Intereting Posts

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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?

- Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+1/3 }{x^3-x+1})^2 \mathrm{d}x$
- Prove that the sum of convex functions is again convex.
- Proving a reduction formula for the antiderivative of $\cos^n(x)$
- Is $f(x,y)=(xy)^{2/3}$ differentiable at $(0,0)$?
- Find the value of $\space\large i^{i^i}$?
- Is the following Harmonic Number Identity true?

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.

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- continuous partial derivative implies total differentiable (check)

I believe this has to do with Jesuit opposition to atomism, rather than their position on infinitesimals. Of course, the two are linked and evolved together in the early 17th century. Today we consider atomism a physical theory, but at the time there was no distinction between a mathematical continuum and the physical continuum, just as there was no distinction between Euclidean geometry and the geometry of the space around us.

Aristotelian physics maintained that time, space, and matter were infinitely divisible, and the Jesuits had sided with this idea. They kept records over various ideas which they had debated and found to be flawed, and atomistic ideas appear here several times throughout the first half of the 17th century.

The idea that the continuum consisted of *finitely* many indivisible particles, each with some physical extension, was considered to be contrary to dogmas about the Holy Communion, and hence particularly offensive. It could be taken to imply that Christ was present in the bread and wine only to a limited degree, corresponding to the number of indivisibles present. This idea was explicitly forbidden in 1608, and in the following years the Jesuit doctrine was refined to forbid atomism also in the case when there were considered to be *infinitely* many indivisibles.

Galileo used some atomistic ideas to explain his new physics. When his Dialogue was published in February 1632, it would be natural to examine these ideas again, and presumably this is what happened in the meeting in August 1632 mentioned by Alexander.

(For some more details, see the chapter by Palmerino in The New Science and Jesuit Science: Seventeenth Century Perspectives. She does not mention the meeting in 1632, though.)

I got the book. An interesting read. It is historical, not a novel.

The “prohibition” mentioned was a prohibition by the Jesuits of what could be taught at Jesuit schools and colleges. Their educational system, probably the best in the world at that time, was based on a unified curriculum. The idea of “indivisibles” was banned, being contrary to Aristotle. “Indivisibles” was the early form of what became the integral calculus years later.

Amir concentrates on two places the contest of whether “indivisibles” should be allowed in mathematics was played out. In Italy, where the two sides were the Jesuits and Galileo’s followers. And in England, where the two sides were Hobbes and Wallis. The book contains a lot of interesting background material on those protagonists (the Jesuits, Thomas Hobbes, John Wallis).

A belated addition: from various reading, which I cannot recall precisely, the Jesuits were unhappy far more about “atomism” than heliocentrism, because “atomism” would seem to strongly indicate that “transubstantiation” (the alleged conversion of wine into Jesus’ blood, or whatever) was impossible.

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