Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World
Matthew DeLong is Professor of Mathematics at Taylor University.
The transition to modernity was shaped by changes in science, politics, religion, economics, and culture. Such changes were contested, and from them emerged a new way of perceiving the world. As the subtitle to Infinitesimal suggests, Amir Alexander makes the startling assertion that ground zero of the battle over the shape of the modern world was a seemingly innocuous and abstruse mathematical claim about the nature of lines. Alexander’s synthesis of interpretive history makes a compelling case, and his book provides an enjoyable and non-technical read for those interested in the history of ideas.
The doctrine of infinitesimals states that the continuum is composed of indivisibles, that is, that “every line is composed of a string of points, or ‘indivisibles,’ which are the line’s building blocks, and which cannot themselves be divided” (9). Ancient Greek mathematicians who accepted this assertion did groundbreaking work on areas and volumes. However, the paradoxes articulated by Zeno and others demonstrated that accepting infinitesimals leads to what seem like logical contradictions. For instance, if the indivisibles that compose a line segment have zero length, then how can the line segment have non-zero length? On the other hand, if the indivisibles have some non-zero length, why is the length of the line segment not infinite?
Following the early successes with infinitesimals by such giants as Archimedes, concern for logical precision meant that for two millennia infinitesimals were shunned by mathematicians who created a Euclidean edifice of mathematics free from paradox and ambiguity that they believed “was fixed, orderly, and eternally true” (78). It was not until the sixteenth century that there was a revival of interest in infinitesimals among European mathematicians. This revival drew battle lines between those who argued for their inclusion into mathematics because of their demonstrable usefulness and those who argued against them on the grounds that they threatened the ordered and reasonable method of mathematics. The infinitesimalists were eventually vindicated by those such as Newton and Leibniz, in whose hands the method led to what we now know of as Calculus. What Alexander offers us is a look at how fraught the question was with cultural significance.
Alexander’s thesis is that the question of indivisibles, which may seem to be of interest only to mathematicians and philosophers, became ground zero for the ideological battles over the nature of European society:
Whether the continuum is made up of infinitesimals seems like the quaintest of questions, and it is hard for us to fathom the passions it unleashed. But when the struggle raged in the seventeenth century, the combatants on both sides believed that the answer could shape every facet of life in the modern world that was then coming into being. (294)
Moreover, it is his contention that in Italy, which had been at the forefront of mathematics and science in the time of Galileo, the victory by the anti-infinitesimalists led to scientific, mathematical, and cultural stagnation. By contrast in England, which had been “a wild and semibarbarous country at the northern edge of European civilization” (292), the subsequent triumph of the infinitesimalists was critical to its transformation into “a leading center of European culture and science and a model for political pluralism and economic success” (292).
Alexander structures the book into two main parts, along with an introduction and an epilogue, with each part telling a similar story but in different settings and with different protagonists and antagonists. Part I, set in sixteenth- and seventeenth-century Italy, pits the anti-infinitesimalist Jesuits, led by mathematician Christopher Clavius, renowned for his mathematical work on the then-new Gregorian calendar, against the infinitesimalist Galileo Galilei and his disciples, most notably Bonaventura Cavelieri and Evangelista Torricelli. Part II takes place in seventeenth-century England, in which the Royalist and philosopher Thomas Hobbes, staunch defender of the traditional anti-infinitesimal position, and the Parliamentarian and mathematician John Wallis, a leading infinitesimalist, waged a two-decade, public, acrimonious feud over the infinitely small.
The Jesuits defended the religious and political order and the hierarchy of the Catholic Church from the threat of the Reformation through, among other things, an extensive system of higher education with a well-defined and centrally controlled curriculum. Some Jesuits, such as Clavius, saw mathematics – particularly Euclidean geometry – as a model discipline in turbulent times, because of its top-down, rational, ordered system. They associated infinitesimals, with their paradoxes and bottoms-up, pragmatic approach to mathematics, with the confusion and disorder that they saw emerging from the proliferation of Protestant sects. They waged an eventually-successful campaign to ban the study and teaching of infinitesimals from Jesuit classrooms around the world. Their chief opponents were followers of Galileo, who championed inductive and experimental approaches to knowledge, and who attempted to show the power of accepting infinitesimals into mathematics.
Both Hobbes and Wallis were motivated in their mathematics and politics by a desire to bring harmony and order in the aftermath of the English interregnum and civil war. For Hobbes, the solution was to model discourse on the deductive methods of Euclidean geometry. By beginning with axioms and working “deductively, step by step, to ever-more-complex but equally certain truths” (286), Hobbes attempted to avoid dissent by constructing an entire philosophy that was “perfectly rational, absolutely transparent, and fully known, a world that held no secrets and whose rules were as simple and absolute as the principles of geometry” (286). The fruit of Hobbes’s labor was, of course, the Leviathan. His adversary Wallis, secretary to the Westminster Assembly of Divines and Savilian Professor of Geometry at Oxford, was a founding member of the Royal Society of London. The society’s members believed that dogmatism and excess certainty led to the disasters of the civil war period, and so they sought to build a model community around experimentation, observation, and free discourse. For Wallis, the method of infinitesimals provided a new kind of mathematics that fit the principles of the society, by being “powerful and effective in the service of science, but serving as a model for tolerance and moderation rather than dogmatic rigidity” (259).
Alexander’s previous two books1 are also concerned with how transitions of eras in mathematical thought occur, and how mathematics both influences and is influenced by its cultural settings. In Infinitesimal, Alexander’s writing is accessible and his storytelling is engaging. His thorough scholarship does not impede the narrative flow, with the nearly 300 endnotes presented without inline citations. To aid the reader with the multitude of historical characters and events, Alexander includes an annotated list of Dramatis Personae and a Timeline.
Writing mathematics for a popular audience is challenging. If it is over-simplified, mathematical precision may be lost, undoubtedly disappointing the mathematical readers. On the other hand, in order to maintain the momentum of lay readers, some effort must be made to make the mathematics accessible. Alexander, who has bachelor’s degrees in Mathematics and History, in addition to his graduate degrees in the History of Science, has nicely walked the middle path. There are approximately 30 pages of mathematical exposition sprinkled throughout the narrative, elucidating some of the relevant work of mathematicians such as Archimedes, Cavalieri, Torricelli, Wallis, and Hobbes. Although the ancient and early modern terminology and methods may make the arguments somewhat challenging for the less mathematically versed to follow, the intriguing narrative in which they are placed should give the squeamish enough motivation to work through them. And doing so helps immerse the reader in the debate.
Because the participants in the intellectual skirmishes described by Alexander are motivated in great measure by theological and philosophical concerns, Christian academics may wonder whether he paints a simplistic, anti-Catholic picture of the church fighting a losing battle against the onslaught of modernity, as is presented in popular but oversimplified accounts of Galileo. This is not the case. Alexander is thorough and balanced in his description of the motivations behind both sides in the contest. Moreover, in Part I the antagonists, or the anti-infinitesimalists, are the religiously motivated Jesuits, whereas in Part II it is the decidedly anti-religious Thomas Hobbes who tries to squelch any introduction of infinitesimals into mathematics. The common thread holding together the religious and political absolutism of the Jesuits and Hobbes, respectively, was the conviction that the deductive method best modeled by Euclidean geometry was the surest way to arrive at incontrovertible truth. Both Hobbes and the Jesuits “tried to use geometry as a model for the proper order of knowledge, society, and the state” (282).
Infinitesimal should be of interest to Christian academics from a variety of disciplines, not just mathematicians or historians, for its contribution to a broader and more unified understanding of the world through the integration of math- ematics, religion, science, philosophy, politics, and history. It could also be used with students from a variety of disciplines. I will be using it as required reading in a course that focuses on how the deductive method in mathematics compares to ways of knowing in other disciplines as well as on how mathematics relates to religion, science, art, and other forms of culture. I heartily recommend the book both for its educational value and because it is an enjoyable read.