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.)
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.
(co-authored by Jennifer Thompson) for half price at Changing Hands in Tempe during my annual AFL trip, I picked it up without a second thought. The book covers a little of the same ground as the Einstein bio, but is primarily a history of superstring theory and the search for a “grand unified theory” (up to 1995, of course) that would bring together the four fundamental forces of physics, building the reader up from the mid-19th century forward through various stops and starts that included the proposal, discarding, and resurrection of string theory from the 1950s to the 1980s. 
for that – but it gave me more than just a superficial explanation along with plenty of the mind-bending stuff that makes theoretical physics seem fun to someone like me.
, another Hugo Award winner.
, the first book in a trilogy that posits a parallel universe where Neanderthals won the evolutionary battle over Cro-Magnons and have since become the dominant species on their version of Earth.
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, a book about the discovery of quantum mechanics and the difficulty the theory’s proponents had in convincing the advocates of the standard model of physics – a group that included Einstein and Bohr – that God does indeed play dice, at least with subatomic particles. The book is thorough, speaking as often as possible through the words of its many characters, while making a complex scientific subject easily accessible to lay readers who, like me, may not have taken a physics class in 20+ years.
*, about the lengthy and difficult question to understand these particular subatomic particles, ones that seemed to also defy conventional wisdom on how such particles should behave. Neutrinos are almost massless and can pass through an entire planet without touching another particle. They also explain the full process involved in beta decay, where an atomic nucleus emits an electron or a positron as well as electron neutrino (or antineutrino, but hold that thought). Without the neutrino to balance the scales, physicists were left with an apparent loss of momentum and energy from beta decay. As it turns out, the Italian physicist Wolfgang Pauli wasn’t just making stuff up when he posited the existence a previously unknown particle, which another Italian physicist, Enrico Fermi, dubbed the “neutrino,” or “little neutron.”
, a description of Copernicus’s earth-shattering (pun intended) discovery that the earth revolves around the sun, not, as the Catholic Church decreed, that the universe revolves around the earth. Copernicus also pointed out that the stars are much farther away from earth than scientists of his era believed them to be. Sobel’s book paled in comparison to her wonderful debut, 
, but also suffers from the paucity of original source material, as Copernicus left little besides his On the Revolutions of the Celestial Spheres, and after his death the work was condemned by the very church he’d once served as a canon.