Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (19 page)

Figure 3.10. Torricelli’s calculation of the slope of an “infinite parabola.”

The significance of Torricelli’s procedure here extends beyond the ingenuity of the proof itself (which is considerable), and to the challenge it posed to the mathematical tradition. Since ancient times, mathematicians had shied away from paradoxes, treating them as insurmountable obstacles, and a sign that their calculations had reached a dead end. But Torricelli parted ways with this venerable tradition: instead of avoiding paradoxes, he sought them out and harnessed them to his cause. Galileo had speculated about the infinitesimal structure of the continuum, but qualified his remarks by admitting that the continuum was a great “mystery.” Cavalieri did his best to avoid paradoxes and to conform to traditional canons, even at the cost of making his method unwieldy. But Torricelli unapologetically used paradoxes to devise a precise and powerful mathematical tool. Instead of banishing the paradox of the continuum from the realm of mathematics, Torricelli placed it at the discipline’s heart.

Despite its clear logical perils, Torricelli’s method made a profound impression on contemporary mathematicians. Although constantly skirting the edges of error, it was also flexible and remarkably effective. In the hands of a skilled and imaginative mathematician, it was a powerful tool that could lead to new and even startling results. In the 1640s it spread quickly to France, where it was developed by the likes of Gilles Personne de Roberval and Pierre de Fermat (1601–65), who corresponded directly with Torricelli. The Minim father Marin Mersenne, who was the central node of the European “Republic of Letters,” also corresponded with Torricelli, and then spread the Italian’s method on to England, where Wallis and Barrow mistakenly attributed it to Cavalieri. Quickly disseminating across the Continent, Torricelli’s radical practice encapsulated the power and the promise, as well as the dangers, of the new infinitesimal mathematics.

Torricelli did not enjoy his newfound prominence for long. On October 5, 1647, he fell ill, and less than three weeks later, on October 25, he was dead at the age of thirty-nine. In an hour of lucidity shortly before his death, Torricelli instructed his executors to deliver his manuscripts to Cavalieri in Bologna, so that he could publish what he saw fit. But it was too late: on November 30, just over a month after Torricelli breathed his last, Cavalieri, too, was dead from the gout that had afflicted him for many years. Within a few short years Italian mathematics was deprived of its guiding light Galileo and of his two chief mathematical disciples. In the span of a few decades these three had transformed the face of mathematics, opening up new avenues of progress, and possibilities that were eagerly seized upon by mathematicians across Europe. A generation later their “method of indivisibles” would be transformed into Newton’s “method of fluxions” and Leibniz’s differential and integral calculus.

In their own land, however, Galileo, Cavalieri, and Torricelli would have no successors. For just as Italian mathematics was being deprived of the leadership of Galileo and his disciples, 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: as 1647 was drawing to a close, the brilliant tradition of Italian mathematics was coming to an end as well. It would be centuries before the land of Galileo, Cavalieri, and Torricelli was once again home to creative mathematicians of the highest rank.

4

“Destroy or Be Destroyed”: The War on the Infinitely Small

THE DANGERS OF THE INFINITELY SMALL

The Jesuit mathematician André Tacquet (1612–60) was, by the standards of his time, a man of the world. Although he may never have left his native Flanders, his network of correspondents spanned Europe’s religious divide, reaching to Italy and France, but also to Protestant Holland and England. Only months before his death he entertained the Dutch polymath Christiaan Huygens, who had traveled to Antwerp with the express purpose of meeting Tacquet, by then regarded as one of the brightest mathematical stars ever to come out of the Society of Jesus. The two spent only a few days together, but got along so well that the Jesuit was convinced that he had managed to lure Huygens to the Catholic faith. (He hadn’t.) But ultimately it was not Tacquet’s personal charm, but rather his mathematical excellence that transcended seventeenth-century prejudices. In England, Henry Oldenburg, secretary of the Royal Society of London and no friend of the Jesuits, spent so much time describing Tacquet’s
Opera mathematica
at the Society’s meeting in January 1669 that he felt compelled to apologize to the fellows for abusing their patience. But it was, he insisted, “one of the best books ever written on mathematics.”

Tacquet’s claim to mathematical fame rested chiefly on his 1651 book
Cylindricorum et annularium libri IV
(“Four Books on Cylinders and Rings”), in which he showed a complete mastery of the full mathematical arsenal available in his day. He calculated the areas and volumes of geometrical figures using both classical approaches and the new methods developed by his contemporaries and immediate predecessors. But when it came to indivisibles, the usually mild-mannered Jesuit turned blunt:

I cannot consider the method of proof by indivisibles as either legitimate or geometrical … many geometers agree that a line is generated by the movement of a point, a surface by a moving line, a solid by a surface. But it is one thing to say that a quantity is generated from the movement of an indivisible, a very different thing to say that it is
composed
of indivisibles. The truth of the first is altogether established; the other makes war upon geometry to such an extent, that if it is not to destroy it, it must itself be destroyed.

Destroy or be destroyed—such were the stakes when it came to infinitesimals, according to Tacquet. Strong words indeed, but to the Fleming’s contemporaries, they were not particularly surprising. Tacquet was, after all, a Jesuit, and the Jesuits were then engaged in a sustained and uncompromising campaign to accomplish precisely what Tacquet was advocating: to eliminate the doctrine that the continuum is composed of indivisibles from the face of the earth. Should indivisibles prevail, they feared, the casualty would be not just mathematics, but the ideal that animated the entire Jesuit enterprise.

When Jesuits spoke of mathematics, they meant Euclidean geometry. For, as Father Clavius had taught, Euclidean geometry was the embodiment of order. Its demonstrations begin with universal self-evident assumptions, and then proceed step by logical step to describe fixed and necessary relations between geometrical objects: the sum of the angles in a triangle is always equal to two right angles; the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of the long side; and so on. These relations are absolute, and cannot be denied by any rational being.

And so, beginning with Clavius and for the next two hundred years, geometry formed the core of Jesuit mathematical practice. Even in the eighteenth century, when the direction of higher mathematics turned decisively away from geometry and toward the newer fields of algebra and analysis, Jesuit mathematicians held firm to their geometrical practice. It was the unmistakable hallmark of the Jesuit mathematical school. If only theology and other fields of knowledge could replicate the certainty of Euclidean geometry, they believed, then surely all strife would be at an end. The Reformation and all the chaos and subversion that flowed from it would never have taken root in such a world.

This vision of eternal order was, to the Jesuits, the only reason mathematics should be studied at all. Indeed, as Clavius never tired of arguing to his skeptical colleagues, mathematics embodied the Society’s highest ideals, and thanks to his efforts the doors were opened at Jesuit institutions for the study and cultivation of the field. By the late sixteenth century, mathematics had become one of the most prestigious fields of study at the Collegio Romano and other Jesuit schools.

Just as Euclidean geometry was, for the Jesuits, the highest and best of what mathematics could be, so the new “method of indivisibles” advocated by Galileo and his circle was its exact opposite. Where 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 and proceeded to generalize from there, reaching for general mathematical principles. In other words, if geometry was top-down mathematics, the method of indivisibles was bottom-up mathematics. 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 infinitesimals were to prevail, it seemed to the Jesuits, the eternal and unchallengeable edifice of Euclidean geometry would be replaced by a veritable tower of Babel, a place of strife and discord built on teetering foundations, likely to topple at any moment. If Euclidean geometry was, for Clavius, the foundation of universal hierarchy and order, then the new mathematics was the exact opposite, undermining the very possibility of universal order, leading to subversion and strife. Tacquet was not exaggerating when he wrote that in the struggle between geometry and indivisibles, one must destroy the other or “must itself be destroyed.” And so the Jesuits proceeded to do just that.

THE CENSORS, PART I

The issue of the structure of the continuum could hardly have been further removed from the minds of the early Jesuit fathers as they faced off against Martin Luther and his followers in a battle for the soul of Europe. The first Jesuit to take any notice of the issue was none other than Clavius’s old nemesis at the Collegio Romano, Benito Pereira. In 1576, at the height of his struggle with Clavius over the proper place of mathematics in the Jesuit curriculum, Pereira published a book on natural philosophy intended to establish the proper principles that should be adopted by the Jesuits. Following the guidelines established by the Society’s founders, Pereira adhered closely to the teachings of Aristotle, and so he also addressed that ancient philosopher’s teachings on the subject of the continuum. In the best tradition of medieval scholasticism, Pereira first posed the thesis that a line is composed of separate points, and presented all the arguments offered in support of the thesis by ancient and medieval masters. He then demolished the arguments one by one, until he was left to conclude, along with Aristotle, that the continuum is infinitely divisible, and not composed of indivisibles. Pereira, it is clear, was not concerned about mathematical innovations or their subversive implications: writing decades before Galileo and his disciples developed their radical mathematical techniques, he had no reason to be. And since he saw no value for the Jesuits in the study of any kind of mathematics, he was unlikely to concern himself with determining the right “kind” of mathematics that should be taught. For him the question of the continuum was merely one more topic to be addressed in a discussion of Aristotle’s natural philosophy.

It was a full two decades before another Jesuit took up the question of the continuum, and this time it was a much more authoritative one: Father Francisco Suárez, the leading theologian of the Society of Jesus. In 1597, Suárez devoted thirteen folios to the question of the composition of the continuum in his
Disputation on Metaphysics
, but like Pereira, he addressed the matter as part of a broader discussion of Aristotelian physics. Unlike Pereira, however, the great theologian did not peremptorily reject the notion that the continuum is composed of indivisibles; admitting that the question is difficult, he gives up any hope of certainty, and seeks only an answer that “appears to be true.” He cites the doctrine that the continuum is composed of indivisibles, and then the complete denial of indivisibles, arguing that both are “extreme” positions. He then proposes some intermediate positions that he thinks more likely, while conceding that a definite solution is beyond reach. For Suárez as for Pereira, the entire question was technical, or what we would call “academic.” Neither one thought that there was much at stake here, except the correct interpretation of Aristotelian physics.

But as the troubled century of Charles V, Luther, and Ignatius was drawing to a close, an unmistakable sense of urgency entered into Jesuit discussions of infinitesimals. At the time, the general superior, Father Claudio Acquaviva, was increasingly concerned with the growing diversity of opinions within the Society. This was undoubtedly the price of success, as the rapid expansion of the Society in those years, in the form of hundreds of colleges and missions across the known world, brought many new peoples into its orbit. But for General Acquaviva this was no excuse for the soldiers of Christ to deviate from the correct teachings of the Church. As far as the Jesuit hierarchy was concerned, the increase in the Society’s numbers and influence was all the more reason it should speak with a single and clear voice. “Unless minds are contained within certain limits,” warned Father Leone Santi, prefect of studies at the Collegio Romano some years later, “their excursions into exotic and new doctrines will be infinite,” leading to “great confusion and perturbation to the Church.” To prevent this, the general superior in 1601 instituted a college of five Revisors General at the Collegio Romano, with the power to censor anything that was taught in the Society’s schools anywhere in the world or published under the Society’s aegis. With the Revisors’ oversight, Acquaviva hoped, only correct doctrine would be taught in Jesuit schools, and books published by Jesuit fathers would speak with a single authoritative voice, approved by the authors’ superiors. It did not take long for the Revisors to begin issuing prohibitions on the teaching and promotion of infinitesimals.