Set theory for poets / Poetry for set theorists

One modern incarnation of the debate between nominalism and realism is to be found in philosophical arguments about sets.  There are two ways of characterizing a set: intensionally, through description (e.g. the set of all inhabitants of London, to use an example of Russell's), and extensionally, which is just a list of the members of the set.

Quine, as nominalist as they come, objected to the "ontological excesses of set theory" when construed intensionally.  Is there really such an entity as "all the inhabitants of London"?  Yes, there are inhabitants, and we, or God, or Facebook could list them.  Each is an entity him- or herself (let's stipulate, because who wouldn't?)

The problem with extensional sets is that the vast, the utterly overwhelming majority of them would be utterly random, by our lights, like the contents of almost any book in Borges's "Library of Babel."  Those books are all (à très peu d'exceptions près) useless, and so too, more or less, would be thinking about things in sets.  The problem with intensional sets is that they may not exist (what is a set and where do I find one?), and even if some do exist, others might turn out to be impossible, despite seemingly innocuous descriptive criteria for membership.

Nevertheless, set theory is not only obviously useful: it's obviously a way that people think about the world and make sense of it (or it's a formalization of how we think and make sense of the world).  "Natural kinds" for example really do rely on a concept of nature not unlike the nature that we live in, that we evolved to survive in.  And it seems too that we find pleasure in finding sets, or figuring out what intensionally-characterized (or -characterizable) sets seemingly random extensional lists belong to.

Just to reiterate: intensional is more or less synonymous with interesting.  To characterize a set intensionally is to say that its members share some interesting property - interesting enough that you don't have to list them.

But here I want to focus on the converse idea as part of human literary or cultural play (as well as work): figuring out from a list what interesting set would embrace the items on that list.  It's true, of course, that a vast number of different interesting sets might embrace them, so we might want some further criteria of economy (this is also how Freud thinks about mental economy) for what the really interesting set is.  (That kind of economy is something like the criterion for a natural kind, and also for Wittgenstein's ideas about rule-following, which is for another post.)

The criteria would not necessarily be pure efficiency, but a balance between specificity and pith.  Pithy specificity is what we're looking for, and we'll know it when we see it.

Example:

{raven, writing desk}.

 Now we're not really asking about this set itself.  We're asking about the set it's a subset of, but we're still looking for a pretty small set.  So items whose names in English start with the phoneme /r/ won't cut it.  Nor, probably will nouns with the letter n, nor objects smaller than an elephant, nor things that don't taste like rhubarb. They both belong to those sets, yes, and to many others too, but still.

The two terms are, as every school child will remember, from a riddle by Lewis Carroll, which the Mad Hatter asks Alice.  He gives no answer, but later Carroll was prevailed upon to solve it.  He wrote:

Enquiries have been so often addressed to me, as to whether any answer to the Hatter's Riddle can be imagined, that I may as well put on record here what seems to me to be a fairly appropriate answer, viz: 'Because it can produce a few notes, tho they are very flat; and it is nevar put with the wrong end in front!' This, however, is merely an afterthought; the Riddle, as originally invented, had no answer at all.

As originally invented, then, it was offered as pure extension.

Now other writers offered later answers.  Martin Gardner and The Straight Dope give some of the best, e.g., Poe wrote on both (Sam Loyd). (Cecil Adams of The Straight Dope also explains the misspelling nevar: it's a palindromic raven.)

So the pleasure of riddles, of this kind of riddle, is the sudden collapse of extension into intension.  Sometimes that will require a reconceptualization of the elements in the extension: not "What's black and white and red all over?" no, but "What's black and white and read all over?"  The extension turns out to be the following set of qualities, denotable by adjectives and adjectival phrases: {black, white, read all over}.

What does this have to do with poetry?  Well, in English, anyhow, rhymes are to be distinguished from inflections.  We don't (really) count unity and disunity as a rhyme; motion and emotion are too close to each other.  As Wimsatt argues, the best rhymes will tend to be different parts of speech, and, as Empson points out, the fact that singular verbs but plural nouns end with -s means that we can't generally or easily rhyme subjects with predicates.  So rhyming words tend to be arbitrarily connected.

Consider the set R = {Mahatma Gandhi, the Coliseum, the time of the Derby winner, the melody from a symphony by Strauss, a Shakespeare sonnet, Garbo's salary, cellophane, Mickey Mouse, the Nile,..., Camembert}.  Extensionally there's nothing unusual about it, even if it is, as the kids say, "kind of random."  Not that random though: these all belong to a somewhat larger set of words that can be formed into subsets consisting of rhymed pairs, e.g. {the melody of a symphony by Strauss, Mickey Mouse}.  Rhyming with a member of some smaller set is the principle of inclusion in the somewhat larger set.

Or to put it another way, rhyming provides a principle of one-to-one correspondence between two sets of entities whose names have at least one rhyme.  That's not how I'm defining those sets: that's how I'm characterizing one of many facts about their members.  So the set R (whose membership I haven't fully listed) is the union of those two sets that are in one-to-one correspondence.

Now that principle, as we've seen, tends to be highly arbitrary in English.  The rhyming dictionary is disconcertingly senseless.  But what a poet does, like a riddler, is to find some intensional principle which defines a set given randomly and extensionally.  In this case that principle is that each member of the set R is a member of the set {things that are the top} (I am simplifying the song a little bit to make my point).

Now this distinction between intension and extension is also a distinction between use and mention.  The principle of membership of the two sets whose union forms R is first of all, that is to say, as a matter of poetic craft, a principle which mentions terms, i.e. selects them for the fact that they rhyme.  (The rhyming dictionary mentions words: it doesn't use them.)  But the job of the poet is to take these mentioned words and use them, which means to say something with them and therefore something about the things they signify or refer to.

The solution isn't just economical (as it is with a riddle), isn't just the sudden lifting of a burden through the sudden glory of an elegant summary of its components.  We shunt back and forth between use and mention, intension and extension, admiring at every moment how they fit together: look it rhymes! look, it's the top!

Studies (e.g. by Ray Jackendoff) of the neural handling of music suggest that different parts of the brain have different access to memory.  Some of the cerebral material we use to process music chunks and forgets immediately, so when a theme or motif is played again, it handles it as entirely new.  But other parts of the brain remember that motif or theme, and therefore experience a different relation to the novelty that is still being felt and processed.  That back and forth, that counterpoint, that complex and differently phased experience of music is the experience of music, or at least a large part of it.

I think the same is true about rhyming (and meter), especially since it appears that music actually recruits the cerebral material that processes sounds: vowels are much lower pitched than consonants, and we put words together from sounds much as we put musical experience together.  So I think that we go back and forth, sometimes putting together the longer-term, more coherent intensional sense of the set of rhymes we're given and sometimes testing the always novel extension of the list, and that the delight in doing so is how the abstract distinctions to be found in set theory play out in the pleasures of poetry, and of math.

(At least that's what struck me today.)