Infinitesimal #1

I read the book Infinitesimal: How A Dangerous Mathematical Theory Shaped The Modern World. In mathematics, infinitesimals are things so small that there is no way to measure them. I thought the history in Infinitesimal – both political-religious and mathematics from about 1500 to 1675 – was very interesting. I believe the author makes the tie between them stronger than what they actually were, but there were parallel ideas - parties opposing one another in two very different realms.

The Society of Jesus, more commonly called the Jesuits, has a prominent role. Before Martin Luther initiated the Protestant Reformation, the Catholic Church was the dominant power in society. Kings and their lower ranking brethren depended on approval by the Catholic clergy. The anti-Reformists believed that the Reformation would bring about disorder and war. The Jesuits became the leading defenders of Catholicism. In large part their success was due to their building of educational institutions.

A leading Jesuit, Christopher Clavius, was almost single-handedly responsible for the adoption of a rigorous mathematics curriculum – Euclidean based -- in an age where mathematics was often ridiculed by philosophers and religious authorities. While Clavius clearly opposed the heliocentric model of Copernicus, it was mainly other Jesuits who opposed infinitesimals.

A leading proponent of infinitesimals was mathematician Bonaventura Cavalieri. He was a Jesuat, which is different from a Jesuit.

Except as noted below, the author summarizes the book's thesis very well as follows.

"Why did the best minds of the early modern world fight so fiercely over the infinitely small? The reason was that much more was at stake than an obscure mathematical concept. The fight was over the face of the modern world. Two camps confronted each other over the infinitesmal. On the one side were ranged the forces of heirarchy and order – Jesuits, Hobbesians, French royal courtiers, and High Chuch Anglicans. They believed in a unified and fixed order in the world, both natural and human, and were fiercely opposed to infinitesmals. On the other side were comparative "liberalizers" such as Galileo, [John] Wallis, and the Newtonians. They believed in a more pluralistic and flexible order, one that might accommodate a range of views and diverse centers of power, and championed infinitesmals and their use in mathematics. The lines were drawn, and a victory for one side or the other would leave its imprint on the world for centuries to come" (p. 8).

Most of the history presented in the book happened before Isaac Newton published his revolutionary Principia  in 1687, so Cavalieri instead of "the Newtonians" arguably fits better.