Infinitesimal #3

For centuries before the Reformation the Roman Catholic Church had reigned supreme in western Europe. Empires rose and fell. There had been invasions and occupations, heresies and plagues, but the Church had survived and thrived. The Church oversaw the lives of Europeans and gave order, meaning and purpose to their existence, and ruled on everything from the date of Easter to the motion of Earth and much else (p. 24). The Reformation was a challenge to all that.

The Jesuits, formed in 1540, were not much interested in mathematics in their first few decades. The founder, St. Ignatius, had little interest in mathematics. But as they built their education system, they became committed to Euclidean geometry. "It was the core of their teaching and the foundation of their mathematical practice. ... [T]he whole point of studying and teaching mathematics was that it demonstrated how universal truth imposed itself upon the world -- rationally, hierarchically, and inescapably. Ideally, the Jesuits believed, the truths of religion would be imposed on the world just like geometrical theorems, leaving no room for avoidance or denial by Protestants or other heretics and leading to the ultimate triumph of the Church. For the Jesuits, mathematics must be studied according to the principles and procedures of Euclid, or it should not be studied at all. A mathematics that ran counter to these practices not only was useless to their purposes, but it would challenge their unconquerable faith that truth, handed down through the hierarchy of the universal catholic Church, would inevitably prevail" (p. 74). Infinitesimals are not part of Euclid's Elements.

"The Jesuits valued mathematics for the strict rational order it imposed on an unruly universe. Mathematics, particularly Euclidean, represented the triumph of mind over matter and reason over the untamed material world, and reflected the Jesuit ideal not only in mathematics but also in religious and even political matters" (p. 91).

"Euclidean geometry was the embodiment of order. Its demonstrations began with the universal self-evident assumptions, and then proceed step by logical step to describe fixed and necessary relations between geometrical objects: the sum of the angles in a triangle is always equal to two right angles ... These relations are absolute, and cannot be denied by any rational being" (p. 119). A group of five Jesuits, the Rectors, had a strong control over what was taught, and it issued prohibitions on the teaching and promotion of infinitesimals (p. 122).

Cavalieri's and Galileo's ideas ran counter to that. They held that lines were comprised of indivisible points, planes of indivisible lines, and solids of indivisible planes. Thus these geometrical objects  were little different from the material objects we see around us (p. 91). Instead of mathematical reason imposing order on the physical world, pure mathematical objects are created in the image of physical ones. They wanted to study the world and find the order within. They were willing to accept some ambiguity and even paradox as long as it led to a deeper understanding (p. 177).

In Discourses (or Two New Sciences) Galileo made use of Aristotle's wheel paradox (link) to arrive "at a radical and paradoxical conclusion: a continuous line is composed of an infinite number of indivisible points separated by an infinite number of minuscule empty spaces. This supported both his theory of the structure of matter and his view that material objects are held together by the vacuum that pervades them" (p. 89-91). How Galileo thought about the wheel paradox is described here.