Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  Guldin’s critique of Cavalieri’s indivisibles is contained in the fourth book of his De centro gravitatis (also called Centrobaryca), published in 1641. He first suggests that Cavalieri’s method is not in fact his own, but was derived from that of two other mathematicians: one was Johannes Kepler, who, though a Protestant, was Guldin’s friend in Prague; the other, the German mathematician Bartholomew Sover. The charge of plagiarism is almost certainly unmerited, and in any case, Guldin was not paying Kepler or Sover much of a compliment, as he soon launches into a harsh and penetrating critique of the method.

  Cavalieri’s proofs, Guldin argues, are not constructive proofs of the kind that classically trained mathematicians would accept. This is undoubtedly true: in the Euclidean approach, geometrical figures are constructed step by step, from the simple to the complex, with the aid of only a straightedge and a compass, for the construction of lines and circles, respectively. Every step in a proof must involve such a construction, followed by a deduction of the logical implications for the resulting figure. Cavalieri, however, proceeded the other way around: he began with ready-made geometrical figures such as parabolas, spirals, and so on, and then divided them up into an infinite number of parts. Such a procedure might be called “deconstruction” rather than “construction,” and its purpose is not to erect a coherent geometrical figure, but to decipher the inner structure of an existing one. Such a procedure, the classically trained Guldin is quick to point out, did not conform to the rigorous standards of a Euclidean demonstration, and should be rejected on those grounds alone.

  Guldin next goes after the foundation of Cavalieri’s method: the notion that a plane is composed of an infinitude of lines, or a solid of an infinitude of planes. The entire idea, Guldin argues, is nonsense: “In my opinion,” he writes, “no geometer will grant him that the surface is, and could in geometrical language be called ‘all the lines of such a figure.’ Never in fact can several lines, or all the lines, be called surfaces; for, the multitude of lines, however great that might be, cannot compose even the smallest surface.” In other words, since lines have no width, no number of them placed side by side would cover even the smallest plane. Cavalieri’s attempt to calculate the area of a plane from the dimensions of “all its lines” was therefore absurd. This then leads Guldin to his final point: Cavalieri’s method was based on establishing a ratio between “all the lines” of one figure and “all the lines” of another. But, Guldin insists, both sets of lines are infinite, and the ratio of one infinity to another is meaningless. No matter how many times one multiplied an infinite number of indivisibles, they would never exceed a different infinite set of indivisibles. In other words, Cavalieri’s supposed ratios between “all the lines” of one figure and those of another violated the axiom of Archimedes, and were therefore invalid.

  When taken as a whole, Guldin’s critique of Cavalieri’s method embodies the core principles of Jesuit mathematics. Clavius and his descendants in the Society all believed that mathematics must proceed systematically and deductively, from simple postulates to ever-more-complex theorems, describing universal relations between figures. Constructive proofs, moving step by logical step from lines and circles to complex constructions, are the embodiment of this ideal. Slowly but surely they build up a rigorous and hierarchical mathematical order which, as Clavius had shown, brings Euclidean geometry closer to the Jesuit ideal of certainty, hierarchy, and order than any other science. Guldin’s insistence on constructive proofs was consequently not a matter of pedantry or narrow-mindedness, as Cavalieri and his friends thought: it was an expression of the deeply held convictions of his order.

  The same was true of Guldin’s criticism of the division of planes and solids into “all the lines” and “all the planes.” Mathematics must be not only hierarchical and constructive, but also perfectly rational and free of contradiction. But Cavalieri’s indivisibles, as Guldin points out, were incoherent at their very core, since the notion that the continuum is composed of indivisibles simply does not stand the test of reason. “Things that do not exist, nor could they exist, cannot be compared,” he asserts with impeccable reasoning. It is therefore no wonder that they lead to paradoxes and contradiction, and ultimately to error. To the Jesuits, such mathematics was far worse than no mathematics at all. If this flawed system were accepted, mathematics could no longer be the basis of an eternal, rational order. The Jesuit dream of a strict universal hierarchy as unchallengeable as the truths of geometry would be doomed.

  In his writings, Guldin does not explain the deeper philosophical reasons for his rejection of indivisibles; nor do Bettini and Tacquet. At one point Guldin comes close to admitting that there are greater issues at stake than the strictly mathematical ones, writing cryptically that “I do not think that the method [of indivisibles] should be rejected for reasons that must be suppressed by never inopportune silence,” but he gives no explanation of what those “reasons that must be suppressed” could be. The three Jesuits’ reticence to disclose any nonmathematical motivations for their stances is, however, quite natural. As mathematicians, they had the job of attacking the indivisibles on strictly mathematical, not philosophical or religious, grounds. Their authority and credibility would only have suffered if they had announced that they were moved by theological or philosophical considerations.

  Those involved in the fight over indivisibles knew, of course, what was truly at stake. When Angeli wrote facetiously that he did not know “what spirit” moved the Jesuit mathematicians, and when Guldin hinted at “reasons that must be suppressed,” they were referring to the Jesuits’ ideological opposition to infinitesimals. Nevertheless, with very few exceptions, these broader considerations were never openly acknowledged in the mathematical debate. It remained a technical controversy between highly trained professionals on which procedures are allowable in mathematics and which are not. When Cavalieri first encountered Guldin’s criticism in 1642, he immediately began work on a detailed refutation. Initially he intended to respond in the form of a dialogue between friends, of the type favored by his mentor, Galileo. But when he showed a short draft to Giannantonio Rocca, a friend and fellow mathematician, Rocca counseled against it. In order to avoid Galileo’s fate, Rocca warned, it was safer to stay away from the inflammatory dialogue format, with its witticisms and one-upmanship that were likely to enrage powerful opponents. Much better, Rocca advised, to write a straightforward response to Guldin’s charges, focusing on strictly mathematical issues, and to refrain from Galilean provocations. What Rocca left unsaid was that Cavalieri, in all his writings, showed not a trace of Galileo’s genius as a writer, nor of his ability to present complex issues in a witty and entertaining manner. It is probably for the best that Cavalieri took his friend’s advice, sparing us a “dialogue” in his signature ponderous and near-indecipherable prose. Instead, Cavalieri’s response to Guldin is included as the third “exercise” of the Exercitationes, and is titled, plainly enough, “In Guldinum” (“On Guldin”).

  In his response, Cavalieri does not appear overly troubled by Guldin’s critique, quickly dismissing the charges of plagiarism before moving on to the mathematical issues. He denies that he posits that the continuum is composed of an infinite number of indivisible parts, arguing that his method does not depend on this assumption. If one believes that the continuum is composed of indivisibles, then, yes, “all the lines” together do indeed add up to a surface, and “all the planes” to a volume; but if one does not accept that the lines comprise a surface, then there is undoubtedly something there in addition to the lines that makes up the surface, and something in addition to the planes that makes up the volume. None of this, he argues, has any bearing on the method of indivisibles, which compares “all the lines” or “all the planes” of one figure with those of another, regardless of whether they actually compose the figure.

  Cavalieri’s argument here may be technically acceptable, but it is also disingenuous. Anyone reading the Geometria or the Exercitationes ca
n have no doubt that they are based on the fundamental intuition that the continuum is indeed composed of indivisibles. Cavalieri’s coyness notwithstanding, the only possible reason for comparing “all the lines” and “all the planes” of figures is the belief that they in some way make up the surface and volume of the figures. The very name that Cavalieri gave his approach, the method of indivisibles, says as much, and his famous metaphors of the cloth and the book make it crystal clear. Guldin is perfectly correct to hold Cavalieri to account for his views on the continuum, and the Jesuat’s defense seems like a rather thin excuse.

  Cavalieri’s response to Guldin’s insistence that “an infinite has no proportion or ratio to another infinite” is hardly more persuasive. He distinguishes between two types of infinity, claiming that “absolute infinity” indeed has no ratio to another “absolute infinity,” but “all the lines” and “all the planes” have not an absolute but a “relative infinity.” This type of infinity, he then argues, can and does have a ratio to another “relative infinity.” As before, Cavalieri seems to be defending his method on abstruse technical grounds, which may or may not be acceptable to fellow mathematicians. Either way, his argument bears no relation to the true reasoning or motivation behind the method of indivisibles.

  That motivation comes to light only in Cavalieri’s response to Guldin’s charge that he does not properly “construct” his figures. Here Cavalieri’s patience is at an end, and he lets his true colors show. Guldin claimed that every figure, angle, and line in a geometrical proof must be carefully constructed from first principles; Cavalieri flatly denies this. “For a proof to be true,” he writes, “it is not necessary to describe actually these analogous figures, but it is sufficient to assume that they have been described mentally … and in consequence nothing contradictory may be deduced if we assume that these figures have been already constructed.”

  Here, finally, is the true difference between Guldin and Cavalieri, between the Jesuits and the indivisiblists. For the Jesuits, the purpose of mathematics was to establish the world as a fixed and eternally unchanging place, in which order and hierarchy could never be challenged. That is why each item in the world must be carefully and rationally constructed, and why any hint of contradictions or paradoxes could never be allowed to stand. It was a “top-down” mathematics, whose purpose was to bring rationality and order to an otherwise chaotic world. For Cavalieri and his fellow indivisiblists, it was the exact reverse: mathematics began with a material intuition of the world—that plane figures were made up of lines and volumes of planes, just as a cloth was woven of thread and a book compiled of pages. One does not need to rationally construct such figures, because we all know they already exist in the world. All that is needed, as Cavalieri says, is to assume and imagine them, and then proceed to investigate their inner structure. Ultimately, he continues, “nothing contradictory can be deduced,” because the fact that the figures exist guarantees that they are internally consistent. If we encounter seeming paradoxes and contradictions, they are bound to be superficial, resulting from our limited understanding, and can either be explained away or used as a tool of investigation. But they should never stop us from investigating the inner structure of geometrical figures, and the hidden relationships between them.

  For classical mathematicians such as Guldin, the notion that you could base mathematics on a vague and paradoxical intuition of matter was absurd: “who will be the judge” of the truth of a geometrical construction, he mockingly asks Cavalieri, “the hand, the eye, or the intellect?” But the charge of practicing an irrational geometry of the hand or eye did little to dissuade Cavalieri, since his method was indeed based on such practical intuitions. To him, Guldin’s insistence that the method must be abandoned because of apparent contradictions was pointless pedantry, since everyone knew that the figures did exist, and it made no sense to argue that they shouldn’t. Such nitpicking, it seemed to Cavalieri, could have grave consequences: if Guldin prevailed, a powerful method would be lost, and mathematics would be betrayed.

  BETTINI’S STING

  By the time Cavalieri published his response to Guldin, the Jesuit had already been dead for three years, and he himself had only months to live. But the death of the two chief protagonists, along with the passing of Torricelli in 1647, did nothing to tamp down the debate. Mathematicians could come and go, but the Society’s determination to extinguish the infinitely small remained the same, and the role of chief critic of indivisibles was simply handed on to another Jesuit mathematician. Mario Bettini, who inherited the mantle from Guldin, did not claim to be a leading mathematical light; nor was he considered one by his contemporaries. His claim to fame was as the author of two very long and eclectic books of mathematical curiosities, which he called the Apiaria universae philosophiae mathematicae (“Beehive of Universal Mathematical Philosophy”), published in 1642, and the Aerarium philosophiae mathematicae (“Treasury of Mathematical Philosophy”), published in 1648. Both of these were fine exemplars of the Jesuit approach to mathematics, highlighting as they did the ways in which geometrical principles pervaded the world. They included mathematical discussions of the flight of projectiles, construction of fortifications, the art of navigation, and so on, all governed by the universal and unassailable principles of geometry. The theory of indivisibles was not a natural fit in these practically oriented and eclectic collections, but it was nevertheless the focus of book 5 of volume 3 of the Aerarium. This was, after all, one year after the publication of Cavalieri’s rebuttal of Guldin, and it was imperative that the Society respond and keep up the pressure on the champions of the infinitely small.

  It is very likely that Bettini and Cavalieri knew each other personally, and there is much to suggest that their relationship was far from friendly. In 1626, Cavalieri was appointed prior of the Jesuat house in Parma, where Bettini was professor at the university, and it is hard to imagine that the two mathematicians did not cross paths in this modest-size city. Cavalieri, it will be recalled, entertained hopes of being appointed to a mathematical chair at the University of Parma, but, as he complained to Galileo on August 7 of that year, it all came to naught. “As for the lectureship in mathematics,” he wrote, “were the Jesuit fathers not here I would have great hope, because of the great inclination of Monsignor Cardinal Aldobrandini to favor me … but as [the university] is under the rule of the Jesuit Fathers, I cannot hope any longer.” There can be little doubt that among the Jesuit Fathers who scuttled Cavalieri’s appointment was their own leading mathematician, Mario Bettini.

  Cavalieri did have a measure of revenge when, in 1629, he became professor of mathematics in nearby Bologna. The appointment of a Galilean to this prestigious chair in the most ancient university in Europe was a stinging blow to the Jesuits, but particularly to Bettini, who was himself a native of Bologna. In the following years, perhaps in response to Cavalieri’s appointment or perhaps out of concern about their limited influence in that city, the Jesuits made plans to transfer their entire faculty from Parma to a new Jesuit college in Bologna. The move was ultimately blocked by the city’s senate, which in 1641 passed an ordinance forbidding the teaching of university subjects by anyone not on the university rolls. It is easy to see Bettini and Cavalieri lining up on opposite sides of this fight: the Jesuit, eager to establish a beachhead for his order in his native city, pushing hard for the new college; the Jesuat, burned by his experience in Parma and grateful to the Bolognese Senate for allowing him to pursue his mathematical work in peace, doing everything in his power to fend off the Jesuit invasion.

  What Bettini lacked in mathematical sophistication he made up for in fervency. Guldin, and later Tacquet, kept the debate largely within the bounds of technical mathematics, but Bettini did not hesitate to use blunt language and warn darkly of dire consequences if his admonitions were not heeded. It is possible that the bitterness of his personal history with Cavalieri led him to go beyond his mandate of a sedate mathematical critique of indivisibles, but whateve
r his personal motivation, Bettini’s attitude was probably more in line with the true tenor of the Jesuit campaign against the infinitely small. He merely gave it voice.

  Mathematically, Bettini added nothing of substance to Guldin’s critique, but hammered away at one point alone: that “infinity to infinity has no proportion,” and it therefore made no sense to compare the infinite lines of one figure with the infinite lines of another. Since this procedure is at the heart of the method of indivisibles, Bettini insisted that it was imperative that students and novices be warned against this tempting but false approach. “In order to set forth the elements of geometry,” he writes, “I point out [these] hallucinations, so that novices will learn to distinguish (as in the proverb) ‘what separates the false coin from the true’ in geometrical philosophy.” Indivisibles, according to Bettini, were a dangerous fantasy that was best ignored if at all possible. Under the circumstances, however, “being pressed, I respond to the counterfeit philosophizing about geometrical figures by indivisibles. Far, far be it from me to wish to make my geometrical theorems useless, lacking demonstrations of truth. Which would be to compare … figures and philosophize about them by indivisibles.” To avoid undermining all demonstrations and subverting geometry itself, one must steer clear of the dangerous hallucination—the method of indivisibles.

  THE COURTLY FLEMING

  In 1651 André Tacquet, the urbane Fleming whose work was celebrated by Catholics and Protestants alike, published his Cylindricorum et annularium libri IV (“Four Books on Cylinders and Rings”), a work dedicated to the study of the geometrical features of these figures and their applications. Befitting a Jesuit publication, the frontispiece shows two angels, bathed in divine light, holding up a ring enclosing the book’s title. On the ground below them, a band of cherubs is busy putting the theory into practice. The implication is clear: divine mathematics, universal and perfectly rational, orders and arranges the physical world to the best possible effect. It is a fetching visual depiction of the Jesuit view of the role and nature of mathematics.