by Andrew Crumey
From Mr Mee (2000)
Jean-Bernard Rosier to Jean le Rond D'Alembert, 3 June 1759.
Sir, you may know that many years ago one of our countrymen was taken prisoner in a remote and barren region of Asia noted only for the savagery of its inhabitants. The man's captors, uncertain what to do with him, chose to settle the issue by means of a ring hidden beneath one of three wooden cups. If the prisoner could correctly guess which cup hid the gold band, he would be thrown out to face the dubious tenderness of the wolves; otherwise he was to be killed on the spot. By placing bets on the outcome, his cruel hosts could enjoy some brief diversion from the harsh austerity of their nomadic and brutal existence.
The leader of the tribe, having hidden his own ring, commanded that the unfortunate prisoner be brought forward to make his awful choice. After considerable hesitation, and perhaps a silent prayer, the wretch placed his trembling hand upon the middle cup. Bets were placed; then the leader, still wishing to prolong the painful moment of uncertainty which so delighted his audience, lifted the rightmost cup, beneath which no ring was found. The captive gave a gasp of hope, and amidst rising laughter from the crowd, the leader now reached for the left, saying that before turning it over he would allow his prisoner a final opportunity to change his choice. Imagine yourself to be in that poor man's position, Monsieur D'Alembert, and tell me, what would you now do?
If the leader, when he turned over the rightmost cup, made his choice at random, then the prisoner now has an even chance of holding the ring beneath his hand. But the leader must have known where the ring was placed, and he may have decided to turn the rightmost cup precisely because he knew the ring was not beneath it. In that case the prisoner's chances, originally one in three, have not been improved by the leader's gesture; instead, it now becomes twice as likely that the remaining cup conceals the ring, so that the prisoner, if he loves life, would be well advised to change his choice!
What the story illustrates, is that the cups can somehow tell whether the leader acts randomly, or out of choice. The probability that the prisoner holds the ring is either one half or one third, depending on whether the leader knows in advance which cup the ring lies beneath; an observation which startled me greatly when I arrived at it, and kept me awake for an entire night as I followed its many implications; for I was led to conclude that the acts of observation, of thought, of consciousness, are inextricably linked to the reality of the world. Nature, I realised, cannot be regarded as consisting simply of cold inanimate matter, proceeding according to laws which you, Monsieur D'Alembert, and your esteemed colleagues, would have us believe you can discover. To understand the world, we must comprehend the human mind and its interaction with all that it perceives and to which it thereby gives existence.
And just as the cup experiment, through many repetitions, provides a means of discovering the leader's strategy, so might we contemplate the possibility of constructing a greater kind of trial, a game against Nature in which would be demonstrated the presence or otherwise of some omniscient consciousness, some cosmic dealer of Fate's cards. Then the laws of physics would truly reveal the mind of God.
What of our prisoner? He accepted the leader's offer, placed his feeble hand upon the leftmost cup, and when it was turned and nothing was found beneath it, his throat was opened without further ceremony. The leader retrieved his bauble from beneath the middle cup, and all that remained of this sad event was a ballad which became popular in the region, and an account of the tragedy which I found in Théodore's Excursions. We could imagine a multitude of worlds, in a third of which the outcome was happier, and neither that book nor this letter might ever have been written.
D'Alembert's reply to Rosier has not been preserved; but we do have a subsequent letter in which Rosier claims to have begun constructing a new philosophy of the Universe based entirely on the laws of chance, which, once completed, would render archaic and redundant the contents of the celebrated Encyclopédie of which D'Alembert, together with Denis Diderot, had been editor.
The experiment was performed at the home of my assistant Monsieur Louis Tissot. A ring belonging to his wife was hidden, without my looking, beneath one of three cups. I then placed my hand upon a cup, Tissot lifted another beneath which the ring was not to be found, and he invited me, if I wished, to alter my choice. We did the experiment more than a hundred times, and I soon discovered that changing my selection was indeed the best strategy. I was elated by this, and would have gone on even longer if Madame Tissot hadn't asked for her ring back.
Next day, inspired by my wonderful discovery, I returned to Tissot's house and proposed a new experiment. A small velvet bag containing an equal number of black and white beads was held open by Madame Tissot, while I and my assistant, blindfolded, each selected a bead. Having noted our choice, Madame Tissot would announce either, "at least one bead is black", or, "at least one bead is white". The two of us, still blindfolded, would then go to opposite ends of the room, from where I would consider the probability that Tissot held black.
So contrary were the results we obtained to what might have expected, that Tissot was led to doubt the accuracy of his wife's record-keeping; and our first run of experiments ended in an acrimonious argument between the couple, which spilled over into a debate about Madame Tissot's mother, about the burning of some cakes, and about various domestic arrangements which I found frankly distasteful. The three of us took a break, and afterwards I proposed that as soon as each of us had chosen a bead, we would hide it in our hand and remove the blindfold before walking to our respective corners, where we would then see what we held and write down its colour, so that Madame Tissot's own records could have independent verification. Madame Tissot wasn't very happy about this, but her husband said it would be a good idea, since he found it very difficult to walk without seeing where he was going, and he was sure he'd set off that injury he'd sustained two months earlier, when he was moving a wardrobe for his mother-in-law and tripped on a step, twisting his ankle in the process. Madame Tissot said he could hardly blame her mother for his sore foot, and I said we'd better just get on with it before another argument broke out between the two of them. When the three of us compared notes after many repetitions of the new procedure, we found that Madame Tissot's data had after all been impeccable.
"You see!" she said. "I told you so!" Tissot wanted to talk about the burnt cakes again, and I left them to it while I began to study the results more closely. Whenever Madame Tissot, for instance, had announced one bead to be black, the probability that either of us held that colour, curiously but indubitably, was not a half, but rather two thirds. Yet if I were then to look at my own bead and found it to be black, Tissot's chances of doing likewise suddenly fell to a half. "Eureka!" I murmured, my voice betraying none of the joy and fear I felt; and looking up, I saw that the Tissots were about to swing fists at each other.
"Stop!" I cried, and ordered the next experiment to begin.
Now, Madame Tissot's reputation having been vindicated, the blindfolds would remain. Madame Tissot folded her arms in triumph; her husband complained.
"But what if I trip and set off my bad ankle?"
"Tread carefully," I told him calmly, "for you walk in the name of Science."
Tissot still protested: "Why should I wear a blindfold if I am to look at the bead as soon as I reach my corner of the room?"
On the contrary, I informed him; Tissot was never once to look at what he held. For I believed that what this experiment showed was that as soon as I saw my own bead, a wave of pure probability flew, instantaneously, from one end of the room to the other. This accounted for the sudden change from two thirds to a half, as a finite quantum of probability (of weight one sixth) passed miraculously between the beads, launched by my own act of observation.
Neither Tissot nor his wife had a clue what I was talking about, but they could do nothing except obey my wishes as, with an excitement bordering almost on obsession, I ordered the experiment to be repeated several hundred times in the course of that momentous summer afternoon. My two colleagues grew exhausted, ultimately somewhat careless, and Tissot stumbled, yelping about his mother-in-law's wardrobe, and about the cakes, while his wife, huffing angrily, gathered up the bag of beads which she said belonged in her sewing chest, and marched briskly out of the room. Still doubting her probity, Tissot would later suggest repeating the experiment, this time with a priest making the announcements which Madame Tissot seemed unable or unwilling to perform correctly, so strange had been the outcome; I, however, knew already how to explain it.
The theory of the Universe which I was to construct, inspired by this experiment and by other observations, is based solely on information and its transmission; reality consists of whatever can be measured. "All is in constant flux," I told Tissot on another occasion. "The world is an endless sea, whose undulations are the chance and necessity which steer us all." Tissot, who by now had moved out from the family home in order to devote himself fully to his duties as my assistant (and also so as to avoid his mother-in-law), listened attentively while I described to him what I call "the measurement problem".
When a man uses a ruler, I explained, he can only achieve whatever degree of accuracy the instrument's markings allow. By introducing finer divisions he can obtain a better value, though still not an exact one. If he were to continue his attempts, making the scale on his ruler twice as fine each time, would he ever reach a final definite answer? I claim not; and since the "actual" length could only be arrived at after an infinite process, it can have no physical meaning. The length of any line does not exist, except as a particular measurement to within some limit of accuracy. This is "Rosier's Uncertainty Principle", named after the genius, myself, who discovered it.
I propose that what exists is only the probability of the line having a particular length. From a thousand measurements, one obtains a distribution of values clustered around some mean; plotting these on a chart as Tissot did, grumbling about his wife and her mother the whole time, one finds a distribution having a bell-shaped appearance, which I call a "probability wave"; and whenever a measurement is made, this wave "collapses" to the single value obtained. In the ball experiment, similar waves had been shown to propagate with infinite speed; their collapse was due to the crucial intervention of human consciousness. As a corollary, I was led to conclude that before I had looked at the ball in my hand, it was neither white nor black; a discovery which led me to a perplexing conjecture.
Imagine, I said to the startled Tissot, a condemned prisoner trapped in a windowless cell. He must choose between two identical potions, one of which is harmless and will enable him to go free, while the other is a deadly poison.
"Does this have anything to do with the story of the cups?" asked Tissot, who obviously hadn't been following my explanations very closely.
"One might call that case an influence, nothing more," I told him. Now the prisoner is locked out of sight of the world while he makes his choice. Until the cell door is opened, according to my theory, we can call him neither live nor dead; he exists instead in a state which is a superposition of both, a ghostly half-life belonging neither to this world nor the next.
As the days passed, and Tissot became ever more preoccupied with his domestic problems, I perfected a philosophy in which pure information serves as the fundamental ingredient of all things. What is the swiftest way for knowledge to be conveyed, I asked Tissot while he sat writing a letter, but I received no answer. I decided that horses must provide the most rapid means for news to be disseminated over a large distance; and prompted by the sight of my sullen disciple, I began to analyse the way in which time might pass at apparently different rates for two people communicating only by way of written messages. Tissot and his wife were by this stage exchanging several angry notes in the course of a single morning; I recalled instead a story concerning a man banished for seducing a princess, whose tender letters to her, though written by him every day, became ever more protracted in their arrival owing to his continually increasing distance from her as he rode from the country. Ultimately, from the point of view of the princess who waited longer and longer for his next greeting, it was as if his days were extending themselves indefinitely. His life was slowing down beneath the weight and torment of his exile, and his final letter promised a sequel which never came. The passage of time for him, as perceived by his lover, came to a halt when he reached the remote northern border of their land, from which letters take forever to arrive. I tried to explain to Tissot my theory that simultaneity and time itself must be regarded as relative concepts, governed by the motion of horses; but we were interrupted by a knock on the door, given by a lad who brought yet another of Madame Tissot's tirades, penned by her only a moment earlier, in which she reaffirmed her willingness to forget their differences as long as Tissot would stop associating with me and give up all this nonsense of his about a bad foot and her mother's cooking. I am pleased that Tissot chose to remain under my roof a little longer; it saddens me that he profited so little from it.
His poor grasp of my theories emerged some days later when, his sister being about to give birth, Tissot payed for a baby girl to be sent into the room, believing it would make the new child twice as likely to be born male. My pupil however gained a niece; and I found no difficulty in explaining the fallacy of his reasoning. Tissot had merely misunderstood my remarkable Paradox of the Twins, which states that if a boy tells you he has a sibling, then the probability of it being a sister is not a half, but two thirds. Tissot showed a similar misunderstanding of my teaching when, exasperated by his continuing moroseness and his near-permanent occupancy of my writing desk, I said to him, "Next week I am going to bring your wife here so that you can speak to her in person and sort out your difficulties. I know you don't want to see her, and so I shall not tell you which day she will arrive; but you can be sure that you'll meet her before the week is out."
Tissot knew his wife would not be brought to confront him next Friday, because in that case he could be certain by Thursday evening that she must be coming, and he could make himself absent. But equally, I would also have to avoid Thursday, since otherwise he would be forewarned when Wednesday passed without a scene. Dismissing every other day in a similar manner, Tissot concluded that his wife could never show up unexpectedly to harangue him; but on Thursday he answered the door to be greeted not only by her, but also by her mother, both of whom boxed him soundly about the ears while I made myself scarce, quietly judging that so poor a logician deserved everything he got.
I continued my investigations unaided while Tissot began to use my desk as his centre of operations for a proposed legal action against his mother-in-law. By now I was fully convinced my new theory would provoke an upheaval among learned men as profound as that caused by Copernicus. I prepared to communicate my ideas to Monsieur D'Alembert, regarding him as one fit to recognise their true worth; but of the shameful treatment I received from him I shall say nothing here, except to indicate it as the reason why I have been so assiduous in finding authority for my philosophy among the numerous writers whose work is assembled here in this Encyclopaedia.
As for Tissot, though I was eventually sorry to lose my assistant, I was pleased he never had recourse to the law. He reached a compromise with his wife, and as he prepared to leave I told him that if there was one thing I hoped he had learned from me, it was that no problem is insurmountable, as long as we can recognise its true nature. I reminded him of a story concerning the Emperor Nero, who is said to have objected to the way in which a certain mountain obstructed his view of the sunset, and who therefore ordered it to be removed, a stone at a time. Ten thousand men laboured for ten years to complete this task; but at what point could the mountain be said to have finally disappeared, when was its final stone taken? Philo of Argos, a Xanthic philosopher, bravely informed Nero that since a mountain cannot have a final stone, it must be either infinite or non-existent, and the emperor's men had toiled to shift an obstacle which dwelt nowhere but in their vain ruler's imagination.
It is said that during the following years Rosier perfected his theory to such an extent, and felt so indignant at the indifference shown to him by the scientific establishment, that he personally undertook a complete rewriting of the Encyclopédie in the light of his doctrines. Of Rosier's Encyclopaedia, however, no known trace survives.