Kepler: “Where There Is Matter, There Is Geometry”

In the autumn of 1601 Kepler was thirty, newly arrived in Prague, and newly promoted. Tycho Brahe had died in October, and the Emperor Rudolf II made Kepler Imperial Mathematician in his place. One of the job's duties was the yearly almanac, and Kepler's for 1602 came out as a short Latin tract of seventy-five theses, addressed “to philosophers”, on what astrology could honestly claim. He kept the parts he thought had physical causes, the planets' light and its angles, and threw out most of the rest.

The line is in Thesis XX, and it is an argument with Aristotle. In the Metaphysics, Aristotle had said that the first contrariety in things is between same and other. Kepler answers that every difference in created things comes from matter, and that where there is matter there is geometry. So the first real contrariety is a geometer's, equal against unequal, more against less, and the varied strengths of the planets are a case of it. For Kepler, geometry was not a tool for describing the world. It was the pattern the world had been made to.

He had said as much five years earlier, in the Mysterium Cosmographicum of 1596, where the gaps between the six planets are set by the five Platonic solids nested one inside the next. He would say it again in 1611, in a New Year's gift to a friend asking why every snowflake has six corners, a pamphlet that happens to contain the first statement of the sphere-packing problem, and in 1619 in the Harmonices Mundi, where he calls geometry coeternal with the mind of God. In between, in 1609, he had found that Mars moves on an ellipse. The man who was sure the world was made of geometry was the one willing to give up the circle.

The sentence is the opposite pole from Atiyah's warning four centuries later that algebra is the devil's offer; for Kepler there was nothing but geometry to begin with. The snowflake pamphlet is on our shelf of the best geometry books, and Galileo, his contemporary and correspondent, drew the same conclusion the other way round: start with mathematics.

Johannes Kepler was born in Weil der Stadt, in the Duchy of Württemberg, a premature child of a mercenary father and an innkeeper's daughter later tried for witchcraft. He studied theology at Tübingen, where Michael Maestlin taught him the Copernican system, and in 1594 was sent to Graz to teach mathematics. In 1600 he joined Tycho Brahe in Prague, and on Tycho's death in 1601 became Imperial Mathematician to Rudolf II.

From Tycho's observations of Mars he derived his first two laws of planetary motion, published in the Astronomia Nova in 1609: the orbits are ellipses with the Sun at one focus, and a planet sweeps out equal areas in equal times. The third law, relating a planet's period to its distance, appeared in the Harmonices Mundi in 1619. He also explained how the eye forms an image, designed the Keplerian telescope, and completed the Rudolphine Tables in 1627. He died in Regensburg in November 1630, at 58, on a journey to collect money he was owed.