An Oxford philosopher has continued a tradition going back to Plato by using a fictional conversation to explore questions about truth, falsity, knowledge and belief in a new book published this month. But unlike Plato, his book is set on a train.
Timothy Williamson, the Wykeham Professor at Logic at Oxford University, has written Tetralogue: I'm Right, You're Wrong, in which four people with radically different outlooks on the world meet on a train. At the start of the journey each is convinced that he or she is right, but then doubts creep in.
In an interview with Arts Blog, Professor Williamson explains his hope that Tetralogue will give a wider audience an insight into the academic philosophy..
Q: What is the aim of the book?
A: Its starting point is the occurrence of radical disagreement, about science, religion, politics, morality, art, whatever. In contemporary society, many people are reluctant to apply ideas of truth and falsity, or knowledge and ignorance, to such clashes in point of view, because they are afraid of being dogmatic and intolerant. But can one really abstain from such distinctions without losing one's own point of view altogether? In a light-hearted way, the book aims to provide readers with the means to think more carefully and critically about such matters, and to avoid common traps and confusions.
Q: Who is your target audience?
A: The book is aimed primarily at people who haven’t studied philosophy academically, but who are interested in philosophical issues like those just mentioned. It might be someone who has been led to worry about them through personal experience of such clashes, or who has trouble handling them in their own research or teaching, or a teenager wondering what it would be like to study philosophy at university. I hope that even academic philosophers may find something to amuse them in it.
Q: What do you hope people take away from the book?
A: I’d like them to take away a nose for when certain fallacies are being committed or certain glib, problematic assumptions are being made. More constructively, I’d like to have empowered them to reason more logically about the sort of difficult issue I’ve mentioned. I also hope that they will have gained a sharper sense of the cut-and-thrust of philosophical argument, but also of the limited power of reason to force anyone to change their mind.
Q: Was it difficult to present philosophical concepts in ordinary dialogue on a train?
A: You find yourself sitting next to strangers on a train for several hours: a chance for a long talk. If Hitchcock is directing the film, the conversation turns to murder. If I’m writing the book, it turns to philosophy. Both are dangerous subjects with roots in ordinary life. Both need to be introduced carefully, because you can’t take much for granted about your audience. You have to start from the beginning. I must admit, when I’m on a train, I rarely speak to strangers, but I often listen in to their conversations. I’d love to hear them discuss murder, or philosophy.
Sarah: It’s pointless arguing with you. Nothing will shake your faith in witchcraft!
Bob: Will anything shake your faith in modern science?
Zac: Excuse me, folks, for butting in: sitting here, I couldn’t help overhearing your conversation. You both seem to be getting quite upset. Perhaps I can help. If I may say so, each of you is taking the superior attitude ‘I’m right and you’re wrong’ toward the other.
Sarah: But I am right and he is wrong.
Bob: No. I’m right and she’s wrong.
Zac: There, you see: deadlock. My guess is, it’s becoming obvious to both of you that neither of you can definitively prove the other wrong.
Sarah: Maybe not right here and now on this train, but just wait and see how science develops—people who try to put limits to what it can achieve usually end up with egg on their face.
Bob: Just you wait and see what it’s like to be the victim of a spell. People who try to put limits to what witchcraft can do end up with much worse than egg on their face.
Zac: But isn’t each of you quite right, from your own point of view? What you—
Zac: Pleased to meet you, Sarah. I’m Zac, by the way. What Sarah is saying makes perfect sense from the point of view of modern science. And what you—
Zac: Pleased to meet you, Bob. What Bob is saying makes perfect sense from the point of view of traditional witchcraft. Modern science and traditional witchcraft are different points of view, but each of them is valid on its own terms. They are equally intelligible.
Sarah: They may be equally intelligible, but they aren’t equally true.
Zac: ‘True’: that’s a very dangerous word, Sarah. When you are enjoying the view of the lovely countryside through this window, do you insist that you are seeing right, and people looking through the windows on the other side of the train are seeing wrong?
Sarah: Of course not, but it’s not a fair comparison.
Zac: Why not, Sarah?
Sarah: We see different things through the windows because we are looking in different directions. But modern science and traditional witchcraft ideas are looking at the same world and say incompatible things about it, for instance about what caused Bob’s wall to collapse. If one side is right, the other is wrong.
Zac: Sarah, it’s you who make them incompatible by insisting that someone must be right and someone must be wrong. That sort of judgemental talk comes from the idea that we can adopt the point of view of a God, standing in judgement over everyone else. But we are all just human beings. We can’t make definitive judgements of right and wrong like that about each other.
Sarah: But aren’t you, Zac, saying that Bob and I were both wrong to assume there are right and wrong answers on modern science versus witchcraft, and that you are right to say there are no such right and wrong answers? In fact, aren’t you contradicting yourself?
William Henry Fox Talbot is best-known today as a Victorian pioneer of photography. But an Oxford researcher has revealed that, for Talbot, photography was a means to an end in deciphering some of the oldest writing in human history.
Talbot's "calotype" process is a direct ancestor of modern imaging technology, and his family archive was acquired by the Bodleian Library last year.
"What's not so well-known about Talbot is that he only photographed for about ten years of his life," said Dr Mirjam Brusius, of the Department of the History of Art at Oxford University. "He worked in optics, botany, politics and other areas. In fact, he developed his photographic process in part to help decipher Mesopotamian artefacts.
"He was drawn to what was difficult and obscure. When he worked in botany, he was interested in mosses, because they're very hard to classify and identify. When he first became interested in antiquity, he worked briefly on Egyptian hieroglyphs - but they were too easy, they had already been largely deciphered, so he moved on to Assyrian tablets, and the cuneiform which was not so well understood. Talbot saw the tablets as a kind of mathematical exercise. He wasn't so interested in their content."
Cuneiform is one of the most ancient writing systems to survive in the world, dating back as far as 3000 BC. Knowledge of the system was completely lost until 19th-century Western archaeologists began to uncover clay tablets while excavating in the Middle East.
"Talbot spent a lot of time trying to propagate interest in photography as a tool for archaeological research," said Dr Brusius. "He put in a huge amount of energy but people were happy to use their existing tools, such as drawing. He did his best to convince the British Museum to take a camera on their expeditions, but the chemicals, the heavy equipment and changeable conditions meant they were not keen.
"Even in 1852, ten years after his invention of the calotype, he was still trying to persuade them to use a camera in the museum, to record the artefacts. He was lucky enough to know Lord Rosse [whom we've covered in a previous Arts Blog]. Rosse was a trustee at the British Museum and put in a good word for him, and the museum eventually hired a freelance photographer. The cuneiform collection was photographed, and Talbot hoped to use the resulting images in his research.
"Unfortunately for him, there was a certain amount of rivalry between him and the professional Assyriologists. Talbot was a wealthy gentleman who did not need to work and they considered him an amateur. Henry Rawlinson, a diplomat, was a major figure in the field. He worked out that Talbot was quite good, and didn't want him to be the first to decipher the inscriptions. Rawlinson's Assyriology was essentially a political enterprise, and there were British imperialist interests at play in the Middle East which were somewhat at odds with Talbot's 'armchair science'. So Talbot did not get the photos at first.
"These were inscriptions which could confirm or disprove the historical accuracy of the Bible. Artefacts might be suppressed if they challenged the prevailing narrative: one tablet contained an account of a flood which was at odds with the biblical story. The authority of the Bible as a historical source was seriously contested, and at a time when religious authority was being challenged in a variety of ways by natural science, this was very significant.
"Working in the Talbot archive at the British Library, I found his handwriting on one of the British Museum photographs. It seems that after another nine years, in about the early 1860s, Talbot finally received the photographs. His relationship with the professional Assyriologists gradually improved, along with his reputation and respect, and he was a founding member of the Society of Biblical Archaeology. They had perhaps come to realise that they needed people like Talbot, as much as he needed their contacts and equipment."
It was standing room only as Marcus du Sautoy and Ben Okri discussed the relationship between narrative and proof at the Mathematical Institute on Tuesday evening (20 January).
The full video of the event to launch The Oxford Research Centre in the Humanities' (TORCH) 'Humanities and Science programme can be seen here.
The speakers have given Arts Blog permission to publish edited extracts of their talks:
Marcus du Sautoy, Simonyi Professor for the Public Understanding of Science, Oxford University
In Borges's short story The Library of Babel the librarian who narrates the story begins with a description of his place of work:
The universe (which others call the library) is composed of an indefinite and perhaps infinite number of hexagonal galleries...From any of the hexagons one can see, interminably, the upper and lower floors.
As befits a library, this vast beehive of rooms is full of books. The tomes all have the same dimensions. 410 pages, each page with 40 lines and each line consisting of 80 orthographical symbols of which there are 25 in number.
As the librarian explores the contents of his library he finds that most of the books are formless and chaotic in nature but every now and again something interesting appears. He discovers a book with the letters MCV repeated from the first line to the last. In another, the cacophony of letters is interrupted on the penultimate page by the phrase Oh time thy pyramids and then continues its meaningless noise.
The challenge the librarian sets himself is to determine whether the library is in fact infinite or, if not, what shape it has. As the story develops a hypothesis about the library is proposed.
The Library is total ... its shelves register all the possible combinations of the twenty-‐odd orthographical symbols (a number which, though extremely large is not infinite): in other words, all that it is given to express, in all languages. Everything.
The library contains every book that it is possible to write. When it was proclaimed that the Library contained all books, the first impression was one of extravagant happiness. But this was followed by an excessive depression. Because it was realized that this library that contained everything in fact contained nothing.
So what is in the mathematican’s library? I think many believe that it is aspiring to be a mathematical Library of Babel. That the role of the mathematicians is to is to document all true statements about numbers and geometry. The irrationality of the square root of 2. A list of the finite simple groups. The formula for the volume of a sphere. The identification of the brachistochrone as the curve of fastest descent.
Mathematics though is very different from simply a list of all the true statements we can discover about number. Mathematicians, like Borges, are story tellers. Our characters are numbers and geometries. Our narratives are the proofs we create about these characters.
Let me quote one of my mathematical heroes Henri Poincaré articulating what it means to do mathematics:
To create consists precisely in not making useless combinations. Creation is discernment, choice...The sterile combinations do not even present themselves to the mind of the creator.
Mathematics, just like literature, is about making choices. What then are the criterion for a piece of mathematics making it into the journals that occupy our mathematical library? Why is Fermat’s Last Theorem regarded as one of the great mathematical opus’s of the last century while an equally complicated numerical calculation is regarded as mundane and uninteresting. After all, what is so interesting about knowing that an equation like xn+yn=zn has no whole number solutions when n>2.
What I want to propose is that it is the nature of the proof of this Theorem that elevates this true statement about numbers to the status of something deserving its place in the pantheon of mathematics. And that the quality of a good proof is one that has many things in common with act of great story telling.
My conjecture if I was to put it into a mathematical equation is that
"proof = narrative."
A proof is like the mathematician’s travelogue. A successful proof is like a set of signposts that allow all subsequent mathematicians to make the same journey. Readers of the proof will experience the same exciting realization as its author that this path allows them to reach the distant peak.
Very often a proof will not seek to dot every i and cross every t, just as a story does not present every detail of a character’s life. It is a description of the journey and not necessarily the re-enactment of every step. The arguments that mathematicians provide as proofs are designed to create a rush in the mind of the reader. The mathematician GH Hardy described the arguments we give as 'gas, rhetorical flourishes designed to affect the psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils'.
What is important for me about a piece of mathematics is not the QED or final result but the journey that I’ve been taken on to get to that point, just as a piece of music is not about the final chord. It is certainly important to know that there are infinitely many primes but the satisfaction comes from understanding why.
The joy of reading and creating mathematics comes from the exciting “aha” moment we experience when all the strands seem to come together to resolve the mathematical mystery. It is like the moment of harmonic resolution in a piece of music or the revelation of who-‐dunnit in a murder mystery.
The element of surprise is an important quality of exciting mathematics. Here is mathematician Michael Atiyah talking about the qualities of mathematics that he enjoys:
I like to be surprised. The argument thatfollows a standard path, with few new features, is dull and unexciting. I like the unexpected, a new point of view, a link with other areas, a twist in the tail.
When I am creating a new piece of mathematics the choices I will make will be motivated by the desire to take my audience on an interesting mathematical journey full of twists and turns and surprises. I want to tease an audience with the challenge of why two seemingly unconnected mathematical characters should have anything to do with each other. And then as the proof unfolds there is a gradual realization or sudden moment of recognition that these two ideas are actually one and the same character.
The importance of the journey to mathematics can be illustrated by a strange reaction that many mathematicians have when a great mathematical theorem is finally proved. Just as there is a sense of sadness when you come out the other side of a great novel, the closure of a mathematical quest can have its own sense of melancholy.
I think that we’d been so enjoying the journey that Fermat’s equations had taken us on that there was a sense of depression that was mixed with the elation that greeted Andrew Wiles’s solution of this 350 year old enigma. That is why proofs that open up the ground for new stories are valued very highly in mathematics.
Ben Okri, author of Flowers and Shadows, The Famished Road (Booker Prize winner 1991), Songs of Enchantment and Infinite Riches
Narrative is woven into the fabric of consciousness as mathematics is woven into the fabric of the world. If it were possible to imagine a consciousness at the heart of all things in stone, in the air, in trees, in mountains, in stars in atoms, in all things that constitute reality, that consciousness would perceive the world in its smallness, as well as in its largeness, as a grand perpetual narrative.
The motion of things is the story of things; the constant change that Heraclitus saw, the mutations the compositions, visions, collisions, growths, deaths, fissions are all part of the infinite narrative that is reality. It would seem that story is implicated in the world, on the condition that there is a consciousness to perceive it. To that degree narrative is woven into the fabric of the mind. Is there a mathematical basis to narration? Is there a mathematics of narration?
We see from Marcus’s excellent paper that there could be said to be a link between proof and narration, that to prove is to narrate. In literature narrative is a kind of proof. It is more than that of course, but it is always a kind of proof. It is the out figuring of an intuition of tension, of the need for psychic resolution, the need to make visible in order to make understandable.
But narrative as proof takes a more intangible, more aesthetic form. The desire to write a story is not merely to prove the existence of the story in the mind, for often the story does not exist until it is written; it exists in an ideal state, as a throb as Nabakov would call it, an impulse, a pressure on the literary glands, an ache in the soul. But as something cannot come from nothing, the existence of story is proof in a way of a previous intangible condition, a sediment, a concretisation of an opalescent state.
There is one way in which narrative shares a profound similarity to mathematics and this in the unavoidable logic of storytelling. The equation must work. Where the story begins, how it evolves, where it goes, must work, it must add up, it must compute. There is mathematics in narration, in the sense that we know when a story's mathematics does not work. This can happen in a quantum sense at the level of sentences, or in a larger sense in terms of the whole. Narration conforms to an aesthetic mathematics, and the best storytellers therefore are profound and rigorous thinkers. To work out the inner maths of a story or a novel is one of the most difficult things that writers do. For on the rightness of that maths, the unfolding of character within the limitation of plot, rests the immortality or oblivion of the text.
Mathematics is not just what mathematicians do, mathematics – the relation of number – is implicated in the structure of reality, in the number of vibrations that make the atom in the periodicity of elements, and the pulsing of quasars and the calyxes of flowers, in the rhythm of all things. In fact it is the relationship between the rhythmic quality of the world and the rhythmic quality of art that so fascinated Leonardo da Vinci. It would seem that the world, its underlying structure, is governed by known and unknown laws of mathematics. That, as Pythagorus implied, number governs the world, the materiality of the world, the manifestation of things.
It is now axiomatic in practical science that by altering the number of the vibrations of a thing you can alter its nature. I have found this to be true in narration as well. There is a novel of mine called Astonishing the Gods, whose nature was changed by altering its underlying beat, its vibration. I will give you an example. In the early draft it began "invisibility is best". The novel was written in that contracted beat. When I came to write it and to re-write the rewriting I realised that there was something not right about that beat for that novel. In the final version it reads "It is better to be invisible".
A great difference in the unfolding of the text emerged from altering the microbeat of the novel. You could call this re-writing but I believe it is something more. I believe that everything exists by virtue of the law of numbers, its underlying vibration. The elements have their rates of vibration, at atomic and subatomic level all is number, all is vibration, this too at the larger levels. The world is what it is by virtue of this quality. This is beyond the scope of our present conversation, but one can conjecture there might also be a sublime mathematics and that someday could reveal that which we now see as transcendent.
Mathematics is woven into the fabric of the world, narrative into the fabric of consciousness; they are both part of the fact of reality. The world exists by virtue of numbers and the fact that the world exists already implies its narrative quality. The curious thing about both is it they need an intelligent consciousness to perceive their existence.
But there are fundamental differences between proof and narration. One is the universality of language, the transparency of narration. Another is the dual nature of narration that it is both of time and timeless. Another is the mimetic quality of narration, that it mirrors the world, it mirrors modes of consciousness, and in its imaginative dimension it contains matrixes of the future.
One could say that the purity of mathematics makes it more akin to music. Both create realms of their own, crystalline and pure, immaculate fantasies with unalterable laws. Another crucial difference is the fact that great narration has higher immeasurable symmetries. The difference between a well plotted detective novel and Hamlet is infinite resonance. This is not a disquisition on lowbrow and highbrow. A murder mystery of Agatha Christie, for example, is satisfying in tying up all the loose ends, but the resolution of Oedipus Rex gives us unending insight into the complexity of fate, the human condition, and the moral law.
The refractions of Oedipus Rex resonates 2500 years later and the chief reason, I believe, is that it lives, and it partakes of higher truths that we still have not fully grasped. And it compels us to contemplate all the dimensions of what it means to be human. This brings me to the greatest difference between narrative and proof, between mathematics and narration. Narration is, at its highest, about the enigma of being human, of being alive, of consciousness itself, in all of its states, in all its realities, in all its unrealities.
Narration is the highest tool the human mind has devised to investigate the mystery of life and the human condition. It is more than a mirror of the world and ourselves. It is more than an order perceived in the chaos. It is a technology we have dreamt up to help us get from our darkness to our progressive light. I believe that the oldest technology is not the wheel, not the discovery of fire, not even the discovery of language itself. I believe it is storytelling.
We tell stories even without language. We tell stories with our faces, in our gestures, in our eyes. We tell stories on cave walls. I believe the impulse of narration has led us to language and not the other way around. Language amplifies the power and scope of narration. Narration is implied in germination, procreation and cessation. Narration begins before birth and continues after death. Archaeology confirms the latter. Ovid’s Metamorphosis hints at the fact of change which is at the heart of narration.
The big bang was an event of incalculable mathematical magnitude, but it was also a singular narrative event. It could also be called the big beginning. In that moment of astonishing singularity was born the mathematics of all things and the narrative of all things - two children of the same mother-father moment, that brings us all here today.
'Narrative and Proof' was held at the Mathematical Institute on 20 January 2015 at 5pm. Sir Roger Penrose and Professor Laura Marcus also spoke at the event, which was chaired by Professor Elleke Boehmer.
Yesterday was 'Museum Selfie Day', where people across the world were encouraged to take self-portrait photographs of themselves visiting a museum and share it on social media.
Oxford University’s museums were used as the location for dozens of 'selfies'. Arts Blog has picked out some of the best.
Rachel, a trainee on the HLF Skills for the Future Museum Education & Outreach scheme at Oxford University Museums & Collections, used a bushy prop for her selfie with a samurai at the Pitt Rivers Museum. This man is from the peaceful years of the Edo period (1603-1868) in which armour was more decorative than functional so he would probably forgive Rachel for her joke.
Hannah, another trainee on the Skills for the Future scheme, took time out from preparing a tour for Oxford Brookes students to take a selfie in front of a painting of Lewis Evans. Mr Evans, who founded the Museum of the History of Science, is holding one of his sundials, which were ‘his greatest interest’.
A visitor to the Ashmolean Museum mimics the pose of Augustus. Do not be fooled by the clever photo crop - Augustus is pointing, not holding a camera.
Gary, a cleaner at the Museum of Natural History, snapped himself through the jaws of a dinosaur at the Museum of Natural History. The dinosaur displays in the Museum include four species from Oxfordshire.
Selfies have been criticised as a symptom of a culture of selfishness and celebrity-worship. But we can assure Arts Blog readers that vanity in Oxford is nothing new, as this Muse on the Clarendon Building shows.
Leading figures from humanities and the sciences will discuss the importance of narrative in scientific proofs at Oxford University today.
The event marks the launch of the 'Humanities and Science' series organised by The Oxford Research Centre in the Humanities (TORCH) and will take place at the Mathematical Institute at the University of Oxford.
Mathematician Marcus du Sautoy, Oxford’s Simonyi Professor for the Public Understanding of Science, will give a presentation on role of narrative in mathematics. He will be joined on a panel by author Ben Okri, Oxford mathematician Roger Penrose and Oxford literary scholar Laura Marcus. The event will be streamed live online from 5pm.
Professor du Sautoy will argue: 'Mathematics is more than just true statements about numbers. Why does a proof of Fermat’s Last Theorem get celebrated as one of the great achievements of 20th century mathematics while an equally complicated calculation is regarded as mundane and uninteresting? Why is the proof more important than the result itself? It is not the QED but the pathway to that QED that mathematicians care about. Is the quality of the narrative journey of the proof actually what elevates a sequence of logically connected statements to be celebrated as mathematics? And what qualities does that narrative share with other narrative art forms?'
The discussion will be chaired by Elleke Boehmer, Professor of World Literature at Oxford. She said: 'Literary narrative and mathematical proof, far from being poles apart, in fact fall into intriguingly similar symbolic patterns: stage by stage sequences, tricky reversals, surprising denouements. Indeed, we might go so far as asking ourselves to what extent proofs are in fact narratives of a kind, and narratives a form of proof.'
TORCH's Humanities and Science series will focus on the relationship between the two disciplines, exploring how new answers can be found and new research questions can be set. It will showcase many of the existing research projects in Oxford that already cross the disciplines and provide an incubation space for new collaborative projects.
The dates for future public events as part of the series are on the TORCH website.