Saturday, 27 November 2010

First Matter

"In the beginning God created the heaven and the earth."
~~Book of Genesis 1:1

~~The NASA Hubble Space Telescope photo of V838 Monocerotis (February 8, 2004)

Before physicists, the men of science, so to speak, were alchemists. Unlike modern theory today, the doctrine of the alchemists was very much centred around theology. To them, God was not some abstract philosphical concept, but rather, God was understood, and accepted as being something from which everything is made. In other words - God is everything - God is the Universe itself. In this sense, the alchemist was more closely related to the mystic than a scientist.

"Alchemists agree with chemists that there is unity in matter, but whereas chemistry teaches that atomic particles are the smallest units in matter, alchemists believe in an ultimate source they call Ether, or the universal fluid. Matter to the alchemist is therefore compact energy, which can be dissolved into free energy or force. For the alchemist energy and matter are the same thing, namely substance. The substance is the Absolute, the One that Hermes Trismegistus describes in the Emerald tablet. This One is divided into three parts: Intelligence or force, Energy and Matter."
~~The complete book of spells, ceremonies, and magic By Migene González-Wippler

~~The Alchymist by Joseph Wright of Derby (1734-97) 1771

Alchemists hoped to move closer to God by developing a greater understanding of God, not only in the spiritual sense, but in a real, physical sense too. If God truly was everything, and everything owed its existence to God, then it must be possible to find the existence of God in any thing, in any place we care to look. What the alchemist was seeking in all material things, and that includes himself, was the very substance of God. There is a real possibility that some alchemists may have found it. Even the word "alchemy" seems to hint at something of the nature of this substance.

"The word alchemy is derived from the words Al and chemia.

Al is an ancient word meaning ‘God’ in the sense of the ‘All’, the ‘Absolute’. As part of the word alchemy it means ‘divine’ or ‘universal’. The word was used in many ancient languages and cultures, including the Egyptian, Mesopotamian, Hebrew and Celtic. Later, the Hebrew form of the word came to be written as El, which in the Christian bible is translated as ‘God’. In Islam the word appears as Allah.

Chemia is from the Greek word khemia, which itself is derived from the ancient Egyptian word kemit, meaning ‘black earth’."

In other words, the term "alchemy" could be interpreted as meaning "God is black earth." These choice of words to describe God may at first seem a bit off-key - one might have expected something a bit more poetic, more holy, more ethereal - such things as "God is mountain dew," or "God is sunshine." Instead, for those seeking the substance of God, the alchemist tells us to ignore all this fluffy idealism, and points only to the dirt at our feet. According to the Bible, this dirt, or dust is also the same foodstuff which the serpent of Eden is condemned to eat forever.

"And the LORD God said unto the serpent, Because thou hast done this, thou art cursed above all cattle, and above every beast of the field; upon thy belly shalt thou go, and dust shalt thou eat all the days of thy life."
~~Genesis 3:14

When God banishes Adam and Eve from the Garden of Eden, God makes a point of telling Adam that the dust of the ground served as the material from which he was created. This at first seems only slanderous, the final insult to be added to Adam's degradation - but this being the word of God, it is told only as a truth. It is intriguing that the only substance required by God to create intelligent life is little more than ordinary dust from the ground.

"By the sweat of your brow you will eat your food until you return to the ground, since from it you were taken; for dust you are and to dust you will return.”
~~Genesis 3:19

Thus, a significant clue to the nature of God, and the powers of creation, are all contained in nothing more than a speck of dust. In searching for the substance of God, it is likely that these same thoughts must have occured to the alchemist. If one could gain knowledge of this biblical dust, then one would know the form which God takes before creation. In other words, the dust mentioned in the Bible, woefully unflattering as it is, could be interpreted as being the definitive description for the uncreated substance of God.

"Uncreated Independent Substance: thing that is not dependent upon the causal power of any other thing in order to exist or to remain in existence, and is not a property of any other thing.(The only thing that satisfies this definition of "uncreated substance" is God.)"

The alchemist believed that all material bodies are formed from only one substance, indeed, "the substance." The entire material of the Universe was said to consist of this one single element - the materia prima (first matter.) For some, the materia prima represented God in its most purest form. It represents the primordial matter from which all things, and especially living things, are made.

Using the Bible, Paracelsus and others, connected prima materia to God; "before Abraham was made, I am." (John 8:58) Since prima materia is supposedly the [philospher's] stone, also, this also demonstrated the stone is without beginning or end. Jung noted many Christians hearing this would not believe their ears, but it was plainly stated in the Liber Platonis quartorum, "That from which things arise is the invisible and immovable God."

If the alchemist was able to gain mastery over this substance, then it should, at least in theory, be possible for them to take dominance over the will of God. The forces of God would be subjucated to the whim of the alchemist, giving him/her the power to fulfil all their earthly desires instantaneously. The alchemist would at last become master of their own fate, and the need for a God will have been entirely displaced.

The goal of Alchemy is the Great Opus or the Great Work which is the purification of the lesser or gross and its elevation to the greater or more refined, whether in plants, metals, or inconsciousness. The ultimate goal of the alchemist is to find the Prima Materia or the First Matter of nature as the dark, passive, unformed and raw virgin and universal stuff of creation. Through the alchemical process the alchemist transforms this Prima Materia into the Philosophers Stone. This accomplishment is most commonly known as the transformation of Lead into Gold, the heaviest, darkest, densest most earthbound, least valuable metal becoming Gold: Incarnated Light; the most glittering, luminous, valuable metal; symbol of the sun and of spiritual attainment and consciousness, spiritual illumination as cosmic consciousness which is the ultimate goal of the human evolution.

Certainly, one of the goals of alchemy was to achieve great wealth (after all, regardless of how close they moved towards God, there was always the danger of getting deeper into debt!). Possessing the philospher's stone would allow the alchemist to turn lead into gold, which quite literally, would give them the power to print their own currency. This remarkable prize however, pales in significance when compared to the true treasure which awaits the seeker. It is the one thing which is universally prized by humans, and desired above all other things, though seldom believed possible - the gift of immortality. The alchemist would finally be allowed to lift the veil, and to enter the world hidden from view, and recieve the key which will give him/her the power, and ultimately freedom, to choose the day they die. To become immortal would signify the alchemist's completion of the Great Work.

This philosopher’s stone is a metaphor – which means that it has both an inner and outer reality, neither of which can be taken for granted or understood exclusively. The development of the philosopher’s stone could only occur through a refinement of the initially untransformed base material of the world – the “prima materia” or black earth, which is simultaneously the alchemist’s own psyche, both conscious and unconscious (Jung, 1967, 1978, 1993) 4, as well as the actual underlying physicality of all the world’s substances.

Descriptions of how the philospher's stone appears physically, are understandably evasive, but that does not mean that the stone, "a stone which is not a stone," is necessarily hidden from view. Quite the contrary, it was sometimes said to be a common substance, found everywhere but unrecognized and unappreciated.

As you can see, the importance of identifying the substance of the materia prima is essential to the creation of the philosopher's stone. So, what exactly is the materia prima? Which one of the elements from the periodic table is meant to be the hidden substance of God?

Many thanks:
Alchemy: an introduction to the symbolism and the psychology By Marie-Luise von Franz

Saturday, 6 November 2010


Who appointed their path to sun and stars?
Who but Thou is it through whom the moon waxes and wanes?
Who set the earth in its place below and the cloudy sky
that it shall not fall?
Who established the waters and the plants?
Who yoked the steeds to wind and clouds?
Who, O Wise One, is the creator of Good Mind?
What artificer made light and darkness?
What artificer sleeping and waking?
Who made morning, midday and night,
to remind the wise man of his task?
Is it as Good Mind that thou hast founded thy Dominion?
Who created Devotion, sacred with Dominion?


All other things have a portion of everything, but Mind is infinite and self-ruled. . . . For it is the finest of all things and the purest; it has all knowledge about everything and the greatest power. And Mind controls all things, both the greater and the smaller, that have life.


European history began with the emergence of Athens as the classical cosmopolis of the northern Mediterranean. Athena won a dispute with Poseidon for protective sovereignty over Attica by giving the area the sacred and nourishing olive tree, and her sacred bird, the owl, became the city's symbol. Doubly blest, Athens became the focus of the converging forces that would in a brief time initiate a culture of such creativity and splendour that subsequent generations in Europe would look to her as the source of science, ethics and knowledge of the soul. Athens first drew ideas to herself, and later attracted thinkers, until she housed the schools and traditions that provided the foundation of European thought. Perhaps even before the time of Homer, an incipient anthropomorphic decadence began to undermine the spiritual appeal of Mycenaean and Minoan mythology. The gods as personifications of intelligent forces in Man and Nature were degraded into beings that reflected only the most superficial human traits. As the Mysteries withdrew increasingly from the desecrating gaze of the public eye, though not from public knowledge, thinkers sought for new ways to vivify the deeper meanings of human existence. Early philosophers emphasized either an understanding of nature shorn of its anthropocentric excesses or the ethical dimension in the structure of the cosmos. Pythagoras and Plato brought this latter concern to sublime fruition, and Anaxagoras laid the foundation for experimental method and theoretical science.

Anaxagoras was born in the port city of Clazomenae in Ionia around 500 B.C., though almost nothing is known about his life or the order of happenings. His father, Hegesibulus, was extremely wealthy, and Anaxagoras apparently devoted the early years of his life to leisurely study. Though a Greek city, Clazomenae had fallen to the Persians, and it is likely that the thoughtful Anaxagoras would have learnt the rudiments of the religion taught by Zarathushtra, whom the Greeks called Zoroaster. Ahura Mazda, the Lord of Light, cannot be described in crude human terms, though Vohu Manah, Good Mind, the first aspect of manifest deity, is seen as that which orders and moves all things. Rather than a human form, Ahura Mazda and its seven aspects are best represented by an eternal fire and the seven-branched candelabrum. From the beginning of his active life, Anaxagoras made fire a central theme in his thought. When he was about twenty years of age, he journeyed to Athens, where his wealth (and perhaps his father's connections) gained him immediate entrance into the highest circles where he became good friends with Themistocles and Pericles. Though Pericles was probably never a student of Anaxagoras in a formal sense, he often acknowledged his debt to Anaxagoras for many of his ideas and policies, and eventually their relationship became famous.

Anaxagoras taught in Athens throughout his adult life. Sometime during that period he became the focus of opponents of Periclean political reform. Satyrus wrote that Thucydides, a long-standing enemy of Pericles, found it impossible to confront his opponent successfully, and so attacked him indirectly by bringing charges of asebeia (impiety) against Anaxagoras. Since the Athenians officially held the sun to be a deity, the view held by Anaxagoras that the sun is like a molten stone was construed by some as impiety. But Thucydides also charged him with treason in the Persian wars. Anaxagoras fled Athens and only later returned with full pardon after Pericles had more firmly secured his own position. Sotion, however, wrote that Anaxagoras had been charged with impiety late in life for his teachings about the sun and that he was forced into exile. Whatever the literal truth, Anaxagoras lived in tumultuous as well as glorious times, and it would not have been impossible for him to be exiled twice. Though vulnerable because of his acquaintances, Anaxagoras neither took part in the political life of the city nor paid any attention to the opinion of others. Much to the dismay of friends and relatives, he allowed his extensive inheritance to waste away, either permitting his farmland to return to pasture or giving it to relatives. His concern lay wholly in the contemplation of heaven and earth.

Anaxagoras was not the first Greek to observe nature studiously, though he was an exceptionally careful scrutinizer of phenomena. For example, painstaking observation led him to the discovery that the image reflected in the dark pupil of the eye constitutes the exact field of vision. Whilst others were teaching that the whole eye saw, he correctly identified the pupil as the active window of perception. Observation alone, however, was insufficient to discover the causes of things. Anaxagoras developed the method of experimentation as a means of confirming explanations of natural processes. His observations of wind blowing through trees convinced him that gases exert pressure, but to convince a sceptical Athenian audience, he filled a bladder with air and sealed it shut, and then invited challengers to crush the bag. Since air was trapped inside, he surmised that the air exerted pressure which manifested as resistance to anyone attempting to flatten the bag. Besides observation and experimentation, Anaxagoras recognized the need for a unity of explanation, a coherence of theory, to account for nature. Though he did not emphasize the mathematical nature of the world as did Pythagoras, Anaxagoras drew together the other elements of scientific endeavour, laying down the broad foundations of scientific method used even to this day.

His concern with explaining the occurrence of natural phenomena in terms of invariable laws and not special circumstances – such as the whimsy of capricious deities – along with his reticent form of life resulted in the status of prophet being attributed to him. On the basis of his theory of the stars and planets, he argued that it was possible for rocky material to fall from them to earth. When a meteorite fell to earth at Aegospotami, perhaps during his lifetime, the legend arose that he had predicted it in detail. His grasp of the nature of eclipses enabled him to inform Pericles of the time of a total solar obscuration, and the statesman used that advance knowledge to reassure the Athenians when it occurred. Ammianus Marcellinus wrote that Anaxagoras had gained such knowledge in Egypt. Plutarch asserted that he had mastered geometry and had successfully squared the circle. Satyrus reported that Euripides greatly admired him and might even have been his student. Theodoretus supported the less likely possibility that Socrates as a youth studied with Anaxagoras.

Anaxagoras followed the tradition of his day and taught students who came to hear him discourse, but he did not establish a permanent school. Nevertheless, like Parmenides, he wrote one book outlining his methodology and explanations of nature. Since Parmenides had taught that change was logically impossible, the work of Anaxagoras, taking change as fundamental to nature, became the standard text for nascent Athenian science. Socrates is made to refer to this work in Plato's Apology and to criticize its approaches in the Phaedo, where he does not say Anaxagoras is wrong but rather that his explanations of behaviour are inadequate when applied to human action. The strong motivation to understand nature so profoundly moved Anaxagoras that he renounced concern for his own welfare, exemplified in his abandonment of his large inheritance, withdrew from the exciting public affairs of the day, and remained aloof from personal involvements of every kind. His detachment from worldly involvements became legendary when he responded to a report of his son's death by saying that he knew that his son was mortal from the moment of his birth. When asked why one should be pleased to have been born rather than not, he answered, "In order to contemplate the heavens and the structure of the world-order as a whole." This constitutes the truly happy life, Anaxagoras said, though most people would not think so.

Anaxagoras lived in Athens for about thirty years, a time spanning the golden age of classical Greek culture. He represented the rational approach to understanding events and things, advocating a calm and detached perception supported by an alert mind, freed from the obscuring clouds of unrestrained emotion and self-seeking bias. Near the end of his life he left Athens, perhaps because he was again charged with impiety, and retired around 433 B.C. to the Hellespontine city of Lampsacus. There he was welcomed, surrounded by students and honoured by the citizens. When he died in about 428 B.C., he was given a public funeral, and the citizenry inscribed a tribute on his tomb:

Here lies Anaxagoras, whose picture
of the order of the universe
came closest to the truth.

Shortly before his death his followers asked him what he would consider an appropriate way to honour his memory. He replied that students should be given the month in which he died as a vacation each year. Anaxagoras was so highly respected in Lampsacus that his death was observed as he wished for well over a century.

The scientific philosophy developed by Anaxagoras rested upon two fundamental convictions: that nature and all its operations are rational, i.e., accessible to understanding through reason, and every account of nature must be consistent with and explain observed phenomena. Affirming that "visible things are a sight of the unseen", he did not think that universal order is limited to the empirical realm, but only that any explanation of Nature must take visible nature into account. His physics began with the theoretical assumption that matter is eternal, for without it one has to explain wholly unobserved processes of matter coming into existence from immateriality. Anaxagoras also recognized that the fundamental feature of every aspect of the universe is change. Whilst Parmenides argued from the fact of ceaseless change in nature to a metaphysical conception of the world in which neither change nor plurality is real, Anaxagoras left aside metaphysics and attempted to explain the principles of change and the maintenance of multiplicity. He did accept the sheer logic of the Parmenidean demonstration that nothing comes into being nor passes away and taught that change is due to separation and combination.

Matter for Anaxagoras is eternal and changeless in its essential nature. Since whatever is cannot cease to be, material substances cannot be reduced to fundamentally different elements. Therefore, matter must be infinitely varied in its irreducible parts to account for the vast variety of things that exist. The doctrine of homoiomeria is simply that the parts have the same nature as the whole, and the ultimate parts are eternal. Since living forms must be composed of parts of the same nature, and since they are nourished by food, every object must contain particles of an infinite variety and life must be part of everything that exists. Just as bone is called bone because of a predominance of bone particles, and rock is so named because of a predominance of mineral particles, so living things are recognized as such when the arrangement of particles allows for the expression of life. Given that "in everything there is a portion of everything", if material particles are of any size and of infinite variety, each thing, for example, a loaf of bread, would be infinite in size. If particles are of no size, however, even an infinite number would constitute no size at all. Therefore, Anaxagoras reasoned, particles are infinitesimal.

For of the small there is no smallest, but there is always a smaller; for it is not possible for what is not to be. But of the great there is always a greater also. And it is equal in number to the small, each thing being in respect to itself both great and small.

This ingenious view explains nutrition, for each part of the body extracts from ingested food that which is like itself, and it requires an infinitely large universe to accommodate an infinitude of particles. There is nothing in the nature of particles, though, that accounts for change. Particles are eternal, but structures composed of them come into existence and pass away. All change is the result of motion, and any rational theory of dynamics must aim to isolate one force or principle of motion which can account for every kind of change. Since the rational order of the universe includes change, the only rational entity which meets the required criteria is Nous or Mind. Materially, the macrocosm and microcosm reflect one another exactly; dynamically, the obvious operation of Nous in man, animals and even plants must reflect the universal operation of cosmic Mind. Mind as a rational principle of change manifests throughout nature as Law. Thus the universe consists of eternal, unchanging particulate matter and eternal, unchanging dynamic Mind. Subsequent philosophers criticized Anaxagoras for describing mechanical processes to explain natural phenomena and for not invoking Mind, but Anaxagoras saw clearly that calling on Mind as the direct cause of everything explained nothing, and that the universality of Mind implies that all mechanical processes are expressions of dynamic Nous. Mind, being as vast as the infinite cosmos, cannot be described in the language of secondary and derivative laws; nevertheless, being rational, it makes possible those describable laws whilst remaining an impenetrable yet undeniable mystery hidden behind them.

Whilst mechanistic explanations are sufficient for many natural phenomena, the laws involved cannot be thought of as simply mechanical. Law is the intelligent activity of Nous. Anaxagoras thought that this was shown by a study of perception. When two fluids of the same temperature are brought together, no alteration in temperature is observed. Only when qualitative opposites are conjoined does change occur. Perception involves change, and it too must follow the general principle that observable changes take place only between bodies whose states differ in some way. Universal Mind pervades every area of nature and is responsible for all change. Therefore Nous is qualitatively different from every material substance, and so is incorporeal yet substantial. This utter difference from everything else allows Mind to perceive and know everything. Nothing is hidden from Mind, which thus operates intelligently.

Armed with these principles of physics, Anaxagoras boldly set forth a theory of origins. Since everything contains everything else, differing only in preponderance of types of particles, Anaxagoras imagined that prior to the activity of Mind, matter abided in a state of chaos in which all particles were evenly distributed. Since nothing would differ from anything in this primordial condition, neither existence nor perception was possible, though matter and Mind were in a kind of timeless Be-Ness. For reasons now impossible to ascertain, Mind began to move matter in an initial rotational motion that tended to gather more dense particles at the centre and lighter particles in the extremes. The vestiges of this initial rotation can be seen in the movements of planets and the turning of the celestial vault. Once the rudimentary separation of material masses had occurred, a second motion, rising and falling, became dominant, as can be seen, Anaxagoras thought, in the convection currents which cause many meteorological phenomena. These motions produced the gross arrangement of the world as presently experienced, and Mind uses numerous more focalized processes to refine the world-order. Mind, being eternal as matter is, cannot perish. Whilst sleep is like death within a lifetime, there is one critical difference between the two.

Sleep comes from weariness of the body. It is a thing undergone by the body, not the soul. And death is the separation of the soul from the body.

Mind is immortal, but observation does not reveal its post-mortem destiny. For Anaxagoras, there is no reason to doubt that Mind continues to experience after the death of the body, remains involved in the ordering of the world, and ever reincarnates.

Since Anaxagoras refused to use Mind as a blanket explanation for every phenomenon, he had to construct explanations of diverse natural processes on the basis of observation and experimentation. Whilst some of his conclusions might appear to be absurd in comparison with contemporary explanations, his fidelity to his rigorous method led him to many remarkable insights. He held that all planets and stars are composed of the same substances of which the earth consists, and that they all follow the same laws. He recognized that thunder and lightning were effects of the same cause, which had to do with fiery etheric forces. He taught that light must travel in straight lines, that lunar eclipses are due to the earth's shadow falling on the moon, and that, unlike the sun which is incandescent, the moon shines by reflected sunlight. Long before Galileo dared to report his telescopic observation, Anaxagoras recognized that the moon is covered with mountains and ravines but is devoid of water. He taught that the Milky Way consists of innumerable stars and that the Nile floods in summer owing to far distant snows melting in the spring. Observation taught him that plants as well as animals breathe. He noted that in humans the father determines the sex of the child. So complete was his commitment to the unity of all life that he could not say the human being differs from animals because he possesses mind: all things are pervaded by Mind. Man is distinguished by his ability to express Nous more fully than other creatures, and this shows up in the human frame. When asked what the human hallmark is, Anaxagoras replied that "Men are the wisest of the animals because they have hands."

Anaxagoras not only exemplified the spirit of the Periclean age; he stood in the forefront of its creators. His faith in reason, reverence for Nature, rigour in method and refinement of experimental techniques earned him a permanent place as one of the early founders of modern science. Because Nous suffused everything, for Anaxagoras the ethical dimension of life is inherent in it, and so he gave no special attention to the subject. Plato saw that ethics had to be made explicit in a society rapidly losing its nearness to Nature, and that Mind in Man, psychology, is also a central study. In refraining from focussing on these critical areas of human understanding, Anaxagoras also adopted an attitude of true scepticism, neither pronouncing upon nor claiming to know anything about them other than that rational enquiry of the loftiest and most precise kind would lead to knowledge here as everywhere else. Whereupon he did focus his gaze, he influenced all subsequent generations of seekers, both in his theoretical methodology and in the experimental practices he laid down. Perhaps it is fitting that historians and philosophers nicknamed him 'Mind'.

Copyright©2010 Theosophy Trust

Prime time

Fame and fortune await the person who cracks the greatest problem in mathematics.
And that could be any day now, says Erica Klarreich

WHEN G. H. Hardy faced a stormy sea passage from Scandinavia to England, he took out an unusual insurance policy. Hardy scribbled a postcard to a friend with the words: "Have proved the Riemann hypothesis". God, Hardy reasoned, would not let him die in a shipwreck, because he would then be feted for solving the most famous problem in mathematics. He survived the trip.

Almost a century later, the Riemann hypothesis is still unsolved. Its glamour is unequalled because it holds the key to the primes, those mysterious numbers that underpin so much of math-ematics. And now whoever cracks it will find not only glory in posterity, but a tidy reward in this life: a $1 million prize announced this April by the Clay Mathematics Institute in Cambridge, Massachusetts.

There are signs that the great prize might soon be claimed, and the most promising approaches come not from pure mathematics, but from physics. Researchers have discovered a deep connection between the Riemann hypothesis and the physical world—a connection that could not only prove the hypothesis, but also tell us something profound about the behaviour of atoms, molecules and even concert halls. One mathematician has followed this lead into a very strange place, seeking a solution in an intricately twisted space with infinitely many dimensions.

Yet the primes seem simple enough at first glance. They are those numbers, like 2, 3, 5 and 7, that are only divisible by 1 and themselves, although 1 isn't included among them. Primes are the atoms of the number system, because every other number can be built by multiplying primes together. Unfortunately there is no periodic table for the primes—they are maddeningly unpredictable, and finding new primes is mostly a matter of trial and error.

In the 19th century, mathematicians found a little order in this apparent chaos. Even though individual primes pop up unexpectedly, their distribution follows a trend. It's like tossing a coin. The result is unpredictable, but after many coin tosses we expect roughly half heads and half tails. The primes get rarer as you look at larger and larger numbers (see Diagram), and mathematicians found that this thinning out is predictable. Below a given number x, the proportion of primes is about 1/ln(x), where ln(x) is the natural logarithm of x. So, for example, about 4 per cent of numbers smaller than ten billion are prime.

So far so good. But that "about" is very vague. Numbers are products of pure logic, and so surely they ought to behave in a precise, regular way. Mathematicians would at least like to know how far the prime numbers stray from the distribution.

Georg Riemann found a vital clue. In 1859, he discovered that the secrets of the primes are locked inside something called the zeta function. The zeta function is simply a particular way of turning one number into another number, like the function "multiply by 5". Riemann decided to see what would happen if he fed the zeta function complex numbers—numbers made from a real part (an ordinary number) and a so-called imaginary part (a multiple of i, the square root of -1). Complex numbers can be visualised as arrayed on the complex plane, with real numbers on the horizontal axis and imaginary numbers on the vertical axis.

Riemann found that certain complex numbers, when plugged into the zeta function, produce the result zero. The few zeros he could calculate lay on a vertical line in the complex plane, and he guessed that, except for a few well-understood cases, all the infinity of zeros should lie exactly on this line.

What does this have to do with the primes? If you plot how many primes exist below a given number (see Diagram above), what you get is a smooth curve with small wiggles added—that is, the 1/ln(x) rule, plus deviations.

According to Michael Berry of Bristol University, you can think of that pattern of deviations as a wave. Just like a sound wave, it is made up of many frequencies. "And what are the frequencies?" asks Berry. "They're the Riemann zeros. The zeros are harmonies in the music of the primes."

Berry isn't speaking in metaphors. "I've tried to play this music by putting a few thousand primes into my computer," he says "but it's just a horrible cacophony. You'd actually need billions or trillions—someone with a more powerful machine should do it."

Riemann worked out that if the zeros really do lie on the critical line, then the primes stray from the 1/ln(x) distribution exactly as much as a bunch of coin tosses stray from the 50:50 distribution law. This is a startling conclusion. The primes aren't just unpredictable, they really do behave as if each prime number is picked at random, with the probability 1/ln(x)--almost as if they were chosen with a weighted coin. So to some extent the primes are tamed, because we can make statistical predictions about them, just as we can about coin tosses.

But only if Riemann's guess was right. If the zeros don't line up, then the prime numbers are much more unruly. As Enrico Bombieri of the Institute for Advanced Study in Princeton writes on the Clay Institute website ( "The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers." And the havoc would spread further. Hundreds of results in number theory begin, "If the Riemann hypothesis is true, then . . ."

This is why mathematicians long to prove the hypothesis. But how do you prove something about an infinity of numbers?

Researchers have used supercomputers to calculate the first 1,500,000,001 zeros above the x-axis, and millions of other zeros higher up, and so far all of them lie on the critical line. If just one of them did not, the Riemann hypothesis would be killed.

This is heartening, but no amount of computer hacking can prove the hypothesis. There are always more zeros to check. And, cautions Andrew Odlyzko of AT&T Labs, who has spearheaded the effort to calculate zeros, "number theory has many examples of conjectures that are plausible, are supported by seemingly overwhelming numerical evidence, and yet are false."

Some deeper insight is needed. Early in the 20th century, mathematicians made a daring conjecture: that the Riemann zeros could correspond to the energy levels of a quantum mechanical system.

Quantum mechanics deals with the behaviour of tiny particles such as electrons. Crucially, its equations work with complex numbers, but the energy of a physical system is always measured by a real number. So energy levels form an infinite set of numbers lying along the real axis of the complex plane--a straight line.

This sounds like Riemann's zeros. The line of zeros is vertical, rather than horizontal, but it is a simple bit of maths to rotate it and put it on top of the real line. If the zeros then match up with the energy levels of a quantum system, the Riemann hypothesis is proved.

For decades, this idea was only wishful thinking. Then in 1972 came a hint that it could work. Hugh Montgomery, at the University of Michigan, had found a formula for the spacings between Riemann zeros. Visiting the Institute for Advanced Study at Princeton, he ran into physicist Freeman Dyson at afternoon tea, and mentioned his formula. Dyson recognised it immediately. It was identical to a formula that gives the spacings between energy levels in a category of quantum systems—quantum chaotic systems, to be precise.

Chaos theory applies to physical systems so sensitive to their starting conditions that they are impossible to predict. In the Earth's chaotic atmosphere, for example, the tiny draught caused by the flap of a butterfly's wings can eventually lead to a tremendous storm. Almost all complicated systems are chaotic.

The quantum versions of these systems have a jumble of energy levels, scattered apparently at random but in fact spaced according to Montgomery's formula. Quantum chaotic systems include atoms bigger than hydrogen, large atomic nuclei, all molecules, and electrons trapped in the microscopic arenas called quantum dots. Could the Riemann zeros fit one of these quantum chaotic systems?

In the late 1980s, Odlyzko picked an assortment of systems, and compared their energy levels with the Riemann zeros. In a discovery that electrified mathematicians and physicists, Odlyzko found that when he averaged out over many different chaotic systems, the energy level spacings fitted the Riemann spacings with stunning precision.

That's still not enough. To prove the Riemann hypothesis, researchers must pinpoint a specific quantum system whose energy levels correspond exactly to the zeros, and prove that they do so all the way to infinity. Which, of all the different systems, is the right one?

Berry and his colleague Jonathan Keating have made one suggestion. In a chaotic system, an object usually moves unpredictably, but sometimes its path will cycle back on itself in a "periodic orbit". Berry and Keating think that the right quantum system will have an infinite collection of periodic orbits, one for each prime number. And last year, Nicholas Katz and Peter Sarnak predicted that the system should have a special kind of symmetry called symplectic symmetry.

Both of these clues should help quantum chaologists zero in on the one system that will prove the Riemann hypothesis. "I have a feeling that the hypothesis will be cracked in the next few years," says Berry. "I see the strands coming together. Someone will soon get the million dollars."

The winner could well be Alain Connes, a mathematician based at the Institute of Advanced Scientific Study in Bures-sur-Yvette, France. Connes has a startlingly direct approach to the problem: create a system that already includes the prime numbers. To understand how, you have to imagine a quantum system not as a particle bouncing around an atom, say, but as a geometrical space. It sounds odd, but it represents one of the weird things about quantum systems: they can be two or more things at once.

Like Schrödinger's cat, which is a peculiar mixture of dead and alive, any quantum object can find itself in a "superposition" of different states. To characterise this messy existence, physicists use what they call a state space. For each kind of possibility (say "alive" and "dead"), you draw a new axis and add a dimension to the space. If there are just two possible states, as is the case for Schrödinger's cat, the space is two dimensional, with three states it is three dimensional, and so on.

Then in the Schrödinger's cat space, you would mark a cross one unit along the x-axis to represent a fully alive cat. Similarly, a stone dead cat would be one unit up the y-axis, and a part-alive, part-dead cat would appear somewhere along an arc between these points.

The "shape" of the space affects how the state moves around in it, and therefore how the system works, including the way its energy levels are arrayed. This depends not just on the number of dimensions, but also on the geometry of how they are stuck together.

Connes decided to build a quantum state space out of the prime numbers. Of course, the primes are a bunch of isolated numbers, nothing like the smooth expanses of space in which we can measure things like angles and lengths. But mathematicians have invented some bizarrely twisted geometries that are based on the primes. In "5-adic" geometry, for example, numbers far apart (in the ordinary way) are pulled close together if they differ by 5, or 15, or 250—any multiple of 5. In the same way, 2-adic geometry pulls together all the even numbers.

To put all the primes in the mix, Connes constructed an infinite-dimensional space called the Adeles. In the first dimension, measurements are made with 2-adic geometry, in the second dimension with 3-adic geometry, in the third dimension with 5-adic geometry, and so on, to include all the prime numbers.

Last year Connes proved that his prime-based quantum system has energy levels corresponding to all the Riemann zeros that lie on the critical line. He will win the fame and the million-dollar prize if he can make one last step: prove that there aren't any extra zeros hanging around, unaccounted for by his energy levels.

That last step is a formidable one. Has Connes simply replaced the Riemann hypothesis with an equally difficult question? Some experts advise caution. "I still think that some major new idea is needed here," says Bombieri.

Berry, for his part, doesn't flinch at the mathematical peculiarity of Connes's system. "I'm absolutely sure that if he's right, someone will find a clever way to make it in the lab. Then you'll get the Riemann zeros out just by observing its spectrum."

Berry and Keating are now turning around this connection with physics, using mathematics based on the Riemann zeta function to predict the behaviour of chaotic systems. Most models of quantum chaos are complicated and difficult to calculate. The Riemann zeros, by comparison, are easy to compute. "We always test our formulae on the Riemann zeta function to see if they work," says Keating.

If Connes or one of the physicists proves the Riemann hypothesis using a quantum system, the link will be firmly established. Then, Berry predicts, the field will blossom. Using the mathematics of the zeta function, scientists will be able to predict the scattering of very high energy levels in atoms, molecules and nuclei, and the fluctuations in the resistance of quantum dots in a magnetic field.

And it turns out that the same mathematics applies to any situation where waves bounce around chaotically, including light waves and sound. So the performance of microwave cavities and fibre optics could be improved, and the acoustics of real concert halls might profit from the music of the primes.

Even so, it is mathematics that will gain the most. "Right now, when we tackle problems without knowing the truth of the Riemann hypothesis, it's as if we have a screwdriver," says Sarnak. "But when we have it, it'll be more like a bulldozer." For example, it should lead to an efficient way of deciding whether a given large number is prime. No existing algorithms designed to do this are guaranteed to terminate in a finite number of steps.

Proving the Riemann hypothesis won't be the end of the story. It will prompt a sequence of even harder, more penetrating questions. Why do the primes achieve such a delicate balance between randomness and order? And if their patterns do encode the behaviour of quantum chaotic systems, what other jewels will we uncover when we dig deeper?

Those who believe that mathematics holds the key to the Universe might do well to ponder a question that goes back to the ancients: What secrets are locked within the primes?

Friday, 5 November 2010

Sri Nisargadatta Maharaj

In the following passages, you will find the unparalleled words of Nisargadatta Maharaj as seen in the book I Am That, translated by Maurice Frydman from Acorn Press.

Only the dead can die, not the living. That which is alive in you is immortal.

In reality there is only the source, dark in itself, making everything shine. Unperceived, it causes perception. Unfelt, it causes feeling. Unthinkable, it causes thought. Non-being, it gives birth to being. It is the immovable background of motion. Once you are there, you are at home everywhere.

In reality there are no others, and by helping yourself you help
everybody else.

That which you are, your true self, you love it, and whatever you do, you do for your own happiness. To find it, to know it, to cherish it is your basic urge. Since time immemorial you loved yourself, but never wisely. Use your body and mind wisely in the service of the self, that is all. Be true to your own self, love yourself absolutely. Do not pretend that you love others as yourself. Unless you have realized them as one with yourself, you cannot love them. Don't pretend to be what you are not, don't refuse to be what you are. Your love of others is the result of self- knowledge, not its cause. Without self-realization, no virtue is genuine. When you know beyond all doubting that the same life flows through all that is and you are that life, you will love all naturally and spontaneously. When you realize the depth and fullness of yourself, you know that every living being and the entire universe are included in your affection. But when you look at anything as separate from you, you cannot love it for you are afraid of it. Alienation causes fear and fear deepens alienation. It is a vicious circle. Only self-realization can break it. Go for it resolutely.

I see what you too could see, here and now, but for the wrong focus of your attention. You give no attention to your self. Your mind is all with things, people and ideas, never with your self. Bring your self into focus, become aware of your own existence. See how you function, watch the motives and results of your actions. Study the prison you have built around yourself, by inadvertence.

The seeker is he who is in search of himself. Give up all questions except one: 'Who am I?' After all, the only fact you are sure of is that you are. The 'I am' is certain. The 'I am this' is not. Struggle to find out what you are in reality. To know what you are, you must first investigate and know what you are not. Discover all that you are not - body, feelings, thoughts, time, space, this or that - nothing, concrete or abstract, which you perceive can be you. The very act of perceiving shows that you are not what you perceive. The clearer you understand that on the level of mind you can be described in negative terms only, the quicker will you come to the end of your search and realize that you are the limitless being.

Nothing is wrong with you, but the ideas you have of yourself are altogether wrong. It is not you who desires, fears and suffers, it is the person built on the foundation of your body by circumstances and influences. You are not that person. This must be clearly established in your mind and never lost sight of.

What is really your own, you are not conscious of. What you are conscious of is neither you nor yours. Yours is the power of perception, not what you perceive. It is a mistake to take the conscious to be the whole of man. Man is the unconscious, the conscious and the superconscious, but you are not the man.Yours is the cinema screen, the light as well as the seeing power, but the picture is not you.

The source of all has all. Whatever flows from it must be there already in seed form. And as a seed is the last of innumerable seeds, and contains the experience and the promise of numberless forests, so does the Unknown contain all that was, or could have been and all that shall or would be. The entire field of becoming is open and accessible; past and future co-exist in the eternal now.

Only when you realize the true peace, the peace you have never lost, that peace will remain with you, for it was never away. Instead of searching for what you do not have, find out what is it that you have never lost? That which is there before the beginning and after the ending of everything; that to which there is no birth, nor death. That immoveable state, which is not affected by birth and death of a body or a mind, that state you must perceive.

Back To Words Of Wisdom

If you seek reality you must set yourself free of all backgrounds, of all cultures, of all patterns of thinking and feeling. Even the idea of being man or woman, or even human should be discarded. The ocean of life contains all, not only humans. So, first of all abandon all self-identification, stop thinking of yourself as such-and-such or so-and-so, this or that. Abandon all self-concern, worry not about your welfare, material or spiritual, abandon every desire, gross or subtle, stop thinking of achievement of any kind. You are complete here and now, you need absolutely nothing.

I find that somehow, by shifting the focus of attention, I become the very thing I look at, and experience the kind of consciousness it has; I become the inner witness of the thing. I call this capacity of entering other focal points of consciousness, love; you may give it any name you like. Love says "I am everything". Wisdom says "I am nothing". Between the two, my life flows. Since at any point of time and space I can be both the subject and the object of experience, I express it by saying that I am both, and neither, and beyond both.

Giving up desire after desire is a lengthy process with the end never in sight. Leave alone your desires and fears, give your entire attention to the subject, to him who is behind the experience of desire and fear. Ask: who desires? Let each desire bring you back to yourself.

Questioner: It will take much time if I just wait for self-realization.
Maharaj: What have you to wait for when it is already here and now? You have only to look and see. Look at your self, at your own being. You know that you are and you like it. Abandon all imagining, that is all. Do not rely on time. Time is death. Who waits--dies. Life is now only. Do not talk to me about past and future--they exist only in your mind.
Questioner: You too will die.
Maharaj: I am dead already. Physical death will make no difference in my case. I am timeless being. I am free of desire or fear, because I do not remember the past or imagine the future. Where there are no names and shapes, how can there be desire and fear? With desirelessness comes timelessness. I am safe, because what is not, cannot touch what is. You feel unsafe, because you imagine danger. Of course, your body as such is complex and vulnerable and needs protection. But not you. Once you realize your own unassailable being, you will be at peace.
Questioner: How can I find peace when the world suffers?
Maharaj: The world suffers for very valid reasons. If you want to help the world, you must be beyond the need of help. Then all your doing as well as not doing will help the world most effectively.

Thursday, 4 November 2010

Are We Ready?

"It's a waste of energy to be angry with a man who behaves badly, just as it is to be angry with a car that won't go."
~~Bertrand Russell

Nowadays, the idea that energy could be a substance is actively discouraged, and very often ridiculed. However, reducing the term "energy" to what amounts to nothing more than an abstract concept, is really only a recent development in terms of human history. Before the advent of modern theory, our knowledge was based on traditions, stemming back to ancient times, which percieved energy as a substance. Indeed, it was supposed that the entire Universe is immersed in it. In an enlightening article taken from Centerpointe's newsletter "Mind Chatter" - "What is Reality (and why should you care?)" - Bill Harris, writes a fascinating account of how the mystics came to interpret "reality." As Harris explains, what the mystics percieved as energy, also had a lot to do with what they saw as the Divine. Below, is an extract taken from the article:

"For thousands of years, mystics have said that there is one energy in the Universe, that the Universe and everything in it is the play, the dance, the vibration, of that one energy. Underneath the seeming multiplicity, they say, everything is made of the same substance. This energy, they say, is everywhere and "everywhen." This principle is sometimes described as Omnipresence or God. The Hindus and Buddhists call this principle, Sat-one energy, everywhere, making up everything, always, past, present, and future.

Quantum mechanical physicists, for several decades, have been saying the same thing. They notice that on the sub-atomic level, particles come into being, seemingly out of nothing, and dissolve, and disappear back into nothing, that two or more particles collide, and one, two, three or more particles, of a different kind, appear from the collision, or all the particles cease to exist. There is a "something" that everything comes out of, and returns to, and which makes up, or is the background of, everything.

The mystics, however, went one step further. In adddition to noting that this one energy is Omnipresent, they also said something else that I think is rather startling. They said that this one energy is aware of itself being everything and everywhere and everywhen: that it is conscious, that it has consciousness. The mystics called this second characteristic of reality Chit."

I suspect that one of the reasons as to why modern theory is so reluctant to imagine energy as a substance, is because admission immediately arouses unwelcome religious fervour. Science wants only to deal with the stuff in the Universe that it can quantify, and substantiate - it's not overly concerned with an invisible, intangible God, nor for that matter, any unseen, immaterial, imponderable substances which might, or might not be energy, or consciousness, or whatever. To science, the questions and answers to these concepts are irrelevant - mere distractions from the job at hand. Entering philosophical and theological discourse will quite often do nothing to enhance their sums.

If science were ever tempted to admit energy as a substance, conceding to the idea that it can be quantified in someway, it would fling the door wide open to suggestions that the stuff of energy, which it is weighing, measuring, and collecting, comes very close to being the pure substance of God. It would allow practically every person on the planet to point out to scientists, that the substance of energy, which they are pouring from their flasks into test tubes, also amounts to unequivocal proof in the existence of God. Now, it is not simply energy which we can reach out and touch, all with our very own fingers, but God's personal Being.

I suspect that there are some physicists who are more than familiar with the God-energy conundrum, but are hesitant to openly discuss it - not because they are wilfully obstructive, or even particularly dispassionate - but because they are aware of the chain of events such an admission will unleash. The question which rises to the forefront most is not: is science willing to admit to the existence of God? The question we should really be asking is: is humanity ready to recieve all that power?