Riemann for Anti-Dummies: Part 6

Happiness as a physical principle

In the Dedication to the “Fourth Book of the Heroic Deeds and Sayings of the Noble Pantegruel”, Francois Rabelais refers to a discovery of Greek father of medicine, Hippocrates.

“The question over which we sweat, dispute, and rack our brains, is not whether the physician’s visage depresses the patient, if he is frowning, sour, morose, severe, ill-humored, discontented, cross, and glum; nor whether he cheers the patient if his expression is joyful, serene, gracious, frank and pleasant. There is no doubt on that score. The real question is whether the patient’s depression or cheerfulness arises from his apprehensions on reading these signs in his physician’s face, and from his consequent deductions of the probable course and issue of his disease; or whether it is caused by the transmission of the serene or gloomy, aerial or terrestrial, joyous or melancholy, spirits from the doctor to the person of the sick man as is the opinion of Plato and Averroes.”

So too is the relationship between investigator and the physical universe. Approach the universe with the joyful, serene, gracious, frank and pleasant expression of a Kepler, Fermat, Leibniz, Kaestner, Gauss, Riemann, or LaRouche, and its secrets are displayed in its reflected smile. Put on the frowning, sour, morose, severe, ill-humored, discontented, cross, and glum, visage of Gallileo, Newton, Descartes, Euler, and Russell, and all you will see is a grid of scowling straight lines.

In point of fact, the history of science, from Kepler on, can be divided into two camps, the smilers and the scowlers. Kepler had demonstrated that Cusa’s principle, of non-uniform physical action, was, in truth, the characteristic of action in the solar system. A flood of new discoveries followed, demonstrating the validity of this principle with respect to other physical phenomena: Fermat’s discovery of the principle of the least-time path for light; Huygen’s discovery of the isochrone, Bernoulli’s discovery of the brachistochrone; Leibniz’ discovery of the catenary; to name but a few.

In each case, as the “Kepler Challenge” showed, the physical principle under investigation, was not susceptible to precise calculation, but, had to be grasped from the standpoint of the higher principles underlying the generation of the phenomena. The example of the “Kepler Challenge” illustrates this problem most clearly. If the entire orbit is known, the time elapsed between any two planetary positions can be determined to an arbitrary degree of precision, by Kepler’s equal area principle. But, if the time elapsed was known, Kepler had no direct means to determine the planetary positions. His “imperfect” solution was to divide the planetary orbit into 360 intervals, calculate the time elapsed for each interval, compile a table of these results, and refer all calculations to the table. The imperfection arose because these small intervals are themselves as non-uniform as larger ones. Only if the planetary motion can be known at each “moment of becoming”, could such a method provide any degree of precision. Yet, there are an infinite number of such moments in any orbit, implying the need for infinite knowledge, and so Kepler made his famous call for future geometers to solve for him this paradox.

This paradox is not merely formal. The individual positions of the planet, are themselves a function of the whole orbit, which, itself, is a function of the whole solar system. Yet, like the shadows in Plato’s cave, the orbit and the solar system do not present themselves to our minds directly, but only through the changing observable positions of the planets. Kepler discovered the nature of the function by which the solar system determined the orbit, and the orbit determined the planetary positions, but he nevertheless required the inversion, to measure the effect of both these functions, at each moment of becoming, i.e., to discern the how the shadows were made, from only the shadows.

For Kepler, this state of learned ignorance was a happy one, albeit not without a certain amount of angst: “And we, good reader can fairly indulge in so splendid a triumph for a little while(for the following five chapters, that is), repressing the rumors of renewed rebellion, lest its splendor die before we shall go through it in the proper time and order. You are merry indeed now, but I was straining and gnashing my teeth,” Kepler tells us in the “New Astronomy”

But, it provoked nothing but rage from Aristoteleans, who desperately sought to limit the world to that which is susceptible to precise calculation. “Wipe that smile off your face,” the Newtonians would say, “We don’t need these complicated, imprecise, convoluted hypotheses, of Kepler. We can get a more precise measure of planetary motion, using the inverse square law.” (“Don’t pay attention to the fact that we have to eliminate everything in the universe except two bodies, for our calculations to work,” they scowl. “We get results.”)

In response to Kepler’s challenge, Pierre de Fermat, set about to overcome those limitations brought to the `fore when one tries to measure action along a path of non-constant curvature, such as an elliptical orbit, or the least-time path of light traveling through different media. He took as his starting point, the earlier Greek investigations into these types of problems. Appolonius, Archytus, Aristaeus, Eratosthenese, among other Platonic thinkers, called these types of investigations theorems of position. (The Greek word used was “topos”, meaning “place”. Today these problems are identified by the Latin word, “locus”.)

Under the Greek concept of locus, geometrical curves are not defined as self-evident entities, but as the concept characterizing those positions resulting from a certain type of action. Proclus re-states this ancient Greek conception as, “those (results) in which the same property obtains over some entire locus, and as `locus’ (topos) the placement (thesis) of a line or a surface which makes one and the same property.”

A couple of examples would be:

1) the locus of all positions equidistant from a given position is a circle.

2) the locus of all positions whose distance from two given positions have a constant sum is an ellipse.

3) the locus of all positions that form an equal angle between two fixed points is a circle.

4) the locus of positions of intersection between a plane and a cone are a circle, ellipse, parabola, or hyperbola, depending on the angle at which the plane intersects the cone.


5) the locus of all positions of a planet, moving about the Sun non-uniformly, such that it sweeps out equal areas in equal times, is an elliptical orbit.

(It should not be overlooked, that examples 1-4 refer to purely geometrical loci, while example 5 refers to a locus in physical space-time.)

Significantly, the Greeks separated loci into several types according to their method of generation, such as planar loci, solid loci, or loci according to means. The limitation of the Greek investigations, according to Fermat, was that such concepts were investigated one by one, whereas Fermat sought a generalized approach.

This is where Fermat began:

“None can doubt that the ancients wrote on loci. We know this from Pappus, who, at the beginning of Book VII, affirms that Apollonius had written on plane loci and Aristaeus on solid loci. But, if we do not deceive ourselves, the treatment of loci was not an easy matter for them. We can conclude this from the fact that, despite the great number of loci, they hardly formulated a single generalization, as will be seen later on. We therefore submit this theory to an apt and particular analysis which opens the general field for the study of loci.”

In essence, Fermat’s proposed generalization was to consider all loci in the following way. Consider two straight line segments intersecting at right angles. While holding one of the segments fixed, allow the other to move along it, both vertically and horizontally, while keeping the intersecting angle right. The locus of positions of the free end of the moving line, will then trace a curve. This curve will be a function of the motion of the moving line.

To visualize this, take two sticks (bamboo skewers work well) and hold them at right angles to each other. Experiment by keeping the vertical stick fixed and slide the horizontal stick over it. The free end of the horizontal stick will trace a line parallel to the vertical stick. Try this again, this time moving the horizontal stick vertically, while at the same time moving it horizontally. If the rate of vertical motion equals the rate of horizontal motion, the free end of the moving stick will trace a diagonal line. Now try the same thing, but this time make the horizontal motion, the square of the vertical. The free end will trace a parabola. If the vertical motion is arithmetic, while the horizontal motion is geometric, the curve will be exponential. (Additional exercises and examples will be forthcoming in the next installment.)

Fermat expressed the relationship of the vertical to horizontal motion by an equation. Rather than deal with each individual locus, he could now consider types of loci, which were described a specific types of equations. Bernoulli would later introduce another, simpler, type of equation, in which the point of intersection remained fixed, while the moving stick rotated about that point, while still able to slide. He could describe these loci by a relationship between an angle and a distance. Gauss, would later generalize this concept further, by replacing the straight lines with any curve whatsoever, and he and Riemann would take this one step further, determining multi-dimensional loci. (These matters will be developed in future installments.)

“Grrrrrr”, snarled the Cartesians, in a fit of inquisitional constipation. (Their Aristotlean minds always got headaches from such thoughts.) “All this relative motion is confusing. Let’s make this simple stupid.” Instead of loci, Descartes and his progeny would create the, unfortunately all too familiar, grid of infinitely many intersecting right lines. Descartes’ equations described, not a locus of motion, but the fixed positions in this grid, where a given geometrical figure intersected the straight-lines.

Nothing exemplifies a frowning visage, and has been responsible for implanting a glum expression on the faces of so many students, than the abominable cage, known today as the “Cartesian co-ordinate system”. It is high time, that this fraud be debunked, once and for all.

Descartes equations were nothing more than a set of instructions about how to connect the dots in the fixed grids of absolute space. For Fermat, like for Kepler, there was no fixed grid in absolute space. These curves were generated by a type of motion, which types would then be susceptible to further investigation.

Fermat’s theory of loci was not sufficient to overcome the Kepler paradox, but it was a necessary intervening step for Leibniz’ happy development of the next phase in the science of the moment of becoming.