Riemann for Anti-Dummies Part 25
SCHILLER AND GAUSS
In his "Aesthetic Estimation of Magnitude", Friedrich Schiller discusses a crucial ontological paradox that confronts science when it tries to exceed existing axiomatic assumptions:
"The power of imagination, as the spontaneity of emotion, accomplishes a twofold business in conceptualizing magnitude. It first gathers every part of the given quantum into an empirical consciousness, which is {apprehension}; secondly, it assembles the {successively collected} parts into a pure self-consciousness, in which latter business, that of {comprehension}, it acts entirely as pure understanding. That concept of "I" (empirical consciousness), in other words, combines with each part of the quantum: and through reflection upon these successively performed syntheses, I recognize the identity of my "I" (pure self- consciousness) in this series as a whole; in this way, the quantum first becomes an object for me. I think A to B to C, and so forth, and while I watch my activity; as it were, I say to myself: in A, as well as in B, and in C, I am the acting subject.
"Apprehension takes place {successively}, and I grasp each partial conception after the other....The synthesis, however, takes place {simultaneously}, and through the concept of the self-identity of my "I" in all preceding syntheses, I transcend anew the temporal conditions under which they had occurred. All those different empirical conceptions held by my "I" lose themselves in a single pure self-consciousness; the subject, which had acted in A, and B, and C, and so forth, is I, the eternally identical self...
"If the power of reflection transgresses this limit, and seeks to bring together mental images, which already lie beyond the limit, into unity of self-consciousness, it will lose as much in clarity as it gains in scope. Between the circumference of the entirety of a mental image and the distinctness of its parts, is an ever insuperable, specific relationship, wherefore in each addition of a large quantum we lose as much backward as we gain forwards and when we have reached the end-point, we see the starting point vanish."
Schiller is not referring to quanta, which have magnitude, simply with respect to quantity, but as Leibniz, Gauss and Riemann did, as {universal principles}:
"Everything which has parts, is a quantum. Every perception, every idea formed by comprehension, has a magnitude, just as the latter has a domain and the former a content. Quantity in general, therefore, cannot be meant, if one speaks about a difference of magnitude among objects. Here we speak about such a quantity as characteristically belongs to an object, that is to say, that which is not simply a {quantum}, but is at the same time a {magnum},"
Think of Schiller's concept with respect to the successive discoveries of Kepler and Gauss concerning planetary motion. If we think of the position and speed of the planet at any given moment, as a quantum, it is indeterminable, except as that quantum is a characteristic of the whole orbit. In that sense, the indeterminable position and speed at the moment, becomes determinable, only as an interval, a part, of the whole orbit. The magnitude associated with that interval, is the area swept out. This, magnitude cannot be measured by the successive addition of the speeds and positions of the planet, which, owing to the non-uniformity of the orbit are indeterminate, but, only as these are grasped as an interval of the whole.
But, the orbits, in turn, are not self-defined, and their magnitudes are indeterminable as individual orbits. Rather, the magnitudes of the individual orbits can only be determined as intervals with respect to the harmonic ordering among all the orbits at once. Inversely, that harmonic ordering cannot be determined by successive addition of each individual orbit, but only as intervals of the whole.
Further, as Gauss' investigation of the asteroids demonstrated, these harmonic orderings are themselves changing, according to a still higher harmonic ordering.
In other words, if we seek to determine the position and speed of the planet at any moment, we are stymied until we are led to the orbit as a whole. And, if we seek to determine the nature of an individual orbit, we are stymied anew, until we are led to all the orbits. And, further, if we try to determine the harmonic ordering of all the orbits, we will be once again stymied, until we are led to the ordering of the harmonic ordering. From this vantage point, the individual position and speed of the planet, which was our first object of investigation, recedes, as the deeper underlying principles come to the fore.
In the terms of Leibniz' calculus, the differential can be known only as a function of the integral. Or, under Schiller's idea, if each principle is thought of as a quantum, it can only be measured with respect to a magnum, which in turn, is a quantum, to a, higher, yet to be discovered magnum. In terms of Riemannian differential geometry, it is the highest principle, which determines all lower ones.
Seen in this way, the principle of Mind, of which Kepler speaks as governing the motion of the planet, is not a simple conception of a mind interacting one on one between the planet and the Sun, but a principle of Mind, as Schiller speaks of above, that comprehends its actions from a higher and higher standpoint, which determine, the seemingly indeterminable action in the small.
Gauss' investigation of bi-quadratic residues, and his and Riemann's further development of differential geometry, provide the pedagogical/epistemological capacity for our minds to grasp this concept.
For Gauss, as for Plato, Fermat, and Leibniz, individual numbers are not self-defined, but are rather defined by a higher principle, which Gauss called congruence. Each modulus, thus, defines a certain indivisible "orbit" in which all the numbers from 1 to the modulus minus 1, are ordered. The ordering within any individual "orbit" is itself a function of the characteristic of the modulus. For example, if the modulus is an odd-even prime number, such as 5, 13, 17, etc. -1 is a residue of the modulus minus 1 power, and √-1 is the residue of the 1/4 the modulus minus 1 power. If the modulus is an odd-odd prime number, √-1 never emerges. However, this characteristic of prime numbers, is not determined by the individual prime numbers, but is rather a function of the, still as yet undiscovered, "orbit", that determines prime numbers.
This characteristic of number led Gauss to search for a higher principle, which he discovered by extending the concept of number from simply-extended magnitudes, to doubly- extended magnitudes, which he called complex numbers.
The significance of this is best grasped pedagogically, by way of an example directly out of Gauss' second treatise on bi-quadratic residues.
Gauss thought of the complex domain as mapped onto a plane that is covered by a grid of equally spaced squares, the vertices of each square signify what Gauss called complex whole numbers. Each complex whole number is of the form a +bi, where i stands for √-1, and a and b are whole numbers. Gauss called a2 + b2 the "norm" of the complex number. Gaussian prime numbers, are those complex whole numbers, whose norms are prime numbers.
Gauss' example uses the complex prime number 5+4i. Taking this as the modulus, the entire complex domain is "partitioned" into diamonds, whose sides are the hypothenuses of right triangles whose legs are 5 and 4. (See last week's pedagogical.) Each diamond encloses 41 (52 + 42) individual complex whole numbers, which are all incongruent to each other, relative to modulus 5+4i.
(You can illustrate this, if you take the diamond whose vertices are the complex numbers 0, 5+4i, 1+9i, -4+5i, as no two numbers within this diamond will be separated by doubly- extended interval greater than 5+4i. Now, construct another diamond whose vertices are 5+4i, 10+8i, 6+13i, 1+9i. Each complex number within this new diamond will all be incongruent to every other within the diamond, but, each complex number of the second diamond will be congruent to that complex number that is in the same relative position in the first diamond, specifically, the number whose difference with it is 5+4i.)
Gauss then takes the complex number 1+2i as a primitive root of 5+4i. To grasp the meaning of this concept, see what happens, geometrically, when 1+2i is raised successively to the powers, in a new type of geometric progression. First you have (1+2i)0 = 1; Next is (1+2i)1 = 1+2i; These two numbers define a triangle whose vertices are 0, 1, 1+2i. This will form a right triangle, whose legs are 1 and 2 with hypotenuse √5. The angle at the vertex 0 will be 63.4349 degrees, the angle at 1 will be 90 degrees and the angle at 1+2i will be 26.5651 degrees. Now construct a similar triangle to this, using the hypotenuse of this first triangle, as the shorter leg, placing the 90 degree angle at the vertex 1+2i. This will define a new vertex at -3+4i, which is (1+2i)2. Repeat this process, constructing another similar triangle, with right angle at -3+4i, and the side 0, -3+4i as the short leg. This defines a new complex number, -11-2i, which is (1+2i)4.
This chain of similar right triangles, is but a general case of the famous chain of right triangles constructed by Theodorus, as reported by Theatetus in Plato's dialogue.
Each new vertex of this chain of similar right triangles, is thus a new, higher, power of 1+2i, and all lie on a unique logarithmic spiral. In other words, as this particular logarithmic spiral winds its way around the complex domain, the complex whole numbers it intersects are the powers of 1+2i. Thus, the powers of 1+2i are determined by a higher principle, of logarithmic spiral action. They are as moments in a orbit, or orbits in a planetary system.
Gauss continued this process, by continuing this spiral, so as to define 41 (52 +42) powers of 1+2i, and investigated these spiral points in a complex domain, "partitioned" into diamonds by modulus 5+4i, with the beginning diamond having 0 at its center. (This is the diamond whose vertices are (-1/2 - 4i), (4 - i), (+4i), (-4 + 1/2 i).) Now think of these diamonds spreading out, partitioning the complex domain, as the spiral winds its way around. Each time the spiral intersects a complex whole number, that number will be a power of 1+2i, and that number will be inside a particular diamond. Gauss showed that the first 40 complex whole numbers the spiral intersects, will each be in a different relative place within their respective diamonds, than any other previous or succeeding one. In other words, each complex whole number the spiral intersects, will be congruent to only one of the complex whole numbers in the beginning diamond. Most importantly, the 10th power of (1+2i) would be congruent to i, the 20th power to -1, the 30th power to -i, and the 40th power to 1. Then the cycle would repeat!
And so, if we begin with individual numbers we soon see these numbers can not be self determined, and we are led to the generating principle of congruence. But, these congruences produce "orbits" which can not be self-determined and we are led to a still higher principle of extended magnitudes. With each successive step, the individual numbers recede and as the higher principles come more to the fore in our minds.
And, yes, there is a still higher principle at work which Gauss discovered was connected directly to the Kepler problem. This was indicated by one of the earliest entries in his diaries that read, "I have discovered an amazing connection between bi-quadratic residues and the lemniscate".
Our investigation of this remark, will have to wait for a future installment.