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

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