My two teachables, the subjects which I will be qualified to teach when I graduate from my education program in May, are mathematics and English. When I tell people this, they usually express surprise, saying something like, “Well, aren’t those very different subjects!”
And it irks me so.
They’re not, not really. Firstly, mathematics and English are both forms of communication. Both rely on the manipulation of symbols to tell a tale. As with writers of English, writers of mathematics have styles: some are elegant yet terse, seemingly expending little effort while getting their point across with an admirable economy of symbols; others are expansive and eloquent, elaborating at some length in order to furnish the reader with an adequate explanation. Secondly, as with English, mathematics is very much grounded in philosophy and history, and it is a subject that is open to deep, almost spiritual interpretation.
If you balk at that last idea, don’t worry. You’ve probably had it drilled into your head since elementary school that in mathematics there is only one correct answer! How could such a reassuringly logical subject be open to interpretation? Despite its apparent objectivity, mathematics is just another human endeavour, and like all our mortal works, it is vulnerable to our flaws, foibles, and fits of passion. Mathematicians can be just as stubborn and argumentative, if not more, than other people. There are many famous follies and feuds in the history of mathematics, and that is one of the reasons I enjoy learning about it so much.
Infinity is one of the mathematical concepts most central to those feuds. It’s one of the areas where math rubs up against the spiritual realm—for, as some mathematicians and philosophers have wondered, what is infinity if not God or some kind of greater being? So it seems natural to look at our shifting views on the infinite along the continuum of the history of maths. In A Brief History of Infinity, Brian Clegg does just that, following the classical, somewhat Eurocentric development of math from Greece to Rome, then zig-zagging down to the Middle East and India before flying back to Britain, France, and Germany.
As with most tricky math concepts, the trouble with infinity begins with its definition. One must be very careful with definitions in math—for example, it is not enough merely to say that infinity means “goes on without end”. After all, the surface of the Earth has no “end”, but that does not mean the Earth has infinite surface area! Rather, the surface of the Earth is unbounded. Grasping the idea of infinity as “not finite” is easy enough, though: there is no “last” counting number, because you can always add one to the largest number you can conceive, and suddenly you have a new largest number. So infinity is a quicksilver of a concept: intuitive and easy to grasp, yet also elusive and far too fluid for some mathematicians to handle. The Greeks, with their mathematics strictly confined to the geometric figure, would have no dealings with the infinite. Infinity confused Galileo, who nevertheless bravely meditated upon it in his final days. And the shadow of infinity hangs over the controversy of the calculus that caused the divide between Newton and Leibniz, and correspondingly, between Britain and the Continent.
The story of infinity gets even more interesting after that. In general, I love the history of mathematics during the 1700s and 1800s. So many brilliant minds pop up during that time: as Newton and Leibniz exit, Euler and Gauss enter. Later, Cauchy and Weierstrass formalize the concept of the limit, which does away with any need for infinity in calculus at all! There are plenty of names and plenty of stories—and this is where A Brief History of Infinity starts to lose its edge.
The first few chapters of this book are fascinating. Clegg devotes a lot more space to the Greek philosophers than others might, going so far as to mention some of the more obscure ones, like Anaxagoras. He provides a considerably detailed development of Zeno’s paradox (well, paradoxes) and a nice, if basic, grounding in the idea of an infinite series. Clegg lays the ground well for what will come in later chapters, all the while emphasizing the reluctance of the Greek philosophers to abandon the solidity of numbers found in the real world.
But as we get closer to those magical two centuries following the great Newton–Leibniz schism, the story of infinity gets more complicated as more people get involved. This book is very similar to Zero: The Biography of a Dangerous Idea. In my review of Zero, I praised the author’s ability to stay focused:
The story intersects with the lives of many famous mathematicians, but the obvious slimness of this book testifies that Seife managed to distill only what was necessary about their lives in his quest to explain the mystery of zero.
To be fair to Clegg, this book is almost as slim as Zero. And although he happens to go off on many a tangent, he at least has the ability to find his way back on track quickly enough—that is, his tangents are interesting and informative. He sometimes seems to go into more detail than is strictly necessary to get the point across, and once in a while he waxes melodramatic—as is the case when he links Cantor’s madness to his study of infinity. Overall, however, Clegg’s writing is crisp and clear.
I’m also impressed by the detail and depth of Clegg’s explanation of the math. He goes so far as to list and briefly elaborate upon each of the axioms of Zermelo-Fraenkel set theory! I was half expecting him to mention the Banach–Tarski paradox after that—he doesn’t quite get there, but he does explain the difference between ordinals and cardinals, develop the continuum hypothesis, and even mention Gödel’s Incompleteness Theorem. He tackles whether imaginary numbers are truly all they’re cracked up to be. And he even discusses nonstandard analysis—we didn’t even learn about that in university.
Don’t let my awe scare you away, though. Rather, think of it like this: if you are not particularly mathematical and read this book, you will gain a wealth of knowledge. You will be fun at parties! If you are particularly mathematical, then depending on how much you like the history of math, you might already be familiar with most of these anecdotes. But the book will still be fun to read, and chances are you will learn at least one or two new things.
So I would recommend A Brief History of Infinity to most people—perhaps not with the same zeal that I do Charles Seife’s Zero, but with a similar hope in mind. I hope this book, or at least my review of this book, demonstrates why I find math, as well as the history of math, so fascinating. It’s not just all about numbers, solving for x, and finding the One True Solution. Mathematics is a subject with a long and storied past, one that is fun to explore by looking at the humans who progressed—or regressed—throughout the centuries. A Brief History of Infinity is a book in this mould. While its organization and its focus leaves something to be desired, its scope and ambition do not.