Review of Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, Amir Alexander, 2014. 368 pp. including diagrams, bibliography, appendix: “Dramatis Personae,” and index. $27 hardcover, ISBN 978-0-374-17681-5. New York: Scientific American / Farrar, Straus and Giroux.

The brotherhood railed at the new mathematical concept. It disturbed their ideals of harmony and mathematical order. It, and its teaching, had to be suppressed. To any student with a vague knowledge of the history of mathematics, this scenario describes the Pythagoreans’ reception to the existence of irrational number. Although the scene and circumstances are similar to the “mathematical scandal” experienced in 5th century BCE Greece, this brotherhood is not the Pythagoreans, it is the Jesuits, members of the Society of Jesus, a Catholic order of priests, the time is the 16th century and the place is Southern Europe. The offensive concept is the “indivisible quantity” or “infinitesimal,” a geometric entity so small that it cannot be divided further. The concept’s originators envisioned all geometric objects as composed of such indivisibles: thus, a plane is composed of tightly packed indivisible lines, like “threads in a cloth”; a solid object is made up by a stack of indivisible planes, like a “pile of cards,” and such indivisibles could be manipulated to arrive at mathematical results. While such a concept is intuitively appealing, it immediately opens itself to a series of contradictions. For example, theoretically, a triangle has no substance, it does not possess matter; lines have no width so to fill a triangle with these indivisible lines, nothing is being filled with nothing therefore there is nothing to manipulate. A resurrected Euclid’s geometry had become the absolute model for geometric existence and logical order in the physical world. Now, this structure was being challenged.

The 16th and 17th centuries were a time of many societal and intellectual upheavals: Copernicus, Kepler and Galileo had reconfigured the universe, displacing the earth as a central focus; Galileo had determined that falling bodies were acted upon by external rather than internal forces, contradicting Archimedean beliefs; rising national and economic movements unsettled the class structure and the place of ultimate worldly authority, whether residing with a king or the encompassing power of the Catholic Church, was questioned. The Protestant Reformation had resulted in Catholic countermeasures, the most effective being the founding of the Jesuit order of priests. These “soldiers of Christ” were chosen men, highly trained in theology and also proficient in worldly disciplines. The Jesuits became missionaries and educators. They and their schools soon dominated the educational landscape of Europe. In particular, Jesuit mathematicians shaped the teaching of mathematics. Led by the well-respected geometer and astronomer Christopher Clavius (1537-1612), Jesuit mathematics professors such as André Tacquet, Paul Guldin and Mario Bettini believed that all mathematics followed a rigid logical structure as evidenced in Euclidean geometry. The idea of a contrived entity such as indivisibles violated their logical worldview and had to be suppressed. Bonaventura Cavalieri (1598-1647), in his Geometria indivisibilibus (1635), and Evangelista Torricelli (1608-1647), Galileo’s former pupil, in his Opera geometrica (1644), had both used this concept to prove their new mathematical results. Indivisibles and their advocates Cavalieri and Torricelli came under a Jesuit attack that was soon backed by the Catholic Church at large. The controversy stagnated Italian mathematical developments. Meanwhile in England John Wallis (1616-1703) experimented with indivisibles in his works, De sectionibus conicis and Arithmetica infinitorum, both published in 1656. While Wallis did not have the opposition of the Jesuits and the power of Rome to overcome, his use of indivisibles came under a similar strong attack by the philosopher and geometer Thomas Hobbes (1558-1679). Hobbes held that true mathematical knowledge began with appropriate definitions and developed by a use of strict logical reasoning – the Euclidean approach. The Hobbs-Wallis debate was prolonged and often bitter. Fortunately for Wallis, the Royal Society, just beginning to exert its scientific influence, encouraged mathematical experimentation and sided with Wallis. So sanctioned, the concept of indivisibles became a new impetus for the development of mathematical analysis in the British Isles.

Author Amir Alexander takes his reader through the intricate web of controversy, conspiracy and intrigue surrounding the introduction of indivisibles into the Age of Enlightenment. His research is impressive, ranging from the testimony of personal correspondence to that of the Vatican Archives. He also attempts to associate this mathematical dispute with the political and social disruptions of the period. This examination of mathematical development as part of larger cultural and societal movements is worthy of attention. Infinitesimal reads like a novel; its depiction of the actors and the playing out of the plot make for high drama. This is a book that every serious teacher of calculus should read. Students of the history of science, in a reading of this book, will find much substance to ponder and possibly to explore further.

See also the review by Judith Grabiner in MAA Reviews.