BOOK REVIEW

Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Reviewed by Slava Gerovitch

Infinitesimal: How a Dangerous Mathematical Theory ing, from the chaos of imprecise analogies to the order Shaped the Modern World of disciplined reasoning. Yet, as Amir Alexander argues Amir Alexander in his fascinating book Infinitesimal: How a Dangerous Scientific American/Farrar, Straus and Giroux, April 2014 Mathematical Theory Shaped the Modern World, it was US$27.00/US$16.00, 368 pages precisely the champions of offensive infinitesimals who Hardcover ISBN-13: 978-0-37-417681-5 propelled mathematics forward, while the rational critics Paperback ISBN-13: 978-0-37-453499-8 slowed the development of mathematical thought. More- over, the debate over infinitesimals reflected a larger clash From today’s perspective, a mathematical technique that in European culture between religious dogma and intellec- lacks rigor or leads to paradoxes is a contradiction in tual pluralism and between the proponents of traditional terms. It must be expelled from mathematics, lest it dis- order and the defenders of new liberties. credit the profession, sow chaos, and put a large stain on Known since antiquity, the concept of indivisibles gave the shining surface of eternal truth. This is precisely what rise to Zeno’s paradoxes, including the famous “Achilles the leading mathematicians of the most learned Catholic and the Tortoise” conundrum, and was subjected to scath- order, the Jesuits, said about the “method of indivisibles”, ing philosophical critique by Plato and Aristotle. Archime- a dubious procedure of calculating areas and volumes by des used the method of indivisibles with considerable suc- representing plane figures or solids as a composition of in- cess, but even he, once a desired volume was calculated, divisible lines or planes, “infinitesimals”. While the method preferred proving the result with a respectable geometrical often produced correct results, in some cases it led to method of exhaustion. Infinitesimals were revived in the spectacular failures generating glaring contradictions. works of the Flemish mathematician Simon Stevin, the This kind of imprecise reasoning seems to undermine Englishman Thomas Harriot, and the Italians Bonaventura the very ideals of rationality and certainty often associated Cavalieri and Evangelista Torricelli in the late sixteenth to with mathematics. A rejection of infinitesimals might look the early seventeenth century. The method of indivisibles like a natural step in the progress of mathematical think- was appealing not only because it helped solve difficult problems but also because it gave an insight into the struc- Slava Gerovitch is a lecturer in history of mathematics at the Mas- ture of geometrical figures. Cavalieri showed, for example, sachusetts Institute of Technology. His email address is slava@ that the area enclosed within an Archimedean spiral was math.mit.edu. equal to one-third of its enclosing circle because the indi- For permission to reprint this article, please contact: reprint- visible lines comprising this area could be rearranged into a [email protected]. parabola. Torricelli, in order to demonstrate the power DOI: http://dx.doi.org/10.1090/noti1367 and flexibility of the new method, published a remark-

May 2016 Notices of the AMS 571 able treatise with twenty-one The Jesuits were largely responsible for raising the different proofs of an already- status of mathematics in Italy from a lowly discipline to a known result (the area inside paragon of truth a parabola). In ten different and a model for proofs, Torricelli used indi- social and po- The Jesuits visibles, producing effective litical order. The explanatory arguments instead Gregorian reform viewed the of cumbersome Euclidean of the calendar of constructions. He called the 1582, widely ac- notion of method of indivisibles “the cepted in Europe Royal Road through the math- across the reli- infinitesimals ematical thicket,” while the gious divide, had Public Domain. as a dangerous Bonaventura Cavalieri traditional Euclidean approach very favorable (1598–1647). deserved “only pity” (p. 110). political ramifi- idea. Due to their calculating power cations for the and explanatory appeal, infini- Pope, and this tesimals quickly gained popu- project endeared larity until they faced stern mathematics to opposition from the Jesuits. the hearts of Catholics. In an age of religious strife and po- The method of indivisibles litical disputes, the Jesuits hailed mathematics in general, had glaring flaws. Compar- and Euclidean geometry in particular, as an exemplar of ing the infinitesimals compos- resolving arguments with unassailable certainty through ing one figure to the infini- clear definitions and careful logical reasoning. They lifted tesimals composing another mathematics from its subservient role well below philoso- could produce different re- phy and theology in the medieval tree of knowledge and sults, depending on the pro- made it the centerpiece of their college curriculum as an cedure used. For example, indispensable tool for training the mind to think in an if one drew a diagonal in a rect- orderly and correct way. Photographic reproduction of public domain work. Evangelista Torricelli angle with a greater horizontal The new, enviable position of mathematics in the Jesu- (1608–1647), by side, it would split into two its’ epistemological hierarchy came with a set of strings Lorenzo Lippi (circa equal triangles, as in Figure attached. Mathematics now had a new responsibility 1647, Galleria Silvano 1 (below). Using the method to publicly symbolize the ideals of certainty and order. Lodi & Due). of indivisibles, however, one Various dubious innovations, such as the method of in- could argue that each hori- divisibles, with their inexplicable paradoxes, undermined zontal segment in the upper triangle was greater than this image. The Jesuits therefore viewed the notion of the vertical segment drawn through the same point on infinitesimals as a dangerous idea and wanted to expunge the diagonal in the lower triangle, and therefore the two it from mathematics. In their view, infinitesimals not only triangles differed in size. Torricelli found a way out of this tainted mathematics but also opened the door to subver- conundrum by claiming that the indivisible lines of the sive ideas in other areas, undermining the established lower triangle were “wider” than the lines of the upper and social and political order. The Jesuits never aspired to built a whole mathematical apparatus around the concept mathematical originality. Their education was oriented of indivisibles of different width. However ingenious, this toward an unquestioning study of established truths, and explanation did not fly with the Jesuits. it discouraged open-ended intellectual explorations. In the first decades of the seventeenth century the Revisors ABGeneral in Rome issued a series of injunctions against in- E finitesimals, forbidding their use in Jesuit colleges. Jesuit F mathematicians called the indivisibles “hallucinations” and argued that “[t]hings that do not exist, nor could they exist, cannot be compared” (pp. 154, 159). The champions of infinitesimals chose different strate- gies to deal with the Jesuit onslaught. In 1635 Cavalieri expounded the method of indivisibles in a heavy volume, filled with impenetrable prose, which even the best math- D GCematicians of the day found hard to get through. He dis- missed the paradoxes generated by his method with long Figure 1. The method of indivisibles can lead to and convoluted explanations aimed to intimidate more contradictions. Here each horizontal segment in than persuade. Later on, when asked about the paradoxes, the upper triangle is longer than the corresponding the defenders of infinitesimals often simply gestured vertical segment in the lower triangle, wrongly toward Cavalieri’s volumes, claiming that he had already implying that the upper triangle has greater area. resolved all of them. Neither the critics nor the support-

572 Notices of the AMS Volume 63, Number 5 ers of the indivisibles dared The Royal Society was ini- to penetrate Cavalieri’s ob- tially suspicious of mathemat- scure fortress. Evangelista ics. Society fellows prized Torricelli, by contrast, found experimental science, public the paradoxes the most fas- demonstrations, and open in- cinating part of the topic tellectual debate as a model for and published several de- peaceful resolution of societal tailed lists of them, believing tensions. Mathematics, with its that a study of such para- reputation as a solitary, private doxes was the best way to pursuit, its claims for incontro- understand the structure vertible truth, its reliance on

of the continuum. For him, obscure professional language, Copyright National Portrait Gallery, London. Copyright National Portrait Gallery, London. Thomas Hobbes John Wallis (1616– the study of paradoxes was and its inaccessibility to lay- (1588–1679), by John 1703), after Sir Godfrey akin to an experiment, for it men, seemed like a poor match Michael Wright, oil on Kneller, Bt oil on pushed a phenomenon to its for the liberal ideals of the so- canvas, circa 1669– canvas, feigned oval, extreme in order to reveal its ciety. Wallis, the only math- 1670, NPG 225. (1701), NPG 578. true nature. ematician among the founders, The dispute over infini- took upon himself the task of tesimals was at the same time a dispute over the nature of reconciling mathematics with the spirit of the society’s mathematics. Do mathematical proofs have to show only ideals. Claiming that “[m]athematical entities exist not in the correctness of a theorem (as in Euclidean geometry) or the imagination but in reality” (p. 263), he put forward a to explain why it is true (as in the method of indivisibles)? new, “experimental” mathematics. In contrast to the Eu- Should one pursue a top-down approach, clidean approach of constructing geometri- starting with universal first principles, put cal objects from the first principles, Wallis mathematical objects in order, and then assumed that geometrical figures already ex- impose this order on the world, or should ...a clash isted in the world. Modifying the method of one build mathematics from the bottom between indivisibles, he viewed a triangle as actually up, beginning with one’s intuition about the composed not of lines but of infinitely thin world and moving up to higher and higher two trapezoids, two-dimensional objects making abstractions? The latter question pitted up the original triangle, just like mountains the Jesuits against Galileo, which led to his visions of formed by geological strata. Studying geo- eventual condemnation and lifetime house metrical objects for him was akin to the work arrest. Harsh administrative measures were modernity. of a scientist probing geological formations. also taken against the remaining stalwarts of His method relied on induction, was open infinitesimals, who lost their jobs and were to discussion, and aimed to persuade the forbidden to teach or publish. The Jesuits reader by examining a series of particular even went so far as to engineer the dissolution of a small cases, much like the laboratory experiments that became monastic order, the Jesuats, which had sheltered the hallmark of the Royal Society’s approach to studying Cavalieri and Stefano degli Angeli, the leading nature. In the eyes of its fellows, this kind of mathematics promoters of the method of indivisibles. The Jesuit was aligned with the society’s epistemological ideals, and mathematicians saw their mission in preserving the its legitimation paved the way for the later transformation eternal truths of Euclidean geometry and in of the method of infinitesimals into calculus in the hands suppressing any threat of potentially disruptive of Isaac Newton. innovation. This led, Alexander argues, to “the slow Alexander persuasively argues that the fight over in- suffocation and ultimate death of a brilliant Italian finitesimals was a reflection of a more fundamental clash mathematical tradition” (p. 165). between what he calls two “visions of modernity.” While The battle over the method of indivisibles played out the Jesuits and Hobbes embodied the desire to achieve differently in England, where the Royal Society proved a societal unity through the imposition of a single truth capable of sustaining an open intellectual debate. One of and suppression of debate, the champions of infinitesi- the most prominent critics of infinitesimals in England mals valued the freedom of discussion and investigation was philosopher and amateur mathematician Thomas and a pluralism of opinions. Their opponents feared that Hobbes. A sworn enemy of the Catholic Church, he intellectual pluralism might lead to political and religious nevertheless shared with the Jesuits a fundamental pluralism and wanted to squash the seeds of instability be- commitment to hierarchical order in society. He fore they produced full chaos. Following the Jesuits’ purge believed that only a single-purpose organic unity of a of creative mathematicians, not only Italy’s mathematical nation, symbolized by the image of Leviathan, could save tradition declined but the country itself became unrecep- England from the chaos and strife sowed by the civil tive to innovation and began falling behind. In England, war. In the 1650s–70s his famously acrimonious by contrast, the support of mathematical novelty by the dispute with John Wallis, the Savil-ian Professor of Royal Society was part of greater openness in intellectual Geometry at Oxford and a leading propo-nent of the and social debates and resulted in rapid scientific and method of indivisibles, again pitted a champion of social order against an advocate of intellectual freedom.

ay 2016 Notices of the aMs 573 technological development, leading up to the Industrial fruitful ideas from nonrigorous fields, such as supersym- Revolution. The author implies that the different fate of metric quantum field theory and string theory. infinitesimals in different countries shaped the fortunes Alexander’s book meaningfully points to a fundamental of these nations in the long run. tension between the popular image of mathematics as Alexander clearly outlines a cultural split between po- a collection of eternal truths which never changes and litical conservatives and knows no debate and its actual practice, filled with uncer- tainty, frustration, failure, and rare glimpses of profound “liberalizers” with respect insight. If, as in the case of the Jesuits, maintaining the Innovation to the method of indivis- appearance of infallibility becomes more important than often grows ibles. His own discussion exploration of new ideas, mathematics loses its creative of Hobbes’s early fascina- spirit and turns into a storage of theorems. Innovation out of tion with infinitesimals, often grows out of outlandish ideas, but to make them however, somewhat chal- acceptable one needs a different cultural image of math- outlandish lenges this overly neat ematics—not a perfectly polished pyramid of knowledge, separation. Despite his but a freely growing tree with tangled branches. ideas. royalist and traditionalist convictions, Hobbes care- About the Author fully read and absorbed Slava Gerovitch teaches Cavalieri’s subversive mathematical treatises. Reinterpret- cultural history of ing the indivisibles as material objects, he developed an mathematics at MIT. His research inter- unconventional geometry in which mathematical objects ests include history were generated by the motion of simpler objects—lines by of twentieth-century motion of points, surfaces by motion of lines, and solids mathematics, cybernet- by motion of surfaces—before he turned against infinitesi- ics, astronautics, and mals in his personal vendetta against Wallis. Well, good computing. His current history of mathematics, like good mathematics, might project explores how occasionally benefit from a paradox or two. the Soviet mathematics

In the 1960s, three hundred years after the Jesuits’ Photo courtesy of Slava Gerovitch. community creatively ban, infinitesimals eventually earned a rightful place in adapted to various political, institutional, and cultural mathematics by acquiring a rigorous foundation in Abra- pressures. He is the author of From Newspeak to Cyber- ham Robinson’s work on nonstandard analysis. They had speak: A History of Soviet Cybernetics, Voices of the Soviet played their most important role, however, back in the Space Program, and Soviet Space Mythologies. A poet and days when the method of indivisibles lacked rigor and was a translator, he has also published Wordplay: A Book of fraught with paradoxes. Perhaps it should not come as a Russian and English Poetry. surprise that today’s mathematics also borrows extremely

574 Notices of the AMS Volume 63, Number 5