MIT historian and ex-Russian Slava Gerovitch reviews the book.

The Jesuits were largely responsible for raising the status of mathematics in Italy from a lowly discipline to a paragon of truth and a model for social and political order. The Gregorian reform of the calendar of 1582, widely accepted in Europe across the religious divide, had very favorable political ramifications for the Pope, and this project endeared mathematics to the hearts of Catholics. In an age of religious strife and political disputes, the Jesuits hailed mathematics in general, and Euclidean geometry in particular, as an exemplar of resolving arguments with unassailable certainty through clear definitions and careful logical reasoning. They lifted mathematics from its subservient role well below philosophy and theology in the medieval tree of knowledge and made it the centerpiece of their college curriculum as an indispensable tool for training the mind to think in an orderly and correct way.I do not accept this. He argues that mathematics must accept nonrigorous work because someone might give it a rigorous foundation 3 centuries later.

The new, enviable position of mathematics in the Jesuits’ epistemological hierarchy came with a set of strings attached. Mathematics now had a new responsibility to publicly symbolize the ideals of certainty and order. Various dubious innovations, such as the method of indivisibles, with their inexplicable paradoxes, undermined this image. The Jesuits therefore viewed the notion of infinitesimals as a dangerous idea and wanted to expunge it from mathematics. In their view, infinitesimals not only tainted mathematics but also opened the door to subversive ideas in other areas, undermining the established social and political order. The Jesuits never aspired to mathematical originality. Their education was oriented toward an unquestioning study of established truths, and it discouraged open-ended intellectual explorations. In the first decades of the seventeenth century the Revisors General in Rome issued a series of injunctions against infinitesimals, forbidding their use in Jesuit colleges. Jesuit 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 battle over the method of indivisibles played out differently in England, where the Royal Society proved capable of sustaining an open intellectual debate. One of the most prominent critics of infinitesimals in England was philosopher and amateur mathematician Thomas Hobbes. A sworn enemy of the Catholic Church, he nevertheless shared with the Jesuits a fundamental commitment to hierarchical order in society. He believed that only a single-purpose organic unity of a nation, symbolized by the image of Leviathan, could save England from the chaos and strife sowed by the civil war. In the 1650s–70s his famously acrimonious dispute with John Wallis, the Savilian Professor of Geometry at Oxford and a leading proponent of the method of indivisibles, again pitted a champion of social order against an advocate of intellectual freedom. ...

In the 1960s, three hundred years after the Jesuits’ ban, infinitesimals eventually earned a rightful place in mathematics by acquiring a rigorous foundation in Abraham Robinson’s work on nonstandard analysis. They had played their most important role, however, back in the days when the method of indivisibles lacked rigor and was fraught with paradoxes. Perhaps it should not come as a surprise that today’s mathematics also borrows extremely fruitful ideas from nonrigorous fields, such as supersymmetric quantum field theory and string theory. ...

If, as in the case of the Jesuits, maintaining the appearance of infallibility becomes more important than exploration of new ideas, mathematics loses its creative spirit and turns into a storage of theorems.

It was only a few years later that Isaac Newton (and Leibniz and others) developed a coherent theory of infinitesimals. The subject was made much more rigorous again with Cauchy, Weierstrauss, and others in the 1800s.