Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

In the 1630s, when the Roman Catholic Church was confronting Galileo over the Copernican system, the Revisors General of the Jesuit order condemned the doctrine that the continuum is composed of indivisibles. What we now call Cavalieri’s Principle was thought to be dangerous to religion. My Solid Geometry teacher didn’t tell us that.

In England a couple of decades later, the mathematician John Wallis (the first to use the symbol \(\infty\) for infinity) and the political philosopher Thomas Hobbes clashed over whether the infinite and the infinitely small were mathematically legitimate. Wallis had used these ideas to find the areas under the graph of \(y=x^n\) for rational \(n\). But Hobbes thought that Wallis’s arguments weren’t good geometry, and that the social order itself depended on sticking to Euclidean rigor.

Alexander’s provocative new book begins with a lively account of these two disputes and their historical context. Let’s first look at that context, and then evaluate his conclusions.

Why did the Church get involved in evaluating the “new math” of indivisibles, infinitesimals, and the infinite? Catholicism had dominated medieval Europe, but by the sixteenth century had been challenged religiously by Protestantism. The Catholic Church responded by firming up church doctrines and institutions. The religious challenges became intertwined with political ones. The Peasants’ Revolt in the 1520s showed how attacks on one kind of authority could spill over into the political realm. Various rulers and nations lined up on one side or another of the religious divide; one result was the Thirty Years War (1618–1648).

All sorts of ideas were judged on the basis of which side they seemed to favor. The doctrine of indivisibles was on the side of Galileo. Besides opposing the Church about whether the earth went around the sun, Galileo treated matter as made of atoms, which are physical indivisibles. Bonaventura Cavalieri, who pioneered indivisible methods in geometry, was among Galileo’s followers. Furthermore, Catholic theology owes much to Aristotle’s philosophy, and Aristotle, arguing for the potentially infinite divisibility of the continuum, had explicitly ruled out both indivisibles and the actual infinite. So it is no wonder that Jesuit intellectuals opposed using indivisibles in geometry.

Even more surprising, the historical context for Hobbes’ philosophical objection to indivisibles was the English Civil War. King Charles I and Parliament clashed, the Army purged Parliament of the king’s supporters, and Charles I was actually executed in 1649. The political dispute over who should rule — an absolute king or a (somewhat) more representative Parliament — was linked to religious disputes between the established Church of England, Catholics, and dissenters like the Puritans. Battles continued off and on until 1688 when Parliament asked William of Orange and his wife Mary to rule the country.

The English Civil War, which resulted in more than 100,000 dead out of a population of five million, provided a major impetus for Hobbes’ political philosophy. In his book Leviathan (1651), Hobbes argued that, without a strong ruler, individuals were condemned to live in a “war of all against all” in which, as he famously said, life was “solitary, poor, nasty, brutish, and short.”

But Hobbes philosophized about more than politics. Though a materialist, he fell in love with the beautiful logical structure of Euclidean geometry. He thought, as had many others before him from Aristotle on, that the certainty of geometry made it a model for all reasoning. More important for our purposes, Hobbes held also that the geometric ideal of certainty was essential to support the all-important social order.

Thus for Hobbes, the non-rigorous mathematics of infinites and infinitesimals, which did not rest on Euclid’s postulates and too often used a sort of enumerative “induction” — if it’s true for 2, for 3, for 4, then it’s true for all numbers — challenged not only the rigor of mathematics but the foundations of society. Wallis, whose political views were more friendly to diverse religious ideas and less friendly to an absolute ruler and who used the new mathematical methods, was an obvious target for Hobbes. But Hobbes, whose geometric publications included squaring the circle, could not and did not win this argument centered on mathematics.

Once Alexander has told these stories, he pulls out all the stops in his last chapter, saying, “From Rome and Florence to London and Oxford, the fight over the infinitely small raged across Western Europe… Consider what the world would have been like without [infinitesimals]. If the Jesuits and their allies had had their way, there would be no calculus, no analysis, nor any of the scientific and technological innovations that flowed from these powerful mathematical techniques.” Of the question of “whether the continuum is made up of infinitesimals,” Alexander says in an epilogue, “Both sides believed that the answer could shape every facet of the modern world.” As a result of these disputes, he concludes, “the world was never the same again.”

Should we be convinced? Of course without calculus it would have been hard to produce modern science and technology. But I think Alexander overestimates the importance of the disputes he describes. He doesn’t show that the Jesuit condemnation of indivisibles was anywhere near as influential as the Church’s condemnation of Galileo’s Copernicanism. Nor does he show that the ideas of Cavalieri and Wallis were widely seen as dangerous and disruptive. And, since a key point of Alexander’s book is the importance of the invention of the calculus, I’d argue that the calculus would still have been invented in the seventeenth century even had the Jesuits convinced everyone in Europe that indivisibles were not rigorous mathematics.

After all, the heuristic use of infinitesimals and indivisibles goes back to Democritus. Euclid and Archimedes gave rigorous proofs, using proof by contradiction, for a wealth of results about areas and volumes discovered using such heuristics. Euclid’s version of what we now call the Archimedean axiom allowed him to demonstrate what such proofs require: that curvilinear areas can be approached as closely as desired by successively closer rectilinear figures. Seventeenth-century mathematicians were aware of the ancient proofs. They were also aware of the vast number of new curves, defined by equations, resulting from the new analytic geometry, and they used all available tools to find the properties of these new curves. Fermat’s algebraic, non-infinitesimalist method of maxima and minima is one example; Descartes’ algebraic method of tangents is another. Newton ultimately abandoned indivisibles in favor of limits, but not because of objections from Aristotle, the Jesuits, or Hobbes; instead, as Newton wrote in the Principia, indivisibles violate “what Euclid had proved concerning incommensurables in the tenth book of his Elements.”

The six-page timeline Alexander provides jumps from Archimedes in 250 BCE to Martin Luther in 1517. This neglects Apollonius and Pappus, whose work profoundly influenced Fermat and Descartes. It also neglects medieval Islamic and Christian predecessors of calculus whose work was known in the Renaissance. For example, ibn al-Haytham used the method of slices to find volumes, and Nicole Oresme interpreted the area under a velocity graph as a distance. The abundance of methods: ancient, medieval, and contemporary; indivisible and infinitesimal; geometric and algebraic; quantities generated by motion and the concept of limit, together with the concomitant progress in the natural sciences, gave seventeenth-century mathematics far too much momentum and too many demonstrable successes to be stopped by philosophical arguments about the nature of the continuum.

Alexander’s book contains occasional imprecise statements. Notably, although the book’s main title is “Infinitesimal,” the Jesuit condemnations he quotes denounce indivisibles, not infinitesimals. Alexander’s discussion of the controversy uses these terms almost interchangeably. But, though Aristotle may have opposed both concepts, they aren’t the same. A parallelogram can be thought of as made up of indivisible lines, each with one dimension, or it can be thought of as made up of an infinite number of infinitesimal rectangles, each having two dimensions.

There are other places where we know what Alexander means, but what he says isn’t quite exact. For instance, Alexander calls the Wallis product for \(4/\pi\) an “infinite series.” He misstates Euclid’s Theorem I.29 as “when a straight line falls on two parallel lines, it creates the same angles with one parallel line as with the other.” He mischaracterizes Zeno’s Achilles paradox, and does not address Zeno’s challenge to the potentially infinite divisibility of the continuum. And he should have proofread this statement about modern mathematical induction more carefully: “It consists of demonstrating a theorem for a particular case, say \(n\), and then proving that if it is true for \(n\), it is also true for the case \(n+1\) (or \(n-1\)), and consequently for all \(n\)’s.” Such details matter, especially in a book aimed at a broad audience.

Alexander’s emphasis on the ways mathematical and scientific achievements and philosophical views about them can influence society should be applauded. The prestige of science can be used in political disputes (witness Wallis vs. Hobbes), and politics or religion can intervene in scientific matters (witness the Jesuits vs. indivisibles). Readers will be able to think of modern parallels, like conservative objections to evolution and climate change, and liberal objections to eugenics and sociobiology. Such parallels help explain the wide popular interest this book has generated.

Still, I think Alexander claims more than he is able to prove. Our response to the book’s provocative subtitle should be tempered by recalling that the forces that shaped the modern world were not just intellectual, but economic, technological, institutional, and political as well. Nonetheless, the stories Alexander tells about these disputes are fascinating, and they deserve to be better known.

Judith V. Grabiner is the Flora Sanborn Pitzer Professor of Mathematics at Pitzer College, where she teaches both mathematics and its history. Her most recent book, A Historian Looks Back: The Calculus as Algebra and Selected Writings, was published by the MAA.