Riemann for Anti-Dummies: Part 5

No person is qualified for political leadership today, or for that matter, even competent to vote, unless they know the significance of what has been called, “The Kepler Problem”. So important is this paradox, and so widespread is its fame, that every significant discovery in physical science since, can trace its origins to it, and every major scientific thinker upon whose shoulders we now stand, can find their first sparks of creative insight in its contemplation. And yet, today, except as a name for an obscure mathematical formula, most know nothing of it, which is a fitting measure of this stage in civilization’s descent.

The problem itself embodies the essential characteristics of any crucial discovery, a paradox of physics and mind. It arises from Kepler’s discovery that non-uniform motion was not merely an appearance, but a physical characteristic of action in the universe, as demonstrated by the eccentric orbits of the planets around the Sun. This discovery immediately presents an ontological problem, that was stated earlier by Cusa, in Book 2 of “On Learned Ignorance”:

“Wherefore it follows that, except for God, all positable things differ. Therefore, one motion cannot be equal to another; nor can one motion be the measure of another, since, necessarily, the measure and the thing measured differ. Although these points will be of use to you regarding an infinite number of things, nevertheless, if you transfer them to astronomy, you will recognize that the art of calculating lacks precision, since it presupposes that the motion of all the other planets can be measured by reference to the motion of the sun. Even the ordering of the heavens with respect to whatever kind of place or with respect to the risings and settings of the constellations or to the elevation of a pole and to things having to do with these is not precisely knowable. And since no two places agree precisely in time and setting, it is evident that judgments about the stars are, in their specificity, far from precise.”

This paradox manifests itself in an eccentric planetary orbit, because no interval of that orbit can measure any other interval of the same orbit. For in every interval of the orbit, no matter how small, the distance from the Sun to the planet, the speed of the planet, the distance travelled along the orbit, and the curvature of the arc, are always changing. (See Figure 5.3 p.27, Summer 1998 Fidelio, for an illustration. The interval between all segments, denoted as “Q” in the figure, exemplifies this characteristic of non-constant curvature.)

This paradox was enough to entice most people to stick with the Aristotelean custom and tradition that banished non-uniform motion from the universe. Why would God, the Aristotelean would argue, create action in the universe that evaded precise calculation?

But, Kepler, on the other hand, understood that this is precisely what God preferred, as the principle of change must be manifest in every part of the universe. It was only when man realized the motion of the planets were non-uniform, that man would inquire into its causes. Kepler overcame the apparent obstacle, not by measuring the motion directly, but by measuring the effect of the motion, as expressed in his principle of equal areas.

However, he could only overcome this obstacle in one direction, so to speak. He could calculate the effect of the planet’s motion in an orbit, if he knew the orbit as a whole, that is, if he knew the position of the Sun relative to the circumference, the eccentricity, the greatest and shortest distances between the Sun and the planet, etc. This was done, as the {Fidelio} article summarizes, by measuring the area swept out during any interval of the planet’s orbit. Intervals that swept out equal areas, corresponded to equal portions of the planet’s total periodic time. [See figure.] But, if he tried the inverse, that is, if he knew one position of the planet, and he needed to calculate the planet’s next position, after a specified area was swept out, he was blocked by what Cusa showed to be the transcendental incommensurability between the arc and the straight line. (See Fidelio p. 28.) In other words, he could determine the time elapsed from the orbit, but he couldn’t determine the orbit from the time elapsed. This is what’s called the “Kepler Problem”.

But, is this merely a problem of calculation, or is it an indication of something more fundamental?

Look again at the paradoxical nature of a non-uniform orbit. On the one hand, the relationship of the Sun to the planet individually, and the relationship of the Sun to all the planets, define a characteristic of the planet’s complete orbit. The planet’s distance from the Sun, it’s speed, and direction, are thus defined for each “moment” of action. It is this global characteristic that Kepler’s equal area principle measures.

The “Kepler Problem” arises, when we try to measure these global characteristics from the standpoint of each of “moment” of action.

“Why even bother?” the Dummy might ask.

“Because, it is the only way human cognition can know anything,” is the short answer. The completed orbit is never seen, nor is it ever known, as a completed orbit. Rather, it is known by its influence on the planet at every “moment” of action. In each of these “moments”, or rather, “moments of becoming”, the planet is ceasing to be what it was, and becoming what it will be next. What it was and what it will be, is the completed orbit. So, at each such “moment of becoming”, the planet is simultaneously in one moment and all moments of the completed orbit.

An analogous paradox can be grasped subjectively, if one thinks back, from present to past, on one’s life, in the context of history. In this direction, it is possible to determine the influence of one’s life’s orbit on each past moment. But, in the other direction, how can one determine the historic curvature of life’s orbit, in this present “moment of becoming”?

The “Kepler Problem” can thus be re-stated as, “How can we measure the influence of the completed orbit at each “moment of becoming”?

Kepler relied on what he admitted was an “imperfect” compromise. He divided the entire area of the orbit into 360 segments, and calculated the areas of each segment, using the equal area principle. The imperfection of this method is obvious. Even these small segments are comprised of an infinite number of “moments of becoming”, so by their very nature they are imprecise.

He knew all too well the imperfection, saying in his “New Astronomy”, “… [I]nsofar as it lacks geometrical beauty, I exhort the geometers to solve me this problem…. It is enough for me to believe that I could not solve this, a priori, owing to the heterogeneity of the arc and sine. Anyone who shows me my error and points the way will be for me the great Apolonius.”

Perhaps “Kepler’s Problem” is better named, “Kepler’s Challenge”. He freed man from the unchanging custom and tradition of perfect circles and straight lines, and left him in a world with principles, but without precise calculation. But, this, “gives rise to a powerful sense of wonder, which at length drives men to look into causes.” Such wonder found form in the minds of Fermat, Leibniz, Kaestner, Gauss, Riemann, and Cantor, where calculation was supplanted by the “science of the moments of becoming.”