Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. Amir Alexander. New York: Farrar, Straus and Giroux, 2014. 354 pp. $27.

Abstract

In the autumn of 1656, the Savilian professor of geometry at the University of Oxford, John Wallis, published an attack on the mathematics of Thomas Hobbes entitled Due correction for Mr Hobbes or schoole discipline for not saying his lessons right. Wallis took aim at several targets presented by Hobbes in his Leviathan, which had appeared five years earlier. These included the use made by Hobbes of the principles of Euclidean geometry, the criticism that Hobbes advanced of the English universities (which he supposed to be stifled by the bonds of Aristotelian or school philosophy), and the attack mounted by Hobbes on the political and intellectual power of the clergy, which was accompanied by suggestions that the author of Leviathan held heterodox ideas about the nature of God. In particular, Wallis focused on the implications of the claim that Hobbes had made elsewhere that he was able to provide a rigorous method of constructing a square equal in area to a given circle. At the heart of the discussion lay the mathematical and philosophical term infinite. What did it mean to suggest that God might not be infinite; how should mathematicians describe the infinite divisions of a line or area; what was implied by talk about an infinite number of terms in a mathematical series? The stance taken by Wallis defined his positions: in defense of a divine being that could not be limited or purely material and in favor of a mathematics based on the principles of arithmetic, as much as those of geometry, and inductive in its reasoning. The quarrel between Wallis and Hobbes (which has been superbly treated in print by Douglas Jesseph) shapes the second part of Amir Alexander's quirky and readable book about the early seventeenth-century history of the use of infinitesimals. The first part concentrates on an Italian story, led by the geometrician Bonaventura Cavalieri, and focusing on the decision in 1632 of the Jesuit order to forbid the teaching of the infinite division of the continuum of material particles into ever smaller units. As in the debate between Wallis and Hobbes, the point of this discussion was not confined to geometry but reached out into questions of the makeup of the physical world, which had profound consequences for teaching. Those consequences extended to issues of the understanding of doctrine as well as nature, not only in terms of the language used about God, but also through consideration of the real presence in the Eucharist. Alexander wants to tell a broad story. His readers are treated to lengthy and digressive potted accounts of the Reformation, the foundation of the Jesuits, and the causes of the

Some Paradoxes of Infinity Revisited

“…This paradox proposed by Evangelista Torricelli (see, e.g., [1,44]) considers a rectangle ABCD that is not a square (see Fig. 1).…”

“…The last paradox we consider here is the Thomson Lamp Paradox described in [63]. Suppose that we have a lamp and start to turn it on for 1 2 minute, then turn it off for 1 4 minute, then on again for 1 8 minute, etc. At the end of one minute, the lamp switch will have been moved infinitely many times (to be precise, countably many times).…”

“…where 1 − 3, 1 − 2, 1 − 1, 1 are infinite natural numbers. Thus, the set of natural numbers will be written in the form (17) instead of the usual record (1). We emphasize that in both cases, we deal with the same mathematical object-the set of natural numbers-that is observed through two different instruments.…”

“…Note that all infinitesimals are not equal to zero. In particular, 1 1 > 0, because it is a result of division of two positive numbers. It also has a clear granary interpretation.…”

“…In the twentieth Introduction of the ideas of the number line, positional number systems, negative numbers, zero, rational and irrational numbers, limits, cardinal and ordinal numbers, continuum hypothesis, problems of consistency and completeness, and non-standard analysis are among the major milestones of these impressive research efforts. Research on these topics continues to be very active nowadays, as well (see, e.g., [1,7,24,25,32,35,36,[39][40][41]49,62] and references given therein).…”