Who would have thought that for a period of decades a student's adherence to certain mathematical methods could get him in trouble with the Inquisition, imprisoned, or even burnt at the stake. Galileo was placed under house arrest for the last two decades of his life, not only for advocating the motion of the Earth, but also for the kind of mathematical analyses he published!

*Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World*, by Amir Alexander, chronicles the development of a "new" kind of mathematics, one that had actually existed alongside Euclidean geometry for centuries, but had been little used and was denigrated by Aristotle and others. It flowered along with the Italian Renaissance, but ran afoul of the reactionary politics of the Jesuits.

To most mathematicians of the early Renaissance, mathematics

**geometry, and all proofs and analyses that proceeded by any method other than straightedge-and-compass derivation from first principles were suspect. It is rather amazing to read how the Society of Jesus, originally rather blind to mathematics because of the proclivities of its founder, Ignatius of Loyola, took up Euclidean geometry as a point of pride within a generation after his death.**

*was*In their to-the-death struggle to throw back the influence of the Protestant Reformation, the Jesuits, brought into being as the Reformation was blossoming throughout Europe, realized that geometrical proofs provided a perfect model for their rigid theology and social structure. The Reformers declared that all persons had a right to know and understand Scripture, and offshoots such as some Anabaptists, and free-land proponents such as the Diggers, began to question the "divine right of the King" and the "natural order" of aristocracy. Dogma was being replaced by opinion. Long-held traditions were in danger of being overthrown. Chaos was imminent. The execution of the English king Charles I emphasized the danger.

If one accepts the validity of the methods of Euclid, there is no room for opinion. A geometrical constructive proof, proceeding by pure deduction, leads step by step to a conclusion that cannot be denied. But it had become evident to the disciples of Pythagoras, nearly a full twenty centuries earlier, that some propositions one could state, could not be proved. They had begun by proclaiming that all problems were subject to "rational" proof; by "rational" they meant using only ratios of whole numbers. An early demonstration that the hypotenuse of a square could not be exactly expressed as a ratio, that it was "incommensurable", led to the breakdown of the Pythagorean system and eventually to the disbanding of the Pythagoreans.

By Aristotle's time, about 200 years later, inductive methods based on "indivisible" quantities had shown some promise, and had been used to demonstrate certain propositions that geometric methods could not solve. But Aristotle, at first intrigued, later decried such methods. Euclid he could understand; the new methods seemed to allow a certain leeway for error. In his way he was as rigid as any Jesuit of the Sixteenth Century.

I have often been astounded that the Medieval Roman Catholic Church based so much of its philosophy on Aristotle, whose only brush with Theism is some vague statements about an "unmoved mover." I was further amazed to read of the process that led to this, via Thomas Aquinas. The Jesuits believed that Aristotle had it right. Mathematical induction by "indivisibles" (also called "infinitesimals" after about 1730) was unreliable. The Church needed …

**NEEDED!**… a rigidly reliable theology and rule of society that disallowed dissent as thoroughly as a Euclidean proof disallows "alternate opinion". Galileo was only the most prominent of a large number of Italian mathematicians to learn of inductive methods, and use them to great effect, so much so that these methods swept through Europe. But over about a century's time the Jesuits drove "indivisibles" out of Italy. Indivisibles and inductive methods flourished elsewhere, in all the countries of Europe.

Reasoning similar to that of the Jesuits led Thomas Hobbes to found his political philosophy on Euclidean geometry. He strongly felt that the chaos following the Reformation simply cried out for a more totalitarian form of government. His exceedingly famous book

*Leviathan*proposes the most profoundly totalitarian political system ever devised. When he learned that three very significant propositions were incommensurable via Euclidean methods, he realized that this left a great loophole in his philosophy.

Three problems: Squaring the Circle (making a square with the same area as a given circle), Trisecting an Angle, and Doubling a Cube (constructing a length that can be used to construct a cube with twice the volume of a given cube). None of these can be done using Euclidean geometric methods. This has been proven, using mathematical methods developed centuries after the time of Hobbes. He spent the rest of his life trying to square the circle, and eventually lost his reputation as a mathematician. He ran afoul of Gödel's Incompleteness Theorem: that

**mathematical system can be used to formulate problems that cannot be solved withing the confines of that system. This includes geometry. But Kurt Gödel was two centuries in Hobbes's future.**

*every*In the opening chapters of the book, it seemed to me that "indivisibles" and "infinitesimals" were described as being in opposition. It took careful reading to understand that they were synonyms separated by a century or two of usage. They form the foundation of The Calculus, as developed by both Newton and Liebnitz. The modern world would not exist without the analytical methods of calculus. From a modest number of "demonstrations" using induction—based on lines being composed of an infinite number of "indivisible" points, planes being composed of indivisible lines, and volumes being composed of indivisible planes—calculus and modern analysis in general have become supercharged, and now include both inductive and deductive methods.

I spent much of my adult life as a working mathematician, and I find it fascinating that such a life-and-death struggle had to be won, and won decisively, for the modern, technological world to appear. I have just touched on a few of the trends and a handful of the players in the saga of Infinitesimals. I have to mention John Wallis, whose 25-year battle with Hobbes "saved" inductive mathematics in England. How much longer would the modern era have been delayed otherwise? He originated the symbol for infinity:

**∞**.

*Infinitesimals*is quite an amazing story, very well told.

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