Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (24 page)

25. The succession continuum and the intensity of qualities are composed of sole indivisibles.

26. Inflatable points are given, from which the continuum is composed.

30. Infinity in multitude and magnitude can be enclosed between two unities or two points.

31. Tiny vacuums are interspersed in the continuum, few or many, large or small, depending on its rarity or density.

Thesis 25 is the broadest of the four, referring to any possible continuum and its composition. The question of the “intensity of qualities” refers to medieval debates in which the intensity of qualities such as “hot” or “cold” existed along a gradient, raising the question of whether there was a finite or infinite number of grades of these qualities forming the continuum. Thesis 26 was a response to widespread speculation in the seventeenth century about what caused a material’s change in density, a question that was considered one of the toughest challenges to an atomic theory of matter. The thesis claimed that matter was composed of innumerable inflatable points, whose size at any given moment determined the degree of density or rarity, but to the Jesuits this was no more acceptable than the simpler doctrine that the continuum was composed of indivisibles. Thesis 30 is the most explicitly mathematical of the four, referring directly to the method of indivisibles practiced by Cavalieri and Torricelli, which relied on the division of finite lines or figures into an infinite number of indivisible parts. Thesis 31 appears to address Galileo’s theory of the continuum as expounded in the
Discourses
of 1638. Relying on analogies with matter and the paradox of Aristotle’s wheel, Galileo concluded that the continuum is interspersed with an infinity of tiny vacuums. Between them, these four propositions encompass the different variations of the method of indivisibles under dispute in the middle of the seventeenth century. All were unequivocally banned.

The
Ordinatio
of 1651 was a turning point in the Jesuit battle against the infinitely small. The prohibition on the doctrine was now permanent, and was backed by the highest authority in the order, the General Congregation. Printed, published, and widely disseminated, the
Ordinatio
was brought to the attention of every Jesuit father teaching at every one of the Society’s institutions across the world. The repeated blasts by the Revisors, who every few years issued their censures of the doctrine, were now at an end. No further condemnations were needed: the prohibition was now permanent and compulsory, and every member of the order knew it well. And so it stood for the next century, as the
Ordinatio
remained the fundamental guide to Jesuit teachings. In fact, the document did much more: it set the tone for intellectual life in the lands where the Jesuits dominated. For the few and lonely mathematicians still defending infinitesimals in Italy, the consequences were devastating.

5

The Battle of the Mathematicians

GULDIN VERSUS CAVALIERI

“The reasoning by indivisibles convinces all the famous geometers brought up here,” wrote Stefano degli Angeli (1623–97), professor of mathematics at the University of Padua, in 1659. Angeli, as was his way, expressed himself with great bravado, but the truth was very different. By the time he was writing, Angeli was likely the last mathematician left in Italy to adopt the method of indivisibles, and one of the even fewer to publish their work in the field. Most of the individuals named in his own list of the “famous” adherents of the method resided north of the Alps, many in France and England. The Italians on the list belonged to an old and dying generation of Galileans, who had stopped publishing on the method decades before. When he wrote those words, Angeli was not in fact reporting on a favorable state of affairs, but rallying the troops for a desperate rearguard action on behalf of infinitesimals, which were in danger of being extinguished in the land where they had once flourished. Who his enemies were, he had no doubt: “the three Jesuits, Guldin, Bettini, and Tacquet” were, he claimed, the only ones who remained unconvinced by the method of indivisibles. “By what spirit they are moved,” he continued, with evident frustration, “I do not know.”

Paul Guldin, Mario Bettini (1584–1657), and André Tacquet were among the most notable mathematicians of the Society of Jesus in the mid-seventeenth century. Tacquet, whom we have already met, was the most original and creative of the three, but Guldin, too, was a widely respected mathematician. Bettini was perhaps less so, known mostly as a prolific author of rambling collections of mathematical results and curiosities, and less as a creative thinker in his own right. But he, too, was known in the Society and beyond as a man of broad learning and considerable authority on things mathematical. Together, Guldin, Bettini, and Tacquet were a formidable trio, exemplifying the intellectual prowess and the cultural and political cachet of the Jesuit mathematical school. And in the 1640s and ’50s, all three were engaged in the same mission: to discredit and undermine the method of indivisibles using sound and incontrovertible mathematical arguments. Theirs was a dimension of the Jesuit war on infinitesimals that was just as critical as the repeated condemnations of the Revisors and the
Ordinatio
of 1651. For if the use of infinitesimals in mathematics was to be permanently abolished, it was not enough to declare them philosophically, theologically, or even morally wrong, and legally banish them. It was also crucial to prove them mathematically wrong.

Guldin, the oldest of the three, was the first to take the field. Born Habakkuk Guldin to Protestant parents of Jewish descent in St. Gall, Switzerland, he may be the first in a long and illustrious line of Jewish (and converted Jewish) mathematicians that continues to this day. Guldin was not raised to be a scholar but an artisan, and he was working as a goldsmith when he began having doubts about his Protestant faith. At age twenty he converted to Catholicism and joined the Jesuits, changing his name in the process from the Old Testament prophet Habakkuk to Paul, the most famous Jewish convert, who preached the Christian faith to the Gentiles. Guldin’s was an eclectic and unusual background for a Jesuit, encompassing as it did many of the religious and ethnic fault lines of the early modern world, but this did not prevent his full acceptance in the Society. Indeed, it is one of the most admirable characteristics of the early Jesuits that despite pressure from the Iberian kingdoms, which placed the highest value on
limpieza de sangre
(purity of blood), the Society remained one of the most welcoming Catholic institutions for converts of all kinds.

The Society of Jesus was also, to a large degree, a meritocracy, and although high-born noblemen such as Marchese Pallavicino enjoyed enviable advantages, there was also a path forward for men of humble origins such as Guldin. As a bright young man with a talent for mathematics, he rose steadily through the ranks of the order, and was ultimately sent to the Collegio Romano to study with Clavius. Guldin spent fewer than three years under Clavius’s tutelage before the old master passed away in 1612. Five years later he was sent to teach mathematics in the Habsburg Austrian lands, and spent the rest of his life at the Jesuit college in Graz and at the University of Vienna. Nevertheless, it is clear from Guldin’s subsequent career that his years with Clavius shaped his mathematical outlook for a lifetime. Guldin was Clavius’s follower in every way: he adhered to the old Jesuit’s view that mathematics lies halfway between physics and metaphysics, he believed in the primacy of geometry among mathematical disciplines, and he insisted on following the classical Euclidean standards of deductive proof. All these positions made him an ideal choice as a critic of the method of indivisibles.

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”).