The problems of solutions
By Owen Gingerich
June 9, 2014
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, By Amir Alexander, Scientific American/Farrar, Straus and Giroux, 352 pp., $27
We all know that an infinitesimal is very tiny, but precisely how small is it? Suppose that we have a supremely sharp knife and we cut infinitesimal in two. We then slice the fragment infinit in half, and so on. We are quickly reminded of British mathematician Augustus De Morgan’s version of Jonathan Swift’s jingle,
Great fleas have little fleas upon their backs to bite ’em,
And little fleas have lesser fleas, and so ad
If the tiny ultimate slice of the letter i has essentially zero width, then an infinite sum of zeros is still zero. However, if the infinitesimal still has a finite width, then an infinity of finite segments is infinite. Hence the paradox.
Astute readers who have advanced beyond elementary algebra and geometry will notice that the paradox smacks of the fundamental idea of the calculus, a mathematical technique that lies at the foundation of Isaac Newton’s Principia (1687). The success of Newton’s physics rapidly quelled any criticism of the techniques, but this was not the case earlier in the 17th century, as UCLA historian Amir Alexander brings to the fore in this brilliantly documented volume. Without doubt Newton’s mathematical underpinning of physics in the Principia has shaped the modern world.
But were infinitesimals and indivisibles a dangerous mathematical theory?
Many of the players in this exotic game certainly thought so. The Jesuits at the Collegio Romano in Rome were a mathematical powerhouse, and under the leadership of Christopher Clavius (who had engineered the Gregorian calendar reform of 1582), they argued that the rigid truth of Euclidean geometry, built up from a handful of basic axioms, was an obvious paradigm for a foundational set of Christian beliefs. As such, mathematics took a place of honor in the elitist Jesuit educational system. Yet the Jesuits also thought that a mathematics built on something as paradoxical as infinitesimals was a recipe for failure. “Destroy or be destroyed” became the banner of the Jesuit war on the infinitely small. And this was closely related to the notion of atoms, equally detested by the Jesuits, but it is beyond the purview of Alexander’s account.
Writing not as a mathematician but as a mathematically savvy historian, Alexander nevertheless fills in a broad canvas. Galileo, who is perhaps best described as an experimental physicist, was mathematically savvy even though he was not particularly involved with infinitesimals. However, among his friends were the leading champions of the mathematics of infinitesimals: Bonaventura Cavalieri and Galileo’s own student, Evangelista Torricelli. In 1616, Galileo had been given a strong slap on the wrists for being too enthusiastic about the radical heliocentric cosmology of Copernicus. Galileo bided his time, and when in 1623 a fellow Florentine, Maffeo Barberini, was elected pope, he sprang into action. Galileo visited Rome, had a series of audiences with the new Pope Urban VIII, and came away with the impression that he had the freedom to write a book on cosmology. His scarcely disguised Copernican defense, the Dialogo of 1632, was the result.
This smattering of Galilaeana is well known; where Alexander shines is with the back story. At that time, the Jesuits were playing a major role in the Counter-Reformation and politically were closely allied with the Habsburgs in their rivalry with the French monarchy. Urban, however, was more enthusiastic about the French court and its cardinals, to the dismay of the Jesuits. The pope was a fan of Galileo’s, and the Jesuits were powerless to constrain his Copernican sentiments and his infinitesimalist friends. But suddenly, in 1631 and ’32, the Catholic forces of the Holy Roman Empire were defeated by Gustavus Adolphus of Sweden. The Thirty Years’ War was approaching its midpoint. Political alliances were shuffled just as Galileo’s book was being printed, and very quickly the Jesuits once again became a power in Italy. In consequence, not only was Galileo tried for “a vehement suspicion of heresy” and ultimately consigned to house arrest, but Italy, which had an enviable start with the new mathematics, was also about to lose out completely.
The scene now shifts to England. The time was still the 17th century. Political alliances were in play, with even a touch of sectarian positions. The field was again fraught with military clashes—the intellectual battle played out against the English Civil War, and Charles I was beheaded in 1649. But Infinitesimal’s story is more bookish, and it becomes essentially an account of a duel between Thomas Hobbes, a Royalist, and John Wallis, a self-trained Oxford mathematician and Presbyterian.
Hobbes’s Leviathan argued for rule by the Commonwealth (that is, by the people, although only the elite qualified), but Hobbes also fancied himself a mathematician. Like the Jesuits, Hobbes believed that Euclidean geometry represented a solid foundation for a consistent theory of government. He noticed, however, that geometry had several flaws, such as the failure to derive a square with the same area as a given circle, and he resolved to clear up this shortcoming. His first “solution” was ruthlessly destroyed by Wallis, who, to everyone’s surprise, was making a name for himself by his mathematical prowess, something that had been almost totally invisible when he was appointed Savilian professor of geometry at Oxford for strictly political reasons. Hobbes returned to the lists, with a revised attempt to square the circle, and in turn Wallis wasted it. Yet again Hobbes tried, and once more Wallis put him down, meanwhile setting forth his own researches on infinitesimals. (Two centuries later, squaring the circle with compass and straightedge was shown to be impossible.)
The Hobbes-Wallis duel is one of the more fun parts of this unusual book. Alexander tries to be evenhanded, but clearly Hobbes was the loser. Wallis went on to become one of the founders of the Royal Society, and his work of striking originality on infinitesimals provided a jumping-off place for Newton’s own towering mathematical achievements. But Newton and the calculus lie beyond Alexander’s compass. Without that sequel, the thesis that a dangerous idea has shaped the modern world falls a little flat.
In a short, final reprise, Alexander writes that “Italy remained frozen in time. The land that for centuries had led all Europe in the arts and sciences … became stagnant, backward, and poor.” In contrast, “England became a land of many voices, where rival views and interests competed openly, relatively free from state repression. And it was in this relative freedom that England discovered its path to wealth and power.”
These conclusions seem obvious, the story of the infinitesimals is fascinating, but the connectivity between them seems scarcely more than coincidental.
Owen Gingerich is professor emeritus of astronomy and of the history of science at the Harvard-Smithsonian Center for Astrophysics.