Everything and More Quotes

“What exactly do ‘motion’ and ‘existence’ denote? We know that concrete particular things exist, and that sometimes they move. Does motion per se exist? In what way? In what way do abstractions exist? Of course, that last question is itself very abstract. Now you can probably feel the headache starting. There’s a special sort of unease or impatience with stuff like this. Like ‘What exactly is existence?’ or ‘What exactly do we mean when we talk about motion?’ The unease is very distinctive and sets in only at a certain level in the abstraction process—because abstraction proceeds in levels, rather like exponents or dimensions. Let’s say ‘man’ meaning some particular man is Level One. ‘Man’ meaning the species is Level Two. Something like ‘humanity’ or ‘humanness’ is Level Three; now we’re talking about the abstract criteria for something qualifying as human. And so forth. 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; the very strangeness of calling something a pen begins to obtrude on the consciousness in a creepy way, like an epileptic aura.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“Maybe even more important than the D.B.P. [Divine Brotherhood of Pythagoras], ∞-wise is the protomystic Parmenides of Elea (c.515-? BCE), not only because of his distinction between the 'Way of Truth' and 'Way of Seeing' framed the terms of Greek metaphysics and (again) influenced Plato, but because Parmenides' #1 student and defender was the aforementioned Zeno, the most fiendishly clever and upsetting philosopher ever (who can be seen actually kicking Socrates' ass, argumentatively speaking, in Plato's Parmenides).”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“that traversing an infinite number of dimensionless mathematical points is not obviously paradoxical in the way that traversing an infinite number of physical-space points is.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“That is, for a mathematical Platonist, what the C.H. proofs really show is that set theory needs to find a better set of core axioms than classical ZFS, or at least it will need to add some further postulates that are-like the Axiom of Choice-both "self-evident" and Consistent with classical axioms. If you're interested, Godel's own personal view was that the Continuum Hypothesis is false, that there are actually a whole (Infinity Symbol) of Zeno-type (Infinity Symbol)s nested between (Aleph0) and c, and that sooner or later a principle would be found that proved this. As of now no such principle's ever been found. Godel and Cantor both died in confinement, bequeathing a world with no finite circumference. One that spins, now, in a new kind of all-formal Void. Mathematics continues to get out of bed.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“Respecting infinite sets, for example, Intuitionism is rabidly anti-Cantor and Formalism staunchly pro-Cantor, even though both Formalism and Intuitionism are anti-Plato and Cantor is a diehard Platonist. Which, migrainous or not, means we're back to metaphysics: the modern wrangle over math's procedures is ultimately a dispute over the ontological status of math entities.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“In other words, Cantor is able to show that real numbers themselves can serve as the limits of fundamental sequences of reals, meaning his system of definitions is self-enclosed and VIR-proof.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“As was foreshadowed in Paragraphs 1 and 4, Cantor, and Dedekind's near-simultaneous appearance in math is more or less the Newton + Leibniz thing all over again, a sure sign that the Time Was Right for (Infinity)-type sets. Just as striking is the Escherian way the two men's work dovetails. Cantor is able to define and ground the concepts of 'infinite set' and 'transfinite number,' and to establish rigorous techniques for combining and comparing different types of (Infinity)s, which is just where Dedekind's def. of irrationals needs shoring up. Pro quo, the schnitt technique demonstrates that actually-infinite sets can have real utility in analysis. That, in other words, as sensuously and cognitively abstract as they must remain, (Infinity)s can nevertheless function in math as practical abstractions rather than as just weird paradoxical flights of fancy.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions or intuitions of space and time, that I consider it an immediate result from the laws of thought.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“For Plato, if two individuals have some common attribute and so are describable by the same predicate-'Tom is a man'; 'Dick is a man'-then there is something in virtue of which Tom and Dick (together with all other referents of the predicate nominative 'man') have this common attribute. This something is the ideal Form man, which Form is what really, ultimately exists, whereas individual men are just temporal appearances of the Form, with a kind of borrowed or derivative existence, like shadows or projected images. That's a very simplified version of the O.O.M., but not a distorted one-and even at this level it should not be hard to see the influences of Pythagoras and Parmenides on Plato's ontological Theory of Forms, which the O.O.M. is an obvious part of.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“Re which, again, please keep in mind that a language is both a map of the world and its own world, with its own shadowlands and crevasses-places where statements that seem to obey all the language's rules are nevertheless impossible to deal with.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“Also essential to math is the sense in which abstracting something can mean reducing it to its absolute skeletal essence, as in the abstract of an article or book. As such, it can mean thinking hard about things that for the most part people can't think hard about-because it drives them crazy.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“The cases of great mathematicians with mental illness have enormous resonance for modern pop writers and filmmakers. This has to do mostly with the writers'/directors' own prejudices and receptivities, which in turn are functions of what you could call our era's particular archetypal template. It goes without saying that these templates change over time. The Mentally Ill Mathematician seems now in some ways to be what the Knight Errant, Mortified Saint, Tortured Artist, and Mad Scientist have been for other eras: sort of our Prometheus, the one who goes to forbidden places and returns with gifts we all can use but he alone pays for. That's probably a bit overblown, at least in some cases. But Cantor fits the template better than most. And the reason for this are a lot more interesting than whatever his problems and symptoms were.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“The unease is very distinctive and sets in only at a certain level in the abstraction process—because abstraction proceeds in levels, rather like exponents or dimensions. Let’s say ‘man’ meaning some particular man is Level One. ‘Man’ meaning the species is Level Two. Something like ‘humanity’ or ‘humanness’ is Level Three; now we’re talking about the abstract criteria for something qualifying as human. And so forth. 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; the very strangeness of calling something a pen begins to obtrude on the consciousness in a creepy way, like an epileptic aura.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“1962 that “No data processing system, whether artificial or living, can process more than 2 × 1047 bits per second per gram of its mass,” which means that a hypothetical supercomputer the size of the earth (= c. 6 × 1027 grams) grinding away for as long as the earth has existed (= about 1010 years, with c. 3.14 × 107 seconds/year) can have processed at most 2.56 × 2092 bits, which number is known as Bremermann’s Limit.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“It's specifically this Z = 2^(Aleph0) that he couldn't prove. Ever. Despite years of unimaginable doodling. Whether it's what unhinged him or not is an unanswerable question, but it is true that his inability to prove the C.H. caused Cantor pain for the rest of his life; he considered it his great failure. This too, in hindsight, is sad, because professional mathematicians now know exactly why G. Cantor could neither prove nor disprove the C.H. The reasons are deep and important and go corrosively to the root of axiomatic set theory's formal Consistency, in rather the same way that K. Godel's Incompleteness proofs deracinate all math as a formal system. Once again, the issues here can be only sketched or synopsized (although this time Godel is directly involved, so the whole thing is probably fleshed out in the Great Discoveries Series' Godel booklet).”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“To make a long story short, Cantorian set theory helps unify and clarify math in the sense that all mathematical entities can now be understood as fundamentally the same kind of thing-a set.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“Cantor's discovery that lines, planes, cubes, and polytopes were all equivalent as sets of points goes a long way toward explaining why set theory was such a revolutionary development for math-revolutionary in theory and practice both.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“His next major proof (which you'll notice still concerns point sets) is an attempt to show that the 2D plane contains a (Infinity symbol) of points that's greater than the 1D Real Line's c in the same way that c is greater than the Number Line's (Aleph0). This is the proof of whose final result Cantor famously wrote to Dedekind "Je le vois, mais je'n le crois pas" in 1877. It's known in English as his Dimension Proof. The general idea is to show that the real numbers cannot be put into a 1-1C with the set of points in an n-dimensional space, here a plane, and hence that the cardinality of the plane's point set is greater than the cardinality of the set of all reals.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“A canny reader here may object that there's some kind of Zenoid sleight of hand going on in the above proof, and might ask why a similar hankie procedure and series couldn't be applied to the irrational numbers to quote-unquote prove that the total % of Line-space taken up by the irrationals is also 2*(Infinitesimally small symbol). The reason such a proof can't work is that, no matter how infinitely or even (Infinity to the Infinity symbol) ly many red hankies you drape, there will always be more irrational numbers than hankies. Always. Cantor proved this, too.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“Imagine you can see the whole Number Line and every one of the infinite individual points it comprises. Imagine you want a quick and easy way to distinguish those points corresponding to rational numbers from the ones corresponding to irrationals. What you're going to do is ID the rational points by draping a bright-red hankie over each one; that way they'll stand out. Since geometric points are technically dimensionless, we don't know what they look like, but what we do know is that it's not going to take a very big red hankie to cover one. The red hankie here can in truth be arbitrarily small, like say .00000001 units, or half that size, or half that half,...,etc. Actually, even the smallest hankie is going to be unnecessarily large, but for our purposes we can say that the hankie is basically infinitesimally small-call such a size (infinitesimally small symbol). So a hankie of size (infinitesimally small symbol) covers the N.L.'s first rational point. Then, because of course the hankie can be as small as we want, let's say you use only a (Infinitesimally small symbol)/2-size hankie to drape over the next rational point. And say you go on like that, with the size of each red hankie used being exactly (1/2) that of the previous one, for all the rational numbers, until they're all draped and covered. Now, to figure out the total percentage of space all the rational points take up on the Number Line, all you have to do is add up the sizes of all the red hankies. Of course, there are infinitely many hankies, but size-wise they translate into the terms in an infinite series, specifically the Zeno-esque geometric series (1/2^0 +1/2^1 + 1/2^2 +1/2^3 +1/2^4 + ...; and, given the good old a/1-r formula for summing such a series, the sum-size of all the infinite hankies ends up being 2*(Infinitesimally small symbol). But (Infinitesimally small symbol) is infinitesimally small, with infinitesimals being (as we mentioned in Section 2b) so incredibly close to 0 that anything times an infinitesimal is also an infinitesimal, which means that 2*(Infinitesimally small symbol) is also infintesimally small, which means that all the infinite rational numbers combined take up only an infinitesimally small portion of the N.L.-which is to say basically none at all-which is in turn to say that the vast, vast bulk of the points on any kind of continuous line will correspond to irrational numbers, and thus that while the aforementioned Real Line really is a line, the all-rational Number Line, infinitely dense though it appears to be, is actually 99.999...% empty space, rather like DQ ice cream or the universe itself.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“What this means is that the (Infinity) of points involved in continuity is greater than the (Infinity) of points comprised by any kind of discrete sequence, even an infinitely dense one. (2) Via his Diagonal Proof that c is greater than Aleph0, Cantor has succeeded in characterizing arithmetical continuity entirely in terms of order, sets, denumerability, etc. That is, he has characterized it 100% abstractly, without reference to time, motion, streets, noses, pies, or any other feature of the physical world-which is why Russell credits him with 'definitively solving' the deep problems behind the dichotomy. (3) The D.P. also explains, with respect to Dr. G.'s demonstration back in Section 2e, why there will always be more real numbers than red hankies. And it helps us understand why rational numbers ultimately take up 0 space on the Real Line, since it's obviously the irrational numbers that make the set of all reals nondenumerable. (4) An extension of Cantor's proof helps confirm J. Liouville's 1851 proof that there are an infinite number of transcendental irrationals in any interval on the Real Line. (This is pretty interesting. You'll recall from Section 3a FN 15 that of the two types of irrationals, transcendentals are the ones like pi and e that can't be the roots of integer-coefficient polynomials. Cantor's proof that the reals' (Infinity) outweighs the rationals' (Infinity) can be modified to show that it's actually the transcendental irrationals that are nondenumerable and that the set of all algebraic irrationals has the same cardinality as the rationals, which establishes that it's ultimately the transcendetnal-irrational-reals that account for the R.L.'s continuity.)”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“What Cantor's Diagonal Proof does is generate just such a number, which let's call R. The proof is both ingenious and beautiful-a total confirmation of art's compresence in pure math. First, have another look at the above table. We can let the integral value of R be whatever X we want; it doesn't matter. But now look at the table's very first row. We're going to make sure R's first post-decimal digit, a, is a different number from the table's a1. It's easy to do this even though we don't know what particular number a1 is: let's specify that a=(a1-1) unless a1 happens to be 0, in which case a=9. Now look at the table's second row, because we're going to do the same thing for R's second digit b: b=(b2-1), or b=9 if b2=0. This is how it works. We use the same procedure for R's third digit c and the table's c3, for d and d4, for e and e5, and so on, ad inf. Even though we can't really construct the whole R (just as we can't really finish the whole infinite table), we can still see that this real number R=X.abcdefhi... is going to be demonstrably different from every real number in the table. It will differ from the table's 1st Real in its first post-decimal digit, from the 2nd Real in its second digit, from the 3rd Real in its third digit,...and will, given the Diagonal Method here, differ from the table's Nth Real in its nth digit. Ergo R is not-cannot be-included in the above infinite table; ergo the infinite table is not exhaustive of all the real numbers; ergo (by the rules of reductio) the initial assumption is contradicted and the set of all real numbers is not denumerable, i.e. it's not 1-1 C-able with the set of integers. And since the set of all rational numbers is 1-1C-able with the integers, the set of all reals' cardinality has got to be greater than the set of all rationals' cardinality. Q.E.D.*”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“But what G. Cantor posits as the defining formal property of an infinite set is that such a set can be put in a 1-1C with at least one of its proper subsets. Which is to say that an infinite set can have the same cardinal number as its proper subset, as in Galileo's infinite set of all positive integers and that set's proper subset of all perfect squares, which latter is itself an infinite set.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“.....but let's emphasize once more here that G. Cantor is, like R. Dedekind, a mathematical Platonist; i.e., he believes that both infinite sets and transfinite numbers really exist, as in metaphysically, and that they are "reflected" in actual real-world infinities.....”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“If you object (as some of us did to Dr. Goris) that Cantor's transfinite numbers aren't really numbers at all but rather sets, then be apprised that what, say, 'P(Infinity to the Infinity +n), really is is a symbol for the number of members in a given set, the same way '3' is a symbol for the number of members in the set {1,2,3}. And since the transfinites are provably distinct and compose an infinite ordered sequence just like the integers,they really are numbers, symbolizable (for now) by Cantor's well-known system of alephs or '(Aleph symbol's). And, as true numbers, transfinites turn out to be susceptible to the same kinds of arithmetical relations and operations as regular numbers-although, just as with 0, the rules for these operations are very different in the case of (Alephs) and have to be independently established and proved.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“Assume that P is a second-species infinite point-set. Cantor shows that P's first derived set, P', can be "decomposed" or broken down into the union of two different subsets, Q and R, where Q is the set of all points belonging to first-species derived sets of P', and R is the set of all points that are contained in every single derived set of P', meaning R is the set of just those points that all the derived sets of P' have in common. Why not take a second and read that last sentence over again. R is the important part, and it's actually how Cantor first defines 'intersection' for sets, here via the infinite sequence of derived sets P', P'', P''',...(the sequence being infinite because P is a second-species-set).”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“Uniform Convergence & Associated Aracana item (d) for exceptional points, which again please recall can also be called 'discontinuities'. (N.B.: Some math classes also use singularity to mean exceptional point, which is both confusing and intriguing since the term also refers to Black Holes, which in a sense is what discontinuities are.)”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“Recall Part 3c's mention of how Cantor took what had been regarded as a paradoxical, totally unhandlable feature of (Infinity)-namely that an infinite set/class/aggregate can be put into a one-to-one correspondence with its own subset-and transformed it into the technical def. of infinite set. Watch how he does the same thing here, turning what appear to be devastating objections into rigorous criteria, by defining a set S as any aggregate of collection of discrete entities that satisfies two conditions: (1) S can be entertained by the mind as an aggregate, and (2) There is some stated rule or condition via which one can determine, for any entity x, whether or not x is a member of S.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“Here are the basic principles of Constructivism as practiced by Kronecker and codified by J.H. Poincare and L.E.J. Brouwer and other major figures in Intuitionism: (1) Any mathematical statement or theorem that is more complicated or abstract than plain old integer-style arithmetic must be explicitly derived (i.e. 'constructed') from integer arithmetic via a finite number of purely deductive steps. (2) 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.”
― David Foster Wallace, Everything and More: A Compact History of Infinity

“In a companion essay to "Continuity and I.N." that's usually translated as "The Nature and Meaning of Numbers," Dedekind evinces a remarkable proof for his "Theorem 66. There exist infinite systems," which runs thus: "My own realm of thoughts, i.e., the totality S of all things which can be objects of my thought, is infinite. For if s signifies an element of S, then the thought s', that s can be an object of my thought, is itself an element of S,..." and so on, meaning that the infinite series ([s] + [s is an object of thought]+['s is an object of thought' is an object of thought] + ...) exists in the Gedankenwelt, which entails that the Gedankenwelt is itself infinite. With respect to this proof, notice (a) how closely it resembles the various Zeno-like VIR back in paragraph 2a, and (b) how easily we could object that the proof establishes only that Dedekind's Gedankenwelt is 'potentially infinite' (and in precisely Aristotle's sense of the term), since nobody can ever actually sit down and think a whole infinite series of (s+s'+s")-type thoughts-i.e., the series is a total abstraction.”
― David Foster Wallace, Everything and More: A Compact History of Infinity