Riemann for Anti-Dummies: Part 22: Your Education was Not Merely Incompetent

Your Education Was Not Merely Incompetent

If you felt a little disconcerted to sit in the same lecture hall with C.F. Gauss, listening to B. Riemann deliver his habilitation address, do not despair. Be happy. You are being afforded the opportunity to discover that your education was not merely incompetent, it was also malicious. Incompetent, in that your teachers were most likely totally ignorant of the most significant original discoveries upon which the human race has depended for survival; malicious, in that the system to which the teachers acquiesced, had no intention of producing individuals capable of making such discoveries. As we now see from the events unfolding around us, a system which does not intend to produce creative individuals, has no intention of surviving. Therefore, rejoice at the occasion to clear your head of the restrictive fixation on facts, laws, opinions, and popularly held beliefs, and set about the task of producing creative thinkers.

Riemann sought to “lift the darkness” that had settled on science for more than 2000 years, by providing for science a general concept of multiply-extended magnitude. A concept in which it was recognized, that magnitude had no {a priori} characteristics, but was itself determined by the nature of the manifold in which it existed which nature was only determined by experiment. Riemann’s taking off point was Gauss’ work on physical geometry and arithmetic, which was itself the revolutionary result of Gauss’ early education in the work of Kepler, Leibniz, Bach, Kaestner, and the scientific achievements of classical Greece. Central to all these discoveries was the desire to discover the principles that generated the objects of investigation, be it physical objects, such as the motions of the planets, living processes, or objects of cognition, the latter being the most fundamental, upon which all other investigations depend.

In this regard, Plato recognized that the mind must be trained to investigate itself, to which end he prescribed the study of geometry, astronomy, music and arithmetic, the latter, because, “thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks `Where is absolute unity?’ This is the way in which the study of the One has a power of drawing and converting the mind to the contemplation of true being … and, because this will be the easiest way for the soul herself to pass from becoming to truth and being….”

That search for the nature of unity underlies Gauss’ arithmetical investigations. Its revolutionary feature being that the nature of unity, is itself not a fixed, but developing and changing. This is what underlies Gauss’ concept of congruence, the ordering of numbers with respect to a modulus. This is based on the principle that numbers are not fixed objects that determine order, but are themselves ordered, according to the principle from which they are generated.

The first principle is the generation of numbers from the juxtaposition of cycles. These juxtapositions form two types of relationships. Either the cycles equally divide one another, such as a cycle of 8 and a cycle of 4, or no such division is generated, such as a cycle of 5 and a cycle of 8. In the latter case, that relationship is called, “relatively prime”. Those cycles, which when juxtaposed to all smaller cycles and One were called simply “prime”, and until Gauss were thought of as absolutely prime, or prime relative to One.

Thus when thinking about numbers from the bottom up, as formed by adding 1 to 1 to 1, the prime numbers are mysterious and arise from an unknown. However, when thought about from the top down, the prime numbers are that from which all numbers are made. The question that Gauss and Riemann contemplated was, “what principle generates prime numbers”. This led to the investigation, not of the numbers, but of the manifolds in which those numbers were generated.

The investigation of those manifolds leads to the second principle of generation. This is the principle which the Greeks called “geometric”, and was examined in last week’s installment. This is where today’s work begins.

Take the example from last week — the investigation of the cycle of residues generated with respect to modulus 11 and compare that to the cycle of residues with respect to modulus 13. For the sake of brevity, I indicate only the cycle with respect to one primitive root. The first row is the index, or power to which the primitive root is raised, and the second row is the corresponding residue. For reasons that will become apparent, we include both the positive and negative residues:

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, half the residues, that is, the residues of even powers, are residues of squares, (quadratic residues). The residues of the other half, the residues of odd powers, are residues of rectangles (quadratic non-residues). In the case of 13, the quadratic residues are the same whether negative or positive. While with 11, the positive quadratic residues are different than the non-residues.

This indicates an at first surprising connection between the ancient Pythagorean discovery of odd and even, which seems to pertain to numbers, and the geometric progression, which seems to pertain to figures in space. That odd and even reflected a deeper principle was described by Cusa in “On Conjectures”:

“It is established that every number is constituted out of unity and otherness, the unity advancing to otherness and otherness regressing to unity, so that it is limited in this reciprocal progression and subsists in actuality as it is. It can also not be that the unity of one number is completely equal to the unity of another, since a precise equality is impossible in everything finite. Unity and otherness are therefore varied in every number. The odd number appears to have more of unity than the even number, because the former cannot be divided into equal parts and the latter can be. Therefore, since every number is one out of unity and otherness, so there will be numbers in which the unity prevails over the otherness, and others in which the otherness appears to absorb the unity.”

It doesn’t stop with the division into even and odd, as both types have a deeper nature. The even numbers can be divided, into those even numbers, such as 10, that, when divided form two odd numbers (5 and 5), and those, such as 12, that form two even numbers (6 and 6). The former are called even-odd, the latter even-even. Likewise odd numbers can be divided into two types. Odd numbers, like 11, that are one more than an even-odd number and are called odd-odd, while odd numbers, like 13, that are one more than an even-even and are called odd-even. (Gauss called odd-even numbers 4n+1, and odd-odd numbers 4n+3.)

Now look at the mid-point of each of the above “orbits” of residues. As we showed at the end of last week’s installment, the midpoint of the orbit is both the arithmetic and the geometric mean. The arithmetic, because it is half the length of the cycle. The geometric, because its half the rotation from the 1 to 1, or the square root of 1. For modulus 11, that residue is either 10 or -1, both of which, when squared, are congruent to 1 modulus 11. For modulus 13, that residue is either 12 or -1, both of which, when squared, are congruent to 1 modulus 13.

Illustrate this in your mind, using Plato’s alternating series of squares and rectangles. In a cycle of 10 squares and rectangles, the 5th action is a rectangle, whose area is 32. That area is the geometric mean between a square whose area is 1 and the square whose area is 1024. Since the residues form a cycle that begins and ends with 1, the residue of the 5th power, mod 11, is the geometric mean between 1 and 1. Similarly, with a cycle of 12 squares and rectangles, the 6th action produces a square whose area is 64. That square is the geometric mean between a square whose area is 1 and a square whose area is 4096. With respect to modulus 11, the geometric mean is a rectangle, while for modulus 13, the geometric mean is a square.

But, there’s a difference between modulus 11 and modulus 13, as 11 is odd-odd, which means the half-way point is an odd number, that is 5. While 13 is odd-even, and is susceptible of further division, into quarters.

The residue of the 1/4 power relative to modulus 13 is either 8 or -5, both of which when squared twice, are congruent to 1 modulus 13. But, when both are squared once, they are congruent to -1 modulus 13. In other words, 8 and 5, -8 and -5, are congruent to the square root of minus 1 modulus 13.

Thus, the square root of -1 has clearly defined existence with respect to an odd-even modulus, while it has no existence in a manifold generated with respect to an odd-odd modulus.

From the naive standpoint, it would appear that the square root of -1 is a product of characteristic of oddness. But, as Cusa states, oddness is a quality in which unity prevails over otherness. So, rather than look for the square root of -1 in nature of oddness, look for the nature of oddness in the characteristic of unity.

This is precisely the way Gauss approached the problem. Rather than think of a manifold of a simply extended unity, he conceived of a manifold of a doubly extended unity, in which the square root of -1 is a “natural product” so to speak. He called this manifold the complex domain.

In his words:

“From this, we had already begun to ponder these objects in 1805, and we soon came to the conviction that the natural source of a general theory be sought in an extension of the field of Arithmetic.

“While higher arithmetic, has until now dealt only with questions pertaining to whole numbers, propositions concerning biquadratic residues appear in their complete simplicity and natural beauty, only if the field of arithmetic is extended to include imaginary numbers, without limitation, the numbers of the form a+bi forms its object, where the customary i denotes the square root of -1 and a and b are all whole numbers between minus infinity and plus infinity.”

Next week we’ll put flesh and bones on this new concept.