Infinitesimal
By Amir Alexander
Scientific American/Farrar, Straus and Giroux, New York, 2014

Amir Alexander proves there is no subject too small to write a book about. Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World is about the mathematics of the infinitely small. It’s about the politics of the infinitely small too, as we’ll see shortly.

Let’s start, as Alexander does, with the math. What exactly are infinitesimals? Well, they’re the smallest possible quantities, infinitely small in fact. If you think back to your elementary school geometry lessons, you might remember being taught that a line is a collection of points. You might even have been told that those points have no size, that is, they have no dimension. They are immeasurably small. Those points are infinitesimal in size. Ringing any bells?

Here’s the problem: If these infinitesimal points have no size, how can they form a line that has any length? After all 0 + 0 = 0. Even if you arranged an infinite number of these infinitely small points in a line, their total length would still be zero. On the other hand, suppose these points are the fundamental building blocks of a line, like quarks are to atoms and molecules. They are very, very small but cannot be divided any further. Well the problem with these indivisible points is that we can always imagine a line only half as long as one of them. Or a tenth, or a billionth. In other words, the indivisible points are not indivisible after all.

So no matter how you look at infinitesimals or indivisibles as they were often called, you end up with paradoxes.

These paradoxes were known to ancient Greek mathematicians like Zeno and Pythagoras. The Greeks, and generations of mathematicians after them, seem to have decided to avoid infinitesimals and their headache-inducing paradoxes and instead focused on developing geometry as a highly systematic discipline based on deductive reasoning from a set of very simple assumptions.

With one exception: Archimedes. Archimedes, possibly the greatest mathematician of the ancient world, used infinitesimals to calculate the volumes of cones, cylinders and spheres. He calculated the volume of a cylinder, for example, by imagining it to be a stack of very thin circles, like a neat pile of salami slices. Now imagine stacking thinner and thinner slices, to the point where there’s an infinite number of indivisibly thin slices (ignoring the paradox for a moment). By adding up the area of each slice, Archimedes was able to arrive at the total volume of the cylinder. He used the same method for cones and spheres.

Archimedes recognized the paradoxes of indivisibles, but he also demonstrated their incredible power. In fact, he was about fifteen centuries ahead of his time: his method was a pre-cursor to integral calculus.

OK, so infinitesimals are paradoxical and possibly controversial within the field of mathematics. But Alexander calls them a dangerous mathematical theory. Why? And to whom?

Now we come to the political part of the story. Actually the political part of the story makes up the bulk of the book. Alexander examines in detail two historical periods; the Counter-Reformation in Italy starting in the mid-1500s, and England following its civil war in the mid-1600’s. In both cases the respective societies had gone through a period of political instability, chaos even, followed by a longing for a return to a more stable, predictable order.

Take Italy. In 1517 Martin Luther nails his 95 theses to the church door in Wittenberg. The Catholic Church faces its greatest challenge, one from which it will never really recover – the Reformation. Not only is the religious authority of the Catholic Church threatened, the entire European social and political order fractures. War and upheaval spread throughout Europe. It’s hard for us today to imagine the scope of change caused by the Reformation. The collapse of the Soviet Union and the end of the Cold War are the nearest analogues in our time, but that changed only the post WW II order that had prevailed for a mere fifty years. The Roman Church had provided structure and order for over a thousand.

In response, the Catholic Church forms the Society of Jesus, better known as the Jesuits in 1540. The Jesuits begin establishing Jesuit schools throughout Europe – some still exist today – to provide rigorous education in philosophy, religion, language and mathematics. But it was a particular kind of mathematics – geometry – that was taught in the Jesuit schools. Geometry was seen as ordered and disciplined, proceeding top-down through logical steps from clear principles to rational conclusions. Perfect for an institution trying to re-establish its hold on society.

Infinitesimals, recently re-discovered and developed by Italian mathematicians like Bonaventura Cavalieri, EvangelistaTorricelli and Galileo were paradoxical, radical, and controversial, a threat to the Church. In Italy their work was opposed and eventually suppressed by the Jesuits, causing, Alexander argues, the decline of Italian leadership in math and science.

Similarly, in England, those seeking to re-establish order after the English Civil War, notably the philosopher Thomas Hobbes, sought to impose a rigorous geometrical approach to almost all aspects of philosophy. Here again, infinitesimals were seen as a threat to this highly ordered worldview, though in England, the idea flourished.

Infinitesimals aroused such strong opposition because, as Amir Alexander says,

“… the infinitely small was a simple idea that punctured a great and beautiful dream: that the world is a perfectly rational place, governed by strict mathematical rules.” p. 13

On the other hand, for those chafing under the constraints of rigid hierarchical approaches to science and mathematics, infinitesimals opened up new possibilities for more flexible, more empirical approaches and, Alexander argues, helped usher in the modern world.

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Throughout the book, Alexander uses the terms “indivisible” and “infinitesimal” interchangeably. Perhaps historically that’s how they have been used. However, I found this terminology quite confusing because it doesn’t match how I learned calculus.

I don’t ever recall hearing the noun “indivisible” when I was a student. Instead we were taught the relatively recent (well, 1800’s) idea of limits. The idea behind a limit is that you can get closer and closer to a particular value or point without ever actually reaching it. You can get arbitrarily close, as close as you need to, within an infinitesimal distance. However, we were never taught that there was a hard, indivisible quark-like distance that could not be subdivided further. Alexander doesn’t address this evolution in terminology so I was a little confused at the beginning of the book, and then a little irritated through the rest of it.

But this is a quibble.

On the whole, I found Infinitesimal to be a really interesting book. It took me through a couple of periods of European history that I knew little about and did so from a unique perspective: mathematics. That said, you’ll be disappointed if you read the book expecting a deep investigation of the mathematics behind infinitesimals. And it certainly is not a history of the development of calculus. Alexander barely mentions Newton and Leibniz and doesn’t discuss limits at all.

The center of gravity of the book really is the history. Amir Alexander teaches the history of math and science at UCLA, and if this book is any guide, he does it well. In fact, Infinitesimal is not so much a history of the development of the theory of infinitesimals, but rather an account of how the theory of infinitesimals was influenced by historical developments. I find it difficult to gauge just how important the tussle over infinitesimals really was in the greater historical struggles of either 16^th Century Italy or 17^th Century England. The book’s subtitle, How a Dangerous Mathematical Theory Shaped the Modern World, seems overblown to me. However, we should always remember that math and science are human endeavors that occur within a historical context which they influence and are influenced by. Infinitesimal is a great illustration of this fact.