Book Review: Everything and More by David Foster Wallace.

Amid complex calculations and intricate graphs, a math history text yearns to emerge. Within that history book, a biography struggles for liberation.

David Foster Wallace's Everything and More: A Compact History of [Infinity symbol] focuses on mathematical developments in the 1800s, primarily regarding transfinite math.

In this study, Wallace highlights a gentleman who evidently held five names, not including titles: Georg F. L. P. Cantor, born 1845—the “acknowledged father of transfinite math.” (p. 5.)

At the outset, Everything and More acknowledges the difficulty of its task: "The aim is to discuss these achievements in such a way that they’re vivid and comprehensible to readers who do not have pro-grade technical background and expertise. To make the math beautiful—or at least to get the reader to see how someone might find it so. Which of course all sounds very nice, except there’s a hitch: just how technical can the presentation get without either losing the reader or burying her in endless little definitions and explanatory asides[.]” (p. 2.) Wallace explains further that he is not content to have the reader "merely knowing about Cantor’s accomplishments” but “appreciating them”. (p. 7.)

Unfortunately, Wallace does not attain his desired balance of achieving understanding without overloading the reader with technical details. I’m not certain how Wallace defines a “pro-grade technical background and expertise”, but this book’s math is basically accessible only to those with math or engineering degrees or a professional background therein. An example: “Since the cardinal number of denumerable sets is [Hebrew letter with numerical subset], it looks as if it would make sense to signify the set of all reals’ cardinality by [Hebrew letter with numerical subset]; but for complicated reasons Cantor designates this set’s cardinal number c, which he also calls ‘the power of the Continuum,’ since it turns out to be the nondenumerability of the reals that accounts for the continuity of the Real Line. What this means is that the [infinity symbol] of points involved in continuity is greater than the [infinity symbol] of points comprised by any discrete sequence, even an infinitely dense one.” (pp. 257-58.)

However, Everything and More’s historical and biographical aspects may be accessed by those who don’t possess technical expertise. These parts also contain interesting revelations, especially as they relate to Cantor. He, for example, “was in and out of mental hospitals for much of his later adulthood and died in a sanitarium in Halle [footnote omitted] in 1918.” (p. 5.) Wallace labors to underscore that Cantor did not derive “his most famous proofs about [infinity] while in an asylum….” (p. 167.) “Cantor’s first hospitalization was in 1884, when he was 39; most of his important work had already been done by then. He wasn’t hospitalized against until 1899. It was in the last 20 years of his life that he was in and out of places all the time.” (p. 167.)

Cantor was “something of a violin prodigy as a child. No one knows why he quit, but after a classical quartet in college there’s no more mention of the violin.” (p. 170.)

Elsewhere Wallace speaks of the high correlation of mathematicians and musicians (p. 181), which probably speaks to left brain functioning rather than a counterintuitive mix of art and science. I found this observation about mathematicians loving music to be especially intriguing because it partially explains my maternal grandfather Lawrence Schoenhals’ achievements in these heretofore seemingly disparate fields: he held a masters in math from the University of Michigan and also conducted an orchestra.

Wallace draws a biographical sketch of another important figure from this era: Karl Weierstrass. Wallace writes: “His early career is spent teaching high school in West Prussia (not exactly a hub), [footnote omitted] and he’s said to have been literally too poor to afford the postage for submitting work to journals. He finally starts publishing in the late 1850s, and sets math on its collective ear, and gets hired by prestigious U. Berlin as a prof—it’s all a long and kind of romantic story. (IYI [“If you're interested”] Weierstrass is also a conspicuous among mathematicians for being physically large, a gifted athlete, an inveterate partier and blowoff in college, indifferent to music (most mathematicians are fiends for music), and a cheery non-neurotic, gregarious, wholly good and much-loved fellow. He’s also widely regarded as the greatest math teacher of the century, even though he never published his lectures or even let his students take notes. [footnote omitted])” (p. 181; italics supplied.)

The book contains an unusual amount of what my English composition teacher in high school disparagingly called “Dear Reader notes”. Perhaps Wallace employs them to help guide the reader through dense verbiage about math (a difficult and imprecise task) or to make it “a piece of pop technical writing” as he asserts is his aim in the “Small But Necessary Foreword. (p. 1.) I however found them too clever by half and a hindrance. Example: “All right.” (p. 246.) That’s a complete sentence in Everything and More. Others: "Soft News Interpolation, Placed Here Ante Rem Because This is the Last Place To Do It Without Disrupting the Juggernaut-like Momentum of the Pre-Cantor Mathematical Context" (p. 167); and “End Q.F.-V.I. Return to §7c, at the ¶ on p. 256 w/ asterisk at end”. (p. 257.)

I recommend the book to those with strong math backgrounds and to others who can parse through the technical language to extract the nuggets of "nape-tingling genius" (p. 248) residing therein.