Riemann for Anti-Dummies: Part 11 : Transcending Euclid

Transcending Euclid

It is crucial for anti-dummies to always bear in mind the groundwork for all modern science, that Nicholas of Cusa teaches us in “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 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. If you subsequently adapt this rule to mathematics, you will see that equality is {actually} impossible with regard to geometrical figures and that no thing can precisely agree with another either in shape or in size. And although there are true rules for describing the equal of a given figure as it exists in its definition, nonetheless equality between different things is {actually} impossible. Wherefore, ascend to the recognition that truth, freed from material conditions, sees, as in a definition, the equality which we cannot at all experience in things, since in things equality is present only defectively.”

Such was the approach of Kepler, Fermat, and Leibniz, who rejected the method of imposing mathematical definitions on the physical universe, but, in doing so, discovered a new higher principle, that led to the creation of a new mathematics. Be it the motion of the planets, the path of refracted light, or the shape of a hanging chain, physical action does not conform to perfect mathematical rules. For example, the planets don’t move in perfect ellipses, nor does a ray of refracted light move in a perfect cycloid, nor does a hanging chain form a perfect catenary. It is in this deviation from the perfect, that the higher, harmonic, principles governing the action can be discovered. Furthermore, these deviations are not expressible in perfect mathematical ratios, such as, for example, the dependence of the planet’s or light’s path on the sine of the angle. These higher principles define the characteristics of what Riemann called “multiply-connected functions”. In other words, it is not the geometry that defines the physical action, it is the physical action that defines the geometry. That geometry, so defined, does not conform to perfect mathematical rules, but is knowable, yet, one must first grasp what is meant by knowable. Cusa’s cited imperfection of the physical universe, thus provoked the discovery of the universe’s perfectability.

The irony is, that if the universe conformed to perfect mathematical rules, it would not be discoverable. As Kepler describes this process in the “Harmonies of the World”:

“… [U]nless the Earth, our domicile, measured out the annual circle, midway between the other spheres changing from place to place, from station never would human cognition have worked its way to the true intervals of the planets, and to the other things dependent from them, and never would it have constituted astronomy.”

Cusa’s method had opened the door, not only to Kepler’s revolutionary discoveries, but, to the revolutionary discoveries of Fermat, who abandoned the path of shortest distance for the path of least time; J.S. Bach, who rejected abstract mathematical notions of musical intervals, creating the well-tempered system through his compositions; and Leibniz, who created the mathematics of non-constant curvature, that he called the infinitesimal calculus, to name but a few.

It was through this doorway, that the 18-year old Gauss walked on March 30, 1796, when he made his first entry into his scientific notebook — his discovery of the constructability of the 17-sided polygon. For more than 2000 years, it was believed that it were impossible to construct a 17-sided regular polygon with straight-edge and compass. On this day, Gauss recalled, after much hard work, the possibility of such a construction appeared to him all at once.

Gauss always considered this one of his most important discoveries, which, accompanied by a provocation from Kaestner, determined the trajectory of his entire creative life. So much so, that he asked that a 17-gon be engraved on his tombstone. (Because an engraved 17-gon would look so much like a circle, a 17-sided star was engraved instead.)

In the next installment, we will work through Gauss’ construction in detail. However, some historical background will be useful to set the stage. From 1792-1795, Gauss received his early education at the classically-oriented Collegium Carolineum in Brunswick-Wolfenbuettel. The curriculum consisted of ancient and modern languages, classical sciences, aesthetics, poetry, music, and art. On Oct. 11, 1795, he left for Goettingen, where he was attracted to the study of philology, under Christian Gottlob Heyne. The philology seminar at Goettingen had been founded by Johann Matthais Gesner, a former colleague of J.S. Bach at the St. Thomas School in Leipzig. Heyne was Gessner’s successor. Also teaching at Goettingen was Leibniz’ defender and Benjamin Franklin’s host, Abraham Gotthelf Kaestner, who had come from Leipzig in 1750, and under whom Gauss studied mathematics.

In a later letter to his Collegium professor, E.A.W. Zimmerman, Gauss said he was particularly attracted to the philology lectures of Heyne and wished to concentrate on the subject. Initially, he thought Kaestner was a dull old man, but, “I have since realized I was in error and that he is quite an extraordinary man.”

In May 1796, Gauss told Zimmerman in a letter, that he had decided to devote himself to mathematics after discovering the constructability of the 17-gon. When he first showed his discovery to Kaestner, the teacher was not at all impressed, and Gauss interpreted this as hostility to something new. He persevered, and when he discussed the concept further, Kaestner was astounded, but responded that the discovery would be of no use, and he (Kaestner) had already developed the basis for the discovery in his, “Beginning foundations of the analysis of finite magnitudes.” Gauss persisted further, obtaining an agreement from Kaestner, to have him review Gauss’ paper. After further contemplation, Gauss realized that he had to separate Kaestner’s criticism concerning the discovery’s practical use, from the rest of Kaestner’s reaction, “but, if I (Gauss) were to be able to give a more general treatment of the subject, it would be of pleasing curiosity and perhaps produce a brighter insight into this area of mathematics.” Gauss told Zimmerman that he took Kaestner’s judgment entirely to heart.

The more general treatment to which Kaestner was guiding Gauss, was toward the solution of the “Kepler challenge”, which concerned the divisibility of the ellipse, a curve of non- constant curvature. In fact, Gauss’ method for the division of the circle was based on the discovery, that the constantly-curved circle, is, actually, a special case of non-constant curvature!

Gauss’ investigation into the division of the circle, was an extension of the study done by Kepler in the first book of the “Harmonies of the World”: “On the regular figures, the harmonic proportions they create, their source, their classes, their order, and their distinction into knowability and representability.”

It is here that Kepler started his elaboration of the harmonic ordering principle that governs the physical universe. “We must seek the causes of the harmonic proportions in the geometrical and knowable divisions of a circle into equal number of parts,” he began.

To divide the circle by geometrical means, according to Kepler, is to determine the ratio of the side of the figure to the diameter. It is in the efforts to divide the circle, that the human mind discovers the different types of harmonic proportions, which Kepler ranked according to degrees of knowability.

The first degree of knowability pertains to those quantities which can be proven equal to the diameter. The second degree of knowability pertains to those quantities that can be proven to be equal to parts of the diameter. The third degree of knowability pertains to those quantities that are inexpressible in length but expressible in square. From this follows those quantities that are inexpressible, or as the Greeks called them, incommensurable. However, the ranking continues with the incommensurables. The fourth degree of knowability pertains to those lengths that are not expressible by squares, but are expressible as rectangles. Kepler continued, following the tenth book of Euclid, the further degrees of knowability of the incommensurable.

What is significant for us, is that Kepler is re-asserting the knowabiblity of incommensurable magnitudes, not only in the context of geometry, but as the very magnitudes by which the physical universe is characterized. Here Kepler carried out a strenuous polemic against Petrus Ramus, a leading Aristotelean of the day, who had sought to ban incommensurable magnitudes, not only from the physical universe, but from geometry as well!

As we will see in the next installment, it is in the division of the circle, that we encounter these different degrees of knowability. This raises the question, if the circle is, at it appears to the eye, a perfectly uniform, constantly curved figure, why, when one attempts to divide it, does one encounter magnitudes of different degrees of knowability? The reader should think of the different types of magnitudes necessary to construct a triangle, square, pentagon, hexagon, and septagon, for example. Each different division of the circle gives rise to different degrees of knowability, and some divisions, such as seven, seem to be unknowable altogether.

This is the question that Kepler investigates in the Harmonies. The question the young Gauss investigated was, “What is the principle that governs the principle of knowability?”