This is in response to Anne’s comment, which is one of the longest comments you’ll ever see on the interwebs. Since this will be longer, and will involve mathematical symbols, I’ll do it in a post.

I’m not sure about the implications for the dynamical sublime vs. the mathematical sublime, or for narrative theory, but maybe if I explain the math more clearly, you’ll get some ideas, Anne?

About infinity. The infinities involved in this discussions have to with the size of a set (its cardinality, rather than an intuitive idea of infinity as a really big number. Consider the graph of the function f(*x*)=1/*x* that’s pictured at the right. Intuitively, we can say that f(*x*) approaches plus infinity as it approaches 0 from the right, and it approaches negative infinity as it approaches 0 from the left. There’s a way to rigorously define that notion of infinity, but that’s not the kind of infinity we’re talking about. A set is a collection of anything. You could think of the set of natural numbers ={1,2,3,…}, or some set called V that’s the set of every Victorianist that ever lived, or a set of the seven days of the week. If it’s a finite set, then its cardinality is straightforward. It would be 7 for the set of the seven days. V is a much “larger” set, but it’s still a finite number. You probably wouldn’t be able to give an exact number for its size, but you could put an upper bound on it. (It must be less than a billion, since there haven’t been a billion academic scholars of any particular variety.) You couldn’t put an upper bound on the size of , though. (Suppose you could: if somebody tells you the size of is *n*, you could say oh, but what about the set {1,2,3,…,*n*+1}, the size of that set is bigger than *n*, and is bigger than that.) So what do you do for talking about the size of infinite sets?

This is where Levinson’s discussion of “matching” comes into play. Mathematicians say that two sets have the same cardinality if there exists a bijection between the two sets. Here’s the definition from wiki:

In mathematics, a

bijection, or abijective functionis a functionffrom a setXto a setYwith the property that, for everyyinY, there is exactly onexinXsuch thatf(x) =yand no unmapped element exists in eitherXorY.

As an example, say X is the set of natural numbers {1,2,3,…}, and Y is the set of even numbers {2,4,6,…}. If f is a function from X to Y is defined as f(*x*)=2*x*, f is a bijection, since for every even number (42, for example), there’s only one natural number such that f(*x*) is that number (21, and nothing else). In addition, every natural number can be doubled, and every even number can be halved, so there are no unmapped elements in either X or Y. In this sense, the set of natural numbers and the set of even numbers are the same “size” (i.e. cardinality). But what about the set of real numbers ?

Emily asked me what they are, and they’re incredibly hard to define. They weren’t defined rigorously until our favourite century! (The whole field of calculus wasn’t rigorously defined until C19). Intuitively, you can think of them as the set of rational numbers (any number that can be put into the form *p*/*q*, where *p* and *q* are integers) , joined up with the set of irrational numbers, which are numbers that, when you put them into decimal form, don’t repeat (the square root of 2, π [the proof that π is irrational *incredibly* complicated–it took three lectures after a whole year of intense math just to outline the proof]). There *doesn’t* exist a bijection between and . There doesn’t exist a bijection between and the set of reals between 0 and 1 either. In that sense, the infinity that’s “between 1 and 2” is “bigger” than the set of natural numbers.

*However!* The set of rational numbers *does* have the same cardinality as . There is a bijection between the two sets. Don’t ask me what that bijection is, but you can imagine a bijection between and the rational numbers that are between 0 and 1. For example, define *g* which is a function that maps onto the interval of between 0 and 1 as follows:

g(1)=0

g(2)=1

g(3)=1/2

g(4)=1/3

g(5)=2/3

g(6)=1/4

g(7)=3/4

g(8)=1/5

…

Okay, so that’s not a definition of *g* that would make mathematicians happy, and I’m not going to prove that it’s a bijection, but you get the idea.

This is where the whole “density” thing gets weird, and why I wanted to get some real math into the picture. Because, mathematically speaking, the rational numbers between 0 and 1 (call this set R) are just as “dense” as the irrational numbers between 0 and 1 (call this set S). Both R and S are considered dense sets in X, where X are the real numbers between 0 and 1. Basically that means that, for whatever element of X that you choose, you can get as close as you would ever want to. (For example, take the square root of 1/2. If you picked some number that’s super-tiny, but not 0, no matter what, you could find a rational number [infinitely many!] between the square root of 1/2 and the square root of 1/2 plus that super-tiny number.) So if you think of S as X with infinitely many “holes” in it, that’s still uncountably infinite, so “bigger” than (i.e. there is no bijection between S and ).

Now, I think this is still in the realm of the mathematical sublime, but it’s still cool. For any set, the power set of it is the set that’s made up of all of its subsets. (For example, the power set of the set of Victorianists would include the set of myself, the set of myself plus Anne, the set of all the Victorianists at the Grad Center,the set of all Victorianists whose last name begins with Y, the set of all Victorianists who aren’t white…) If you take the power set of , denoted P(), (which would include {1,2,3}, {123}, the even numbers, the odd numbers, the prime numbers, the prime numbers which have at least 200 digits, etc.), there’s no bijection between that and . I don’t think after all those semesters of math I took I ever got there, but you can prove that P() is the same size as (i.e. there exists a bijection connecting the two). But then what about P()? That turns out to be “bigger” than . And you can take the power set of that again. And again. And again….

November 13, 2009 at 10:24 am

Wow. I think my brain has gone soft from all these years in the humanities. (As evidenced by my pathetic attempts to explain quantum physics and deconstruction to my 8am students yesterday — they asked!)I am going to read this post N more times and see what clicks. But thank you. (I think.)

Also, my comment wasn’t THAT long….

November 13, 2009 at 3:43 pm

I suppose your comment wasn’t so long compared to the comments other academics leave on blogs. As for quantum physics, is it Heisenberg’s uncertainty principle that’s the most applicable to deconstruction?

November 15, 2009 at 3:15 pm

Yeah, I brought up the uncertainty principle and the wave-partical duality. All while the rabbit-duck image was projected on the board behind me. One of my students got kind of freaked out about the whole thing and really they just wanted me to end class early so they could register for their spring classes. It was sort of a ragged day all around and I’m not sure I’m going to assign Jankelevich again next year.