Charting Infinity's Elusive Past
Mathematics alone make us feel the limits of our intelligence. For we can always suppose in the case of an experiment that it is inexplicable because we dont happen to have all the data. In mathematics we have all the data . . . and yet we dont understand.
Loyal fans of David Foster Wallaces fiction are likely to be disappointed in his new, condensed treatise on the history of infinity if they’re looking for their fix of his burnished prose and psychological insight. The common denominator between his fiction and his latest book, Everything and More: A Compact History of Infinity, is complexity. Wallace, ever the lover of technical jargon, is a logical choice for Nortons Great Discoveries series to explore the concept of infinity and how mathematicians grappled with and evaded its explanation for over two thousand years, until Georg Cantor theoretically proved its existence in the late 19th century.
Wallaces signature linguistic flair, characterized by sentences fancifully scattered with words culled from unabridged dictionaries and compendia of medical terminology, finds itself inverted here. His prose deciphers complex concepts, rather than acts the cipher, out of necessity to simplify the mathematical ideas he unfolds. While this may be Wallaces attempt at mastering the technical book, its didactic purpose and instructional slant limit his ability to revel in nuances of language and lexicon. But as the pianist who adeptly plays a concerto with one hand tied behind his back, Wallace comes out looking all the more gifted for having pulled off the feat.
Despite the aforementioned restraints, Everything and More stands apart from most educational primers on math in its play with language and by interspersing levity as the conceptual load grows dense. At points, Wallace’s attempt at humor seems forced—such as with his peculiar assertion that advanced math is sexy. The repetition of the adjective “sexy” in reference to terminology and equations becomes a tic that you either find endearing or annoying, as is typical with offbeat, eccentric teachers whose devotion to their subject inspires both awe and dismay.
Wallace realizes that the mental labor required to understand abstract mathematics and the dry language needed to explain its concepts could cause ceaseless yawning and migraines, preventing a considerable portion of readers from finishing this book. The self-consciousness Wallace employs in his fiction pays off here, as it’s used to appeal to the reader, to make the material more accessible and enticing. What other serious mathematical text could pull off the metaphor “a towering baklava of abstractions and abstractions and abstractions” or apologize for its necessary, yet confounding verbiage with “Sorry about the occluded math-prose”?
These are serious merits, considering the book takes on a level of mathematics that many would preferably avoid. As Wallace acknowledges, most popular explorations of leading mathematical figures and movements adopt a humanistic approach, often exploring facets of the psychological instability that can accompany mathematical genius, rather than tackling the genius’s actual contribution to math. For example, the crux of Darren Aronofsky’s film Pi depends on the intricate interconnection of numbers, language, and life, yet it only loosely draws from theory to stage a fictional account of a mathematician’s search for a life-governing equation. Everything and More similarly draws connections between mathematics and language, but here the analogy’s purpose is to give greater insight to the realm of mathematics, to show that math is a language made up of numbers and equations that express and explain the composition of the world and life in general.
Wallace’s decision to go spelunking in the caverns of abstract math—an exploration that includes discussions of surds, transfinite numbers, convergence, and degrees of infinity—means that his audience requires a pre-established foundation to understand what he’s talking about. Wallace is noble, but also a bit naïve in the area of mathematical instruction, for attempting to engage readers at various levels of expertise. Though he posits in the book’s first pages that >“no particular experience or recall of college math is actually required for this booklet,” he simultaneously acknowledges that “it seems only reasonable to assume that some readers will have strong math backgrounds, and only polite to acknowledge this from time to time.” This predicament is akin to an elementary school teacher trying to contend with a mixed class of TAG and remedial students.
Interpolations, emergency glossaries, and IYI (If You’re Interested) sections abound in an attempt to remedy the situation with both positive and negative results. It is possible to skim through the optional theorems and breeze by the more complicated mathematical deductions, but doing so turns the book into a choose-your-own-adventure text that is interesting, but not necessarily complete.
The book’s strongest point, and where Wallace really shines, comes as he navigates the history of mathematics and the series of advancements that each supplied a piece of the mathematical puzzle. Wallace, steeped in the knowledge of all things philosophical and Greek (even the book’s dedication to his parents is in Greek), thoroughly lays the groundwork in both philosophical and mathematical thinking to provide the reader with respect for why a mathematical understanding of the infinite was so elusive for so long.
The ancient Greeks geometrical understanding of math, coupled with the rise in popularity of Aristotelian thought and its denial of the infinite, prevented the move to mathematical abstraction needed to explain infinity. Thus the status quo went unchallenged for over a thousand years until the advent of calculus and the Industrial Revolution turned math’s focus to the abstract. The quest for a better understanding of the infinite culminated in Georg Cantor’s development of set theory, providing him with the tools necessary to mathematically prove the existence of infinity.
This account of infinity, in itself, is enough to make a head reel, as paradoxes, mind games, and riddles that left the greatest minds in mathematics stumped for ages are packed into a matter of a few hundred pages. From elucidating Zeno’s Paradox (the problem of getting from here to there if you have to cross an infinite number of halfway points to reach your intended destination) to explaining that infinity itself has various degrees (infinity refers to both the infinitesimally small and the infinitely large, and that even once infinity can be defined, there are still infinite numbers that are larger than the base infinity), the book demands an exponentially larger amount of mental exertion than its length would suggest. Fortunately, even if your mathematical skills aren’t refined enough to follow Wallace’s explanations in mathematical terms, he supplies enough background, guidance, and brain teasers to respect the genius of Cantor’s breakthrough and to duly appreciate the ancient Greeks’ aversion to anything abstract that evaded definition.