POWER AND CURVATURE
In his 1854 habilitation lecture, Bernhard Riemann spoke of the twofold task involved in lifting more than 2,000 years of darkness that had settled on science:
“From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor by the philosophers who have labored upon it. The reason of this lay perhaps in the fact that the general concept of multiply extended magnitudes, in which spatial magnitudes are comprehended, has not been elaborated at all. Accordingly, I have proposed to myself at first the problem of constructing the concept of a multiply extended magnitude out of general notions of quantity. From this it will result that a multiply extended magnitude is susceptible of various metric relations and that space accordingly constitutes only a particular case of a triply-extended magnitude. A necessary sequel of this is that the propositions of geometry are not derivable from general concepts of quantity, but those properties by which space is distinguished from other conceivable triply-extended magnitudes can be gathered only by experience. There arises from this the problem of searching out the simplest facts by which the metric relations of space can be determined, a problem which in the nature of things is not quite definite; for several systems of simple facts can be stated which would suffice for determining the metric relations of space; the most important for present purposes is that laid down for foundations by Euclid. These facts are, like all facts, not necessary but of a merely empirical certainty; they are hypotheses….”
To grasp the significance of Riemann’s “Plan of Investigation,” it must be recognized that the 2,000 years of darkness of which he spoke, was, like the foundations of Euclidean geometry, not necessary. The Romantic cult-belief that the definitions, axioms, and postulates of Euclid, were the {a priori}, fixed, immutable and necessary condition of the universe, never had any basis in truth. It was a false doctrine imposed by an imperial system, which required the widespread acceptance of the belief that the universe was ruled by forces beyond human comprehension and control, and that these forces could only be administered by an oligarchical authority. The edicts of this oligarchy, like the definitions, axioms, and postulates of Euclidean geometry, were laid down as given, not requiring, nor susceptible of, proof. They were simply, “the way things are.”
This view was expressed succinctly by the hoaxster, Claudius Ptolemy, the hatchet-man who imposed the knowingly false, fixed, geocentric conception of the solar system. Ptolemy, agreeing with Aristotle, justified his attack on Aristarchus’ provably true heliocentric conception, as a necessary consequence of his view of Man. In the introduction to his {Almagast}, Ptolemy stated that knowledge of both God and the physical universe was impossible. The only knowledge accessible to man was, what Ptolemy called “mathematical,” that is, knowledge which follows logically from a given set of axioms, definitions, and postulates. Those axioms, definitions and postulates, themselves can not be proven. As such, their authority resides not in demonstrable truths, but in the arbitrary power of whoever decrees their primacy. The evil lies not with the axioms, postulates, and definitions themselves, but in the acceptance of the method that knowledge can be derived only from them.
The popular acceptance of the darkness ushered in by the dominance of this Aristotelean method was a tragic degeneration from a higher concept of man and the universe developed in Classical Greece from Pythagoras until the murder of Archimedes. Euclid’s {Elements}, in a strange way, demonstrate this themselves. Read in their customary order, the {Elements} proceed from the definitions of point, line, surface, and solid, as objects of, respectively, 0, 1, 2, and 3 “dimensions,” and certain postulates about the unlimitedness of these objects. From there, a set of theorems is developed that elaborate the possible actions in a universe that conforms to the restrictions contained in the opening definitions, axioms, and postulates.
Yet read backwards, Euclid’s {Elements} begin to reveal a completely different comprehension of the universe. The {Elements} end where they should begin–with the construction of the five regular (Platonic) solids from the characteristic of spherical action. This investigation leads to the discovery of magnitudes of different powers, as exhibited in the problem of doubling the line, square, and cube. The relationships among these powers, give rise to the proportions called the arithmetic, geometric, and harmonic means, and to the prime numbers and the relationships among them. Only then do the investigations concern the reflection of these relationships in a plane. Only at the end, should we arrive at the point, line, surface, and solid. Seen in this way, these objects are concepts arising from a higher principle–the action that produced the five regular solids from a sphere–not as objects created by arbitrary decree from below, in the form of axioms, definitions, and postulates.
(It is from this standpoint that Kepler begins his {Harmony of the World} with a strong denunciation of Petrus Ramus, the leading Aristotelean of the day, who sought to ban books 10 through 13 of Euclid.)
This principle is similarly demonstrated by the Pythagorean/Platonic investigations of doubling the line, square, and cube. As discussed in previous pedagogicals, each object is generated by magnitudes of successively higher powers. The relationship among these higher powers is reflected by the arithmetic and geometric proportions. Initially, it appears that each power is associated simply with an increase in extension. For example, the magnitude that doubles the square is incommensurable with the magnitude that doubles the line, but it is produced from within the square. Yet, when the problem of doubling the cube is considered, the sought-after magnitude is not generated anywhere in the cube. Both the constructions of Archytus and Menaechmeus demonstrate, that the magnitude that has the power to double the cube is produced by the higher form of action represented by the cone, torus, and cylinder. While that action has a causal effect on the generation of cube, it is not produced anywhere in the cube. In other words, it is not produced by an increase in extension from two to three “dimensions.”
Another principle is involved. As emphasized in last week’s pedagogical discussion, the principle that generates the magnitude that doubles the cube, is expressed in a change of “curvature.”
As Riemann stated in his habilitation paper, the determination of extension is only the first step:
“Now that the concept of an n-fold extended manifold has been constructed and its essential mark has been found to be this, that the determination of position therein can be referred to n determinations of magnitude, there follows as second of the problems proposed above, an investigation into the relations of measure that such a manifold is susceptible of, also, into the conditions which suffice for determining these metric relations.”
To illustrate this pedagogically, perform the following experiment. Stand in the corner of a room and mark one point on the ceiling above your head, a second point on the wall directly to your right, and a third point on the other wall directly to your left. Now, in your mind connect these three points. If you point to these points in succession, the motion of your arm will define three right angles, implying that these three points all lie on the surface of a sphere. However, if you connect these points, in your mind, with straight lines, the points now lie on a flat surface, forming the triangular face of an octahedron. On the other hand, if you connect the three points to one another by hanging strings between them, the surface thus formed will be bounded by catenaries, and thus be negatively curved. These three points form three different triangles, which in all three cases, are doubly-extended magnitudes. Yet, each is very different from the other. The difference lies not in the degree of extension, but in the curvature of the surface on which the triangle lies. Thus, the lines that form the sides of these triangles, are defined by the nature of the surface in which they exist. The Euclidean definition of a line as “breadthless length,” cannot distinguish the side of the spherical triangle from the flat or negatively curved one; nor can the Euclidean definition of surface as, “that which has length and breadth only,” distinguish the three triangles from one another.
The curvature of these three surfaces can be measured by the sum of the angles of the triangles formed on each. On the spherical triangle, the sum of the angles is greater than 180 degrees. On the flat one, the sum of the angles is exactly 180 degrees. On the “catenary” triangle, the sum of the angles is less than 180 degrees.
Now, think, as Gauss and Riemann did, of a manifold that encompasses all three curvatures. Begin first with a positively curved surface such as a sphere. Here the sum of the angles of a triangle is always greater than 180 degrees. The larger the triangle, the greater the sum, until a maximum is reached when the triangle covers the whole sphere. As these triangles become smaller, the sum of the angles approaches, but never reaches 180 degrees, for when the sum of the angles reaches 180 degrees, the surface becomes flat. On a negatively curved surface, just the opposite occurs. As the triangle becomes smaller, the sum of the angles of a triangle gets larger, approaching, but never reaching 180 degrees.
These three surfaces form a manifold of action, in which the flat plane of Euclid is only the momentary transition between a negatively and a positively curved surface.
Gauss saw in this the possibility of a physical determination of geometry.
“It is easy to prove, that if Euclid’s geometry is not true, there are no similar figures. The angles of an equal-sided triangle, vary according to the magnitude of the sides, which I do not at all find absurd. It is thus, that angles are a function of the sides and the sides are functions of the angles, and at the same time, a constant line occurs naturally in such a function. It appears something of a paradox, that a constant line could possibly exist, so to speak, {a priori}; but, I find in it nothing contradictory. It were even desirable, that Euclid’s Geometry were not true, because then we would have, {a priori}, a universal measurement, for example, one could use for a unit of space [{Raumeinheit}], the side of an equilateral triangle, whose angle is 59 degrees, 59 minutes, 59.99999… seconds.”