Numbered Days: Literature, Mathematics and the Deus Ex Machina
Think French. Think genius. Think rebellious, tormented, iconoclastic. Finally, think dead tragically young in the nineteenth century… And if you’re thinking of anyone at all, I think you’ll be thinking of Rimbaud.
And you’d be right to do so. But only half-right. Because there were in fact two rebellious, tormented, iconoclastic French geniuses who died tragically young in the nineteenth century. One was called Arthur Rimbaud (1854-91) and the other Évariste Galois (1811-32). Rimbaud is still famous, Galois never has been. At least not to the general educated public, though on all objective criteria – but one – you might expect his fame to be greater. In every way – but one – Galois has the more powerful appeal.
Because if Rimbaud had a stormy adolescence, alienating friends and relatives by his arrogance and ill-temper, Galois had a stormier, being expelled from school and imprisoned for his radical politics. And if Rimbaud died tragically and senselessly young, of cancer exacerbated by self-neglect at 37, Galois died more tragically and more senselessly younger, in a duel over “a girl he hardly knew” at 20.1 From which you might deduce that Galois was hetero where you probably already knew that Rimbaud was homo.2 Do you see what I mean about Galois having the more powerful appeal? In every way but one, that is. That’s because Galois was a mathematician where Rimbaud was a poet.
And that’s also, of course, all you need to explain why Rimbaud is famous and Galois is not. All the rest counts for nothing, because Rimbaud created literature and Galois created maths. The general educated public is only interested in one of those things.3 Why is that? Well, in part I think it’s because maths is harder than literature, but even if the two were equally easy or difficult I still think people would be more interested in literature than in maths. And I think that’s because maths holds a mirror up to the universe and literature holds a mirror up to the self.
You need to know something about Rimbaud’s life to understand and appreciate his writing, and the more you know, the more you understand and appreciate it. You don’t need to know anything at all about Galois to understand and appreciate his mathematics, and some books on what is now called Galois theory do not even explain who Galois was. In art, the creator and his creation are indissolubly linked and each sheds light on the other. In mathematics, the creator and his creation are distinct, and neither necessarily sheds light on the other. That is why, despite the strong parallels between their lives, Rimbaud is famous and Galois is not.
And I have listed only a few of those parallels, in which Rimbaud always seems a paler reflection of Galois. For Rimbaud too was shot in a lovers’ quarrel, in his case by his own lover Verlaine, but suffered only a superficial wrist wound. Galois was shot in the stomach by a rival for the affections of an “infamous coquette”,4 and died a day later of the wound. Rimbaud rejected Christianity by scrawling graffiti like “Merde à Dieu” – “Fuck God” – on park benches.5 Galois “refused the office of a priest” on his death-bed.6 Rimbaud’s muse fell silent after he turned twenty, but he lived on to receive, and ignore, the acclaim of the French literary world. Galois, as already remarked, was dead at twenty, and only then achieved the fame he had longed for. The parallels cease with what was perhaps Rimbaud’s seduction, perhaps his rape, by soldiers at a barracks in Paris, but that again is only to Galois’s advantage, for such an incident would not appeal directly to many people.
What does appeal directly to many people, of course, is the way in which the violent emotion aroused by it was turned into literature. Rimbaud’s poetry is strong meat:
Puisque les sens infects m’ont mis de leur victimes,
Je confesse de l’aveu des jeunes crimes!…
Pourquoi la puberté tardive et le malheur
Du gland tenace et trop consulté? Pourquoi l’ombre
Si lente au bas de ventre? et ces terreurs sans nombre
Comblant toujours la joie ainsi qu’un gravier noir?
Moi, j’ai toujours été stupéfait! Quoi savoir.
Because my infected senses have made me their victim,
I confess and affirm my youthful crimes! …
Why puberty so late, and the misery
Of a nagging prick too often attended? Why the darkness
So thick down in my guts? And the numberless fears
Always burying my joy like a black landslide?
I’ve always been bruised by life, never known what to do.7
Compare now a passage of Galois’s maths:
Suppose that K -> K’ is an isomorphism of fields. Let f be a polynomial over K and let S be any splitting field for f over K. Let L be any extension field of K’ such that i(f) splits over L. Then there exists a monomorphism j: SL such that j|k = i.8
I don’t understand that. If you do, you will not, I’m sure, find anything of yourself in it: there is no emotion there to recognize or reconstruct from your own experience, because mathematics, unlike poetry, is not built of emotion. Which is not to say that it cannot arouse emotion:
Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.9
Rimbaud’s poetry is great art, but not in the sense meant by Bertrand Russell there: it is beautiful but soiled, stained, encrusted, like gems and gold dredged up from black silt.10 It appeals to the weaker part of our nature because its themes – sex, fetishism, hallucination, psychosis – were inspired by the weaker part of Rimbaud’s nature. In fact, he deliberately weakened and tortured himself, describing his poetic technique in one letter as a conscious déreglement de tous les sens – “derangement of all the senses”.11 Hence his use of hashish and absinthe to twist and re-shape his mind and with it his language.
The language of poets and other writers can survive and be enhanced by hallucination and psychosis: other examples are too numerous and familiar to need restating here. This is not true for mathematicians, who can be and often have been unbalanced or insane, but who cannot allow this to disturb their completed work. Mathematics is built of logic and reason, and however valuable intuition and insight are in the discovery of mathematical truth, they can never be sufficient: mathematics is about what but also about why and how, and seeks to communicate the universal. Poetry, like other forms of art, is only about what, and seeks to communicate the personal.
Which is another reason for mathematics not appealing to the general educated public. We like an artist’s expression of the personal because we can feel that it is expressed on our behalf as human beings or, better still, as particular types of human being: Rimbaud’s poetry can be appreciated by many but will be most appreciated by young, rebellious, unhappy French homosexuals. Politics differs from art because it does not combine this appeal to the personal with an appeal to our aesthetic sense – or not very often, at least. It’s no coincidence, for example, that Nazism, the most powerful of all political movements, was created by a frustrated artist and mystic: Nazism combined art, politics and religion. The appeal of all three is, in a different way, to the ego.
The way of art, at first hand for the artist or second hand for the audience, is that of creation, and the ego that is flattered is that of the creator, the one who makes something that did not exist before and that, without him, could never have existed at all. Unless it is directly plagiarized, any work of art, however inferior, is in some way unique. This is not true in mathematics, where there are many examples of work being unconsciously duplicated by mathematicians sometimes working decades or continents apart. The independent invention of calculus by Newton and Leibniz is one example; the independent development of non-Euclidean geometry by Gauss, Bolyai, and Lobachevsky is another;12 the work of Galois himself is a third, because “some of his results had been independently obtained by Niels Henrik Abel”.13
Abel was a Norwegian mathematician who was born in 1802 and died tragically young in 1829. He also duplicated another mathematician’s work, “develop[ing] the concept of elliptic functions independently of Carl Gustav Jacobi”.14 Jacobi was a German mathematician who was born in 1804 and died tragically young in 1851. The early age at which Abel, Jacobi and Galois died probably reflects the medically primitive century in which they were born. The intermeshing of their work certainly reflects the field in which they worked. Parallel stories can be told of scientists who obtained identical independent results, and for the same reason: that science, like mathematics, is based on explorations of and discoveries in an objective, external world.15
It seems likely, for example, that intelligent life anywhere in the universe, however much it may differ from human beings in its biology and culture, will have developed mathematics in a recognizable and perhaps even identical way to us. We can draw this conclusion by extrapolation from what we know of our own planet. The algorithm that underlies the Fibonacci sequence – xn = xn-1 + xn-2 – is consciously exploited by human beings, and unconsciously exploited by plants. The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…
The daisy’s spiral ratio of 21:34 corresponds to two adjacent Fibonacci numbers, as do the pine cone’s 5:8 and the pineapple’s 8:13 – and the same is true of many other plants with a spiral leaf-growth pattern.16
That’s natural history: ordinary history demonstrates the universality of mathematics too: the theorem that a2 + b2 = c2 in a right-angled triangle was known to ancient cultures as different as the Greek and the Chinese. But an even more striking proof that mathematics is in some way independent of environment and culture was provided much more recently by Srinivasa Ramanujan (1887-1920), yet another mathematical genius who died tragically young.
Ramanujan was born in Madras, southern India, and was largely self-taught. He might have remained entirely unknown had it not been for the insight of the English mathematician G.H. Hardy (1877-1947), who recognized a potential in work Ramanujan sent to him that had been overlooked by two of his colleagues at Cambridge. Parochialism and snobbery were probably responsible in part for this earlier rejection, but it is also true that the potential of the work was difficult to recognize because it was so original and seemed unconnected with anything that had gone before. For the most part, that is: perhaps a third of Ramanujan’s results, obtained in almost total isolation and with little or no support and encouragement, were rediscoveries of results already published by mathematical giants like the Swiss Leonard Euler (1707-83) and the German Karl Friedrich Gauss (1777-1855). In other words, much of Ramanujan’s efforts had gone for nothing.
Though that was perhaps necessary for him to obtain his other, wholly original results: he was not familiar with what had gone before and so was not conditioned by it. His work was a rediscovery not only of actual mathematical history but also of potential: the unknown results he obtained were missed by Euler, Gauss, and his other predecessors because they had been conditioned by their familiarity with what had gone before and with the work of their own contemporaries. Ramanujan’s now-famous notebooks are proof both of the interconnectedness of mathematics and of its vastness, for they show that a single genius working in isolation can both recapitulate work in a millennium-old intellectual tradition and create work that the tradition has entirely overlooked. And this is not because Ramanujan’s original work was trivial or shallow: some of his results are still being exploited today, more than seventy years after his death. Some, indeed, are still waiting for other mathematicians to learn how to exploit them.17
The same is potentially true of Galois’s work, which was not created in such isolation as Ramanujan’s, but was certainly touched by genius in a similar way. In the document that Galois wrote on the eve of the duel that killed him
he sketched the connection between groups and polynomial equations, stating that an equation is soluble by radicals provided its group is soluble. But he also mentioned many other ideas, about elliptic functions and the integration of algebraic functions, and other things too cryptic to be identifiable.18
But not too cryptic to be rediscovered by other mathematicians. Several of Galois’s manuscripts were lost for good during his lifetime, perhaps through the malice of his enemies. But the regret we feel on learning this is tempered and lessened by the knowledge that they were mathematical texts, not literary or autobiographical ones. Even if they contained wholly original work not preserved elsewhere, they are not irreproducible: as remarked previously, mathematical results are taken from an objective, external world and remain there even when their symbolic representations on paper or in the brains of their discoverers are destroyed. But suppose, on the other hand, that wholly original manuscripts by Rimbaud were known to be lost. Our regret here would be acute, because the work here is irreproducible: no-one will ever feel or write like Rimbaud again because no-one will ever be Rimbaud again.
Similarly, but even more strongly, we mourn the lost work of the ancient Greek poetess Sappho (fl. 6th century BC) more than the lost work of the ancient Greek mathematicians. No-one will ever be Rimbaud again, but many millions still speak his mother tongue. No-one will ever be Sappho again, and no-one will ever speak her mother tongue again: it was an extinct dialect, ancient Lesbian, of an extinct language, ancient Greek. The work of mathematicians contemporary to Sappho is now of historical interest only, for their speciality has progressed year on year to become something far more powerful and far more extensive. Has poetry progressed in the same way since Sappho or Rimbaud? Some would say it has done the reverse, but in either case it is still treating the same themes and expressing the same emotions and still holding a mirror up to the self in a way that is still far more appealing than mathematics. Je n’ai pas le temps, Galois scribbled on the eve of his duel – “I don’t have the time.” He was right, but also wrong. His work will outlast and have vastly more influence than Rimbaud’s, but only because, in the deepest sense, it wasn’t his: it belongs to every mind capable of understanding and applying it, and could have been created by any similar genius – and perhaps, elsewhere in the universe, has been.
For the objective, external world of mathematics could also be described as a world of Platonic archetypes accessible to any mind of sufficient power, no matter what its physical substrate. The physical world is a world of imperfections and rough edges: the mathematical world is not, unless it chooses to be. Mathematics and language are both symbolic, but their symbols relate to different realities – or rather, to different levels of reality. Mathematical symbols are reduced to an essence, and that is why they are in fact far less complex and far easier to manipulate than the symbols of language. A mathematician understands far more of what he is manipulating than a writer does. That is why the greatest chess-player on earth is now a computer and the greatest writers are still human beings: chess can be defined very rigorously and precisely in mathematical terms, language, as yet, cannot be.
This seems paradoxical because chess, like other forms of obviously mathematical endeavour, is what might be called undemocratic. As many linguists have pointed out, any normal human being can use language with endless creativity; few human beings, normal or otherwise, can be creative in chess or mathematics. More, of course, would be able to do so if chess and mathematics were taught better and more widely, but it would still remain true that some people had a special aptitude and insight that separated them in a qualitative way from the majority of us. To the majority of us, doing maths does not come naturally.
Doing maths explicitly, that is. Implicitly, we and every other form of life do it all the time simply to stay alive. What mathematics as an intellectual discipline has done is to take this implicit activity and formalize it, creating a symbolic language in which it can be expressed in a clear and easily manipulatable way. It is also an artificial language in the sense of having been deliberately and consciously created to meet certain fixed and well-defined ends. That is why, again with seeming paradox, it is both far less rich than natural language and far more powerful. Mathematicians can use their symbolic language far nearer to the limits of its potential and their own ability because mathematics, as a conscious creation, is in some sense an embodiment of their ability, rather as the creations of visual artists are embodiments of their ability. Though perhaps a better simile would be drawn from engineering, for the creations of engineers, like those of mathematicians, are tools designed to manipulate and process. An engine literally manipulates and processes time, energy, and matter; geometry manipulates symbolic representations of these things.
And in both cases, when understanding outstrips ability, tools can be re-designed or made more powerful or created anew. At no time have the tools of engineering or mathematics advanced very far beyond the understanding of engineers or mathematicians, because the tools currently used in each field are a reflection of the current understanding of those who work there.19 This is not true of language: its “tools” have always been far beyond – or below – the conscious understanding of speakers or writers, who have never, except in trivial, specific senses, created those tools. Languages have unconsciously evolved; mathematics and engineering, though subject to evolutionary or quasi-evolutionary processes, have always been consciously created.
This may mean that language has never been used to anywhere near its full potential: perhaps no writers, however great, have ever manipulated their symbols with the power and effect achieved by mathematicians with theirs. Language may have too many interacting variables and combine too many distinct functions to have been well-controlled by any human being who has so far lived. We may think we have seen geniuses of language fit to compare with geniuses of maths – a Rimbaud to set against a Galois, a Dickens to set against a Gauss – but the truth may be that human beings have never ascended the heights of language as they have ascended the heights of maths. Language has been too rich for any past or present mind to have truly mastered, and will always remain so until we find some means of overcoming the limitations that cripple us: our weak memories, our weak powers of concentration and analysis. Advances in computing and neuro-science seem to promise that we will indeed find some means of overcoming these limitations. The prospect dazzles like the sun – or the sun-god.
The chess-master Gary Kasparov is one of the greatest who have ever lived. He described the computer that beat him as having “played like a god”. If so, it was an idiot god, employing Herculean powers of strength and endurance, not Apollonian powers of subtlety and insight. But one day computers will play chess like Apollo – and will speak and write like Apollo too. But by then the distinction between human beings and computers may have vanished, and we may literally be men like gods. Man is doomed: the D.E.M., or Deus Ex Machina, is fast approaching. The machinery and electronics that create the D.E.M. will depend on mathematics and mathematicians like Galois will have made our apotheosis possible, not writers like Rimbaud. But that, in mathematical or quasi-mathematical terms, is a function of their differing kinds of genius and says nothing about them as individuals. Unlike language, mathematics has, till now, said very little or nothing about individual human beings, because individual human beings are too complex. The paradox is that by the time mathematics is able to discuss them, they, like language, will almost certainly have disappeared, because they, like language, will almost certainly have changed beyond recognition.
Completed c. 2000, revised 2013.
3. There is a fashion for films and books about maths at the moment, but it seems to be a pretty superficial thing and I think what I say in the rest of the article still applies, particularly to the past.
10. These tropes seemed for a moment to have occurred to me spontaneously; then I realized that they were echoing something I had read the day before in C.S. Lewis’s book Miracles (1947): “Or one may think of a diver, first reducing himself to nakedness, then glancing into mid-air, then gone with a splash, vanished, rushing down through green and warm water into black and cold water, down through the increasing pressure into the death-like region of ooze and slime and old decay; then up again, back to colour and light, his lungs almost bursting, till suddenly he breaks surface again, holding in his hand the dripping, precious thing that he went down to recover” (ch. XIV, “The Grand Miracle”, pp. 115-6 of the 1960 Fontana paperback). The ‘gems and gold’ seem to come from a slightly less recent reading of a Robert E. Howard Conan story called “The Treasure of Tranicos” (originally “The Black Stranger”) reprinted in Conan the Usurper (*).
19. Compare, for example, the early use of calculus with contemporaneous literature: mathematicians and writers at the time were both ignorant of precisely what they were doing to achieve their ends, but the gap in understanding was far wider for writers, and while calculus was properly understood in time, language is still waiting to be centuries later.