Riemann for Anti-Dummies Part 23
THE CIVIL RIGHTS OF COMPLEX NUMBERS
As the unfolding of current history demonstrates, it is reality that determines policy, not the other way around. This should come as no surprise to a scientific thinker knowledgeable in the method of Plato, Cusa, Kepler, Leibniz, Fermat, Gauss, Riemann and LaRouche. It is, however, shocking for anyone unfortunate enough to have accepted, wittingly or unwittingly, the delusion of Aristotle, Kant and Newton, that extensible magnitude exists outside the domain of universal physical principles.
This is the standpoint from which Gauss introduced his concept of the complex domain, beginning with his doctoral dissertation on the fundamental theorem of algebra, his Disquisitiones Arithmeticae, his treatises on geodesy and curvature, and his second treatise on biquadratic residues. From his earliest work, Gauss adopted the standpoint of his teacher Kaestner, and Leibniz before him, that the characteristic of extensible magnitude is a function of the manifold out of which those magnitudes were created.
It is in this light that one must view the discussions in the previous week’s installments. Gauss has rejected any {a priori} conception of magnitude, and instead derived the characteristic of numbers from a set of generating principles. First, by generating numbers from the juxtaposition of simple cycles, and then from the standpoint of a geometric cycle of cycles. As such, the relationships among numbers can not be found in the numbers themselves, but only in the relationship of those numbers to the manifold in which they exist. Like Leibniz’ monads, numbers don’t relate to each other directly, but only through the manifold from which they are created.
A quick review from last week illustrates the point. Take the “orbit” generated by the residues of the powers of the primitive root of 11 and 13.
Modulus 11:
Index: 0, 1, 2, 3, 4, 5
Residue: {1,-10},{2,-9},{4,-7},{8,-3},{5,-6},{10,-1}
Index: 6, 7, 8, 9, 10
Residue: {9,-2}, {7,-4},{3,-8},{6,-5},{1,-10}
Modulus 13:
Index: 0, 1, 2, 3, 4, 5, 6
Residue: {1,-12},{2,-11},{4,-9}, 8,-5}, {3,-10},{6,-7},{12,-1}
Index: 7, 8, 9, 10, 11, 12
Residue: {11,-2},{9,-4}, {5,-8},{10,-3},{7,-6}, {1,-12}
In both cases the orbit begins with 1 and ends with 1, ordering all the numbers between 1 and the modulus minus 1, according to a principle. That principle, at first does not appear obvious, but on further investigation, it reveals itself to be highly ordered. At the halfway point of the “orbit,” (the 5th power for 11 and the 6th power for 13), the residue is -1, which when squared equals 1.
In the case of 13, a further division by half is possible. This gets us to the 3rd power, whose residue is 8, which, when squared is congruent to -1 modulus 13. In other words, -1 is at half the orbit; the square root of -1 is at half of the half.
This phenomenon hints at a paradox that reveals the underlying geometry of the ordering principle that generates the numbers. In the above example we were “experimenting” with positive and negative whole numbers. Naive sense-certainty indicates that these numbers can be represented completely as equally spaced intervals along an infinitely extended straight line, with positive numbers lining up in one direction and the negative numbers in the other. However, under such a conception, the square root of -1 does not exist as a magnitude, yet, its existence was just discovered as the biquadratic root of 1, modulus 13. (8 = ?-1 mod 13; 82 = -1 mod 13; 84 = 1 mod 13.)
In other words, a species of magnitude exists, that can not be logically deduced from a manifold of one dimension. Euler concluded that such magnitudes were, therefore, “impossible.” Gauss, on the other hand, would not be restricted to a one-dimensional manifold, when an anomaly required an extension into two dimensions, in which such “impossible” magnitudes become “possible.” Not only were such magnitudes possible, but Gauss proclaimed, they deserved “complete civil rights.” As he stated in his announcement to the second treatise on biquadratic residues:
“It is this and nothing other, that for the true establishment of a theory of bi-quadratic residues, the field of higher arithmetic, that otherwise extends only to the real numbers, will be enlarged also to the imaginary, and these must be granted complete and equal civil rights, with the real. As soon as one considers this, these theories appear in an entirely new light, and the results attain a highly surprising simplicity.”
In a manifold of two dimensions, the relationship among objects is not restricted to the back and forth relationship of objects along a line, but also includes a relationship of up and down, so to speak. Be careful, this is not two separate relationships, back-forth and up-down. Rather it is one, doubly-extended relationship. As Gauss stated:
“Suppose, however, the objects are of such a nature that they can not be ordered in a single series, even if unbounded in both directions, but can only be ordered in a series of series, or in other words form a manifold of two dimensions….”
The root of this conception lies not in mathematics, but in physical geometry. In a fragmentary note, “On the Metaphysics of Mathematics,” Gauss described a doubly-extended relationship using the metaphor of a carpenter’s level. The bubble in the level can only move back and forth, if the ends of the level move up and down. Furthermore, Gauss repeatedly noted, such concepts as back and forth, up and down, left and right, can not be known, as Kant claimed, mathematically. Instead, such concepts are only known with respect to real physical objects.
This type of action is represented geometrically by two-dimensional magnitudes which Gauss called complex numbers. Gauss represented these numbers as the vertices of a grid of equally spaced squares on a plane. Be mindful. It is not the grid that generates the numbers. It is the {idea} of a doubly-extended manifold, that generates doubly-extended magnitudes, that form the grid. As in the case of the bubble in the carpenter’s level, any relationship between two complex numbers is a combination of horizontal and vertical action along the grid.
This geometrical representation of complex numbers flows easily from the geometry of the “orbits” generated by the residues of powers. For example, take the case of 13, (or any odd-even prime number modulus) as illustrated above. Think of the cycle of residues as a closed orbit, beginning with 1 and returning to 1. Halfway around the orbit is -1. One quarter the way around the orbit is the square root of -1. Three quarters around is minus the square root of -1.
This is the geometrical relationship that is reflected in the characteristics of the residues, and is nothing more than a generalization of the principle that Plato presents in the {Meno} and {Theatetus}, for the special case of squares. In that case, the diagonal of the square, which forms the side of a square whose area is double the original square, is called the geometric mean. The diagonal has the same relationship to the two squares, as -1 does to 1, and the square root of -1 does to -1.
Gauss described the manifold of complex numbers this way:
“We must add some general remarks. To locate the theory of biquadratic residues in the domain of the complex numbers might seem objectionable and unnatural to those unfamiliar with the nature of imaginary numbers and caught in false conceptions of the same; such people might be led to the opinion that our investigations are built on mere air, become doubtful, and distance themselves from our views. Nothing could be so groundless as such an opinion. Quite the opposite, the arithmetic of the complex numbers is most perfectly capable of visual representation, even though the author, in his presentation has followed a purely arithmetic treatment; nevertheless he has provided sufficient indications for the independently thinking reader to elaborate such a representation, which enlivens the insight and is therefore highly to be recommended.
“Just as the absolute whole numbers can be represented as a series of equally spaced points on a line, in which the initial point stands for 0, the next in line for 1, and so forth; and just as the representation of the negative whole numbers requires only an unlimited extension of that series on the opposite side of the initial point; so we require for a representation of the complex whole numbers only one addition: namely, that the said series should be thought of as lying in an unbounded plane, and parallel with it on both sides an unlimited number of similar series spaced at equal intervals from each other should be imagined, so that we have before us a system of points rather than only a series, a system which can be ordered in two ways as series of series and which serves to divide the entire plane into identical squares.
“The neighboring point to 0 in the first row to the one side of the original series corresponds to the number {i,} and the neighboring point to 0 on the other side to -i and so forth. Using this mapping, it becomes possible to represent in visual terms the arithmetic operations on complex magnitudes, congruences, construction of a complete system of incongruent numbers for a given modulus, and so forth, in a completely satisfactory manner.
“In this way, also, the true metaphysics of the imaginary magnitudes is shown in a new, clear light….”
Consequently, the domain of whole numbers has been extended beyond simply positive and negative numbers, to numbers of the form “a+bi“, where “i” stands for the square root of -1. These numbers are represented as points on a plane, in which “a” expresses the horizontal action while “b” the vertical action. For example, 2+3i would be represented by a point 2 to the right of 0 and 3 up from 2; 5+4i would be represented by 5 to the right of 0 and 4 up from 5. The difference (interval) between 2+3i and 5+4i would be 3+i, which is the combined amount of horizontal and vertical action required to move from 2+3i to 5+4i.
In Gauss’ complex domain, the fundamental characteristics of numbers are re-defined. Of particular importance is prime numbers. Here, numbers that are prime in a simply-extended manifold, are no longer prime in the complex domain. For example, 5, can be factored into the complex numbers (1+2i)(1-2i); or 13 into (3+2i)(3-2i). Gauss showed that all odd-even prime numbers are no longer prime in the complex domain, while all odd-odd prime numbers are still prime. Gauss went on to discover a new type of prime number that he called complex primes, which are now called “Gaussian primes.” These are complex numbers of the form a+bi, where a2 + b 2 is a prime number. (The geometrical demonstration of this principle will be developed in a subsequent installment.)
Thus, prime numbers, the “stuff” from which all numbers are made, are themselves not primary. Instead, they are defined by the nature of the manifold in which they exist. A one-dimensional manifold produces a certain set of prime numbers, whose “primeness” is absolute within a one-dimensional manifold, but relative with respect to a two-dimensional manifold. In turn, a two-dimensional manifold produces prime numbers whose characteristic “primeness” is different from what constitutes “primeness” in a one-dimensional manifold. The characteristic “primeness” of one-dimensional prime numbers can be derived from the characteristic of “primeness” of two dimensions, but not vice versa. Implicit in this, is a hierarchy of dimensionality, in which the singularities of n-dimensions are subsumed and transformed by manifolds of higher dimensions. Gauss himself anticipated such an idea stating:
“The author reserves the possibility of treating these matters, only barely touched upon in this paper, more fully at a later date, at which time we shall also answer the question, why such relations between things as form manifolds of more than two dimensions might not provide additional species of magnitudes to be admitted in general Arithmetic.”
This is only a taste of the manifold of ideas manifest in the minds of the hearers of Riemann’s habilitation lecture. The more this manifold begins to order your mind’s thoughts, the more lively Riemann’s ideas will become.