Galileo hung in the balance, didn’t the Revisors General of the Society of Jesus have greater concerns than whether a line is composed of separate points?
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 reveal that the structure of the continuum was one of the most persistent of this body’s concerns. From 1606 to 1651 they issued a steady stream of decrees denouncing every conceivable variation of the doctrine.
What was it about the indivisibles that was so abhorrent to the Jesuit Revisors? The Jesuits after all were a religious order, whose purpose was saving souls, not resolving technical philosophical questions. To understand this we need to go back to the founding days of the order, at the time of the gravest crisis in the history of the Catholic Church. In the summer of 1540 the Protestant Reformation was sweeping across Europe like an irresistible tide. First provinces and cities, then entire kingdoms were severing their ties with Rome and embracing the teachings of Luther, Calvin, and a host of other reformers. The spiritual unity of medieval Europe was shattered, replaced by religious struggle, incessant war, and the ever-present threat of social revolution. The days of the Roman Church seemed numbered.
But on September 27, 1540, Pope Paul III approved a petition from a group of ten priests to form a religious company dedicated to serving the Pope and the Church. Though little noted at the time, it may have been the single most important step taken by the Papacy to save the Roman Church from dissolution. In his bull announcing the new order, Paul also approved the name requested by the group for their new association: They called it “The Society of Jesus.”
The Society of Jesus, or more commonly, the Jesuit Order, was the creation of the Spanish nobleman Ignatius of Loyola. It would be, he announced, the Pope’s own army, and so it proved. By the time of Ignatius’ death in 1560, the original 10 priests grew to 1000, and by 1615 to 13,000. Despite its rapid growth the Society never compromised on the quality of its trainees or the rigor of their training, and it became, in the assessment of Montaigne, a “nursery of great men.”[ii] Far more than an association of impressive individuals, they were a highly trained and disciplined collective, organized in a strict top-down hierarchy and honed into a powerful instrument in a single-minded quest: to spread the teachings of the Catholic Church, expand its reach, and bolster its authority. With skill, dedication, and an energetic enterprising spirit, they led a stunning Catholic resurgence that not only halted the spread of the Reformation but won back for the Pope many territories that had seemed lost forever. As a grateful Pope Gregory XIII acknowledged to the general congregation of the Society in 1581, “there is no single instrument raised up by God against heretics greater than your holy order.”[iii]
Mathematics, however, played little role in the early history of the Society, as the Jesuit fathers saw little connection between the abstract truths of the field and their mission to save the Roman Church. And so things would likely have remained were it not for the work of one man who made it his life’s mission to bring mathematics to the center of the Jesuit curriculum. Thanks to his efforts the Jesuits became not only skilled teachers of mathematics, but leading scholars of the field, numbering among their own some of the most prominent mathematicians in all of Europe. His name was Christopher Clavius, Professor of Geometry at the Collegio Romano.
As a champion of the unpopular mathematical sciences, Clavius was progressing slowly through the Jesuit ranks. But in 1575, when he was summoned to serve on Pope Gregory XIII’s calendar reform committee, Clavius played a key part in one of the greatest triumphs of the Catholic Church. By reforming the calendar the Pope was exercising his universal authority to correct a problem that had troubled all Christians for more than a millennium. In a display of near Godlike power, the Pope transformed the year, the religious festivals, and the seasons for millions across the globe. Even Protestants had no choice but to acknowledge the binding power of the Pope’s proclamation, and his unrivaled ability to reorder the universe. What a contrast between the bog of indecision in other areas of theological dispute and the glorious clear-cut victory afforded the Roman Church by the reform of the calendar! If only the secret of the calendrical triumph could be infused into those other fields, the ultimate victory of Pope and Church would be assured. Clavius believed that he knew what this secret was: mathematics, which alone among the fields of learning forces its truths upon the audience. One can dispute the Catholic doctrine of the sacraments, but one cannot deny the Pythagorean theorem. Mathematics, Clavius believed, held the key to the ultimate triumph of the Church.
Rigorous, orderly, and irresistible, mathematics was for Clavius the embodiment of the Jesuit program. By imposing truth and vanquishing error, it established a fixed order and certainty in place of chaos and confusion. But when Clavius was speaking of “mathematics,” he had something quite specific in mind: Euclidean Geometry. No other mathematical field, he believed, captured the power and Truth of the discipline in its most distilled form as did geometry.
Composed around 300 BCE, Euclid’s Elements was to Clavius the embodiment of the mathematical method. It begins with a series of definitions and postulates that are so simple as to be self-evidently true. According to one definition, “A figure is that which is contained by any boundary or boundaries”; according to one postulate, “all right angles are equal to one another”; and so on. From these seemingly trivial beginnings, Euclid moves step by step to demonstrate increasingly complex results: that the base angles of an isosceles triangle are equal; that in a right triangle, the sum of the squares of the two sides containing the right angle is equal to the square of the third side (the Pythagorean theorem); that in a circle, the angles in the same segment are all equal to one another; and so on. At each step, Euclid does not just argue that his result is plausible or likely, but demonstrates that it is absolutely true and cannot be otherwise. In this manner, layer by layer, Euclid constructs an edifice of mathematical truth, composed of interconnected and unshakably true propositions, each dependent on the ones that precede it.
Euclid’s method, Clavius concluded, had succeeded in doing precisely what the Jesuits were struggling so hard to accomplish: imposing a true, eternal, and unchallengeable order upon a seemingly chaotic reality. It was an expression of the highest Jesuit ideals and provided a clear roadmap for the Society as it struggled to build a new Catholic order. Consequently, Clavius argued, mathematics could no longer languish as an afterthought in the Jesuit empire of learning, but must become a core discipline and a key component in the formation of Jesuits. By the 1590s he had not only established mathematics as a core discipline in the Jesuit colleges, but had also become the leader of a group of young mathematicians who were not only competent teachers, but brilliant mathematicians in their own right. And at the heart of their practice was Euclidean geometry.
But in the very years that Clavius was establishing the powerful Jesuit school of mathematics, a very different mathematical practice was gaining ground that would put all of his cherished principles to the test. Where the Jesuits insisted on clear and simple postulates, the new mathematicians relied on a vague intuition of the inner structure of matter; whereas the Jesuits celebrated absolute certainty, they proposed a method rife with paradoxes; and whereas the Jesuits sought to avoid controversy at all cost, the new method was mired in intractable controversies seemingly from its very inception. It was everything that the Jesuits thought mathematics must never be, and yet it flourished, gaining new grounds and new adherents. It was known as The Method of Indivisibles.
For a taste of the new method, consider the following simple proof by Bonaventura Cavalieri, the man whose name became inextricably bound with indivisibles:
‘If in a parallelogram a diagonal is drawn, the parallelogram is double each of the triangles constituted by the diagonal.’
Bonaventura Cavalieri, Exercitationes geometricae sex (Bologna: Iacob Monti, 1647), p. 35, proposition 19
This means that if a diagonal FC is drawn for the parallelogram AFDC, the area of the parallelogram is double the area of each of the triangles FAC and CDF. If one approaches the proof in a traditional Euclidean manner, then it is almost trivial: it can easily be shown that the triangles FAC and CDF are congruent, and therefore equal in area. It follows that the area of the parallelogram is double that of each of them.
Cavalieri, of course, knew all this very well, but he proceeded differently. He assumed that the area of the triangle FAC is composed of innumerable parallel lines ranged side by side, and the same is true for the triangle CDF. He then showed that each of the parallels composing FAC, such as the line BM, is equal to one of the parallels composing CDF, such as EH, and vice versa. Consequently the sum of all the lines composing FAC is equal to the sum of all the lines composing CDF, and their areas are therefore equal. QED.
The point of Cavalieri’s proof is not to show that the theorem is true – which is obvious – but rather why it is true: the two triangles are equal because they are composed of the same number of identical indivisible lines, placed side by side. And it is precisely this materialist take on geometrical figures that distinguishes Cavalieri’s approach from the classical Euclidean one. The Euclidean approach can rightly be called “top-down,” since it orders geometrical objects, and ultimately the world, through its universal first principles and its irrefutable logical method. Cavalieri’s approach, in contrast, begins with a material intuition of the world as we find it, and then proceeds to broader and more abstract mathematical generalizations. It can rightly be called “bottom-up” mathematics. Furthermore, whereas Euclidean Geometry was hailed for its logical rigor and incontrovertible results, the Method of Indivisibles was anything but. The premise that the continuum is composed of distinct indivisibles ran counter to the paradoxes of Zeno, which had been known for two millennia, as well as the problem of incommensurability, first identified even earlier by the Pythagoreans. It was not difficult to show that the premise, though effective and powerful in the hands of Cavalieri and his followers, could also lead to absurd and contradictory results.
For all the radicalism of his approach, Cavalieri was by nature a cautious mathematician, who added cumbersome restrictions that made his method nearly unusable for fellow mathematicians. It consequently fell to Evangelista Torricelli, Galileo’s disciple and heir in the Medici court, to demonstrate the full potential of the method of indivisibles. One of his most influential works was entitled “De dimensione parabolae” and offered 21 different proofs of the area of a parabola —11 proofs using indivisibles, and 10 proofs by the traditional Euclidean method. The work was eagerly read by mathematicians north of the Alps, including Roberval, Fermat, Wallis, and Barrow, and set the tone for mathematical developments for decades to come. In their own land, however, Cavalieri and Torricelli would have no successors. For even in the 1640s, as they were publishing their most influential works on indivisibles, the tide in Italy was turning decisively against their brand of mathematics. The Society of Jesus, which had long viewed the method of indivisibles with suspicion, had swung into action. In a fierce decades-long campaign the Jesuits worked relentlessly to discredit the doctrine of the infinitely small and deprive its adherents of standing and voice in the mathematical community. Their efforts were not in vain.
To the Jesuits, the new “method of indivisibles” advocated by Cavalieri was what mathematics should never be. Whereas Euclidean geometry began with unassailable universal principles, the new approach began with an unreliable intuition of base matter. Where geometry proceeded step by irrevocable step from general principles to their particular manifestations in the world, the new methods of the infinitely small went the opposite way: they began with an intuition of what the physical world was like (i.e. composed of minuscule atoms), and proceeded to generalize from there. Most damaging of all, whereas Euclidean geometry was rigorous, pure, and unassailably true, the new methods were riddled with paradoxes and contradictions, and as likely to lead one to error as to truth. If Euclidean geometry was, for Clavius, a buttress for universal hierarchy and order, then the new mathematics was the exact opposite—a fount of subversion and disorder. To preserve mathematics and the ideals of truth and order that it represented and supported, the Jesuits believed that the method of indivisibles must be destroyed. And so, they proceeded to do just that.
The Jesuit campaign against the infinitely small proceeded on several fronts. The first was a judicial and disciplinary track, in which the Jesuits authorities worked to stamp out the doctrine in their own institutions and beyond. This duty fell primarily to the Revisors General, who were charged with policing what was taught in the Jesuit colleges, as well as to the superior generals who set and enforced the order’s policies, and punished transgressors. Between 1606 and 1651 the Revisors sent a steady stream of injunctions to the provinces, prohibiting all possible variations of the doctrine of the infinitely small. Those who disobeyed suffered the consequences.
Once the Revisors pronounced their verdict, notices of their decision were immediately sent out to every Jesuit institution around the world. The process was routine, and usually the responsibility of the scribes and secretaries, but when it came to indivisibles this was sometimes deemed insufficient. In more than one case, the order to cease and desist from teaching, holding, or even entertaining the doctrine came directly from the Superior General of the Jesuit Order. On one occasion Superior General Vincentio Caraffa was informed that Marchese Father Sforza-Pallavicino, Professor of Philosophy at the Collegio Romano, had taught his students that indivisibles could not be entirely ruled out. Caraffa personally reprimanded the Marquis, and forced him to publicly retract his words in front of his students. Finally, in 1651, the Society published an authoritative list of philosophical doctrines that were ruled erroneous and could under no circumstances be taught or disseminated. No less than four of them dealt directly with infinitesimals.
Alongside their judicial campaign against infinitesimals, the Jesuits conducted an equally fierce mathematical campaign, seeking to demolish the mathematical credibility of the method. While practically all Jesuit mathematicians expressed hostility to indivisibles, the job of attacking them directly fell to three of them in particular: Paul Guldin, Mario Bettini, and André Tacquet, who in the 1640s and 1650s published savage critiques of the new mathematics. It was the most distinguished among them, gentlemanly and mild-mannered Tacquet, who best summarized their goal: The notion that a quantity is composed of indivisibles, he wrote “makes war upon geometry to such an extent, that if it is not to destroy it, it must itself be destroyed.”[iv] Destroy it or be destroyed: For the Jesuits, when it came to indivisibles, there was no middle ground.
There is much we do not know about the Jesuits’ decades-long campaign against indivisibles: how many mathematicians who privately supported the method of indivisibles chose to keep quiet for fear of the Jesuit reprisals? How many were denied university appointments because of their suspected allegiance to the forbidden doctrine? How many aspiring mathematicians simply turned away from the infinitely small, fearing that their professional prospects would suffer if they supported it? This hidden facet of the Jesuit campaign, conducted through personal interaction, private correspondence, and institutional pressure, is very hard to pin down with any certainty. But we do know this: after the death of Cavalieri and Torricelli in 1647, the brilliant Italian mathematical tradition came to a sudden end. Their students corresponded with each other, and sometimes wrote long tracts on indivisibles which they hoped would someday see light. Some wrote admiring biographies of their mentors, and even styled themselves as latter-day Galileans. But with one notable and short-lived exception, none of them attained a university or courtly post, or published any work on indivisibles.[v] By the 1670’s Italy was a land cleansed of indivisibles. The Jesuits had won.
The Jesuits did not wage war on indivisibles out of pettiness or spite, or merely to humiliate their opponents. They fought because they believed that their most cherished principles, and ultimately the fate of Christendom, were at stake. Forged in the crucible of the Reformation, the Jesuits’ overriding mission was to reverse the catastrophe and ensure that it would never recur. In place of the chaotic pluralism of competing truths, they would establish a single unchanging and incontestable truth, flowing seamlessly though a single hierarchy, and accepted by all without hesitation or doubt. Since the time of Clavius mathematics, in the guise of Euclidean geometry, had served this purpose, providing both a tool and a model for establishing hierarchical, incontestable truth. When the new infinitesimal methods threatened this seamless mathematical hierarchy, the Jesuits struck back. They were defending not just their mathematical tradition, but the Christian world order they were toiling tirelessly to establish.
A note by the S&P editors: Want more? Read Amir Alexander’s Infinitesimal: How A Dangerous Mathematical Theory Shaped The Modern World (New York: Scientific American / Farrar, Straus, and Giroux, 2014).
Endnotes
[i] The Revisors’ decree is preserved as manuscript FG 657, p. 183, in ARSI (Archivum Romanum Societatis Iesu), the archive of the Society of Jesus in Rome.
[ii] On Montaigne’s comment following his visit to the Collegio Romano see William V. Bangert, A History of the Society of Jesus (St. Louis, MO: Institute of Jesuit Sources, 1972), p. 56.
[iii] Pope Gregory XIII’s address to the Jesuits can be found in Bangert, History of the Society of Jesus, p. 97.
[iv] André Tacquet, Cylindricorum at annularium libri IV (Antwerp: Iacobus Meurisius, 1651), p. 24.
[v] The exception was Stefano degli Angeli (1623-1697), who was a student of Cavalieri and fellow member of the religious order of the “Jesuati” (as distinct from the Jesuits), as well as professor of mathematics at the University of Padua from 1662 onwards. Between 1658 and 1667 Angeli published no less than 9 books defending indivisibles and denouncing the Jesuits. His activities came to an end in 1668 when the order of the Jesuati was dissolved by the Pope with no warning or explanation. Though he retained his position in Padua until his death 29 years later, Angeli never again published a single word about indivisibles.