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Representing Indeterminacy in Art and Science
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George Darby

Introduction[1]

Everyone knows there’s something strange about Schrödinger’s cat. Shut in a box with an atom that may or may not have undergone radioactive decay and thus triggered the release of a poison, it has somehow ended up in a state where it is both dead and alive. Or was it neither dead nor alive? Or indeterminately dead and alive; or did it no longer even make sense to think of it being dead or alive, at least until we bother to look, at which point our looking decides its fate? However it’s understood, uncertainty and observer involvement make a good source of metaphors and literary devices.

In a key passage of Jean-Philippe Toussaint’s novel Monsieur, the eponymous protagonist narrates the story of the cat, noting that one ought to be able to say whether it was alive or dead, but that one cannot. Then, he concludes with a popular interpretation:

le simple fait de le regarder altérait de façon radicale la description mathématique de son état, le faisant passer de l’état de limbes à un nouvel état, où il était soit positivement en vie, soit positivement mort, c’était selon. Tout était selon.[2]
The cat’s tale has some significance for Monsieur – he brings it up more than once. Perhaps he identifies with its predicament. But the passage does more than that: Monsieur is an example of a genre characterized by indeterminacy of plot, involvement of the reader in constructing the story for themselves, and an interest in self-reference. In this context the story of the cat, underlined by the repetition of the key message – Tout était selon – seems to play a particular stylistic role: a mise-en-abîme, a story-within-the story used to reflect the nature of the text itself. Indeterminacy of plot is reflected in the indeterminacy of dynamical properties of a quantum system; involvement of the reader in constructing a story for themselves is reflected in the involvement of the observer in determining the result of a quantum measurement; the self-referential device evokes quantum indeterminacy to draw attention to this.

The workshop at which this paper was presented concerned the relationship between the philosophy of art and the philosophy of science. A central area of overlap concerns representation, and one instance would be the representation of uncertainty and indeterminacy. Understanding indeterminacy in quantum mechanics is a key question in the philosophy of physics (see, for example, Busch and Jaeger [2010] and Isham and Butterfield [1998]). On the other hand, understanding “worldly” indeterminacy in general is a growth area in analytic metaphysics (see, for example, Rosen and Smith, [2004], Akiba [2004] and Barnes and Williams [2010]). And there are interesting questions about the relationship between the physics and the metaphysics (see, for example, Skow [2010]).

So here’s a question: What does the philosophy of literature (and the philosophy of art more generally) have to say about the representation of indeterminacy? There will be various ways in which we might cash out the idea that the plot of a novel is indeterminate, or multiple, or essentially constructed by the reader. Then we can ask whether these kinds of indeterminacy, multiplicity and observer-dependence map onto various ways in which, historically, quantum mechanics has been thought of as involving indeterminacy, or multiplicity, or a crucial role for conscious observers. The focus here is on the indeterminacy part, with Toussaint’s mise-en-abîme as inspiration (from the point of view of exploring the connection, it’s not crucial, of course, whether this is precisely what Toussaint intends).

Indeterminacy in Art

A mise-en-abîme might reflect something other than indeterminacy. La Jalousie (Robbe-Grillet [1969]) contains the following passage. The main character is concerned that his wife (known only as ‘A’) is having an affair with Franck; it is not in the nature of the work to confirm whether the suspicion is correct. Here the text concerns another novel, the ‘African novel’, which, as it happens, concerns suspicions of adultery.
Le personage principal du livre est un fonctionnaire des douanes. Le personage n’est pas un fonctionnaire, mais un employé supérieur d’une vielle compagnie commerciale. Les affaires de cette compagnie sont mauvaises, elles évoluent rapidement vers l’escroquerie. Les affaires de la compagnie sont très bonnes. Le personnage principal – apprend-on – est malhonnête. Il est honnête, il [… continues in this vein]
The inconsistency here reflects the nature of La Jalousie, which in turn reflects (or perhaps creates – see Robbe-Grillet [1969, p.xxii]) the confusion of its main character. Unlike a more traditional novel, of course, the confusion will never be resolved: this passage occurs shortly before the end of the book.

Here again the use of mise-en-abîme forces us to meditate on the fact that we are reading a text, moreover a text that does not aspire to give a faithful representation of some determinate story. This time it is inconsistency, not indeterminacy, that is at play. But the devices are similar. In creating the Nouveau Roman, Robbe-Grillet ([1961, p.119]) gives up on a “ready-made meaning”, notes the loss of old certainties, among other things blaming then-recent discoveries in science, and sets out a new template:
The meanings of the world around us are no more than partial, provisional, even contradictory, and always contested. How could the work of art pretend to illustrate a meaning known in advance, whatever that is? The modern novel, as we said in the beginning, is an experiment, but one that creates for itself its own meanings, as necessary. Does reality have a meaning? The contemporary artist cannot answer this question: he knows nothing of it. All he can say is that this reality will perhaps have meaning after its passing, in other words once the work is finished.
His novels and films reflect this. In fact, various features of the work prevent the reader from reading it in a straightforwardly realist way, as we’ve seen.

In the case of Toussaint’s Monsieur, we learn certain things about the character (he hurts his wrist in a fight; he’s a commercial director for Fiat), but others are strangely unresolved (he goes to call a doctor, but comes back having phoned his boss, and it never does become clear whether he has an X-ray; he seems to be good at his job, but it never becomes clear whether he really is). It’s not just that we don’t get told: at the places where we’re led to expect to be told, we instead get something incongruous. These and other elements of Monsieur chime with Robbe-Grillet’s departure from certain “obsolete notions”, for example that the character should have a proper name (Robbe-Grillet [1961, p.27]). Monsieur’s inertia, and lack of a direction in the narrative (though there are many events) is reflected in Robbe-Grillet’s playing down the importance of determinate plot (Robbe-Grillet [1961, pp. 29-32]). The present idea is to explore the use of the Schrödinger Cat story as metaphor for this idea.

Indeterminacy in Science?

The indeterminacy we’re after appears in the way in which the representational apparatus used to capture properties in quantum mechanics differs from that used for classical physics. Consider, for the sake of an example of the physical system to be described, the character from the African Novel. First we’ll think of him classically. There are two observables of interest: Job and Honesty. There are two possible values for Job: Customs Official, and Senior employee of an old trading company. Likewise there are two possible values for Honesty: Honest and Dishonest. We will represent this system, with its two observables, in Figure 1.

This abstract space in Figure 1 has one segment for each possible state. For any state, the system has a definite value for Job, and also a definite value for Honesty: just read them off the representation. Here our system is Dishonest and a Customs Official. This a character in a Classical story, so to speak. Using the classical representational apparatus, the theory will always supply definite values for all of the observables concerning the system.



Figure 2 shows the beginnings of a vector space representation for the properties of our physical system. This is a different kind of apparatus, with different representational capabilities. Now we are thinking of the same character from the story, but this time as a quantum system. Two vectors are labelled, representing the possible values for Honesty. If the system’s state vector happens to be the one pointing up, then the value of the Honesty observable is Honest. If the system’s state vector happens to be the one pointing right, then the value of the Honesty observable is Dishonest.

But this representational apparatus allows other states, which reflect the philosophical departure from classical physics. The diagram also shows one of these, the weighted sum 1/√3|Honest| + √2/√3|Dishonest|. Is it a state where our system is Honest, or a state where it is Dishonest? Quantum mechanics doesn’t say. All it gives us is a probability for finding the system to be Honest or Dishonest (probability 1/3 and 2/3, respectively), were we to perform a measurement of its honesty. A state like this is a superposition of Honest and Dishonest. At this point we sometimes hear things like:

1. A system in such a superposition is neither honest nor dishonest.
2. A system in such a superposition is both honest and dishonest.
3. If a system is in such a superposition, it is indeterminate whether it is honest or dishonest.
4. If a system is in such a superposition, it makes no sense to ask whether it is honest or dishonest.

One of the main philosophical questions is to disentangle these, to clarify what it means is to be in a superposition – how could something be both, or neither, or how should we think about indeterminacy?

Things get better still when we represent another observable using this representational apparatus (see Figure 3). The Job observable is represented in the same vector space, using a different basis. The |Customs official| vector, the one in which we would say that the system does have that definite value for Job, happens to be that same vector that was a superposition for Honesty, as is the other definite Job state. And vice versa: |Honest| and |Dishonest| are (different) superpositions for Job, so the definite Honesty states are states of indeterminate Job. So if quantum mechanics assigns a definite value to Honesty then it won’t assign a definite value to Job. This is essentially the same as Heisenberg’s famous Uncertainty Principle: if quantum mechanics assigns a definite position then it doesn’t assign a definite momentum.

One thing to note about the Uncertainty Principle is its relational nature: it says that the more certain is the value of one observable, the greater the uncertainty in the other. Another is that, despite its name, it is not, or at least not straightforwardly, about uncertainty. That term implies that there’s something to be uncertain about. But if the theory says everything there is to say (in other words, if it is complete), and yet fails to say definitively whether a man is honest or dishonest, then a natural conclusion is that there is nothing definitive to know – it really is indeterminate whether the man is honest.

Come to think of it, though, doesn’t that suggest another natural conclusion? If the theory fails to say whether a system is here or there, when that system is of a type that clearly can be here or there, and will be seen to be determinately either here or there if an observation is made, then that’s not a feature of the quantum world, it’s a defect of the theory. A theory that fails to answer such questions is incomplete; it does not after all tell you everything there is to know. A more complete theory would give a verdict, for every state, on whether the man was honest or not.

The suspicion of incompleteness has historically played a central role in the foundations of quantum mechanics. Einstein famously thought that the theory was incomplete, and demonstrably so, by its own lights. As it happened, things didn’t go Einstein’s way: von Neumann produced an argument (based on dubious assumptions), later strengthened by others (by making the assumptions less dubious) to the effect that no theory could fill in the gaps. If those arguments are correct then the incompleteness (or rather indeterminacy) is not in the quantum mechanical theory but in the world, so to speak.

Connecting the Two

For each of the various ideas – incompleteness, inconsistency, indeterminacy, and so on – different questions may be asked from different angles: How is the relevant phenomenon produced in literature? How should it be thought about philosophically? And how, if at all, does it relate to similar concepts in science?

Incompleteness is the least exciting: What is it for a fiction to be incomplete? Well, all fictions are incomplete. An author doesn’t have to work to produce incompleteness, since it arises whenever something is not mentioned, usually because irrelevant to the story. Already, though, there is an interesting connection with analytic philosophy of fiction: A prominent approach to truth in fiction (Woodward [2011]) uses the philosopher’s device of possible worlds, (maximally specific) ways things might have been. The boring fact of incompleteness is a minor obstacle to this analysis. Possible worlds, unlike fictions, are not incomplete, so truth in a fiction can’t be identified with truth at one possible world. That can be solved simply by associating fictions with sets of possible worlds, the “story worlds” compatible with the fiction. Then something is true in a fiction if it is true in every one of its story worlds, and something is false in the fiction if it is false in every one of its story worlds. So it’s true in the fiction that Bilbo is a Hobbit, and it’s false in the fiction that Gandalf is an Elf, but it’s neither true in the fiction nor false in the fiction that Sauron enjoys cricket. And those are right.

What is it for a story to be inconsistent? Well, the most straightforward device would be for the text to contain sentences that jointly imply a contradiction. Perhaps the passage earlier from La Jalousie is an example (“perhaps”, because this relies on a face-value reading – one might ask whether the story is really inconsistent, and whether inconsistency spreads from the African Novel to the wider novel; this depends on how the mise-en-abîme works, and so on.). Things get more interesting with the attempt to translate impossible fictions into the story-worlds talk. The standard view says that the story worlds are all the possible worlds compatible with everything in the passage. But no possible worlds are compatible with everything in the passage. (Because that’s contradictory, and so impossible). So everything is true at all worlds compatible with the passage (trivially, because there are no such worlds). So everything is true in that fiction. And this is the wrong result. Getting round this problem is a current challenge for this approach to truth in fiction (see Woodward [2011, pp. 159-161]).

But our concern is not with inconsistency (if it were then perhaps Gödelian themes might be more relevant than physics, but that would be a separate project – thanks to Otávio Bueno for this suggestion). Our concern is with what it would be for a story to be indeterminate, and in particular, how indeterminacy differs, and may be distinguished through literary devices, from incompleteness.

So, to generate a true “Quantum text” (as Robbe-Grillet’s work has been described – see Broad [2008]) will require at least certain features. There is nothing particularly quantum mechanical about being uncertain about something, nor even about absolute, or inescapable, lack of knowledge. The plot would have to be indeterminate altogether, and it’s not clear how that could be achieved. If a detail is left out of the plot, or if two different possible ways of filling in the detail are offered, then it will always be open that the text is simply incomplete; and if the text explicitly includes both alternatives between which things are indeterminate then it starts to look simply inconsistent. What is needed is some device for highlighting the fact that there is more than one interpretation, and that the competing interpretations are all equally right, without collapsing into incompleteness or inconsistency. Even then, for an analogue of the Uncertainty relations, there would have to be interaction between the things that are certain or uncertain. Perhaps if it were settled whodunnit, then it would have to be unsettled what they’d done (or where, when, why or how). It would require something quite radical to achieve this. But it’s not necessary to think that Toussaint and Robbe-Grillet do so exactly (they certainly go beyond mere uncertainty) to ask interesting questions about the representation of indeterminacy.

One such question is how indeterminacy should be thought about on the possible worlds approach to truth in fiction. The natural thing to try would be this: Something is indeterminate in the fiction if it is true in some of the story-worlds and false in others. After all, that is how indeterminacy is captured in recent metaphysics: the things that are indeterminate are those that are true at some of the worlds that are candidates for actuality, and false at others (Barnes and Williams [2010]). Then it’s indeterminate in the fiction whether Monsieur had his hand x-rayed. And we wanted that. But it’s also indeterminate in the fiction whether it was raining in London at the time. And we didn’t want that – now we’ve conflated indeterminacy and incompleteness. So, where the standard approach to truth in fiction runs into trouble with properly representing inconsistency, we have a parallel problem properly representing indeterminacy and separating it from mere incompleteness.

As far as I am aware, there is no detailed treatment of this that would parallel the attention to impossible fictions. What there is, however, is a variety of work on the general idea of metaphysical indeterminacy, and it will be interesting to link this to the fiction case. But perhaps the most significant point here is that a straightforward application of possible worlds apparatus (indeterminacy as truth in some worlds and falsity in others), as is found in (Barnes and Williams [2010]), runs into trouble when applied to what we get from quantum mechanics (Skow [2010] sees this as a simple refutation of the metaphysical project – though I would prefer to call it “interestingly problematic”.)

Conclusion

We have begun to explore some ideas about how metaphysicians, literary theorists and philosophers of physics, in their different ways, think about indeterminacy, but of course we have barely scratched the surface. From the point of view of the philosophy of science, the immediate aim is to explore the metaphysics of indeterminacy, specifically as it arises in quantum mechanics. The metaphysical work mentioned above is part of this, and the aim is to connect also to the representation of indeterminacy as might be approached in the analytic philosophy of literature. In the other direction, collaboration between philosophers of science, philosophers of literature, linguists and literary theorists adds depth to the study of the way in which indeterminacy functions in works such as the Nouveau Roman and is reflected in devices such as Toussaint’s mise-en-abîme.

Notes

1. A version of this paper was presented at the workshop “What Can the Philosophy of Science do for the Philosophy of Art (and Vice Versa)?” on October 19, 2012 at the University of Leeds. Thanks to the participants for feedback, especially Steven French, Dean Rickles, Otávio Bueno and Juha Saatsi, and to James Fowler for discussions on the Nouveau Roman.

2. Toussaint [1986, p.27]. “…the mere fact of looking altered the cat’s state, changing it from a state of limbo to a new state, either definitely alive or definitely dead…”; “selon” is “according to”, but according to what?

References

Akiba, K. [2004]: “Vagueness in the World”, Noûs 38(3), pp. 407-429.

Barnes, E. and Williams, J.R.G. [2010]: “A theory of metaphysical indeterminacy”, Oxford Studies in Metaphysics 6, 2010.

Broad, L.K. [2008]: “Prisoners of Possibility: Robbe-Grillet’s La Belle Captive as ‘Quantum Text’”, Senses of Cinema 48, pp. 1-10.

Busch, P and Jaeger, G. [2010]: “Unsharp Quantum Reality”, Foundations of Physics, pp. 1-27.

Isham, C.J. and Butterfield, J. [1998]: “Topos perspective on the Kochen-Specker theorem: I. Quantum states as generalized valuations”, International Journal of Theoretical Physics 37(11), pp. 2669-2733.

Robbe-Grillet, A. [1961]: Pour un Nouveau Roman. Paris: Minuit.

Robbe-Grillet, A. [1969]: La Jalousie. London: Methuen.

Rosen, G. and Smith, N.J.J. [2004]: “Worldly Indeterminacy: A Rough Guide”, Australasian Journal of Philosophy 82(1), pp. 185-198.

Skow, B. [2010]: “Deep Metaphysical Indeterminacy”, The Philosophical Quarterly, 60, pp. 851-858.

Toussaint, J-P. [1986]: Monsieur. Paris: Minuit.

Woodward, R. [2011]: “Truth in Fiction”, Philosophy Compass 6(3), pp. 158-167.

2014 © George Darby

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