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Philosophical Sweep
To understand the fiction of David Foster Wallace, it helps to have a little Wittgenstein.

By James RyersonPosted Tuesday, Dec. 21, 2010, at 10:42 AM ET

The following is adapted from "A Head That Throbbed Heartlike: The Philosophical Mind of David Foster Wallace," an introduction to Wallace's undergraduate honor thesis in philosophy, which has just been published by Columbia University Press as Fate, Time, and Language: An Essay on Free Will<http://www.amazon.com/Fate-Time-Language-Essay-Free/dp/0231151578/ref=sr_1_4?s=books&ie=UTF8&qid=1292269695&sr=1-4>.

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[David Foster Wallace's book "Fate, Time, and Language: An Essay On Free Will"]I. "A special sort of buzz"

When the future novelist David Foster Wallace was about 14 years old, he asked his father, the University of Illinois philosophy professor James D. Wallace, to explain to him what philosophy is, so that when people would ask him exactly what it was that his father did, he could give them an answer. James had the two of them read Plato's Phaedo dialogue together, an experience that turned out to be pivotal in his understanding of his son. "I had never had an undergraduate student who caught on so quickly or who responded with such maturity and sophistication," James recalls. "This was this first time I realized what a phenomenal mind David had."

The experience seems to have made an impression on David as well. Not long after he arrived at Amherst College in the early 1980s, he developed a reputation among his professors as a rare philosophical talent, an exceptional student who combined raw analytical horsepower with an indefatigable work ethic. He was thought, by himself and by others, to be headed toward a career as a professor of philosophy. Even after he began writing fiction, a pursuit he undertook midway through college, philosophy remained the source of his academic identity. "I knew him as a philosopher with a fiction hobby," Jay Garfield, a professor now at Smith College who worked with Wallace at the time, remembers. "I didn't realize he was one of the great fiction writers of his generation with a philosophy hobby."

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For most of college, Wallace's main philosophical interests were in the more technical branches of the subject, such as mathematical logic and the philosophy of language. One semester, he took a seminar on the philosophy of Ludwig Wittgenstein, whose early work grapples with the writings of Gottlob Frege and Bertrand Russell, two of the founders of modern logic. As Wallace recollected in 1992 in a letter to the novelist Lance Olsen, he was "deeply taken" in the seminar with Wittgenstein's first book, Tractatus Logico-Philosophicus (1921). Along with its controversial arguments about the nature and limits of language, the Tractatus introduced some indisputable formal innovations, including a method of analyzing the propositions of modern logic by way of "truth tables." To some, the book might have seemed forbiddingly spare and exacting; Wallace remembered being moved by its "cold formal beauty." When the seminar moved on to Wittgenstein's so-called late philosophy, in which he repudiates the ideas and austere methodology of the Tractatus in favor of new assumptions and a looser, less mathematical style, Wallace was not immediately impressed. He wrote to Olsen that at first he found Philosophical Investigations, the crowning statement of the late philosophy, to be "silly."

Wallace would later identify his attraction to technical philosophy in aesthetic terms: It was, he suggested, a craving for a certain kind of beauty, for the variety of imaginative experience characteristic of formal systems like mathematics and chess. In an interview with the literary critic Larry McCaffery published in 1993, Wallace explained that as a philosophy student he had been "chasing a special sort of buzz," a flash of feeling whose nature he didn't comprehend at first. "One teacher called these moments 'mathematical experiences,' " he recalled. "What I didn't know then was that a mathematical experience was aesthetic in nature, an epiphany in Joyce's original sense. These moments appeared in proof-completions, or maybe algorithms. Or like a gorgeously simple solution you suddenly see after filling half a notebook with gnarly attempted solutions. It was really an experience of what I think Yeats called 'the click of a well-made box.' The word I always think of it as is 'click.' "

For his honors thesis in philosophy, Wallace continued to chase the click, writing a highly specialized, 76-page work on the metaphysical doctrine of fatalism (which holds, quite radically, that human actions and decisions have no influence on the future). Brace yourself for a sample sentence: "Let Φ (a physical possibility structure) be a set of distinct but intersecting paths ji – jn, each of which is a set of functions, L's, on ordered pairs <t, w> (<time, world situation>), such that for any Ln, Lm in some ji, Ln R Lm, where R is a primitive accessibility relation corresponding to physical possibility understood in terms of diachronic physical compatibility." There are reasons that he is better known for an essay about a cruise ship.

[David Foster Wallace's book "The Broom of the System."]II. An "artistic and religious crisis"


One of the many impressive aspects of Wallace's work on the thesis was that he was able to sustain his philosophical focus long after having begun a countervailing transformation: from budding philosopher to burgeoning novelist. The transition was set in motion toward the end of his sophomore year, when a bout of severe depression overcame him. He left school early and took off the following term. Wallace would suffer from depression for much of his life, and he tended to avoid public discussion of it. On a rare occasion in which he did allude publicly to his hiatus from Amherst, in his interview with McCaffery about a decade later, he described the episode as a crisis of identity precipitated by mounting ambivalence about his future as a philosopher. "I was just awfully good at technical philosophy," he said, "and it was the first thing I'd ever been really good at, and so everybody, including me, anticipated I'd make it a career. But it sort of emptied out for me somewhere around age twenty."

A debilitating panic followed. "Not a fun time," he went on. "I think I had a kind of midlife crisis at twenty, which probably doesn't augur well for my longevity." He moved back home to Illinois, "planning to play solitaire and stare out the window," as he put it—"whatever you do in a crisis." Though he now doubted that he should devote his life to philosophy, he was still drawn to the topic and found ways to engage with it, even dropping in on a few of his father's lectures at the university, where he monopolized the discussion. "He came to some of my classes in aesthetics, and tended to press me very hard," James Wallace told me. "The classes usually turned into a dialogue between David and me. The students looked on with 'Who is this guy?' looks on their faces."

During this time, Wallace started writing fiction. Though it represented a clean break from philosophy, fiction, as an art form, offered something comparable to the feeling of aesthetic recognition that he had sought in mathematical logic—the so-called click. "At some point in my reading and writing that fall I discovered the click existed in literature, too," he told McCaffery. "It was real lucky that just when I stopped being able to get the click from math logic I started to be able to get it from fiction." When he returned to Amherst, he nonetheless resumed his philosophical studies (eventually including his work on Taylor's "Fatalism"), but with misgivings: he hoped he would ultimately be bold enough to give up philosophy for literature. His close friend Mark Costello, who roomed with him at Amherst (and also became a novelist), told me that the shift was daunting for Wallace. "The world, the reference, of philosophy was an incredibly comfortable place for young Dave," he said. "It was a paradox. The formal intellectual terms were cold, exact, even doomed. But as a place to be, a room to be in, it was familiar, familial, recognized." Fiction, Costello said, was the "alien, risky place."

Wallace's solution was to pursue both aims at once. His senior year, while writing the honors thesis in philosophy, he also completed an honors thesis in creative writing for the English Department, a work of fiction nearly 500 pages long that would become his first novel, The Broom of the System<http://www.amazon.com/gp/product/0143116932?ie=UTF8&tag=slatmaga-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=0143116932>, which was published two years later, in 1987. Even just the manual labor required to produce two separate theses could be overwhelming, as suggested by an endearingly desperate request Wallace made in a letter to William Kennick, the Amherst professor who had taught his Wittgenstein seminar. "Since you're on leave," he wrote, "are you using your little office in Frost library? If not, does it have facilities for typing, namely an electrical outlet and a reasonably humane chair? If so, could I maybe use the office from time to time this spring? I have a truly horrifying amount of typing to do this spring—mostly for my English thesis, which has grown Blob-like and out of control—and my poor neighbors here in Moore are already being kept up and bothered a lot."

Despite the heavy workload, Wallace managed to produce a first draft of the philosophy thesis well ahead of schedule, before winter break of his senior year, and he finished both theses early, submitting them before spring break. He spent the last month or so of the school year reading other students' philosophy theses and offering advice. "He was an incredibly hard worker," Willem deVries, a philosopher now at the University of New Hampshire and the principal adviser on Wallace's thesis, told me, recalling the bewilderment with which he and his fellow professors viewed Wallace. "We were just shaking our heads." By the end of his tenure at Amherst, Wallace decided to commit himself to fiction, having concluded that, of the two enterprises, it allowed for a fuller expression of himself. "Writing The Broom of the System, I felt like I was using 97 percent of me," he later told the journalist David Lipsky, "whereas philosophy was using 50 percent."

Given his taste for experimental fiction, however, Wallace didn't assume, as he prepared to leave Amherst, that he would be able to live off of his writing. He considered styling himself professionally after William H. Gass, the author of Omensetter's Luck<http://www.amazon.com/gp/product/0141180102?ie=UTF8&tag=slatmaga-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=0141180102> (a novel Wallace revered), who had a Ph.D. in philosophy from Cornell and whose "day job" was teaching philosophy at Washington University in St. Louis. Wallace toyed with applying to Washington University for graduate school so he could observe Gass firsthand. But in the end, he chose to attend the University of Arizona for an M.F.A. in creative writing, which he completed in '87, the same year he published The Broom of the System and sold his first short-fiction collection, Girl with Curious Hair<http://www.amazon.com/gp/product/0393313964?ie=UTF8&tag=slatmaga-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=0393313964>.

Even with those literary successes, however, Wallace soon suffered another serious crisis of confidence, this time centered around his fiction. He later described it as "more of a sort of artistic and religious crisis than it was anything you might call a breakdown." He revisited the idea that philosophy could provide order and structure in his life, and that year he applied to graduate programs at Harvard and Princeton Universities, ultimately choosing to attend Harvard."The reason I applied to philosophy grad school," he told Lipsky, "is I remembered that I had flourished in an academic environment. And I had this idea that I could read philosophy and do philosophy, and write on the side, and that it would make the writing better."

Wallace started at Harvard in the fall of '89, but his plan quickly fell to pieces. "It was just real obvious that I was so far away from that world," he went on. "I mean, you were a full-time grad student. There wasn't time to write on the side—there was 400 pages of Kant theory to read every three days." Far more worrisome was the escalation of the "artistic and religious crisis" into another wave of depression, this time bordering on the suicidal. Late that first semester, Wallace dropped out of Harvard and checked into McLean Hospital, the storied psychiatric institution nearby in Massachusetts. It marked the end of his would-be career in philosophy. He viewed the passing of that ambition with mixed emotions. "I think going to Harvard was a huge mistake," he told Lipsky. "I was too old to be in grad school. I didn't want to be an academic philosopher anymore. But I was incredibly humiliated to drop out. Let's not forget that my father's a philosophy professor, that a lot of the professors there were revered by him. That he knew a couple of them. There was just an enormous amount of terrible stuff going on. But I left there and I didn't go back."

[Roger Federer.]Roger FedererIII. "INTERPRET-ME fiction"

Though Wallace abandoned it as a formal pursuit, philosophy would forever loom large in his life. In addition to having been formative for his cast of mind, philosophy would repeatedly crop up in the subject matter of his writing. His essay "Authority and American Usage," about the so-called prescriptivist/descriptivist debate among linguists and lexicographers, features an exegesis of Wittgenstein's argument against the possibility of a private language. In Everything and More, his book about the history of mathematical ideas of infinity, his guiding insight is that the disputes over mathematical procedures were ultimately debates about metaphysics—about "the ontological status of math entities." His article "Consider the Lobster" begins as a journalistic report from the annual Maine Lobster Festival but soon becomes a philosophical meditation on the question, "Is it all right to boil a sentient creature alive just for our gustatory pleasure?" This question leads Wallace into discussions about the distinction between pain and suffering; about the relation between ethics and (culinary) aesthetics; about how we might understand cross-species moral obligations; and about the "hard-core philosophy"—the "metaphysics, epistemology, value theory, ethics"—required to determine the principles that allow us to conclude even that other humans feel pain and have a legitimate interest in not doing so.

Those are just explicit examples. Wallace's writing is full of subtler philosophical allusions and passing bits of idiom. In Infinite Jest<http://www.amazon.com/gp/product/0316066524?ie=UTF8&tag=slatmaga-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=0316066524>, one of the nine college-application essays written by the precocious protagonist, Hal Incandenza, is "Montague Grammar and the Semantics of Physical Modality"—a nod to Wallace's own philosophy thesis. A story in his short-fiction collection Oblivion, "Philosophy and the Mirror of Nature," shares its title with the 1979 book of anti-epistemology by the philosopher Richard Rorty. The story "Good Old Neon" invokes two conundrums from mathematical logic, the Berry and Russell paradoxes, to describe a psychological double bind that the narrator calls the "fraudulence paradox." At the level of language, Wallace's books are peppered with phrases like "by sheer ontology," "ontologically prior," "in- and extensions," "antinomy," "techne."

Perhaps the most authentically philosophical aspect of Wallace's nonfiction, however, is the sense he gives his reader, no matter how rarefied or lowly the topic, of getting to the core of things, of searching for the essence of a phenomenon or experience. His article on the tennis player Roger Federer<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer.html?scp=1&sq=wallace%20federer&st=cse> delves into the central role of beauty in the appreciation of athletics. His antic recounting of a week-long Caribbean cruise penetrates beneath the surface of his own satirical portrait to plumb a set of near-existential issues—freedom of choice, the illusion of freedom, freedom from choice—that he saw lurking at the heart of modern American ideas of entertainment. "I saw philosophy all over the place," DeVries, his former professor, said of Wallace's writings. "It was even hard to figure out how to single it out. I think it infuses a great deal of his work."

As far as Wallace's fiction is concerned, the most philosophically intriguing text is the novel he wrote when his own philosophical efforts were most intense: The Broom of the System. In some way—though it's not obvious at first in what way—the book is clearly supposed to be "about" Wittgenstein's philosophy. The plot follows a young switchboard operator named Lenore Stonecipher Beadsman as she searches for her great-grandmother, a former student of Wittgenstein's at Cambridge University who has disappeared from her nursing home. Gramma Beadsman had been a dominant and intellectually bullying figure in Lenore's life, forever hinting that she would prove to Lenore "how a life is words and nothing else"—a haunting suggestion that seems to be the source of Lenore's persistent anxiety that she herself might be just a character in a novel. Gramma has left behind in her desk drawer several objects that are potential clues to her disappearance, including a copy of Philosophical Investigations.

The Broom of the System takes its title from a philosophical lesson that Gramma Beadsman once imparted to Lenore's younger brother, LaVache. While sweeping the kitchen floor with a broom, Gramma asked LaVache "which part of the broom was more elemental, more fundamental," the handle or the bristles? LaVache replied that the bristles are the essence of a broom. But Gramma corrected him, insisting that the answer depends on the use to which the broom is being put: if you want to sweep, the bristles are the essence—in effect, the meaning—of the broom; if you want, say, to break a window, its essence is the handle. "Meaning as use," Gramma intoned. "Meaning as use." The reader familiar with Wittgenstein will recognize in Gramma's words the governing slogan of his late philosophy: "the meaning of a word," he wrote in the Investigations, "is its use in the language."

In his letter to Lance Olsen, Wallace revealed that Gramma Beadsman was "based loosely" on Alice Ambrose, "a very old former Smith professor who lived near me"—Smith College is part of the Five Colleges consortium to which Amherst belongs—"and had been one of the students whose notes were comprised by Witt's Blue and Brown books." Though Wittgenstein's late philosophy was published posthumously, parts of it were available during his lifetime in the form of two sets of students' notes known as the "Blue Book" and the "Brown Book"; the "Brown Book" notes were dictated to Ambrose and another student, Francis Skinner, during classes at Cambridge in 1934–35. As the great-granddaughter of Alice Ambrose/Gramma Beadsman, Lenore, like Wallace himself, is the descendent of a philosopher with an amanuensis-like connection to Wittgenstein: James Wallace's mentor, Norman Malcolm, served as the sounding-board and assistant for the writing of Wittgenstein's final philosophical work, On Certainty.

By the time Wallace started writing Broom, he had developed a serious interest in Wittgenstein's late philosophy. As his relationship with technical philosophy cooled, he became increasingly curious about approaches to philosophy that, for all their differences with one another, were united in their opposition to the kind of work with which he previously self-identified. He was intrigued not only by Wittgenstein's late philosophy but also by J. L. Austin's "ordinary language" philosophy and even Jacques Derrida's radical conception of philosophy as a metaphysically arrogant form of literature.

Those new curiosities about the relation of language to reality mark another point of connection between Wallace and his character Lenore, who worries that language suffuses reality to the point of constituting it. Indeed, at the simplest level, Lenore just is Wallace, and The Broom of the System is just a fictionalized retelling—a "little self-obsessed bildingsroman," Wallace called it—of the intellectual struggles he was then undergoing, struggles not only between philosophy and literature but also between technical philosophy and its philosophical alternatives. "Think of The Broom of the System," he told McCaffery, "as the sensitive tale of a sensitive young WASP who's just had this mid-life crisis that's moved him from coldly cerebral analytic math to a coldly cerebral take on fiction and Austin-Wittgenstein-Derridean literary theory." This transformation, he explained, had a disturbing side effect, shifting the young WASP's "existential dread from a fear that he was just a 98.6-degree calculating machine to a fear that he was nothing but a linguistic construct." Lenore, with her apprehension that she may be nothing more than a character in a novel, is giving voice to Wallace's own anxieties about crossing into a wholly new relationship with language.

Understanding The Broom of the System as an autobiographical roman à clef is a useful first step in grasping Wallace's literary-philosophical aims, but his engagement with Wittgenstein's philosophy was a more profound and lasting affair than that reading alone suggests. In both his early and his late work, Wittgenstein addressed the doctrine of solipsism, the philosophical position that holds (in its most radical form) that nothing exists apart from your own mind and mental states. Like fatalism, solipsism is an extreme and counterintuitive view that is nonetheless difficult to disprove. Also like fatalism, it was an idea that bewitched and bothered Wallace, absorbing his intellect and artistic imagination and becoming a lifelong fascination. In his interview with McCaffery, Wallace said that "one of the things that makes Wittgenstein a real artist to me" is the handling of solipsism in his work. In Broom, Wallace sought to do some measure of novelistic justice to this aspect of Wittgenstein's thought.

Broom, then, belongs to the genre of the novel of ideas—books like Voltaire's Candide and Jean-Paul Sartre's Nausea, which all but instruct the reader to interpret them in light of certain schools of thought. (Candide is usually read as a parody of Leibnitz's metaphysics, Nausea as a vision of Sartre's existentialism.) In his essay "The Empty Plenum," published in 1990, Wallace called this genre of writing "INTERPRET-ME fiction" and argued that it had a special role to play in the life of the mind. As he knew from chasing the "click" in math and technical philosophy, there are areas of inquiry that might seem remote from the concerns of everyday life but that can, in fact, offer an array of intimate emotional and aesthetic experiences. Even for the reader with an appetite for it, however, a theoretical work can be so intellectually taxing, so draining of one's mental energies, that what Wallace called the "emotional implications" of the text are overlooked. The novel of ideas is at its most valuable, he contended, not when making abstruse ideas "accessible" or easy to digest for the reader, but rather when bringing these neglected undercurrents to the surface.

Wallace wrote "The Empty Plenum" in Boston in the summer of 1989, as he readied himself to begin the philosophy program at Harvard. The essay is an extended appreciation of David Markson's novel Wittgenstein's Mistress<http://www.amazon.com/gp/product/1564782115?ie=UTF8&tag=slatmaga-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=1564782115> ("a work of genius," in Wallace's estimation), which came out in '88, a year after The Broom of the System, and which was also "about" Wittgenstein's philosophy. It was an emotional reckoning, as Wallace read it, with the discussion of solipsism in Wittgenstein's early work. Wallace felt that Markson's novel had succeeded in uniting literature and philosophy in a way that he, in Broom, had tried but failed to do. (Wallace pronounced Broom "pretty dreadful.") The circumstances in which Wallace was writing the essay only underscored for him the importance of Markson's accomplishment. As Wallace prepared to seek a renewed merger of philosophy and fiction in his own life, at Harvard, he celebrated Markson as a novelist who, with the utmost artistry, had already fused the two. In defiance of "the rabid anti-intellectualism of the contemporary fiction scene," Wallace wrote, Markson had demonstrated the still-vital role of the novel of ideas in joining together "cerebration & emotion, abstraction & lived life, transcendent truth-seeking & daily schlepping." Markson had delivered on Wallace's literary-philosophical ideal of "making heads throb heartlike."

[David Markson's book "Wittgenstein's Mistress."]IV. "A kind of philosophical sci-fi"

To understand the philosophical ambitions of Broom it is worth first looking in detail at what Wallace thought Markson had done. Markson's novel, a work of experimental fiction with a lean style reminiscent of Samuel Beckett, is narrated by a painter named Kate, who appears to be the last person alive and who has been alone on earth for many years by the time the novel opens. Kate doesn't so much narrate (for she has no audience) as write into the void, tapping out on a typewriter declarative statement after declarative statement in simple paragraphs of just one or two sentences. Unlike many novels of ideas, Wittgenstein's Mistress doesn't feature cerebral characters or lofty discussions. Though Kate makes highbrow allusions, her grasp of history and literature and philosophy is idiosyncratic and shaky. As Wallace noted, in Kate's hands intellectual ideas are "sprayed, skewed, all over the book."

After many years roaming the earth, futilely looking for anyone else, Kate has retired to a beach house, where she is writing out her thoughts. She does so with a peculiar controlled indirection, free-associating but looping back again and again to a recurring set of personal preoccupations—compulsively trying to keep straight the memory of what has been lost, organizing and reorganizing scattered memories of her own life and her piecemeal knowledge of the world to which she once belonged:

I do remember sitting one morning in an automobile with a right-hand drive and watching Stratford-on-Avon fill up with snow, which must surely be rare.

Well, and once that same winter being almost hit by a car with nobody driving it, which came rolling down a hill near Hampstead Heath.

There was an explanation for the car coming down the hill with nobody driving it.

The explanation having been the hill, obviously.

That car, too, had a right-hand drive. Although perhaps that is not especially relevant to anything.

The possibility increases that Kate's narration is unreliable, that she is mentally unhinged, as it becomes clearer that the onset of her peculiar experience of the world coincided with a profound personal loss. The book imparts a double-layered feeling of loneliness and isolation: Kate's is the voice of a writer trapped not only inside her own head but also inside a world that now exists only through her own continual reconstructing of it. The text she types, Wallace wrote, "is itself obsessed & almost defined by the possibility that it does not exist, that Kate does not exist."

What does any of this have to do with Wittgenstein? Part of the achievement of Markson's novel, one of the ways in which it avoids the pitfalls of many novels of ideas, is that it doesn't require any understanding of Wittgenstein. The novel operates on its own terms. But the allusion to Wittgenstein in its title, its repeated citation of the first sentence of the Tractatus ("The world is all that is the case"), and its stylistic affinity with that book (the Tractatus is also composed of short aphoristic paragraphs) all invite the reader versed in philosophy to wonder what Markson is up to. "This isn't a weakness of the novel," Wallace stressed. "Though it's kind of miraculous that it's not."

Wallace had read the Tractatus, of course (he wrote to Lance Olsen that he thought its first sentence was "the most beautiful opening line in western lit"). He knew that Wittgenstein's book presented a spare and unforgiving picture of the relations among logic, language, and the physical world. He knew that the puzzles solved and raised by the book were influential, debatable, and rich in their implications. But as a flesh-and-blood reader with human feelings, he also knew, though he had never articulated it out loud, that as you labored to understand the Tractatus, its cold, formal, logical picture of the world could make you feel strange, lonely, awestruck, lost, frightened—a range of moods not unlike those undergone by Kate herself. The similarities were not accidental. Markson's novel, as Wallace put it, was like a 240-page answer to the question, "What if somebody really had to live in a Tractatusized world?" Pronouncing the novel "a kind of philosophical sci-fi," Wallace explained that Markson had staged a human drama on an alien intellectual planet, and in so doing he had "fleshed the abstract sketches of Wittgenstein's doctrine into the concrete theater of human loneliness."

[Ludwig Wittgenstein in 1947.]Ludwig WittgensteinV. "The loss of the whole external world"
The particular form of "human loneliness" to which Wallace was attuned was the sense of seclusion suggested by solipsism. Kate, Markson's narrator, seems to be in a situation like this, her world constituted entirely by her mental states. She shares this predicament with the traditional metaphysical subject of epistemology—the knowing consciousness, the "I" of Descartes's "I think, therefore I am"—who begins his intellectual journey trapped in his own mind, concerned that everything might just be a figment of his imagination (though he ultimately builds his way out of those confines to reach the external world). Wittgenstein, in the Tractatus, runs into the concern that his argument leads to solipsism—and his striking response is to agree, after a fashion, that it does. "There's a kind of tragic fall Wittgenstein's obsessed with," starting with the Tractatus, Wallace explained to McCaffery. "I mean a real Book-of-Genesis-type tragic fall. The loss of the whole external world."

How did Wittgenstein get to this point? The Tractatus is concerned with a disarmingly basic question: How is language possible? When we consider the world around us, everything seems to interact with everything else causally, in accordance with the laws of nature. The exception is a certain strange thing we call language, which somehow manages to interact with other things in the world in an entirely different way: it represents them meaningfully. The ability to represent things allows us to communicate, enables us to deal with things that are not actually present to us, and provides the fabric of our mental life, our daily thoughts. But how is it, exactly, that language produces meaning?

In the Tractatus, Wittgenstein argues that for words to represent things, for sentences to stand for states of affairs, language and reality have to share something in common. To explain what this commonality is, he introduces his so-called picture theory of meaning. An ordinary spoken or written sentence, he contends, when properly analyzed or disassembled into its component bits, reveals an elementary structure of logical parts and factual parts. This elementary structure, he argues, literally pictures reality: objects in the world correlate with the words in the sentence, and the relations among and between objects in the world correlate with the relations among and between the words in the sentence. A sentence has a certain elementary structure; things in the world can stand to one another in a certain structure; the identity of these two structures simply is meaning. A meaningful sentence depicts a possible state of affairs in the world; a meaningful and true sentence depicts an actual state of affairs in the world; anything in language that does not depict a possible state of affairs—that is, anything that does not depict possible fact—is, strictly speaking, meaningless.

Wittgenstein draws from the picture theory of meaning some arresting philosophical conclusions. The Tractatus regards as nonsensical, as literally meaningless, any claim that cannot be reduced to discrete facts about things in the world—for instance, any statements about ethics or aesthetics ("goodness" and "beauty" don't refer to actual things or properties). Another such type of nonsense, according to Wittgenstein, are metaphysical statements, claims about the supernatural, say, or the nature of the world as a whole. How language relates to reality—the very subject of the Tractatus—is itself, however, a concern about the world as a whole. This is the central irony of the Tractatus: its own claims are, strictly speaking, meaningless. They can be used only to try to show, but never to state, anything true. (This is the source of Wittgenstein's famous parting image of his book as a ladder that his reader must "throw away" after "he has climbed up it.")

For Wallace, the most disquieting feature of the Tractatus was its treatment of solipsism. Toward the end of the book, Wittgenstein concludes, "The limits of my language mean the limits of my world." This is a natural corollary of the picture theory of meaning: Given that there is a strict one-to-one mapping between states of affairs in the world and the structure of sentences, what I cannot speak of (that is, what I cannot meaningfully speak of) is not a fact of my world. But where am "I" situated in this world? By "I," I don't mean the physical person whom I can make factual reports about. I mean the metaphysical subject, the Cartesian "I," the knowing consciousness that stands in opposition with the external world. "Where in the world," Wittgenstein writes, "is a metaphysical subject to be found?"

On the one hand, the answer is nowhere. Wittgenstein can't make any sense of the philosophical self—any talk of it is, strictly speaking, nonsense. On the other hand, Wittgenstein can get some purchase on this question. He draws an analogy between the "I" (and the external world) and the eye (and the visual field): Though I cannot see my own eye in my visual field, the very existence of the visual field is nothing other than the working of my eye; likewise, though the philosophical self cannot be located in the world, the very experience of the world is nothing other than what it is to be an "I." Nothing can be said about the self in Wittgenstein's philosophy, but the self is made manifest insofar as "the world is my world"—or, as Wittgenstein more strikingly phrases it, "I am my world." This, he declares, is "how much truth there is in solipsism."

"I am my world" is what Wallace had in mind when he spoke of "the loss of the whole external world" in the Tractatus. There is no difference, ultimately, for Wittgenstein between solipsism and realism (solipsism "coincides with pure realism," he writes). For Wallace, this was a harrowing equation, the dark emotional takeaway of the Tractatus's severe anti-metaphysics. This was also, for Wallace, what Markson had rendered imaginatively in his novel. Without ever raising these ideas explicitly, Markson had conveyed them with a special kind of clarity. Wittgenstein's Mistress, by echoing the Tractatus's brusque, dreamlike sentences and placing Kate in a cold, lonely, self-as-world cosmos, had managed, as Wallace put it, to "capture the flavor both of solipsism and of Wittgenstein." What's more, Wallace felt Markson had done something that even Wittgenstein hadn't been able to do: he humanized the intellectual problem, communicating "the consequences, for persons, of the practice of theory; the difference, say, between espousing 'solipsism' as a metaphysical 'position' & waking up one fine morning after a personal loss to find your grief apocalyptic, literally millennial, leaving you the last and only living thing on earth." That was something only fiction, not philosophy, could do.

Solipsism, sometimes discussed as a doctrine but also evoked as a metaphor for isolation and loneliness, pervades Wallace's writing. "Plainly, Dave, as a guy and a writer, had a lifelong horror/fascination with the idea of a mind sealed off," Mark Costello told me. "His stories are full of sealed-off people." The self-obsessing narrator of "Good Old Neon," who has committed suicide and addresses the reader from beyond the grave, says "you're at least getting an idea, I think, of what it was like inside my head," of "how exhausting and solipsistic it is to be like this." The high-school students at the tennis academy in Infinite Jest wrestle with the question, "how we can keep from being 136 deeply alone people all jammed together?"—a problem that one of them diagnoses in intellectual terms ("Existential individuality, frequently referred to in the West. Solipsism") and another in emotional ones ("In a nutshell, what we're talking about here is loneliness"). The novelist Jonathan Franzen, one of Wallace's close friends, has said that he and Wallace agreed that the fundamental purpose of fiction was to combat loneliness. The paradox for Wallace was that to be a writer called for spending a lot of time alone in one's own head, giving rise to the feeling, as he wrote in "The Empty Plenum," "that one's head is, in some sense, the whole world, when the imagination becomes not just a more congenial but a realer environment than the Big Exterior of life on earth."

[David Foster Wallace.]David Foster WallaceVI. "The single most beautiful argument against solipsism that's ever been made."

Could solipsism be overcome? In The Broom of the System, Norman Bombardini, a very wealthy and very overweight man who owns the building in which Lenore works, bemoans what he calls "the Great Horror": the prospect of "an empty, rattling personal universe, one where one finds oneself with a Self, on one hand, and vast empty lonely spaces before Others begin to enter the picture at all, on the other." He devises a solution, a kind of spoof of the Tractatus's line "I am my world," which is to keep eating until he grows to infinite size, making himself coextensive with the world. (He calls the scheme "Project Total Yang.") Bombardini is only a minor character in the novel, and fittingly so, for the bulk of The Broom of the System is concerned not with the solipsism of early Wittgenstein but rather with the philosophy of the later Wittgenstein—who roundly rejected solipsism. Just as Markson conjured the solipsism of the Tractatus into an artistic creation, so too did Wallace hope to summon, in Broom, the anti-solipsistic worldview of Philosophical Investigations.

The Investigations offers a conception of language that is diametrically opposed that of the picture theory of the Tractatus. In Wittgenstein's early work, language is something sublime, logical, abstract—something with a defining structure or essence that, if you think hard enough, you can puzzle out in your head. In the Investigations, by contrast, language is seen as a messy human phenomenon, part of social reality—a rich variety of everyday practices that you figure out the way a child does, by publicly engaging in them, getting the hang of the unspoken rules by which communities use them. The shift in imagery is from language as a picture to language as a tool. This is the point of the Wittgensteinian mantra "meaning as use": If you want to understand the meaning of a word or phrase or gesture, you don't try to figure out what it represents; you try to figure out how to use it in real life. Wittgenstein called the rule-governed social practices that determine meaning "language games."

As Wallace was delighted to discover when he immersed himself in the Investigations later in college, the implications of this view for solipsism are potentially devastating. Given Wittgenstein's conception of language as a public phenomenon, whereby words get their meaning only by virtue of their shared use, what are we to make of the notion of a strictly private language, the voice of a solipsistic "I" who is speaking only to himself, in his own unique tongue, reporting private sensations and entertaining private thoughts in an otherwise barren world—the voice of a person living entirely in his own head? Wittgenstein's answer was that this idea, though seemingly viable, at least as a thought experiment, is in fact incoherent. The meaning of words is their use; the use of words is a matter of following rules; and following rules is entirely a social affair. There cannot be thought apart from the use of language—and language can operate only within a set of social practices. Thus there is no private thought without a corresponding public reality. "An 'inner process,' " as Wittgenstein put it, "stands in need of outward criteria." To phrase it in Cartesian terms: I think, therefore I am part of a community of others.

Wallace told McCaffery that Philosophical Investigations was "the single most beautiful argument against solipsism that's ever been made." Though the anti-private-language argument has been extraordinarily controversial, Wallace heralded it as though it were an indisputable mathematical proof. "The point here," he wrote in "Authority and American Usage," while giving a summary of Wittgenstein's argument, "is that the idea of a private language, like private colors and most of the other solipsistic conceits with which this reviewer has at various times been afflicted, is both deluded and demonstrably false." Solipsism was dead. Loneliness—at least that image of loneliness—was an illusion.

The defeat of solipsism was half of what Wallace sought to capture in Broom. But while Wittgenstein may have "solved" solipsism for Wallace, there was a catch—a final entangling conundrum with its own frightening implications—which Wallace also wanted to convey. On its face, the account of language in the Investigations seems pleasantly, reassuringly everyday: language is an ordinary, familiar, social, custom-bound human activity. But in other respects the account is quite extreme. Because all language and thought take place inside some language game or other, there is no transcendent, non-language-game standpoint from which you can step back, as it were, and see if any language game is better than any other—if one of them, for instance, does a better job of mirroring reality than another. Indeed, the question of whether any language game accurately represents reality can be asked only within some other language game, which operates according to its own set of nonevaluable conventions. In his early work Wittgenstein was in the business of stepping back from language, appraising its relation with reality, and pronouncing which uses connected us with something real and which did not; the Investigations is in another business altogether, describing without judging, merely "assembling reminders for a purpose," in Wittgenstein's phrase.

In Wallace's view, Wittgenstein had left us, again, without the possibility of contact with the outside world. As he told McCaffery, the Investigations "eliminated solipsism but not the horror." The only difference between this new predicament and that of the Tractatus was that rather than being trapped alone in our private thoughts, we were trapped together, with other people, in the institution of language. This was warmer than solipsism, but, as another form of being sealed-off from reality, it was cold comfort. Explaining this disheartening realization, Wallace said that "unfortunately we're still stuck with the idea that there's this world of referents out there that we can never really join or know because we're stuck in here, in language, even if we're at least all in here together."

In The Broom of the System, these two dueling emotional reactions—the fear of being trapped in language and the relief that at least we're all trapped in it together—are given playful expression. Lenore suffers from a fear, as she explains to her psychiatrist, that Gramma Beadsman is right that "there's no such thing" as "extra-linguistic anything." (Wallace's metafictional joke is that, for Lenore, as a character in a novel, there really isn't any reality of than language.) Lenore's boyfriend, a magazine editor named Rick Vigorous, soothes her throughout the book by compulsively telling her stories. Each of his stories is a not-so-thinly veiled allegory of the problems in their relationship, so that, even within the confines of the novel, Lenore and Rick become characters joined together in a reality constituted entirely by language. In the novel's climactic scene, a televangelist-charlatan named Reverend Sykes provides another image of this same double bind: escaping loneliness together in a language game, but sealed off from a higher reality. He asks the members of his TV audience to lay their hands on their TV screens in unison in order to commune with God—to join together in what he calls a "game" that will give everyone the consoling impression of making contact, together, with the ultimate transcendent referent. "So friends," Sykes says, "laugh if you will, but tonight I have a game for us to play together. A profoundly and vitally important game for us to play together tonight." His patter culminates in a three-sentence exhortation, the lines of which invoke the ideas of "meaning as use," language games, and the struggle against loneliness: "Use me, friends. Let us play the game together. I promise that no player will feel alone." Compared to the artful techniques of Markson's novel, these devices may seem clunky, but the intellectual aspiration was much the same.

It is worth noting that, in his discussions of Markson, Broom, and solipsism, Wallace was engaging throughout in what you might call a "strong misreading" of Wittgenstein's work. His explications of Wittgenstein's philosophy are not always convincing or strictly true. Highly questionable, for instance, is his assertion of what he called "the postmodern, poststructuralist" implications of the Investigations, which entail that we can't make true claims about the real world (a popular reading of Wittgenstein that many scholars hotly dispute). More straightforwardly wrong is Wallace's claim that Wittgenstein shared Wallace's own horror of the picture of the world in the Tractatus. Wallace told McCaffery that the reason Wittgenstein "trashed everything he'd been lauded for in the Tractatus" and developed the philosophy of the Investigations was that he "realized that no conclusion could be more horrible than solipsism." Wallace also contended, in "The Empty Plenum," that the impoverished role granted to ethics, aesthetics, and spiritual values in the Tractatus was "a big motivation" for its disavowal.

In truth, however, the biographical literature suggests that Wittgenstein was perfectly at ease with the solipsism of the Tractatus, as well as oddly, even mystically consoled by its suggestion that ethical, aesthetic, and spiritual truths are unutterable. As for the development of the late philosophy, it seems to have had its origins not in a fear of solipsism but rather in two deeply resonant objections: a technical criticism that the British mathematician Frank Ramsey made in 1923 about the Tractatus's treatment of the matter of "color-exclusion" and a playful challenge, posed by the Italian economist Piero Saffra, that Wittgenstein provide the "logical form" of a meaningful hand gesture.

It's possible that Wallace's own anxieties about being "trapped" in his own head colored or confused his reading of Wittgenstein—that he projected them, in philosophical terms, onto the Tractatus and the Investigations, resulting in an overemphasis on solipsism and giving Wittgenstein's treatment of the doctrine an alarmist, even hysterical cast. But given Wallace's otherwise sure-handed feel for philosophical texts, it seems likely that his distortions were at least in part intentional, offered in the service of artistic and emotional "truths." That would certainly be consistent with the ideal of fictionalized philosophy that he strove for in Broom and venerated in Wittgenstein's Mistress—a kind of writing that blended scholarly command and poetic reimagining.


Whatever the explanation for his preoccupation with solipsism in Wittgenstein, Wallace never abandoned his fixation on sealed-off people. Few readers of Infinite Jest will forget the lonely fate of the Hal Incandenza, who becomes so alienated from the world that his speech becomes unintelligible to others, or the lifeless zombiehood that befalls anyone who watches the novel's eponymous film, which is so entertaining that its viewer becomes incapable of doing anything other than watch it. But Mark Costello pointed out to me an important irony: for someone as obsessed with isolation as Wallace, he was "obviously a social novelist, a novelist of noticed details, on a near-encyclopedic scale." Where other novelists dealing with solipsism, like Markson and Beckett, painted barren images with small compressed sentences, Costello observed, "Dave tackled the issue by massively overfilling his scenes and sentences to comic bursting"—indeed to the point of panicked overstimulation. There was a palpable strain for Wallace between engagement with the world, in all its overwhelming fullness, and withdrawal to one's own head, in all its loneliness. The world was too much, the mind alone too little. "You can't be anything but contemptible living for yourself," Costello said, summing up the dilemma. "But letting the world in—that sucks too."

It's not exactly what you'd call an intellectual conundrum. But it was the lived one.



Federer as Religious Experience

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By DAVID FOSTER WALLACE

Published: August 20, 2006

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Federer at Wimbledon is meticulous, down to the hang of his blazer.

Correction Appended

Almost anyone who loves tennis and follows the men's tour on television has, over the last few years, had what might be termed Federer Moments. These are times, as you watch the young Swiss play, when the jaw drops and eyes protrude and sounds are made that bring spouses in from other rooms to see if you're O.K.

The Moments are more intense if you've played enough tennis to understand the impossibility of what you just saw him do. We've all got our examples. Here is one. It's the finals of the 2005 U.S. Open, Federer serving to Andre Agassi<http://topics.nytimes.com/top/reference/timestopics/people/a/andre_agassi/index.html?inline=nyt-per> early in the fourth set. There's a medium-long exchange of groundstrokes, one with the distinctive butterfly shape of today's power-baseline game, Federer and Agassi yanking each other from side to side, each trying to set up the baseline winner...until suddenly Agassi hits a hard heavy cross-court backhand that pulls Federer way out wide to his ad (=left) side, and Federer gets to it but slices the stretch backhand short, a couple feet past the service line, which of course is the sort of thing Agassi dines out on, and as Federer's scrambling to reverse and get back to center, Agassi's moving in to take the short ball on the rise, and he smacks it hard right back into the same ad corner, trying to wrong-foot Federer, which in fact he does — Federer's still near the corner but running toward the centerline, and the ball's heading to a point behind him now, where he just was, and there's no time to turn his body around, and Agassi's following the shot in to the net at an angle from the backhand side...and what Federer now does is somehow instantly reverse thrust and sort of skip backward three or four steps, impossibly fast, to hit a forehand out of his backhand corner, all his weight moving backward, and the forehand is a topspin screamer down the line past Agassi at net, who lunges for it but the ball's past him, and it flies straight down the sideline and lands exactly in the deuce corner of Agassi's side, a winner — Federer's still dancing backward as it lands. And there's that familiar little second of shocked silence from the New York crowd before it erupts, and John McEnroe<http://topics.nytimes.com/top/reference/timestopics/people/m/john_mcenroe/index.html?inline=nyt-per> with his color man's headset on TV says (mostly to himself, it sounds like), "How do you hit a winner from that position?" And he's right: given Agassi's position and world-class quickness, Federer had to send that ball down a two-inch pipe of space in order to pass him, which he did, moving backwards, with no setup time and none of his weight behind the shot. It was impossible. It was like something out of "The Matrix." I don't know what-all sounds were involved, but my spouse says she hurried in and there was popcorn all over the couch and I was down on one knee and my eyeballs looked like novelty-shop eyeballs.

Anyway, that's one example of a Federer Moment, and that was merely on TV — and the truth is that TV tennis is to live tennis pretty much as video porn is to the felt reality of human love.

Journalistically speaking, there is no hot news to offer you about Roger Federer<http://topics.nytimes.com/top/reference/timestopics/people/f/roger_federer/index.html?inline=nyt-per>. He is, at 25, the best tennis player currently alive. Maybe the best ever. Bios and profiles abound. "60 Minutes" did a feature on him just last year. Anything you want to know about Mr. Roger N.M.I. Federer — his background, his home town of Basel, Switzerland, his parents' sane and unexploitative support of his talent, his junior tennis career, his early problems with fragility and temper, his beloved junior coach, how that coach's accidental death in 2002 both shattered and annealed Federer and helped make him what he now is, Federer's 39 career singles titles, his eight Grand Slams, his unusually steady and mature commitment to the girlfriend who travels with him (which on the men's tour is rare) and handles his affairs (which on the men's tour is unheard of), his old-school stoicism and mental toughness and good sportsmanship and evident overall decency and thoughtfulness and charitable largess — it's all just a Google search away. Knock yourself out.

This present article is more about a spectator's experience of Federer, and its context. The specific thesis here is that if you've never seen the young man play live, and then do, in person, on the sacred grass of Wimbledon, through the literally withering heat and then wind and rain of the '06 fortnight, then you are apt to have what one of the tournament's press bus drivers describes as a "bloody near-religious experience." It may be tempting, at first, to hear a phrase like this as just one more of the overheated tropes that people resort to to describe the feeling of Federer Moments. But the driver's phrase turns out to be true — literally, for an instant ecstatically — though it takes some time and serious watching to see this truth emerge.

Beauty is not the goal of competitive sports, but high-level sports are a prime venue for the expression of human beauty. The relation is roughly that of courage to war.

The human beauty we're talking about here is beauty of a particular type; it might be called kinetic beauty. Its power and appeal are universal. It has nothing to do with sex or cultural norms. What it seems to have to do with, really, is human beings' reconciliation with the fact of having a body.(1<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote1>)

Of course, in men's sports no one ever talks about beauty or grace or the body. Men may profess their "love" of sports, but that love must always be cast and enacted in the symbology of war: elimination vs. advance, hierarchy of rank and standing, obsessive statistics, technical analysis, tribal and/or nationalist fervor, uniforms, mass noise, banners, chest-thumping, face-painting, etc. For reasons that are not well understood, war's codes are safer for most of us than love's. You too may find them so, in which case Spain's mesomorphic and totally martial Rafael Nadal is the man's man for you — he of the unsleeved biceps and Kabuki self-exhortations. Plus Nadal is also Federer's nemesis and the big surprise of this year's Wimbledon, since he's a clay-court specialist and no one expected him to make it past the first few rounds here. Whereas Federer, through the semifinals, has provided no surprise or competitive drama at all. He's outplayed each opponent so completely that the TV and print press are worried his matches are dull and can't compete effectively with the nationalist fervor of the World Cup.(2<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote2>)

July 9's men's final, though, is everyone's dream. Nadal vs. Federer is a replay of last month's French Open final, which Nadal won. Federer has so far lost only four matches all year, but they've all been to Nadal. Still, most of these matches have been on slow clay, Nadal's best surface. Grass is Federer's best. On the other hand, the first week's heat has baked out some of the Wimbledon courts' slickness and made them slower. There's also the fact that Nadal has adjusted his clay-based game to grass — moving in closer to the baseline on his groundstrokes, amping up his serve, overcoming his allergy to the net. He just about disemboweled Agassi in the third round. The networks are in ecstasies. Before the match, on Centre Court, behind the glass slits above the south backstop, as the linesmen are coming out on court in their new Ralph Lauren<http://topics.nytimes.com/top/reference/timestopics/people/l/ralph_lauren/index.html?inline=nyt-per> uniforms that look so much like children's navalwear, the broadcast commentators can be seen practically bouncing up and down in their chairs. This Wimbledon final's got the revenge narrative, the king-versus-regicide dynamic, the stark character contrasts. It's the passionate machismo of southern Europe versus the intricate clinical artistry of the north. Apollo and Dionysus. Scalpel and cleaver. Righty and southpaw. Nos. 1 and 2 in the world. Nadal, the man who's taken the modern power-baseline game just as far as it goes, versus a man who's transfigured that modern game, whose precision and variety are as big a deal as his pace and foot-speed, but who may be peculiarly vulnerable to, or psyched out by, that first man. A British sportswriter, exulting with his mates in the press section, says, twice, "It's going to be a war."

Plus it's in the cathedral of Centre Court. And the men's final is always on the fortnight's second Sunday, the symbolism of which Wimbledon emphasizes by always omitting play on the first Sunday. And the spattery gale that has knocked over parking signs and everted umbrellas all morning suddenly quits an hour before match time, the sun emerging just as Centre Court's tarp is rolled back and the net posts driven home.

Page 2 of 5)

Federer and Nadal come out to applause, make their ritual bows to the nobles' box. The Swiss is in the buttermilk-colored sport coat that Nike's gotten him to wear for Wimbledon this year. On Federer, and perhaps on him alone, it doesn't look absurd with shorts and sneakers. The Spaniard eschews all warm-up clothing, so you have to look at his muscles right away. He and the Swiss are both in all-Nike, up to the very same kind of tied white Nike hankie with the swoosh positioned above the third eye. Nadal tucks his hair under his hankie, but Federer doesn't, and smoothing and fussing with the bits of hair that fall over the hankie is the main Federer tic TV viewers get to see; likewise Nadal's obsessive retreat to the ballboy's towel between points. There happen to be other tics and habits, though, tiny perks of live viewing. There's the great care Roger Federer takes to hang the sport coat over his spare courtside chair's back, just so, to keep it from wrinkling — he's done this before each match here, and something about it seems childlike and weirdly sweet. Or the way he inevitably changes out his racket sometime in the second set, the new one always in the same clear plastic bag closed with blue tape, which he takes off carefully and always hands to a ballboy to dispose of. There's Nadal's habit of constantly picking his long shorts out of his bottom as he bounces the ball before serving, his way of always cutting his eyes warily from side to side as he walks the baseline, like a convict expecting to be shanked. And something odd on the Swiss's serve, if you look very closely. Holding ball and racket out in front, just before starting the motion, Federer always places the ball precisely in the V-shaped gap of the racket's throat, just below the head, just for an instant. If the fit isn't perfect, he adjusts the ball until it is. It happens very fast, but also every time, on both first serves and second.

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Besting Jonas Bjorkman, who said he was pleased to "have the best seat in the house."

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A between-the-legs return against Nicolas Kiefer at last year's U.S. Open.

Nadal and Federer now warm each other up for precisely five minutes; the umpire keeps time. There's a very definite order and etiquette to these pro warm-ups, which is something that television has decided you're not interested in seeing. Centre Court holds 13,000 and change. Another several thousand have done what people here do willingly every year, which is to pay a stiff general admission at the gate and then gather, with hampers and mosquito spray, to watch the match on an enormous TV screen outside Court 1. Your guess here is probably as good as anyone's.

Right before play, up at the net, there's a ceremonial coin-toss to see who'll serve first. It's another Wimbledon ritual. The honorary coin-tosser this year is William Caines, assisted by the umpire and tournament referee. William Caines is a 7-year-old from Kent who contracted liver cancer at age 2 and somehow survived after surgery and horrific chemo. He's here representing Cancer Research UK. He's blond and pink-cheeked and comes up to about Federer's waist. The crowd roars its approval of the re-enacted toss. Federer smiles distantly the whole time. Nadal, just across the net, keeps dancing in place like a boxer, swinging his arms from side to side. I'm not sure whether the U.S. networks show the coin-toss or not, whether this ceremony's part of their contractual obligation or whether they get to cut to commercial. As William's ushered off, there's more cheering, but it's scattered and disorganized; most of the crowd can't quite tell what to do. It's like once the ritual's over, the reality of why this child was part of it sinks in. There's a feeling of something important, something both uncomfortable and not, about a child with cancer tossing this dream-final's coin. The feeling, what-all it might mean, has a tip-of-the-tongue-type quality that remains elusive for at least the first two sets.(3<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote3>)

A top athlete's beauty is next to impossible to describe directly. Or to evoke. Federer's forehand is a great liquid whip, his backhand a one-hander that he can drive flat, load with topspin, or slice — the slice with such snap that the ball turns shapes in the air and skids on the grass to maybe ankle height. His serve has world-class pace and a degree of placement and variety no one else comes close to; the service motion is lithe and uneccentric, distinctive (on TV) only in a certain eel-like all-body snap at the moment of impact. His anticipation and court sense are otherworldly, and his footwork is the best in the game — as a child, he was also a soccer prodigy. All this is true, and yet none of it really explains anything or evokes the experience of watching this man play. Of witnessing, firsthand, the beauty and genius of his game. You more have to come at the aesthetic stuff obliquely, to talk around it, or — as Aquinas did with his own ineffable subject — to try to define it in terms of what it is not.

One thing it is not is televisable. At least not entirely. TV tennis has its advantages, but these advantages have disadvantages, and chief among them is a certain illusion of intimacy. Television's slow-mo replays, its close-ups and graphics, all so privilege viewers that we're not even aware of how much is lost in broadcast. And a large part of what's lost is the sheer physicality of top tennis, a sense of the speeds at which the ball is moving and the players are reacting. This loss is simple to explain. TV's priority, during a point, is coverage of the whole court, a comprehensive view, so that viewers can see both players and the overall geometry of the exchange. Television therefore chooses a specular vantage that is overhead and behind one baseline. You, the viewer, are above and looking down from behind the court. This perspective, as any art student will tell you, "foreshortens" the court. Real tennis, after all, is three-dimensional, but a TV screen's image is only 2-D. The dimension that's lost (or rather distorted) on the screen is the real court's length, the 78 feet between baselines; and the speed with which the ball traverses this length is a shot's pace, which on TV is obscured, and in person is fearsome to behold. That may sound abstract or overblown, in which case by all means go in person to some professional tournament — especially to the outer courts in early rounds, where you can sit 20 feet from the sideline — and sample the difference for yourself. If you've watched tennis only on television, you simply have no idea how hard these pros are hitting the ball, how fast the ball is moving,(4<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote4>) how little time the players have to get to it, and how quickly they're able to move and rotate and strike and recover. And none are faster, or more deceptively effortless about it, than Roger Federer.

Interestingly, what is less obscured in TV coverage is Federer's intelligence, since this intelligence often manifests as angle. Federer is able to see, or create, gaps and angles for winners that no one else can envision, and television's perspective is perfect for viewing and reviewing these Federer Moments. What's harder to appreciate on TV is that these spectacular-looking angles and winners are not coming from nowhere — they're often set up several shots ahead, and depend as much on Federer's manipulation of opponents' positions as they do on the pace or placement of the coup de grâce. And understanding how and why Federer is able to move other world-class athletes around this way requires, in turn, a better technical understanding of the modern power-baseline game than TV — again — is set up to provide.

Wimbledon is strange. Verily it is the game's Mecca, the cathedral of tennis; but it would be easier to sustain the appropriate level of on-site veneration if the tournament weren't so intent on reminding you over and over that it's the cathedral of tennis. There's a peculiar mix of stodgy self-satisfaction and relentless self-promotion and -branding. It's a bit like the sort of authority figure whose office wall has every last plaque, diploma, and award he's ever gotten, and every time you come into the office you're forced to look at the wall and say something to indicate that you're impressed. Wimbledon's own walls, along nearly every significant corridor and passage, are lined with posters and signs featuring shots of past champions, lists of Wimbledon facts and trivia, historic lore, and so on. Some of this stuff is interesting; some is just odd. The Wimbledon Lawn Tennis Museum, for instance, has a collection of all the various kinds of rackets used here through the decades, and one of the many signs along the Level 2 passage of the Millennium Building(5<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote5>) promotes this exhibition with both photos and didactic text, a kind of History of the Racket. Here, sic, is the climactic end of this text:

(Page 3 of 5)

Today's lightweight frames made of space-age materials like graphite, boron, titanium and ceramics, with larger heads — mid-size (90-95 square inches) and over-size (110 square inches) — have totally transformed the character of the game. Nowadays it is the powerful hitters who dominate with heavy topspin. Serve-and-volley players and those who rely on subtlety and touch have virtually disappeared.

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It seems odd, to say the least, that such a diagnosis continues to hang here so prominently in the fourth year of Federer's reign over Wimbledon, since the Swiss has brought to men's tennis degrees of touch and subtlety unseen since (at least) the days of McEnroe's prime. But the sign's really just a testament to the power of dogma. For almost two decades, the party line's been that certain advances in racket technology, conditioning, and weight training have transformed pro tennis from a game of quickness and finesse into one of athleticism and brute power. And as an etiology of today's power-baseline game, this party line is broadly accurate. Today's pros truly are measurably bigger, stronger, and better conditioned,(6<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote6>) and high-tech composite rackets really have increased their capacities for pace and spin. How, then, someone of Federer's consummate finesse has come to dominate the men's tour is a source of wide and dogmatic confusion.

There are three kinds of valid explanation for Federer's ascendancy. One kind involves mystery and metaphysics and is, I think, closest to the real truth. The others are more technical and make for better journalism.

The metaphysical explanation is that Roger Federer is one of those rare, preternatural athletes who appear to be exempt, at least in part, from certain physical laws. Good analogues here include Michael Jordan<http://topics.nytimes.com/top/reference/timestopics/people/j/michael_jordan/index.html?inline=nyt-per>,(7<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote7>) who could not only jump inhumanly high but actually hang there a beat or two longer than gravity allows, and Muhammad Ali<http://topics.nytimes.com/top/reference/timestopics/people/a/muhammad_ali/index.html?inline=nyt-per>, who really could "float" across the canvas and land two or three jabs in the clock-time required for one. There are probably a half-dozen other examples since 1960. And Federer is of this type — a type that one could call genius, or mutant, or avatar. He is never hurried or off-balance. The approaching ball hangs, for him, a split-second longer than it ought to. His movements are lithe rather than athletic. Like Ali, Jordan, Maradona, and Gretzky, he seems both less and more substantial than the men he faces. Particularly in the all-white that Wimbledon enjoys getting away with still requiring, he looks like what he may well (I think) be: a creature whose body is both flesh and, somehow, light.

This thing about the ball cooperatively hanging there, slowing down, as if susceptible to the Swiss's will — there's real metaphysical truth here. And in the following anecdote. After a July 7 semifinal in which Federer destroyed Jonas Bjorkman — not just beat him, destroyed him — and just before a requisite post-match news conference in which Bjorkman, who's friendly with Federer, says he was pleased to "have the best seat in the house" to watch the Swiss "play the nearest to perfection you can play tennis," Federer and Bjorkman are chatting and joking around, and Bjorkman asks him just how unnaturally big the ball was looking to him out there, and Federer confirms that it was "like a bowling ball or basketball." He means it just as a bantery, modest way to make Bjorkman feel better, to confirm that he's surprised by how unusually well he played today; but he's also revealing something about what tennis is like for him. Imagine that you're a person with preternaturally good reflexes and coordination and speed, and that you're playing high-level tennis. Your experience, in play, will not be that you possess phenomenal reflexes and speed; rather, it will seem to you that the tennis ball is quite large and slow-moving, and that you always have plenty of time to hit it. That is, you won't experience anything like the (empirically real) quickness and skill that the live audience, watching tennis balls move so fast they hiss and blur, will attribute to you.(8<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote8>)

Velocity's just one part of it. Now we're getting technical. Tennis is often called a "game of inches," but the cliché is mostly referring to where a shot lands. In terms of a player's hitting an incoming ball, tennis is actually more a game of micrometers: vanishingly tiny changes around the moment of impact will have large effects on how and where the ball travels. The same principle explains why even the smallest imprecision in aiming a rifle will still cause a miss if the target's far enough away.

By way of illustration, let's slow things way down. Imagine that you, a tennis player, are standing just behind your deuce corner's baseline. A ball is served to your forehand — you pivot (or rotate) so that your side is to the ball's incoming path and start to take your racket back for the forehand return. Keep visualizing up to where you're about halfway into the stroke's forward motion; the incoming ball is now just off your front hip, maybe six inches from point of impact. Consider some of the variables involved here. On the vertical plane, angling your racket face just a couple degrees forward or back will create topspin or slice, respectively; keeping it perpendicular will produce a flat, spinless drive. Horizontally, adjusting the racket face ever so slightly to the left or right, and hitting the ball maybe a millisecond early or late, will result in a cross-court versus down-the-line return. Further slight changes in the curves of your groundstroke's motion and follow-through will help determine how high your return passes over the net, which, together with the speed at which you're swinging (along with certain characteristics of the spin you impart), will affect how deep or shallow in the opponent's court your return lands, how high it bounces, etc. These are just the broadest distinctions, of course — like, there's heavy topspin vs. light topspin, or sharply cross-court vs. only slightly cross-court, etc. There are also the issues of how close you're allowing the ball to get to your body, what grip you're using, the extent to which your knees are bent and/or weight's moving forward, and whether you're able simultaneously to watch the ball and to see what your opponent's doing after he serves. These all matter, too. Plus there's the fact that you're not putting a static object into motion here but rather reversing the flight and (to a varying extent) spin of a projectile coming toward you — coming, in the case of pro tennis, at speeds that make conscious thought impossible. Mario Ancic's first serve, for instance, often comes in around 130 m.p.h. Since it's 78 feet from Ancic's baseline to yours, that means it takes 0.41 seconds for his serve to reach you.(9<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote9>) This is less than the time it takes to blink quickly, twice.

The upshot is that pro tennis involves intervals of time too brief for deliberate action. Temporally, we're more in the operative range of reflexes, purely physical reactions that bypass conscious thought. And yet an effective return of serve depends on a large set of decisions and physical adjustments that are a whole lot more involved and intentional than blinking, jumping when startled, etc.

Successfully returning a hard-served tennis ball requires what's sometimes called "the kinesthetic sense," meaning the ability to control the body and its artificial extensions through complex and very quick systems of tasks. English has a whole cloud of terms for various parts of this ability: feel, touch, form, proprioception, coordination, hand-eye coordination, kinesthesia, grace, control, reflexes, and so on. For promising junior players, refining the kinesthetic sense is the main goal of the extreme daily practice regimens we often hear about.(10<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote10>) The training here is both muscular and neurological. Hitting thousands of strokes, day after day, develops the ability to do by "feel" what cannot be done by regular conscious thought. Repetitive practice like this often looks tedious or even cruel to an outsider, but the outsider can't feel what's going on inside the player — tiny adjustments, over and over, and a sense of each change's effects that gets more and more acute even as it recedes from normal consciousness.(11<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote11>)

The time and discipline required for serious kinesthetic training are one reason why top pros are usually people who've devoted most of their waking lives to tennis, starting (at the very latest) in their early teens. It was, for example, at age 13 that Roger Federer finally gave up soccer, and a recognizable childhood, and entered Switzerland's national tennis training center in Ecublens. At 16, he dropped out of classroom studies and started serious international competition.

It was only weeks after quitting school that Federer won Junior Wimbledon. Obviously, this is something that not every junior who devotes himself to tennis can do. Just as obviously, then, there is more than time and training involved — there is also sheer talent, and degrees of it. Extraordinary kinesthetic ability must be present (and measurable) in a kid just to make the years of practice and training worthwhile...but from there, over time, the cream starts to rise and separate. So one type of technical explanation for Federer's dominion is that he's just a bit more kinesthetically talented than the other male pros. Only a little bit, since everyone in the Top 100 is himself kinesthetically gifted — but then, tennis is a game of inches.

This answer is plausible but incomplete. It would probably not have been incomplete in 1980. In 2006, though, it's fair to ask why this kind of talent still matters so much. Recall what is true about dogma and Wimbledon's sign. Kinesthetic virtuoso or no, Roger Federer is now dominating the largest, strongest, fittest, best-trained and -coached field of male pros who've ever existed, with everyone using a kind of nuclear racket that's said to have made the finer calibrations of kinesthetic sense irrelevant, like trying to whistle Mozart during a Metallica concert.

Page 4 of 5)

According to reliable sources, honorary coin-tosser William Caines's backstory is that one day, when he was 2½, his mother found a lump in his tummy, and took him to the doctor, and the lump was diagnosed as a malignant liver tumor. At which point one cannot, of course, imagine...a tiny child undergoing chemo, serious chemo, his mother having to watch, carry him home, nurse him, then bring him back to that place for more chemo. How did she answer her child's question — the big one, the obvious one? And who could answer hers? What could any priest or pastor say that wouldn't be grotesque?

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It's 2-1 Nadal in the final's second set, and he's serving. Federer won the first set at love but then flagged a bit, as he sometimes does, and is quickly down a break. Now, on Nadal's ad, there's a 16-stroke point. Nadal is serving a lot faster than he did in Paris, and this one's down the center. Federer floats a soft forehand high over the net, which he can get away with because Nadal never comes in behind his serve. The Spaniard now hits a characteristically heavy topspin forehand deep to Federer's backhand; Federer comes back with an even heavier topspin backhand, almost a clay-court shot. It's unexpected and backs Nadal up, slightly, and his response is a low hard short ball that lands just past the service line's T on Federer's forehand side. Against most other opponents, Federer could simply end the point on a ball like this, but one reason Nadal gives him trouble is that he's faster than the others, can get to stuff they can't; and so Federer here just hits a flat, medium-hard cross-court forehand, going not for a winner but for a low, shallowly angled ball that forces Nadal up and out to the deuce side, his backhand. Nadal, on the run, backhands it hard down the line to Federer's backhand; Federer slices it right back down the same line, slow and floaty with backspin, making Nadal come back to the same spot. Nadal slices the ball right back — three shots now all down the same line — and Federer slices the ball back to the same spot yet again, this one even slower and floatier, and Nadal gets planted and hits a big two-hander back down the same line — it's like Nadal's camped out now on his deuce side; he's no longer moving all the way back to the baseline's center between shots; Federer's hypnotized him a little. Federer now hits a very hard, deep topspin backhand, the kind that hisses, to a point just slightly on the ad side of Nadal's baseline, which Nadal gets to and forehands cross-court; and Federer responds with an even harder, heavier cross-court backhand, baseline-deep and moving so fast that Nadal has to hit the forehand off his back foot and then scramble to get back to center as the shot lands maybe two feet short on Federer's backhand side again. Federer steps to this ball and now hits a totally different cross-court backhand, this one much shorter and sharper-angled, an angle no one would anticipate, and so heavy and blurred with topspin that it lands shallow and just inside the sideline and takes off hard after the bounce, and Nadal can't move in to cut it off and can't get to it laterally along the baseline, because of all the angle and topspin — end of point. It's a spectacular winner, a Federer Moment; but watching it live, you can see that it's also a winner that Federer started setting up four or even five shots earlier. Everything after that first down-the-line slice was designed by the Swiss to maneuver Nadal and lull him and then disrupt his rhythm and balance and open up that last, unimaginable angle — an angle that would have been impossible without extreme topspin.

Extreme topspin is the hallmark of today's power-baseline game. This is something that Wimbledon's sign gets right.(12<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote12>) Why topspin is so key, though, is not commonly understood. What's commonly understood is that high-tech composite rackets impart much more pace to the ball, rather like aluminum baseball bats as opposed to good old lumber. But that dogma is false. The truth is that, at the same tensile strength, carbon-based composites are lighter than wood, and this allows modern rackets to be a couple ounces lighter and at least an inch wider across the face than the vintage Kramer and Maxply. It's the width of the face that's vital. A wider face means there's more total string area, which means the sweet spot's bigger. With a composite racket, you don't have to meet the ball in the precise geometric center of the strings in order to generate good pace. Nor must you be spot-on to generate topspin, a spin that (recall) requires a tilted face and upwardly curved stroke, brushing over the ball rather than hitting flat through it — this was quite hard to do with wood rackets, because of their smaller face and niggardly sweet spot. Composites' lighter, wider heads and more generous centers let players swing faster and put way more topspin on the ball...and, in turn, the more topspin you put on the ball, the harder you can hit it, because there's more margin for error. Topspin causes the ball to pass high over the net, describe a sharp arc, and come down fast into the opponent's court (instead of maybe soaring out).

(Page 5 of 5)

So the basic formula here is that composite rackets enable topspin, which in turn enables groundstrokes vastly faster and harder than 20 years ago — it's common now to see male pros pulled up off the ground and halfway around in the air by the force of their strokes, which in the old days was something one saw only in Jimmy Connors.

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Connors was not, by the way, the father of the power-baseline game. He whaled mightily from the baseline, true, but his groundstrokes were flat and spinless and had to pass very low over the net. Nor was Bjorn Borg a true power-baseliner. Both Borg and Connors played specialized versions of the classic baseline game, which had evolved as a counterforce to the even more classic serve-and-volley game, which was itself the dominant form of men's power tennis for decades, and of which John McEnroe was the greatest modern exponent. You probably know all this, and may also know that McEnroe toppled Borg and then more or less ruled the men's game until the appearance, around the mid-1980's, of (a) modern composite rackets(13<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote13>) and (b) Ivan Lendl, who played with an early form of composite and was the true progenitor of power-baseline tennis.(14<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote14>)

Ivan Lendl was the first top pro whose strokes and tactics appeared to be designed around the special capacities of the composite racket. His goal was to win points from the baseline, via either passing shots or outright winners. His weapon was his groundstrokes, especially his forehand, which he could hit with overwhelming pace because of the amount of topspin he put on the ball. The blend of pace and topspin also allowed Lendl to do something that proved crucial to the advent of the power-baseline game. He could pull off radical, extraordinary angles on hard-hit groundstrokes, mainly because of the speed with which heavy topspin makes the ball dip and land without going wide. In retrospect, this changed the whole physics of aggressive tennis. For decades, it had been angle that made the serve-and-volley game so lethal. The closer one is to the net, the more of the opponent's court is open — the classic advantage of volleying was that you could hit angles that would go way wide if attempted from the baseline or midcourt. But topspin on a groundstroke, if it's really extreme, can bring the ball down fast and shallow enough to exploit many of these same angles. Especially if the groundstroke you're hitting is off a somewhat short ball — the shorter the ball, the more angles are possible. Pace, topspin, and aggressive baseline angles: and lo, it's the power-baseline game.

It wasn't that Ivan Lendl was an immortally great tennis player. He was simply the first top pro to demonstrate what heavy topspin and raw power could achieve from the baseline. And, most important, the achievement was replicable, just like the composite racket. Past a certain threshold of physical talent and training, the main requirements were athleticism, aggression, and superior strength and conditioning. The result (omitting various complications and subspecialties(15<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote15>)) has been men's pro tennis for the last 20 years: ever bigger, stronger, fitter players generating unprecedented pace and topspin off the ground, trying to force the short or weak ball that they can put away.

Illustrative stat: When Lleyton Hewitt<http://topics.nytimes.com/top/reference/timestopics/people/h/lleyton_hewitt/index.html?inline=nyt-per> defeated David Nalbandian in the 2002 Wimbledon men's final, there was not one single serve-and-volley point.(16<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote16>)

The generic power-baseline game is not boring — certainly not compared with the two-second points of old-time serve-and-volley or the moon-ball tedium of classic baseline attrition. But it is somewhat static and limited; it is not, as pundits have publicly feared for years, the evolutionary endpoint of tennis. The player who's shown this to be true is Roger Federer. And he's shown it from within the modern game.

This within is what's important here; this is what a purely neural account leaves out. And it is why sexy attributions like touch and subtlety must not be misunderstood. With Federer, it's not either/or. The Swiss has every bit of Lendl and Agassi's pace on his groundstrokes, and leaves the ground when he swings, and can out-hit even Nadal from the backcourt.(17<http://www.nytimes.com/2006/08/20/sports/playmagazine/20federer-footnotes.html?pagewanted=all#footnote17>) What's strange and wrong about Wimbledon's sign, really, is its overall dolorous tone. Subtlety, touch, and finesse are not dead in the power-baseline era. For it is, still, in 2006, very much the power-baseline era: Roger Federer is a first-rate, kick-ass power-baseliner. It's just that that's not all he is. There's also his intelligence, his occult anticipation, his court sense, his ability to read and manipulate opponents, to mix spins and speeds, to misdirect and disguise, to use tactical foresight and peripheral vision and kinesthetic range instead of just rote pace — all this has exposed the limits, and possibilities, of men's tennis as it's now played.

Which sounds very high-flown and nice, of course, but please understand that with this guy it's not high-flown or abstract. Or nice. In the same emphatic, empirical, dominating way that Lendl drove home his own lesson, Roger Federer is showing that the speed and strength of today's pro game are merely its skeleton, not its flesh. He has, figuratively and literally, re-embodied men's tennis, and for the first time in years the game's future is unpredictable. You should have seen, on the grounds' outside courts, the variegated ballet that was this year's Junior Wimbledon. Drop volleys and mixed spins, off-speed serves, gambits planned three shots ahead — all as well as the standard-issue grunts and booming balls. Whether anything like a nascent Federer was here among these juniors can't be known, of course. Genius is not replicable. Inspiration, though, is contagious, and multiform — and even just to see, close up, power and aggression made vulnerable to beauty is to feel inspired and (in a fleeting, mortal way) reconciled.

Correction: Aug. 27, 2006

An article in PLAY magazine last Sunday about the tennis player Roger Federer referred incompletely to a point between Federer and Andre Agassi in the 2005 United States Open final and incorrectly described Agassi's position on the final shot of the point. There was an exchange of groundstrokes in the middle of the point that was not described. And Agassi remained at the baseline on Federer's winning shot; he did not go to the net.

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David Foster Wallace is the author of "Infinite Jest," "Consider the Lobster" and several other books.

Berry paradox
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The Berry paradox is a self-referential<http://en.wikipedia.org/wiki/Self-referential> paradox<http://en.wikipedia.org/wiki/Paradox> arising from the expression "the smallest possible integer<http://en.wikipedia.org/wiki/Integer> not definable by a given number of words<http://en.wikipedia.org/wiki/Word>." Bertrand Russell<http://en.wikipedia.org/wiki/Bertrand_Russell>, the first to discuss the paradox in print, attributed it to G. G. Berry (1867–1928),[1]<http://en.wikipedia.org/wiki/Berry_paradox#cite_note-0> a junior librarian<http://en.wikipedia.org/wiki/Librarian> at Oxford<http://en.wikipedia.org/wiki/Oxford>'s Bodleian library<http://en.wikipedia.org/wiki/Bodleian_library>, who had suggested the more limited paradox arising from the expression "the first undefinable<http://en.wikipedia.org/wiki/Definable_number> ordinal<http://en.wikipedia.org/wiki/Ordinal_number>".

Contents
[hide<http://en.wikipedia.org/wiki/Berry_paradox#>]

* 1 The paradox<http://en.wikipedia.org/wiki/Berry_paradox#The_paradox>
* 2 Resolution<http://en.wikipedia.org/wiki/Berry_paradox#Resolution>
* 3 Formal analogues<http://en.wikipedia.org/wiki/Berry_paradox#Formal_analogues>
* 4 Relationship with Kolmogorov complexity<http://en.wikipedia.org/wiki/Berry_paradox#Relationship_with_Kolmogorov_complexity>
* 5 See also<http://en.wikipedia.org/wiki/Berry_paradox#See_also>
* 6 Notes<http://en.wikipedia.org/wiki/Berry_paradox#Notes>
* 7 References<http://en.wikipedia.org/wiki/Berry_paradox#References>
* 8 External links<http://en.wikipedia.org/wiki/Berry_paradox#External_links>


[edit<http://en.wikipedia.org/w/index.php?title=Berry_paradox&action=edit&section=1>] The paradox

Consider the expression:

"The smallest positive<http://en.wikipedia.org/wiki/Positive_number> integer<http://en.wikipedia.org/wiki/Integer> not definable in under eleven words."

Since there are finitely<http://en.wikipedia.org/wiki/Finite_set> many words, there are finitely many phrases of under eleven words, and hence finitely many positive integers that are defined by phrases of under eleven words. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words. By the well ordering principle<http://en.wikipedia.org/wiki/Well_ordering_principle>, if there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property; therefore, there is a smallest positive integer satisfying the property "not definable in under eleven words". This is the integer to which the above expression refers. The above expression is only ten words long, so this integer is defined by an expression that is under eleven words long; it is definable in under eleven words, and is not the smallest positive integer not definable in under eleven words, and is not defined by this expression. This is a paradox: there must be an integer defined by this expression, but since the expression is self-contradictory (any integer it defines is definable in under eleven words), there cannot be any integer defined by it.

[edit<http://en.wikipedia.org/w/index.php?title=Berry_paradox&action=edit&section=2>] Resolution

The Berry paradox as formulated above arises because of systematic ambiguity<http://en.wikipedia.org/wiki/Ambiguity> in the word "definable". In other formulations of the Berry paradox, such as one that instead reads: "...not nameable<http://en.wikipedia.org/wiki/Name> in less..." the term "nameable" is also one that has this systematic ambiguity. Terms of this kind give rise to vicious circle<http://en.wikipedia.org/wiki/Vicious_circle_principle> fallacies. Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal.[2]<http://en.wikipedia.org/wiki/Berry_paradox#cite_note-1> To resolve one of these paradoxes means to pinpoint exactly where our use of language went wrong and to provide restrictions on the use of language which may avoid them.

This family of paradoxes can be resolved by incorporating stratifications of meaning in language. Terms with systematic ambiguity may be written with subscripts denoting that one level of meaning is considered a higher priority than another in their interpretation. The number not nameable0 in less than eleven words' may be nameable1 in less than eleven words under this scheme.[3]<http://en.wikipedia.org/wiki/Berry_paradox#cite_note-2>

[edit<http://en.wikipedia.org/w/index.php?title=Berry_paradox&action=edit&section=3>] Formal analogues

Using programs or proofs of bounded lengths, it is possible to construct an analogue of the Berry expression in a formal mathematical language, as has been done by Gregory Chaitin<http://en.wikipedia.org/wiki/Gregory_Chaitin>. Though the formal analogue does not lead to a logical contradiction, it does prove certain impossibility results.

George Boolos<http://en.wikipedia.org/wiki/George_Boolos> (1989) built on a formalized version of Berry's paradox to prove Gödel's Incompleteness Theorem<http://en.wikipedia.org/wiki/G%C3%B6del%27s_Incompleteness_Theorem> in a new and much simpler way. The basic idea of his proof is that a proposition<http://en.wikipedia.org/wiki/Proposition> that holds of x if x = n for some natural number n can be called a definition for n, and that the set {(n, k): n has a definition that is k symbols long} can be shown to be representable (using Gödel numbers<http://en.wikipedia.org/wiki/G%C3%B6del_number>). Then the proposition "m is the first number not definable in less than k symbols" can be formalized and shown to be a definition in the sense just stated.

[edit<http://en.wikipedia.org/w/index.php?title=Berry_paradox&action=edit&section=4>] Relationship with Kolmogorov complexity
Main article: Kolmogorov complexity<http://en.wikipedia.org/wiki/Kolmogorov_complexity>

It is not possible in general to unambiguously define what is the minimal number of symbols required to describe a given string (given a specific description mechanism). In this context, the terms string and number may be used interchangeably, since a number is actually a string of symbols, i.e. an English word (like the word "eleven" used in the paradox) while, on the other hand, it is possible to refer to any word with a number, e.g. by the number of its position in a given dictionary, or by suitable encoding. Some long strings can be described exactly using fewer symbols than those required by their full representation, as is often experienced using data compression<http://en.wikipedia.org/wiki/Data_compression>. The complexity of a given string is then defined as the minimal length that a description requires in order to (unambiguously) refer to the full representation of that string.

The Kolmogorov complexity is defined using formal languages<http://en.wikipedia.org/wiki/Formal_language>, or Turing machines<http://en.wikipedia.org/wiki/Turing_machines>, that allow to avoid ambiguities about what string results from a given description. After defining that function, it can be proved that it cannot be computed. The proof by contradiction shows that if it were possible to compute the Kolmogorov complexity, then it would also be possible to systematically generate paradoxes similar to this one, i.e. descriptions shorter than what the complexity of the described string implies. That is to say, the definition of the Berry number is paradoxical because it is not actually possible to compute how many words are required to define a number, and we know that such computation is not feasible because of the paradox.

Russell's paradox
From Wikipedia, the free encyclopedia
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Bertrand Russell series
________________________________

[Russell in 1907]<http://en.wikipedia.org/wiki/File:Russell1907-2.jpg>

Russell in 1907


________________________________

* Bertrand Russell<http://en.wikipedia.org/wiki/Bertrand_Russell>
* Views on philosophy<http://en.wikipedia.org/wiki/Bertrand_Russell%27s_views_on_philosophy>
* Views on society<http://en.wikipedia.org/wiki/Bertrand_Russell%27s_views_on_society>
* Russell's paradox
* Russell's teapot<http://en.wikipedia.org/wiki/Russell%27s_teapot>
* Theory of Descriptions<http://en.wikipedia.org/wiki/Theory_of_descriptions>
* Logical atomism<http://en.wikipedia.org/wiki/Logical_atomism>


In the foundations of mathematics<http://en.wikipedia.org/wiki/Foundations_of_mathematics>, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell<http://en.wikipedia.org/wiki/Bertrand_Russell> in 1901, showed that the naive set theory<http://en.wikipedia.org/wiki/Naive_set_theory> of Richard Dedekind<http://en.wikipedia.org/wiki/Richard_Dedekind> and Frege<http://en.wikipedia.org/wiki/Gottlob_Frege> leads to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo<http://en.wikipedia.org/wiki/Ernst_Zermelo> but he did not publish the idea, which remained known only to Hilbert<http://en.wikipedia.org/wiki/David_Hilbert>, Husserl<http://en.wikipedia.org/wiki/Edmund_Husserl> and other members of the University of Göttingen<http://en.wikipedia.org/wiki/University_of_G%C3%B6ttingen>.

Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox. Symbolically:

[\text{let } R = \{ x \mid x \not \in x \} \text{, then } R \in R \iff R \not \in R]

In 1908, two ways of avoiding the paradox were proposed, Russell's type theory<http://en.wikipedia.org/wiki/Type_theory> and the Zermelo set theory<http://en.wikipedia.org/wiki/Zermelo_set_theory>, the first constructed axiomatic set theory<http://en.wikipedia.org/wiki/Axiomatic_set_theory>. Zermelo's axioms went well beyond Frege's axioms of extensionality<http://en.wikipedia.org/wiki/Extensionality> and unlimited set abstraction<http://en.wikipedia.org/wiki/Set_builder_notation>, and evolved into the now-canonical Zermelo–Fraenkel set theory<http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory> (ZF).

Contents
[hide<http://en.wikipedia.org/wiki/Russell's_paradox#>]

* 1 Informal presentation<http://en.wikipedia.org/wiki/Russell's_paradox#Informal_presentation>
* 2 Set-theoretic responses<http://en.wikipedia.org/wiki/Russell's_paradox#Set-theoretic_responses>
* 3 History<http://en.wikipedia.org/wiki/Russell's_paradox#History>
* 4 Applied versions<http://en.wikipedia.org/wiki/Russell's_paradox#Applied_versions>
* 5 Applications and related topics<http://en.wikipedia.org/wiki/Russell's_paradox#Applications_and_related_topics>
* 5.1 Russell-like paradoxes<http://en.wikipedia.org/wiki/Russell's_paradox#Russell-like_paradoxes>
* 6 Related paradoxes<http://en.wikipedia.org/wiki/Russell's_paradox#Related_paradoxes>
* 7 See also<http://en.wikipedia.org/wiki/Russell's_paradox#See_also>
* 8 Notes<http://en.wikipedia.org/wiki/Russell's_paradox#Notes>
* 9 References<http://en.wikipedia.org/wiki/Russell's_paradox#References>
* 10 External links<http://en.wikipedia.org/wiki/Russell's_paradox#External_links>


[edit<http://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&action=edit&section=1>] Informal presentation

Let us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all squares. That set is not itself a square, and therefore is not a member of the set of all squares. So it is "normal". On the other hand, if we take the complementary set that contains all non-squares, that set is itself not a square and so should be one of its own members. It is "abnormal".

Now we consider the set of all normal sets, R. Attempting to determine whether R is normal or abnormal is impossible: If R were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if it were abnormal, it would not be contained in the set of normal sets (itself), and therefore be normal. This leads to the conclusion that R is both normal and abnormal: Russell's paradox.

[edit<http://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&action=edit&section=2>] Set-theoretic responses

In 1908, Ernst Zermelo<http://en.wikipedia.org/wiki/Ernst_Zermelo> proposed an axiomatization<http://en.wikipedia.org/wiki/Axiomatic_system> of set theory that avoided the paradoxes of naive set theory by replacing arbitrary set comprehension with weaker existence axioms, such as his axiom of separation<http://en.wikipedia.org/wiki/Axiom_of_separation> (Aussonderung). Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel<http://en.wikipedia.org/wiki/Abraham_Fraenkel>, Thoralf Skolem<http://en.wikipedia.org/wiki/Thoralf_Skolem>, and by Zermelo himself resulted in the axiomatic set theory called ZFC<http://en.wikipedia.org/wiki/ZFC>. This theory became widely accepted once Zermelo's axiom of choice<http://en.wikipedia.org/wiki/Axiom_of_choice> ceased to be controversial, and ZFC<http://en.wikipedia.org/wiki/ZFC> has remained the canonical axiomatic set theory<http://en.wikipedia.org/wiki/Axiomatic_set_theory> down to the present day.

ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts that given any set X, any subset of X definable using first-order logic exists. The object R discussed above cannot be constructed in this fashion, and is therefore not a ZFC set. In some extensions of ZFC<http://en.wikipedia.org/wiki/Von_Neumann-Bernays-Godel_set_theory>, objects like R are called proper classes<http://en.wikipedia.org/wiki/Proper_class>. ZFC is silent about types, although some argue that Zermelo's axioms tacitly presuppose a background type theory.

In ZFC, given a set A, it is possible to define a set B that consists of exactly the sets in A that are not members of themselves. B cannot be in A by the same reasoning in Russell's Paradox. This variation of Russell's paradox shows that no set contains everything.

Through the work of Zermelo and others, especially John von Neumann<http://en.wikipedia.org/wiki/John_von_Neumann>, the structure of what some see as the "natural" objects described by ZFC eventually became clear; they are the elements of the von Neumann universe<http://en.wikipedia.org/wiki/Von_Neumann_universe>, V, built up from the empty set<http://en.wikipedia.org/wiki/Empty_set> by transfinitely iterating<http://en.wikipedia.org/wiki/Transfinite_recursion> the power set<http://en.wikipedia.org/wiki/Power_set> operation. It is thus now possible again to reason about sets in a non-axiomatic fashion without running afoul of Russell's paradox, namely by reasoning about the elements of V. Whether it is appropriate to think of sets in this way is a point of contention among the rival points of view on the philosophy of mathematics<http://en.wikipedia.org/wiki/Philosophy_of_mathematics>.

Other resolutions to Russell's paradox, more in the spirit of type theory<http://en.wikipedia.org/wiki/Type_theory>, include the axiomatic set theories New Foundations<http://en.wikipedia.org/wiki/New_Foundations> and Scott-Potter set theory<http://en.wikipedia.org/wiki/Scott-Potter_set_theory>.

[edit<http://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&action=edit&section=3>] History

Russell discovered the paradox in May or June 1901.[1]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-0> By his own admission in his 1919 Introduction to Mathematical Philosophy, he "attempted to discover some flaw in Cantor's proof that there is no greatest cardinal".[2]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-1> In a 1902 letter,[3]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-2> he announced the discovery to Gottlob Frege<http://en.wikipedia.org/wiki/Gottlob_Frege> of the paradox in Frege's 1879 Begriffsschrift<http://en.wikipedia.org/wiki/Begriffsschrift> and framed the problem in terms of both logic and set theory, and in particular in terms of Frege's definition of function<http://en.wikipedia.org/wiki/Function_(mathematics)>; in the following, p. 17 refers to a page in the original Begriffsschrift, and page 23 refers to the same page in van Heijenoort 1967:

There is just one point where I have encountered a difficulty. You state (p. 17 [p. 23 above]) that a function too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite flows. Therefore we must conclude that w is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a definable collection [Menge] does not form a totality.[4]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-3>

Russell would go to cover it at length in his 1903 The Principles of Mathematics<http://en.wikipedia.org/wiki/The_Principles_of_Mathematics> where he repeats his first encounter with the paradox:[5]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-4>

Before taking leave of fundamental questions, it is necessary to examine more in detail the singular contradiction, already mentioned, with regard to predicates not predicable of themselves. ... I may mention that I was led to it in the endeavour to reconcile Cantor's proof...."

Russell wrote to Frege about the paradox just as Frege was preparing the second volume of his Grundgesetze der Arithmetik.[6]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-5> Frege did not waste time responding to Russell, his letter dated 22 June 1902 appears, with van Heijenoort's commentary in Heijenoort 1967:126-127. Frege then wrote an appendix admitting to the paradox,[7]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-6> and proposed a solution that Russell would endorse in his Principles of Mathematics,[8]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-7> but was later considered by some unsatisfactory.[9]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-8> For his part, Russell had his work at the printers and he added an appendix on the doctrine of types<http://en.wikipedia.org/wiki/Type_theory>.[10]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-9>

For his part, Ernst Zermelo<http://en.wikipedia.org/wiki/Ernst_Zermelo> in his (1908) A new proof of the possibility of a well-ordering (published at the same time he published "the first axiomatic set theory")[11]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-10> laid claim to prior discovery of the antinomy<http://en.wikipedia.org/wiki/Antinomy> in Cantor's naive set theory. He states: "And yet, even the elementary form that Russell9 gave to the set-theoretic antinomies could have persuaded them [J. König, Jourdain, F. Bernstein] that the solution of these difficulties is not to be sought in the surrender of well-ordering but only in a suitable restriction of the notion of set".[12]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-11> Footnote 9 is where he stakes his claim:

91903, pp. 366-368. I had, however, discovered this antinomy myself, independently of Russell, and had communicated it prior to 1903 to Professor Hilbert among others.[13]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-12>

A written account of Zermelo's actual argument was discovered in the Nachlass of Edmund Husserl<http://en.wikipedia.org/wiki/Edmund_Husserl>.[14]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-13>

It is also known that unpublished discussions of set theoretical paradoxes took place in the mathematical community at the turn of the century. van Heijenoort in his commentary before Russell's 1902 Letter to Frege states that Zermelo "had discovered the paradox independently of Russell and communicated it to Hilbert, among others, prior to its publication by Russell".[15]<http://en.wikipedia.org/wiki/Russell's_paradox#cite_note-14>

In 1923, Ludwig Wittgenstein<http://en.wikipedia.org/wiki/Ludwig_Wittgenstein> proposed to "dispose" of Russell's paradox as follows:

The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition 'F(F(fx))', in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(f(x)) and the outer one has the form Y(O(fx)). Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write '(do) : F(Ou) . Ou = Fu'. That disposes of Russell's paradox. (Tractatus Logico-Philosophicus<http://en.wikipedia.org/wiki/Tractatus_Logico-Philosophicus>, 3.333)

Russell and Alfred North Whitehead<http://en.wikipedia.org/wiki/Alfred_North_Whitehead> wrote their three-volume Principia Mathematica<http://en.wikipedia.org/wiki/Principia_Mathematica> (PM) hoping to achieve what Frege had been unable to do. They sought to banish the paradoxes of naive set theory<http://en.wikipedia.org/wiki/Naive_set_theory> by employing a theory of types<http://en.wikipedia.org/wiki/Type_theory> they devised for this purpose. While they succeeded in grounding arithmetic in a fashion, it is not at all evident that they did so by purely logical means. While PM avoided the known paradoxes and allows the derivation of a great deal of mathematics, its system gave rise to new problems.

In any event, Kurt Gödel<http://en.wikipedia.org/wiki/Kurt_G%C3%B6del> in 1930–31 proved that while the logic of much of PM, now known as first-order logic<http://en.wikipedia.org/wiki/First-order_logic>, is complete<http://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem>, Peano arithmetic<http://en.wikipedia.org/wiki/Peano_axioms> is necessarily incomplete if it is consistent<http://en.wikipedia.org/wiki/Consistent>. This is very widely – though not universally – regarded as having shown the logicist<http://en.wikipedia.org/wiki/Logicist> program of Frege to be impossible to complete.

[edit<http://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&action=edit&section=4>] Applied versions

There are some versions of this paradox that are closer to real-life situations and may be easier to understand for non-logicians. For example, the Barber paradox<http://en.wikipedia.org/wiki/Barber_paradox> supposes a barber who shaves all men who do not shave themselves and only men who do not shave themselves. When one thinks about whether the barber should shave himself or not, the paradox begins to emerge.

As another example, consider five lists of encyclopedia<http://en.wikipedia.org/wiki/Encyclopedia> entries within the same encyclopedia:

List of articles about people:

* Ptolemy VII of Egypt<http://en.wikipedia.org/wiki/Ptolemy_VII_of_Egypt>
* Hermann Hesse<http://en.wikipedia.org/wiki/Hermann_Hesse>
* Don Nix<http://en.wikipedia.org/wiki/Don_Nix>
* Don Knotts<http://en.wikipedia.org/wiki/Don_Knotts>
* Nikola Tesla<http://en.wikipedia.org/wiki/Nikola_Tesla>
* Sherlock Holmes<http://en.wikipedia.org/wiki/Sherlock_Holmes>
* Emperor Kōnin<http://en.wikipedia.org/wiki/Emperor_K%C5%8Dnin>
* Tim Tebow<http://en.wikipedia.org/wiki/Tim_Tebow>
List of articles starting with the letter L:

* L<http://en.wikipedia.org/wiki/L>
* L!VE TV<http://en.wikipedia.org/wiki/L!VE_TV>
* L&H<http://en.wikipedia.org/wiki/L%26H_(disambiguation)>

...

* List of articles starting with the letter K
* List of articles starting with the letter L
* List of articles starting with the letter M

...
List of articles about places:

* Leivonmäki<http://en.wikipedia.org/wiki/Leivonm%C3%A4ki>
* Katase River<http://en.wikipedia.org/wiki/Katase_River>
* Enoshima<http://en.wikipedia.org/wiki/Enoshima>
List of articles about Japan:

* Emperor Showa<http://en.wikipedia.org/wiki/Emperor_Showa>
* Katase River<http://en.wikipedia.org/wiki/Katase_River>
* Enoshima<http://en.wikipedia.org/wiki/Enoshima>
List of all lists that do not contain themselves:

* List of articles about Japan
* List of articles about places
* List of articles about people

...

* List of articles starting with the letter K
* List of articles starting with the letter M

...

* List of all lists that do not contain themselves?


If the "List of all lists that do not contain themselves" contains itself, then it does not belong to itself and should be removed. However, if it does not list itself, then it should be added to itself.

While appealing, these layman<http://en.wikipedia.org/wiki/Layman>'s versions of the paradox share a drawback: an easy refutation of the Barber paradox seems to be that such a barber does not exist, or at least does not shave (a variant of which is that the barber is a woman). The whole point of Russell's paradox is that the answer "such a set does not exist" means the definition of the notion of set within a given theory is unsatisfactory. Note the difference between the statements "such a set does not exist" and "it is an empty set<http://en.wikipedia.org/wiki/Empty_set>". It is like saying, "There is no bucket", or "The bucket is empty".

A notable exception to the above may be the Grelling–Nelson paradox<http://en.wikipedia.org/wiki/Grelling%E2%80%93Nelson_paradox>, in which words and meaning are the elements of the scenario rather than people and hair-cutting. Though it is easy to refute the Barber's paradox by saying that such a barber does not (and cannot) exist, it is impossible to say something similar about a meaningfully defined word.

One way that the paradox has been dramatised is as follows:

Suppose that every public library has to compile a catalog of all its books. The catalog is itself one of the library's books, but while some librarians include it in the catalog for completeness, others leave it out, as being self-evident.

Now imagine that all these catalogs are sent to the national library. Some of them include themselves in their listings, others do not. The national librarian compiles two master catalogs - one of all the catalogs that list themselves, and one of all those that don't.

The question is now, should these catalogs list themselves? The 'Catalog of all catalogs that list themselves' is no problem. If the librarian doesn't include it in its own listing, it is still a true catalog of those catalogs that do include themselves. If he does include it, it remains a true catalog of those that list themselves.

However, just as the librarian cannot go wrong with the first master catalog, he is doomed to fail with the second. When it comes to the 'Catalog of all catalogs that don't list themselves', the librarian cannot include it in its own listing, because then it would belong in the other catalog, that of catalogs that do include themselves. However, if the librarian leaves it out, the catalog is incomplete. Either way, it can never be a true catalog of catalogs that do not list themselves.

[edit<http://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&action=edit&section=5>] Applications and related topics
[edit<http://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&action=edit&section=6>] Russell-like paradoxes

As illustrated above for the Barber paradox, Russell's paradox is not hard to extend. Take:

* A transitive verb<http://en.wikipedia.org/wiki/Transitive_verb> <V>, that can be applied to its substantive<http://en.wikipedia.org/wiki/Substantive> form.

Form the sentence:

The <V>er that <V>s all (and only those) who don't <V> themselves,

Sometimes the "all" is replaced by "all <V>ers".

An example would be "paint":

The painter that paints all (and only those) that don't paint themselves.

or "elect"

The elector (representative<http://en.wikipedia.org/wiki/Group_representation>), that elects all that don't elect themselves.

Paradoxes that fall in this scheme include:

* The barber with "shave"<http://en.wikipedia.org/wiki/Barber_paradox>.
* The original Russell's paradox with "contain": The container (Set) that contains all (containers) that don't contain themselves.
* The Grelling–Nelson paradox<http://en.wikipedia.org/wiki/Grelling%E2%80%93Nelson_paradox> with "describer": The describer (word) that describes all words, that don't describe themselves.
* Richard's paradox<http://en.wikipedia.org/wiki/Richard%27s_paradox> with "denote": The denoter (number) that denotes all denoters (numbers) that don't denote themselves. (In this paradox, all descriptions of numbers get an assigned number. The term "that denotes all denoters (numbers) that don't denote themselves" is here called Richardian.)

[edit<http://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&action=edit&section=7>] Related paradoxes

* The liar paradox<http://en.wikipedia.org/wiki/Liar_paradox> and Epimenides paradox<http://en.wikipedia.org/wiki/Epimenides_paradox>, whose origins are ancient
* The Kleene–Rosser paradox<http://en.wikipedia.org/wiki/Kleene%E2%80%93Rosser_paradox>, showing that the original lambda calculus<http://en.wikipedia.org/wiki/Lambda_calculus> is inconsistent, by means of a self-negating statement
* Curry's paradox<http://en.wikipedia.org/wiki/Curry%27s_paradox> (named after Haskell Curry<http://en.wikipedia.org/wiki/Haskell_Curry>), which does not require negation<http://en.wikipedia.org/wiki/Negation>
* The smallest uninteresting integer<http://en.wikipedia.org/wiki/Interesting_number_paradox> paradox