137: Jung, Pauli, and the Pursuit of a Scientific Obsession Part 4

It was Pythagoras who pioneered the quest for a link between numbers and the cosmos. Pondering the hidden meanings of the world around him as he played on his lyre he began to wonder whether the laws of harmony depended on numbers. He found that to play tones an octave apart, the length of the strings needed to be in specific ratios: the keynote of the octave sounded when the ratio was 1:2; a fifth required a ratio of 2:3; and a fourth, 3:4.

Perhaps numbers might belong to a world beyond perception, which could only be fully apprehended by thought. His striking conclusion was that the numerals 1, 2, 3, and 4 represented all known objects: 1 represents a point; 2 points can be connected by a line; 3 points make a triangle, in particular a perfect equilateral triangle; and 4 points make a tetrahedron, a pyramid of three perfect triangles. From these could be constructed the five "Pythagorean" solids (later "Platonic" solids after Plato): the tetrahedron, cube, octahedron (eight equilateral triangles), dodecahedron (twelve pentagons), and icosahedron (twenty equilateral triangles). Each could be circ.u.mscribed by a sphere, with each point of the solid touching its surface, and each could also contain a sphere whose surface touched each of its sides.

Represented as dots, 1, 2, 3, and 4 form an equilateral triangle set out in four rows, known as the tetraktys (tetras is Greek for "four"): Pythagoras's tetraktys.

To Pythagoras this a.n.a.lysis made sense of our world, in which he recognized four elements (earth, water, air, and fire), four seasons, four points of the compa.s.s, and four rivers of paradise (the Pishon, the Gihon, the Tigris, and the Euphrates). His followers swore an oath "by him who has committed to our soul the tetraktys, the original source and root of eternal Nature." The sum of the numbers that made up the tetraktys is ten, which Pythagoras considered the perfect number. Once we have counted to ten, we return to one, the number of creation.

Pythagoras's claim was that numbers were the fabric of our universe and existed independently of us. Numbers were the keys through which could be heard the harmony of the cosmos.

The Kabbalah.

The Egyptian G.o.d Thoth, known to the Greeks as Hermes Trismegistus ("Thrice-great Hermes"), was credited with a huge number of writings on philosophy, astrology, and magic. Over the centuries Hermetic literature incorporated elements of whatever science existed as well as the teachings of Pythagoras.

In Kepler's time Hermetic literature was enthusiastically embraced as an antidote to the rational approach of Greek philosophy and science. It was full of mystery and magic and spoke in terms of a vital or living force at the heart of the cosmos. Hermetic literature also included kabbalistic texts.

Versions of the Kabbalah had begun to appear in the thirteenth century. A princ.i.p.al theme was how one might see the invisible in the visible and the spiritual in matter. The Kabbalah discussed the clash between opposites like light and darkness to produce the world in which we live. Someone like Kepler, who was interested in the teachings of Proclus, was naturally drawn to the Kabbalah with its similar theme.

A central notion of kabbalistic philosophy is the Sephirot. The Sephirot is usually represented as the tree of life with ten branches rooted in the earth and extending to Heaven, signifying the earth as a microcosm reflecting the universe, the macrocosm. It is made up of five pairs of opposites-beginning and end, good and evil, above and below, east and west, and north and south-and thus has ten emanations, ten being a holy number in Judaism as well as in Pythagoreanism.

By the end of the fifteenth century the Kabbalah had been integrated into Christian theology, though the Christian Kabbalah emphasized the Trinity rather than the Sephirot. Christian thinkers were especially fascinated by the Gematria, which a.s.signed numbers to letters of the Hebrew alphabet. This concept of numbers for language opened up the possibility of a.s.signing numbers to the various names of G.o.d, thereby further revealing His celestial powers and His mystery. Thus the Kabbalah became identified with magic and numerology. (Until the nineteenth century the Hebrew alphabet had no numbers; letters were used for numbers. Thus in Roman times 666 happened to be the letters for Nero's name.) By Kepler's day the Christian Kabbalah was considered one of the "handmaidens" of true wisdom, along with alchemy and astrology. But all this clashed with the onset of a new, materialistic science that claimed to be able to predict the course of cannonb.a.l.l.s and planets using mathematics, but only if a division were made in nature between dead and live matter. For mathematics could be applied only to the former, not to the latter.

Kepler's model of the universe.

When Kepler was growing up, there was a flood of astrological, kabbalistic, and alchemical texts being published. Anything attributed to Hermes Trismegistus was hailed as a revelation. They held readers spellbound, the vaguer the better. Kepler was hooked; his enormous imagination was sparked.

Why was the world as it was? Why were there six planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn)? Why were they at certain distances from the sun? What was the relationship between their distances from the sun and their speeds? Might the answers to these questions lie in certain arrangements of geometrical figures?

By this time Kepler was a district mathematician, teaching mathematics and astronomy at the Protestant Seminary in Graz. During one of his cla.s.sroom lectures on geometry he happened to draw an equilateral triangle. Inside it he drew a circle touching all three sides and around it another circle touching its points, just as Pythagoras had described. Suddenly it all fell into place. It was a model of the universe.

Clearly the reason why there were six planets was because there were five perfect solids symbolizing the five intervals between the planets. The planets moved on spheres that circ.u.mscribed the five solids. Calculating the distances of the planets from the sun, Kepler drew up a new image of Copernicus's universe with the sun at the center and nested planetary spheres on which the planets moved. The sphere of Mercury inscribed an octahedron, that of Venus an icosahedron, that of Earth a dodecadron, that of Mars a cube, and that of Saturn a tetrahedron, which the orbit of Jupiter circ.u.mscribed.

Kepler's 1596 model of the universe. (Kepler, Mysterium Cosmographic.u.m [1596].).

Kepler attributed this revelation to "divine ordinance." He had "always prayed to G.o.d [that] Copernicus had told the truth." In his diary, he noted the fateful day when G.o.d spoke to him: July 19, 1595.

He was convinced he had discovered G.o.d's geometrical plan of the cosmos, which G.o.d had made in his own image. He published his work in 1596 in a book ent.i.tled Mysterium Cosmographic.u.m (The Mystery of the Universe).

However, his model was not in total agreement with Copernicus's data, especially the data for the orbit of Mercury. Despite his mystical leanings, Kepler was a new breed of scientist. He required theories to be supported by data. He decided that what he needed was more precise data. Copernicus's were not good enough.

From circles to ellipses.

Among the people to whom Kepler sent his new book was the greatest observational astronomer of the day, Tycho Brahe. Kepler by now was a handsome twenty-five-year-old with a high forehead, immaculate goatee, aquiline nose, and a look of piercing intelligence. Tycho, as he was always known, was twice Kepler's age. He sported a mustache so immense that it looked like a walrus's tusks and was famous for his prosthetic nose, having had his real one cut off in a duel. He had a copper nose for everyday and a gold and silver one for special occasions.

Tycho achieved his world-renowned accuracy by making all his observations from his monstrous-looking observatory, Uraniburg, on the island of Hveen, off the coast of Denmark. But to Kepler's annoyance, Tycho refused to reveal his data to anyone until he had refined his own model of the universe-in which every planet except the earth orbited the sun and the entire a.s.semblage, in turn, orbited the earth.

Impressed with Kepler, Tycho offered him a position as his a.s.sistant so that he could help him with the mathematics of his model, little realizing that Kepler simply wanted to lay his hands on his data. Kepler accepted the offer and joined Tycho in Prague, where he was imperial mathematician to the court of Emperor Rudolf II.

Johannes Kepler.

Tycho Brahe.

Tycho set Kepler to work to improve his observations of Mars, the most difficult of the planets due to its p.r.o.nounced retrograde motion. Astronomers described the orbit of Mars as having a large "eccentricity"-the distance that the sun had to be moved from the center of Mars's...o...b..t to improve agreement with Tycho's data of the complicated system of circles rolling on circles. This displacement was a mathematical device used in every model of the universe-in Ptolemy's it was the earth whose position was displaced from the center of the universe. In reality, of course, in Copernicus's system, the sun was at the center of the universe. The models of Ptolemy, Copernicus, and Tycho could not deal adequately with Mars's eccentricity. Kepler bet his colleagues that he could straighten it out in eight days. In fact it took him eight years.

In October 1601 Tycho suddenly died at the age of fifty-five and Kepler was appointed imperial mathematician. He inherited all of Tycho's data and more important, no longer had to waste time fiddling with Tycho's model of the universe.

To start with, Kepler a.n.a.lyzed Tycho's data on the orbit of Mars, trying to preserve the old model of the universe by explaining the orbits of the planets in terms of circles. Taking Tycho's best data for Mars, he used the mathematical device of displacing the sun from the center of Mars's...o...b..t by a certain distance to allow for eccentricity. Then, by adroit mathematics, he moved himself from the earth to Mars and found that the earth also moved in an orbit similar to Mars's, with varying speeds.

Supposing the orbit of Mars was not a circle but an oval? Kepler spent 1604 struggling with the mathematics of an oval. That year was full of problems. Both he and his wife fell ill; when he became short of money his wealthy wife refused to dip into her funds; and she also gave birth to yet another child whom Kepler saw as yet another problem. And an ominous new star appeared in the sky-the nova of 1604.

Then he tried replacing the oval with an ellipse. An ellipse is a circle that has been squashed at its north and south poles. It has two centers, or focii, neither of which is in the middle. When the two centers are moved together the ellipse becomes a circle. This worked perfectly. The curve went through all of Tycho's data points for the orbit of Mars and also fitted Mars's measured eccentricity. Kepler had discovered his first law of planetary motion: that every planet moves in an ellipse with the sun at one of its centers. The sun is no longer at the center of the universe but at one of the ellipse's foci.

Soon after, he discovered his second law: that a line drawn from the sun to a planet sweeps out equal areas in equal times. This meant that a planet's speed varied as it traveled in an ellipse around the sun: the planet sped up as it neared the sun and slowed down as it moved away.

Kepler had overthrown the two-thousand-year-old a.s.sumptions that the complicated orbits of planets could only be explained by adding circles moving on circles in uniform circular motion and that the planets move with a uniform speed. He published his new laws of astronomy in his 1609 Astronomia nova (The New Astronomy).

But what kept the planets from escaping altogether and flying off into the void? Perhaps there were tentacles emanating from the sun, grasping a planet and whipping it around in its...o...b..t. Kepler imagined the attraction to be magnetism. Newton would later discover that it was gravity. Kepler, however, could only conceive of it as some sort of vital or living force.

As Pauli points out, Kepler was caught between two worlds. His laws of planetary motion were an accurate description of the paths of the planets around the sun, but they emerged from mathematical calculations that wrenched the sun out of its true place at the center of the universe. Using mathematics meant he had to treat the earth as dead matter. However, according to his Renaissance beliefs the earth was not dead at all, it had a soul, an anima terrae, akin to the human soul. It was a living organism. Sulphur and volcanic products were its excrement, springs coming from mountains its urine, metals and rainwater its blood and sweat, and sea water its nourishment. Kepler's attempts to link such animistic beliefs to scientific data made him a new breed-a scientific alchemist. He had no choice but to compartmentalize his work: ellipses were confined to the scientific side of his life, circles and spheres to the religious and alchemical side.

Kepler's third law.

In 1611 Emperor Rudolf abdicated. To escape the dangerous political intrigues that followed, Kepler moved to Linz, the capital of Upper Austria, a charming city on the Danube. Before leaving Prague, however, his wife fell ill with typhus and died.

Kepler's marriage had not been happy. Nevertheless, after her death he was lonely. He also had three young children to look after, two girls and a boy. He looked around for another wife in the same way he had discovered his two laws-by trial and error. He ended up with eleven choices, some of whom he had advertised for, others whom he had tried out, sometimes boarding his children with them to see if they all got along. One was attractive but too young, another fat, another was of poor health. Kepler finally settled on number five and she gave him the peace of mind to resume his scientific research.

His first two laws had been essentially geometrical-number was missing. Now he turned his attention to numbers. If the sun controlled the planets, he thought there had to be a relationship between the planets' distances from the sun and their speeds.

Meanwhile Europe was heading for the Thirty Years' War. Troops were on the move causing famine, havoc, and plague. Then one of his daughters died. In his grief he turned inward to "contemplation of the Harmony," which he believed to exist in nature. Thinking of the musical harmonies explored by Pythagoras, he pondered the eternal reality of numbers, which revealed the very essence of the soul.

How did this numerical harmony relate to the planets in a sun-centered system? Kepler tried to find a way to work out whether harmonious ratios could be formed out of the planets' periods of revolution, their volumes, their sizes, or their velocities when they were furthest from and closest to the sun. But he failed. Then he thought of examining the ratios of a planet's angular velocities at its extremes from the sun, that is, its change in angle at any period as it moved across the sky. And finally the astral music of the Divine Composer began to emerge.

Little by little Kepler worked out the ratios that produced the melodies played by the planets as they moved in their elliptical orbits. It was a heavenly symphony "perceived by the intellect, not by the ear," he wrote. But for Kepler it was much more. To him the planets sang in "imitation of G.o.d" in different voices-soprano, contralto, tenor, and ba.s.s. But on the earth there was only discord: "The Earth sings Mi-Fa-Mi, so we can gather even from this that Misery and Famine reign on our planet," he wrote despondently.

Kepler's third law a.s.serts that the following two quant.i.ties are proportional: the time needed for the earth to go once around the sun, multiplied by itself (that is, squared); and the earth's average distance from the sun, multiplied by itself three times (that is, cubed). It completed for him what had been the goal of Pythagoras: to explain the universe in terms of geometry and number. He scoured tables of numbers until he found the pattern but he never revealed precisely how he had discovered this capstone of his life's work. He recorded the date: March 8, 1618. "At first I thought I was dreaming," he wrote in the book he published the following year, Harmonices Mundi.

Sure that G.o.d had spoken through him, he wrote that he did not mind if his book had to "wait a hundred years for a reader. Did not G.o.d wait six thousand years for one to contemplate His works?"

All this took place at a time of great personal difficulty. Kepler's mother, Katharina, had been put on trial for witchcraft. Her sister had been burned at the stake as a witch, and this, together with her husband's disappearance, rendered her very suspicious to the gullible populace. In old age she was far from lovely and had a nasty temperament that made her an easy target in the witch-hunting mania in Germany of the early seventeenth century, so much so that she came close to sharing her sister's fate. In 1615, she was in the middle of a feud with another old woman. This neighbor persuaded an influential relative to accuse Katharina of making her extremely ill by feeding her a witch's potion. Others soon began to remember becoming seriously ill after having accepted drinks from Katharina.

Not only was his mother in danger, but so was the family name. Kepler had to take time off from pondering the universe to defend his mother for whom he felt affection and pity, despite his horrendous childhood. The proceedings took over six years. At one point jailers flourished instruments of torture and execution in front of Katharina's face, as was customary. Unusually for the time, the story has a happy ending. Kepler finally succeeded in obtaining her release.

Robert Fludd.

Robert Fludd-a universe made up of fours.

Two years before he finished Harmonices, in 1617, Kepler happened to see a highly ill.u.s.trated book at the Frankfurt book fair: A Metaphysical, Physical and Technical History of the Macro-and the Micro-Cosm, by the English physician and Rosicrucian, Robert Fludd.

While Kepler's family was low cla.s.s, Fludd's was n.o.ble. His father, Thomas Fludd, had been knighted by Queen Elizabeth I for his services as war treasurer in the Netherlands and paymaster to English troops in Provence. In portraits Fludd looks rather plump and well fed, with a pointed goatee. He holds his two middle fingers pressed together, perhaps in some sort of secret sign. In one portrait he has fingernails as long as a mandarin's.

Fludd studied at Oxford and became intrigued by Greek philosophy. As was the custom for wealthy young gentlemen, he toured France and Germany, meeting and sometimes tutoring n.o.bility. In Germany he became acquainted with a secret society who called themselves Brothers of the Rosy Cross-Rosicrucians. They called for a reform of knowledge in preparation for Armageddon and claimed access to deep secrets and truths in medicine, philosophy, and science. Governments deemed their mysticism and apocalyptic message dangerous and they were often charged with heresy and religious innovation, serious offenses in those days.

When Fludd's enemies at the court of King James I accused him of collaborating with them, he argued persuasively that the Rosicrucians were innocent of heresy. James was so impressed that he became Fludd's patron.

In his book Fludd a.s.serted that "the true philosophy...will sufficiently explore, examine and depict Man, who is unique, by means of pictures." In other words, he intended the sumptuous ill.u.s.trations in his books not merely as decoration but as saying something very definite about the world. Kepler, too, used diagrams, but of a scientific character-optical constructions made up of rays of light, a sphere with light emanating from its center as straight lines, or an image of planets moving in ellipses around a sun displaced from the center of the universe.

Both agreed that there was an invisible realm of qualities and powers, as well as a harmonics of nature. But while Fludd's world was one of astral powers and invisible spiritual illumination, Kepler's was of invisible magnetic forces, archetypal images, and hidden astrological meanings.

In his Harmonices, Kepler included a devastating critique of Fludd's book. Fludd immediately sprang to the defense. To start with, Kepler derided Fludd's extensive reliance on pictures; for what interested Kepler was mathematics. In reply, Fludd ranked him with "vulgar mathematicians" who concern themselves only with "quant.i.tative shadows." Philosophers like himself, Fludd wrote, "comprehend the true core of natural bodies" rather than stripping nature bare with cold mathematics. Kepler replied, "In Fludd's method is the business of alchemists, hermetists and Paracelsians; mine is the task of the mathematician."

In his drawings Fludd represented the text of Genesis using images based on alchemy, astrology, and the Kabbalah, with light playing a central role. To him the mundane world was the mirror image of the invisible world of the Trinitarian G.o.d. He represented this as two equilateral triangles placed together and wrote beside the upper one: "That most divine and beautiful Object [G.o.d] seen in the murky mirror of the world drawn underneath." The upper triangle contains the four Hebrew characters -the tetragrammaton-for the ineffable name of G.o.d, YHVH, set within another perfect triangle. The triangle beneath it is the "reflection of the incomprehensible triangle seen in the mirror of the world," Fludd wrote.

The divine and mundane triangles. (Fludd, Utriusque Cosmi Maioris scilicet et Minoris, Metaphysica, Physica atque technica Historices mundi [1621].) To depict the creation Fludd used an image of interpenetrating triangles. One triangle ascends from the earth. It is dark at the base and becomes brighter as it moves toward heaven. The inverted triangle, meanwhile, has its apex on the earth. The former culminates in the perfect triangle, the symbol of G.o.d, while the latter emanates from it. They mirror each other precisely and thus represent the constant struggle of polar opposites: the triangle rising from the earth represents the dark principle, or matter, while the other is the light principle, or form. Matter and form, light and darkness are the polar principles of the universe. This is reminiscent of the Kabbalah where these opposites are called antipathy and sympathy.

Interpenetration of material and formal pyramids. (Fludd, Utriusque Cosmi Maioris scilicet et Minoris, Metaphysica, Physica atque technica Historices mundi [1621].) Fludd emphasizes that the world about us results from a struggle between dark and light by placing the sun at the intersection point of the two pyramids, where the opposing principles counterbalance each other. This also signals his belief that the unity of G.o.d Himself is symbolized in the mystery of the alchemical wedding in which opposites are fused together.

Placing the apex of the light triangle on the earth symbolizes the withdrawal of light and the appearance of matter. In his a.n.a.lysis of all this, Pauli was particularly interested in the Lurianic story of Creation, as revealed by the sixteenth-century mystic and kabbalist Isaac Luria, of whom Fludd was aware. Luria reported that his soul often traveled to divine realms to study the secrets of existence and claimed that he could not write his visions down because they gushed so rapidly from his mind. Others recorded them in what became known as the Lurianic Kabbalah. Some of his disciples a.s.serted that his early death, at thirty-eight, was G.o.d's retribution on him for revealing forbidden knowledge.

Luria asked questions such as, Why everything? Why did creation occur? What is the meaning of everything? Fludd's inverted triangles contain his replies. Luria called Tsimtsum-the withdrawal of light and thus of G.o.d to create matter-one of the most important notions in kabbalistic thought. The problem is, if G.o.d is everywhere, how can there be a world? How can there be anything that is not G.o.d? To accomplish this separation G.o.d must have had to abandon a region within Himself to create a "kind of mystical primordial s.p.a.ce from which He withdrew in order to return to it"-or so the kabbalistic scholar Gershom Scholem, a friend of Pauli's, wrote.

Once darkness, or Nothing, could be visualized, then the act of creation-Let there be light!-followed, or so Fludd believed. To express this he drew a dark square. In another image he drew rays of light emanating from a dark core and terminating at a circular periphery with darkness outside it-light, dark, and spirit, the Trinity. From this triad, according to Fludd, the four elements emerged and the struggle among them began. This cosmogony was the blueprint for all natural processes in that they were bases for all subsequent alchemical transformations among the four elements.

Thus the Pythagorean tetraktys emerged out of Fludd's version of how G.o.d created the cosmos. First comes the unity (one) culminating in darkness, followed by the duality of light and dark (two), then by the Trinity (three), culminating in the four elements and the four seasons, and all the other sets of fours that make up the world as we know it, Fludd argued. Pauli wrote appropriately, "His goal is the coniunctio of light and darkness: not the spiritualization of matter.... This is alchemy in the best sense."

Kepler versus Fludd.

Kepler scoffed at Fludd's attempt to seek harmonies "from the interpenetration of his Pyramids which he privately carries around in his mind as a world drawn in pictures." Kepler conversely claimed to have found harmonies in the motions of the planets within a scheme based on mathematics, and that fit astronomical observations and measurements. Without mathematics, he wrote, "I am like a blind man." While Fludd claimed to take his lead from the "Ancients," Kepler followed "Nature herself."

All the same, Kepler's Harmonices was full of astrological, alchemical, Pythagorean, and mystical concepts. Even though Kepler had fulfilled Pythagoras's dream of explaining the universe through geometry and number, he was not satisfied. He was torn between the irresistible pull of his three laws, which postulated that the sun was not at the center of the universe, and the archetypal Trinitarian view of a spherical cosmos with the sun at its center. They did not mirror each other.

He fretted over the division between inert and live matter. Mathematics seemed to apply only to the former; but surely matter had a soul? He could not derive his laws with the mathematics available to him and they did not make much sense without the concept of there being something that tied the planets to the sun.

Fludd published the full text of the Macrocosm two years after Kepler's Harmonices. "Spurred on by the insolence of" Kepler, Fludd gave the usual Pythagorean reasons as to why the key number of the universe was four: its importance for geometry and music, its role in the "mystery of the seven days of creation: the sun was created on the fourth day." Four plus three, he pointed out (the quaternary plus the Trinity) adds up to the magic number seven.

He then referred to the four letters that made up the name of Yahweh--the tetragrammaton. The double "He," he wrote, signified the progression from the Father to the Son.

To this he added the "hieroglyphic monad"-the four symbols representing the sun, the moon, the four elements, and fire. These are depicted as the crescent moon on the round sun, connected by the "quaternary of the cross, four lines being arranged so as to meet in the common point" to the symbol for fire. All of these, according to ancient beliefs and also the beliefs held at the time-such as Hermeticism, alchemy, the Kabbalah, and the Rosicrucians-are responsible for the cycle of transformations that produce our world.

Kepler looked at all this and realized he was wasting his time. He decided to cease communicating with Fludd, "I have moved mountains; it is astonishing how much smoke they expel," he wrote.

All coherence gone.

Among Kepler's last projects was the completion of Somnium, Sive Astronomia Lunaris-Dream or Astronomy of the Moon, a science fiction story about a journey to the moon. In it he imagined what the universe would look like to someone standing on the moon. It was a bold notion that had been important to his discovery of his three laws.

The Somnium in its fragmented form sparked the curiosity of many readers, including the poet John Donne. Donne visited Kepler in 1619 as part of an English delegation dispatched by King James I to Germany. He was interested also in Kepler's book on new stars, De Stella nova. Donne was struck by the implications of the new astronomy: stars no longer immutable, the earth no longer at the center of the universe and, worst of all, the universe most likely of infinite extent, making Heaven far away while h.e.l.l was just beneath our feet. "Tis all in peeces, all cohaerence gone [sic]," he wrote.

At this point Kepler and his family were living in an apartment in the wall surrounding the city of Linz, which was constantly under siege. They often had to admit soldiers to fire their guns through their windows. When the siege was lifted in 1626, Kepler finally left. He died in Regensburg on November 15, 1630. The cemetery in which he was buried was obliterated in the Thirty Years War.

Fludd died seven years later in London. Like a lightning rod his ideas had attracted sharp controversies, most notably with Kepler. His will stated that all those at his funeral should return to the local pub and entertain themselves at his expense.

Three or four?

To Pauli, Kepler and Fludd were a study in opposites. At first he sided with Kepler but over time came to realize that Fludd's worldview included science, music, religion, and the mind. For Fludd four was "the eternal fountainhead of nature." For Kepler the perfect number was three. "I hit upon Kepler as trinitarian and Fludd as quaternarian-and with their polemic, I felt an inner conflict resonate within myself. I have certain features of both," Pauli wrote.

Like Kepler, Pauli brooded about his work, suffering over problems he couldn't solve, far removed from the world of ordinary people. In 1924, when he discovered the exclusion principle, perhaps like Kepler he felt that he had tapped into something that went beyond science. Moving from three to four quantum numbers was a momentous step. It meant a complete break with the iconic imagery of the Bohr atom as a miniscule solar system. It was a step into the unknown, into a world without any visual images. Perhaps Pauli had in mind one of Bohr's favorite sayings from the eighteenth-century German poet Friedrich Schiller: Only fullness leads to clarity.

And truth lies in the abyss.

In his day Kepler stopped short at the number three, basing this decision on the three-dimensionality of s.p.a.ce, on the one hand, and the Holy Trinity on the other. The deep mysteries of alchemy with its emphasis on the number four overwhelmed him.

Pauli, Heisenberg, and the Great Quantum Breakthrough.

EVERYONE AGREED with Pauli that there should be four not three quantum numbers. His exclusion principle had shown that no two electrons in an atom could have the same four quantum numbers. Beyond that his colleagues could see that the principle must have huge implications. But no one could yet see what they were.

By the beginning of 1925 it was clear that Bohr's theory of the atom as a miniature solar system no longer provided even an adequate basis for understanding the atom, let alone for the exclusion principle or the anomalous Zeeman effect. Bohr's theory by now was under attack from all sides.

The demise of Bohr's theory of the atom.

Pauli, despite his best intentions, had been one of the key wielders of the knife. As he had discovered, the theory had failed to produce a realistic model of either a hydrogen-molecule ion or a helium atom. Then new data appeared showing that the hydrogen atom did not respond to being hit by light as if it were a tiny solar system. This model produced spectral lines for the struck light that did not agree with those found in the laboratory.

Bohr fought back with a variation of his theory in which the invisible orbits of the invisible electrons were replaced by invisible electrons on springs, each emitting light at the frequency of an observed spectral line. To emphasize that these invisible electrons were an intermediate kind of reality, he referred to them as "virtual oscillators." Pauli wanted nothing to do with them. He had had enough of bizarre models and was totally discouraged.

Heisenberg, who was twenty-four, thought otherwise. Throughout spring 1925 he pushed Bohr's theory of virtual oscillators to its limits. But it failed. It seemed that Bohr's theory barely worked even for the hydrogen atom and even then no one really understood why. Atomic physics lay in ruins.

Many physicists spoke of their despair. Pauli did not respond well to crises and was becoming more and more depressed. He joked bitterly that physics was all wrong and wrote to Kronig, "I wish I were a film comedian or something similar and had never heard of physics." He hoped, he added, that "Bohr will rescue us with a new idea."

Around this time, Pauli wrote to Bohr about Heisenberg, "I always feel strange with him. When I think about his ideas, they seem dreadful to me and inwardly I swear about them. For he is very unphilosophical, he pays no attention to expressing clearly the fundamental a.s.sumptions and their connection with existing theories. But when I talk to him he pleases me very much and I see that he has all sorts of new arguments.... I believe that some time in the future he will greatly advance science." Pauli was to be proved right.

Unlike Pauli, Heisenberg thrived in periods of chaos. Far from despairing, he would go all out to find a solution. He welcomed the stretch of the imagination required by Bohr's virtual oscillators. He used his immense experience in every aspect of atomic physics, together with his natural audaciousness, spurred on by Pauli's critical comments-among them that he should deal only with quant.i.ties that can be measured in the laboratory, such as the energy and momentum of electrons, and avoid abstract concepts such as...o...b..ts of electrons. "We must adjust our concepts to experience," was the approach Pauli suggested. Heisenberg worked day and night and came up with a whole new atomic physics that was to become known as quantum mechanics. Full of excitement, Pauli wrote that Heisenberg's work gave him "new hope and a renewed enjoyment in life."

"We must adjust our concepts to experience"

Like every highly creative scientist of his era, Pauli was a philosophical opportunist. He picked and chose from whatever philosophy had to offer to tackle the problem at hand. Scientists use philosophy when they ask the deepest of questions, such as What const.i.tutes a scientific theory? What sort of physical objects should it consider and how should it treat them? What is physical reality?

At the beginning of the twentieth century these questions became crucial when scientists had to contend with objects-such as electrons and atoms-that they could not actually see. Cla.s.sical ways of understanding the world suddenly seemed insufficient. An intellectual tidal wave-the avant-garde-swept across Europe.

Scientific concepts, ways of thinking, and ways of knowing were all being re-examined. Einstein did so when he discovered his special theory of relativity in 1905. This upheaval in thinking pervaded the world outside science too. In 1907 Pablo Pica.s.so launched cubism with his "Les Demoiselles d'Avignon" and in 1910 Wa.s.sily Kandinsky unveiled abstract expressionism. In 1913 Igor Stravinsky ruptured all the conventions of cla.s.sical ballet with his "Rite of Spring." The postwar 1920s produced the twelve-tone music of Arnold Schonberg, Bauhaus architecture, and James Joyce's extraordinary novels, which encompa.s.sed everything from relativity to cubism. Meanwhile Freud and Jung were investigating the unconscious.

Pauli first encountered this ferment of ideas through his G.o.dfather, the positivist Ernst Mach. As a boy he was spellbound by the scientific equipment in Mach's apartment. Its ultimate purpose, said Mach, was to eliminate unreliable thinking-to demonstrate that the only thing that was really out there was what you can experience with your senses. The rest was all metaphysics-quite literally beyond physics and not worth considering, mere illusion.

Atoms could not be experienced with the senses. Did that mean they were merely "metaphysical," in Mach's pejorative sense? Were they not part of the elaborate scientific theories which made predictions that could be proved in the laboratory? According to Einstein's theory of relativity-Pauli's first scientific love-time turned out to depend on the motion of a clock and our world was four-dimensional, not three as everyone had always thought. The message of relativity theory seemed to be that scientists should look beyond what was immediately perceptible by the senses. It was to Einstein's disappointment that he failed ever to convince Mach to accept relativity theory.

In the light of relativity theory Mach's view seemed too restrictive. A group of young philosophers with strong scientific backgrounds began to meet in the coffeehouses of Vienna to discuss how to correct this situation, how to bring positivism into line with relativity theory. They called themselves the Vienna Circle and came up with a sophisticated version of positivism that they dubbed "logical positivism." Then they renamed it "logical empiricism": the word "empiricism" refers to experimental data (empirical data). Logical empiricism emphasized the role of mathematics in that a theory required a consistent logical or mathematical structure. Mach, on the other hand, regarded mathematics as merely an economical way to summarize experimental data.

In the view of the Vienna Circle a scientific theory had to be built on empirical data with the help of mathematics and had to generate predictions that could be tested in the laboratory. Science was a two-way street, beginning with data and ending with predictions that could be verified by data in the laboratory. Logical empiricism also insisted that every concept in a scientific theory must be measurable. Distance could be measured with a ruler, time by clocks, and so on. Thus they claimed that Einstein's discovery of relativity theory was actually in accordance with positivism.

As for atoms, this was just a name for a list of experimental results. The rays emerging from cathode-ray tubes-primitive television tubes-were a.s.sumed to be a sort of light ray with an electric charge. Actually, every scientist knew that cathode rays were made up of electrons. Both Mach and the logical empiricists declared that atoms were not real as they could not be seen or measured individually. But the logical empiricists were able to see a way around Mach's rejection of Einstein's theory of the relativity of time in that it emerged from a consistent mathematics and experiments had been done to ill.u.s.trate it in the laboratory. Mach's philosophical heirs made the important point that the criterion "to observe something in the laboratory" had to be replaced by "to ascertain it or measure it in the laboratory."

Pauli was well read in philosophy and introduced himself to the then-doyen of the Vienna Circle, the German-born Moritz Schlick. Schlick was twice his age and an esteemed professor at the University of Vienna, where he had taken over Mach's position. Schlick was impressed with Pauli's philosophical ac.u.men. Pauli did not let the fact that he was a mere postdoctoral student hinder him from giving Schlick his blunt a.s.sessment of positivism. He had no objection to it, he wrote in 1922, "But, of course, it is not the only [philosophical approach]."