Printer Friendly

Possible worlds, mathematics, and John Mighton's possible worlds.

Do possible worlds really exist? Or is there one (actual) world, and infinitely many (im)possibilia? Despite sounding like something out of science fiction fantasy or theoretical physics, depending on who you run with, the question of possible worlds has been debated in certain philosophical circles since the turn of the twentieth century, ranging from set theory in mathematics to modal philosophy to semiotics. (1) The link between the three disciplines lies in the ontological fiat, or performative, that brings a possible world into "existence," however that existence may be defined. For Umberto Eco, possible worlds are at base "cultural constructs," but, then again, so is the real world; for David Lewis, the founder of modal realism, possible worlds are just as real as the real world; and for David Hilbert, one of the early developers of mathematical set theory, the set-theoretical universe, which underpins possible worlds theory, quite simply is paradise. (2)

My project in this essay is to detect if (and how) the theory of possible worlds may underpin certain aspects of the narrative of drama and performance, and more specifically, the 1990 play, Possible Worlds, by Canadian playwright John Mighton. In order to do so, I will interrogate the seminal article Eco published in 1978 on textual semiotics, "Possible Worlds and Text Pragmatics: 'Un drame bien parisien'," and consider how the theory of possible worlds is transferable to a narratology of drama and performance. In the course of this discussion, though, I also want to peel back two further layers and consider: (1) the way Eco's work depends upon and departs from the possible worlds theories of philosophy, especially that of Lewis, and (2) the way Lewis and Eco depend upon mathematical set theory. Finally, I hope to put the whole thing back together by appraising how the play Possible Worlds navigates the theory(s) of possible worlds. (3)

POSSIBLE WORLDS SEMIOTICS

Umberto Eco is well known for his work in the semiotics of theater as well as texts. (4) In "Possible Worlds and Text Pragmatics," he follows the work of logicians and modal philosophers such as Jaakko Hintikka, Janos S. Petofi, and David Lewis to develop a theory of possible worlds for a semiotics of literary narrative. Interestingly, though, he does not extend the theory of possible worlds to a semiotics of performance genres. Nevertheless, as Keir Elam points out in The Semiotics of Theatre and Drama, the "as if" quality of drama on the page or in performance clearly depends on hypothetical world-creating properties, as any text does, which makes the theories of possible worlds developed by logical semanticists and semioticians especially pertinent and fruitful to the study of drama (99-102). Eco's particular angle on possible worlds theory and how it may unpack the structural workings of texts is concerned not only with the "as if" construct of the (actual) text as itself, but also the "as if" constructs posited by the text within itself.

In "Possible Worlds and Text Pragmatics," Eco outlines three "worlds of the fabula": 1) the possible world imagined and asserted by the author; 2) the possible sub-worlds imagined by the characters of the fabula; and, 3) the possible sub-worlds imagined by the "Model Reader" (46-47). Eco is following, and I will as well, terminology initiated by the Russian formalists. The term "fabula" refers to the chronology of the narrative and is independent of the strategic ordering of the sequence of scenes, the "sjuzet" or "plot." What is notable about Eco's taxonomy is that the ontological ground of all three worlds of the fabula, regardless of their serial embeddedness, is conjectural and imagined, based on the beliefs and wishes of someone, whether that someone is author, reader, or character within the text. And the possible worlds of the text include possible states of affairs, possible individuals, possible events, and so on, and can become "actual" or not, depending on the way the narrative trajectory of the text develops.

Eco's taxonomy can be recognized in any fictional text, and certainly in plays as dramatic text or performance, though the embodied nature of performance adds a complication. Since it interrogates the theory of possible worlds in a highly complex fashion, I want to avoid, for the moment, using Mighton's Possible Worlds as an example. Nevertheless, his play A Short History of Night will work to demonstrate Eco's three possible worlds of the fabula: 1) the life of astronomer Johannes Kepler as imagined by Mighton; 2) sub-worlds imagined by the characters, such as whether Kepler will become the soldier of his father's wishes; and 3) sub-worlds suggested by the text and imagined by the reader or audience--whether Kepler's wife, Barbara, will be burned at the stake, for example. As the narrative progresses, the world of the fabula actualizes; the number of possible worlds becomes fewer as sub-worlds are yielded by characters and audience to narrative fact. Kepler becomes an astronomer not a soldier, Barbara survives the witch hunt. Certainly, as Herbert Blau puts it in The Eye of Prey, in a performance this process is also alternating between what is expected and what is observed (178), with expectation gradually yielding to observation.

The possible world emerging by the end of the text, the fabula, is considered actualized within the logic of the fiction, and thus forms the ontological ground of all other possible sub-worlds in the narrative. It is virtual as well as global. But what is the relation of the actualized fabula to "the real world," the world the rest of us inhabit? To Eco, the hypothetical status of the fabula presumes necessarily the existence of a real world, the world wherein resides the author and the Model Reader or audience, the "reality" that logicians refer to as actualized as opposed to possible. This presumption is made explicit in performance, for any object on the stage of the play world, Kepler's quadrant for instance, is also necessarily an object existing offstage in the real world, having gone through a process of semiotization. In Eco's schematic, such an actualized presumption of the real world is necessary in order to provide some way for the fabula to come into its own, since its existence requires stipulation, or "furnishing," by statements, which in performance would also be semioticized objects, from somewhere outside itself. The fabula cannot be created ex nihilo. The actualized world (our world) works to provide the world-creating performative for the fabula--"Let there be light!"

Eco's taxonomy of the fabula is clearly indebted to theories of possible worlds developed by the logicians of philosophy. To deal with issues of referentiality and "truth," logicians have devised a way of generating complete, formal worlds that are stipulated by statements. Thus, a possible world might consist of statements {a, b, c, d}. Additional possible worlds could be generated out of the statements, {a, b}, {b, c}, {a, c}, {a}, {b}, {c}, {d}, and so on. To the possible world(s) could be stipulated properties such as red, black, blue, brown, which would be used to form subsequent possible worlds according to combinations of statements and properties: {a, red}, {a, b, black}, etc. To return to A Short History of Night and applying Eco's taxonomy, the global possible world (the fabula) of the play is stipulated as the cast of characters, setting, events, qualities and properties as well as the various possible subworlds, which themselves are stipulated as sundry combinations of inclusive or exclusive statements: i.e. {Kepler, soldier}, {Kepler, not- soldier}.

Nevertheless, the possible world of the fabula is certainly a much more complicated beast than the possible worlds created by logicians, even more so in performance. For one thing, one has to consider how much stuff is in the possible world. To logicians, it can be as little as {a}. {a} constitutes a complete possible world and needs no other recourse to the "real" world other than being stipulated from it (and borrowing a symbol from the alphabet). In contrast, as Eco points out, the possible world of the fabula relies on a vast amount of information from outside its borders that is not formally stipulated, information garnered from a "system of codes and subcodes" outside the text, what he calls the reader's "encyclopedia" (9). The fabula requires a contextual encyclopedia simply because it is not possible to furnish completely the possible world of a (dramatic) text, just as it is not possible to itemize everything in the real world (see Eco 31). The sets are just too big to be fully articulated by finite beings like ourselves.

Moreover, in performance, much of the "furniture" is "real"; that is, embodied statements occur on-stage in the form of properties, actors, sets, and so on that must come from the real world. Thus, in a play like A Short History of Night, Mighton need only stipulate the statement, "there is a man, Kepler," for the performance to take from the real world an actor who will play Kepler in the fabula, and thus begin to launch a highly furnished possible world. Further, the play relies on the audience's encyclopedia (from the real world) to fill in historical information on Kepler as well as vast cultural information on astronomy, sixteenth century Europe, Western culture, and so on. And the reliance on this encyclopedia continues throughout the play as more furniture is added. Halfway through, for instance, the audience must conjure up information about early modern witch hunts.

Acknowledging reliance on a real-world encyclopedia brings the discussion back to the relation of the possible world(s) of the drama to the real world of the audience. Here, Eco considers the issue of accessibility (from one world to another) in a manner that suggests an early apprehension of the cultural constructivism so prevalent in literary, performance, and cultural studies today. To logicians, accessibility from one world to another simply means that if a world W1 can generate another world W2, then W2 is accessible to W1 (Eco 42). For example, of the possible worlds discussed above, W2 {a, b} is accessible to W1 {a, b, c, d} because the world of W1 can generate the complete world of W2. On the contrary, W1 is not accessible to W2. If W1 is the real world, then another world W2 {astronomers} is accessible to W1, but not the other way around. This is because the real world can generate the set of astronomers, but the set of astronomers cannot generate all the people, let alone everything else, in the real world. Certainly, accessibility helps describe how a play comes into performance. Objects in the real world of W1 are used to generate the performance of W2, making the performance accessible to the real world. A real quadrant from W1 can be placed on stage to become a quadrant in the W2 performance.

But how, Eco wonders, can a (generated) possible world be compared in any legitimate way to a world that is already given, a world that is always already "real." How can a fictive performance be legitimately compared to everyday life? (This is like trying to compare apples and oranges.) To overcome this problematic, Eco first assumes a constructivist approach to the ontology of the possible worlds, and labels the possible worlds, "cultural constructs" (32); the possible worlds are obviously constructed rationally out of statements. For instance, in A Short History of Night, the possible world of the fabula is a cultural construct containing the statement, "the earth does not move". While this statement is untrue to an audience in our real world of the third millennium, it would have been true for an audience in the real world of sixteenth century Europe. In Eco's terms, only the encyclopedia makes the difference. Therefore, Eco asserts, the encyclopedia of the real world, which contains all our knowledge about the real world, must also be a cultural construct (32), contingent and changeable according to the operative epistemological model. The real world posited by Eco must be just as much a possible world, based on stipulations (or beliefs) as any generated possible world. Eco's next move is nothing short of brilliant: "our commitment to a possible world is an 'ideological' rather than an ontological matter" (32).

Of course, the next question to Eco is: if the real world is a cultural construct, an ideological rather than ontological matter, how does it get stipulated? What is outside it? What is "off-stage"? Or does it come into existence ex nihilo? Eco admits that his constructivist approach has a Kantian drift to it (33). Indeed, I would suggest that it is drifting toward the theological, a sort of postmodern Calvinism in which the real world has no ontological tangibility outside the shadow of discursive structures. In fact, this position has become something of a sticking point in current discussions of constructivism, with some theorists, most notably those involved in Foucauldiantype sexual politics such as Monique Wittig, Judith Butler, and Susan Bordo, asserting the world's discursivity "all the way down," and others such as philosopher Kate Soper and science critic Evelyn Fox Keller holding out for a recognition of the facticity of the real world, the "material structures and processes that are independent of human activity" (Soper 132) or "the world of prelinguistic and pretheoretical phenomena, constraints and opportunities in which we reside" (Keller 4).

For instance, Butler grounds her claims about the discursivity of the world on the premise that a material world cannot "exist" before being posited in language (67). Oddly enough, while taking her cue from Foucault, Butler chides Foucault for (perhaps unconsciously) yielding to an admittance of the reality of the world (Paradox 308). But Bordo, now modifying her earlier position, has acceded to a notion of materiality: "The body's materiality ... is first and foremost about concreteness, and concrete (and limiting) location" (92). Soper has perhaps the most thorough rebuttal to extreme constructivist theories in her book What is Nature? Here, she argues for a dialectical approach, which recognizes the priorness of the real world but sees it in a mutually transformative relationship with culture (47). Much of the discussion of cultural constructivism and the various theories of possible worlds hinges, then, on what position one takes toward the real--is it real or is it Memorex?--an ontological (in)decision that harbors one of the current conundrums of the Western academy.

POSSIBLE WORLDS, MODAL REALISM

If one does not want to follow Eco's recommendation--that the real world is a cultural construct--philosophical Realists offer an alternative. The theory of Realism simply asserts the ontological existence of something, whatever that something is (no relation to literary Realism). According to John Searle in The Construction of Social Reality, Realism does not demand a description of reality, or even an epistemology of reality (155). Realism is an axiomatic theory, with its only axiom being the givenness of a real. All else may be generated from this axiom as cultural construct or what Eco calls encyclopedia. If this position on the givenness of the real is acceptable, then what happens when we consider the theory of possible worlds? Are we not returned to Eco's problematic of how to compare two incomparable things, something constructed with something given, apples with oranges? Enter David Lewis.

Lewis handles this problematic by going in the opposite direction from Eco. Rather than claiming that all possible worlds, including the real world, are cultural constructs, Lewis asserts that all possible worlds, as well as the real world, exist (2-3). Incredible as this statement may seem, it does make (logical) sense. Like Realism itself, modal realism simply depends on accepting the axiom of givenness, though in this case Lewis does admit that the ontology is controversial. Because his position is so hard to take, I want to stress that Lewis really does mean the real existence of possible worlds and not possible worlds as abstract entities or narratives which, of course, are real but in a different (constructed) way. He calls such theories, proposed by Richard Jeffrey and Robert Stalnaker, "ersatz modal realism" (136, 142). Lewis explains his perspective in a number of examples, one of my favorites being about the "beer in the fridge." Is the beer in the fridge any more real than all the beer outside the fridge? Why, Lewis demands, would one want to limit oneself to just the beer in the fridge, and ignore all the beer that there is? In fact, Lewis describes such quantification--inside or outside--as "restricted speaking": "When we quantify over less than all there is, we leave out things that (unrestrictedly speaking) exist simpliciter" (3).

To apply modal realism to a semiotics of texts would suggest the existence simpliciter of the possible world of the fabula as well as all the possible sub-worlds in the fabula. Contrary to Eco, the actualization of the narrative trajectory would have no impact on the existence of all the possible sub-worlds generated by the characters or reader. They would not become fewer or yield existence to narrative fact. They would simply continue to be as they are. Moreover, a play in performance under these rules is just as existentially real as the real world. In fact, following Lewis, the fabula, the performance, and the real world of the audience would not differ at all in manner of existing; the only difference would lie in such things as where they exist and what stuff they have in them. Thus, a possible world in which Kepler is a soldier is as existentially real as a possible world in which Kepler is an astronomer, which are both as existentially real as the performance and the world of the people watching the play. It is simply a matter of not limiting oneself to the beer in the fridge.

And in terms of considering all the stuff in the possible world of the play, Lewis's theory offers a way to account for the furnishing of dramatic possible worlds (very large sets) by finite beings that does not depend on an encyclopedia such as Eco's. This is how it would work: the audience does not limit itself to quantifying over less than all there is, less than all there is in the dramatic worlds of the play or the performance. What Eco would call the stipulations for the possible world of A History of Night, for instance, Lewis would call restricted speaking. To Lewis, A History of Night would (always) be more than the sum of its stipulations for character, setting, location, and dialogue, and its performance always more than what it embodies on stage. Whereas Eco's theory would ask the audience to refer to its encyclopedia to furnish further the dramatic world, Lewis's theory would ask the audience to consider all that there is. The encyclopedia is a cultural construct, whereas all that there is, is.

How would modal realism account for the problem of accessibility? How would one get from one world to another, from the audience's world to the play world, for instance? How would a quadrant in the real world become a quadrant in the performance? To Lewis, accessibility can be accounted for by "universals" rather than by making claims, as most logicians do, for the generative abilities of worlds. Again, Lewis views accessibility relations as restricted quantification over all there is, and moreover, restricted quantification from a standpoint that is invariably hegemonic, the standpoint of our world, or for a play the audience's world. To Lewis, worlds are "accessible" to each other if they share certain properties, such as quadrants, or laws, such as the laws of physics. Contrary to Eco, the real W1 and the performance W2 are accessible to each other if they both contain quadrants. In other words, at every world that obeys the physical laws of our world, the earth moves around the sun. Yet there may be many possible worlds that have entirely different laws of physics, where spirits instead of quarks govern the universe, or possible worlds where some of the laws of physics are the same--gravity operates in A History of Night for example--and some are not. Two possible worlds may be nomologically accessible to each other in some parts but not in other parts (7). In the parts where they are not accessible, they differ in kind (not in existence)--true or not true according to the standpoint of our world. How would we account for statements that are true in one world but untrue in another, statements like "the earth does not move"? Simply, that A Short History of Night is not accessible to the audience's world at a certain nomological point--whether the earth moves. But such inaccessibility is, then, according to Lewis read as an untruth: A Short History of Night is a possible world with an untrue statement in it (even though in sixteenth century Europe, conventional wisdom would have taken the statement as true).

This version of accessibility works because of modal realism: "every way that a world could possibly be is a way that some world is" (Lewis 2). The audience would have no trouble accessing the play world because the play world simply is another way that a world could be and, therefore, is. To Lewis, possible worlds are not imaginary, as they are to Eco. They are not hypothetical, "as if" constructs. They are not of our own making. As Lewis puts it: we may describe and make representations that apply to possible worlds, but we do not cause possible worlds (3). Of course, as I mentioned above, this viewpoint is underpinned by axiomatic Realism and derives directly from the willingness to accept the givenness of the real. And in accepting this ontology, Lewis is able to steer clear of a Kantian or theological drift. Still, Lewis concedes that the acceptance (or not) of (modal) realism is nothing less than a choice.

MATHEMATICAL SET THEORY

Remarkably, in On the Plurality of Worlds Lewis does not insist that the reader embrace his position on ontology. Despite acknowledging his own acceptance of Realism he thinks of the real world as a big physical object (1)--Lewis continually reminds us that such a position is axiomatic. The reader has the privilege of choosing whether to accept the first axiom. Rather, he bases his defense of modal realism on what seems at first glance to be merely practical terms: the theoretical gains of modal realism make worthwhile the acceptance of the controversial ontology of the existence of possible worlds (3-4). What is perhaps most surprising in this approach is that Lewis specifically takes his lead from mathematical set theory, the theory which, according to Charles C. Pinter, is considered the "'unifying' branch of mathematics" (20). (Of course, Lewis gets terrific mileage out of linking his theory with the intellectual authority of mathematics.) As far as Lewis is concerned, if set theory creates a paradise for working mathematicians, why should philosophers deny themselves a paradise of possible worlds? Mathematical set theory is also axiomatic, based, as modal realism is, on the initial acceptance of an ontology.

An axiomatic theory is one that unabashedly begins with the acknowledgment of a premise or premises from which other statements are logically derived. Perhaps the example most familiar to Westerners is Euclid's geometry. Euclidean geometry requires the acceptance of a small number of axioms--there is a point, there is a straight line, and so forth--from which the entire possible world of Euclidean geometry is then built. Such a method involves an ontological choice from the outset. One may choose not to accept the axioms, but then, of course, one does not get to play the game (or plays a different game). As Pinter points out, axioms are not universal truths, but whatever statements we wish to use as premises. I would add that they are performatives, speech acts that bring something into existence, like the geometry of Euclid.

Let x be an element of A. This statement is the basic axiom, or performative, of mathematical set theory. The rest of set theory, as well as the rest of mathematics, can be derived from the acceptance of this simple, existential statement. The reader will realize by now that I have already alluded to certain aspects of set theory in my above discussion of the semiotics of possible worlds. Recall that Eco, following logicians, stipulates his possible worlds out of statements such as: there exists a possible world A containing the statements {a, b, c, d}. To translate into the terminology of set theory, one would proclaim: Let a, b, c, d be elements of the set A. At bottom, possible worlds are nothing more than the sets of set theory, and indeed, are descended from the ideas of such nineteenth century mathematicians as Georg Cantor and Richard Dedekind.

For instance, what Lewis and Eco call possible worlds are sets, each brought into existence (real or constructed, again depending on one's position on the real) by the acceptance of a performative fiat. Eco's fabula, for instance, and its possible subworlds, is a set containing subsets. The same can be said of Lewis's possible worlds--also sets and subsets. What Eco calls accessibility is the generation of subsets from a set: i.e. the subset {a, b} is accessible to the set {a, b, c, d}. What Lewis calls universality is the intersection of two sets, one of which may be a subset of the other: i.e. the intersection of the set {a, b, c, d} and its subset {a, b} is {a, b}. (In fact, in set theory any set may be considered a subset of itself.) In terms of Lewisean universals, the inaccessibility of two possible worlds at some point--{ the earth does not move}, for instance--means that the two possible worlds do not form an intersection at that point: {c, d} falls outside the intersection of {a, b, c, d} and {a, b}. Considering drama and performance in this light, it is easy to see that, as I have implied above, the "creation" of a dramatic or performance possible world is the "creation" of a very large set, brought into existence by a performative fiat (Let x be an element of A), that in turn contains all sorts of subsets: a {the characters}, b {the setting}, c {the properties}, d {the subworlds (Eco) or subsets (Lewis)}, even e {the set A itself}. Functions such as accessibility or universality may be accounted for by performing arithmetic on the sets.

Interestingly, early in its development set theory ran into the very same problem with large sets that Eco and Lewis have with the size of possible worlds, perhaps because Cantor had already been working on the properties of infinity. Intuitively, we accept that the integers, for instance, form an infinite set. In fact, as mathematician Robert Kaplan shows in The Nothing That Is, set theory can generate the sets of all the number systems, which is the property that makes it the underlying theory of mathematics, from the integers to the reals to the complex numbers. Here is how it works: out of one performative fiat, Let there be a set with nothing in it, we form the empty set [phi]. Call this set, "zero". Then form the set with the empty set in it {[phi]}. Call this set, "one", since it contains one thing. Form the next set out of the previous sets {[phi],{[phi]}}. Call this set, "two". And so on. Then the integers can be used to form additional number systems (211). But here's the hitch: since we are finite beings, how can we form infinitely many elements in this ponderous and all too human fashion? We just don't have all the time in the world!

In 1904, Ernst Zermelo "solved" this paradox for set theory by coming up with the Axiom of Choice. (5) The Axiom of Choice is existential, rather than constructivist. It states that a "choice function" exists for every set A, whether we can actually produce a choice function or not. (The term "a choice function" just means a way to make a choice.) For example, we can choose to represent a set by one element in the set. For an infinite set like the integers, we can choose the number n+l in the set to represent the "entire" set. The long version looks as follows: {0, 1, 2, 3, ..., n, n+1, ...}. As such, the Axiom of Choice avoids the impossibility of producing step-by-step an infinite or even very large set, because it does not ask that a sequence of choices be carried out. (6) It just asserts the existence of certain mathematical objects. And, according to Pinter, since the Axiom of Choice involves infinite sets and is therefore outside our experience, it cannot be empirically confirmed or denied (114).

What is so intriguing about the Axiom of Choice for our discussion of the possible worlds of drama and performance is its position on ontology. Again, the Axiom of Choice is an existential statement, not a constructivist one (Pinter 114). To set theorists, one cannot be skeptical about the ontology of sets, no matter how large the sets are. If one wants to play the game, one must accept the ontological ground of the Axiom of Choice else the entire edifice of mathematics comes tumbling down (as it nearly did in the late nineteenth century). In fact, the basic function of the Axiom of Choice is to prevent the razing of mathematics! By accepting the Axiom of Choice, mathematicians are enabled to furnish very large, even infinite sets. Indeed, something like the Axiom of Choice on a literary scale works for explaining how an audience can manage the size of a dramatic possible world. Whether one uses the position of Eco (encyclopedia) or Lewis (givenness), the furnishing of a dramatic possible world beyond what has been provided in the text or in the performance depends upon agreeing to let representations be representations, to let n+1 stand for the whole thing.

The stage directions for the Setting of A Short History of Night, for instance, state: "A Castle and locations nearby in Bohemia. Late 16th Century" (86). The formal statements (elements) of this possible world (set) are clearly representative of a much larger dramatic possible world. In fact they are synecdoches: "the castle" stands in for all the buildings, "locations" stand in for an entire territory, "late 16th century" stands in for the aggregate of cultural moments in time; as well a whole range of metonymies ("castle" also associates the characters with a class status, for instance). Of course, the play cannot actually produce the entirety of the dramatic world--it is far too big. It is enough just to let the representations of the possible world do their work. In other words, we can assert the existence (whether constructed or real) of the dramatic world by letting one castle represent the set of all the castles, a few locations represent the set of all the locations in Bohemia, the late sixteenth century represent the set of all the cultural moments at that particular time. Of course, a performance achieves this effect by even more abbreviated representations: a wall, for instance, may stand in for the castle.

Moreover, something like an Axiom of Choice, certainly its existential quality, is behind the controversial ontology to which Lewis refers in On the Plurality of Worlds. It is one thing to accept the premise that an object (or world) exists, quite another to accept that there are infinitely many such objects (or worlds). Indeed, Lewis sees the parallel between set theory and modal realism as follows: if mathematicians can accept a controversial ontology, why cannot modest philosophers (4), or I might add, modest literary critics? Lewis's theory of possible worlds is an axiomatic system: its premises are existential. Possible worlds exist. Period. The performative fiat becomes the axiom of a possible world, and the possible world exists. The difference between Lewis and Eco's systems, even though both arise from set theory, is that Eco's does not accept axiomatization. Eco's system does not have an existential premise. To him, all premises are constructed, including the premises of the "real" world. The performative fiat will always be outside the possible world, creating the possible world from the outside. In Eco's case, the performative fiat would have to be something like Lacan's transcendental signifier or Derrida's supplement.

POSSIBLE WORLDS BY JOHN MIGHTON

So, which theory of possible worlds does John Mighton's 1990 play, Possible Worlds, demonstrate? Eco's or Lewis's? Fortunately, the play is not nearly that definitive, simple, or straightforward. There are a number of intriguing twists and turns to the title referent (most notably at the play's ending) that suggests Mighton is interrogating theories of possible worlds, especially as to how they might play out in drama and performance, rather than just employing one conjecture or the other. To find this sort of interest should not be surprising in a playwright who holds an MA in Philosophy from McMaster University and PhD in Mathematics from the University of Toronto. Possible Worlds, which premiered in 1990 in Toronto and was made into a film in 2000, is set up (or stipulated as Eco would say) as a detective story. The play opens with two detectives at the scene of a grisly homicide in which the victim's head has been incised and the brain stolen. In the course of introducing the fabula, we learn that this particular case corresponds to a rash of unsolved homicides wherein the victim's brain (and nothing else) has gone missing. This detective story will reappear at numerous points in the play, specifically Scenes One, Five, Seven, Ten, Twelve, Fourteen, and Seventeen, and forms what Eco would call the first possible world of the fabula, that is, the possible world imagined and stipulated by the author from outside the parameters of the play itself, generated by the performative fiat of our world: Let there be a possible world, P.

And, in keeping with Eco, P has been furnished not only by statements in the play text, specifically stage directions such as, "a body covered by a blood stained sheet" (11), but also by action and dialogue, which of course would compose all the stipulations in a stage production or the filmic performance. On-stage, the statement would be embodied by a real actor. The dialogue in Scene One furnishes P with properties such as the existence of the two detectives, the name of the victim and his occupation (George Barber, stockbroker), the nature of the homicide, suspicions directed at the university clinic, and so on. Such statements are performatives that bring the dramatic world into existence, for every sign in a performance, from dialogue to costume to props, is a stipulation in P. (7) Moreover, Scene One can also be seen to manifest Eco's notion of a necessary reliance on the audience's encyclopedia to furnish more completely a large P, a set too large to be stipulated by the performatives. The audience can be expected to furnish P with properties relating to a number of literary conventions, adding to the detective story genre the horror story genre (excised brain) and the Frankenstein genre (university clinic), as well as legal and cultural furniture about murder, police work, human anatomy, and so on.

If we could just consider the detective story and ignore the rest of the play, Eco's theory would work well. P is clearly a hypothetical, "as if" construct, a set generated by the real world and accessible to it. Moreover, Eco's second and third possible worlds of the fabula, the subwodds imagined by the characters and by the Model Reader or audience are operating even by the end of Scene One: the detectives and the audience are imagining subworlds of possible gang or mad scientist activity, the audience may even be imaging other possible subworlds like alien intruders (a subworld that is indeed proposed in Scene Ten), and these subwodds should begin to give way to narrative fact once the fabula moves farther along its trajectory. Presumably, in the detective story genre one eventually finds out who-done-it, and all other possibilia fall away.

But, of course, to ignore the rest of the play, if that were even possible, would be a practice amounting to Lewis's notion of restricted quantification or restricted speaking, that is, ignoring all that there is. And, to be sure, focusing simply on the detective story would restrict an appreciation of the play to the specter of the hegemony of our world, to a perspective derived from the real world of the audience and what is considered accessible to it. For not very far into the play, even by the end of Scene One and during Scene Two, the issue of possiblia in terms of modal realism has been intimated, and certainly by Scene Four we can no longer easily correlate Eco's theory of possible worlds to the play. This situation arises because Scene Four has properties stipulated in it that contradict the properties already stipulated in Scene Two.

In Scene Two, George and Joyce meet in a cafeteria, possibly set at the university clinic, where Joyce is a research neurologist. In Eco's taxonomy, we could explain the logic of this scene in terms of instigating a subworld imagined by the audience and drawing on the audience's encyclopedia of literary and dramatic conventions which rationalizes this phase of the plot as having undergone a flashback, for George is surely alive, head intact, in this scene. And through Joyce's occupation the mad scientist subworld seems to be encouraged. This scene also stipulates Joyce and George, both single, as virtual strangers, with Joyce behaving coldly to George's romantic overtures. They have no history together other than a vague recollection of attending the same high school in the same small Canadian town years ago. In the notation of set theory, we could consider Scene Two a subset of P, the detective story. Let's call it [S.sub.2] (for subset, Scene Two) and note a few of its properties: {Joyce, scientist, not married, George, stockbroker, not married}.

However, it is much more difficult to rationalize Scene Four with Scene Two according to the conventions of plot, certainly if we use the standard definition of plot in narrative theory. Recall the difference between "sjuzet" and "fabula": the sjuzet signifies the temporal ordering of information in the narrative, whereas the fabula constructs the chronology of the narrative. In Scene Four, Joyce and George meet again, this time in a popular bar; oddly enough, they are still strangers despite their encounter in Scene Two. Moreover, Joyce's subject position seems to have undergone a sea change. She is no longer a cooly detached research scientist, but a stockbroker and libertine. The set notation of a few of the properties for Scene Four would be [S.sub.4]." {Joyce, not scientist, not married, George, stockbroker, not married}. [S.sub.4] can only be made logically acceptable to an audience in Eco's terms if the audience were now willing to throw out [S.sub.2] as a flashback, because [S.sub.2] and [S.sub.4] do not fit together in the temporal logic of the plot. [S.sub.2] has to be considered a purely hypothetical "as if" construct (a subset) within P that must now be completely rejected as a subworld in terms of the narrative fabula--Joyce was never a scientist. Now, it would appear that [S.sub.2] was wrong, and that [S.sub.4] is right.

However, it is impossible to ignore the explanation for the discrepancies between [S.sub.2] and [S.sub.4] that George provides in Scene Four, an explanation that immediately makes better (logical) sense than the above Eco-based rationalization: "Each of us lives in an infinite number of possible worlds" (23). It is interesting to note that in this scene Mighton establishes the basis for George's claim of possible worlds in mathematics, and even in set theory as I will shortly argue. (8) George tells Joyce how he discovered this reality of multiple worlds while doing a math problem in grade school. He could see two ways of solving the problem. Half way through working it one way, he suddenly found himself in a different world working it the other way (24). The film takes this example further than the playtext, by actually showing George working a problem in geometry, despite being somewhat facile in showing the "other way" as simply a reversed, mirror image. The problem asks for the area of a small square nested inside six larger squares:

Since each square is formed by cutting out half the area of the previous square, George calculates the answer as 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 64 = 1. The area of the smallest square is 1. (Rather than showing a mirror reversal of the calculations as the "other way", the film could have shown George actually doing the problem another way, for example by halving the sides of every other square and taking the area, continuing on to the smallest square: 8 x 8 = 64; 4 x 4 = 16; 2 x 2 = 4; 1 x 1 = 1.)

[ILLUSTRATION OMITTED]

What I find really interesting, though, about the appearance of this particular math problem in the play is that it stands as an icon for possible worlds theory. Each of the squares can be seen as representing a possible world, complete with the accessibility relations theorized by Eco--the smallest square is accessible to the next largest square, and so on up the squares, and each square, or possible world, can have stuff in it, ranging from properties such as George and Joyce, to the points on a Cartesian graph, to simply nothing at all but the previous square. In set theoretical terms, the icon is a set of nested subsets, with each square being a subset of each larger square. Indeed, if we consider the subsets as possible worlds with nothing in them except themselves, the icon becomes a geometrical representation of the integers generated by set theory: the smallest square would be the set with nothing in it [phi], or 0; the next square would be the set with [phi] in it, {[phi]}, or 1; the next square would be the set with [phi] and {[phi]} in it, {[phi],{[phi]}}, or 2; and the largest square would be the set with [phi] {[phi]) and {[phi], {[phi]}} in it, {[phi], {[phi]},{[phi], {[phi]}}}, or 3. Theoretically, this practice could be carried on forever by generating ever smaller squares within the smallest square, thus forming the infinite set of the integers. But, thanks to the Axiom of Choice, we don't actually have to undertake this impossible task. Representing it as an icon is good enough. Indeed, the math problem is really a geometrical representation of the play, Possible Worlds, with the squares representing scenes.

Moreover, the geometrical representation of the play not only confirms the set theoretical basis of possible worlds theory and the play itself, it also provides the argument for accepting an axiomatic system without expecting the audience to know anything about set theory. As I noted above, geometry is axiomatic. Presumably, a well-educated audience could be relied upon to remember that much of geometry from elementary school. And now the play, with this accounting in place, can reinstate [S.sub.2] as a possible world, or subset within P. As George tells Joyce in [S.sub.4], he already knew her name even though they are strangers, because he had met her in another world, very possibly the world of [S.sub.2]. And, even if not in [S.sub.2], the existence of another world ([S.sub.n]) in which they have met, does not deny the existence simpliciter of [S.sub.2] or any other S. As Lewis puts it, any way that a world could possibly be is a way that a world is. [S.sub.2] and [S.sub.4] are both right--Joyce is a scientist and not a scientist.

In fact, Lewis becomes more useful (and Eco less useful) as the play proceeds. This is because Lewis's theory of possible worlds allows the plot of the play to unhinge itself from the logic of temporal ordering. It turns out that there are six distinct possible worlds in the play (by my count), not including the detective story, P, and these possible worlds occur "randomly" throughout the course of the play. In other words, they do not form a cause and effect temporal trajectory, and therefore appear to contradict each other, at least if we only think in Eco's terms. I have covered above possible worlds one and two ([S.sub.2] and [S.sub.4]). Possible world three consists of two men, some stone blocks, and the words "slab", "block", and "hilarious" (42-43). In possible world four, George begins to think he's trapped in some kind of container. A possible world five occurs close to the end of the play in which Joyce has, once again, never met George. And a sixth possible world exists where George and Joyce are married to each other.

One reason Lewis's theory makes better sense than Eco's at this point is Lewis's take on accessibility among possible worlds. Recall that Eco's notion of accessibility is generative. One subset is accessible to another if it can be generated from it. In other words, [S.sub.2] is accessible to P, because all that is in [S.sub.2] is already an element of P. The fact that Eco's subworlds of the fabula must give way, as narrative fact is established in the plot as cause and effect and the fabula is actualized, means that each subsequent subworld, if it is not to be ejected, must be accessible to the previous one--that is, [S.sub.2] must be able to generate [S.sub.4] and all the other possible worlds I have described above. Clearly, in this play, such an exercise cannot be performed. A subset {Joyce, married, George, married} cannot generate a subset (Joyce, not married, George, not married}. They are two distinct critters, and if taken together, are in contradiction.

In Lewis's theory, to the contrary, we do not run into this problem. Possible worlds theory, here, is not generative; it is existential. As Lewis maintains, possible worlds may differ from each other, but not in manner of existing (3). Two distinct subsets such as {Joyce, married, George, married} and (Joyce, not married, George, not married} do not logically contradict each other because they are not generative. Accessibility in this instance would simply refer to what is universal (a comparison of difference), an intersection between two subsets in the arithmetic of mathematical set theory. {Joyce, George} is the intersection of the two subsets {Joyce, married, George, married} and (Joyce, not married, George, not married). Married or not married simply falls out of the equation. In other words, according to Lewisean theory, possible worlds don't have to worry about generative accessibility because they are given worlds, not constructed worlds.

Certainly, with all of these possible worlds going on at once in Possible Worlds, it seems likely that Mighton is interrogating some ontological issues about dramatic and performative worlds. What, exactly, is in existence in a play? Is a play a figment of someone's imagination (Eco), or is there something real about the play world (Lewis)? Should a play, and its possible worlds, be considered constructed (Eco) or given (Lewis)? So far, it looks as if a play world, and everything in it, would have to be considered given, if only for the reason that in this particular play Lewis is working better than Eco. In Lewisean terms, possible worlds are not hypothetical, "as if" constructs. They are not of our making. We may describe and make representations that apply to possible worlds, but we do not cause them (Lewis 3). On the other hand, for a play world to work under Eco's taxomony, the possible worlds would have to be imagined, or caused, by the characters or by the audience, conforming to items 2 and 3 of Eco's three worlds of the fabula. In Possible Worlds, the possible worlds are not presented in this fashion. They don't yield to narrative fact. They continue to exist simultaneously.

Just as the real world of the audience continues to exist simultaneously along with P, [S.sub.2], [S.sub.4] as well as the four other possible worlds enumerated above. Possible Worlds is the sort of play that offers the audience an opportunity to reflect on the make-up of its own world. (Interestingly, we have to take our key from the play itself, a sort of reverse grounding!) The play will ground our view of the real world, rather than the usual way round, a recirculation through the postmodern of the aesthetic movement's "life imitates art" According to the parameters of the play, is the real world a cultural construct as Eco would have it, or a given world as claimed by Lewis? If we accept the axiom provided by George, "Each of us lives in an infinite number of possible worlds" (23), as well as Eco's injunction against comparing two incomparable things, Possible Worlds at this point must lead us to conclude that the real world is given. Moreover, since the possible worlds of the play make more sense according to the theory of modal realism than cultural constructivism, the conclusion that the actualized world is given is nothing less than an acceptance of the axiom of Realism. Something, whatever that something is, exists.

In other words, the real world would have to be a set (with lots of stuff in it), just as Possible Worlds and its possible worlds are sets. There is no difference among them, as Lewis would say, in "manner of existing," only in what they may or may not contain. The performative fiat, Let there be a world with elements in it (or mathematically notated, Let x be an element of A), would be the first axiom of a real world (an axiom like those axioms in Euclid's geometry) that requires an ontological choice. In fact, Mighton has provided the axiom in the words of George, "Each of us lives in an infinite number of possible worlds" (23), as well as the geometry problem in the film as an example of axiomatization to back it up. And because the system is axiomatic, we avoid the Kantian problematic that devolves from Eco's refusal to axiomatize. The performative fiat under Realism is not stipulated from outside. The performative fiat is the axiom of a possible world, in this case the real world, and the possible world exists. All the things we know about the real world may be cultural constructs, but they are built, like Euclid's geometry, on the back of the first axiom. Interestingly, Possible Worlds, like Lewis's theory, relies on an act very similar to what is entailed in accepting the Axiom of Choice in mathematics. The Axiom of Choice simply asserts the existence of mathematical objects and a way to represent them. More generally, the Axiom of Choice states that such a choice can be made.

THE ENDING

The ending of the play, though, makes a narrative twist that renders questionable the acceptance of modal realism as the foundation of the play's possible worlds, forcing a re-reading of what has previously been presented as narrative logic. In fact, the play appears to make an eleventh-hour reinstatement of Eco's theory of the fabula with its three subworlds over Lewis's. It turns out that of the six possible worlds I enumerated above, number four is the "right one," the one together with P (the detective story) that is actualized as the virtual or global ontological ground. In Scene Sixteen we find out that George, or more accurately his brain, is indeed suspended in a container of chemical soup at the university's "mad scientist lab," where he/it has become part of a macabre experiment. (The play exonerates Joyce, who turns out to be actually married to George, or what's left of him.)

Interestingly, George's brain-in-a-vat condition points directly to radical epistemological skepticism, an anti-realist, idealist philosophy that claims that the external world cannot be known to exist. How do we know we are not brains-in-a-vat with sensory input about the external world being provided by a mad scientist, computer (The Matrix), or evil demon (Descartes)? The philosopher Hilary Putnam actually coined the phrase "brain-in-a-vat" to refute skepticism through reference: "Suppose we...are and always were 'brains in a vat'. Then how does it come about that our word 'vat' refers to noumenal vats and not to vats in the image?" (127). Of course, one could say regarding Possible Worlds that we can tell number four is right, because we are outside George's vat, but in fact this point of view offers nothing about whether we are also (please bear with me) brains-in-vats perceiving a brain-in-a-vat. In other words, the brain-in-a-vat image in Possible Worlds supports Eco's theory that the real world is a cultural construct, in terms of the play perceivable only through the chemically-induced support system of the lab.

Furthermore, with respect to the narrative of the play, four conforms to Eco's first world of the fabula, the world stipulated by the author from the real world, and it also works with P, the detective story--here is the solution to the murder mystery. In fact, four is accessible to P under Eco's definition of accessibility. It can be generated from P as the development of the narrative trajectory. But what of the other five possible worlds in the play that not only contradict each other, but also contradict the narrative trajectory, all the possible worlds that appear to be given, free-standing worlds? They become the second category subworlds of Eco's schemata, the subworlds imagined by the characters of the fabula--in other words, of George's brain. George's brain has apparently imagined all of the possible worlds, including the possible world that establishes the axiom for a Lewisean possible worlds theory: "Each of us lives in an infinite number of possible worlds" (23). In fact, recall the geometry problem that was offered in this scene to establish the set theoretical basis for Possible Worlds. Ironically, it can now be seen as representative of Eco's theory of possible worlds, because each square is a proper subset of each previous square and thus accessible to it.

Curiously, the critical bit of information about George's true condition does not enter the play until Scene Sixteen, just two scenes shy of the ending. Why wait so long? In terms of the temporal ordering of the plot, whose logic (generative and otherwise) has now been reinstated as essential to the fabula, why not place this scene earlier? Two reasons: conflicts among the possible worlds throughout the play have been piquing audience interest by compelling the audience to "fill in the gaps;" and, secondly, conflict between the two theories of possible worlds have created the two narrative twists, the first when Lewis takes over from Eco in Scene Four, and the second when Eco is reinstated in Scene Sixteen. In fact, the reinstatement of Eco sets the stage for the play's penultimate scene, the solving of the mystery by the two detectives, which provides the matching bookend to the play and thus completes the fabula of E And consistent with P, the final scene is a coda in which George continues to dream a (Lewisian) possible world in his vat, Thus, time and space dovetail at the end: the temporal ordering of the plot in exactly the way it is presented not only makes the play teleological, but also perfectly geometrical according to the mathematical icon.

Yet is this ending to the play something of a disappointment? After all, Lewis's take on possible worlds is more daring, more controversial, sexier than Eco's (I think), and it would be fascinating to see Mighton carry it through the ending. Upon reflection, though, I think not. The attempt to achieve closure and resolve the mystery at the play's ending does not (or cannot) evacuate the destabilizing effects of Lewis's theory on Eco's presented in the middle. In fact, I want to reassert the position I started with: Possible Worlds is an interrogation of possible worlds theory and narrative practice, not a polemic of one over another. That the play can sustain two theories more or less at once, as well as play them off each other, is a remarkable achievement. Certainly, Possible Worlds is its own take on the usefulness of such theories to dramatic and narrative worlds, offering the opportunity to consider just how the play comes into (what sort of) existence, how it might work itself out in narrative time, and how it relates to the world of the audience. And if the purpose of a play is to engage the audience about issues, whether social, moral, or philosophical, the play succeeds well.

WORKS CITED

Aczel, Amir D. The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity. New York: Washington Square Books, 2000.

Blau, Herbert. The Eye of Prey: Subversions of the Postmodern. Bloomington: Indiana Univ. Press, 1987.

Briemont, Jean. "Postmodernism and its problems with science?' Lecture given in Helsinki at the invitation of the Finnish Mathematical Society, 6 Mar. 2002. 16 Nov 2004. <http://dogma.free.fr/txt/JBPostmodernism.pdf>

Bordo, Susan. "Bringing Body to Theory." In Body and Flesh: A Philosophical Reader, edited by Donn Welton, 84-97. Oxford: Blackwell, 1998.

Butler, Judith. Bodies That Matter: On the Discursive Limits of "Sex". New York: Routledge, 1993.

--. "Foucault and the Paradox of Bodily Inscriptions" In The Body: Classic and Contemporary Readings, edited by Donn Welton, 307-313. Oxford: Blackwell, 1999.

Eco, Umberto. "Possible Worlds and Text Pragmatics: 'Un drame bien parisien:'" Versus 19/20 (1978a): 5-72.

Keller, Evelyn Fox. Secrets of Life, Secrets of Death: Essays on Language, Gender and Science. New York: Routledge, 1992.

Mighton, John. Possible Worlds and A Short History of Night. Toronto: Playwrights Canada Press, 1992. Lewis, David. On the Plurality of Worlds. Oxford: Blackwell, 1986.

Pinter, Charles C. Set Theory. Reading: Addison-Wesley Publishing Company, 1971.

Possible Worlds. Videocassette. Directed by Robert Lepage. Montreal: Alliance Atlantis. 2000.

Putnam, Hilary. Meaning and the Moral Sciences. London: Routledge & Kegan Paul, 1978.

Rotman, Brian. Mathematics as Sign: Writing, Imagining, Counting. Stanford: Stanford Univ. Press, 2000.

Searle, John R. The Construction of Social Reality. New York: The Free Press, 1995.

Soper, Kate. What is Nature? Culture, Politics and the non-Human. Oxford: Blackwell, 1995.

ENDNOTES

(1.) The philosophical notion of possible worlds can actually be traced back to the seventeenth-century metaphysician Gottfried Leibniz, who theorized that God had initially created an infinite number of possible worlds and then chose the perfect one to bring to actuality.

(2.) It is impossible to verify where Hilbert made this statement since it seems to have been made verbally. But it is quoted widely by mathematicians, who clearly believe in its descriptive accuracy: "No one will expel us from the paradise that Cantor has created?'

(3.) Eco's article is one of the earliest efforts to link possible worlds theory with literary theory. See also Thomas G. Pavel, Fictional Worlds (Harvard, 1986), Ruth Ronen, Possible Worlds in Literary Theory (Cambridge 1994), and Marie-Laure Ryan, Possible Worlds, Artificial Intelligence, and Narrative Theory (Indiana 1991).

(4.) See for instance "Semiotics of Theatrical Performance," Drama Review 21 (1977): 107-117.

(5.) In their 1998 book, Fashionable Nonsense, scientists Jean Bricmont and Alan Sokal take to task several postmodernist theorists for misusing scientific ideas. They rightly criticize Julia Kristeva for mangling the Axiom of Choice in her book, Semeiotike. My understanding of the Axiom of Choice as an existential rather than constructivist principle is the same as Bricmont's: "the axiom of choice allows to prove the existence of sets that one cannot 'construct' "(7). See the 2002 lecture given by Jean Bricmont in Helsinki, "Postmodernism and its problems with science."

(6.) Brian Rotman in Mathematics as Sign offers an intriguing alternative to the problem of infinity. He proposes a system of "transfinites", very large numbers (as large as you like) produced by the functions of arithmetic rather than by counting. That is, one can produce a number by multiplying 10 by 10, without actually having to count to it, and one can keep multiplying "indefinitely." See pages 126-36. The problem with the transfinites is that they cannot be used to calculate instantaneous speed (differential calculus), and thus have limited applicability for physics and engineering. The real numbers are necessary for getting arbitrarily close to the limit.

(7.) Drama and performance studies have benefited greatly from the speech act theory first theorized by Austin and Searle.

(8.) It would, of course, be appropriate to investigate the play's use of possible worlds theory according to quantum mechanics, specifically the claims first put forward by physicist Hugh Everett in 1956 and developed more recently by quantum physicist David Deutsch. Nevertheless, I take my lead from the statement made by George in the play that points specifically to mathematics.

Elizabeth Klaver is professor of English at Southern Illinois University Carbondale. She is author of Performing Television: Contemporary Drama and the Media Culture (2000), Sites of Autopsy in Contemporary Culture (2005), and the editor of Images of the Corpse from the Renaissance to Cyberspace (2004).
COPYRIGHT 2006 Ohio State University Press
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2006 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Klaver, Elizabeth
Publication:Narrative
Article Type:Critical essay
Date:Jan 1, 2006
Words:10239
Previous Article:Narrativity, self, and self-representation.
Next Article:Conjectural history and Satire: narrative as historical argument from Mandeville to Malthus (and Foucault).
Topics:

Terms of use | Privacy policy | Copyright © 2021 Farlex, Inc. | Feedback | For webmasters |