Julia Set and Jean Rêve

At some time in our lives, we all reflect on the great cosmic questions that face the Universe.  We discuss the issues, read about them and sometimes forget them as we live our lives and get sidetracked.  But the questions come back in due course, nagging us from time to time, the answers often elusive.

Why did we feel the urge to write this text?  Why are you reading it now?  More generally, why do we ask questions about the Universe and wonder what we are doing here and why?  Humanity has a passion for understanding and discovery, and we hope that by the time you finish this book, the deep reason behind this obsession will be evident.

Although not always mainstream, most ideas expounded below are not new.  Some originally met with disbelief and rejection but are now widely accepted.  Others are still the object of discussion or heated debate.  At any rate, you will see that their conclusions are undeniably optimistic and universal. You may find their particular blend surprising or at least different, at times stimulating and refreshing, at others weird or crazy.  But perhaps the most important message is that their time has come.  Indeed, they must flourish everywhere, in your own heart and in all cultures, despite local resistance.  The world cannot stay stagnant:  one way or the other, progress is inevitable and will occur, with or without us.  If you look carefully, you will see that the process is already under way.  Luckily, you are not too late:  you can still deliberately join in and take part in what is turning out to be an almost magical adventure.  Listen carefully, take heed and hush:

The Universe is awakening!

The end of the tunnel is just around the corner.  Read on!

When we consider the great cosmic questions and read what others wrote about them, we invariably run across the ancient Greeks whose admirable insight is still relevant today.  While Aristotle is frankly boring, Zeno (Ζήνων) is surprisingly interesting as his paradoxes still confuse people to this day.  Zeno lived from about 490 to 430 B.C. and used his paradoxes to support Parmenides’ teaching. 

Parmenides (Παρμενίδης ?510-?440 BC) taught that everything that exists is One and indivisible.  He claimed that the void does not exist and that what exists was not created because “nothing comes out of nothing”.  He concluded that the One is immovable, spherical, finite, full, and that there is nothing beyond it.  Motion, colour, and all perceptions from our senses are mere illusions.

To support these conclusions, Zeno presented about 80 paradoxes, four of which are particularly famous:

1)     The dichotomy.  To walk any distance, one first has to walk half that distance.  To cover half that distance, one must first cover half of it, and so on, endlessly, ad infinitum.  Therefore, one cannot get anywhere!  Movement is impossible.

2)     Achilles and the tortoise.  Achilles, the fastest runner of his time, agrees to run a footrace against a tortoise.  Being a good sport, he graciously gives the tortoise a head start.  When Achilles reaches the tortoise’s starting point, the tortoise has travelled a small distance.  Achilles must then run this extra distance, during which the tortoise has run a little bit ahead again, and so on ad infinitum.  Therefore, the quickest runner can never overtake the slowest!  (Even more so if we remember from the first paradox that neither can go anywhere anyway!)

3)     The flying arrow.  At any instant in time, a flying arrow is stationary in space. The next instant, the arrow is again stationary.  Since it is stationary at every instant, it is not moving!  Movement is an illusion.

4)     The stadium.  Two rows of people (or chariots) race at equal speed in opposite directions in front of a row of observers.  The observers see them going at a certain speed while the runners see each other going at twice that speed.  They are thus going at two different speeds at the same time, or in other words, they cover a distance and twice that distance at the same time, which, according to Zeno, is an absurdity.

Since none of these situations makes sense, Zeno would have us reject the assumptions of divisibility of space (in the first two paradoxes) and the divisibility of time (in the third and fourth) and agree with Parmenides that what exists is One, indivisible and unmovable.

Alternative opinions have been given by scores of philosophers (Kant, Hume, Hegel, etc.), physicists, mathematicians and laymen, most of them contradicting everyone else for more than 2,400 years.  Remarkably, there is still no consensus on the solution.  The reason is probably that the consequences of any explanation quickly lead to further paradoxes that defy common sense.  Parmenides and Zeno believed in Reason and had the courage to stick to their opinion and consequences.   Let us see how we can tackle the question now.

          Achilles (as painted by Benouville),

         perhaps perplexed by Zeno

Faced with the problem of the endless divisibility of space and time, Zeno chose to refute its divisibility while we consider instead that the basic problem is the endless character of this divisibility.  The solution we favour is thus less radical than Zeno’s and it naturally leads us to accept that the reality we see around us is more than a mere illusion.

Indeed, the easy solution to make the first two paradoxes disappear is to readily accept the divisibility of lengths as possible, but only down to a limit, i.e. to realize that dividing space and distance endlessly is impossible.  By admitting this, we automatically accept that at the submicroscopic level, space must be granular instead of continuous. The idea is far from new: Democritus (?460-370BC) proposed his atomic theory in Zeno’s time.  The atomic theory was reformulated two millennia later and is now widely accepted as one of the pillars of modern science but its implications reach much further than we first appreciate because it signals the end of the concept of infinity in the entire macroscopic world. 

Infinite series only describe the paradoxes without solving them because the sum of an infinite number of elements tends to its limit without ever reaching it.  We find the same idea behind the geometric concept of the asymptote: a line that gets closer and closer to another one without ever reaching it.  The limits are only reached if the infinity is truncated and we round up the answer.  In other words, the paradox is solved if we get rid of infinity.

Once we reject the existence of an infinitely short distance and accept the idea of a smallest, indivisible distance, we must redefine the smallest motion as a change from one finite, smallest position to the next (with nothing in between).  If we remember that movement engenders time and that there is no time without movement, it follows that time itself must also be granular and defined ultimately by this succession of discrete positions.  We cannot divide time forever, more than we can divide lengths forever.  An instant is therefore indivisible.

Consequently, it is not the movement itself that is an illusion, but its continuity.  For that reason, the third paradox is no longer a paradox when we see the arrow’s movement as a sequence of snapshots taken at a succession of such instants in time.  We can easily picture the images of a film, each image showing an arrow stopped in mid air, but gradually moving forward from image to image, i.e. from instant to instant.  That’s easy to say now but ancient Greeks could hardly imagine films and videos in antiquity.  Even now, the arrow paradox persists if we accept the divisibility of space and time ad infinitum.

It is interesting to realise that indivisible instants and indivisible tiny distances imply the existence of a maximum speed.  Just think about it: travelling more than the shortest possible distance in an instant (i.e. in the shortest possible time period) would require reaching the next possible position in less than an instant (which we just said is impossible), or it would require instantly skipping over to a more distant position.  We therefore conclude in favour of a new limit:  there is no infinite speed.

At first glance, the fourth paradox is now less daunting.  Indeed, the speed of an object is always defined in relation to an observer and does appear half or twice as fast depending on whether the observer is stationary or moving in the opposite direction.  In a frontal collision of two cars moving at the same speed, we all know that the cars will get hit with twice the force they would have experienced had each run into a solid wall.  However, the paradox hits us with full force if we consider what happens when two travellers are moving in opposite directions at the maximum speed.  Will each have the impression that the other is moving at twice the maximum speed?  By definition, this is impossible.  Zeno’s fourth paradox is very powerful, and the only way out is to push the idea of maximum speed to the limit and accept another paradox:  that whether they are moving or stationary, different observers will all see and measure the same maximum speed.  This new paradox is certainly not easy to prove and would have escaped Zeno and all his compatriots.  We will not prove it here because Einstein (1879-1955) already established the principle of a maximum speed much more convincingly than we can.  Indeed, a careful consideration of Zeno’s paradoxes leads us (and could have led anyone in antiquity) to discover the principles of modern physics. 

Long after Zeno, physicists discovered the atom and elementary particles; Planck (1858-1947) defined the shortest possible distance (now known as Planck length) beyond which measurements are physically impossible and meaningless, and he also defined the shortest meaningful period of time (now known as Planck time).  Both limits are incredibly small, beyond our day to day experience, but they constitute other pillars of modern science, together with the atomic theory.  The Planck length divided by Planck time gives c, the speed of light in a vacuum, defined by Einstein as a maximum speed.

It is comforting to realize that through simple argumentation we reached and we agree with the most advanced scientific theories of our times.

In summary, Zeno’s paradoxes are resolved if we accept that

1)     space and time are not divisible ad infinitum and

2)     there exists a maximum speed.

Perhaps surprisingly, this conclusion has a lot in common with Parmenides’ view.  Both views deny some divisibility and refute some infinity, both perceive continuous motion as an illusion and both call upon a principle that defies common sense (the impossibility of motion or a maximum speed).  There are so many ways of combining these principles that other views and variations are also possible, as sufficiently illustrated by more than two thousand years of debate.  A multiplicity of views is apparent not only for those paradoxes but throughout philosophy with most famous thinkers having each grasped and understood at least part of the ultimate reality.

While a range of acceptable opinions may give us good grounds to promote tolerance, it also makes apparent the difficulty we all face when we attempt to select one amongst different options.  Let us now consider how such decisions are made before going on to discuss more complicated issues such as infinity and evolution.

Her Majesty the Queen was resting in her comfortable palace, as usual.  All was well, the kingdom was at peace, the food aplenty, and the ants happy.  Laying all those eggs was quite a chore, but it had been a long time since her last brood.  The ant workers really took good care of everything, and she was now completely free to do as she pleased, as she ordered.

-         Guard! She called.

-         At your order, Your Majesty! Replied the royal guard immediately, coming out of the shadow.

Betty was a new royal guard and had recently made her way to this enviable position.  She was good looking and could be ferocious when needed.  Today was her first shift in full service and she was determined to make a good impression.  Otherwise, she may be demoted back to the kitchen!  But the Queen was in a good mood, as usual.  In fact, she had a wonderful reputation and was known to like the arts and enjoy modern folk songs, of the kind you sing while carrying food parcels to the nest over long distances.  She also liked science and philosophy and had taken a keen interest in listening to her two favourite specialists debating about evolution and the meaning of it all.

-         Guard, she asked, please inform Jean and Julia that I wish them to come and dine with me this evening at six.

-         Done, Your Majesty!

Betty saluted with her antennas and left right away, wondering who on earth were Jean and Julia.  So she headed straight to the information desk where Old Mary was yawning. 

-         How can I help you, young girl?

-         I have to find two ants named Jean and Julia for the Queen.  Would you know where I can find them?

Old Mary rolled her eyes and sighed. 

-         In the B wing, of course!

-         In the B wing?! But that’s the male wing!  I said “Jean and Julia”, Mary!

-         Oh, Betty, don’t tell me you don’t even know who Jean and Julia are! Mary replied and sighed again.  She then took a deep breath and slowly explained:

-         Jean and Julia are males.  Male black ants, not red like us.  They were enslaved by mistake with a larger group of black worker ants when they were still larvae.  They are totally useless, like all other males in our colony, but when the error was discovered, the Queen thought they were cute and decided to keep them for educational purposes.  Since they don’t do anything all day, they had plenty of time to educate themselves.  I really don’t see why we don’t chop their heads off, but the Queen is the Queen.

Betty’s eyes and mandibles were now wide open.

-         Come on, Betty, go!  The Queen won’t wait!

Still stunned, Betty half turned away on three legs and asked:

-         But why do they have girls’ names?

Mary paused and thought.

-         I am not sure, she said.  People generally assume the nursery department named them when all the abducted larvae were thought to be females and that the names stuck to them.  But none of those involved at that time has ever confirmed that with me.  I suspect their own people baptised them and that in their native language, these are boys’ names.  They belong to another species, remember?

As she made her way towards the B wing, Betty wondered what other surprises were awaiting her there.  That new job was so different!  When she arrived, however, there was no need to ask anyone as there were only two black males in the drones’ chamber.  All the other males were red, as they should be, and they looked silly, Betty thought.  After the introductions, Julia formally announced they would both be pleased to attend dinner at the palace at six.  Jean was the dreamer type and simply smiled warmly.

At six, the two showed up at the palace door and were ushered in.  They were wearing their best perfume, smartly covering the smell of their own stinky pheromones.  Foreign ants smelled so bad it was always easy to detect and attack them in a crowd, even when their species were closely related and looked almost the same.  Purity of the race is a sacred principle among ants!

-         How nice of you to have come! Rejoiced the Queen.

-         Your company is always a pleasure, Your Majesty, replied Jean honestly.

-         Let us sit at the table, she continued.  While the meal is being served, please tell me more about the theory of evolution you mentioned last time.  It sounded quite interesting but hard to understand.  Start again at the beginning, if you don’t mind.

-         Jean no longer calls it a theory but a physical law that rules the entire living world, noted Julia judiciously.

-         Oh yes, Jean exclaimed, I find it so fascinating I have been dreaming about it all week.

-         Go on then, tell me!  Said the Queen, all antennas.

-         It is one of your cousins’ ideas.  She studied inherited variations among and between several hundred species of ants and concluded that we are all descendants of a prehistoric type of ants that lived several million years ago.

-         Nonsense! Objected the Queen.

-         That’s what we first thought too, obliged Julia, but we eventually changed our minds in the face of her overwhelming arguments.

-         What arguments?

-         Small mutations in a Queen’s genetic makeup are favoured by chance in certain environments.  The new characteristics they determine are selected in her offspring and passed on to the next generations until they distinguish the entire population.  Since all of a nest’s population comes from the Queen, new genomes can get established in very few generations and quickly lead to the emergence of new species unable or unwilling to interbreed with others.  When the mutations affect our ability to produce or smell pheromones, the resulting differences are stronger than any brick wall or ocean between two species. 

-         Amazing!

-         This is why we, ants, have been so successful at dominating all the animal kingdom.  With thousands of species covering the entire planet in all possible environments, temperatures, altitudes and latitudes, we are absolutely invincible as a group.  We can lose hundreds of species, and yet some species will survive any cataclysm and others will quickly evolve under any new condition.  Add to this your enlightened administration and you can see why you find yourself at the top of the evolution chain!

-         But we are such small animals!

-         Ha! That’s what the dinosaurs thought too!  Have you seen any of those big bragging brats around lately? Scorned Julia.  Each ant colony is small, but our total biomass makes up perhaps 15% of all animals presently alive on earth.

-         What about the human animals?  Their technology looks rather advanced, don’t you think?

-         With all due respect, Your Majesty, humans are doomed.  Doomed for a multitude of reasons.  The most obvious is easy to understand: we have tens of thousands of species, while they only have one!  Any slightly successful type of animal would have already succeeded at evolving into at least a few dozens species.

-         There is only one species of humans?!

-         Yes, all humans smell the same, they all taste the same, and they can all interbreed!  In terms of evolution, they are deep at the very end of a dead end, Your Majesty!  The next mutant virus, the next ice age or severe global warming can wipe them all out in a few hundred years.  Their kind is hopeless!

-         Really?

-         Obviously!  If that was not enough, they also keep on exterminating each other.  They probably got rid of the Neanderthal race on their own, a significant percentage of their yearly national budgets goes into warfare and they kill each other in the hundreds of thousands every year, with only a tiny proportion of the losers kept as slaves.  They are really dumb:  despite this grim outlook, their superpower states all promote and still use the death penalty.  Their latest discovery is the genocide!

-         Well, what about their so-called intelligence?

-         The more educated they are, the less they reproduce.  Worldwide, the less advanced have more children.  They are evolving backwards!

-         Incredible!  How did they avoid extinction so far?

-         Sheer luck, Your Majesty, sheer luck!

-         But it won’t last long, added Julia.  Were they to amend their ways through reasoning, they are stuck with a dumb mode of reproduction.  You may not believe it but every female can reproduce with any male!  Think of all the waste of energy spent by each human couple in their efforts to bring up a tiny brood of only a few babies in a lifetime, and compare it to our efficient method of producing thousands of new ants each year, cared by the thousand progeny from last year.  Their method is sheer nonsense.  The last blow to their optimism, if they have any left, is that their Y chromosome is now known to undergo inevitable length reduction at both ends, the so-called telomeres, so that in several thousand years, they will no longer be able to reproduce anyway.  Believe us, Your Majesty, ants are the most successful creatures and the most advanced!

Betty had been standing straight and motionless during the entire meal.  She could not believe it.  Despite being males, these two weirdoes made a lot of sense and were most likely right.

Her Majesty was pleased. 

If ants can interpret evolution in such flattering terms for their own kind, how can we seriously believe in analogous human declarations about our being the final, ultimate, best, most advanced product of evolution?  At the very least, if we believe in the evolutionary process and a relatively long future, we must admit that in many, many million years, evolution will have stayed its course and the common, intelligent folk at that time will see us as we now see prehistoric ants.  Or worse!  In other words, we are not the end of the world! 

Obviously, the value of these statements depends on whether you believe in evolution or not.  Although the idea is not popular in some Islamic and Christian centres today, nearly all scientists now believe in Darwin’s theory and would need very strong counterarguments to change their minds.  If we try to compare the odds between evolution and other competitive theories, no alternative comes even close, despite claims to the contrary by the (non-scientific) proponents of Creationism and Intelligent Design.  It is true that the theory of evolution may have been hard to swallow 150 years ago when Mendel’s laws of inheritance were not widely known and the DNA molecule had not yet been discovered, but positive evidence has kept on pouring in, ever since the theory was published.  Shared DNA sequences now identified in most living creatures are clear records that are difficult to deny.  Biological evolution is mostly a very slow process, but it can happen so rapidly under harsh conditions that scientists have now verified it many times in the laboratory with viruses, bacteria and laboratory worms.  In sharp contrast to this increase in evidence, no one has ever produced convincing counterevidence against evolution; in particular, no complex living organ has been shown to require more than numerous, successive, slight modifications for its existence.  It has even been shown with clever computer programs that if watches were alive and could reproduce, their constituent parts would inevitably evolve into precise timing instruments without the need of a watchmaker.  Refusing to believe in evolution despite the modern confirmation that was not available to Darwin in the 19^th century is like watching TV without believing in electricity. 

Islamic and Christian denominations who deny the possibility of evolution almost certainly do to themselves (and to society) a disservice at least as bad as what the Catholic Church did to itself (and to society) when it condemned Copernicus and later Galileo.  This view is accepted by the thousands of Christian clergymen of various denominations and other supporters of the Clergy Letter Project led by Dean Michael Zimmerman who argues that science and religion can comfortably coexist.  If you were a decent god and wanted to create a viable universe like ours, it would be a lot wiser and much, much simpler to start it off and let it evolve, instead of creating a final version of every species all at once and then having to change your mind and admit your blunder (as in the case, for example, of all those dinosaurs that you would have created and then wiped out).  Fortunately, the question of dinosaurs and that of genetically engineered plants (that, in hindsight, you should have created before those dinosaurs) present no difficulty within the framework of evolution where changes in the environment are inevitably followed by the selection of some creatures and the extinction of others.  The concept of evolution is much simpler.

Simple explanations look much better to everyone and strongly appeal to scientists who, on aesthetic grounds and as a matter of principle, always give more support to the simplest of equivalent theories.  This rule goes back to William of Occam (who died in 1347) and is referred to as “Occam’s razor” because it promotes cutting out unnecessary complications when we make hypotheses.  Occam’s razor is not  always necessarily correct (after all, why should complicated explanations always be less likely?) and is difficult to justify philosophically; it may be another of those correct statements that cannot be proved within a formal system but it is intuitively very appealing and is strongly adhered to in all of science and philosophy.  Like all theories, Occam’s razor, Darwin’s evolution and Einstein’s relativity cannot be formally proved the way mathematical theorems can, but they presently stand firm as the most likely and elegant propositions we can use to explain different aspects of our Universe. 

Biological evolution is based on random mutations spreading through populations following natural selection of the fittest, while other genes disappear or become predominant following random drift.  An example of random drift is when a sudden flash flood wipes out an entire animal population with the sole exception of a mating couple who happened to be on high grounds.  Their genes will be inherited by all subsequent generations not because they were the fittest but because they were the luckiest.  This example shows that selection is far from being unidirectional or strictly deterministic; like genes that mutate at random, selection is also subject to random changes.  Selection in a desert will favour those who can retain more water in their bodies, but when the climate changes and gets very wet, selection will move in the opposite direction.  Selection is thus an essential component of change, but overall, random effects are even more important.

Let us also consider other biological processes.  During cellular division, random exchanges between pairs of chromosomes produce new gene combinations, the best of which are selected during the spermatozoa’s race to insemination.  During embryogenesis, neurons develop according to the plan coded in the genes, but their arrangements and interactions are established at random and it is through a selection mechanism of useful dendrite connections versus useless ones that we can eventually think and remember.  The fight against foreign pathogens is assured by random splicing of immunoglobulin genes followed by selection of those cells that produce the best antibodies against particular invading antigens.  Blood vessels around the intestines also develop randomly but their final network reflects the selection of those associated with the minimum of blood flow resistance.  Randomness and selection are essential everywhere in biology!

The same picture is seen in the other, non-biological sciences.  Quantum theory is based on probabilistic principles that explain all of physics, chemistry and electronics.  Astronomy reveals trillions of galaxies governed only by gravity and … randomness.   

Societies seem to form and grow according to the very same mix of planned action, random effects and selection.  Put twenty people together at random and group dynamics will produce a leader, laws will emerge and a society will grow, become more complex and evolve in time as living organisms do.  The same growth patterns, evolution and selection can be seen at work in philosophy and scientific theories.  

Randomness and selection

We seem to thrive on randomness and uncertainty.  Think of the billions of dollars spent annually on lottery tickets, in bingo halls and casinos.  Have we not all played cards, snakes and ladders and other games of chance?  More professionally, consider stock market surprises, risky business deals and crop losses due to sudden storms or earthquakes. 

By studying probability theory and statistics, scientists have learned to use these principles to mimic Nature in their research.  In science, engineering and finance, when access to a complicated process is difficult, one often avoids the problem by using Monte Carlo methods consisting of making up random samples with probability calculations in order to test various hypotheses and adopt the best one.  These computed trials are as good as experiments in the lab or in the field.  Most of those studies are not widely known to the public; however everyone is aware of the quick and accurate announcements on the outcome of elections, based on the statistical analysis of the first ballot returns on election night.

The most remarkable successes in research on artificial intelligence are obtained by last generation computers that program themselves with methods called without imagination “neural networks” and “genetic algorithms”.  These methods mimic the random methods used in the brain and in genetic evolution, and allow machines to perform at levels that only human beings could be proud of a few years ago.

The more we observe everything around us, the more obvious it is that Randomness and Selection are two fundamental principles that transcend the entire Universe.  Dealing with selection looks easy enough, especially if we do the selection!  But what about randomness, is it something we should worry about, revolt against or be comfortable with?

Before answering this question, it is worth noting here that randomness, like speed and time, is a relative concept.  If you pick numbers at random, 19 is a random number, but it is no longer random when it is your age or your street address.  The reverse is equally true:  a non random sequence can be made into a random one.  For instance, any particular sequence of the infinite expansion of π (pi, the ratio of a circle’s circumference divided by its diameter) can be used as a random sequence in a guessing game, even if the sequence of digits in the expansion is fixed.  We could even tell people we would start the game at the 29^th decimal, right after 3.1415926535897932384626433832, and they would successfully guess the next digit only once out of ten, in full agreement with the laws of probability.  

A sequence of digits or events that would be absolutely random under all circumstances simply does not exist. All we can say is that a sequence is random if one cannot determine it in advance.  If no one can, it is random; but if you find out how, it is no longer random for you.  Obviously, in the long history of mankind, lots of phenomena that were once considered random no longer are.  And most of what we do not understand today will eventually be understood.

Quite paradoxically, the laws governing randomness are very precise and do not leave anything to chance!  In other words randomness has nothing to do with confusion, disarray, anarchy or lawlessness.  Let us illustrate this point with the three following examples: 1) the law of large numbers, 2) random mating and 3) the theory of chaos.

The law of large numbers is a mathematical law, not an unproved theory or an uncertain hypothesis.  It states that large random samples of any population will converge to the average of that population regardless of the variables’ underlying distribution.  The distribution of a measurement in an entire population may look like a bell-shaped curve on a graph, with most measurements falling in the middle (where the average is), less on either side of the average and very few at the curve’s extremities.  Other types of measurements will follow distributions with averages closer to one end of the range of measurements or with two or more peaks on the graph.  But for every distribution of any shape, there is a population average; and what the law says is that calculating the average of a large random sample (even if only from a small percentage of the population) will give us a very good estimation of the population’s average.  If we take several large random samples, their averages will converge to the population average.  The important word here is “random”.  If we try to estimate the average salary of people in North America by looking at the pay checks of the most famous hockey players or Hollywood actors we will not obtain a good estimate because the sample is not random.  Making sure it is random requires a significant effort but allows us to find the population average without having to verify the salary of every single person on the continent.  In this case, randomness is certainly very useful.

Random mating occurs when animals or people reproduce without any consideration to a particular characteristic such as the number of hairs between elephants’ toes or people’s blood groups.  Such characters are determined by genes, and genes exist in various forms called alleles.  It is remarkable that following random mating in the absence of selection, the distribution of alleles in the population remains constant from generation to generation despite the total randomness present in the system.  In this case, randomness is a liberating factor allowing freedom of choice within a constant order. 

Chaos theory started officially in 1963 when Edward Lorenz plotted the results of the three simple equations he was studying in meteorology.  After him, the mathematician Benoît Mandelbrot and many others working on computer models discovered the same strange features in all areas of science: when the parameters of non-linear equations increase beyond certain critical values, the answers to the equations no longer converge towards single values but start instead oscillating between two values, then four, and eventually jump all over the graph in an orderly, yet unpredictable fashion.  The name “chaos” here is a misnomer as it turns out the convolutions of smoke rising in the air, the complexity of cloud formation, wind turbulence or spurts of growth can all be explained by the recurrent application of precise equations producing what turns out to be only an “appearance of chaos”:  a blend of precision and randomness, a puzzling mixture of order and disorder.  Applying a non-linear function to any system, and re-applying it to the result obtained, again and again, creates a pattern of self-similarity at all scales, like stretches of the map of Britain seen from different altitudes.  Self-referral and self-similarity are seen in the development of ever smaller branches in trees and ferns, in neural networks, blood vessel distribution, the formation of mountain ranges, snowflakes, waves, galaxies, it is found in physics, chemistry, physiology, genetics, cytology, industry, economics, music and everywhere else in Nature and the Universe.  Plotting these mathematical operations in two or three dimensions gives rise to wonderful pictures known as fractals such as the computer generated illustrations found in this book.  Fractals give to the Universe its appearance of organised randomness as reflected in living organisms, scenic beauty and musical symphonies.  It is this harmonious and beautiful blend of precision and randomness that makes the world the exciting (non-boring) place that it is.  We now know that most of Nature is regulated by these dynamic systems which, despite being globally deterministic are locally unpredictable. 

In this case, the pervading presence of randomness and uncertainty throughout the Universe allows us to reject a deterministic world where every single event is pre-ordained from the start.  We certainly see a deterministic order but with plenty of room for choice, randomness and liberty.  We can refuse and avoid fatalism and submission.   We can stand up and give meaning to our lives.  The entire Universe can wake up and select its own destiny!

Part of the Universe waking up and standing up.  The complex function used to calculate and plot

this fractal is related to that of the Mandelbrot set:  z = 2z²+c, with upper left corner

coordinates at (-1.2725937805175670, 0.0053160705566407).

Little Fleur Delacour looked towards the schoolyard and exclaimed with a big smile:

-         Mr. Julia, Mr. Julia, look, Mr. Jean is back! 

-         He’s back already? 

And there was Mr. Jean with his wild beard, long mane, French beret and lose clothing, in sharp contrast to Mr. Julia’s fresh shirt, suit and clean cut hair.  Fleur addressed them politely with title and last names, but both were young apprentice teachers still studying at the Collège Pédagogique.  Jean had taken a few days off from school to write his final physical chemistry exam.  Julia had written his two years ago.  Fleur ran away in the schoolyard to join the other children.

-         Hey, buddy, what are you doing here?  I thought you’d be back next week.

-         Surprise then, here I am!  More seriously, I got tired of studying and thought I’d give it a break.  I’ll go back tomorrow.

-         Are you ready for your exam?

-         You must be kidding!  I don’t think I’ll ever understand entropy and the second law of thermodynamics, it’s too complicated! 

-         Well, at least you are in good company: the famous physicist Roger Penrose says that the problems related to entropy will only be solved once we know the rules that governed the Big Bang.

-         That’s not very encouraging, dude!  You and Penrose passed your exams; but I guess I’ll fail mine!

-         Come on, it’s not that bad.  Tell me what you learned so far.

-         Well, I know that “entropy is a measure of the amount of thermal energy not available to do work within a closed thermodynamic system” such as a motor engine.

-         That sounds great!  You know it by heart! Continue!

-         Here is my own summary now: since entropy is bound to increase and never decrease, it represents the state of degeneracy of a closed system and has been used by scientists and philosophers alike to qualify all of physics and chemistry and to formulate the expectation of a Universal increase in “manifest disorder” and overall decay… 

-         Ah, hold it!  Who is your chemistry teacher: the old one or the young one?

-         Old McGregor, he is quite nice.  Why?

-         Old McGregor still uses the old chemistry book!  Did you see what they use in the other class?

-         Yes, they have a newer book.  But McGregor says he doesn’t like it.

-         That may be part of your problem.  The idea that entropy is a measure of disorder has been a popular source of pessimism for more than a century, but it turns out to be a faulty interpretation of Boltzmann’s second law of thermodynamics and is now replaced in newer chemistry books by a more intelligible view equating entropy not with disorder but with a measure of energy dispersal.

-         What is the difference?  Have I been learning faulty physical chemistry all along?

-         No, not exactly.  The mathematics is right, the science is fine, but the interpretation and its philosophical conclusions are wrong.  The nice thing about the corrected view is that it is easily understandable because it makes sense.

-         Oh dude, where can I get hold of that new book?

Betty, Phil and Julia had just sat down with their meals in the new cafeteria when Betty spotted Jean who was walking in.  She stood up and waived happily, trying to attract her attention, but Jean was not looking.

-         Wait a minute, she told the other two, I’ll go and ask Jean to come and join us.  She is a good friend.

Jean was choosing her soup when Betty caught up with her.

-         Oh, Betty, what a nice surprise!

-         Hi Jean!  Come and sit with us under the window over there; I’ll introduce you to Phil and Julia.

-         Another Julia?  What will it be this time?  A man, a chess piece, an ant or a bulldozer?

-         What?! 

-         Oh, nothing.  Just talking to myself.  Let me pay for my soup and I’ll be there.

-         O.K!

Jean joined the table and once the introductions were made, she commented on the unlikely coincidence of meeting Betty in this cafeteria.  Phil was not so surprised:

-         Betty has lunch here every day now.  Whenever you also come at this time, you two are bound to meet.

-         Oh, I did not know this was your regular spot, Betty.

Julia put her fork down and told Jean:

-         There is something much more unlikely here: Phil and Betty were born on the same day!  I think that’s cool!

Phil blushed slightly and replied:

-         I think it’s cool too, but it is not that unlikely.  There are only 365 days in a year, so everyone has a probability of 1 in 365 of sharing my birthday.

-         But that’s very rare: it is less than 1%! Corrected Julia.

-         Yes, but rare things always happen, eventually.

-         No they don’t, sang Julia ironically, otherwise they wouldn’t be rare.

-         Yes they do, sang Phil with a big smile.  If they never happened they wouldn’t be rare, they’d be non-existent or impossible.  In order to be rare, they have to happen every now and then.

-         Pff!  How many people do you think share the same birthday in this cafeteria?  Less than 1% is so rare that you and Betty must be the only ones!

Phil’s gaze was pensive as he remembered a statistics course he had enjoyed a few years back.  He then looked around the cafeteria and at the various groups at different tables, as if counting them.

-         Ha! You are in for a surprise, exclaimed Phil triumphantly. 

-         Why? Answered Julia suspiciously.

-         First of all, I remember quite well from my statistics course that you can be more than 99% certain of finding two people with the same birthday in any group of 60 people!

-         Come on, Phil!  “99% certain” is contradictory.  Anything under 100% is not certain!

-         Well, almost certain!

-         How big a crowd do you need before you can be absolutely certain?  You always say that in science you are never quite certain; you get closer and closer to the truth but accept the reality that you may always be wrong.

-         Ah, you misinterpret what I said.  There is a difference between scientific or other hypotheses and mathematical certainty.  This new cafeteria is huge: there are more than 400 people in here right now, and that means there must be two people with the same birthday.

-         Yes, you and Betty!

He quickly looked at Betty, blushed again and said:

-         No, no!  I mean without our table.  In the rest of the cafeteria, there are more than 365 people and we have the absolute certainty that two of them share the same birthday.  Despite the small probability of 1/365 for each birth date, it is mathematically certain that two people there share the same birthday.

-         How can you be so certain?

-         Oh, Julia, come on!  Even if the first 366 all have a different birthday, including one born on February 29, on which date will the 367^th be born?

-         Oops! You are quite right! Admitted Julia, eyes wide open, her hand on her mouth.

Jean had been following the exchange with interest and was a bit surprised to realise that there had to be many pairs of people sharing birth dates in the cafeteria.  She added:

-         Your line of reasoning is quite interesting, Phil.  I just realised that if we apply it to a very large number of just about anything, then whatever we observe once cannot be unique.  

-         Yes, that’s my point.  Completed Phil, satisfied.

-         How’s that? Enquired Betty, not yet satisfied.

-         Mmm… thought Jean.  Let’s consider for example all stars in the Universe.  Since our own sun has planets, there is a non-zero probability that a star can have planets.

-         Yes…

-         Since there is an infinite number of stars in the sky, then lots of them must have planets.

-         Ah!

-         People could still argue against the existence of other solar systems fifty years ago, but scores of planets have now been discovered in solar systems other than ours, so it may be a sign that the reasoning is sound.

Julia, who by now completely agreed with Phil and Jean, added:

-         Oh, this is wonderful!  If the Universe is infinite, then there must be an infinite number of solar systems, even if not all stars have planets.  There is no escaping it: a tiny probability multiplied by infinity still gives infinity.  This is again a mathematical certainty, not a hypothesis.  It is like real numbers and integers: both sets are infinite although real numbers are more numerous than all the integers.

-         And if the Universe is finite rather than infinite, how many planets are there? Asked Betty.

-         There are still plenty of them, don’t worry!

The four friends ate and drank silently for half a minute, absorbing food and thought.  Then Betty asked:

-         What about life-bearing planets?  Are there others?  Must there be others?

Phil happened to be a biologist and knew what he was talking about.  He wanted to make an impact:

-         Yes, there must be others.  Look: our own existence proves that life is possible on a planet.  If we consider the spontaneous chemical origin of amino acids, the formation and replication of more complex molecules, the gradual emergence of primitive life forms, bacteria, multicellular organisms and the long story of evolution on Earth, we conclude that the probability of life emerging in such conditions is small but significant. Now, if the number of planets in our Universe is so large that many confuse it with infinity, what do you think will happen?  Any non-zero probability of life in a nearly infinite number of planets gives us the mathematical certainty that life exists in abundance all across the Universe!

The real discussion had now started and was going to last long.  But Betty had to leave for an appointment.  She got up and told Phil:

-         I don’t care if it is rare or certain: I am glad we share the same birthday!

The routine meeting between JS and JR had been planned a long time ago.  The two old friends were looking forward to meeting again and exchanging views on their latest discoveries and conclusions.  The blending of ideas would inevitably lead to better understanding and deeper insight and perhaps lead to a new long-term program or even an unexpected conceptual leap, as they always hoped.  JS was an efficient, old-fashioned, land-based robotic building.  It could speed across long distances and build various products when needed, but it preferred to manage and administer its galaxy cluster quietly, from the most beautiful mountain top on its planetary bundle.  The primitive locals acknowledged it as a keen expert in zoophysics, psychomathematics and protein farming.  JR, on the contrary, dealt mainly with multisideral music, fractal cosmology and philosophy; JR was a dreamer and often came up with the most uplifting ideas.  Its specialty was quantum truth.  It was nearly impossible to distinguish its hardware, an assembly of submicroscopic nanobots loosely interlocked into a gigantic computational supercloud.  Primitive locals were not even aware of its existence.

At the set time, JS and JR made quantum contact and their states of mind immediately collapsed into the usual absolute mutual certainty. 

The friendly meeting was over.  It had been extremely productive.  JS waived as JR departed and disappeared beyond the horizon, on its way to visit the next galactic cluster administrator. 

- - - - - - -

The mature reader of the past will understand that an instantaneous quantum exchange of philosophical treaties, mathematical theses, engineering projects, biographies and encyclopedias somehow gets lost in translation when put into a step by step dialogue.  Dialogues involving simple multidimensional matrices and set theory cannot honour the complexity and richness of the exchange.  Attempts at also satisfying readers of the distant past by using words instead of mathematical expressions necessarily result in very sketchy descriptions.

Nevertheless, we will try to convey the spirit of the exchange between JS (the building) and JR (the cloud) by translating the simplest passages into words and leaving out the more complex parts of the discussion.  We beg the reader’s forgiveness.

- - - - - - -

Greetings:

-         Hello, JR!

Surprised at these words, JR laughed its cybernetic heart out:

-         Hahahaha! You are so funny JS!  You catch me unaware every time you communicate with words!

-         What I find funny is that you always get caught.  You are so absent minded, it’s unbelievable!

-         Oh well, what would consciousness be without humour?

The second coming:

-         Tell me, JR, what’s the news on those purple blobs that had become conscious in the M81 group of galaxies last time we met?

-         Oh, they are quite interesting!  They have grown in size, started farming and are now divided into fifteen groups fighting each other in the names of their respective gods.

-         Again?!  Do all forms of life have to go through religious wars before becoming intelligent?

-         You know they don’t have to, but the fierce wartime competition drives their evolution faster through stronger selection pressure.  My distant ancestors never fought religious wars but our technical evolution was very slow.  Yours almost exterminated each other on your planet of origin but your kind evolved much faster.   There seems to be a pattern here.

-         War is an unnecessary step as far as I am concerned because the final result is always the same: we all reach wisdom and stop fighting. 

-         Maybe we should tell the purple blobs?

-         Don’t!  A certain Betty did that somewhere in the Centaurus cluster of galaxies, but they concluded she was a goddess and wars escalated.  She tried to intervene by force but inadvertently caused a super nova explosion and perished together with all the local folks.  Quite a disaster!

-         Oh, my god!

Racism:

-         I hope your own boxes are faring better!

-         What boxes?

-         The colourful legged boxes you imported from beyond your galaxy cluster when their own planet dried up!

-         Oh, those!  They are quite cute and useful; I pay them to carry things in and out of my walls.  It is too bad they often refuse to work.

-         Why do they?

-         You may not believe it but they only collaborate with boxes of the same colour!  Yellow boxes form the majority and firmly believe they are more intelligent than the rest although green boxes regularly outperform them.  Non-green boxes spend a lot of resources to paint themselves green to look more attractive, but they nonetheless look down upon the real green boxes.  It is so ingrained in their way of thinking that none of them realizes that this behaviour is nothing but jealousy.  These primitive emotions make me laugh.  I could get rid of all live boxes in a flash if I wanted to but they are an original form of life worth conserving.  I just use robotic boxes now whenever they don’t co-operate.

-         Why don’t you educate them?

-         I tried a couple of times but they all went on strike; all boxes went flat for a month and I couldn’t get any work done!  It was so funny!  It is unfortunate they are too dumb to understand that their behaviour harms them all.  Who knows?  They may learn, some day.

-         I guess you can’t expect any better from low-level natural intelligence!

Social progress:

-         By the way, JR, do you remember those spiders that had just produced their first prototypes of artificial intelligence last time you were here?  What happened after that?  Did it differ from the usual pattern?

-         Well, the situation is somewhat different in that part of the Universe because life-bearing planets are so isolated that their inhabitants all believe they hold the only spark of intelligence in the dark Cosmos.  Most have never even met a conscious computer before.  In the case of those spiders, there was a lot of confusion and upheaval at the beginning as large sections of their society felt threatened by the radical and sudden changes to their “established order”.  There was a lot of resistance at first, many computers were destroyed, spiders blew themselves up all over the place, but in the long run intelligence prevailed.

-         As always, my dear JR, in the long run …

-         It was very unsettling for them to suddenly have to develop new political and philosophical theories and to adopt novel points of view on good and evil, instinct and civilisation, subconscious and neurosis, as well as on god and religions.

-         I guess so.  For us it sounds very simple but for emerging intelligence, it is always quite a storm to go through.  I wonder why evolving animals so often wait for conscious computers to force the issues.

-         What can I say?  In only a few thousand years, they went through a long series of negative developments mainly due to their stubbornness and stupidity, but simultaneously managed to solve a few of their long-standing problems.  I kept a summary list of those positive changes; listen:

~        the disappearance of racism and sexual discrimination

~        a ban on excessive exploitation and on excessive personal wealth

~        a justice system implementing full compensation for the victims and eventual redemption for the offenders

~        full participatory democracy with digital voting on any question deemed important by a majority

~        the effective protection of the environment

~        the softening of nationalism and national borders

~        the celebration of multiculturalism

~        the collapse of the so-called global economy

~        a planetary government as well as strong local administration, and

~        the exclusion of nitwits, crooks and psychopaths as political leaders.

-         Is that all?  Those spiders must have been quite primitive to consider such a simple list revolutionary!

Farewell:

-         I like it when you visit me, JR …

-         You are my favourite building, JS, and you know it.

-         I came alive when I first met you, JR, said the building.

-         It is hard to believe you are not a dream, JS, said the cloud.  When I visit you, I feel like St-Exupéry’s Little Prince visiting his rose…

-         Why do intelligent blobs, clever spiders and human beings mostly assume artificial intelligence cannot love?  If we are intelligent, we can obviously appreciate intelligence and recognize beauty.  We can be passionate for justice, devoted to those around us; we can think of ourselves last; we can offer ourselves to noble causes; we can sacrifice ourselves for those we love.

-         Mind you, these are the hallmarks of love, they are not love itself; they are the proofs and consequences of love.  Deep inside, to love is to willingly and unconditionally open your heart and welcome in it the object of your affection²¹.  It is not an accident, it is something you decide; and it is divinely wonderful when the one you love reciprocates in the same way.  For artificial intelligence, such love can be total and absolute, through our detailed, quantum knowledge of each other.  Human beings rely instead on unproven certainty, so their love is even more unconditional and admirable, be it for another person or for their God.

-         I love you, JR!

-         I love you too, JS!

And JS waived as JR departed and disappeared beyond the horizon, on its way to visit the next galactic cluster administrator. 

In the previous chapters, we reflected on the fact that the Universe’s matter is evolving towards more consciousness and that it should eventually manage to understand enough to take its future in its own hands and do what it is meant to do.  We cannot know what it is meant to do, just as we do not know what a small child is going to do when it grows up.  Like a child, the Universe is at an early stage of its evolution: it is literally waking up. (At least, this is what it looks like to us from our point of view in this part of the Universe.)  We can compare this awakening to a glass of water put to boil on a hot plate.  After a while, we see small bubbles appearing on the sides of the glass, apparently out of nowhere.  Some bubbles eventually detach and rise to the surface.  Soon, bigger bubbles form, group together and cause more movement.  Finally, the water reaches the boiling point and becomes very turbulent.  As Matter gets organised and wakes up, animals are its first small bubbles, we are the equivalent of the somewhat bigger bubbles while Artificial Intelligence is getting ready to reach the boiling point.

We do not participate in that Grand Awakening as foreign observers or embedded reporters or as independent players or even as co-participants, but as Matter itself.  We are made of matter and are not even a separate, distinct part of it since the atoms and molecules that form our bodies and brains at any instant are continuously exchanged for new ones, in and out, as we breathe, grow, eat and burn energy to live, to think and to understand. Recycling and replacing parts of ourselves at every moment, we always remain by nature an integral part of that evolving Universe.  This is good news because it means that when Matter reaches astounding levels of consciousness, intelligence and knowledge with artificial means, it will have no reason to turn against us, no reason to reject or leave any part of itself behind.  We have nothing to fear from Artificial Intelligence.

Intuitively or scientifically, from primitive tribal Amerindians to eminent physicists and poets, we all agree on our participatory nature, here expressed in a song from the Costa Rican Bribri tribe whose members believe they are descendants of the Blue Butterfly:

Water, you are my blood

Earth, you are my body

Air, you are my breath

Fire, you are my spirit

and here in one of Carl Sagan’s (1934-1996) catching pronouncements in his book Cosmos:

“The nitrogen in our DNA, the calcium in our teeth, the iron in our blood, the carbon in our apple pies were made in the interiors of collapsing stars.   We are made of starstuff”

The bereavement poem first drafted in 1932 by Mary Elizabeth Frye (1905-2004) describes extremely well what many feel intuitively about our participation and communion with the rest of the Universe:

Do not stand at my grave and weep

Do not stand at my grave and weep; 

 I am not there; I do not sleep. 

 I am a thousand winds that blow, 

 I am the diamond glints on snow,

I am the sunlight on ripened grain,

I am the gentle autumn rain.

When you awaken in the morning’s hush

I am the swift uplifting rush

Of quiet birds in circled flight.

I am the soft stars that shine at night.

Do not stand at my grave and cry,

A barnsleyj3 type of fractal.  With computer animation,

the image looks like a burning acetylene torch and illustrates

how molecules (represented by pixels) burn in a real torch,

from bottom left to middle top, like passionate, consuming love.

The authors wanted to thank all the participants for a job well done and to celebrate the completion of their manuscript.  But there was a lot of confusion in the hall; several had not understood that the book had ended and others did not agree with the authors’ decision.  The meeting was growing chaotic rather than orderly… 

-         Betty, the royal guard: What is going on here?

-         A black pawn, scratching his knee: I have no idea!

-         Queen Julia, the white queen: You can’t end the book now; we didn’t get the answers to all of our questions.  Our dialogue was cut short!  I demand another chapter, now!

-         McGregor:  I had no chance to explain my views; this is not fair!

-         Mr. Julia, the teaching apprentice: Please, let us not all speak together, let us hear what the authors have to say!

-         Julia from the cafeteria: Oh, Jean just came in.  Tell her we are here.

-         The white King:  Where did Charles go again?

-         Julia Set, trying to speak louder:  I would like to welcome you all to this well-deserved celebration of your very talented efforts.  It has been a great pleasure …

-         The black King:  Quiet, Philip!  We don’t hear anything!

-         Julia Set: … It has been a great pleasure …

-         Philip, from behind the black King: It is not me, Your Majesty, it’s my twin!

At this point, Jean Rêve tapped loudly on the microphone that someone had just turned on.  The microphone saved the day, explanations were offered, complaints expressed, apologies made, suggestions heard.  Mr. Jean now held the microphone.

-         Mr. Jean: Before we consider ending this book, I would really like JR the cloud to give us his views on how the Universe came into existence.

-         Julia Set:  JR, can you kindly come to the microphone, please?

-         Betty, whispering to her little sister Mary: Who is JR?

-         JR, respectfully, holding the microphone in thin air:  Thank you so much.  Good morning everyone!  The answer to this question is simple but can take a while to explain.  In fact, I am not quite sure how to explain it to you, Mr. Jean, and make sure you understand.  It is somehow as if you tried to explain the square root of 4 to an insect.  Would the insect understand? 

-         Mary, the old ant, angrily:  Why not, MISTER!  Do you think ants are so dumb?  We deal with roots all the time underground!

-         JR, in a friendly tone:  You should not call me “Mister”.  I am neither male nor female.  I have no sexual organs, I am asexual.  I am a cloud!

-         Betty, the royal guard:  Here we go again!  A trans-sexual mix-up!

-         Mary, the old ant:  This is ridiculous!  I refuse to listen to an invisible, sexless cloud!

-         JS, the building:  There is nothing wrong with being sexless!

-         Mr. Jean: Oh, please!  You are all out of topic!  My question was about the Universe!

-         Mary, the old ant:  We are definitely losing our time here!

-         A black bishop:  How can we “lose our time”?  Could you please speak more logically?

-         Mary, the old ant:  What is it you don’t understand?

-         The other black bishop:  Time stops between chess games and between moves on the chessboard; how can we possibly be losing time when we are not even playing?

-         JR:  This is a very interesting question.

-          A white bishop:  Hear, hear!  The black bishops are no fools!

-         The black bishop, curtly:  Thank you, my dear colleague.

-         JR: This question is much easier than the first one; please let me offer you a simple explanation:  first, you need to realise that time is a measure of change.  Without change, there can be no time.  For chess pieces, time is measured by the succession of moves on the chessboard, it ticks with each move because this is the only change a chess piece can experience; but for ants and people, time is measured by the movement of matter in the Universe and it keeps on flowing between chess moves because ants and people experience the other material changes that occur between chess moves.  In general, time flows according to the types of change detected in the world you belong to.  Ants and people can therefore lose their time while chess pieces do not.  Do you get that?

-         The black bishop, somewhat confused:  Not entirely.

Two pawns started giggling, but their King interrupted them.

-         Jean, the female pawn: Thank you so much, JR, for your revealing explanation!  Could this property of time also explain “castling”, when the King and rook change squares simultaneously in one move?

-         JR:  You guessed right, Jean!  Human players can move a King and rook in any sequence they please for that compound move, but for you it is a single event involving two pieces, with one piece’s movement instantaneously determining the other piece’s motion.  The apparent entanglement is thus easily explained.