Tag Archives: Everything and More

The remainder

Just in case you want to continue the discussion of E & M, here are some things that I’d be super interested to discuss:

1) Math as a parallel to life (Tom, I think it was you who ended the conversation on this note, which I think is brilliant!). Wallace’s E & M certainly draws out parallels between math and life. For example, the lack of solutions and finality. Another example, that not everything can be proven. And another–there are real numbers, irrational numbers, rational number, imaginary numbers–do these types of numbers perhaps mirror a few general categories of people?

Precisely what is the relationship between the math world and the human world? Can this relationship be pinpointed, or is the solution / answer to this question indefinite like infinity? Is there a map / function / correspondence between these two worlds? Are there common elements that both care about? For instance, existence seems to play a huge role in E & M as it does in The Broom. But in what ways is the issue or urgency of existence different in these two (con)texts?

2) I also wanted to ask, “What did you guys learn about math from E & M?”

3) Also, Wallace definitely dramatizes the history of math in E & M, I think. But it’s interesting how he does it–he seems to portray the history of math as a battle–the battle between Intuitionists and Platonists, the battle between the everpresent existence vs. the created existence of math, etc.

Some questions

In effort to prepare for our presentation on E&M on Monday, our group decided to ask some questions on the blog over the weekend so that you can start thinking about them!

1. What makes this text so difficult to talk about?

2. Recently on the blog, a few conversations have returned to The Broom. Does E&M figure math as a game? In what ways? And what are the implications?

3. Wallace’s question throughout the book:   In what way do abstract entities exist?

The proof

According to the author, what does the mathematical proof do? How does the proof function in Everything and More?

The concept of a math proof is fascinating to me, and in its discussions of proofs, Everything and More inquires about the nature and power of the proof.

Please keep these questions in mind as you read this blog:   “In what way can we say a unicorn exists that is fundamentally different, less real, than the way abstractions like humanity or horn or integer exist? Which is once again the question:   In what way do abstract entities exist, or do they exist at all except as ideas in human minds-i.e., are they metaphysical fictions?….Are mathematical realities discovered, or merely created, or somehow both?” (20).

The proof holds enormous power in the world of math and in Everything and More. The author proposes that through a proof, we can definitively prove the truth. In discussing abstractions in the context of set theory, the author insists that “we are proving, deductively and thus definitively, truths about the makeup and relations of such things” (256). Furthermore, the writing suggests that the proof literally give existence to abstractions:   “So one thing to appreciate up front is that, however abstract infinite systems are, after Cantor they are most definitely not abstract in the nonreal/unreal way that unicorns are” (205). This quote conveys the author’s faith in the power of the proof. The proof, according to the author, confers existence to infinite systems. The author does not assert that infinite systems have always existed. He does, however, express that “after Cantor, they are most definitely not abstract in the nonreal/unreal way that unicorns are.” What does the author mean by “after Cantor”? What does Cantor do that convinces the author of the abstract existence of infinite systems? Cantor proofs it. Thus, the proof has the power to make previously “nonreal/unreal” entities real or at least confirm their reality.

I mentioned that by writing “after Cantor, they are most definitely not abstract in the nonreal/unreal way that unicorns are” (205), the author suggests that the proof somehow creates the abstract existence of infinite systems. In another moment, however, the author rescinds his earlier insinuation that the proof creates existence and scorns that entire notion. In a footnote response to an editor asking, “we can always create new ones [subsets]?” (273). The author responds, “The ‘we can always create new ones’ part is deeply, seriously wrong:   we’re not creating new subsets; we’re proving that there do exist and will always exist some subsets” (273). Thus, here, the proof becomes an instrument of finding, not creating, and here, the author insists on the distinction between the two.

In another instance, however, the author does not seem to mind the conflation between proving that “there do exist” and “creating.” He explains that for the Constructivists, “The only valid proofs in math are constructive ones, with the adjective here meaning that the proof provides a method for finding (i.e., ‘constructing’) whatever mathematical entities it’s concerned with” (225). The author points out an interesting English-language coincidence in the footnote, where he notes that “the word ‘constructive’ for us can mean ‘not destructive’. As in good rather than bad, building up rather than tearing down” (225). The author, however, does not point out the other interesting English-language nuance that he does call attention to earlier:   the difference between finding and creating. In the beginning, the author asks, “Are mathematical realities discovered, or merely created, or somehow both?” (20). For the Constructivists, discovering and creating become synonymous, and by not calling out this distinction here, the author indicates his acceptance of this conflation, which seems odd considering his previous tirade on the difference between the proof as creating existence and the proof as merely showing existence.

Nonetheless, the existence led by mathematical entities, whether created by proofs or already there, seems more tragic than celebrated. The author laments, “And, as true numbers, transfinites turn out to be susceptible to the same kinds of arithmetical relations and operations as regular numbers” (243). He writes as if existence were a disease, rendering these entities “susceptible.”

What, then, is the relationship between mathematical existence and human existence, our existence? As Will proposes in his post (Math as Communication), math can be read and understood as a language, and Wallace claims that “language is both a map of the world and its own world” (30). Does the world of math ever intersect our world?

One point of intersection occurs in Cantor’s inability to prove the Continuum Hypothesis. Although the math world eventually proves the unprovability of the C.H., the human world witnesses the actualization of this unprovability, for Cantor’s inability to prove the C.H. performs its unprovability. Another point of intersection crystallizes when Wallace writes, “Mathematics continues to get out of bed” (305). Mathematics here, is personified. Mathematics here, becomes human, even if just for a moment.

Appreciating Infinity

In class there was a little bit of discussion regarding whether Everything and More even needed to include all the technical math terms, symbols,  and proofs.   Since DFW  seems to be trying to illuminate the historical/aesthetic aspect of infinity, why not just stick to descriptive writing, right?   For most of the people in class,  it seemed that  the math became a big  headache.    Some math majors felt that  the technical aspect of the book came off as reductive and serous.   For most of the humanities people, the math jargon was either ignored or dismissed as way too abstruse.   I’m going to submit, however, that the technical breakdowns are necessary for Wallace to communicate the beauty and profundity he sees in  the history of infinity.    

(Note: whether Wallace conveys this effectively isn’t what this post is addressing.   I’m just saying if  this is what the book is going for, technical explanations are necessary)

A key component of the aforementioned communication is our ability to appreciate the subject Wallace is dealing with.    Wallace needs to show us not just why, but how, to appreciate Cantor’s discovery.   Since appreciation, I think, derives from experience, we need to some how experience the buildup toward  infinity.

Analogy: We appreciate Michael Jordan’s jump shot because we know how difficult it is to make a basket over a defender.   We appreciate Wallace’s essays because we know how hard it is to write well, and we’ve read lots of mediocre writing.   In the case of Everything in More, we need to experience the abstractness and difficulty in thinking about infinity to truly appreciate the astounding discoveries of Cantor and company (another  subject of our appreciation is  the historical context:  mathemeticians being dismissed as heretics, their battle against the beliefs of time, etc., but that’s for another post).  

Hence Wallace’s opens the book by saying that “abstractions ha[ve] all kinds of problems and headaches built in, we all know” (11).   He’s already starting to get us to think about this stuff in a broad sense.   When we actually get to specifics like derivatives, integrals, and number lines, we accordingly need more specific explanations.   It’s arguable that broad definitions could get the job done, e.g., to just say that an integral is the “area bound by a curve” – and Wallace does this in the emergency glossary (p. 109)  – but it doesn’t do much in communicating serious meaning, I think.  

If one actually sits down and fights through one of these proofs, however,  one might actually gain deeper apprecation for the ingenious mind behind it.  For most non-math people, these concepts are not easy – that’s the point.   If we can feel the difficulty of these abstract ideas, we can begin to form  some sense  of the sheer brilliance of Cantor, Dedekind, et al.   It’s more than paying lip-service to their genius because they’ve been exalted by history; it’s more like feeling it.    To know, empirically, that they  have done something incredible.  And as far as I can tell, beauty and profundity always manifest as feelings.

The Return of the Barber

While finishing up Everything and More, I came across a familiar paradox:

“Russell also has a famous way to set up his Antinomy in natural language, to wit: Imagine a barber who shaves all and only those who do not shave themselves-does this barber shave himself or not?” (Everything 278 IYI2).

Sound familiar?

“Lenore nodded. ‘Gramma really likes antinomies. I think this guy here,’ looking down at the drawing on the back of the label, ‘is the barber who shaves all and only those who do not shave themselves'” (Broom 42).

In The Broom of the System, Wallace uses the barber paradox as more of a plot device than anything else. Lenore Senior leaves the drawing of “a person, apparently in a smock. In one hand was a razor, in the other a can of shaving cream. Lenore could even see the word ‘Noxzema’ on the can. The person’s head was an explosion of squiggles of ink” (Ibid.) for Lenore Junior as a clue to the whereabouts of Lenore Senior. While Lenore Senior is indeed big on words and antinomies, the barber paradox is not really all that important to the story in the end.

In Everything and More, on the other hand, Wallace uses the paradox as it was originally used by Bertrand Russell in an effort to explain Russell’s Paradox. Russell’s Paradox, from what I can glean, is a long proof that eventually ends with the paradox that the set of normal sets is both normal and abnormal (sorry if I’m butchering this, math friends). Basically, the paradox here serves to help readers gain a better understanding of Russell’s Paradox through more concrete language than the original theorem offers.

My question, then, is why? Why does Wallace use the barber paradox in both The Broom of the System and Everything and More? I know this is backtracking to the beginning of the semester, but reading the paradox in its “original” state in Everything and More makes me wonder all the more exactly what it’s doing in The Broom of the System. Also (and here comes some sort of fallacy, I’m sure), I think it’s really interesting to note that The Broom of the System was Wallace’s senior thesis, which came just a few short years after his abrupt “click” away from the world of math. It’s interesting to see that math still plays a role in his first novel, no matter how subtle it may be. Furthermore, Wittgenstein apparently tried to prove that Russell’s Paradox was incorrect and should be disposed of. I don’t know if this movement got much of a following, but it is worth noting, especially given Wallace’s many Wittgenstein-like characteristics.

When I read The Broom of the System at the beginning of the semester, I had no idea that the barber paradox was as famous as it is-in fact, I had never heard of it before. But after having read Everything and More, it makes me wonder if the paradox plays a larger role in The Broom of the System than as a mere plot device, or something to just ponder over while reading page 42 and then forget about. Any thoughts?

Math as Communication

Reading through Everything and More with DFW’s McCaffery interview discussion of the purpose of art in mind, I’m tempted to treat mathematics as a language, and assess its value as a method of communication. Now, in some ways, math’s a great language: It’s a formal system. As DFW went over in §1, this basically means that it starts with a set of axioms and then deductively derives all other expressions from the relationships and properties of these axioms, or from previously deduced expressions. The driving force of the expansion (or to use DFW’s term abstraction) of a formal system is going to be the purification of that system, the rationalization of relationships and aspects of the system which seem paradoxical, or irreconcilable. A perfect formal system would be one that was totally free of contradiction or paradox at every level of abstraction, each statement being deductively provable as consistent with the system or inconsistent, true or false; Infinity proves to be such an interesting subject because it has, throughout history, been a chink in mathematics’ armor, it has been an imperfection in the formal system of mathematics since Zeno (Mathematics being a formal system aspiring to perfection, the attempted solutions to the problems of infinity literally are the history of mathematical progress). Anyway, because math is a formal system, two different people who understand math will understand it in close to exactly the same way, all proven propositions can be understood by anyone who reads through the proof. An almost exactly mutual body of knowledge is shared by people at the same level of mathematical knowledge. This is great communication-wise because it means that we’re all very much on the same page, all vocabulary is singularly and perfectly denotative, all symbolic representation singularly interpretable, and thus the knowledge contained within each symbolic representation is communicated perfectly by that representation.  

But exactly what kind of knowledge is communicated by mathematics? Pure math hopes to get at a more perfect understanding of the formal system of mathematics, a refinement of the language so to speak. Basically the knowledge pure mathematicians hope to communicate is knowledge about math itself; what does and doesn’t fit into the system, how and why. This makes pure math a bit esoteric (which tends to be the complaint of most high school pre-cal students, who find themselves asking why they need, or even would want, to understand trigonometric identities) but also makes it kind of the metafiction of mathematics, and the type of math that advances mathematics itself most consistently. Case in point, historically, pure math has tended to invent ideas which are only later discovered to be ‘useful.’

Which brings us to applied mathematics. It turns out that anything that can be described quantitatively can also be described mathematically, and so the powers of the formal system, namely unambiguity, can be brought to bear on the real world. In this way, the project of modern mathematical science can be thought of as the description of the world in the terms of a formal system. Indeed, the divergence of science and philosophy came when science embraced the language of mathematics to describe the world while philosophy remained grounded in traditional verbal language. Their projects remain the same, their methods are all that’s diverged. But when you start trying to describe the real world you’re intrinsically limited by the language you’re using, and this is important to keep in mind. It’s easy to see why mathematical platonists see some higher world of mathematical relationships at the core of experience, but isn’t this kind of like claiming the word tree created the thing, that grammatical relationships are at the core of experience? In the end when you describe the world, you’re fitting the world to your language more than your language to the world, so does it really matter what language you’re speaking? is the whole project moot in the face of solipsism? Does the project of pure math succeed, as metafiction attempted, in being modest enough so as to be truly achievable?

What is this madness?

In Infinite Jest, Wallace writes about Tony Krause in the process of Withdrawal: “He’d naively assumed that going mad meant you were not aware of going mad; he’d naively pictured madmen as forever laughing” (303).

The fact that Tony “naively pictured” and “naively assumed” suggests that going mad does not mean forever laughing and that going mad does not mean you are not aware of the process (of going mad). Then what does going mad mean?

Suppose that Tony is in fact going mad. Then for Tony, madness certainly entails an awareness of itself. The text, however, does not explicitly state that Tony is aware of his madness. Instead, the text defines madness by what it does not mean: madness does not mean laughing forever, does not mean being oblivious of one’s own madness. The text leaves the reader to infer what madness actually means. What does the text leave the reader?

The text leaves the reader with an absence or a void–in particular, one that results from an extraction (like a dentist extracting a tooth and leaving you with this hole where your tooth used to be). Immediately before discussing Tony’s naïve assumptions about going mad, Wallace discloses that “He [Tony] was haunted by the word Zuckung, a foreign and possibly Yiddish word he did not recall ever before hearing. The word kept echoing in quick-step cadence through his head without meaning anything” (303). In the first sentence, Wallace declares “Zuckung” as the word that haunts Poor Tony. In the second sentence, however, Wallace does not use the word Zuckung again. He does not say, “Zuckung kept echoing.” Instead, he simply writes: “The word kept echoing.” That way, the word becomes a void, emptied of its content, just as it becomes emptied of its meaning: “The word kept echoing in quick-step cadence through his head without meaning anything.” Incidentally, Wallace does not provide the reader with the meaning of Zuckung (according to the Internet, Zuckung means twitch, spasm, or convulsion in German).

Another effect of “The word kept echoing in quick-step cadence through his head without meaning anything” is that the vagueness and generality of “The word” gives the reader space to fill in another word, a different word–in which case, for me, the word would doubtlessly be “time.” Throughout the paragraph in which this sentence is embedded, “time” crops up repeatedly; the word “time” keeps echoing in quick-step cadence through the reader’s head and perhaps, “without meaning anything.”

With each repetition of “time,” time takes on another form; in the end, time ceases to mean anything at all. Wallace writes, “Time was being carried by a procession of ants….time itself seemed the corridor, lightless at either end. After more time time then ceased to move or be moved or be move-througable….time with a shape and an odor….time had become shit itself” (302-303). In the last mention of “time” in the paragraph, Wallace explains that “Poor Tony had become an hourglass: time moved through him now” (303). In this metaphor, Tony becomes an hourglass, a device that measures time, but time does not move through an hourglass; sand does. This warp draws attention to time’s absence. Here, Tony becomes a metaphor for an hourglass and time becomes a metaphor for sand, but nothing becomes a metaphor for time. Nothing can do that. By rendering time a metaphor for a cornucopia of different objects and not allowing time to be a reference for any other metaphor (does this make sense?), this paragraph strips time of its denotation. Time itself becomes empty and extracted.

Perhaps this is what it means to go mad: to be emptied and to be aware of it–the emptiness.

A bridge to Everything and More: Apparently, according to the dictionary, “to extract” and “to abstract” are synonymous. Check out this quote from Everything and More: “Thinking this way can be dangerous, weird. Thinking abstractly enough about anything…surely we’ve all had the experience of thinking about a word–‘pen,’ say–and of sort of saying the word over and over to ourselves until it ceases to denote” (12). Compare it with “The word kept echoing in quick-step cadence through his head without meaning anything.”

Ah, Math.

David Foster Wallace’s Everything and More: A Compact History of Infinity is about math. An obvious statement, yes, but also the reason that I had read all of Wallace’s work except for this booklet prior to the start of this class. I was always decent at math, but I never enjoyed it; some people think in numbers, but I am not one of them.

Before reading the first half of this booklet, I wondered whether Wallace’s signature style would shine through all the equations and theorems, or if the book would be more textbook-like than Wallace-like. Fortunately, I think that even among all the numbers, Wallace’s sense of humor and attention to detail are still present. One light-hearted moment that I particularly enjoyed came in Wallace’s description of the Number Line, which “infinitely dense though it appears to be, is actually 99.999 . . .% empty space, rather like DQ ice cream or the universe itself” (90).  The same attention to detail that is present in Wallace’s other works like A Supposedly Fun Thing I’ll Never Do Again (particularly the titular essay) is also evident here; I personally never would have made the connection between the Number Line’s density and DQ ice cream, but now that Wallace has, I see it and think that it’s a pretty good comparison.

The footnotes in this booklet, too, are still humorous and clever (despite being math-related). While discussing the Fallacy of Equivocation, Wallace inserts an example via a footnote:

“As in:

(1) Curiosity killed the cat.

(2) The World’s Largest Ball of Twine is a curiosity.

(3) Therefore the World’s Largest Ball of Twine killed the cat (57).”

Even though it has absolutely nothing to do with math, it still proves the point he’s trying to make, and allows the reader a humorous interlude among all the numbers. I feel like Wallace is greatly attuned to the fact that many of his readers might not be mathematically savvy, and is always trying to make the math more interesting for those of us who don’t share his love of numbers. At one point he even becomes a bit defensive in a footnote, stating, “Sorry if this is confusing; we’re doing the best we can” (116). Because Wallace himself was so mathematically talented, it must have been difficult to have to make compromises between the difficulty of the material and having the reader actually understand something.

Yet Wallace does succeed in his quest to make this booklet reader-friendly: I found that, despite only going through Calc I and not having taken math in two years, I still understood most of what he was trying to explain throughout the course of the first 157 pages. While talking about the “weirdness” of mathematics and the wide variety of symbols used, Wallace addresses this by talking about the Humanities:

This difficulty, despite what Humanities majors often think, is not because of all the heavy-looking notation that can make flipping through a college math book so intimidating. The special notation of analysis is actually just a very, very compact way to represent information. There aren’t that many different symbols, and compared to a natural language it’s ridiculously easy to learn. The problem isn’t the notation-it’s the extreme abstractness and generality of the information represented by the symbolism that makes college math so hard. Hopefully that makes sense, because it’s 100% true (146).

I found that I agreed with some of this analysis. Yes, the abstractness and generality of the information represented by the symbolism is complex and difficult to understand if you are not particularly math-inclined. But at the same time, there are several different symbols, and many of them look remarkably similar to one another (like X, and the x used to indicate multiplication). I think that here, Wallace is perhaps underestimating the complexity of the symbols (and maybe overestimating my math capabilities!).