Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

From today’s perspective, a mathematical technique that lacks rigor or leads to paradoxes is a contradiction in terms. It must be expelled from mathematics, lest it discredit the profession, sow chaos, and put a large stain on the shining surface of eternal truth. This is precisely what the leading mathematicians of the most learned Catholic order, the Jesuits, said about the “method of indivisibles”, a dubious procedure of calculating areas and volumes by representing plane figures or solids as a composition of indivisible lines or planes, “infinitesimals”. While the method often produced correct results, in some cases it led to spectacular failures generating glaring contradictions. This kind of imprecise reasoning seems to undermine the very ideals of rationality and certainty often associated with mathematics. A rejection of infinitesimals might look like a natural step in the progress of mathematical thinking, from the chaos of imprecise analogies to the order of disciplined reasoning. Yet, as Amir Alexander argues in his fascinating book Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, it was precisely the champions of offensive infinitesimals who propelled mathematics forward, while the rational critics slowed the development of mathematical thought. Moreover, the debate over infinitesimals reflected a larger clash in European culture between religious dogma and intellectual pluralism and between the proponents of traditional order and the defenders of new liberties. Known since antiquity, the concept of indivisibles gave rise to Zeno’s paradoxes, including the famous “Achilles and the Tortoise” conundrum, and was subjected to scathing philosophical critique by Plato and Aristotle. Archimedes used the method of indivisibles with considerable success, but even he, once a desired volume was calculated, preferred proving the result with a respectable geometrical method of exhaustion. Infinitesimals were revived in the works of the Flemish mathematician Simon Stevin, the Englishman Thomas Harriot, and the Italians Bonaventura Cavalieri and Evangelista Torricelli in the late sixteenth to the early seventeenth century. The method of indivisibles was appealing not only because it helped solve difficult problems but also because it gave an insight into the structure of geometrical figures. Cavalieri showed, for example, that the area enclosed within an Archimedean spiral was equal to one-third of its enclosing circle because the indivisible lines comprising this area could be rearranged into a parabola. Torricelli, in order to demonstrate the power and flexibility of the new method, published a remarkSlava Gerovitch is a lecturer in history of mathematics at the Massachusetts Institute of Technology. His email address is slava@ math.mit.edu.