Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Read Online Free Page B
Author: Amir Alexander
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.
THE EMPEROR AND THE MONK
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
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.
THE EMPEROR AND THE MONK
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
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