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.)

Punishment and resentment

Nevertheless, despair is veritably a self-consuming, but an impotent self-consuming that cannot do what it wants to do.  What it wants to do is to consume itself, something it cannot do, and this impotence is a new form of self-consuming, in which despair is once again unable to do what it wants to do, to consume itself; this is an intensification, or the law of intensification. --Kierkegaard [Hong translation of The Sickness Unto Death, p. 18]

Kierkegaard is interested in the disanalogy that the analogy between despair and vertigo suggests:

Upon [the spirit] rests the responsibility for all despair at every moment of its existence, however much the despairing person speaks of his despair as a misfortune and however ingeniously he deceives himself and others, confusing it with [the] case of dizziness, with which despair, although qualitatively different, has much in common, since dizziness corresponds, in the category of the psychical, to what despair is in the category of the spirit, and it lends itself to numerous analogies to despair. [p. 16]

Robert Frank, in Passion within Reason, discusses some aversive experiments with rats.  Cause them pain when you give them a certain food, and they'll stop eating that food, of course.  But you can reverse this experience by reversing the modalities.  If you cause them pleasure, later, when they are forced to eat that food, they'll unlearn their avoidance.  On the other hand, if you cause them nausea when they eat a certain food (even several hours later), they'll never unlearn their aversion.

Nausea (which Derrida points out is a central affect for the purposes of judgment [as Freud suggests in his great essay "Negation"], but one that Kant fails to discuss in the Critique of Judgment) and vertigo are differently aversive from pain.  How?  Well, for one thing, they are somehow totalizing.  Pain is something that just happens to one, and in most cases it is possible to sustain some clarity of thought even in pain - clarity of thought about the pain it may be (as I know from having had a kidney stone), but still clarity of thought.  Nausea and dizziness are different, all-engrossing, panic-inducing, mind-eroding.

Now it seems to me that our fantasies of revenge (and people's actual acts of revenge) are massively more about causing pain (including the psychic pain of grief, whose analogy to physical pain J.B. Pontalis has brilliantly defended) than about causing nausea or vertigo.  It's not that pain is worse (though of course it may be).  It's that pain is more in tune with the idea of punishment than nausea or vertigo are.

I think that this is because punishment is not so much about the gratification of seeing someone else suffer as it is about making that person understand why he or she is justly suffering.  Resentment is about making those you resent see why you resent them, where that "why" has the full force of justice behind it (cf. P.F. Strawson's great essay on "Freedom and Resentment").  Pain is external, or feels external.  (Elaine Scarry points out that the vocabulary for pain is always instrumental: burning, stabbing, piercing, even throbbing, as though being squeezed.)  Pain speaks to a relation to the outside world, to external things that injure us.  So injuring another is being in a relationship to them.  In French injurier means insult; etymologically injury (from injuria) means counter to the law, so counter to justice: injustice or wrong.  So injury in English suggests one person doing something to another: causing them pain and the damage that pain represents.  It attacks their bodily integrity (or the integrity of their world), but leaves their mind whole.

And so too does punishment.  Thomas Harris's fantasy of Mason Verger's fantasy of revenge on Hannibal Lecter would have Hannibal watching himself being eaten alive by hogs, kept alive and alert to his own consumption.  I think punishment can go only a little farther (as it does in ancient mythology): being forced to eat yourself (or your children), not because the idea is nauseating but because it brings the externalized destruction of body integrity to its zenith, while preserving the mind whole.  (Harris probably goes a little to far and loses a bit of his force when Clarice and Hannibal eat the brain of the conscious Paul Krendler from his open skull, carpaccio by carpaccio.  Here though we might have a fantasied rediscovery of Leibniz's view that all is surface, that the stripping away of each surface of the brain still constitutes and act of bodily harm and can still be perceived by the mind facing its own imminent death.  But I don't think Harris sells it, and the scene is nauseating rather than gratifying.  Though that may be the point; but still it doesn't feel like punishment.)

So why do we want to keep the mind whole?  For the same reason that we fantasize our enemies in  hell knowing their own sinfulness, knowing what they've done to deserve this.  It's because punishment aims at teaching -- this is what's crucial to the idea of altruistic punishment, even though not spelled out in most of its analysts -- at reform and correction, and so there's an acknowledgement in punishment of the humanity of its objects.  That's why Ugolino is conscious and aware of what he's doing in eating Ruggieri's head: Ugolino is still human, and his punishment is also a means of communication.  The communication is two-way: punishment communicates his crime to him, and makes him feel it; he can describe how he feels to the rest of us, because punishment is still a way of keeping others in the nexus of the human. Causing nausea or vertigo lacks that essential criterion for punishment.

So what about despair then?  Despair, we could say, is something like the successful vector towards correction that punishment aims at.  It affects the soul even as it keeps it whole, mindful of its sins and conscious of them.  Despair is the the borderline between punishment and aversiveness which no longer belongs to the ethical world of punishment.  If we can get someone to despair, we've succeeded. Punishment has had a moral success.  Anything more, like vertigo or nausea, has nothing to do with punishment, and becomes pointlessly dreadful.  Pain always seems structured as though it has a point.  (Obviously I'm talking about how we meditate or fantasize about these things, not about the truth of real pain.)  Despair is where that point or purpose disappears: it is a vanishing point or moment of transition.

It's not that pain causes despair directly (though it can cause despair, it can't cause what Kierkegaard means by despair directly).  It's that pain can remind you of what you've done to others, and that can bring home to you a reason for remorse (etymologically: eating yourself up alive!) and despair.

(Although I'm not so sure that this connection works elegantly, what's important about it is how it helps distinguish despair from vertigo.)

In which I comment on Philip Roth's pretty hilarious open letter to Wikipedia in The New Yorker

[Roth's "Open Letter to Wikipedia"]

I don't think Roth gives the plot of The Human Stain quite accurately.  It's Zuckerman who thinks all these things about Les Farley, but he never knows them. The scene where he talks to Farley at the end-- but is it Farley? Zuckerman writes: "I came to notice, parked at the edge of a wide field I would otherwise have shot right by, the dilapidated gray pickup truck with the POW/MIA bumper sticker that, I was sure, had to be Les Farley's.  I saw that pickup, somehow knew it was his...", but a couple of pages later is still putting this in the conditional: "if  this was Les Farley..."--is a masterpiece of anxious paranoia: he may be talking to Farley, Farley may be a psychotic murderer.  But he may not be, and he may not be. Zuckerman doesn't know, any more than he knows the truth of Swede Levov's life.  It's interesting that the great care Roth takes to make that point, over and over again in the trilogy, is absent here.  One might wonder to what extent Roth, in this lovely rant, is playing the sly game he trademarked so many years ago, tempting us to wonder what exactly we're supposed to understand as the truth behind his official claims, but refusing ever to show his hand.  That doesn't mean that Broyard was a source for Silk--I think he wasn't--but that Roth is enjoying the serendipitous opportunity not to settle the issue once and for all.

Nathan Zuckerman, of course, would be a good secondary source for the Wikipedia article, if only he were still writing about Roth.

"...which thou must leave ere long"

I feel that I finally get Sonnet 73:

That time of year thou mayst in me behold When yellow leaves, or none, or few, do hang Upon those boughs which shake against the cold, Bare ruined choirs, where late the sweet birds sang. In me thou see'st the twilight of such day As after sunset fadeth in the west; Which by and by black night doth take away, Death's second self, that seals up all in rest. In me thou see'st the glowing of such fire, That on the ashes of his youth doth lie, As the deathbed whereon it must expire, Consumed with that which it was nourished by. This thou perceiv'st, which makes thy love more strong, To love that well which thou must leave ere long.

I always had a too clever by two (2x2 clever) reading of that last line: that the word "leave" also meant putting out leaves or putting leaves on to, and not just "abandon" or "depart from." I didn't like this facile vulgarization, but it was the only way I could think of to approach the paradox that it's the young man who's forced to leave at the end of the sonnet, not the dying Shakespeare.

Facile vulgarization, yes, because the words to have in mind are Antony's sublime "Let that be left which leaves itself." Being left, leaving: the abandonment there is what matters and is only blighted by imposing a stupid pun (as opposed to a kind of homonymic echo) on to the word.

But what would the young man be leaving? I think we have here a precursor to Wordsworth's Resolution and Independence. It is the nature of life, or of an aesthetic life, the life of those who would agree with Deleuze when he says, "An indescribable joy always rushes out of great books, even when they speak of ugly, hopeless, or terrifying things,” that poets in their youth begin in the gladness of being able to feel this joy, the gladness of anticipating horror and writing about it with all the gusto or brio that's the obverse of even the most melancholy intensity. "Thereof in the end come despondency and madness," Wordsworth comes to realize, as despondency comes to seem real to him. Mary Shelley will note something similar in her preface to the third edition of Frankenstein, after Percy's death, and the death of so many of her little ones:

And now, once again, I bid my hideous progeny go forth and prosper. I have an affection for it, for it was the offspring of happy days, when death and grief were but words, which found no true echo in my heart. Its several pages speak of many a walk, many a drive, and many a conversation, when I was not alone; and my companion was one who, in this world, I shall never see more. But this is for myself; my readers have nothing to do with these associations.

Her hideous progeny is the book, not the monster. She describes death and grief but is immune to it when writing the book: these are words which find no echoes and so they yield a Deleuzean joy and Wordsworthian gladness. Does it prepare you for the future, this flower-guided dallying with death (cf. Frost, Coleridge)? That's the question in these poems. Wordsworth dallies with the very fact that the gladness he feels in knowing that life is a life of despondency and madness is going to be self-defeating, self-undermining. That's a glad realization too, to the poetic, rejoicing figure of youth he was at the time.

What about the young man then? I think that the last line of the sonnet describes the merger of aesthetic and real experience. By aesthetic experience I mean what Wordsworth means: the inspiration to poetry through the contemplation of the miseries of life since those miseries measure the possibilities and depths of human dignity and human experience. So Shakespeare shows the young man the curtailments of time (life, year, day, fire at night) and what the young man can "behold and see" (the sonnet echoes Antony and Cleopatra) can yield authentic perception.

Such perception makes the young man's love of life which, like everyone, he must leave ere long ("All lovers young, all lovers must / Consign to thee, and come to dust") the stronger. Stronger not because it is more precious, or not only because it is more precious, for its scarcity. But stronger because the grimness of life gives it depth, and the depth of life is what we love about it.

Or love at any rate about literature, about the depth that literature can achieve. That achievement, in Sonnet 73, redounds to the merit of its speaker (or poet, to the merit of Shakespeare). In telling the young man that he too must die, and telling it in such a way that the young man loves the world which produces poems like Sonnet 73, Shakespeare shows us two ways of thinking deeply: writing well, as the poet does; reading well, as the young man does. Naturally this is the structure of drama as well, and there's a sense in which all the sonnets are reflections on Shakespeare's thought about play-writing: "As a decrepit father takes delight / To see his son perform the deeds of youth," e.g. (though there the positions are reversed).

Resolution and Independence, then, would be the young man's answer. (I am certain it is. I am certain that an enormous number of Wordsworth's poems are haunted by Sonnet 73. Consider: "Seals up all in rest" and "A slumber did my spirit seal.") Here too the young man loves what he must leave ere long, but it's the young man who's the poet, and the leech-gatherer who's just a leech-gatherer. The burden that Wordsworth takes up, or has taken up, is that the youth is both poet and audience of his own coming dilapidation. It's not that dilapidation will put him in a position to speak of such things to an audience that will thereby fall in love with the depths of life. It's that falling in love with the depths of life is something we poets do in our youth. We fall in love with the fact that futurity holds despondency and madness, and we speak of such things as part of our youthful vocation (no danger that Shakespeare's young man has such a vocation), a vocation which leads to just these things.

Writing, to use the word Beckett and Blanchot hit upon independently, is a serious task, and only youth woud be foolhardy enough to undertake it. Only youth or Shakespeare, thinking dramatically, thinking about what he owes to a younger generation that will need a measure of the depth of human sadness, in order to be able to offer such a measure themselves when they too come to be old.

Derrida, Austin, Quotation, Evolution

Derrida's critique of Austin isn't an unhelpful one, since it evinces just the sort of metaphysical thinking that Austin and Wittgenstein are concerned to think past.  Basically, Derrida's claim in "Signature Event Context" is that quotation can't be "parasitic" on performative utterances because to engage in such an utterance requires a citation or quotation of the appropriate formula.  (Like everyone else, and I do mean everyone Derrida neglects Austin's distinction between "hollow" and "void" performative utterances, but that's for another post.)  So performative utterances are logically dependent on a practice of quotation.

This is wrong.  It would be far better to say that the very idea of quotation arises out of performative utterances.  Performative utterances -- or Wittgenstein's language games -- come first.  It may be possible to formulate the rules of such games, but those formulations are descriptive, not prescriptive, attempts to formalize what we do.  Performative utterances, and moves in language games more generally, are practices before they are more-or-less-successful attempts to be adequate to some set of rules governing them.

For Wittgenstein these are practices which arise out of what he calls "agreement in forms of life." The supersubtle mechanisms by which such agreement could evolve (see, for example, Robert Axelrod's Evolution of Cooperation) can't and won't have presupposed something so crude as Derridean citation. Quotation in the Paris-is-Burning Jennie Livingston sense, possibly, where quotation is something closer to biological mimesis as Roger Caillois understands it: interaction, gaming, self-exposure to the spatial world. But this has almost nothing to do with Derridean formalism.

But what we could see, and say, is that citation, verbatim quotation, the idea of the verbatim, Quinean inscription, arise from performatives.  Formulae are fossilized performatives, and the idea of a formula (which is of course manifold in rituals and rites themselves dependent on prior belief in the performative power of utterance, a belief raised to a magical pitch) can give rise (see Homer and Milman Parry) to the idea of quotation itself.

So the great, Emersonian literary device of quotation is secondary to performative utterance.  That's what makes it literature: the evocation of a fictive world, where the performances aren't real, and all the more haunting for that reason.  Maybe I should say something about the hollow vs. the void.  In Austin hollow performatives are those which are not "meant," are those which the utterer performs without any intent to back them up.  Void performatives are those which have no standing, no matter how passionately they are uttered.  Fiction or the literary is the region, then, of the void, not of the hollow.  The poet nothing lieth because he nothing affirmeth, but instead gives us some sense of what the void is, next to which our loquacious selves are so precariously perched (to allude to Kenneth Burke).