Everything and More:
A Compact History of Infinity
By David Foster Wallace
Norton, 319 pages, $36
David Foster Wallace's new book, Everything and More: A Compact History of Infinity, is representative of a new trend in publishing: novelists and other writers attempting to move into the field of popular science writing. On the face of it, there is nothing wrong with such a development, and the infusion of talent from fiction and drama into what may otherwise be perceived as a dry area should be a welcome development. But science writing does require knowledge about the subject area.
The best science and mathematics writers have always been scientists themselves. Brian Greene proved it with his book about string theory, The Elegant Universe, as have Roger Penrose, Stephen Hawking and Alan Guth, all of whom have penned bestsellers about scientific developments they themselves have contributed to.
A writer who is not a scientist or a mathematician should at least have a deep understanding of -- and respect for -- the subject. In the words of one renowned scientist, such understanding should be "nine levels above that of your readers." And nowhere is such mastery of the material as important as it is in abstract mathematics. For in order to explain complex mathematics to non-experts, the writer should have both the knowledge and the teaching experience of a professor in the field.
Unfortunately, novelist David Foster Wallace has neither. And what he lacks in knowledge and presentation ability, Wallace makes up for with glibness and a kind of unwarranted irreverence that do not do justice to the great mathematicians whose work he writes about. To a professional mathematician, much in this book will raise eyebrows, to say the least. Wallace's book is full of statements like: "In general, the situation of mathematics after 1700 is intensely weird." Three hundred years of immensely important mathematical developments, including the calculus, abstract algebra, topology, number theory and so on are summed up as "intensely weird"? And what about mathematics before 1700? Is it also "weird," or "not weird"?
Wallace attempts to describe the development of modern set theory -- one of the most abstract and complicated mathematical theories ever developed, and one with deep philosophical implications and paradoxes that required tackling by some of the greatest minds of the 20th century, including Bertrand Russell and Kurt Gödel. But he never gets to address any of the deep issues involved. Instead, he always goes back to his high school and college freshman experiences, to the point where the reader suspects that this is all he knows.
The subject of this book is infinity, which is the most perplexing and frustrating concept in the history of science. The ancient Greek philosophers and mathematicians from Pythagoras (fifth century B.C.) to Zeno and to Euclid (third century B.C.) were obsessed with this concept and developed some of the early theory to account for it. Wallace devotes some early chapters to ancient Greek contributions to the theory of infinity. He then addresses the Middle Ages, saying: "c. 500-c.1200 CE Nothing much going on in Western math thanks to Rome, Aristotle, Neoplatonism, Church, etc. The real action now is in Asia and the Islamic world." However, he only devotes one paragraph to "Asia and the Islamic world," moving right to 1260 and the work of Thomas Aquinas. He devotes a few pages to the history of mathematics from the 14th century to the 17th, glossing over some very important work -- including that of Galileo, who made a key discovery about infinity.
Wallace's sense of time and feel for chronology evidently leave something to be desired, for he switches back and forth between the 19th-century mathematician Georg Cantor and other mathematicians who lived earlier. Discussing the work of Bernhard Bolzano, who lived before Cantor, he writes: "Bolzano's approach to Galileo's paradox is purely abstract, and Cantorian." A mathematician would argue with the truth of this statement; but assuming it true, wouldn't it then be more correct to call Cantor's (later) work as "Bolzanian"?
Wallace soon arrives at the work of Cantor himself, a German mathematician who died in an insane asylum in Halle in 1918. Cantor is one of the most important and intriguing figures in modern mathematics, and there are several biographies of him. Wallace gives us some details of Cantor's life, but probably not enough to satisfy the reader's curiosity. He soon addresses the work Cantor did on the concept of infinity -- most importantly, the discovery that there are several different kinds of infinity. This surprising development is today understood by students of mathematics; but during Cantor's lifetime, he suffered from persecution by colleagues for his views. Perhaps this continuous harassment contributed to his mental illness. Overall, the treatment of the important mathematical concepts in Wallace's book is not satisfying. I was often left with questions that any reader would have about the material -- perhaps because Wallace lacks the ability to address these deep issues properly. The biographical details are sketchy at best. The book is full of acronyms: IYI, DBP, LEM, NL, VSP, CST, DE, EGI, and my favourite: GCPFS. Virtually every page is sprinkled with such meaningless letters, making the reading even more difficult. Perhaps the purpose of this device is to give an impression of the author's being in the know, although no one else uses these acronyms.
Overall, this book is very disappointing. I found mathematical misinterpretations, such as Wallace's statement that the ancient Greek philosopher Zeno somehow knew various kinds of infinity (a discovery made by Cantor two millennia later). I also found many mathematical statements that are patently wrong, such as the claim that complex functions are functions of functions.
There is great beauty to be found in mathematics. And a lot can be done to whet the general reader's appetite to find out more about the wonderful truths of pure mathematics and about the fascinating lives of the men and women who work to uncover these treasures of sheer knowledge. But Wallace is not the right expositor of these ideas.
Amir D. Aczel is a professor of mathematics and the author of10 books about mathematics and theoretical physics. His latest book is Pendulum: Leon Foucault and the Triumph of Science.