Infinitesimal #4

A intellectual war involving math and politics also occurred between Thomas Hobbes and John Willis. Near the end of the English Civil War (1642-1651) Hobbes wrote Leviathan. In it Hobbes argued for a social contract and rule by an absolute sovereign. He wrote that civil war and a brutal state of nature ("the war of all against all") could only be avoided by strong, undivided government.

Hobbes also was a geometer of some repute. Similar to the Jesuits, he believed that the answer to uncertaity and chaos was absolute certainty and eternal order. They believed the key to both was Euclidean geometry. He set about trying to "square the circle" and solve two other long-standing geometry problems. "Square the circle", or "quadrature the circle," means construct a square with area exactly equal to the area of a given circle. (It can be done with great, but not perfect, precision.) Under the traditional restrictions of using only a compass and straight edge, this had been proven impossible. Hobbes tried anyway. Mathematician John Willis was well prepared to discredit any solution Hobbes proposed. Willis' political attitudes also reflected the chaotic years in England, but he believed in a state that would allow for a plurality of views and wide scope for dissent. Willis also sided with those who supported the use of infinitesimals.

Hobbes was also a sharp critic of the mathematical works of Willis. For Hobbes the infinitely small was an unwelcome intruder in mathematics. In contrast Willis considered practically all the features of the infinitely small to be clear advantages. His math was for investigating the world as it is. The world could be a little mysterious, unexplored, and ambiguous, but it invited new investigation and new discovery (p. 287).

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.

Infinitesimal #2

What are infinitesimals? That term was coined around 1670. A related term is indivisibles.  The author of Infinitesimal says: "To understand why the struggle over indivisibles became so critical, we need to take a close look at the concept itself, which appears deceptively simple but is in fact deeply problematic. In its simplest form the doctrine states that every line is composed of a string of points, or "indivisibles," which are the line's building blocks, and which cannot themselves be divided" (p. 8-9).

Some mathematicians, especially Cavalieri, held that a line is composed of an "infinite" number of points, a plane is composed of an "infinite" number of lines laid parallel and a solid is composed of an "infinite" number of planes laid parallel. Of course, for any polygon, sets of parallel lines can be drawn in different directions, e.g. vertical or horizontal or diagonal, which gives a different number of lines.

Infinitesimal mentions that the math symbol used for infinity, ∞, was invented by John Wallis (who has a prominent role in the book). Wikipedia confirms this. Wallis used 1/∞ for an infinitesimal.

One conundrum is how many indivisibles (or points) there are in a given line and how small they are. Assume a given line has a huge, yet finite, number, we can call a gazillion. That suggests each indivisible is 1/gazillion in size. Problems start arising when considering other lines of different length. A "deep" problem -- discovered by the Greeks -- is that some lines can not be subdivided into an integer number of uniform-sized units, no matter how small they are. For example, the square root of two length units (e.g. inches) cannot be converted to an integer number of anything, because the square root of two is an irrational number. Ditto for pi, the circumference of a circle with a diameter of 1 length unit, and Euler's number e, the base base of natural logarithms.

A related topic is the continuum. The usual meaning of continuous is “unbroken” or “uninterrupted.” Thus a continuous entity—a continuum—has no “gaps.” Aristotle addressed it in book 6 of his Physics. He concluded that the concept of infinitesimals was erroneous, and that continuous magnitudes can be divided ad infinitum (p. 10). The Jesuit Benito Pereia proposed the thesis that a line is composed of separate points and presented all the arguments in its favor by others. He then demolished them one by one, and concluded, like Aristotle, that the continuum is infinitely divisible, and not composed of  indivisibles (p. 121). The Jesuits thought Cavalieri's were untenable.

The Epilogue of Infinitesimal describes how the use of "infinitesimals" later grew to have a prominent part of mathematics. I believe it would have been improved if it explained the concept of a limit as used in math, such as in calculus. Said concept was developed in the 19th century. It involves, but is not identical to, "infinitesimals" as described in the book. The similarity is stronger in integral calculus.

A much longer history of infinitesimals and related topics is here. The historical period the book is mainly about is a very small part of it.

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.