Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?

The jury is still out on whether Einstein was right (QM cannot be considered complete) or Bohr (yes it can). There is a remarkable proof by Bell that no local hidden variables theory will be able to reproduce the predictions by Quantum Mechanics, and this proof seems to favour Bohr's point of view. After Bell, others, such as Clauser et al. have made similar claims that to a lesser degree rely on the assumptions that underlie Bell's proof, such as the soundness of counterfactuals.

In cosmology, before you are able to know the distance between two celestial objects or the time difference between the Big Bang and Now, and before you do understand the angular distortions of milky ways that lie behind other milky ways, you have to know almost everything else that can be known. The concept of distance is fantastically theory-laden. The simple, even childish question: "How big is the Universe?" can even today not be answered.

How enormously different is our thinking about geometry on the scale of our close neighbourhood, down here on the Earth's surface. We take it as a given that distances, angles, and time lapses can be measured with measuring sticks, protractors, and clocks. The founding fathers of the Relativity theory and of that other great theory, Quantum theory, even insist that we describe our physical experiments in the words that we know from Classical Physics, which is the physics before the advent of the Quantum theory. The implication of this insistence on classical wordings is that we describe our experiments as though they take place against an Euclidian background, where objects have coordinates that we can reveal at any time by looking at the tick marks on measuring sticks, protractors, and clocks.

Sometimes, especially in the Quantum realm, phenomena stick out because they do not seem to be describable in classical terms. Such phenomena are said to display "entanglement" and prove that "non-locality" is real. So we end up with a classical language that is stained by a couple of words that lack a meaning in the classical sense. Something is wrong. Nobody can rightfully claim to fully understand this language.

This text takes the reader out on a journey where we forget that we have measuring sticks, protractors and clocks. But we do not venture in non-classical language, if such a thing exists at all. Instead, we give ourselves the freedom to not say anything about geometrical whereabouts until we have understood almost everything else, just like we do in cosmology.

Bell's and others' writings about hidden variable theories almost always are rather unpronounced as to what space-time, the backcloth of Bell's experiment, or rather of the Bohm-Aharonov experiment, is. This is surprising from a General Relativistic point of view. Read for example Hans Reichenbach's The Philosophy of Space and Time and then try to apply what you have learned as you read about the EPR experiment. But notice: the Special Theory of Relativity is already taken into account in all discussions about the EPR experiment, and is even seen as a necessary prerequisite. It is only as you step up to studying the foundations of General Relativity that new and relevant considerations come into view.

With the following story I want to illustrate my point. The story, although very unlikely to happen, is realistic. Care has been taken to let nothing a-causal slip in. Although the characters may seem to act in ways that are more based on superstition than reason, the fiction is not in conflict with known physical laws. As a special case in point, the validity of the quantum formalism is not disputed. This is all old school physics and not about yet another loophole.

A Cosmic EPR Experiment

Bart Jongejan

A catastrophe

Long, long time ago and now almost forgotten, there were two planets that went by the names of Bryiks and Heolaam. They were populated by intelligent species of which very little is known. These peoples' true names have been lost and therefore we call them Bryiksians and Heolaamites.

Bryiksians were making a living of agriculture, cultivating their small fields to make the best out of it. Bryiksians did not have much success planning the proper time for seeding and harvesting, and therefore they sought advice from wise men.

Many light years from Bryiks, the Heolaamites spent their days as hunters of the shy animals as they went to their drinking place during twilight. The life of the Heolaamite hunters was just as miserable as that of the Bryiksians. It was never to say whether the animals came to the drinking place at dawn or at dusk. The hunters' life style was an impediment for lying in wait at both times, so they often went to sleep with a hollow feeling in their belly. To improve their chances, the Heolaamites had subjected themselves to a cast of priests.

One day, an enormous explosion blew apart the far away star Nektuura Alpha. Nektuura Alpha was one of the two stars controlling the Nektuura solar system. Nektuura's inner planets turned into dead planets, but against all odds the people on the fifth planet, Cricsis, survived. These people were highly developed, the result of their willingness to bet and take risks. There were rumours that the explosion was the result of a nuclear experiment that had run out of control, but nobody was sure. After the explosion the complete scientific staff of the Campana Institute for Quantum Experiments was missing and the Campana laboratory fell into a deplorable state. Its walls begun to crumble and, without anybody taking notice of it, puffs of smoke and malodorous gas leaked away through holes in the roof. Nobody did really care, because taking risks was in high vogue on Cricsis.

Ask heaven

By the time the intense radiation from Nektuura Alpha had traveled many light years through outer space and reached Bryiks, the explosion was seen by the wise men. Their power over the Bryiksians increased overnight when they told the farmers that the heavens had given a sign, because many simple minds also had seen something very extraordinary in the skies.

The Bryiksian wise men built a temple. Only a small group of wise men was allowed access to the temple. Nothing ever became known of what was inside the temple to anybody outside this small circle of wise men.

The temple was not a place for contemplation but rather a place where the farmers could receive advice from the wise men - rumours went that there was an oracle inside the temple. An applicant for advice had to give up an agricultural tool that was dear to him and leave it behind on the doorsteps to the temple. The wise men kept all those utensils in a safe place inside the temple, and quite likely they became part of the oracle. All those who had sacrificed a utensil would receive an engraved seal as proof that they were entitled to receive advice in all future ceremonies of the Oracle.

Time went by. Once in a while the wise men announced that the heavens were to give advice to one single Bryiksian. Each Bryiksian farmer in need of advice would plow the wise men's fields during the night. At sunrise the farmers measured how much each of them had plowed and he who had tilled most was pointed out as the coming receiver of advice. The wise men took notice of the engraving on the lucky person's seal and then withdrew into the temple. When they came out, they addressed the grateful farmer during a short ceremony. The advice was proclaimed aloud and written down on a paper roll, and the farmer's seal was set next to it. The advice was nothing more than a simple "do it" or "don't do it", so the winning farmer should in advance have thought of something he could choose to do or not to do, and then he could apply the wise men's advice.

The paper roll started with a sign that symbolised the vision of an exploding star. After this symbol followed a list with two columns. Each ceremony of the oracle concluded with the addition of a new line to the list, the first column containing the seal of the farmer who had won the advice contest and the other column containing the wise men's advice, a V for "do it" and an O for "don't do it".

Some seals occurred considerably more often on the paper roll than average, and that was because not all farmers were in a like position to work for the wise men, and some might not always be willing to spend the whole night outside. As to the given advises, the farmers knew well that the chances of getting a V or a O were the same and that to predict the answer was just as hopeless as predicting 'head' or 'tail' when tossing a coin. Some farmers went one step further and said that the oracle was no better at giving useful advise than a coin, but such heretics stood quite isolated from the other farmers.

On Heolaam, far, far away from the Nektuura solar system, and even farther away from Bryiks, the explosion was also seen. The priests, like the wise men on Bryiks, were inspired by the celestial event to build a temple. The priests let nobody enter and leave the temple. They told the hunters that the temple concealed a secret power that could answer questions of a practical nature, and they urged the hunters to bring a personal trophy to the temple if they wanted to enjoy the oracle power of the temple. The priests would keep the trophies in a safe place inside the temple. As a proof of ownership, each Heolaamite hunter who had brought a trophy to the temple, received an iron ring with a distinctive inlaid design of ivory.

Like the Bryiksian wise men, the Heolaamite priests held ceremonies at times that only they knew about in advance. The oracle spoke for only one hunter at a time, so to find out who was going to receive advice, the priests released the Aima, a snow rat (rattus nix) that the priests had raised especially for this occasion. The hunter who brought back the Aima was the winner. The hunters were mostly interested in knowing whether it was best to hunt just after dawn or to hunt before dusk, so the priests' answer was either "dawn" or "dusk". To put down the verdicts for eternity, the priests used animal skins that they stitched together to form a journal. The design on the elected hunter's ring was carefully copied to the parchment. The verdicts were represented by holes cut in distinctive patterns next to the elected hunter's ring design. A "dawn" verdict was depicted by five holes in the corners of a regular pentangle and a "dusk" verdict by three equidistant holes on a row. Perhaps some hunters were more successful at bringing back the Aima than others, but each hunter was just as likely to get the "dawn" answer as the "dusk" answer, and there were no tendencies in the sequence of answers that allowed prediction of the next answer any better than by flipping a coin.

These temple ceremonies went on for generations. The Bryiksian seals and Heolaamite rings were handed from generation to generation, and generation after generation followed the advice given by the wise men and the priests. The list on the Bryiksian paper roll contained more entries than most inhabitants on the planet Bryiks were able to count. And the Heolaamite priests had to stitch ever more parchments together so that all ceremonies could be properly performed with the required punching of ever more holes, until ...

Visitors

One day, captain Wejurch of the Alic Wee flipped the deceleration switch. The ionic front thrusters spew a blinding plasma and the Cricsian flying saucer coasted into orbit around Heolaam. Wejurch and his crew looked at the world below and then descended to a place that had drawn their interest. They landed a few hill tops away from the temple, or rather, what had been the temple. Just at that moment a revolt was taking place against the priests. In their anger, the hunters had completely shattered the building. All that was left were rubbles. Silence fell when the foreigners came nearer. A priest held the journal of parchments in the air and with a blank gaze stared at Wejurch, who seemed to be the leader of the visitors. Wejurch understood why the priest looked at him and he nodded to indicate his consent. While the priest put the parchments in Wejurch's custody, his crew tried to find out what had been happening. They understood that the priests as of late had been unable to give advice, as though whatever was in the temple was on strike. The hunters explained by gesticulating and pointing at their engraved rings that the priests had broken a deal. Wejurch thought he knew why the ceremonies didn't occur anymore, but there was nothing he could do to reconcile the parties. The visitors returned to their space ship and begun a long journey back to Cricsis, the parchments in a safe place aboard.

At about the same time when the Alic Wee arrived at Cricsis, another space ship, the Bo Cowzay, returned from Bryiks. Its crew told about the turmoil they had met when they visited the Bryiksian farmers. Other than descriptions from farmers, no trace of the wise men was left. The wise men's temple was burnt to ashes. By chance, the Cricsian space farmers noticed how the wind was playing with what looked like a fluffy stick. It was the paper roll, partly buried under the loose sand of the temple's parvis and already disintegrating at the unprotected free end. Since the Bryksians clearly had lost all interest in the document, the Cricsians picked it up and brought it to their own planet for a closer analysis.

Cause and effect

Apart from the scant accounts that the farmers and hunters had been able to give the visitors during their short visits, the only information about what had happened during the temple séances had to be found in the paper roll and the journal of parchments. But nothing did the Cricsians know about what had been happening inside the temples. Or did they? How could it be that both space ship crews were witnessing the strikingly similar consequences of religious upheaval? Had the timing of their missions anything to do with the timing of those tragic events? Was there a common cause for all this? And had the common cause to be found on Cricsis?

Since the explosion of Nektuura Alpha the planet Cricsis had orbited around the remaining central star, Nektuura Beta, in a trajectory much more stable and predictable than previously. Reflecting the changed circumstances, the people of Cricsis had evolved into a race seeking certainty and predictability, forsaking excursions into the unknown and speculative. As the ages went by the Cricsian power agency, during a rigorous check of the finances, discovered that part of the energy production at the electricity utility seemed to disappear in the blue. There was a leak. It turned out that delivery of electricity to the Campana laboratory, although it had been unmanned since that day when Cricsians woke up under one, instead of two suns, had not been stopped. The electricity utility sent inspectors to the laboratory and they found two robotic instruments that seemed to be aimed at two distant constellations that we know as the Bryiks and Heolaam solar systems. On the floor lay a piece of paper, almost crumbled beyond recognition by the forces of time and an oxidizing atmosphere, with a handwritten notice saying

spukhafte Fernwirkung?

Spooky action at a distance?

The note with the mysterious words was later recognised as the start of a new scientific endeavour in the strange world of causation where no causation should exist. At that time, the inspectors did not know that, and they summarily pulled the chord to the robotic machine. For completing the financial statement of the utility with a full account of the lost energy, space ships were sent off to Bryiks and Heolaam to find out what the energy had been used for. Would the travellers be able to find any evidence of events that could be related to the activities of the robotic instruments under the roof of the Campana laboratory? Since the instruments had been silenced before the space ships lifted off, the crews knew that good places to look for such evidence were areas with visible traces of the recent sudden termination of old traditions.

When the space ships had returned, the secretary general of the Cricsian Academy of Science, Tuelleor Raxep, decided to do a comparative study of the of the book and the paper roll. Strikingly, the number of ceremonies in the journal and in the paper roll were the same. Obviously at least something had been coordinated between the two planets. Was this the spooky action at a distance that the note hinted at? Or was there a more natural explanation, namely that the robotic machine in all those years had dictated when a session, involving ceremonies on each of the two planets Bryiks and Heolaam, had to take place? But why would the scientists in the Campana laboratory in their days have been interested in dictating ceremonious events on distant planets, like otherworldly toastmasters? Tuelleor Raxep decided to compile a new data set, consisting of all the data in the journal, ceremony-wise paired with all the data in the paper roll.

Seeing that there were two possible verdicts in the journal as well as in the paper roll, he choose to denote them with '+' and '−'. The V's in the paper roll and the five hole patterns in the journal were transcribed as '+' and the O's and three hole patterns were transcribed as '−'. He also gave all farmers and hunters unique names by inventing names for each seal and for each ring design.

This is how a tiny part of the data set looked like. A thin line separated a pair of ceremonies - a session - from the next and a thicker line separated the ceremonies from each other, a vague reminiscence of the enormous spatial separation at which the ceremonies had taken place.

The committee adopted Sorbil Fig's Book of Missives as a valid scheme to explain how the combination of the book of parchment and the paper roll could display correlations between non-interacting civilisations. This solution gave a completely causal explanation of the correlations between the verdicts on the two planets. The committee agreed that no spooky action at a distance had been in play. But to be completely sure, they had to check that the real data could be fit into a correlation separation table along the lines of the scheme. Yrfes Fig was appointed to this task.

The failure of the Book of Missives

After filling out all cells the table looked markedly different from the table constructed by using Sorbil Fig's book-of-missives scheme, Table 5. It was immediately clear that homogeneity was not exact: in some columns and rows some values were lacking, while others occurred more than once. But it seemed unlikely that this was an essential difference. More disturbing was that the fractal appearance of Table 5 was completely absent in Table 7c. If you looked closely at Table 7c you saw an irregular and grainy pattern of dark and light cells that was lost at larger scales. Yrfes demonstrated this for himself by selecting a lot more farmers and hunters, 1024 of each, and plotting their correlations with just one pixel for each farmer-hunter pair on the interactive wall in the meeting room. (See Table 8)

A game of Chapshap

The four brave committee members, their high expectations being shattered by Yrfes Fig's conclusion, took a break to play a card game called Chapshap.

Exit Book of Missives

To get a feel of why the Book of Missives did not produce any Poirot, the committee decided to work backwards from hypothetical Poirot data to a Book of Missives that produces such data, and to see where the attempt got off the track.

To start, the committee made up a set of four correlations that would guarantee the upper Poirot value.

They decided that Abrafo and Jun always would receive the same verdict (+ + or − −), but that Abrafo and Gina, Dede and Jun, and Dede and Gina always would receive opposite verdicts (+ − or − +). These correlations are summarized in Table 14.

From Table 14 one immediately sees that Abrafo and Jun must have the same number in a fictitious Book of Missives: they are each other's perfect twins. One also sees immediately that Abrafo and Gina and Jun and Dede are each other's inverse twins. However, because Abrafo and Jun have the same number, their inverse twins also must have the same number, which is the same as to say that these inverse twins are twins for each other. But this contradicts the assumption that Dede and Gina always should obtain opposite verdicts!

Sorbil tried to defend the Book of Missives by pointing out that the real data were not quite like the hypothesized Poirot producing Table 14. The real data contained "Campana Wins", but no Poirots were ever found in the real data either. The other three members of the committee, reminding Sorbil of the fact that the Book of Missives not even could generate Campana Wins, almost fell over each other to explain to Sorbil that the Book of Missives was a lost case.

Beyond the Book of Missives

The Book of Missives was clearly not the way by which the outcomes of the ceremonies on Bryiks and Heolaam had been causally coordinated, but there might be another way to causally coordinate spatially widely separated events. Only if it could be proven that there are no such ways, the committee would have to conclude that the existence of Campana Wins in the real data, as opposed to their absence in the data produced by the Book of Missives mechanism, had no causal explanation and that they therefore had to be ascribed spooky action at a distance.

Tuelleor Raxep noted that the Book of Missives did not distinguish between the farmer or hunter that is elected for receiving a verdict and all other farmers and hunters. If the wise men and the priests really produced their verdicts from a book of missives, they were free to elect any number of farmers and hunters in the same session and give them all their own personalised verdict. But neither the Bryiksian paper roll nor the Heolaamite parchments gave any hint that more than one farmer or one hunter was ever elected for any ceremony.

Oeg Fig wondered whether the elected farmer and hunter could be incorporated into a causal explanation without compromising the freedom of the farmers and hunters to decide how much they wanted to invest into a coming competition. The nice thing about the Book of Missives hypothesis was that it allowed a clear separation between the common cause on the one hand - the production of the Book of Missives at some time preceding the sessions - and the election of the farmers and hunters at later times and in spatially separated regions. Any alternative solution should exhibit the same separation of roles, because otherwise the election of a farmer or a hunter, instead of playing an independent role in the determination of a verdict in a far away part of the universe, would itself be determined by a common cause in the common past of the elections.

Oeg Fig proposed a direction in which to look for a more sparse solution than the Book of Missives that might incorporate the good quality of the Book of Missives without producing a plethora of verdicts. His idea was to replace the book containing verdicts with something that only was a seed for giving a verdict. The determination of the verdict would be postponed until the point in space and time where a farmer or hunter was elected. When a farmer or a hunter was elected, a verdict for the elected person would materialise, while all other farmers and hunters only would have been allowed to speculate about what verdict they would have received, had they been elected. They would have been free to imagine that the verdicts they didn't receive were real and "out there", but not known or even knowable, not even to their spiritual leaders. They might also nurture the opinion that the not received verdicts had no existence. It wouldn't make any difference which opinion they held.

This sounded all very vague, but Oeg Fig presented an analogy that at least partly illustrated his point: the measurement of the size of an object. The size of an object is something inherent to the object and yet the outcome of a measurement cannot be determined unless a unit length is chosen. So in a sense, the object is or carries the seed for the outcome of a measurement, but it does not determine the outcome on its own. However, this example does not preclude the simultaneous or successive selection of different units of lengths and the measurement of the object's size in terms of each of these units.

The problem with being able to do many measurements on the same seed is that the wise men and priests might lie several times about whom was elected, combine each counterfactually elected person with the seed and in that way derive a table that, like a book of missives, prescribes the verdict for any farmer or hunter before any farmer or hunter was elected, and then we would again be unable to explain the Campana Wins in the data. Therefore Oeg Fig added that the solution, if it existed, had to preclude this possibility.

We normally assume that measurements can be repeated, unless a measurement does something irreversible to an object. But if you think about it, it is clear that all measurements that leave a trace in a memory are irreversible. Reverting a measurement would also involve wiping the trace in the memory, which is utterly contraproductive. So if the wise men or priests had plans to secretly create a Book of Missives, they would necessarily change the seed every time a new verdict was written in the book. In the end, the verdict given to the really elected farmer or hunter would be seeded by a combination of the original seed and all the measurements that the wise men or priests had performed. Conversely, if the wise man and priests were aware of this conundrum and did revert each measurement so as to undo the change inflicted to the seed, the last created verdict in the book would have to be wiped out before a new verdict could be measured, making it impossible to fill out a complete column in the book. And if the wise man and priests had peeked at some of the data, they would have to forget these results before performing the next measurement. So it seems natural to think of measurements as firework crackers that consume the stuff that causes the visual display and to think of repeatable measurements as not quite precise and in need of an explanation.

A distanced look

The correlation separation table constructed by means of the scheme proposed by Sorbil Fig - the Book of Missives - reminded Yrfes Fig of a distance table, for example a table from which you can read all the walking distances between Cricsis' main cities. Yrfes wanted to know whether the superficial similarity was incidental, or whether the correlation separations had anything whatsoever to do with walking distances.

Yrfes wrote down what he knew for sure about distances:

Neighbourhood

If you walk a small distance away from home, you are still in your neighbourhood. Everybody in your neighbourhood has only a small walking distance to everybody else in your neighbourhood.

Triangle Inequality

If you live in Ab and you travel a long distance to your friend in Tuv, you may want to sleep in a hostel somewhere on your way, lets say in Nop, before you reach Tuv. The next day you can travel the other part of the journey from Nop to Tuv. Now suppose that the next time you visit your friend in Tuv, you manage to do the journey from Ab to Tuv in one day, because you need fewer stops for looking at the map or taking pictures, say. Under the second journey you have travelled the same distance as under the first journey, or perhaps less, because the first time you probably had to deviate from the planned route to get to the hostel in Nop. So whereas the night at the hostel may have lengthened the trip, you would not expect ever to be able to shorten your trip by staying at hostels.

Degrees of Freedom

If you can travel in all directions, as long as you stay to the ground, you can enlarge your visitable world about fourfold by traveling twice the distance. If you can dig into the ground, dive into the see, and fly in the air, your world will even be enlarged eightfold. In the other extreme, if you are confined to travelling in only one direction (and back again), your known world grows at the same rate as the distance you travel.

If your world is finite but endless, such as the surface of a planet, there is a maximum walking distance you can get away from home. You will also notice that as you are making trips further and further away from home, the part of the world that is knowable to you will grow slower and slower if your world is of a finite size. When all of the world is knowable to you, you are able to walk the maximum distance from home.

Not all numbers that grow as you move away from home are good walking distance measures. For example, during the first day of your first journey from Ab to Tuv with a stop at Nop you travelled a distance corresponding to crossing the size of a city with 9 000 000 people and the second day you travelled the distance corresponding to crossing a city with 4 000 000 inhabitants. The sum of these numbers is 13 000 000, so 13 000 000 would be the maximum population size of the city that you would have crossed in both days together if population size was a good distance measure. However, If you had travelled from Ab to your destination Tuv in one day, you would have crossed a distance corresponding to a city with many more than 13 000 000 inhabitants, namely an enormous city with a population size of about 25 000 000. Because the Triangle Inequality says that you cannot shorten the total walking distance by staying at hostels, the population size of a city is not a good measure of walking distance. However, it can still be an indicator that only needs a translation step to turn into a measure of walking distance. For example, the square root of the city population size would be a fine walking distance measure, because

$$\sqrt{9000000}+\sqrt{4000000}=3000+2000=5000=\sqrt{25000000}$$

Why would it be interesting to use population size as a measure of distance? As we saw earlier, the correlation separation table that was built from the true observations (Table 7c) exposed a property we called homogeneity: in each column and each row, most correlation separation values occurred once. The left column and the top row show that as you let the correlation separation grow from 0 to the maximum value, you add, apart from only a few exceptions, one person for each notch up the separation scale. That is, in the real data a given correlation separation is proportional to the size of the population consisting of the persons with that correlation separation or a correlation separation that is less than the given correlation separation:

$$\mathrm{homogeneity\; condition}\Rightarrow \mathrm{population\; size}\propto \mathrm{correlation\; separation}$$

Because the committee's assignment was to explain the relation between the data sets that were brought back from Bryiks and Heolaam, the committee decided to reject any explanation that did not fulfil the homogeneity condition, and thus accepted that correlation separation had to be proportional to population size in every proposal for a solution they would come up with.

A walk trough the Book of Missives

To see how distances could play a role in the correlation separations brought about by the Book of Missives, Yrfes Fig first tried to simply equate the correlation separations implied by the Book of Missives with walking distances. All three properties of distance mentioned above seemed to be fulfilled. The farmers and hunters with numbers 5, 6, 7 and 8, according to Table 5, seemed to form a neighbourhood with a greatest walking distance of 3 from one member in the neighbourhood to any other member. Also, it never was the case that the sum of two correlation separations with a stop at a hostel was less than the correlation separation between the start and the end of the journey. For example, the correlation separations between number 1 (home) and number 5 (the hostel) is equal to 5 and the correlation separation between 5 (the hostel) and your friend at number 16 is equal to 11, giving a total correlation separation of 15, which so happens to be the correlation separation between 1 (home) and 16 (your friend).

The example showed that 5 (hostel) seemed to be on the route from 1 to 16. Was that a coincidence? Yrfes tried other hostels. A hostel further from home was number 12 with a correlation separation of 11 to home. But again, the total separation added up to 15, because the separation between 12 and 16 was 4.

This suggested that home was at one end of the world and 16 at the other, and that all other numbers lined up in between at regular correlation separations. That would mean that there was just one degree of freedom, the number of visitable persons growing in proportion to the distance from home. But there was something that didn't fit in this picture: the correlation separation between number 2 and number 15 was also 15 and the correlation separation between number 3 and number 14 likewise, so there were many pairs of numbers that seemed to occupy the two ends of the world, while the other numbers lined up between each of these pairs. The configuration that came to mind was that all numbers were arranged on a circle in such a way that 1 and 16 were opposite each other and likewise 2 and 15, 3 and 14, and so on.

This idea at first seemed to work right. Yrfes selected the numbers 1, 2, 3 and 4 and he arranged them on a circle in such a way that places were separated further if the correlation separations were greater. He did the same with other quartets of person numbers. See Fig. 1.

Yrfes realised that all persons not only were at one of the end points of a diagonal through the circle, but that they also were at the top of a right triangle, and that therefore the distance along one of the sides always was dictated by the distances along the other two sides by the relation: $${\left(\mathrm{distance\; along\; short\; side}\right)}^2+{\left(\mathrm{distance\; along\; other\; short\; side}\right)}^2={\left(\mathrm{distance\; along\; long\; side}\right)}^2$$

This could only be the case if not the correlation separations themselves, but their square roots were equated to distances, and two (adjacent) trajectories always were perpendicular. With such a definition of walking distance, doubling your walking distance quadruples the number of people you can reach, as the following example shows. Let us assume that you are number 1 and that you are going to visit number four. That is a walking distance √3. Within that distance you can visit the three persons numbered 2, 3 and 4. Doubling this distance, √3 + √3 = √12, you have found the walking distance to person with number 13, and indeed, as you can test in Fig. 1, you can reach four times as many, 12, different persons at shorter walking distances than person 13.

But that should not let us conclude that there is a two-dimensional world underlying the Book of Missives. Interestingly, if the visitable population size is proportional to the correlation separation, you can, at least in the case of the correlations inherent to the Book of Missives, argue for any number of dimensions, because setting the walking distance equal to the correlation separation to the inverse power of the number of dimensions is tautologically equivalent to computing the population size by taking the walking distance to the power of the number of dimensions, for any non-zero number of dimensions. $$\begin{array}{ccc}\left.\begin{array}{c}\mathrm{walking\; distance}=\sqrt[\mathrm{dimensions}]{\mathrm{correlation\; separation}}\\ \mathrm{population\; size}\propto \mathrm{correlation\; separation}\end{array}\right\}& \Rightarrow & \mathrm{walking\; distance}\propto \sqrt[\mathrm{dimensions}]{\mathrm{population\; size}}\\ & \Rightarrow & \mathrm{population\; size}\propto {\left(\mathrm{walking\; distance}\right)}^{\mathrm{dimensions}}\end{array}$$

A spherical take on correlation numbers

Because we saw that homogeneity, in the case of the Book of Missives, implied that we cannot deduce the dimensionality of an underlying world, we might also be led to think that we cannot say anything about the dimensionality of a world underlying the real data. Nothing could be more wrong.

Yrfes had noticed that the maximum separation having the value '15' (see Table 7a-c and Table 12) was just a consequence of him having taken a look at a sample of 15+1=16 farmers and hunters. Factoring out the sample size minus one, the normalized correlation separation would vary between 0 and 1 where the value 0 corresponded to full correlation and the value 1 to full anticorrelation.

$$\mathrm{normalized\; correlation\; separation}=\frac{\mathrm{correlation\; separation}}{\mathrm{sample\; size}-1}$$

Even if all data were taken into account, the value of the normalized correlation separation would always be somewhere between 0 and 1. Furthermore, the normalized correlation separation that followed from the data that were brought back from Bryiks and Heolaam pertaining to two specific persons, was not influenced by how many other persons were taken into account.

Because the correlation separation between two persons seemed to be a property of some underlying world and not of the number of inhabitants in the world, Yrfes's idea was to evenly spread a two planets' worth of people over a closed, finite sized area, a spherical surface for simplicity. He had not decided whether that should be a surface with 1, 2, 3 or even more dimensions - in other words whether the people would be confined to a circle, a globe or a higher dimensional surface. Of course, he himself didn't have the power to shuffle people around, but he could do some modelling in a computer.

First Yrfes tried a very simple model of the world: a circle. The populations were evenly spread out over the circle. Yrfes defined "home" as the position of one of these people on one of the planets. Then Yrfes divided this world in concentric "rings" (which become pairs of points if the world is a circle) centred on "home" in such a way that between a ring and the next ring there are always the same number of people. If he had 1000 people in all, 500 'Bryiksian farmers' and 500 'Heolaam hunters', and between each neighbouring pair of rings there would be 20 people, about ten from each planet, he would create 49 rings, the last one of which would encircle a 'polar cap' around the point opposite "home".

Figure 2. Populations on spheres in one or more dimensions. If the populations can spread out in more than two dimensions, all points are projected in a way that suggests how the number of visitable persons grows with growing separation from your "home" at the "north pole", the orange coloured dot in the middle of the northern polar cap. For each person, only persons with the other colour are visitable.

0 ? 0.5*Math.PI : -0.5*Math.PI; return phi; } //http://www.filosophy.org/post/35/normaldistributed_random_values_in_javascript_using_the_ziggurat_algorithm/ function Ziggurat(){ var jsr = 123456789; var wn = Array(128); var fn = Array(128); var kn = Array(128); function RNOR(){ var hz = SHR3(); var iz = hz & 127; return (Math.abs(hz) < kn[iz]) ? hz * wn[iz] : nfix(hz, iz); } this.nextGaussian = function(){ return RNOR(); } function nfix(hz, iz){ var r = 3.442619855899; var r1 = 1.0 / r; var x; var y; while(true){ x = hz * wn[iz]; if( iz == 0 ){ x = (-Math.log(UNI()) * r1); y = -Math.log(UNI()); while( y + y < x * x){ x = (-Math.log(UNI()) * r1); y = -Math.log(UNI()); } return ( hz > 0 ) ? r+x : -r-x; } if( fn[iz] + UNI() * (fn[iz-1] - fn[iz]) < Math.exp(-0.5 * x * x) ){ return x; } hz = SHR3(); iz = hz & 127; if( Math.abs(hz) < kn[iz]){ return (hz * wn[iz]); } } } function SHR3(){ var jz = jsr; var jzr = jsr; jzr ^= (jzr << 13); jzr ^= (jzr >>> 17); jzr ^= (jzr << 5); jsr = jzr; return (jz+jzr) | 0; } function UNI(){ return 0.5 * (1 + SHR3() / -Math.pow(2,31)); } function zigset(){ // seed generator based on current time jsr ^= new Date().getTime(); var m1 = 2147483648.0; var dn = 3.442619855899; var tn = dn; var vn = 9.91256303526217e-3; var q = vn / Math.exp(-0.5 * dn * dn); kn[0] = Math.floor((dn/q)*m1); kn[1] = 0; wn[0] = q / m1; wn[127] = dn / m1; fn[0] = 1.0; fn[127] = Math.exp(-0.5 * dn * dn); for(var i = 126; i >= 1; i--){ dn = Math.sqrt(-2.0 * Math.log( vn / dn + Math.exp( -0.5 * dn * dn))); kn[i+1] = Math.floor((dn/tn)*m1); tn = dn; fn[i] = Math.exp(-0.5 * dn * dn); wn[i] = dn / m1; } } zigset(); } var z = new Ziggurat(); function initRandomUnitVector(dim,oldvector,olddim) { var sum2 = 0.0; var e; if(olddim == 0) e = []; else e = oldvector; var k; for(k = olddim;k < dim;++k) { /* var d = Math.pow(2, -32); var U = Math.random()+d; var V = Math.random()+d; var root = Math.sqrt(-2.0 * Math.log(U)); var XXd = root * Math.cos(2.0*Math.PI*V); e[k] = XXd; */ e[k] = z.nextGaussian(); } if(olddim == 0) { var a = 0.8 * e[0] - 0.6 * e[1]; var b = 0.6 * e[0] + 0.8 * e[1]; e[0] = a; e[1] = b; } return e; } function normalised(e,dim) { var sum2 = 0.0; var k; var ret = []; for(k = 0;k < dim;++k) { var X = e[k]; sum2 += X * X; } var f = Math.sqrt(sum2); if(f > 0.0) { for(k = 0;k < dim;++k) { ret[k] = e[k]/f; } } return ret; } function planetNo(p) { return p - 2 * Math.floor(p/2); } function Accent(p) { var planet = planetNo(p); var plan = plnt[planet]; return (plan[0] == p || plan[1] == p) ? 1 : 0; } function ei(dim,cx,cy,r) { this.dim = dim; var svg = this; var g = document.createElementNS(svgns,"g"); var homenode = document.createElementNS(svgns,"circle"); this.homenode = homenode; homenode.setAttribute("cx",850); homenode.setAttribute("cy",100); homenode.setAttribute("r",8); homenode.setAttribute("fill","red"); g.appendChild(homenode); var moredimensions = document.createElementNS(svgns,"circle"); this.moredimensions = moredimensions; moredimensions.setAttribute("cx",400); moredimensions.setAttribute("cy",150); moredimensions.setAttribute("r",20); moredimensions.setAttribute("fill","red"); g.appendChild(moredimensions); var fewerdimensions = document.createElementNS(svgns,"circle"); this.fewerdimensions = fewerdimensions; fewerdimensions.setAttribute("cx",400); fewerdimensions.setAttribute("cy",250); fewerdimensions.setAttribute("r",20); fewerdimensions.setAttribute("fill","blue"); g.appendChild(fewerdimensions); this.svg = g; this.globes = null; this.root2 = document.getElementById("svgwortel"); this.egg = function(r,x,y,angle,Z,stroke,fill) { var R = Math.sqrt(1.0 - Z*Z); // circle diameter (long axis of ellipse when circle is projected) var rx = Math.floor(r * R); var ry = Math.floor(r * R * Math.sin(angle)); var cx = Math.floor(x); var cy = Math.floor(y - r * Z * Math.cos(angle)); var ellipse = document.createElementNS(svgns,"ellipse"); ellipse.setAttribute("cx",cx); ellipse.setAttribute("cy",cy); ellipse.setAttribute("rx",rx); ellipse.setAttribute("ry",ry); var hstroke = ("000000" + Math.floor(stroke).toString(16)).substr(-6); var hfill = ("000000" + Math.floor( fill).toString(16)).substr(-6); ellipse.setAttribute("style","stroke:#"+hstroke+"; fill:#"+hfill); this.globes.appendChild(ellipse); this.ellipse = ellipse; } this.spot = function(r, x,y,theta,phi,R,stroke,fill,p,rl) { // theta == 0.0 : top // theta == pi : bodem // phi == 0.0 : midden // phi < 0.0 : links // phi > 0.0 : rechts /* A/a (elliptical arc) rx, ry x-axis-rotation large-arc-flag sweep-flag x, y Draws an elliptical arc from the current point to (x, y). The size and orientation of the ellipse are defined by two radii (rx, ry) and an x-axis-rotation, which indicates how the ellipse as a whole is rotated relative to the current coordinate system. The center (cx, cy) of the ellipse is calculated automatically to satisfy the constraints imposed by the other parameters. Large-arc-flag and sweep-flag contribute to the automatic calculations and help determine how the arc is drawn. the arc starts at the current point the arc ends at point (x, y) the ellipse has the two radii (rx, ry) the x-axis of the ellipse is rotated by x-axis-rotation relative to the x-axis of the current coordinate system. */ var X, Y, RX, RY; var large_arc_flag = 1; var sweep_flag = 1; var dx = r*Math.sin(theta)*Math.sin(phi); var dy = -r*Math.cos(theta); var cx = x + dx; var cy = y + dy; var R2 = dx*dx + dy*dy; var R1 = Math.sqrt(R2); var x_axis_rotation = 180.0/Math.PI*Math.asin(dx/R1); // positief: draaiing met de klok mee var flat = Math.sqrt(r*r - R2)/r;// cosine of angle away from direction out of the screen. if(Math.cos(theta) < 0) x_axis_rotation = -x_axis_rotation; X = Math.floor(cx - (flat*Math.abs(dx)-dy)*R/R1); Y = Math.floor(cy); RX = Math.floor(R); RY = Math.round(R*flat); var path = document.createElementNS(svgns,"path"); //--X; path.setAttribute("d","M"+X +","+Y +" A"+RX +","+RY +" "+Math.floor(x_axis_rotation) +" "+large_arc_flag +" "+sweep_flag +","+X +","+(Y+1) ); var tjek = path.getAttribute("d"); var hstroke = ("000000" + Math.floor(stroke).toString(16)).substr(-6); var hfill = ("000000" + Math.floor( fill).toString(16)).substr(-6); path.setAttribute("style","stroke:#"+hstroke+"; fill:#"+hfill); path.setAttribute("id",p+"."+rl); this.globes.appendChild(path); } this.slice = function(r,x,y,Zd,Zu,stroke,fill) { var Rd = Math.sqrt(1.0 - Zd*Zd); var xdL = Math.floor(x - r * Rd); var ydL = Math.floor(y - r * Zd); var xdR = Math.floor(x + r * Rd); var ydR = Math.floor(y - r * Zd); var R = Math.floor(r); var path = document.createElementNS(svgns,"path"); if(Zu <= 1.0) { var Ru = Math.sqrt(1.0 - Zu*Zu); var xuL = Math.floor(x - r * Ru); var yuL = Math.floor(y - r * Zu); var xuR = Math.floor(x + r * Ru); var yuR = Math.floor(y - r * Zu); path.setAttribute("d","M"+xdL+","+ydL +" A"+R+","+R+" 0 0,1 "+xuL+","+yuL +" H"+xuR +" A"+R+","+R+" 0 0,1 "+xdR+","+ydR +" H"+xdL ); } else { path.setAttribute("d","M"+xdL+","+ydL +" A"+R+","+R+" 0 0,1 "+xdR+","+ydR +" H"+xdL ); } var tjek = path.getAttribute("d"); var hstroke = ("000000" + Math.floor(stroke).toString(16)).substr(-6); var hfill = ("000000" + Math.floor( fill).toString(16)).substr(-6); path.setAttribute("style","stroke:#"+hstroke+"; fill:#"+hfill); this.globes.appendChild(path); } this.integral = function(x,n) { if(n > 1) return -1.0/n * Math.pow(Math.sin(x),n-1) * Math.cos(x) + (n-1)/n * this.integral(x,n-2); else if(n == 1) return 1 - Math.cos(x); else return x; } this.setMinMaxIntegral = function(dimensions) { this.maxintegral = this.integral(Math.PI,dimensions - 2); this.minintegral = this.integral(0.0,dimensions - 2); } this.expectationValueA = function(angle,dimensions) { var ret; if(dimensions == 3) ret = Math.cos(angle)/*inp*/; else if(dimensions == 2) { ret = 1.0 - (angle / Math.PI*0.5); } else { var inp = this.integral(angle,dimensions - 2); ret = 1.0 - 2.0*(inp - this.minintegral)/(this.maxintegral - this.minintegral); } return ret; } this.expectationValue = function(p1,p2,dimensions) { console.log('p1:'+p1); console.log('p2:'+p2); var A = points[p1]; var B = points[p2]; console.log('A:'+A); console.log('B:'+B); var inner = 0; var normA2 = 0; var normB2 = 0; console.log('dimensions:'+dimensions); for(i = 0;i < this.dim;++i) { normA2 += A[i]*A[i]; normB2 += B[i]*B[i]; inner += A[i]*B[i]; } inner /= Math.sqrt(normA2*normB2); console.log('inner:'+inner); var angle = Math.acos(inner); console.log('angle:'+angle); exp = this.expectationValueA(angle,dimensions); console.log('exp:'+exp); return exp; } this.angle = function(expectectationvalue) { var mina = 0.0; var maxa = Math.PI; var mine = this.expectationValueA(mina,this.dim); var maxe = this.expectationValueA(maxa,this.dim); if(expectectationvalue == mine) return mina; else if(expectectationvalue == maxe) return maxa; else { var h = 0.0; var i; for(i = 0;i < 100 && maxa - mina > 0.0000000001;++i) { h = (mina+maxa)/2.0; var e = this.expectationValueA(h,this.dim); if(e > expectectationvalue) { mine = e; mina = h; } else if(e < expectectationvalue) { maxe = e; maxa = h; } else break; } return h; } } this.fz = function(Z) { var ret; if(Z <= -0.999) { return -1; } if(Z >= 0.999) { return 1; } switch(this.dim) { case 2: ret = Math.cos((1.0 - Z) * Math.PI*0.5); break; default: var a = this.angle(Z); ret = Math.cos(a); } return ret; } this.density = function(Z) { var FZ = this.fz(Z) if(FZ != 0) return Z / FZ; else return 1; } this.col = function(Z,maxdensity,mindensity) { var dens; if(maxdensity > mindensity) dens = (this.density(Z) - mindensity) / (maxdensity - mindensity); else dens = 0.5; var allbits = 0xffffff; var idens = ((1.0 - (dens)) * 255.0); return (idens << 16) + (idens << 8) + idens; } this.setDim = function(dim) { this.setMinMaxIntegral(dim); this.dim = dim; var p; for(p = 0;p < 1000;++p) { var e = initRandomUnitVector(this.dim,points[p],DIMENSIONS); points[p] = e; } } this.setTop = function(top) { console.log("setTop:"+top); var planet = planetNo(top); console.log("planet:"+planet); var plan = plnt[planet]; if(plan[0] != top) { plan[1] = plan[0]; plan[0] = top; console.log("planN:"+plan[0]+" "+plan[1]); plnt[planet] = plan; } /* for(p = 0;p < 1000;++p) { rot[p] = points[p]; } */ var d; for(d = this.dim-1;d > 0;--d) { ptop = points[top]; var a = ptop[0]; var b = ptop[d]; var Determinant_ = Math.sqrt(1.0/(a*a + b*b)); for(p = 0;p < 1000;++p) { var e = points[p]; var X = Determinant_ * ( a*e[0] + b*e[d]); var Y = Determinant_ * (-b*e[0] + a*e[d]); e[0] = X; e[d] = Y; points[p] = e; } } var pl,pt; for(pl = 0;pl < 2;++pl) for(pt = 0;pt < 2;++pt) { if(plnt[pl][pt] == -1) return; } var separations = []; var sumseparations = 0; var biggestseparation = 0; var dBiggest; for(d = 0;d < 4;++d) { separations[d] = 1 - this.expectationValue(plnt[0][parseInt(d/2)],plnt[1][parseInt(d%2)],this.dim); console.log('d:'+d+' : '+separations[d]); sumseparations += separations[d]; } var minchsh = sumseparations - 2*separations[0]; var maxchsh = minchsh; var mind = 0; var maxd = 0; for(d = 0;d < 4;++d) { var chsh = sumseparations - 2*separations[d]; if(chsh < minchsh) { minchsh = chsh; mind = d; } if(chsh > maxchsh) { maxchsh = chsh; maxd = d; } } console.log('chsh: '+minchsh+' '+maxchsh); var difmin = -minchsh; var difmax = maxchsh - 4; dBiggest = difmin > difmax ? mind : maxd; var iA = parseInt(dBiggest/2); var iB = parseInt(dBiggest%2); var A = plnt[0][iA]; var B = plnt[1][1-iB]; var C = plnt[0][1-iA]; var D = plnt[1][iB]; console.log('AB:'+separations[2* iA +1-iB]); console.log('CB:'+separations[2*(1-iA)+1-iB]); console.log('CD:'+separations[2*(1-iA)+ iB]); console.log('AD:'+separations[2* iA + iB]); } this.Egg = function() { if(this.globes != null) { var y = this.root2.getElementById("globes"); y.parentNode.removeChild(y); } var globes = document.createElementNS(svgns,"g") globes.setAttribute("id","globes") this.globes = globes; this.svg.appendChild(this.globes) var yy = this.root2.getElementById("globes"); var desc; switch(this.dim) { case 2: desc = "Circle mapped onto 3-D sphere"; break; case 3: desc = "Normal 3-D sphere"; break; default: desc = "Higher dimensional sphere mapped onto 3-D sphere"; } var Z = 0.0; var inc = 0.04; var maxdensity = 0.0; var mindensity = 1.79769e+308; var rings = 0; for(Z = -1.0;Z < 1.0; Z += inc) { var temp = this.density(Z+0.5*inc); if(temp > maxdensity) maxdensity = temp; if(temp < mindensity) mindensity = temp; } if(maxdensity < 1.001 * mindensity) maxdensity = mindensity; var tilt = 1.0; for(Z = -1.0;Z <= 1.0; Z += inc) { // bin[Slice++] = 0; this.egg(160.0, 200.0, 200.0, tilt, this.fz(Z), 0, this.col(Z+0.5*inc,maxdensity,mindensity)); } /* var MaxSlice = Slice; */ for(Z = -1.0;Z < 1.0; Z += inc) { ++rings; this.slice(160.0, 600.0, 200.0, this.fz(Z), this.fz(Z+inc), 0, this.col(Z+0.5*inc,maxdensity,mindensity)); } var sintilt = Math.sin(tilt); var costilt = Math.cos(tilt); var colours = [0xff8000,0x00ff00]; for(p = 0;p < 1000;++p) { // var e = points[p]; planet = planetNo(p); var e = points[p]; var n = normalised(e,this.dim); // Normalized in dim dimensions! Not in three! var Y = n[0]; var Z = n[1]; var X = 0; /* var Slice = 0; var ZZ = -1.0; while(Slice < MaxSlice) { if(this.fz(ZZ) >= Y) { bin[Slice - 1] += 1; break; } ++Slice; ZZ += inc; } */ var theta = Math.acos(Y); //This ensures that the apparent // angle from a pole to another point is the true angle. // All other angles are distorted! // So we take the Y from the n-dimensional sphere as the Y // of the three-dimensional sphere. var phi; if(this.dim > 2) { X = n[2]; phi = Atan(X,Z); } else if(Z > 0) phi = -0.5; else phi = Math.PI - 0.5; if(Z > 0) { this.spot(160.0, 600.0, 200.0, theta, phi, 4*(1+Accent(p)), 2, colours[planet],p,0); } else if(Accent(p) == 1) { this.spot(160.0, 600.0, 200.0, theta, phi, 4*(1+Accent(p)), 2, (colours[planet]|0xaaaaaa),p,0); } var sintheta = Math.sin(theta); var X1 = Math.sin(phi)*sintheta; // NOT the same as X! var Z1 = Math.cos(phi)*sintheta; // NOT the same as Z! var Y2 = costilt*Y - sintilt*Z1; var Z2 = sintilt*Y + costilt*Z1; if(Z2 > 0) { var thetatilt = Math.acos(Y2); var phitilt = Atan(X1,Z2); this.spot(160.0, 200.0, 200.0, thetatilt, phitilt, 4*(1+Accent(p)), 0, colours[planet],p,1); } else if(Accent(p) == 1) { this.spot(160.0, 200.0, 200.0, thetatilt, phitilt, 4*(1+Accent(p)), 0, (colours[planet]|0xaaaaaa),p,1); } } // this.spot(160.0, 600.0, 200.0, 0, 0, 4.0, 0, 0xff8000,0,2); // this.spot(160.0, 200.0, 200.0, tilt, 0, 4.0, 0, 0xff8000,0,3); DIMENSIONS = this.dim; return 0; } homenode.addEventListener("click",clickhome,false); moredimensions.addEventListener("click",clickmore,false); fewerdimensions.addEventListener("click",clickfewer,false); this.setDim(this.dim); this.setTop(theTop); this.Egg(); var ip2 = document.getElementById("ip2"); ip2.appendChild(this.svg); document.querySelector('#svgwortel').addEventListener('click', function(event) { if (event.target.tagName.toLowerCase() === 'path') { console.log("klik:"+event.target.id.split('.')[0]); var the = parseInt(event.target.id.split('.')[0]); if(!isNaN(the)) { theTop = the; Ei.setTop(theTop); Ei.Egg(); } } }); }; var Ei; var dim = 2; function clickhome(evt) { Ei.setTop(theTop+1); Ei.Egg(); } function clickmore(evt) { document.getElementById("dim").firstChild.nodeValue = dim + ' dimension' + ((dim > 1) ? 's' : ''); dim += 1; Ei.setDim(dim); Ei.setTop(theTop); Ei.Egg(); } function clickfewer(evt) { if(dim > 2) { dim -= 1; Ei.setDim(dim); Ei.setTop(theTop); Ei.Egg(); } document.getElementById("dim").firstChild.nodeValue = (dim-1) + ' dimension' + ((dim > 2) ? 's' : ''); } function main(evt) { Ei = new ei(dim,100,100,45); } ]]> 1 dimension

Yrfes defined the correlation separation between the person at "home" and another person as the number of rings someone would have to cross to walk from "home" to the other person, divided by the total number of rings, thus: $$\mathrm{correlation\; separation}=\frac{N\left(\mathrm{rings\; crossed}\right)}{N\left(\mathrm{all\; rings}\right)}$$ Each person could only visit people with the other colour, so each person could make 500 different visits. This was how Yrfes modelled the circumstance that the bare data exhibited correlations between persons from different planets, but not between persons from the same planet.

Yrfes computed the correlation separation for each farmer-hunter pair in which "home" was a specific farmer, and built the the first column of a new correlation separation table in the same way as described for Table 7a, taking care to sort the rows in order of increasing correlation separation (decreasing correlation), or increasing number of rings to cross. This procedure was repeated for all farmers inside the first ring, but without sorting the rows. From now on, each person had its own row, defined by the sorting order of the very first iteration. When all persons inside the first ring had been "home", Yrfes picked a person outside the first ring, but inside the second ring and declared him to be the new "home". Yrfes repeated the computation for this second person and filled in the second column. And so he went on until all persons had been "home". This came out of it if the world was a circle:

Figure 3. correlation separation table for population on a N-Dimensional sphere. (From left to right and top to bottom, N = 2,3,4,5,6,7,8,10 and 20)

Spin

Yrfes Fig endorsed his uncle's idea that something that was not expressible in numbers was sent from Cricsis to the planets. If these things were feeble enough, the wise man and priests would have only one chance to let these things "talk" and decide the verdict, because any attempt to listen to the message would change or destroy it. Examples of such things could be pairs of tops spinning in opposite directions, which would be able to operationally define parallelism at a distance, without the need to invoke coordinates (which are numbers) and even without the need to worry about the space-time structure. The real data apparently allowed one to find out which farmer somehow was parallel with which hunter, the angle between a farmer and a hunter being defined by the angle between the points on the sphere that represented them. It was difficult to envisage other operational definitions of parallelism at a distance than by way of the constancy of the angular momentum of a material object.

Several committee members thought this would not work for the following reason. Once it had been established how the temples on Bryiks and Heolaam were spatially aligned with respect to each other, using the aforementioned pairs of spinning tops, it was possible to define a system of coordinates encompassing both temples. Knowing that angular momentum can be denoted by an axial vector, it would seem natural to imagine that during every session, the mediating spins could be decomposed in components, using the coordinate system as the framework for defining a set of basis vectors along which to take the components. But then, these committee members said, we would be back to the missives-in-a-book model. If an angular momentum can be fully described by its components (three numbers), then we can do without the angular momenta and just send the numbers. On these grounds they did not think that the angular momentum was any improvement over the book and they rejected Yrfes Fig's idea.

Yrfes Fig countered this "reductio ad absurdum" by saying that you cannot say much about the individual angular moments' components if you do not know the metric structure of space in detail. If you assume a flat or otherwise nice structure, then you can indeed use the parallelism of the temples to define components at all times, but how can we know that space time has such a simple structure? Especially if the angular momenta themselves and individually influence the metric structure, arguing like these committee members becomes similar to arguing that all particles in a gas have the same velocity, as defined by the temperature of the gas.

Yrfes Fig also said that his opponents' reasoning would lead to the absurd conclusion that sending numbers could help establish and maintain a system of coordinates incorporating both temples. Did his opponents think that numbers alone could define a coordinate system? If not, what else was needed and how much of that?

To explain the data, the physical objects that had been used for coordinating the verdicts on the two planets should somehow resist decomposition, because such decomposition would make it possible to create a Book of Missives. One way (perhaps the only way!) to understand how a physical object can escape decomposition is by assuming that there is no frame of reference to decompose into.

Remains the question how a model that is agnostic as to counterfactual verdicts can be shown to produce Campana Wins. Indeed, not assigning verdicts to persons that did not win the competition is a prerequisite, as the failure of the Book of Missives has shown, but not sufficient.

Think now, imagine later

1. Think of the farmers being represented by the item each of them brought to the temple. Likewise think of the hunters being represented by the items they brought to their temple.
2. Instead of manipulating symbols, the farmers' wise men and hunters' priests undertook something that cannot be replaced by consultation of a book of prescripts: they let something point at an item, the item belonging to the elected farmer or hunter, establishing a direction in doing so.
   • If you think that the pointing at an item can be fully described by symbols, then think again: Can you ensure that an alien intelligent race can construct a copy of an object that is sitting in front of you by following ever so detailed instructions that you sent to them? Wouldn't the reconstruction have a 50% chance of being mirrored? Wouldn't it be necessary to also send them a physical reference?
3. Instead of a book with prescriptions of how to decide the outcome of ceremonies, think of directions being sent, one direction for each session. During each session, the two temples receive the same direction. For the moment, do not try to picture a direction. Especially, don't make a projection of a direction on the model of physical space that you carry around in your mind. Also, the 'sameness' of directions is not meant in the usual sense in which two lines can be parallel in Euclidean space.
   • Do not think of the received direction as pointing at something or other, for example the item a fellow hunter or farmer brought to the temple. This explanation requires that you accept that the 3D world we perceive is a construction based on innumerable measurements of the kind involved in the cosmic experiment.
   • The sameness of the directions received at the temples is an after-the-fact attribute, given because there were correlations between hunters and farmers.
4. Between the direction received at a temple and the direction of the elected hunter's/farmer's item is an angle. You may think of that angle as something real, 'out there', but not measurable in practice, except for a single bit of information, telling whether the angle is acute or obtuse. There exists machines that can elicit this single bit of information. To obtain the bit of information that you want, as part of the preparation of the machine you have to turn in way so that it points in the direction of the elected person's item. More precise information about this angle, as everything else about the received direction, is hidden.
5. The answer to the hunter or farmer would be an arbitrary (but consistently adhered-to) mapping to the 'yes/no' like answers they receive from the priests/wise men.
6. Although distributions of angles at the temples seem to indicate complete randomness, the three directions involved in one session - the received direction, the direction of the farmer and the direction of the hunter - are not independent. That is, whereas any two of the involved angles are uncorrelated, the third angle cannot be chosen completely free.
   • This is easy to see with an example: if the farmer and the hunter have almost the same direction, then the angle between the hunter and the received direction and the angle between the farmer and the received direction must be almost the same.
7. Free will is allowed to decide the hunter's and farmer's directions. Free will has nothing to do with the received direction.
8. There are innumerable distributions of three vectors, two of which can be chosen at random, while the third is more or less restricted, all distributions having marginal distributions of angles that indicate randomness.
9. Find the distribution that reproduces the CHSH-statistics.
10. Even though it is possible to construct a complete model for what was going on in the temples it is not possible to do a replay of the cosmic experiment in a virtual world, involving real players with free will and real big distances between them to ensure that one ceremony's outcome cannot be used to make a last moment adjustment of the other ceremony's outcome. All virtual worlds consist of manipulations of dimensionless symbols, and as we have seen, the messages that have to be sent cannot be embodied by symbols.
11. Explanatory models that can reproduce the statistical correlations of the cosmic experiment must themselves utilize the kind of messages that were employed in the cosmic experiment. In other words: such models are not really explanatory, but pedagogical at best.

Definitions

ceremony

An event localised in space to a planet (Bryiks or Heolaam) and in time to about one morning.

correlation

If expressed numerically, a correlation of 1 means perfect agreement between the verdicts of a hunter and a farmer over a long run of ceremonies. A correlation of -1 also means a perfect agreement, but verdicts are always found to be the opposites of each other. A correlation of 0 indicates that a farmer verdict corresponds as many times with the same and the opposite hunter verdict.

correlation separation

An alternative way to numerically express a correlation. A correlation separation of 0 is the lowest possible correlation separation and corresponds to perfect correlation (1). If normalized, the maximum correlation separation is 1, corresponding to a correlation of -1. In some contexts correlation separation is expressed as a natural number.

session

A pair of ceremonies, one taking place on Bryiks, the other taking place on Heolaam. Pairing is done by combining the n-th ceremony in the paper roll with the n-th ceremony in the journal of parchments.

farmer

An animate agent with free will, inhabiting the planet Bryiks and acting independently of far away events on other planets, such as decisions taken by hunters (on Heolaam) and laboratory experiments (on Cricsis).

hunter

An animate agent with free will, inhabiting the planet Heolaam and acting independently of far away events on other planets, such as decisions taken by farmers (on Bryics) and laboratory experiments (on Cricsis).

item

An inanimate object representing a farmer or hunter. Items are, in their turn, uniquely represented by symbols.

priest

An agent following strict procedures, with no free will. Local of the planet Heolaam.

wise man

An agent following strict procedures, with no free will. Local of the planet Bryiks.

selection

Procedure by which, during a ceremony, a farmer or hunter, and therefore an item, is singled out. No two items can be selected during the same ceremony. Selection is decided by those who are nearby and have free will.

symbol

An element of meaning that an animate agent can be reminded of when observing a shape, rhythm or item. The same symbol can have more than one physical representation, thus establishing symbolic links between physical objects.

verdict

The outcome of a ceremony: one of two symbols

identity

An emergent property of things physical. When an animate agent is reminded of the same symbol on different occasions, the mind tends to ascribe unique identity to the physical causes of the aroused memory.

laws

A sine qua non for the existence of symbols and identity. And without symbols and identity, no laws.

log book

A memory of all ceremonies. Contains, for each ceremony, a symbol denoting the selected item and a symbol denoting the verdict.

Epilogue

Space-time may be emergent, an epiphenomenon. The kind of experiment that Bell envisaged is one way to get a closer view at what is underlying. This closer view is perceived as entanglement. As an analogue, spreading rings on a surface of water are clearly coordinated if seen as a deviation from quietness, but the dynamical laws of water molecules are the same. The rings are not any more correlated than the quietness. It is characteristic of the smallest constituent of a thing that it is behaving in a different way than the thing. A water molecule does not behave like a volume of water, a letter does not behave like a text, a corn of sand does not behave like a dune and a coin does not behave like a budget. Spin half particles are not oddly behaving guests in a normal world. They are the essence of the world.

YOU CANNOT GET SUCKED INTO A TRON WORLD AND GET BACK AGAIN

Why?

As you enter a tron world inside a computer, you loose any sense of orientation, for good. Left and right cannot be expressed in bits. Even if there were a system of coordinates, in terms of which you always had x, y, z, and t coordinates and you were heading in direction theta, phi, with your feet pointing to xi zeta, you would not be able to use your coordinates when you wanted to return. A computer is a Turing machine, a long tape and a machine moving back and forth, with no concept of orientation.

The CHSH inequality

It is fairly easy to understand that the sum of the shortest separations between A and B, between B and C and between C and D cannot be shorter than the separation between A and D. $$\overline{\mathrm{AD}}\le \overline{\mathrm{AB}}+\overline{\mathrm{BC}}+\overline{\mathrm{CD}}$$ That is the triangle inequality applied twice. But the CHSH inequality is about two, not one, limit. What is the other limit? The separations CHSH look at have a maximum. While the smallest separation corresponds to the highest correlation, the largest separation corresponds to the highest anticorrelation. But we could just as well have defined separation in terms of anticorrelation, the smallest separation corresponding to the highest anticorrelation and the largest separation to the highest correlation. If separation has a range from 0 to 2, we have the simple conversion rule $$\mathrm{anticorrelation\; separation}=2-\mathrm{correlation\; separation}$$ By writing down the CHSH inequality for the anticorrelation separation and then substituting the above formula, we get $$\left(2-\overline{\mathrm{AD}}\right)\le \left(2-\overline{\mathrm{AB}}\right)+\left(2-\overline{\mathrm{BC}}\right)+\left(2-\overline{\mathrm{CD}}\right)$$ or $$\overline{\mathrm{AB}}+\overline{\mathrm{BC}}+\overline{\mathrm{CD}}-4\le \overline{\mathrm{AD}}$$ or, combined $$\overline{\mathrm{AB}}+\overline{\mathrm{BC}}+\overline{\mathrm{CD}}-4\le \overline{\mathrm{AD}}\le \overline{\mathrm{AB}}+\overline{\mathrm{BC}}+\overline{\mathrm{CD}}$$

Events, labels, regions, manifolds

TODO: formulate a la: "There are events e(r,n,l,o) where region r in {H,B}, number n in {0...N}, label l in {L}, outcome o in {yes,no}, instructions lambda(n) such that ..."

Two dimensional analogy

Draw two concentric circles of different radius. The centre of the circle is the parameter lambda. each circle is a measurement, which can be binary, conditioned on the radius of the circle. The distance between the circles is the angle between the measuring instruments. Two of the three numbers can be chosen freely. Once two numbers are chosen, the third is limited to two possible values.

(Will be edited and continued)

Last Modified: 8 March 2016

References

• Barrett, 2005: J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, D. Roberts, Nonlocal correlations as an information-theoretic resource , Phys. Rev. A 71, 022101 (2005).
• Bell, J. S. On the Einstein-Podolsky-Rosen Paradox. Physics 1, 195-200, 1964.
• Bohm, D. and Aharonov, Y. (1957) Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky , Phys. Rev. 108, 1070 - 1076
• Bohr, N. (1935) Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , Phys. Rev. 48, 696 - 702
• Clauser, 1969: J. F. Clauser, M.A. Horne, A. Shimony and R. A. Holt , Proposed experiment to test local hidden-variable theories , Phys. Rev. Lett. 23, 880-884 (1969).
• Einstein, A., Podolsky, B. and Rosen, N. (1935) Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , Phys. Rev. 47, 777-780
• S. Popescu and D. Rohrlich (1994), Quantum Nonlocality as an Axiom Found. Phys. 24 , 379.