IT IS PRINCIPLES, NOT NUMBERS, THAT COUNT

As we continue the investigations into the "hints" from Gauss, to which Riemann referred in his 1854 habilitation lecture, it is vitally important to maintain the perspective of a member of the audience in the lecture hall that June day when Riemann delivered his revolutionary address. Don't be a fearful, passive observer. Go in. Take the open seat next to the 77 year old Gauss and hear these living ideas, not only as they were spoken then, but as they are today, alive and transformed in the mind and work of LaRouche, for which Riemann provides brief hints.

Listen as Riemann boldly proposes "to lift the darkness" that has existed for more than 2000 years, by elaborating a "general concept of multiply-extended magnitude". But, before you can even begin to lift that darkness, you must first realize that the lights aren't on.

That is the basis on which the preliminary exercises into the investigations of the geometry of numbers was begun last week.

Gauss is training the mind to give up all deductive, a priori, notions of number. Instead of investigating numbers, we investigate what generates them. It is the principle of generation to which we must turn our thoughts, aided by concepts from Classical art. The numbers are simply players, guides to what's in between.

The first principle of generation of numbers, to which Gauss points, is the generation of numbers by the juxtaposition of three cycles. While this concept was introduced in a new form in the Disquisitiones Arithmeticae, by the concept of congruence with respect to a modulus, the principle underlying it is perhaps the earliest, and most elementary concept of number. In this case, no number exists on its own. Rather, all numbers exist as players, whose parts are a function of their relationship to one another and a One, which Gauss called a modulus. Thus, all numbers are ordered according to the characteristics of the modulus. Those characteristics are themselves determined by an underlying generating principle, which will become more clear below. The so-called, "natural", counting numbers are only the special case, of numbers ordered with respect to the modulus 1.

The second principle of generation to which Gauss turned his attention, is generating numbers from a cycle of cycles, specifically, the "geometric" cycle. Here each cycle is generated by the function described by Plato in the Theatetus dialogue, as reflected in the alternating series of squares and rectangles produced by some repeated action, such as doubling or tripling, etc. However, Gauss, as Plato, Kepler, Leibniz, Bernoulli and Fermat before him, understood that the alternating series of squares and rectangles, was itself only a shadow of a higher principle of generation, that had to be discovered.

Naive sense certainty says this geometric progression is not a cycle at all, but open ended and continuously growing. Yet, as the experiments at the end of last week's installment illustrate, if each "stage" of the geometric progression is thought of as a cycle, and each such cycle is juxtaposed to a third cycle (modulus), an underlying periodicity is revealed, indicating the characteristics of the cycle that generated each stage.

The principle of generation of that underlying cycle, is best investigated by experiment. Hopefully, you carried out the experiment indicated last week. If so, you will have no trouble producing the necessary geometric constructions.

Construct a chart of the residues of powers with respect to modulus 11 by first making a row of numbers from 0 to 10. These denote the powers. Then make a separate row of the residues of the powers of 2 through 10, writing each residue under the corresponding power. The result should be the following:

Powers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10 
     2: 1, 2, 4, 8. 5, 10,9, 7, 3, 6, 1 
     3: 1, 3, 9, 3, 4, 1, 3, 9, 5, 4, 1 
     4: 1, 4, 5, 9, 3, 1, 4, 5, 9, 3, 1 
     5: 1, 5, 3, 4, 9, 1, 5, 3, 4, 9, 1 
     6: 1, 6, 3, 7, 9, 10,5, 8, 4, 2, 1 
     7: 1, 7, 5, 2, 3, 10,4, 6, 9, 8, 1 
     8: 1, 8, 9, 6, 4, 10,3, 2, 5, 7, 1 
     9: 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1 
    10: 1,10, 1, 10,1, 10,1, 10,1,10, 1 

Now have some fun. Obviously, this action produces, from the open and growing geometric cycle, a regular structured periodicity. The question Fermat, Leibniz, and Gauss investigated was, "What generating principle produces this?" To begin to answer this, they looked for the paradoxes, within the seemingly regular structure.

First, it is clear that each period begins and ends with 1 and there are three types of periods. Those periods, such as for the powers of 2, 6, 7, and 8, that are 10 numbers long and include all the numbers between 1 and 10. Gauss called these numbers, 2,6,7 and 8 "primitive roots" of 11. The second type of period, for the powers of 3, 4, 5, and 9 are 5 numbers long. The third type of period is the powers of 10 which is only 2 numbers long. Thus, the residues of powers with respect to modulus 11 permits only certain size "orbits" so to speak, which are restricted to the size of 10 and the prime number factors of 10, that is, 2 and 5.

Now, even though each period puts the numbers in a different order, this order is highly determined. To see this, hunt through the whole chart and circle the primitive roots, 2, 6, 7, and 8, wherever they appear. You should discover that these numbers do not appear as residues, except in the periods that are 10 numbers long. Also, even though they appear in different places in each period, they always are residues of the powers 1, 3, 7, or 9, which are the numbers that are relatively prime to 10.

This begins to reveal the nature of the underlying generating principle of these "orbits", as the characteristics of the number 10, specifically its prime factors and its relative primes, determine the ordering of the periods!

Keeping this in your mind's eye, draw an alternating series of squares and rectangles, first by doubling, then by tripling, and label each according to the residue from the powers of 2 and 3 respectively to which it corresponds. This will reveal that the even powers correspond to squares and the odd powers correspond to rectangles. Notice how the residue 1 only appears on a square and the residue 10 only appears on a rectangle. Also, the squares always correspond to even numbered powers, while the rectangles correspond to odd numbered powers.

Thus, the quality of odd and even, reflect a geometric characteristic, not a numerical property of numbers. For these geometrical reasons, Gauss called the residues of the even powers, "quadratic residues" and the residues of the odd powers, "quadratic non-residues". Gauss paid special attention to this characteristic, for its investigation opened the door to some of the most profound principles. In the next installment we will explore this more fully.

But, before closing, look at one more anomaly. Notice that the residue at the halfway point, that is the residue of the power 5, is always 1 or 10. Since 10 is congruent to -1, the residue of the middle power is always 1 or -1. While 5 is the arithmetic mean between 0 and 10, its residue, 1 or -1 is the geometric mean between 1 and 1! In other words, 1 and 10 are the square roots of 1 relative to modulus 11.

Look back over the preceding investigations from the perspective of a classical drama. Think of the foregoing as a drama of 10 characters. Each character has several roles, in which they wear the same costume, but do different things. The playwright has deliberately chosen this device so that the audience can be broken from judging these characters by naive sense-certainty. This helps convey an idea that could not be expressed in words by any of the characters, but only by the totality of all their actions taken as a whole, and the ironies revealed when the same actor does something completely different, without changing his costume. Each number from 1 to 10 has a different function, whether it's a power, a residue, or a base. In some roles, the obvious characteristics of the number, such as odd or even, factor or relative prime, seem to affect its function, but in other cases, such as the primitive roots, these obvious characteristics seem to have no bearing. Only when all the roles are played out can we begin to taste the intention of the playwright.

Gauss could see that these anomalies could not be derived from a concept of number, in the naive sense of an object that counts things, but, rather, these anomalies revealed an underlying {geometric} generating principle, that shone through the numbers themselves. But, to bring out that light, required a complete revolution in the way people thought about number. As he said in the beginning of the first Treatise on Bi-quadratic Residues, "we soon came to the knowledge, that the customary principles of arithmetic, are in no way sufficient for the foundation of a general theory, and that it is very much necessary, that the region of higher arithmetic be, so to speak, infinitely much more extended."