Many of us may not remember when we first learned to count, but perhaps all of us remember learning to write our numbers, first from one to ten, and then as we learned the pattern, from one to one hundred, and beyond. At some time, some of us may even remember sitting down with a sheet of paper and a pencil with the goal of writing down all of the numbers. We labored for hours, or perhaps only minutes, before we realized that it was going to take more than one, or even a few, sheets of paper to finish the task. When I was in first grade, Larry (my best friend at the time) and I probably spent a week on the task, each night writing down more and more numbers, in a race to see who could write the most. At some point — I don't remember who was winning — we got bored and switched to Roman numerals, a change that slowed us down substantially. After writing the first few hundred or so Roman numerals, the game lost all its luster and we moved on to other things. That was my first exposure to the concept of infinity, and I never really gave it much thought until my calculus courses many years later when I was introduced to the ∞ symbol and learned to use it in formulas such as
By this time, I knew, of course, that there were an infinite number of numbers, but I was still troubled with questions such as what was ∞ + 1 and does ∞/∞ = 1? This book, Infinity: The Quest to Think the Unthinkable by Brian Clegg, helped clear up some of those questions for me.
In Infinity, Clegg takes the reader on a tour of the infinite, from the time of the ancient Greeks, through the early Christians, Europe in the middle ages, and finally the 20th century. In the journey, we meet a veritable who's who of philosophers, mathematicians, and scientists (Archimedes, Aristotle, Wallis, Galileo, Newton, Leibniz, Cantor, Gödel, to name a few) who have pondered, studied, and in some cases, have purportedly even been driven mad by the subject. Fortunately, Clegg makes the material understandable and approachable. This is clearly a general interest book, which is written for the lay person, and should be accessible to anyone with a basic high school algebra background, although knowledge of calculus would be beneficial. Throughout the book, basic mathematical topics are explained as needed. For example, Clegg explains power notation (32 = 9), the Pythagorean discovery that the square root of 2 is not a rational number (along with an informal proof), the irrationality of π, the definition of a function, Newton's and Leibniz's discovery of the calculus and their famous feud, set representation using Venn diagrams, and the number i. The mathematicians may find this fundamental material tiresome, but then again, they are not necessarily the target audience for this book, and could easily skip over those sections. With an undergraduate-level background in mathematics (I teach computing), I already knew much of this material, but what I found most interesting in Infinity is Clegg's emphasis on the history concerning the topic — the stories and biographical backgrounds of the major players. This is more a "history of mathematics" look at infinity than a rigorous mathematical exploration.
The book starts in ancient Greece with Zeno and four of his paradoxes which relate to the sequence of numbers: 1, 1/2, 1/4, 1/8, ..., and its sum, where today we use the notation "..." to mean "and so on without limit." The Greeks found such series troubling, and Clegg tells us the closest word they had to our concept of infinity was apeiron which roughly means without bounds in an uncontrolled and messy sort of way. Another example is the sum of the sequence 1, -1, 1, -1, 1, -1, ... . Parenthesized two different ways, we get (1 + -1) + (1 + -1) + (1 + -1) + ..., which clearly equals 0, but 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ..., is obviously 1. So which is it? 0? Or 1? Or does 0 = 1? From this simple example, we conclude right away that when we start pondering the infinite, things are destined to become weird.
Aristotle argued for the concept of potential infinity as a way to tame apeiron. Potential infinity is infinity that neither is nor isn't, because the word is can refer to something that is in existence at this exact point in time (e.g., there is a tree in my yard) or it can refer to something that has the potential to exist in the future. Aristotle uses the example of the Olympic Games. Certainly the games exist, because the concept exists, but the actual games themselves only exist at the time they are being held. So, when Aristotle says there are Olympic games, he means both that the Games do exist — at the time they are being held — and that they have the potential to exist. For Aristotle then, potential infinity exists as a concept, but unlike the Games, never arrives or can be reached. For him, and for mathematicians for centuries, this concluded the matter.
As sometimes happens when dealing with a subject which is somewhat vague and difficult for the human mind to grasp, some philosophers approached the infinite from a spiritual direction. The Greek Plotinus was one of the first to establish a correspondence between his god, the One, and the infinite, arguing that if the One was not infinite then there had to be something beyond, which was untenable to him. Clegg discusses St. Augustine, who argued that if the Christian God had such unlimited creative power, it would be difficult to believe that He could not conceive of anything larger than a certain number, no matter how large the number. St. Thomas Aquinas puts forth the argument that nothing this God could create could be infinite because such entities would essentially have to be specific and countable, and since there are no infinite numbers, the entities could not be counted. Unfortunately for Aquinas, Clegg tells us, the theory of countably infinite sets which was to arise in the 19th century ruined this bit of logic.
About a third of the way through the book, we've worked our way up to the mid 17th century and John Wallis, who is reported to have been the first to use the ∞ symbol to represent infinity. Noting that Wallis never discussed why he chose to use this particular symbol, Clegg reports that some historians of mathematics have suggested that it originates from the old Roman sign for 1,000, or perhaps from a closed up version of the last letter of the Greek alphabet, ω. Some have noted that the symbol somewhat resembles a Möbius strip, although Möbius was not born until long after Wallis had died, so we know Wallis was not influenced by his work. It's doubtful we will ever know.
In Chapter 11, "Set in Stone", which introduces the reader to basic set theory, we first meet Cantor, one of the most famous mathematicians supposedly driven mad by his work. It seems quite likely that Cantor had other problems that contributed to his breakdowns — including faulty genes — but it makes for a more interesting story to say that trying to comprehend infinity could drive a person to madness. As might be expected, a significant number of the latter chapters of the book are devoted to Cantor and his work on set theory. For those unfamiliar with the story, and even for those who are, it will make for fascinating reading (many more details of Cantor's tragic life can be found in the book Georg Cantor by Joseph Dauben). Clegg discusses not only Cantor's work, but also his feud with Kronecker and reasons for his breakdowns. After Cantor, near the end of the book, we encounter the story of Kurt Gödel, another famous mathematician supposedly "driven mad" by his work on infinity and the continuum hypothesis.
After spending most of the book dealing with the infinitely large, in Chapter 16, "Infinitesimally Small", Clegg ponders smaller and smaller numbers and the non-standard analysis of Abraham Robinson. He discusses briefly how this branch of mathematics finally allowed Brownian motion to be properly modeled.
If space and time are infinite, then is it also possible for them to be infinitely small? The space we can observe seems to exhibit that property. Molecules are composed of atoms, which are composed of subatomic electrons, protons, and neutrons, which are theorized to be composed of smaller quarks. Doesn't it seem logical to surmise that quarks — which Clegg tells us have never actually been observed — may be composed of smaller particles, each of which may be composed of smaller particles, and so on. There are no answers here, of course, but Clegg gives the reader a brief introduction to fractals, which are an example of a shape with an infinitely long outline that can be drawn in normal, finite space.
The final chapter wraps things up with an interesting discussion of a mathematical construction called Gabriel's Horn, which I was not familiar with. Gabriel's Horn is constructed by considering the function f(x) = 1/x and rotating it about the x-axis to construct a 3-dimensional structure. It turns out, although Clegg does not prove it, that the volume of this structure is a finite number, π, but the surface area is infinite. This is another example of the bizarreness that we encounter in the land of infinity, and it makes for a nice finish to the book.
This was an entertaining and interesting book. The formal mathematics is kept to a minimum, and the only place where I actually felt a little confused was in the chapters on Cantor and the discussion of aleph-0 and the continuum hypothesis, but a more careful reading cleared that up. All in all, Clegg did a great job on making this book an interesting and (relatively) easy read on a topic which is definitely fascinating.
Kevin R. Burger is Assistant Professor of Computer Science at Rockhurst University in Kansas City, Missouri. His primary scholarly interests are in undergraduate computer science education, teaching programming, discrete mathematics, and algorithms. Outside of work, he enjoys throwing pottery on the wheel, lifting weights at the gym, hiking, listening to music, and drinking coffee at the Broadway Café.