Amir Alexander is a historian of mathematics. His new book is entitled "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". See here. Two questions:
(1) In what sense are these dangerous?
(2) The ban on infinitesimals and the trial against Galileo's alleged endorsement of heliocentrism date from the same year: 1632 (and in fact occurred within a month of each other). Is there any reason for such a coincidence?
What I find particularly interesting is Alexander's comment that infinitesimals were officially declared forbidden by catholic clerics on 10 august 1632. The reason this is interesting is because the date 1632 falls precisely in a critical period in Fermat's mathematical activity. Fermat originally introduced his techique of adequality in 1629, but it was first made known to a wider audience in the late 1630s. In the meantime infinitesimals have been declared persona non-grata. This may explain Fermat's legendary reluctance to talk about infinitesimals. In this he may have been more affected than for instance Wallis who spoke freely about infinitesimals. Wallis was not catholic but a presbyterian.
Note 1. I have edited the question to address the concern of critics. Interested readers are invited to click on the "reopen" button below.
Note 2. Wiki reports that the original heliocentric ban dates from 1615. Furthermore, In September 1632, Galileo was ordered to come to Rome to stand trial. He finally arrived in February 1633 and was brought before inquisitor Vincenzo Maculani to be charged. Thus the infinitesimal ban from august 1632 seems to be a separate development.
Note 3. Here is Amir Alexander's own description of his historical work: I am currently working on a new book, provisionally entitled Infinitely Small, which examines the interconnections between mathematics and political and social order. Mathematics, at its most abstract, is the science of order, and it follows that different conceptions of mathematics have been associated with different views of proper social arrangements. In particular, the book will examine a sequence of historical instances in which mathematical infinitesimals acquired political significance, showing that even the purest mathematics can at times serve to buttress or undermine a political order. See here.
Note 4. Paulos provides a hint of an answer in the following terms: To the Jesuits, tradition, resoluteness and authority seemed bound up with Euclid and Catholicism; chaos, confusion and paradoxes were associated with infinitesimals and the motley array of proliferating Protestant sects. See here.
Note 5. See also this NPR review.
Note 6. The latest review is in the Notices of the American Mathematical Society by Slava Gerovitch.