137 - Arthur I. Miller [92]
Pauli laid all this out as a mandala in the shape of a cross. From this he deduced that the laws of physics are a projection onto the psyche (the conscious/unconscious) of an archetypal association of ideas arising from the collective unconscious: in other words, a clash of the four opposing concepts that he depicts at opposite ends of the two crosses.
Pauli’s preliminary mandala showing the collective unconscious and events in space and time.
The mandala lays out the fundamental complementarity at the heart of both psychology and quantum physics. Bohr had spoken of “the general difficulty in the formation of human ideas, inherent in the distinction between subject and object.” In quantum physics, the person making the measurement and the measuring apparatus affect whatever is being measured. Similarly in psychology, the psychologist can never really know the unconscious through psychoanalysis. He must always interpret the results of his questions and inevitably he himself will affect his conclusions. Data can never be understood except through the lens of a theory.
In 1948, around the time of the spring equinox, Pauli had two dreams. For him the equinoxes, he said, were times of “relative psychic instability, which can manifest itself both negatively and positively (creatively).” The dreams that arose at those times were always of particular significance.
His dreams were full of mathematical symbols. In one of them i appears, i being the square root of –1: . i is an “imaginary number” because it is not one of the numbers we use in daily life—the so-called real numbers. Nevertheless, i often functions to unify complicated formulas by making them more compact.
In one of his dreams a woman brings Pauli a bird. It lays an egg that then divides into two eggs. Then he notices that he has another egg in his hand, which makes three. Suddenly the egg in his hand divides into two. He now has four; a quaternity has appeared. Before his eyes the four eggs morph into four mathematical symbols, in two groups, side by side:
“cos” (cosine) and “sin” (sine) are quantities from trigonometry (a form of mathematics that deals with triangles) while ““(delta) is the angle formed by two sides of a triangle. These four symbols coalesce into a single expression, unified by the symbol i:
This expression is well known to mathematicians.
In his dream he turns this expression into an equation:
where e, the “base of natural logarithms,” has a numerical value of 2.71828…(referred to as an “irrational” number in that the group of numbers “1828” never repeats); and has a magnitude of 1. The insertion of i into these sets of four has created a unity.
Reflecting on the eggs in his dream, Pauli realized that it was precisely what Maria Prophetissa, the early practitioner of alchemy, had described some seventeen centuries earlier: “One becomes two, two becomes three, and out of the third comes the One as the fourth.” This transformation, he noticed, “typically comes about for me through mathematics.”
Pauli’s interpretation of this whole dream is far removed from mathematics. Describing it to Jung, he explains that is a number that always lies on a circle of radius 1. Through the power of the mathematical symbol i, a mandala has appeared in the form of a circle. In Pauli’s dream i “has the irrational function of uniting pairs of opposites”—the cosine and sine functions arranged in two groups of opposites—“and thus producing wholeness.” But e too is “irrational,” it is an irrational number. This shows,