Infinite Wisdom

For several decades in the early 20th century, it looked like
mysticism, mathematics, physics, philosophy, and religion would become
cozy roommates. Physicists, excited by the new quantum arena, were
looking for God in the particles. Mathematicians were pondering the
infinite. The best minds from Einstein down were embracing the idea of
God. It didn’t last long, and science and math have returned to being
secular endeavors, but a spate of new books on this time when
scientists were pondering the infinite suggests that perhaps the
seemingly separate worlds are coming back together.

The story begins with a simple new way to organize mathematics. Set theory, developed by Georg Cantor in 1870s Germany, seems innocuous enough to the layperson. A set collects similar objects into one group, and then it is given a name. All of the books authored by Ray Bradbury would be a set, for example, and we could name it “Ray Bradbury’s books.” This set has a finite number of elements, and it is immediately obvious that this set is unique from a set that would collect all the varieties of tea grown in India.

Yet in the early 20th century set theory was troubling some of the leading French mathematicians, including Henri Lebesgue, Henri Poincaré, and Émile Borel. Birthed in the 1870s in Germany, set theory tries to provide an underlying structure to all of mathematics, but it almost accidentally awakened long sleeping questions about infinity and the nature of the universe. By 1910, several of the French mathematicians studying set theory were blaming it for nervous problems and for the eventual suicide of one of their colleagues. Around the same time, physicists were moving from the realm of classical Newtonian laws of nature to the more curious laws of relativity. It was a time of massive upheaval, and many of the questions asked then are still waiting for answers.

The problems arose when the sets were used to collect groups with infinite numbers, as explained in Loren Graham and Jean-Michel Kantor’s Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity. Cantor’s sets could provide a framework for infinity by creating a set of all the integers. No matter what number a person can think of, one more can always be added, which creates a potential infinity. They could also create one infinity that is larger than another. A line fragment contains an infinite number of points, and so a set of those points would create an infinity. But a set of all of the points on the entire line, which is larger than the simple fragment, would arguably create a larger infinity. As Graham and Kantor explain it, Cantor’s set theory created “a whole hierarchy of infinities, an infinity of different infinities.” The French, who mostly followed the Aristotelian approach to mathematics, did not like to think of mathematics as an abstract philosophy; they preferred that their mathematics be rooted in the material world, like the movement of planets. As a result, they could only take set theory so far, and the strain caused mental breakdowns in many who worked in the field.

Mathematics has a long history entangled with mysticism and religion. Pythagoras, Pascal, and Anaximander of Miletus and others saw mathematics as a way of understanding the divine. Hermann Weyl, who worked in the early 20th century, wrote, “[P]urely mathematical inquiry in itself, according to the conviction of many great thinkers, by its special character, its certainty and stringency, lifts the human mind into closer proximity with the divine than is attainable through any other medium. Mathematics is the science of the infinite, its goal the symbolic comprehensive of the infinite with human, that is finite, means.”

The leading mathematicians of Russia leaned more in this mystical direction than the French. Several of the mathematicians who worked with set theory were involved with the Name Worshippers, a heretical sect of monks who would go into trances by meditating on the names of Jesus and God. They believed that by naming God, they were worshipping and in some way creating God. Set theory, which created mathematical concepts by naming them, fit right into their worldview. Many of these mathematicians hid their association with the Name Worshippers, as this was the time when the Soviets were rounding up priests and executing them, but Pavel Florensky — whose work in mathematics, art, and theology has led some to refer to him as the Russian da Vinci — boldly presented papers at conferences with his long beard and wearing clerical gowns. Graham and Kantor write:

Like many members of the Russian intelligentsia of this time, Florensky believed that all intellectual life is a connected entity, and that ideas in mathematics and philosophy could be extended to the social and moral realms… Florensky was convinced that intellectually the 19th century had been a disaster, and he wanted to identify and discredit what he saw as the governing principle of its calamitous effects. He saw that principle in the concept of ‘continuity,’ the belief that one could not make the transition from one point to another without passing through all the intermediate points.

At the time, evolution was theorized to be a gradual, slow, methodical process, with certain traits being developed step by logical step. It wasn’t until the 20th century that biologists began to prove that mutations were erratic, sudden, and surprising. There would be nothing for generations, and then a mutation would burst through. This was the same with mathematics, psychology, religion, and other fields. Florensky believed the progress of the world was discontinuous and patterned with intense fragments. As a result, Florensky and the other mathematicians from the Moscow school were able to advance set theory far beyond the French, and with fewer emotional side effects.

Similar rumblings were heard in the physics world as well. The 19th century was ruled by positivism, the belief that the world, the mind, and basically everything else could be reduced to its material nature and explained through Newtonian physics and chemistry. It wasn’t until Einstein and his theory of relativity that this started to change. Wolfgang Pauli was one of the leading physicists working in this realm, although his name is unfamiliar to most these days. Pauli — the subject of Arthur I. Miller’s Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung — was very influential during his lifetime. But his habit of not publishing his theories and giving his ideas to others to prove — for example, he sent a letter to Heisenberg outlining an idea that one could never be certain both of the momentum and the location of a particle, which Heisenberg then proved and published as the Heisenberg Uncertainly Principle — means that today only the hardcore nerds know who he is.

As Pauli was working on problems like Bohr’s structure of the atom and the inexplicable response of atoms in a weak magnetic field, called the anomalous Zeeman effect, the entire world of physics was being turned upside down by the theory of relativity. Pauli’s own life was following suit, and he found himself with a Jekyll and Hyde personality. In the lab he was studious and respected. In his personal life, he was drawn to Zurich’s red light district and becoming increasingly erratic. He entered into analysis with Carl Jung, and through studies of alchemy, dream interpretation, and intense therapy, Pauli was able to intuit his way through several scientific breakthroughs — including the existence of neutrinos, which he hypothesized 20 years before they were discovered in the laboratory, and the requirement of a fourth quantum number for electrons.

Just as the Russians looked to the past, to mathematics’ origins, as a process to understand infinity and the eternal nature of the divine and were therefore able to advance set theory, Pauli looked to science’s roots in alchemy to further his understanding of relativity. He studied Johannes Kepler and Robert Fludd, who argued over whether the number three or four was the “key number necessary to explain the workings of the universe.” Alchemists believed that everything material was made of four elements — air, fire, earth, and water — and that bringing the four together could produce the quintessence, the fifth element, also known as the “philosopher’s stone.” Pauli saw modern science as having cut off the fourth element — the feminine, intuitive element — and its restoration was required to move forward. It was during this time that he determined that electrons had to have a fourth quantum number, and it solved a problem that had been plaguing physicists for years.

Back then Carl Jung was a superstar, “dangerously famous,” as he described it. Now his ideas of archetypes and synchronicity, so groundbreaking at the time, have fallen out of favor even while “the collective unconscious” remains in our cultural vocabulary. The Moscow school of mathematics no longer has its religious tinge, after years of the instructors being round up by the Soviets. But maybe Jung was right about certain ideas creeping from the unconscious to the consciousness of the collective at the same time, and this sudden interest by scholars and readers in this period when God was in the math and the laws of physics is its own form of synchronicity. • 1 May 2009