There’s a difference between the truth and the part of the truth that can be proved. In fact this is one of Tarski’s corollaries to Godel’s theorem,” said Seldom. “Of course, judges, pathologists, archaeologists all knew this long before mathematicians. Think of any crime with only two possible suspects. Both of the suspects know the part of the truth that matters, i.e. it was me or it wasn’t me. But the law can’t get to that truth directly; it has to follow a laborious, indirect route to gather evidence: interrogations, alibis, fingerprints and so on. All too often there isn’t enough evidence to prove either one suspect’s guilt or the other suspect’s innocence. Basically, what Godel showed in 1930 with his Incompleteness Theorem is that exactly the same occurs in mathematics. The mechanism for corroborating the truth that goes all the way back to Aristotle and Euclid, the proud machinery that starts from true statements, from irrefutable first principles, and advances in strictly logical steps towards a thesis-what we call the axiomatic method-is sometimes just as inadequate as the unreliable, approximative criteria applied by the law.”
Seldom paused for a moment and leaned over to the neighbouring table for a paper napkin. I thought he was going to write out a formula on it, but he simply wiped his mouth quickly and went on: “Godel showed that even at the most elementary levels of arithmetic there are propositions that can neither be proved nor refuted starting from axioms, that are beyond the reach of these formal mechanisms, and that defy any attempt to prove them; propositions which no judge would be able to declare true or false, guilty or innocent. I first studied the theorem as an undergraduate, with Eagleton as my tutor. What struck me most-once I had managed to understand and above all accept what the theorem was really saying-what I found so strange, was that mathematicians had got by perfectly well, without upsets, for so long, with such a drastically mistaken intuition. Indeed, at first, almost everyone thought that Godel must have made a mistake and that someone would soon show that his proof was flawed. Zermelo abandoned his own work and spent two whole years trying to disprove Godel’s theorem. The first thing I asked myself was, why do mathematicians not encounter, and why over the centuries had they not encountered, any of these indeterminable propositions? Why, even now after Godel, can all the branches of mathematics still calmly follow their course?”
We were the last two people left at the long Fellows table at Merton. Facing us in an illustrious row hung portraits of distinguished former alumni of the college. The only name I recognised on the bronze plaques beneath the portraits was T.S. Eliot. Around us, waiters discreetly cleared away the plates of dons who had already gone back to their lectures. Seldom grabbed his glass of water before it was removed and had a long drink before continuing.
“In those days I was a fervent Communist and was very impressed by a sentence of Marx’s, from The German Ideology, I think, which said that historically humanity has only asked itself the questions it can answer. For a time I thought this might be the kernel of an explanation: that in practice mathematicians might only be asking the questions for which, in some partial way, they had proof. Not, of course, unconsciously to make things easier for themselves but because mathematical intuition-and this was my conjecture-was inextricably linked with the methods of proof, and directed in a Kantian way, shall we say, towards what can either be clearly proved or clearly refuted. That the knight’s moves involved in the mental operations of intuition were not, as was often believed, sudden dramatic illuminations but modest, abbreviated versions of what could always be reached eventually with the slow, tortoise-like steps of a proof.”
“I met Sarah, Beth’s mother, at that time. She had just started studying physics and she was already engaged to Johnny, the Eagletons’ only son. The three of us would go bowling or swimming together. Sarah told me about the uncertainty principle in quantum physics. You know what I’m referring to, of course: that the clear, tidy formulas governing physical phenomena on a large scale, such as the motion of celestial bodies, or the collision of skittles, are no longer valid in the subatomic world of the infinitesimal, where everything is far more complex and where, once again, logical paradoxes even arise. It made me change direction completely. The day she told me about the Heisenberg Principle was strange, in many ways. I think it’s the only day of my life that I can recall hour by hour. As I listened, I had a sudden intuition, the knight’s move, so to speak,” he said, smiling, “that exactly the same kind of phenomenon occurred in mathematics, and that everything was, basically, a question of scale. The indeterminable propositions that Godel had found must correspond to a subatomic world, of infinitesimal magnitudes, invisible to normal mathematics. The rest consisted in defining the right notion of scale. What I proved, basically, is that if a mathematical question can be formulated within the same ‘scale’ as the axioms, it must belong to mathematicians’ usual world and be possible to prove or refute. But if writing it out requires a different scale, then it risks belonging to the world-submerged, infinitesimal, but latent in everything-of what can neither be proved nor refuted. As you can imagine, the most difficult part of the work, and what has taken up thirty years of my life, has been showing that all the questions and conjectures that mathematicians from Euclid to the present day have formulated can be rewritten at scales of the same order as the systems of axioms being considered. What I proved definitively is that normal mathematics, the maths that our valiant colleagues do every day, belongs to the ‘visible’ order of the macroscopic.”
“But that’s no coincidence, I think,” I interrupted. I was trying to link the results that I had presented at the seminar with what I was now hearing and find where they fitted in the large figure that Seldom was now drawing for me.
“No, of course not. My hypothesis is that it is profoundly linked to the aesthetic that has been promulgated down the ages and has been, essentially, unchanging. There is no Kantian forcing, but an aesthetic of simplicity and elegance which also guides the formulation of conjectures; mathematicians believe that the beauty of a theorem requires certain divine proportions between the simplicity of the axioms at the starting point, and the simplicity of the thesis at the point of arrival. The awkward, tricky part has always been the path between the two-the proof. And as long as that aesthetic is maintained there is no reason for indeterminable propositions to appear ‘naturally’.”
The waiter returned with a pot of coffee and filled our cups. Seldom remained silent for a time, as if he was unsure whether I’d followed what he was saying, or was perhaps a little embarrassed at having talked so much.
“What I was most struck by,” I said, “the results that I presented in Buenos Aires, were in fact the corollaries that you published a little later on philosophical systems.”
“Actually, that was much easier,” said Seldom. “It’s the more or less obvious extension of Godel’s Incompleteness Theorem: any philosophical system which starts from first principles will necessarily have a limited scope. Believe me, it was much easier piercing through all the philosophical systems than through that single thought matrix to which mathematicians have always clung. Because all philosophical systems are simply too ambitious. Basically, it’s all a question of balance: tell me how much you want to know and I’ll tell you with how much certainty you’ll be able to state it. Butt at the end, when I’d finished and I looked back after thirty years, it seemed that that first idea that Marx’s sentence had suggested to me hadn’t been so misguided after all. It had ended up, as the Germans would say, both eliminated from and included in the theorem. Indeed, a cat doesn’t simply assess a mouse, it assesses it as a prospective meal. But the cat doesn’t assess all animals as prospective meals, only mice. Similarly, historically, mathematical reasoning has been guided by a criterion, but that criterion is, deep down, an aesthetic. I found this to be an interesting and unexpected substitution with regard to necessity and a priori Kantians. A condition that is less rigid and possibly more elusive, but which also-as my theorem had shown-was substantial enough to be able, still, to say something and make waves. As you see,” he said, almost apologetically, “it isn’t easy to be free of such an aesthetic: we mathematicians always like to feel that we’re saying something that is meaningful.
“However that may be, I have devoted myself ever since to studying what I privately call the aesthetic of reasoning in other spheres. I began, as always, with what seemed like the simplest model, or at least the closest: the logic of criminal investigations. I found the parallels with Godel’s theorem very striking. In every crime there is undoubtedly a notion of truth, a single true explanation among all the possible explanations. On the other hand, there are also material clues, facts that are incontrovertible or at least, as Descartes would say, beyond reasonable doubt: these would be the axioms. But then we’re already in familiar territory. What is a criminal investigation if not our old game of thinking up conjectures, possible explanations that fit the facts, and attempting to prove them correct? I began systematically reading about real-life murders, I went through public prosecutors’ reports for judges, I studied the method of assessing evidence and of structuring a sentence or an acquittal in a court of law. Just as when I was a teenager, I read hundreds and hundreds of crime novels. Gradually I began to find a multitude of interesting little differences, an aesthetic inherent in criminal investigations. And errors, too. I mean theoretical errors in criminology, which were potentially much more interesting.”
“What kind of errors?”
“The first, and most obvious, is attaching too much importance to physical evidence. Just think of what’s happening now in this investigation. If you recall, Inspector Petersen sent one of his officers to retrieve the note I received. Here once again the same insurmountable gap opens up, between that which is true and that which is provable. I saw the note, and that’s the part of the truth that the police can’t get to. My statement isn’t much use as far as police procedure goes; it doesn’t carry the same weight as the little piece of paper itself. Now, the officer, Wilkie, completed his task as conscientiously as he could. He questioned Brent and got him to go over what he knew several times. Brent clearly remembered seeing a piece of paper folded in two at the bottom of my wastepaper basket, but it hadn’t occurred to him to read it. Brent remembered too that I’d asked him if there was any way of retrieving the paper, and he told Wilkie what he told me: that he’d tipped the contents of the basket into an almost full refuse bag, which he’d put out soon after. By the time Wilkie arrived at Merton, the refuse lorry had been and gone almost half an hour earlier. When Petersen called me yesterday to ask me to describe the handwriting to their artist, I could tell that he was very disappointed at not finding the note. He’s considered to be the best police inspector we’ve had in years. I’ve had a look at the complete notes to several of his cases. He’s thorough, meticulous, implacable. But he’s still an inspector. I mean, he was trained in accordance with police procedure: you can predict the way his mind is going to work. Unfortunately, people like him follow the principle of Ockham’s Razor: as long as there’s no physical evidence to the contrary they always prefer a simple hypothesis to a more complicated one. That’s the second error. Not just because reality tends to be naturally complicated but mainly because, if the murderer really is intelligent and has prepared the crime carefully, he’ll leave a simple explanation for all to see, a smokescreen, like a conjuror leaving the stage. But in the stingy logic of the economy of hypothesis a different reasoning prevails: why assume something strange and out of the ordinary, such as a murderer with intellectual pretensions, if they have more immediate explanations to hand? I could almost physically feel Petersen step back and re-examine his hypotheses. I think he would have started suspecting me, if he hadn’t already checked that I was teaching between one and three that afternoon. I expect they checked out your statement too.”
“Yes. I was in the Bodleian Library when it happened. They went to enquire about me there yesterday. Luckily, the librarian remembered me because of my accent.”
“So you were consulting books at the time of the murder?” Seldom raised his eyebrows sardonically. “For once, knowledge really is freedom.”
“Do you think Petersen will pounce on Beth now? She was terrified yesterday after they questioned her. She thinks the inspector is after her.”
Seldom thought for a moment.
“No, I think Petersen is cleverer than that. But consider the dangers of Ockham’s Razor. Suppose for a moment that the murderer, wherever he is, decides that he doesn’t have a taste for murder after all, or that the business with the blood and the police getting involved have ruined his fun; suppose that, for some reason, he decides to disappear from the scene. I think Petersen would then go after Beth. I know he questioned her again this morning, but this may simply have been a diversionary tactic, or a way of provoking the murderer, acting as if they don’t know about him, as if this were an ordinary case, a murder in the family, as the newspaper suggested.”
“But you don’t really think the murderer is going to quit the game, do you?” I asked.
Seldom pondered my question much more seriously than I’d expected.
“No, I don’t,” he said at last. “I just think he’ll try to be more…imperceptible, as we said before. Are you free at all now?” he asked, glancing at the dining room clock. “Visiting hours at the Radcliffe are about to start, and I’m heading there. If you’d like to come along, there’s someone there I’d like you to meet.”