Everything and More
Everything and More: A Compact History of Infinity is the first in the series of W.W. Norton's "Great Discoveries" series, which are math- and science-focused books written mostly by authors outside the scientific field. Throughout this "booklet" (3), as the narrator calls it, Wallace explains and examines the complex history of infinity in the context of mathematics. In particular, he focuses on nineteenth-century mathematician Georg Cantor, the inventor of set theory. Wallace traces the concept of infinity (as well as resistance to it) from the Greeks through the twentieth century, paying special attention to the beauty and often counter-intuitive nature of mathematics. Although Wallace states that Everything and More is a "piece of pop technical writing" (1) designed to appeal to literature enthusiasts, he also includes phrases titled "IYI" ("If You're Interested") to appeal to those with more of an interest in or knowledge of mathematics.
Everything and More exists as a fusion of historical account, mathematical textbook, and story-telling. Wallace combines the story of math’s journey toward understanding infinity with textbook-style symbols, glossaries, explanations, and proofs—all wrapped up in a delightfully conversational tone, as if Wallace were the teacher, the text his classroom, and the readers his students.
Everything and More does not include an explicit definition of infinity, but does bestow infinity a central position in the story of mathematics and figures infinity as a problem, a solution, a controversy—whose story parallels the story of human progress.
Wallace identifies infinity as both a primary source of problems for mathematics as well as an integral solution that moves math forward. On the one hand, Wallace claims that “nothing has caused math more problems—historically, methodologically, metaphysically—than infinite quantities” (32). In fact, according to Wallace, the story of math is the history of infinity-related problems (32). On the other hand, a careful parsing of the different types of infinity—infinity as quantity and infinity as procedure—leads to solutions and advancement in mathematics. For instance, discerning the nuances of infinity led mathematicians to a solution for Zeno’s Paradox: “The Dichotomy’s central confusion is now laid bare: the task of moving from point A to point B involves not [an infinite-number] of necessary subtasks, but rather a single task whose ‘1’ can be validly approximated by a convergent infinite series” (195).
Furthermore, in expounding “math’s battle over [infinity]” (205), Everything and More captures infinity as controversial, as a concept whose existence was constantly threatened throughout the history of math in spite of, or perhaps because of, all the problems and solutions it created. While Cantor’s work attests that “actually-infinite sets can be understood and manipulated, truly handled by the human intellect” (205), Wallace nonetheless posits infinity as “a concept it [math] couldn’t really ever handle anyway” (68)—because with each paradox that infinity solves and each testament to its prospect of being “handled,” infinity introduces a new wave of paradoxes and problems that dashes previous hopes.
Wallace summarizes: “the Story of [infinity’s] overall dynamic, whereby certain paradoxes give rise to conceptual advances that can handle those original paradoxes but in turn give rise to new paradoxes, which then generate further conceptual advances, and so on” (278). The way Wallace figures it—the story of infinity mirrors the story of human progress, where problems lead to solutions that lead to new problems and so on, infinitely.
Infinity and Mental Illness
Everything and More forges a strong link between infinity and mental illness and argues that the origin of this link lies in abstraction. Wallace observes that “Historians and pop scholars tend to spend a lot of time on Cantor’s psychiatric problems and on whether and how they were connected to his work on the mathematics of [infinity]” (6), but he goes further to investigate the reasons for this connection—between psychiatric problems and the mathematics of infinity. Wallace concludes that the instigator of madness is the abstract way of thinking. He qualifies G.K. Chesterton’s claim that logic induces madness and instead asserts that “what Chesterton’s really trying to talk about is one of logic’s main characteristics—and mathematics’. Abstractness. Abstraction,” for “logic is just a method, and methods can’t unhinge people” (8). In so doing, Wallace implies that abstraction can “unhinge people.” Moreover, Wallace uses the ability to end abstract thinking as a means to categorize “sane, functional people…from the unhinged” (17). By conjecturing that “the dreads and dangers of abstract thinking are a big reason why we now all like to stay so busy and bombarded with stimuli all the time,” since “abstract thinking tends most often to strike during moments of quiet repose” (13), Wallace conveys that the seeds of mental illness are not inherent within us, but rather planted by abstract thinking. Thus, Wallace warns the reader against the dangers of abstract thinking and its ability to “unhinge,” but on the other hand, he does not portray mental illness or the state of being “unhinged” (17) as negative. In fact, throughout the booklet, Wallace glorifies the Mentally Ill Mathematician as a hero of our times.
The Language or Map of Math
The abstractness of math renders math ostensibly a world of its own, but Everything and More illuminates math as more than just a solipsistic inhabitant of its own world. Everything and More conceives of math as a map—a map that translates problems, ideas, and concerns from “the world we all really live in” (71) to a world of symbols and abstractions. In this process of translation, math becomes more than just a set of isolated symbols and abstractions; math becomes a language—a language that maps one world into another. After acknowledging the status of math as a real language, Wallace proceeds to remind the reader to keep in mind the principal traits of language: “both a map of the world and its own world” (30).
Math acts as a map by connecting and translating the problems of humanity into its own problems. Wallace explains that “most of math’s definitions are built up out of other definitions; it’s the really root stuff that has to be defined from scratch. Hopefully…that scratch will have something to do with the world we all really live in” (71). Through the vocabulary of math presented in Everything and More, one can see that the “root stuff” of mathematics is indeed defined by the world we all really live in. Intersections between the vocabulary of math and our daily vocabulary abound—for example, real, imaginary, rational, irrational, identity, existence, uniqueness, and sensitivity—just to name a few. In fact, Everything and More portrays the function of math as the endeavor to represent real world; the text identifies calculus as “a seminal advance” because calculus gives math the “ability to represent continuity and change and real-world processes” (126). The roots of math are therefore intertwined and scratched in the world we all really live in. The language of math seems to capture the same issues that our world cares about. Sometimes, math simplifies the issues that we struggle with day-to-day and offers complex concepts the well-defined and stable type of definitions that we cannot manage to offer; other times, math mirrors or complicates these concepts.
An instance of math mapping principles from our world to its own world occurs in the battles between the Platonists and Intuitionists. During these battles, notions of morality slither into the realm of math. In using words such as “delusory” and “evil” to describe continuous functions (216), Wallace renders the world of math reflective of the morality, politics, and divisions extant the human world.
While we know the humanities for the endless and answerless questions it asks, we know math for its ability to provide each problem with a solution. In Everything and More, however, Wallace adduces a counterintuitive instance, where math, like the humanities, raises questions it cannot answer: “In the words of one math-historian, Fourier’s techniques ‘raised more questions than he was interested in answering or capable of solving.’ Which is both tactful and true” (163). Thus, at times, like in the humanities and in the rest of our world, there is no answer in math.
Furthermore, math, like any other language, cannot escape the difficulties in communication—in expressing the self and in translating the self into a language that others can understand. In spite of its ambition to be crystal clear and “clean” (34), math falls prey to the inescapable inability to communicate. In particular, math suffers from an inability to articulate its difficulties. Wallace quotes Russell, “To state clearly the difficulties involved, was to accomplish perhaps the hardest part of the philosopher’s task” (52) and then revises Russell’s claim into a stronger one: “Stating these difficulties clearly is, in fact, the whole and only difficulty involved here” (53). Just as the inability to communicate becomes central to nearly every one of Wallace’s characters, math is no exception. Wallace reminds us that even though “the power and maybe even whole raison d’etre of the language of math is that it’s designed to be so clean and nonconnotative that it avoids ambiguities,” the attempt to “express numerical quantities and relations in natural language—to translate mathematical propositions into English and vice versa—often causes trouble” (34). In a way, Everything and More exists as Wallace’s attempt to tackle this difficulty of translation—between numerical relations and our natural language, between the world of math and “the world we all really live in.”
Footnotes and Abbreviations
As is typical to much of Wallace's writing, he uses footnotes regularly throughout Everything and More. One of the most frequent footnotes that he employs is the "If You're Interested" note, abbreviated "IYI." According to Wallace, "the boldface 'IYI' designates bits of material that can be perused, glanced at, or skipped altogether if the reader wants. Meaning skipped without serious loss . . . most IYI-grade chunks, though, are designed for readers with strong technical backgrounds,or unusual interest in actual math, or preternatural patience, or all three; they (the chunks provide a more detailed look at the stuff that the main discussion glosses or breezes through" (2-3). In fact, this type of footnote is so prevalent throughout the booklet that Wallace states, "over half the document's footnotes are probably IYI, as well as several different paragraphs and even a couple subsections of the main text" (2). One purpose of IYI is to make the actual book more readable for those with weaker mathematical backgrounds--they can merely skip the notes that do not, in fact, interest them. Another purpose is due to the "strange stylistic problem in tech writing, which is that the same words often have to get used over and over in a way that would be terribly clunky in regular prose" (3), thus resulting in the need for an abbreviation in order to "achieve any kind of variation at all" (3).
The Beauty of Math
One of the more prevalent themes in Everything and More is the notion that beauty is an integral aspect to math. On the first page of the forward, for example, Wallace writes that the booklet's subject is "a set of mathematical achievements that are extremely abstract and technical, but also extremely profound and interesting, and beautiful" (1). In the same paragraph, he states that the aim of Everything and More is to "make the math beautiful--or at least to get the reader to see how someone might find it so" (2). Although Wallace clearly recognizes that not all readers will inherently see the beauty of math, he does hope to make readers see the possibility of beauty in the more complex achievements he details throughout the booklet. Wallace also frequently refers to math as being "sexy" throughout the course of Everything and More; the first example comes when he states that the "sexy math terms don't matter for now"(5). Thus, Wallace attempts to transform numbers and equations into things with aesthetic value, such as beauty or sexiness, to illustrate one way by which to view mathematics and make sense of complex problems.
A Word on Sexy Math
Wallace’s use of the word sexy in descriptions of math concepts stands out to some degree and becomes a memorable minutia of the text. For some readers, the use of sexy stimulates a newfound interest in math or causes them to pay more attention to the math; for others, it does not. It is possible that the use of sexy to describe math is just another element of Wallace’s distinct voice and humor.
It is also possible that the use of "sexy" can be interpreted in another way. In Everything and More, sexual imagery mingles with mathematical concepts. Throughout the text, Wallace litters variations of the word sex in his descriptions of math, and at one point, the booklet even deploys sex as a metaphor for math.
To illustrate, after compiling a list of terms, Wallace declares that “the sexy math terms don’t matter for now” (5). When explaining Zeno’s paradox, Wallace insists, “Put a little more sexily, the paradox is that a pedestrian cannot move from point A to point B without traversing all successive subintervals of AB” (49). When describing the difference between the Number Line and the Real Line, Wallace mentions, “For sexy technical reasons that we’ll get to, the Number Line is more properly called the Real Line if it also maps the irrational numbers” (73). Why are these terms or explanations or reasons sexy? The text provides no clear answer. The text, however, does imply that the language of math can be translated into the language of sex. Wallace explains the derivative: “In sexual terms, it’s an expression of the rate of change of a function with respect to the function’s independent variable” (116). Although the text does not clarify the way in which these explanations are sexual, the text does establish an explicit connection between math and sex. In one case, the text even figures sex as a mathematical activity: “The thing to appreciate about the humble Number Line’s marriage of math and geometry is that it’s also the perfect union of form and content” (72). With the “marriage of math and geometry”—a “perfect union of form and content”—Everything and More imbues mathematical activities with images of sex. In one revealing footnote, Wallace informs the reader: “If you’re innocent of college math, this def. will make more sense when we look at classical calc” (116). In this phrase, sex becomes a metaphor for math. The phrase insinuates that doing college math is like having sex; not having done college math, like being a virgin. But how can doing math possibly be like having sex?
Perhaps they have a similar function? If sex serves to propagate the human species, then math may serve to propagate another species. In Everything and More, Wallace claims “there is more than one species of infinity” (72). Thus, infinity in Everything and More occupies the status of a species. In fact, infinity is the species that mathematics, throughout the course of its history (and through Georg Cantor), has disseminated. Then based on this analysis, math becomes a metaphor for sex, and infinity a metaphor for the human race.
In what way do abstractions exist?
Throughout Everything and More, Wallace posits this question. Wallace wonders about different types of existence. For instance, does a unicorn exist in the same way that people or integers exist? Do abstract entities exist outside of human minds? Are they discovered, or created, or somehow both (20)?
The text suggests that abstractions owe their existence to mathematical proofs. Essentially, abstractions are constructed out of proofs. In discussing abstractions in the context of set theory, Wallace insists that “we are proving, deductively and thus definitively, truths about the makeup and relations of such things” (256). The necessity of proofs indicates that without proofs, the “truths about the makeup and relations of such things” falter with uncertainty. Further, the writing suggests that the proof literally gives existence to abstractions: “So one thing to appreciate up front is that, however abstract infinite systems are, after Cantor they are most definitely not abstract in the nonreal/unreal way that unicorns are” (205). Wallace does not assert that infinite systems have always existed; to the contrary, by writing “after Cantor, they are most definitely not abstract in the nonreal/unreal way that unicorns are,” Wallace suggests that Cantor’s mathematical discoveries (i.e. proofs) somehow confer a more real existence to infinite systems. Through proofs, mathematicians construct and validate the existence of abstract entities.
The actual existence led by these entities, however, seems to be replete with distress, haunted by illness. Just as Wallace’s other works capture the pain and suffering enmeshed in human existence, Everything and More detects the vulnerability of abstract existence. Wallace laments, “And, as true numbers, transfinites turn out to be susceptible to the same kinds of arithmetical relations and operations as regular numbers” (243). The word “susceptible” renders the existence of abstractions as a pathological and tragic one.
Interestingly, as “both a map of the world and its own world” (30), the abstractions of mathematics exist both on their own in their own world and as a copy of themselves in our world. In the final line of the booklet, Wallace declares, “Mathematics continues to get out of bed” (305), and in so doing, conveys that in spite of the coming and going of mathematicians, mathematics, on its own, continues to live, to exist. On the other hand, for everything in the land of abstractions, there exists a copy of it in our land. For instance, while the math world eventually constructs a proof of the unprovability of the Continuum Hypothesis, the human world receives a version of that proof in the form of Georg Cantor’s inability to prove the C.H. Everything and More narrates a connection, a correspondence between the existence of abstractions and existence in our world.
Everything and More and Classroom Discussion
With our class, Everything and More has proven to be not conducive to class discussion. Why not? Here is one theory. The definitions of abstraction proffered by the text in many cases also describe the text itself. Wallace spends over four pages explaining the meaning of abstraction. He writes that the O.E.D. has nine major definitions of the adjective abstractus, but he quotes only the most relevant ones, which include “‘Withdrawn or separated from matter, from material embodiment, from practice, or from particular examples’” (8). Like this definition of abstract, Everything and More, by virtue of being a book, is separated from the actual matter, material embodiment, and practice of math. If “abstraction proceeds in levels, rather like exponents or dimensions” (11), and if math is already abstract, then Everything and More, in being a book about math, becomes one level more abstract. Everything and More also presents the definition of abstract as a verb: “Abstracting something can mean reducing it to its absolute skeletal essence” (10). The definition matches precisely the way in which the text treats its mathematical concepts and proofs. In covering the entire history of infinity, the text shrinks much of the history, the processes, and the details. In catering to the general reader, the text distills the complexity of the actual concepts and proofs. Essentially, the text reduces the mathematical concepts and proofs to “its absolute skeletal essence.” Finally, the definition of abstract as “‘Abstruse’” (8) is clearly fitting; the text seems to cause “lots of confusion” (from class handout—needs citation) for not just the common readers, but also the PhDs of math. How does its abstractness contribute to this text’s non-conduciveness to discussion? Wallace suggests with respect to abstractions, that “the confusion comes when we try to consider what exactly they [abstract concepts like motion and existence] mean” (11). Coincidentally, English class discussions tend to ask the same question: what exactly does this mean? In the case of Everything and More, the “this” often refers to an abstraction sitting on top of multiple levels of abstractions, and thinking about these abstractions means, in Wallace’s words, “thinking hard about things that for the most part people can’t think hard about—because it drives them crazy” (10), and that is why Everything and More might not be the text most conducive to classroom discussion.
Brief Introduction to Criticism on Everything and More
The writer claims the aim of the booklet to be not only “to discuss these [abstract and technical mathematical] achievements in such a way that they’re vivid and comprehensible to readers who do not have pro-grade technical backgrounds and expertise,” but also “to make the math beautiful—or at least to get the reader to see how someone might find it so” (2).
Given this aim, how successful is this booklet? In a class discussion of what constitutes the beauty in math, one student suggested that the beauty of math derives largely from the actual process and experience of doing math—from the moments in which we figure something out, make a breakthrough, or even advance one tiny step along a problem with a seemingly endless array of steps. Consequently, as a booklet that mainly details another’s mathematical discoveries, Everything and More inevitably denies its readers that experience—the experience of figuring out math—and therefore, a part of math’s beauty. On the other hand, the writer does embed a few problems into the text in which he does not immediately give away the answer but instead gives readers a chance to solve the problem on their own—for instance, missing dollar word problem (34).
Critiques of Everything and More
One frequent complaint about Everything and More is that Wallace's writing style diminishes the mathematical aspects of the booklet, and vice-versa. Because of the difficulty of his subject matter, Wallace puts his fiction-writing skills to work in order to make the material more approachable to those readers with weaker backgrounds in math. Wallace attempts to create the best of both worlds in Everything and More: both a book about serious, complex math, and an easily readable work. Unfortunately, this aim results in phrases like "Let's pause to consider the vertiginous levels of abstraction involved here. If the human CPU cannot apprehend or even really conceive of infinity, it is now apparently being asked to countenance an infinity of infinities, an infinite number of individual members of which are themselves not finitely expressible, all in an interval [0-1] so finite- and innocent-looking we use it in little kids' classrooms. All of which is just resoundingly weird" (80), which is far too informal for a math textbook and too syntactically awkward for a "readable" work.
Another criticism about Everything and More is that the abbreviations are unnecessary and distracting. Indeed, the list of abbreviations in the forward contains 37 terms (3-4), not including the ubiquitous IYI. Although Wallace states that the "booklet's abbreviations are contextualized in such a way that it ought to be totally clear what they stand for" (3), some readers experience difficulty with this claim, and therefore must frequently flip to the front of the book to reread what each abbreviation stands for, or skip that section of the booklet, neither of which is a productive option.
The IYIs have been criticized as being somewhat insidious. Because the reader has bought, and is reading 'Everything and More', the reader is presumably already interested in the book. Therefore it may seem that the If-You're-Interested notes aren't so much considerate asides, but devices to grab the reader's attention.
Errors in Everything and More
A substantial portion of the backlash on Everything and More consists in complaints on the abundance of mathematical errors in the text. The mathematical community has pointed out dozens of errors, ranging from computational errors to mathematical statements that “make no sense” and produce “lots of confusion” (from in class handout—Mathematical errata for Everything and More). There are different ways to interpret these errors. One critic, for example, views the cornucopia of errors as detrimental to the author’s credibility and insists that “any time errors are found in a text that bring the author’s credibility into question, the critic/reader can take two paths, justifying the error’s using the intentional fallacy (or maybe the affective fallacy, or maybe both) or admit the author’s fallibility and humanity.”
Everything and More itself provides another way of looking at mathematical errors. As Wallace enumerates Cantor’s repeated attempts to prove the Continuum Hypothesis, he exposes the unintentional nature and heartfelt grief in Cantor’s errors: “Cantor’s attempts to prove the C.H. went on through the 1880s and ‘90s, and there are some heartbreaking letters to Dedekind in which he’d excitedly announce a proof and then a couple days later discover an error and have to retract it” (293). This passage discerns the inevitability and tragedy in errors: sometimes we make mistakes, and it’s sad when we do—because these mistakes are born not necessarily out of sloppiness, but out of an attempt to reveal some truth. As readers, we can extend and apply Wallace’s outlook on Cantor’s errors to our own outlook on Wallace’s errors. Unlike the above critic, we do not have to let an author’s errors devastate his or her credibility.
David Foster Wallace is a recognized mathematical critic as well. For Science Magazine (22 December 2000: Vol. 290. no. 5500, pp. 2263 - 2267) he reviewed two books, one titled The Wild Numbers, by Philibert Schogt, and one titled Uncle Petros & Goldbach's Conjecture, by Apostolos Doxiadis. This is not directly related to Everything and More, but serves to showcase Wallace's mathematical talents and inclinations in a more comprehensive manner. He also muses on how a new wave of mathematical work, perhaps a whole new genre, called "Math Melodrama" by Wallace, is invading the commercial sphere. The whole thing can be read at http://www.sciencemag.org/cgi/content/full/290/5500/2263