Colin Fraser

There is a particular problem in a branch of mathematics called Ramsey theory. The problem is somewhat difficult to explain clearly as it involves something called a hypercube, which is just your basic cube (well, square, really) generalized to n dimensions. This is a 4-cube (a 4 dimensional hypercube) projected into three-dimensional space (then projected into 2 dimensional space, since you’re looking at it on a 2 dimensional computer screen):

Anyway, that’s not really the hilarious part. The Ramsey theory problem is to find the smallest number of dimensions for which connecting each vertex of an n-cube by a line segment and then colouring each line segment will force you to have a single-coloured 4-vertex planar complete subgraph. Don’t worry if you didn’t follow that; you’ll still understand the funny part. But just to give you an idea of the problem, this is Wikipedia’s example of what the problem means with a 3-cube (the regular type of cube):

The thing under the cube is a single-coloured 4-vertex planar complete subgraph—it’s just a square with an X inside of it that is all one colour (don’t worry, the funny part is coming). So again, the problem is to try to figure out what is the smallest dimension N of the hypercube for which it is impossible to not have a single-coloured square with an X in it. It turns out that it’s possible to construct such a 3-cube, so N>3.

So the problem isn’t solved, per se, but in 1971 a fellow named Ronald Graham determined an upper bound on the solution. The upper bound is known as Graham’s number, and it is so large that you cannot express it with regular notation. There is literally not enough space in the universe; if you tried to write it out in a normal digital representation (like 4938293… etc), and each digit took up the smallest amount of space that we know how to think about, the representation would take up more space than there is in the universe. Incidentally, we somehow know how to calculate the last digits of Graham’s number—they are 2464195387.

So that’s the upper bound on the solution. There is also a lower bound. Would you like to know what the lower bound is? It’s 13. So, to sum up, the solution to the above problem is known to exist. Let’s call it N. Here’s what mathematics has been able to prove about N:

13 ≤ N ≤ some number that is considerably larger than the number of atoms in the Universe.

This is what mathematicians do for a living.

The thing that I like about David Foster Wallace’s prose, aside, of course, from his in-your-face-erudition—which many might find off-putting but which I find endearing, if a little desperate, as though he wants so badly to show that he can speak intelligently about any topic under the sun that he is willing to postpone advancement of the actual story for lengths spanning sentences (spanning David Foster Wallace sentences, that is, which are not short) or paragraphs or pages in favor of spending that time flexing his scholarly muscles (I get that)—is that I believe his loopy, meandering sentences, with long parenthetical asides and clauses playfully nested within clauses (that are playfully nested within other clauses still)—sentences which, while challenging to navigate for the reader, stubbornly refuse to violate any syntactic rules and commit no formal grammatical error—are, rather than a simple artifact of his aforementioned in-your-face-erudition, a comment on the absurdity of the notion that the complexity and texture of human thought and emotion can be properly transmitted through this clumsy combinatorial system of symbols called language by simply following the rules, being that it (i.e., language) cannot even seem to keep straight what is meant by which pronoun in a sentence, a fact that he often forces to the forefront of the reader’s mind by unabashedly highlighting pronominal ambiguities, following them (i.e., the ambiguities) with a bracketed explanation of what precisely the ambiguous pronoun was meant to mean at all (e.g. this quotation from The Depressed Person: “The depressed person shared how she could remember, all too clearly, how at her third boarding school she had once watched her roommate talk to some boy on their room’s telephone as she (i.e., the roommate) made faces and gestures of entrapped repulsion and boredom with the call, this popular, attractive, and self-assured roommate finally directing at the depressed person an exaggerated pantomime of someone knocking on a door until the depressed person understood that she was to open their room’s door and step outside and knock loudly on it so as to give the roommate an excuse to end the call.”—emphasis mine), a trick which, in addition to being helpful to the reader (his (ie. David Foster Wallace’s) sentences are difficult to follow, after all) is meant to remind him or her that words on page are, as a representation, or worse, as a wannabe high-fidelity reproduction, of pure visceral thought, simply as good as it gets and nothing more, regardless of the erudition of their (i.e., the words’) author.

I just wanted to try that.

This paper gives an example of a formula that works when you plug in 1, 2, 3, 4, 5, … all the way up to about 10^102832732165, and then it stops working.