A Brief History of Infinity.

Infinity is a big topic, to put it mildly. The mere concept of a limitless quantity has vexed mathematicians, philosophers, and theologians for over two centuries. The Greeks developed some of the first infinite series, some divergent (they approach infinity) and some convergent (they approach a finite number), with Zeno making use of these concepts in some of his famous paradoxes. Galileo is better known for his observations in astronomy and work in optics, but he developed an early paradox that he argued meant that we couldn’t compare the sizes of infinite sets in a meaningful way, showing that, although we know intuitively that there are more integers in total than there are integers that are perfect squares, you can map the integers to the perfect squares in a 1:1 ratio that appears to show that the two sets are the same size. Georg Cantor later explained this paradox in his development of set theory, coining the aleph terminology for infinite sets, and then went mad trying to further his theories of infinity, a math-induced insanity that later afflicted Kurt Gödel in his work on incompleteness. There remain numerous – dare I say infinite? – unsolved problems in mathematics that revolve around infinity itself or whether there are an infinite number of some entity, such as primes or perfect numbers, in the infinite set of whole numbers or integers.

Science writer Brian Clegg attempts to make these topics accessible to the lay reader in his book A Brief History of Infinity, part of the Brief History series from the imprint Constable & Robinson. Rather than delving too far into the mathematics of the infinite, which would require more than passing introductions to set theory, the transfinite numbers, and integral calculus, Clegg focuses on the history of infinity as a concept in math and philosophy, going back to the ancient Greeks, walking through western scholars’ troubles with infinity (and objections from the Church), telling the well-known story of Newton and Leibniz’s fight over “the” calculus, and bringing the reader up through the works of Cantor, Gödel, and other modern mathematicians in illuminating the infinite both large and small. (It’s $6 for the Kindle and $5 for the paperback as I write this.)

Infinity can be inconvenient, but we couldn’t have modern calculus without it, and it comes up repeatedly in other fields including fractal mathematics and quantum physics. Sometimes it’s the infinitely small – the “ghosts of departed quantities” called infinitesimals that Newton and Leibniz required for integration – and sometimes it’s infinitely large, but despite several millennia of attempts to argue infinity out of mathematics, there’s no avoiding its existence and even the necessity of using it. Clegg excels when recounting great controversies over infinity from the history of math, such as the battle between Newton and Leibniz over who invented the calculus, or the battle between Cantor and his former teacher Leopold Kronecker, who disdained not just infinity but even the transcendental numbers (like π, e, or the Hilbert number) and actively worked to prevent Cantor from publishing his seminal papers on set theory.

Clegg’s book won’t likely satisfy the more math-inclined readers who want a crunchier treatment of this topic, especially the recent history of infinity from Cantor forward. Cantor developed modern set theory and published numerous proofs about infinity, proving that there are at least two distinct sets of infinities (the integers, aleph-null, are infinite, but not as numerous as the real numbers, aleph-one; aleph notation measures the cardinality of infinities, not the quantity of infinity itself). I also found Clegg’s discussion of Gödel’s incompleteness theorems rather … um … incomplete, which is understandable given the theorems’ abstract nature, but also meant Gödel earned very little screen time in the book other than the overemphasized parallel between his own descent into insanity and Cantor’s. I was disappointed that he didn’t get into Russell’s paradox*, which is a critical link between Cantor’s work (and Hilbert’s hope for a resolution in favor of completeness) and Gödel’s finding that completeness was impossible.

Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then it must be a member of R … but that produces a contradiction by the definition of R.

Clegg does a much better job than David Foster Wallace did in his own book on infinity, Everything and More: A Compact History of Infinity, which tried to get into the mathier stuff but ultimately failed to make the material accessible enough to the reader (and perhaps exposed the limits of Wallace’s knowledge of the topic too). This is a book just about anyone who took one calculus class can follow, and it has enough personal intrigue to hold the reader’s attention. My personal taste in history of science/math books leans towards the more technical or granular, but I wouldn’t use that as an indictment of Clegg’s approach here.

Next up: I’m reading another Nero Wolfe mystery, after which I’ll tackle Michael Ondaatje’s Booker Prize-winning novel The English Patient.