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.)

Lewis on Fictional Worlds

My friends assure me that D.K. Lewis was completely serious about his belief in the real existence of possible worlds. For me they're a useful heuristic, but I guess it helps to go whole hog, since to do that commits you to a more robust view of truth: it really is true in a possible world that something possible happened although it didn't happen here.  Lewis's idea makes truth more robust by shifting the adjective possible from modifying truth to modifying a set of worlds.  Then not only are possible things true in some worlds, but it is true, pure and simple, that in all worlds taken together every possible thing is true.

Which can be a depressing idea, depending on the vividness of your imagination, and depending also on your views of counterparts and zombies and the like.  (I myself would want instead to assume that all my counterparts are zombies, and that I escape death at every moment by threading the maze of worlds where my counterparts, are, at every moment, dying.  I could believe in a kind of asymptotic immortality this way, or a kind of perpetual standing wave of reincarnation into some still-living version of myself.  Or, really, a kind of packet-switching of the entity or identity consisting of myself plus all my zombie counterparts; me, I'd always exist in one of the packets getting through to the future, and not in the packets that weren't.  This suggests, by the way, a reason to be a one-boxer: I want to live in one of the one-box worlds.  Lewis, a two-boxer, himself admits that the one-boxers would do better than he would, as a matter of fact.   He'd still pick one box though, since that couldn't causally change his outcome.  But to resume the thread of my discourse....)

If Lewis is serious, and if he's right, he would justify, and also give a spooky and discomfiting authority to, a kind of literary fiction (and fan fiction too) which it's harder to justify if there's just one world, "la vie, la seule qu'on a." Here's how Lewis says we should think about fictional worlds:

The worlds we should consider, I suggest, are the worlds where the fiction is told, but as known fact rather than fiction.  The act of story-telling occurs, just as it does here at our world; but there it is what here it falsely purports to be: truth-telling about matters whereof the teller has knowledge.

Lewis doesn't say why the fiction has to be told, perhaps because he thinks it's obvious: the world of the fiction is one in which the events occur and in which the fictional narrative also exists, since it is, at least implicitly, part of the fictional narrative that it exists.  Now, I myself don't think that it is an implicit part of a narrative that it exists in the world it purports to describe, for reasons I've mentioned in previous posts, but I don't think this actually matters very much for Lewis's view.  I'll note though that he does seem to worry a similar issue at the very end of the essay, when he raises a question about how to treat a fictional character who in the fiction is telling a fiction. But I think all we need to accept is that any consistent fictional world, as long as it is possible, really exists somewhere. Some of those worlds will indeed include true accounts of the events in the world that are counterparts of the fictional accounts of those events in our world.  But I don't see that they need to. (In fact if they did need to, it would seem that there might be an infinite regress to worry about, since we'd have to add to any narrative that it is a true account of the world, which would change it and require another such assertion, etc. Lewis seems to prefer to go the way of collapsing iterations rather than allowing for this infinite regress, but sees the collapse as problematic too, since now it would be hard to keep separate fiction and the things that are fictional within that fiction.)

So every consistent and possible fictional world really exists, somewhere.  Then the work of the writer of "realistic" fiction (where realistic means something like "possible") is a kind of discovery of truth, the truths consistent with the writer's possible and therefore true premises.  It's like math, or like chess, or like their intersection: given a possible state of affairs, here is an array of incidents, events, characters, interactions, and outcomes consistent with it.  Since possible states of affairs are all of them true somewhere, so are all the incidents, event, characters, interactions, and outcomes consistent with them.  The fiction writer discovers truths which we might be glad to know, or be horrified to know.  But they are true, somewhere (unless you go my zombie route, in which case their truth won't matter as much: they'll just really be avatars -- in the gaming sense, not the strongly counterpartial sense in which they'd have something of the meaning they have as reincarnations - coincarnations, perhaps, of real souls.)

So reading fiction is reading about something that's true somewhere.  If there are inconsistencies, then your reading about multiple truths, all of which may obtain in different places, even if not simultaneously.  And the kind of literary criticism so popular a century or so ago, where critics hazard and argue for strange back-stories: all that's true too.  Hamlet was a woman (in many many worlds, in all the worlds in which she existed); was Claudius's son; was his daughter; was old Hamlet's murderer.  Desdemona did betray Othello with Cassio (in many, many worlds), and Emilia did sleep with Othello in many of them.  Fan fiction consistent with the authoritative fiction it riffs on is all true too, somewhere.

Yes, all consistent fan fiction is true, and all consistent back-story mongering is true, but I am making a stronger claim than that.  It can be, is somewhere, true of our Hamlet, our Desdemona, that is to say, Shakespeare's Hamlet and Desdemona, if it is (possibly) consistent with what we know about them.  But if it is consistent with what we know about them, well then these are things that are true, and true in possible worlds as close as you could want them to be to our own. ("Closeness" of possible worlds is how Lewis handles overall resemblance or divergence between worlds.  The worlds in which Hamlet has three eyes and is green are farther from us than the worlds in which he looks like Jude Law.)

Maybe literary critical arguments are another way of arguing for or against closeness of possible worlds. "Only an idiot would think Claudius was innocent" really means "The possible worlds in which Claudius is innocent are each much farther away than its almost perfect replica where he's guilty."  Maybe.  But that world still exists, and that story, that solution to the chess problem the play presents to its critics, is still true.

If Lewis is right, just imagine the Sartrean seriousness of the responsibility of the critic, or the fan-fictionist, or any audience member guessing at what happens after the story ends.  It's not that we cause those worlds to exist by imagining them.  It's that we come to know that they exist, and whatever claim those events have on our emotions is a claim we now can't dodge.

Writing would be like writing on an Etch-a-Sketch or frosted window: we expose fragments of what's behind it in our writing, and then can't unlearn what we've learned.  As pure epistemological beings, we can know in theory that all possible things are true.  But as empirical humans, the true things that we learn explicitly about have a very great claim on our attention and care, our Sorge. Which suggests we should be cautions about the truths we learn through exposing ourselves to fiction, or through exercising our own imaginations.

Luckily I can't believe any of these possible worlds are real, as the possibly still living and thus-believing Lewises do (or as our this-worldly Lewis was said to).  But it's possible that they are.  And then all you could do is hide.

(The whole thing is Lemian or Borgesian. Lewis would be committed to think, I am sure happily, that the whole Library of Babel really existed, extensionally. For Quine, the whole library may be found condensed, amply, in the alphabet.)