The following interview between Apostolos Doxiadis and David Herman was conducted on July 20, 2007. The interview as it appears here was transcribed and edited only slightly for clarity.
Herman: Could you talk a bit about the “photos-and-movies language” that you use in your paper for the symposium? How does this language help us think about the family resemblances between stories and proofs?
Doxiadis: I do a lot of lectures, mostly to students to make them understand ideas about narratives or about mathematics or about the mixture of both. In one lecture, on Andrew Wiles’ proof of Fermat’s Last Theorem, I wanted to have a language that could be used to describe the proof as containing two types of objects. I said it really contains photos and movies which are ways of connecting them. It was the first time I used this. And because I was talking about mathematics, I wanted to have a very soft language that could relate not dots and arrows and vertices and edges, but rather photos and movies. You freeze and you move. I used the example of freezing and fast-forwarding your DVD player.
Herman: Just to put that in brackets for a moment…Is there any sense of (this) clashing with the graph languages, which is an earlier incarnation of the project (which you delivered at the Mykonos symposium)? Do you feel like you’ve reconciled these languages?
Doxiadis: No. I didn’t think it was clashing at all because one of the problems of the previous language was that both the dots and the arrows involve action. That was a bit confusing because you had an elementary atomism with two kinds of objects and both were made of the same substance: action. That was a bit confusing.
Herman: So, the photo is?
Herman: And the movie is?
Doxiadis: The movie as a rule is invisible. In comics, the movies as a rule are in the gutter. I want to comment on this in another way. Let me leave it aside for a moment. In prose, the photos would be the action sentences. The movies could be, again, invisible…Do you know this poem by Jacques Prevere? I don’t remember exactly how it goes, but it’s something that all students of French learn because it’s so very elementary in language. It’s a great poem. It goes something like:
“He came to the table.
He opened the cupboard.
He took out a cup.
He put in the coffee.
He put two pieces of sugar in the coffee.
He picked up the spoon.
He mixed the sugar.
He added some milk.
He drank a sip of coffee.
He put the cup down.
He got up.
He put on his raincoat.
He put on his hat.
He opened the door.
And he left.
And I sat down and cried.”
Almost verbatim. In slightly more elevated French. I may be missing some words. It’s interesting because there is no moving, except in the last sentence. Although there is a mere conjunction there its function is like that of the key element in Forster’s definition of plot vs. story: “The king died and the queen died…of grief.” It is the equivalent of the “of grief”. And that is a case where you just have your photos and…in fact, it’s very interesting because all the first photos are connected with temporal links. There’s no direct causality there until suddenly the narrator, the woman (whom we understand from the last line to be the narrator), signals that this is about the end of a love affair. It is this which is causal. This is the power of the poem. In this series of temporal links suddenly a causal one has entered. So, for many years, I was trying to find a parallel for all this – and you know, this is relevant to the photos-and-movies language. If you have two elementary entities, why would they be made of the same substance? Why not call them one entity? I was trying to find a functional definition of narrative that was good enough for my purpose of developing a model for comparing proofs and stories; not for all eternity; not to be the cornerstone of narratology.
I actually went about these lectures in two ways. First, I went the atomic way. Let’s talk about the elementary. Simple narratives. Simple narratives are descriptions of simple actions. Action sentences. I didn’t use that expression but it was about that. Or molecular actions. A panel of dialogue, a sort of mini-action. Then, I said, from that you build longer narratives. And that seriality is always there. You meet more complex narratives and I said that it’s a series of descriptions of actions, which are ordered in some meaningful way, which may have, if they are of a more advanced form, a beginning, a middle and an end, and a definition–well, I don’t want to use the word ‘focused’. Let’s not get into that. I have a very precise definition of that.
I just want to take one line from that definition which surprised me when I read it but I didn’t like it as a definition. I wanted it to be cleaner than that but I couldn’t get past the sentence which said “..ordered serially in time and often also causally….” “Serially in time” was not enough; clearly for a narrative, there is something more than that. This is why “causally” is needed. If we take the Prevere poem, however, sometimes things are not causally linked. They are just temporally sequenced. Sometimes you see that when you describe passages from scene to scene, you will advise a novice director or writer to cut the in-between. But sometimes, some of the real world life of action is necessary to ground your action in the real world. You know, there are some forms of narrative where you have an excessive amount of causal connection. And when you do that, it is artificial. The narrative form that relies excessively on causality is, for example, the Victorian melodrama or the Hollywood action blockbuster. 24 Hours. Have you ever seen that series? I was bored by it because it was all so causally determined. The tiniest little thing was of momentous importance. Only events of tremendous importance occurred and everything was significant. It was a paranoid vision applied to narrative all the way down. They took Chekhov’s maxim that if you see a shotgun in the first half of a play, it has to be used by the third to its total extreme.
To me, that is the sort of an example that is intensely causal. Intensely temporal are narratives like the ones little children tell. ‘Tell us how you passed your Sunday.’ ‘Well, I got dressed, I ate breakfast…’ And any English teacher would say, ‘we don’t need all of these things because they are just temporal’. So, in my definition, I had it as “temporally ordered with frequent but not constant causal connections.” So, because of those two words I could not throw out, to my mathematical instincts this definition was not very elegant. How could I get around this ‘sometimes but not always causal connections’ element. That’s silly. But I couldn’t get away from that.
So, it helped to think of the functions of the gutter in comics, and how it mediates between what I call photos versus movies. So, we just have action. Photos are just action. And when I looked at that slide I showed to kids with Wiles’ proof, I saw that anything mathematically significant was in photos. If you look at that slide, you can understand all the movies because none of them are mathematically significant. Movies are in Greek. We first observe that. You understand that. Now, from the construction, we made__________, you don’t understand what the result it but you don’t need to. There is a result which is by… It follows formally. Forget the “formally.” It means, follows mathematically, logically. All of these are very simple things. And the reason you can understand them all is that they do not all involve mathematics. They are just connectors. So, when you start to think about what types of things these things are and whether they are temporal or not, you start to see that one means for advancing a story is “and then” and another is “and so”. You begin to see that there are two kinds of movies and two kinds of passages.
And then I realized that these relationships could be captured in the non-linear structure of the graph. If a movie is “and then”, the link has one arrow going in and then one going out. In terms of the story as narrated, the sjuzhet, it just has one photo leading to this photo. If you speak of causality there’s always the example from Forster, namely, “The king died and then the queen died of grief.”
Let me give another example here. It involves a non-narrated connection: “Will you go to Paul’s party?” “I have to work.” What is that?
Herman: That would involve Grice’s Maxim of Relation, I think. The idea is that your interlocutor would assume that there is a relevant connection–would assume that there is a reason for mentioning the need to work in this context.
Doxiadis: What do you call it?
Herman: Grice’s term is conversational implicature.
Doxiadis: So, if we look at the Forster example, the interesting thing about “The king died and then the queen died of grief” is that people assume it to have more causality that real life. That is, if you’re telling them something (like in a murder mystery) you don’t just combine clues, you combine author strategies. Yes, but would an author like Agathe Christie tell us these obvious clues, or is she trying to mislead us–to make us think that if she’s showing us all those clues that it can’t be him? Anyway, with the example of “Then the king died and then the queen died,” most readers already think that the two are already related; but let’s assume the (author) wants us to read that as a neutral case of temporal order.
Now, if you say “The king died and then the queen died of grief,” there is again a sense that the king died and then the queen died. Then if you focus on that link and want to see the full non-linear structure, you will see another movie coming there, you will see, the queen loved the king very much. That is the only thing that had made her die. Otherwise, it’s a non-sequitur kind of thing. It’s an absurd kind of story. In the context of the story, you have to know that she loves him. In that sense, I realized it was useful to talk about photos and movies—because it gave you a kind of linear structure for time, which was what I call I-junctions, and a V- or Y-junction, where you have at least two things coming in.
In turn, that same structure applies quite naturally to proofs. In proofs, you have times where you are just doing one thing: I-junctions. For example, take the statements: ABC is a right triangle. ABC is a triangle. Moving from one statement to the other is an I-junction. That is what Kant would call an analytic statement, if ever there was one. The most usual analytic statement is a statement which contains all of the information. The most usual example of an analytic statement in a philosophy book is ‘A red rose is red’. Now, in my example, you have the implication from ABC is a right triangle to ABC is a triangle, which may be eminently useful you see. And it’s not trivial. It’s perhaps trivial to construct an analytic sentence but in the progress of a proof, remembering that ABC is a right triangle does not stop it from being a triangle. It may be eminently useful to talk about concepts of triangleness, e.g., that the sums of the angles of a triangle add up to 180 degrees. But, going the other way, is not a simple relation. You cannot say that because ABC is a triangle, ABC is a right triangle. It’s not necessarily true.
And then, because I was trying to find similarities of proofs to stories at the atomic level, when I use the photos-and-movies language, I got to that and I said ‘that’s it.’ We have something that’s strong. We’ve taken our optical instruments and we’ve moved to using telescopes—from using our eyes, to using a lens like Sherlock Holmes, to using a primitive sort of microscope, to using a huge electron microscope…and then we got down to the elements. And there, we saw something. One of my beloved sentences, I think it’s used in Logicomix, I don’t remember, is when people found fault with Frege’s model of a logical language, he replied: “I have invented a microscope. You cannot hold it against a microscope that it is not a good instrument with which to look at the stars.” I have the sense that at the microscopic level, I’ve gone down all the way to connections, and there are two kinds of connections in stories and proof: one is linear and the other is non-linear.
Herman: And so, as you’re working out these connections, how does that effort bear on (or derive from) your practice as a creator of narratives? Are you drawing on narratological theory or rather on ideas that are part of your repertoire as an artist? Was it your practical, hands-on experience with creating narratives that led you to make these refinements?
Doxiadis: Yes, but, I must open up a parenthesis here and then perhaps we can talk a bit about narratological theory. One thing I have not spoken at all about is my theoretical background. I have spoken about mathematics. I have spoken about stories and my work as a creative writer. But, I haven’t spoken about theory of any kind. I have to say that already at the level of my Masters degree, I moved from pure mathematics to mathematics applied to cognition to the study of the nervous system.
Herman: Is that right?
Doxiadis: My Masters was a micro-level study of the nervous system. I became immensely interested in cognitive theory, what today you would call artificial intelligence, cognitive psychology, neurophysiology. And that stayed with me no matter what I was doing. So, I first read Propp when I was 20 or so and I was so impressed by that I said, “Oh my God, this guy is really onto something.” I wasn’t thinking very much about stories then but I knew about fairly tales (and had loved them since I was a kid) and this was about the time I was doing my Masters and trying to look at models that went beyond the nervous system. I was very interested in structure, and I was shopping around for ideas in all sorts of things to help me combine a mathematical tendency for models and structures with cognitive models. And I saw that in Propp, so that when I started to operate as a writer, I started to think about it again. I went back to Propp and via Propp I first hit upon the word narratology and I started to read very selectively and in a very utilitarian way things that were relative to what I was doing. I did not read narratology as a writer and the things narratology told me as a writer, I found either evidently true or needlessly complicated. There was the odd occasion where I said ‘I know that. I have been doing that for many years but no one has put a name to it’.
But problem with Propp is that the moment you construct a graph of more than three nodes, you are already in the billions of possibilities. What can you do with that? And then, when I got into even more complex models, I would always think, ‘That’s too complex.’
Herman: One of my personal interests is in the distribution and dissemination of narratological research in different countries. Is there a tradition in Greece of modern-day narratological research that’s built on the original foundations? And, did you draw on anybody locally?
Doxiadis: No. Look, we are on the Continent. There is a lot of knowledge and interest about the French theorists; Genette is very well known. Barthes, Derrida, people like that are very well known and read. Of the more analytical, sober nuts-and-bolts kind of narratology, to which you belong, I would say that there is practically zero.
What I myself most got from narratology was a climate and I think that is huge. The climate of a feeling of narrative can be analyzed rationally, using tools which are tools of reason and logic, Aristotelian methodology. In fact, I see narratologists as children of the Poetics. When I do research, when I write novels, I constantly read around them. When I write about the Greek Junta, I read endlessly about Modern Greek history. When I work on Logicomix, I read about the Vienna Circle, I read about Belle Epoque Paris, and so on.. As I was doing this work, I kept my mind alive by reading narratology.
Herman: Let me ask you some more specific questions about the proof/quest/narrative connection. I’m still having a little trouble understanding where you’re locating the parallelism or the analogy or the commonality between proof and quest. And I guess, therefore, the commonality betwteen proof and quest narrative. In your account, the parallelism sometimes seems as though it pertains to the experience of doing the proof, i.e., the experience of following the proof – especially for the knowledgeable mathematician who has familiarity with the context and can almost reawaken the original effort that went into the formation of the proof. Even in the formulation of the your key propositions early in your paper, you’ve located the quest structure in the quest for the proof. “The quest for a mathematical proof is precisely a quest” (5). This makes the quest the search for the proof, the proof as discovered rather than the proof as published. But at other points in your argument, you want to attribute to the proof as published a quest-like structure in its own right.
Doxiadis: A mathematical proof is a quest—and not just a quest for a mathematical proof. I could give you two propositions, and both are true and both are related. The first is that the quest for a mathematical proof is a quest. And the second is that a proof is a quest.
Herman: A quest for the mathematician?
Doxiadis: A quest for QED. A quest in graph language. One thing that my analysis does as it advances is to begin from a more natural language which is intuitive and has to do with stories of proof as they happen and the historical reality of mathematics, then proceed to the graph language, and from there, once the focus on movies of graph language is established, to start to using increasingly only that more technical language to explain things. A good example is in the fifth section of my paper, which discusses the birth of proof and gives verbal examples. I am only seeing those in the graph language of photos and movies. I use these statements to create a sense of a huge realm of possibility. Once I do that, a proof itself is a quest in “P-space” and it is a quest in P-space, because with this diagram I have defined the requirements to have a proof. To have a graph-setting and a graph-setting in the proof is P-space. It is to have an A, a searcher and in the proof the searcher is the person reading the proof, the person moving the graph space. It is an entity moving graph-space. In fact, it is the entity which has to move temporally: A to B to C. An initial position is what we know, what a mathematical proof would begin with, namely, axioms and previously known propositions. The destination is the QED, the theory to be proven.
So, although I use “quest” in the everyday sense, I use it in order to build a graph language and once I build it, I restrict all my discourse–I can restrict it when I want, to designate just a proof and not the human quest. It is a quest in the sense that you have a place on the graph to be reached, a place on the graph to begin from and a way to get there. You might object that an enhanced tree is not a quest. But insofar as I’m focusing the idea of the published proof, I might be correct–in the sense that it’s an account of a journey.
Herman: Which is actually the passage I was going to point to. You talk about the ant and you put us in the position of the ant, as would be the case with an internally focalized narrative. We are looking at the steps of the proof as seen through the eyes of the ant. But then you say, because you are a very obsessive ant, you write down the record of your journey which then would suggest that it is a quest after the fact—that it can only be conceptualized as a quest after you’ve taken the journey.
Doxiadis: Keep endless notes that you report this to your superiors, to the deities. They will keep only the reports that are interesting. In a sense it’s like that story about the monkeys typing away and eventually typing all of Shakespeare’s sonnets. These are searchers who do their searches in often very random ways. A proof is whenever they start from an interesting proposition. If they get from there to a highly graded proposition that was previously unknown, they write that down as a significant journey. So, you generalize the language of quests to journeys, and you define a journey as a trivial quest in which there are not too many problems.
Herman: What I am actually trying to do is search for disanalogies between these categories of proof and stories.
Doxiadis: There are huge disanalogies. You know I am not a lunatic! The most basic disanalogy, if we are not talking about the formal language but the language of proofs and stories as they occur in life, is that proofs and stories are imitations of actions, human or anthropomorphic, whereas proofs are progressions in the world of mathematical composition. It’s a huge difference.
Herman: So, you are still comfortable using ‘quest-story’ given that disanalogy.
Doxiadis: Quest-story is a metaphor, as is the photos-and-movies language. When I say photos, I am not carrying a camera. When I say movies, I am not carrying a camera recorder. Quest is really a metaphor that helps with understanding. It is understandable by all and it’s applicable at many levels. What it boils down to in this analogy is a graph language. This is the profit of using models where you move from quotidian terms (such as “love,” “quest,” “human being,” “freedom,” etc.) whose meaning you could argue about indefinitely, to a more sparse symbolic language where meaning can be agreed on a la Euclid, because it’s obvious, or because, a la Hilbert and Formalists, that’s all there is. This world only exists in the rules I give for it. When we are talking about human beings, we are talking about actual quests. When we are talking about P-space, we are talking about graphs. I am happy to talk of quests at every level because most of the things we talk about are quests.
Herman: This is would allow you overcome a point I raised in my written objections, concerning particularity and generality—i.e., the way stories are ground in the specific.
Doxiadis: Mathematics is about generality; but proofs are not about generality. Let me give you an example. A general mathematical statement is the Pythagorean theorem. When you get into the nitty-gritty of a proof, every step is at the level which is much more analytical and much more detailed than such general statements. Take for example Euclid’s proof of the infinity of primes. The thing to be arrived at, that primes are infinite, is a very general statement. But the way Euclid constructs his proof has him at one point assuming that primes are n, a precise number, multiplying all these numbers and getting a very precise number, and then adding 1 to it and attempting to divide that number by each of the n primes and seeing that there is a residue of 1. Conversely, narratives can involve generalities. A general statement at the human scale is that people must obey a divine decree. In a story, particularly ancient ones, you often see statements like that. Especially in tragedy, you have this sense of the narrative moving through a world of divine decrees. You can have both levels of realities—divine and human–and the story itself tracks a course between the general and the particular, which makes it dramatic.
Herman: If we have time, I had a few questions about your concept of outlines.
Doxiadis: Let’s use outlines to address the concept of independence of levels. As a practicing writer I do outlines, like a painter who is painting big paintings; a novel, too, is big in scope in temporal or spatial scope, and like a painter when I want to understand a composition I cannot do that working on the detail but have to move away. In Arts school students are constantly told to move away, to hang the canvas upside down or turn down the lights or do other things that give them another perspective. Likewise, when I do long texts, I always do outlines and I have discovered that there is nothing like the experience of outlines to show to you certain aspects of a text that cannot be seen when you are working on the detail.
Let’s take as an example a crucial scene from a known narrative—say, the killing of Clytemnestra in The Libation Bearers by her son Orestes. If I say that in the central scene of that play Orestes kills Clytemnestra, I have produced a very rough outline. The way that scene is written will not change that outline. If we go into Sophocles’ Electra, which is a legend, that scene is not in the play. We hear about it but we don’t see it. But in the fabula, there is a scene called “Orestes kills Clytemnestra.” That does not constrain either Aeschylus or Sophocles when it comes to writing it–not even the decision of whether or not to include it visually in the play. So, if we get into the writing of that scene, there are other choices. There is a very moving moment in The Libation Bearers where Clytemnestra bears her breast and says “strike over this breast which fed you when you were a baby”. Orestes was not moved. Sophocles was so far removed from this level of detail as not to include the scene. The independence of levels would tell us that any such decision pertaining to the writing of this scene doesn’t affect its structural importance as a basic element of the play. So outlining, here outlining the fabula, tells you something that is important in itself, and much more important than any of the details realized there. This is a precise example of the independence of levels.
The independence of levels is perhaps best incarnated in my paper by the discussion in section 5.2 of moving from one city to the next. These two cities are only connected by a line. When you are leaving here to go there, you are only searching to get to this highway. What is important is to get to the highway. If you get to this highway, you are done because you’ll waste two hours going to another city. Once you are there, it doesn’t matter how long you searched to find it. All that matters is that you are on it. It depends on levels. This whole approach comes from my experience as a writer. Often, I have a sense that a given problem is at a detailed level, at a medium level, or at a rough outline level. Of course, this is a statistical. Because, if at the moment when Orestes is about to put his knife into Clytemnestra’s chest, Orestes changes his mind, and says ‘Ok, live Mother’ and departs, that is a very detailed action. It’s two lines. Yet because of it the whole play would come apart: the whole structure would be affected. That kind of detail is carried over in any outline. But I should note that this is an exception vis-à-vis independence of levels. Usually, levels operate independently in story. And the same in proofs.
Herman: Could you just define levels in this context?
Doxiadis: Yes. Degree of resolution. The cinema is the best example. Close shot. Long shot. If for example, a close shot is a very analytical level showing reactions and facial expressions; however, if you moved away in a battle scene, you would just see the two battles, not the soldier’s eyes. I would define it perhaps as the ratio of the degree of detail in what is narrated to the degree of detail in the original fabula.
Herman: And how exactly would this notion map onto proofs?
Doxiadis: Let’s not talk about proofs in the abstract but in the contextualized proof, just as we are talking about specific stories. I think in your definition of story, in the talk accompanying your own paper for the volume, you suggest that narratives involve events that must be interpreted in a light of a particular context of telling. Let’s speak of a proof in the same sense because often when people talk of proofs it’s only an ideal, platonic sense of proof, which never occurs in reality. In fact, one is always dealing with a proof article, one mathematician talking to another mathematician. Whatever.
Herman: So, proofs are interactionally tailored.
Doxiadis: Yes, when they are published. At this level, mathematicians may assume a certain level of sophistication, and also some shared knowledge and basic principles. Plus, once it’s published, I do not have to repeat the proof. My interlocutor or reader can go and look it up. In short, when you are explaining a proof the degree of knowledge available to your audience can vary. In developing a proof, you can say, ‘Look. There’s a crucial step here where we jump from modular forms to elliptic curves called the Taniyama – Shimura Conjecture, which Wiles proved in an immensely complex and difficult proof. OK. Forget it; you cannot understand it. Let’s use it to go from here to there.” That tale scopes twenty years of the history of the field to one line. You do the same thing in narratives all the time. A narrator might say, “She looked at the window. And a tear flowed down her cheek. She closed shut the window. Ten years passed. One morning in October, a bell rang.” You can talk of those ten years in two seconds. And we have the ability both to compose and interpret a story at all levels at once. It’s amazing that a child, having listened to a very detailed story, can describe it to you in a few words. It’s the same with proofs. I can tell you an outline and if I know the details of Wiles’ proof, I could detail it to you in one page or 20 pages or 1,000 pages.