Mathematics And Literature
by Andrew Crumey
Published in MathKnow
(proceedings of the conference on Mathematics, Applied Science and Real Life; Politecnico di Milano, Italy, May 2008)
ed. Michele Emmer and Alfio Quarteroni (Springer, 2009)
Abstract. Euclid's Elements is not a novel - but it could have been. Mathematics differs in obvious ways from conventional artistic literature, yet there are also similarities, explored here through writers including Plato, Galileo, Edgar Allan Poe and Lewis Carroll. By considering possible definitions of what a novel is - using ideas from E. M. Forster, Mikhail Bakhtin and Gérard Genette - it is argued that the fundamental difference between conventional mathematical and artistic literature is one of form rather than content.
When we compare mathematics and literature we can immediately think of differences. Mathematics is typically seen as abstract, remote from everyday experience and emotion; whereas literature is viewed as the opposite. Mathematics is logical and analytic; literature is intuitive and expressive.
To most people, the difference is apparent simply from comparing the appearance of a mathematics text with a literary one. There is a "language" of mathematics with its own symbols and terminology, mysterious to non-specialists, while most literary works are written in language which, if not always of the "everyday" kind, is at least familiar. In artistic texts such as novels or poetry, we find that the particular words a writer uses are of great importance to the aesthetic effect: it is often remarked that poetry, in particular, loses something in translation. With mathematics, the situation is quite different: we could even say that mathematics is concerned precisely with those things that are invariant under linguistic translation. In that sense there is not really a "language" of mathematics; rather, mathematics is an abstraction of whatever can be said equally well in any natural language.
So much for differences. Yet when we look at mathematics and literature as human activities, there are obvious similarities. Writers, like mathematicians, spend a lot of time sitting at their desk, trying to come up with a good idea. They wrestle with problems existing only in the mind, have moments of inspiration, try to work out the implications that follow, and find yet more ideas. Both disciplines have a corpus of "classic" works which we can go and find in a library.
There is a deeper and more philosophical connection, and it concerns the existence of those objects that the writer or mathematician deals with: the question of ontology. Hamlet, for instance, in Shakespeare's play, sees the ghost of his father, and we can ask: is the ghost real? One answer is yes: the play is set in a world where ghosts exist. Another is no: the play is set in our world, and the ghost is an illusion. Another is that there is no ghost, and no Hamlet - none of the characters is real. And what about those other characters who inhabit mathematics, the number 2, say? Is that a real thing, or else (as Bertrand Russell observed), an idealisation of the property common to a pair of socks, a married couple, a brace of pheasants and a deuce of hearts?
We should also examine more closely the notion that mathematics is purely logical while literature is intuitive. Mathematicians themselves have long taken issue with this: mathematics can also be intuitive, as Henri Poincaré (1854-1912) emphasised. Poincaré reckoned there are two sorts of mathematician :
The one sort are above all preoccupied with logic... The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests... Though one often says of the first that they are analysts and calls the others geometers, that does not prevent the one sort from remaining analysts even when they work at geometry, while the others are still geometers even when they occupy themselves with pure analysis. It is the very nature of their mind which makes them logicians or intuitionalists... The mathematician is born, not made, and it seems he is born a geometer or an analyst.
Poincaré maintained that great mathematicians could be of either sort: as intuitionalists he cited Lie and Riemann; his logicians included Hermite and Weierstrass. But Poincaré made a special plea for intuition, saying that rigour alone could not suffice.
[In] becoming rigorous, mathematical science takes a character so artificial as to strike every one; it forgets its historical origins; we see how the questions can be answered, we no longer see how and why they are put.
This shows us that logic is not enough; that the science of demonstration is not all science and that intuition must retain its role as complement, I was about to say, as counterpoise or as antidote of logic.
So among mathematicians there has always been a sense of needing to strike a balance between intuition and logic; and because outsiders tend to see only the logical side of mathematics, mathematicians themselves are quite keen to highlight the intuitive aspect.
Now what about literature? There is a long history of seeing the irrational, the intuitive, as being the essence of literary genius, often viewed as a kind of contact with divine forces, even verging on madness: the poet as prophet. Immanuel Kant defined genius as "the innate mental predisposition (ingenium) through which nature gives the rule to art" . In other words, genius is the way in which some people can directly apprehend truths about nature, without the need for logical deduction.
In this way of thinking, logic can take us only so far, then genius has to take over. Using a sequence of logical steps it's possible to produce a work of art that is beautiful, but genius can carry us beyond beauty. This state beyond beauty is called the sublime.
According to Edmund Burke (1729-97), beauty is what gives us pleasure, but the sublime is associated with fear . Well-tended gardens, properly proportioned buildings - these are beautiful. A storm at sea, wild mountains, the infinity of space, the thought of death - these are sublime. Artists of the Romantic movement became preoccupied with these sublime themes, and artists themselves were increasingly seen as heroic figures, delving into areas of experience unavailable to lesser mortals. This was the image of Byron or Beethoven - people with wild hair, furiously scribbling away at divinely inspired work penetrating the deepest mysteries of the universe.
The popular view of the artist as entirely intuitive was naturally one that some artists would react against. One of these was Edgar Allan Poe (1809-49), who was very keen to point out the degree of calculation that can be involved in creating art. Poe was a psychologically troubled alcoholic whose work was largely neglected during his lifetime (his one real success was his poem 'The Raven'), and to that extent he fits a certain kind of artistic stereotype. But Poe was also fascinated by the idea of logical deduction - we remember him as the inventor of the detective story. In the last two years of his short life he became passionately interested in astronomy and cosmology, and his final book, Eureka, presents his theory of the universe, which Poe saw as being made of both "matter" and "spirit" . His concern with the latter immediately put his book into the category of crank science, and it was ignored by the scientific community.
In fact, however, Poe had come up with what is now recognised as the first valid explanation for why the night sky is dark. It was thought at the time that the universe was infinitely old, filled with infinitely many randomly distributed stars. But in that case, wherever you look in the sky, your line of sight should be directed at a star, and the whole sky should glow with starlight. The problem was first noticed by Kepler, who guessed that the glow was too faint to notice, but Olbers calculated that it ought to be very bright indeed, so the problem is known as Olbers' Paradox. The standard explanation in Poe's day was that there must be dust blocking the starlight: it wasn't realised that this dust would simply heat up and re-radiate the light.
Poe's solution was that the universe must be of finite age, so that there are stars still too far away for their light to have reached us, and he proposed that it all began from an explosion - so we could call him the father of the Big Bang. Poe's work was forgotten by astronomers, and it was not until 1987 that his contribution began to be acknowledged in scientific literature, with the publication of Edward Harrison's Darkness At Night .
Nineteenth-century astronomers were of course right to dismiss his theory: it implied that the universe was expanding, but in the 1840s there was absolutely no evidence of that. Furthermore, Poe suggested that the force that caused the expansion was light - something that didn't match with physics as it was then understood. But his theory was not simply rejected for those reasons. It was also ignored because it wasn't presented in the right style.
A couple of years before Eureka, Poe wrote a much shorter essay that in some ways is just as remarkable. This is 'The Philosophy Of Composition' , in which he describes how he composed his poem 'The Raven'.
It is my design to render it manifest that no one point in its composition is referable either to accident or intuition - that the work proceeded step by step, to its completion, with the precision and rigid consequence of a mathematical problem.
This is a complete reaction against the idea of art as being intuitive: Poe claims his poem was carefully calculated, down to the smallest detail. He goes on to describe how he arrived at his chosen length:
[The] extent of a poem may be made to bear mathematical relation to its merit... for it is clear that the brevity must be in direct ratio of the intensity of the intended effect - this, with one proviso - that a certain degree of duration is absolutely requisite for the production of any effect at all.
Holding in view these considerations... I reached at once what I conceived the proper length for my intended poem - a length of about one hundred lines. It is, in fact, a hundred and eight.
Many people have wondered if Poe was being serious, or if the whole essay should be treated as an elaborate hoax. But although there is certainly a great deal of irony and mischief in the essay, I think Poe's underlying point is a valid one. Literary composition is not simply a matter of inspiration; there is also something deliberately calculated and logical about it.
So we have two nicely contrasting cases: Poe highlighting the role of logic in literary art, and Poincaré, half a century later, emphasising the importance of intuition in mathematics. And by Poincaré's day, it wasn't only artists who were seen as wild-haired seers, probing the ineffable mysteries of existence: Einstein came to be seen that way too.
We cannot split mathematics and literature neatly apart by saying that one is logical and the other intuitive. And of course there is a "literature" of mathematics, containing works such as Euclid's Elements. So we want to ask how a work like that differs from what we ordinarily regard as artistic literature, meaning novels, poetry and so on. And since I am a novelist and not a poet, I shall stick to asking how Euclid's Elements differs from a novel.
E. M. Forster defined a novel to be "a fiction in prose of a certain extent", suggesting 50,000 words as the lower limit . Something that just makes it past this mark is F. Scott Fitzgerald's The Great Gatsby (50,061 words), which everyone certainly thinks of as a novel, while Henry James's The Turn Of The Screw, always called a novella, duly pitches in at a mere 43,380. So perhaps Forster positioned the bar more accurately than he realised. But is Joseph Conrad's Heart Of Darkness (51,011 words) a novel or a novella? And what about the paltry 32,535 words of H.G. Wells's The Time Machine?
The simplest and perhaps best way of thinking about length was put forward by Poe, in 'The Philosophy Of Composition', and another essay, 'The Poetic Principle' . There are some things that we can read from start to finish in a single sitting, and others that we need to leave and come back to. We can't put a definite word-count on it, but we can say that novels are a kind of book meant to be read in more than one sitting.
Euclid's Elements is long enough to be a novel, but is it "prose"? Mathematics nowadays looks nothing like ordinary written language, but the symbols of mathematics are relatively recent. The equal-sign was first introduced by Robert Recorde in 1557.
The earliest surviving manuscript of Euclid's Elements dates from 888AD (over a thousand years after Euclid wrote it), and the text is written continuously, with diagrams added in the margins by the copyist as an aid to understanding. The numbering and cross-referencing of definitions, propositions etc. was done by later editors: the only symbolism Euclid employed was the use of letters to label unknown quantities. So the first proposition of Book One runs :
On a given finite straight line to construct an equilateral triangle. Let AB be the given finite straight line. Thus it is required to construct an equilateral triangle on the straight line AB. With centre A and distance AB let the circle BCD be described; again, with centre B and distance BA let the circle ACE be described; and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined. Now, since the point A is the centre of the circle CDB, AC is equal to AB. Again, since the point B is the centre of the circle CAE, BC is equal to BA. But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB. And things which are equal to the same thing are also equal to one another; therefore CA is also equal to CB. Therefore the three straight lines CA, AB, BC are equal to one another. Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB, as required.
This is prose, so Euclid's Elements would appear to pass at least two out of our three tests of novel-hood.
Now let's consider another Greek writer, Plato, who wrote fifty to a hundred years before Euclid. The earliest surviving source for much of Plato's work is the Clark Codex (in the Bodleian Library), dating from 895AD.
There are no diagrams in Plato, but we do find some mathematics. The dialogue Meno consists of a conversation between Socrates and Meno, starting with Meno asking whether virtue is something innate or learned. In the course of the discussion, Socrates asks to speak to a slave boy in Meno's household, and he poses a mathematical problem. Socrates draws a square on the ground (of side 2 units, i.e. area 4 units2) and asks the boy to draw a square with double the area (i.e. 8 units2). At first the boy guesses that he has to double the sides of the square, but Socrates draws a diagram to show that this is wrong. We aren't shown the diagram in the text; all we get is the ensuing conversation. Here is part of it :
Socrates: Here, then, there are four equal spaces?
Socrates: And how many times larger is this space than this other?
Boy: Four times.
Socrates: But it ought to have been twice only, as you will remember.
Socrates: And does not this line, reaching from corner to corner, bisect each of these spaces?
Socrates: And are there not here four equal lines which contain this space?
Boy: There are.
Socrates: Look and see how much this space is.
Boy: I do not understand.
Socrates: Has not each interior line cut off half of the four spaces?
Socrates: And how many spaces are there in this section?
Socrates: And how many in this? Boy: Two.
Socrates: And four is how many times two?
Socrates: And this space is of how many [square] feet?
Boy: Of eight [square] feet.
Socrates: And from what line do you get this figure?
Boy: From this.
Socrates: That is, from the line which extends from corner to corner of the figure of four [square] feet?
As with Euclid's Elements, it's a lot simpler if we can see the diagram:
The point to notice, though, is that Plato presents the argument as a dialogue between two people. Euclid, writing decades later, could have written the whole of the Elements in the same style, but he chose not to. Plato gives us a dramatic representation of ideas; Euclid gives us something more like a lecture. Plato's style is "dialogic", Euclid's is "monologic".
The twentieth-century literary theorist Mikhail Bakhtin (1895-1975) maintained that the real defining feature of novels is dialogism, as opposed to monologism . Euclid's Elements, by this reckoning, is not a novel; Plato's Meno is (except perhaps for the fact that it's rather short).
Does mathematics have to be presented monologically? We've already seen from Plato that the answer is no. Nevertheless, Euclid's monologic style became standard in mathematics and physics; Ptolemy's Almagest was written in the Euclidean way . But the work that eventually helped overturn Ptolemy was a fictional dialogue: Galileo's Dialogue On The Two Chief World Systems, which portrayed the real-life figures Salviati (a Copernican) and Sagredo, along with the Ptolemaist Simplicio .
Why did Ptolemy choose a monologic style and Galileo a dialogic one? The monologic style says: here is the true and final answer. The dialogic style says: here is something which might be true - but you have to make up your own mind. Galileo no doubt thought it safer to use the latter style.
Let's look at a bit more of Meno. The slave boy is led through the geometrical problem with a little prompting from Socrates, but essentially he is encouraged to work it out for himself - he can see that the construction works. Socrates then speaks to Meno :
Socrates: What do you say of him, Meno? Were not all these answers given out of his own head?
Meno: Yes, they were all his own.
Socrates: And yet, as we were just now saying, he did not know?
Socrates: But still he had in him those notions of his - had he not?
Socrates: Then he who does not know may still have true notions of that which he does not know? Meno: He has. Socrates: And at present these notions have just been stirred up in him, as in a dream; but if he were frequently asked the same questions, in different forms, he would know as well as any one at last?
Meno: I dare say.
Socrates: Without any one teaching him he will recover his knowledge for himself, if he is only asked questions?
Socrates: And this spontaneous recovery of knowledge in him is recollection?
Socrates is using exactly the same method of questioning on Meno, only this time what he is drawing out is an admission that the boy must somehow already have known the geometrical solution:
Socrates: ...Now, has any one ever taught him all this? You must know about him, if, as you say, he was born and bred in your house.
Meno: And I am certain that no one ever did teach him.
Socrates: And yet he has the knowledge?
Meno: The fact, Socrates, is undeniable.
Socrates: But if he did not acquire the knowledge in this life, then he must have had and learned it at some other time?
Meno: Clearly he must.
Socrates: Which must have been the time when he was not a man?
Socrates: And if there have been always true thoughts in him, both at the time when he was and was not a man, which only need to be awakened into knowledge by putting questions to him, his soul must have always possessed this knowledge, for he always either was or was not a man?
Socrates: And if the truth of all things always existed in the soul, then the soul is immortal.
So we are offered a "proof" of immortality, exactly like the proof that the constructed square had twice the area of the first. This should certainly make us suspicious of dialogism - even when Galileo uses it to make arguments about physics. In his Dialogue On The Two Chief World Systems, we find this discussion of tides :
Simplicio: Lately a certain clergyman has published a small treatise in which he says that, as the moon moves through the sky, it attracts and raises toward itself a bulge of water which constantly follows it...
Sagredo: Please, Simplicio, do not tell us any more, for I do not think it is worthwhile to take the time to recount them or waste words to confute them...
Salviati: I am calmer than you, Sagredo, and so I will expend fifty words for the sake of Simplicio... To that clergyman you can say that the moon every day comes over the whole Mediterranean, but that the waters rise only at its eastern end and here for us in Venice.
The clergyman was Marcantonio de Dominis (1566-1624), Archbishop of Split, and he was of course right. Galileo wrongly believed Earth's tides to be caused by the planet's motion, and took the lack of significant tides in the Mediterranean as disproof of any lunar influence.
So we might say that the reason why mathematicians prefer the monologic style is that dialogue is rhetorical and untrustworthy. But what about Euclid's Elements and its "self-evident" postulates, such as the notorious Fifth, saying that parallel lines never meet? As we know, there can be non-Euclidean geometries in which they do. Monologism appears more authoritative, but need not actually be so.
Dialogue was basic to Socrates' way of doing philosophy (at least as we understand it from Plato). One of his favourite rhetorical techniques was to feign ignorance, asking questions that he pretended not to have an answer to. The Greek word for this was eironeia, from which we get the word "irony".
In everyday usage, irony is saying one thing while meaning something else, so that both meanings are conveyed, as in, "what a fine state you're in!" (meaning you're in a very un-fine one). Irony is associated with multiple mental states. Dialogism represents these multiple states, whereas monologism presents a single state. Euclid's Elements cannot be read ironically, whereas Plato's Meno can. We don't know if the real-life Socrates would have agreed with the ideas put into his mouth by Plato, and we don't even know if Plato believed them. We are put into a state of puzzlement - aporia. What we usually find in mathematics is a different kind of puzzlement - if we can't understand it then the fault must be our own stupidity, not the problem itself.
Even so, we can find irony and aporia in mathematics. The first master of this was Zeno of Elea (c490-c430BC), with his famous paradoxes, such as the one about Achilles' race with the Tortoise. Achilles gives the Tortoise a head-start, but before he can catch up with the Tortoise he must reach the half-way point between them. By the time he reaches it, the Tortoise has already moved further; and so on. The paradox is easily resolved if we admit the summation of infinite series - but what if we don't?
Zeno's spirit disappeared from mathematics until the late nineteenth century, when new paradoxes of logic and set theory began to emerge. One of these was proposed by Charles Lutwidge Dodgson (1832-98), lecturer in mathematics at Christ Church, Oxford, who is better known as Lewis Carroll, author of Alice In Wonderland.
Let's look again at Euclid's first proposition, quoted earlier. This was Lewis Carroll's starting point, in a dialogue published (under his real name) in the journal Mind in 1895, with the title 'What the Tortoise Said To Achilles' .
"That beautiful First Proposition of Euclid!" the Tortoise murmured dreamily... "Well, now, let's take a little bit of the argument..."
(A) Things that are equal to the same are equal to each other.
(B) The two sides of the Triangle are things that are equal to the same.
(Z) The two sides of this Triangle are equal to each other.
"Readers of Euclid will grant, I suppose, that Z follows logically from A and B... [However] I want you [Achilles]... to force me, logically, to accept Z as true."
"I'm to force you to accept Z, am I?" Achilles said musingly. "And your present position is that you accept A and B... but you don't accept:
(C) If A and B are true, Z must be true."
"That is my present position," said the Tortoise.
"Then I must ask you to accept C."
"I'll do so," said the Tortoise, "as soon as you've entered it in that note-book of yours... Now write as I dictate:
(A) Things that are equal to the same are equal to each other.
(B) The two sides of this triangle are things that are equal to the same.
(C) If A and B are true, Z must be true.
(Z) The two sides of this Triangle are equal to each other."
..."If A and B and C are true, Z must be true," the Tortoise thoughtfully repeated. "That's another Hypothetical, isn't it? And, if I failed to see its truth, I might accept A and B and C, and still not accept Z, mightn't I? ... [But] I'm quite willing to grant it, as soon as you've written it down. We will call it
(D) If A and B and C are true, Z must be true."
We can see where it goes: Achilles can never state all the logically necessary steps of the argument, because there are infinitely many of them. It's just like Zeno's paradox. Bertrand Russell answered it by making a distinction between implication (if p, then q) and inference (p therefore q) - but what if we don't accept that distinction?
We could say that Lewis Carroll's paradox shows there is always a step beyond logic, a step that has to be intuitive - a leap from premise to conclusion. And if we cannot make this step intuitively, we simply accept it, on the writer's authority. But in that case, buried inside even the most rigorous mathematics, it would appear that there is a rhetorical element, appealing to our beliefs. Then mathematics is not so completely remote from other forms of literature as we might suppose.
Poincaré had no problem with the idea of intuition in mathematics, but his style encouraged a lack of rigour, against which there was a reaction in France by the self-styled "Bourbaki" group, whose aim was to create a series of textbooks that were completely rigorous, formal and abstract.
Curiously, alongside this desire for formalism in mathematics, there was also a school of formalism in the theory of the novel. These Formalists were active particularly in Russia, and included Vladimir Propp (1895-1970), who wrote a book called Morphology Of The Folk Tale, in which he analysed the structure of Russian oral stories, finding standard character types and plot sequences that he could classify . Propp was interested in narrative structure, not the particular way in which narrative is presented, but another Formalist, Roman Jakobson (1896-1982) looked specifically at language, isolating its "functions" and classifying them according to how they are oriented (for example towards a listener, towards oneself, towards establishing contact, etc.) .
The Formalist view was that language has structure, and that literature has analogous structures at a higher level. This way of thinking influenced people in other disciplines, in particular the anthropologist Claude Lévi-Strauss (b1908), who initially looked for structures in the culture of Amazonian tribes-people. Jakobson and Lévi-Strauss are considered the founders of Structuralism, which took the idea of linguistic structure and applied it to culture in general.
At the same time, the Bourbaki group were trying to build up mathematics from the basic structures of set theory. A leading member of the group was André Weil (1906-1998), and he met Lévi-Strauss in New York in 1943, where both had fled from German-occupied France. They worked together on the kinship structures of a group of aboriginal Australians called the Murngin, which had a four-caste system with rules of who could marry whom. Weil did an analysis of it using group theory, published as an appendix to Lévi-Strauss's book The Elementary Structures Of Kinship .
In literature, Gérard Genette (b1930) looked for structures in the work of Marcel Proust, presenting his work in Narrative Discourse . We can see the simplest of these structures by looking at four sentences:
1. I left at dawn and got back at dusk.
2. "Hello, Sally, it's lovely to see you!"
3. The bomb exploded in a brilliant blue-white flash that sent a searing wave of pressure hurtling across the room.
4. You can't fit an elephant inside a Mini.
These four sentences each treat time in a different way. The first three represent the passing of a certain amount of time while the fourth doesn't represent time at all; it states a "timeless fact". Let's denote by dC the "duration of character time", and by dR the "duration of reader time". Then we can analyse the sentences as follows:
1. dC/dR > 1 (Event takes longer to happen than to describe).
2. dC/dR = 1 (Takes the same time to narrate as to happen).
3. 0 < dC/dR < 1 (Description lasts longer than the event).
4. dC/dR = 0 (No passing of character time).
Genette used different terminology and notation (which I have modified for unity and compactness), and although he was not the first person to observe the difference between what I have called character time and reader time, he was, I think, the first to present a systematic classification - one I find useful in the teaching of creative writing. I don't show my students groups of inequalities, but I do tell them that passages of type 1 are called "summary", type 2 is "scene", type 3 is "slow-motion" and type 4 is "pause". Variation between different types creates narrative "rhythm", with type 2 tending to be a "showing" (or mimetic) mode, imitative of real time, while types 1 and 4 tend to be more "telling" (or diegetic), and type 3 is something of a "special effect". What we usually find in conventionally written novels is a rhythm that moves between these types.
It is also possible to leap forward discontinuously in time; for example, "I went on holiday. I came back." This is called ellipsis, and in our notation it corresponds to:
5. dC/dR = ∞ (Passing of character time left undescribed: dR = 0).
What Genette did not explicitly note, but which is clear from the notation introduced here, is that narrative rhythm is determined by variations in the ratio of the C- and R- timescales; we can think of this ratio as a scale factor. And while Genette's analysis covered all positive values of this ratio (the five cases given above), we could also wonder about negative values. These would arise if the story were told "backwards" - a special effect rarely used in fiction, though an example of "negative summary" occurs in Kurt Vonnegut's novel Slaughterhouse-Five . Far more common are various kinds of "flashback"; for example, "Today I am happy. Yesterday I was sad". We could see the discontinuity between the two sentences as "negative ellipsis" with dC/dR = -∞.
As well as duration, Genette partially classified another important time aspect. Consider the following:
1. I went to bed early.
2. Every Sunday night I go to bed early.
3. I went to bed. Yes, I went to bed early. I had to go to bed early as it was Sunday.
These sentences differ with respect to frequency. Again introducing notation different from Genette's, let's define fC to be the number of times that a given event occurs in character time, and fR the number of times it is narrated. Then we can analyse the sentences as:
1. fC = fR = 1 (It happens once and is narrated once).
2. fC > 1, fR = 1 (It happens many times but is narrated once).
3. fC = 1, fR > 1 (It happens once but is narrated many times).
The names for types 1 and 2 are "singulative" and "iterative" narration; type 3 can simply be termed "repetition". Genette's interesting observation was that Proust's style is typified by sentences of type 2: Proust's multi-volume novel À la recherche du temps perdu begins, "Longtemps, je me suis couché de bonne heure"  ("For a long time I used to go to bed early" ). Many other novels and stories begin iteratively (e.g. Don Quixote, and most fairy-tales), but usually they become singulative as soon as the "action" begins. What is unusual in Proust is that he sustains iteration, representing the past in a radically new way.
As with duration, we could extend our consideration of frequency to include other possibilities not explicitly covered by Genette. Most are pathological (e.g. fC = 0, fR > 0, it is narrated but never happened); however, one case is of practical interest:
4. fC > 0, fR = 0 (It happens but is not narrated).
This could be seen as simply another way of classifying ellipsis; but the implication here of repeated instances of an undescribed event suggests it is ellipsis of a different kind. I suggest it corresponds to what Genette termed "paralipsis"; not simply a skipping of time, but a "putting aside" of information (such as, in Proust's novel, the narrator's first experiences of love with a girl in Combray, omitted from the chronological narrative and only alluded to retrospectively).
We see that narration is a kind of "mapping" from "character world" to "reader world", giving rise to the representation of time. (Dubbing the "map" M, we could write M(dC) = dR, M(fC) = fR.) We might even see this as a basic defining feature of narrative - not only novels, but also folk tales, films etc. Euclid's Elements does not represent time in any obvious way, whereas the mathematical proof in Meno does.
Yet novels represent lots of other things apart from time. The most obvious of these are character, setting, plot - but long before we reach such "high-level" representations, there are still many at a more elementary level. For example:
1. She looked inside the box and was horrified by what she saw.
2. There was a snake inside the box. Mary unwittingly opened it.
In (1), Mary knows what is inside the box but the reader does not. In (2), the reader knows but Mary doesn't. If we denote by kC and kR the character's knowledge and the reader's knowledge, then we can informally write:
1. kC > kR.
2. kC < kR.
Type 1 creates "suspense", type 2 is a kind of "omniscient narration". We might choose to call these two types of knowledge representation "hypergnosis" and "hypognosis" - but narrative theory already has more than enough terminology, and I don't propose to add to it. In any case, what usually happens in narratives is that we get far more complex representations of knowledge. For example:
Sally met Mary's boyfriend, John. "I'm going to play a trick on Mary," he said. He was going to hide a toy snake in a box. Sally thought - he doesn't know how terrified of reptiles his stupid girlfriend really is - the shock could kill her. Hmm, what if the snake were not a toy...?
There are three states of knowledge represented here: those of Sally, John and Mary. Each of them knows some things that the others don't, but which we can figure out from the text:
John knows: he has a girlfriend Mary and a toy snake...
He doesn't know: Mary is mortally terrified of snakes, Sally hates Mary...
Mary knows: she has a boyfriend John...
She doesn't know: John wants to play a trick on her, Sally wants to kill her...
Sally knows: Mary hates snakes; Sally hates Mary...
Sally doesn't know (initially): John is going to play a trick.
Notice that there is something special about Sally's state of knowledge in relation to the others: the text is oriented around the change in this state. The idea that changes of knowledge are crucial to the way plots progress was first pointed out by Aristotle ; but the idea that different people's states of knowledge can be represented to differing degrees was only first stated explicitly in the nineteenth century by Henry James . In the narrative above, we see things from Sally's "point of view". In James's terminology, Sally is the "reflector", or, in Genette's terminology, the "focaliser".
Could we generalise kC to deal with this? We would need kSally, kMary and kJohn; and we would need to take into account the objects of their knowledge (snakes, tricks, murderous impulses). It would all get very complicated - and we can see that the reason for this complication is that although narrative represents human knowledge and consciousness, human consciousness is itself a form of representation. Even in a passage as trivial as the one above, it is not enough to analyse only the representations of time and knowledge ("pace" and "view"); there is another kind of representation visible in the text, that of "voice", meaning linguistic register, or more generally, language itself. "Hmm, what if the snake were not a toy", is clearly meant to be Sally's unspoken words, not those of an objective narrator. This aspect of novel writing is the one that Bakhtin particularly emphasised. According to Bakhtin, there are some literary forms that present a single stable narrative voice, while others present many voices. And among those that present many, there are some in which the voices are merely quoted (as in a newspaper article), while in others the voices assume full and equal authority, permeating the fabric of the text. This is what Bakhtin meant by "dialogism", and we can see it in a typical passage from Jane Austen's Pride and Prejudice . Here, Mr Bennet is visited by Collins, an obsequious clergyman in the employ of the imperious Lady Catherine:
During dinner, Mr. Bennet scarcely spoke at all; but when the servants were withdrawn, he thought it time to have some conversation with his guest, and therefore started a subject in which he expected him to shine, by observing that he seemed very fortunate in his patroness. Lady Catherine de Bourgh's attention to his wishes, and consideration for his comfort, appeared very remarkable. Mr. Bennet could not have chosen better. Mr. Collins was eloquent in her praise. The subject elevated him to more than usual solemnity of manner, and with a most important aspect he protested that he had never in his life witnessed such behaviour in a person of rank -- such affability and condescension, as he had himself experienced from Lady Catherine. She had been graciously pleased to approve of both the discourses which he had already had the honour of preaching before her.
Pace, view and voice are all represented here in deliciously subtle ways. The opening is clearly summary ("during dinner"), but then there is a sudden slowing down, almost approaching scene. There is now something markedly mimetic about the writing, and it comes not so much from the time aspect as from the other two, knowledge and register. The passage appears oriented around Mr Bennett's knowledge, his "point of view", but what about, "The subject elevated him [Collins] to more than usual solemnity of manner"? The fulsomeness of register is Collins', not Bennett's - so are we reading Bennet's view of Collins, or Collins' view of Collins, or Austen's view of him, or "the narrator's" view? Lady Catherine "had been graciously pleased to approve": are those Collins' words, or Lady Catherine's? As with Hamlet's ghost, the interpretations are endless, there is no final answer. This instability - or rather, richness - is what shows Austen to be a great writer, even in a passage as short as this - one that the average reader will skip through in seconds, subliminally aware of its beauty without necessarily understanding how it arises.
Bakhtin said that monologic narratives (such as epic poetry) were "Ptolemaic", whereas dialogic novels (such as Austen's) gave rise to a "Copernican revolution". Mixing his metaphors, Bakhtin said the text becomes "relativised". What we have seen is that this "relativism" is basic to artistic literature (in particular, novels), and it is made possible because of the "representational" character of the art form. If we want to define what fiction is, we see that it is about something more fundamental than plot, character or setting (things we find in movies, biographies and even newspaper articles). Nor is fiction simply a matter of being untrue (lies can be found everywhere). No, the essence of fiction, I have argued, is that involves the representation of pace, view and voice.
By that reckoning, Euclid's Elements is a prose work of novel-length, but it is not a novel. Nor can any subsequent mathematical work written in the Euclidean manner be considered a novel. But that is a matter of historical choice, not necessity. The paradoxes of Zeno or Lewis Carroll; the dialogues of Plato or Galileo; the thought-experiments of Einstein and Bohr - all of these indicate that mathematical or scientific thinkers can, when they choose, adopt dialogic narrative rather than monologic discourse, depending on rhetorical need.
Mathematics, according to an often-repeated remark attributed to Gauss, is the "queen of sciences"; and when D'Alembert classified scientific knowledge in the Encyclopédie, he placed mathematics at the foundation. The claim of theoretical physics to be the most "fundamental" science rests on its being the most mathematical, the most abstract, the most remote from everyday experience - as far as possible, in fact, from what we might consider the normal domain of artistic literature. Yet for Bakhtin, the novel is the most universal art form, precisely because of its ability to absorb all linguistic genres. We can find poems inside novels, or discourses on history, politics, topography, cookery... There is no reason at all why we shouldn't find mathematics there too.
1. Aristotle, Heath M. (trans.): Poetics. Penguin, London (1996).
2. Austen, J.: Pride and Prejudice. Penguin, London (2003).
3. Bakhtin, M.M., Holquist, M. (ed.), Emerson, C. (trans.): Dialogic Imagination, University of Texas Press (1982).
4. Burke, Edmund: A Philosophical Enquiry into the Origin of Our Ideas of the Sublime and Beautiful, Oxford World's Classics (1998).
5. Dodgson, C.: What the Tortoise Said to Achilles, Mind, NS, Vol IV (April 1895), pp. 278-280. Reprinted in Wilson, R.: Lewis Carroll in Numberland. Allen Lane, London (2008).
6. Euclid, Heath T.L. (trans.): Elements. Cambridge University Press (1926).
7. Forster, E. M., Stallybrass, O. (ed.): Aspects Of The Novel. Penguin, London (1990).
8. Galileo, Finocchiaro, M.A. (trans., ed.): Galileo on the World Systems. University of California Press (1997).
9. Genette, G., Lewin, J. E. (trans.): Narrative Discourse: An Essay in Method, Cornell University Press (1983).
10. Harrison, E.: Darkness at Night: A Riddle of the Universe. Harvard University Press (1987).
11. Jakobson, R.: Linguistics and Poetics. In Sebeok, T. (ed.) Style in Language, pp. 350-377. M.I.T. Press, Cambridge, MA (1960).
12. James, H., Edel, L. (ed.): Literary Criticism: Essays on Literature, American Writers and English Writers. Library of America, New York (1984).
13. Kant, I., Meredith, J.C. (trans.): Critique of Judgment, Oxford University Press (2007).
14. Lévi-Strauss, C.: The Elementary Structures of Kinship. Beacon Press, Boston MA (1977).
15. Plato, Jowett, B. (trans.): The Dialogues of Plato. Oxford University Press (1924).
16. Poe, E.A.: Eureka. Hesperus, London (2002).
17. Poe, E.A., Thompson, G.R. (ed.): Essays and Reviews. Library of America, New York (1984).
18. Poincaré, H., Halsted, G.B. (trans.): The Value Of Science. Random House, New York (2001).
19. Propp, V.I., Wagner, L.A. (ed.): Morphology of the Folk Tale. University of Texas Press (1968).
20. Proust, M.: À la recherche du temps perdu. Gallimard, Paris (1987).
21. Proust, M., Scott Moncrieff, C. K. (trans.): Remembrance Of Things Past, Volume One. Wordsworth, London (2006)
22. Ptolemy, Toomer, G.J. (trans.): Almagest. Princeton University Press (1998).
23. Vonnegut, K.: Slaughterhouse-Five. Vintage, London (1991).