Riemann for Anti-Dummies: Part 20 : Gauss’ Attack on Deductive Thinking

Gauss’ Attack on Deductive Thinking

In his 1854 habilitation dissertation, Bernhard Riemann referred to two “hints” as preliminary to his development of an anti-Euclidean geometry–specifically Gauss’ second treatise on bi-quadratic residues and Gauss’ essay on the theory of curved surfaces. It is but one more testament to the ignorance of all so-called experts today, (not to mention those who wish to qualify as educated citizens) that direct knowledge of these two works by Gauss, let alone a working understanding of Riemann himself, is virtually non-existent.

It should not be surprising that in a lecture focused on ridding science of “ivory tower” mathematics, Riemann would refer to the climactic conclusion of Gauss’ investigation of whole numbers. Riemann, like Gauss and Leibniz before him, began his scientific education by confronting the paradoxes that emerge from an anti-deductive investigation of whole numbers. At an early age, Riemann was given a copy of Legendre’s “Theory of Numbers”, and within one week he returned the 600 page book saying, “This is wonderful book. I know it by heart.”

Plato prescribed such investigations as necessary for the development of competent leadership, because it forced the mind out of realm of sense-certainty and into the realm of paradoxes where, “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.,,,we must endeavor to persuade those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study {until they see the nature of numbers with the mind only;} nor again, like merchants or retail-traders, with a view to buying or selling, but for the sake of their military use, and of the soul herself; and because this will be the easiest way for her to pass from becoming to truth and being….”

And as LaRouche pointed out in “Marat, DeSade & Greenspin”:

“Since the beginning of the Nineteenth Century, Carl Gauss’s {Disquisitiones Arithmeticae}, inspired by his teacher, the great founder of anti-Euclidean geometry, Abraham K„stner, had been the standard for competent mathematics instruction. This masterpiece should be the recognized standard, even today, for basic secondary and higher education in mathematics. The result of replacing that standard with “the new math” program, should have reminded any literate professional of Jonathan Swift’s famous description of education as practiced on the allegorical floating island of Laputa.”

To ameliorate this pitiable condition of mankind, and save any readers of these pedagogicals from being condemned to perpetual flatulence on Laputa, the next installments of this series will begin to acquaint the reader with the basic conceptions of these two works by Gauss, as a prerequisite to looking more deeply into Riemann’s work itself.

To begin, you must, as Gauss does, give up all deductive notions of number. Instead of thinking of whole numbers as self-evident things in themselves, think of numbers as being generated by a principle. Gauss took an experimental approach to numbers, designing experiments that revealed paradoxes with respect to a known principle. The resolution of that paradox required the introduction of a new principle. Gauss himself described the approach to be taken in our investigations:

“The questions of higher arithmetic often present a remarkable characteristic which seldom appears in more general analysis, and increases the beauty of the former subject. While analytic investigations lead to the discovery of new truths only after the fundamental principles of the subject (which to a certain degree open the way to these truths) have been completely mastered; on the contrary in arithmetic the most elegant theorems frequently arise experimentally as the result of a more or less unexpected stroke of good fortune, while their proofs lie so deeply embedded in the darkness that they elude all attempts and defeat the sharpest inquiries…. These truths are frequently of such a nature that they may be arrived at by many distinct paths and that the first paths to be discovered are not always the shortest. It is therefore a great pleasure after one has fruitlessly pondered over a truth and has later been able to prove it in a round-about way to find at last the simplest and most natural way to its proof.”

The opening motivic idea of the Disquisitiones, is to identify numbers as being generated by an interval, or modulus, much the same way as musical notes are generated by intervals. If the interval between two numbers is divisible by the modulus, Gauss called those numbers, “congruent”. For example, 2, 7, 12, 17, 22, etc, are all congruent to each other relative to modulus 5. Relative to modulus 7, 2 is congruent to 9, 16, 23,etc.

Gauss’ use of the term congruence is consistent with Kepler’s use of that concept in Book II of his “Harmonies of the World”. For Kepler the word “congruentia” was the Latin equivalent to the Greek word, “harmonia”, which means to fit together. Thus, it is not the numbers on which the mind must focus, but the way they fit together.

Gauss’ concept of congruence reflects the actual nature of numbers more truthfully than the so-called “natural” ordering of numbers that seemed so commonsensical when you learned it in school. This is because, contrary to such common sense certainty notions, the concept of number does not arise from counting things. Rather, it arises from the juxtaposition of cycles, such as, for example, astronomical cycles. Each cycle is a One, but when juxtaposed to each other these cycles give rise to a multiplicity.

As Leibniz puts it in his doctoral dissertation, “On the Art of Combinations”:

“Furthermore, every relation is either one of union or one of harmony. In union the things between which there is this relation are called parts, and taken together with their union, is a whole. This happens whenever we take many things simultaneously as one. By one we mean whatever we think of in one intellectual act, or at once. For example, we often grasp a number, however large, all at once in a kind of blind thought, namely, when we read figures on paper which not even the age of Methuselah would suffice to count explicitly.

“The concept of unity is abstracted from the concept of one being, and the whole itself, abstracted from unities, or the totality, is called number.”

Any two cycles can be known in relation to each other only by a third. For example, the cycle discovered by the Greek astronomer Meton who attempted to resolve the lunar month and solar year cycles into a One. One solar cycle contains 12 lunar cycles, plus a small residue, so in Gauss’ words, the lunar cycle is {incongruent} with the solar one. However, Meton discovered that 19 solar years contains 235 lunar months with no residue. So, while one lunar month is not congruent to one solar year, one lunar month is congruent to 19 solar years. The relationship between the solar cycle and lunar cycle can be known with respect to this 19 year Metonic cycle, which defines the modulus under which the solar and lunar cycles are congruent.

To get familiar with this concept play with some more examples. Consider two cycles one of which is three times longer than the other. These cycles would be congruent to each other relative to modulus three. Examples of this relationship expressed in numbers would include: 3 is congruent to 9 relative to modulus 3, or 9 is congruent to 27 relative to modulus 3.

Now consider cycles that don’t fit exactly, such as a cycle of 4 and a cycle of 9. The smaller cycle of 4 will fit into the larger cycle of 9 twice with a residue of 1. Under Gauss’ concept, 9 is congruent to 1 relative to modulus 4. On the other hand, a cycle of 4 fits into a cycle of 10 twice with a residue of 2. Under Gauss’ concept, 10 is congruent to 2 relative to modulus 4. Continuing, a cycle of 4 fits into a cycle of 11 with a residue of 3. Thus, 11 is congruent to 3 relative to modulus 4. Further, a cycle of 4 fits into a cycle of 12 with 0 residue, and into a cycle of 13 with a residue of 1. Thus, a cycle of 4 will fit into any cycle with a residue of 0, 1, 2, or 3.

If you play around with this idea, trying cycles of different relationships, you will discover for yourself, that any modulus defines a period, from 0 to the modulus minus 1. This will probably strain your brain, as you will be forcing yourself to think in terms of relationships instead of things, but that is precisely why all great thinkers, from Plato onward, struggled to free themselves from the straight jacket of deductive relationships by investigating the nature of numbers.

This would probably be enough to get you started, but in order to speed up our pursuit of the concepts in Gauss’ second treatise of bi-quadratic residues, we should push ahead.

After developing the concept of congruence in the beginning of the Disquistiones, Gauss turns to an investigation of what he called, “residues of powers”. Here you must leave completely the world of sense certainty and deductive reasoning.

By “powers” Gauss meant the concept developed by Plato in the Theatetus dialogue. These are the magnitudes associated with action in what Riemann would call a doubly extended manifold. Shadows of these magnitudes are represented by Plato as the successive doublings, triplings, etc. of squares. When these magnitudes are expressed in whole numbers it generates a geometric series such as: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, etc. or: 1, 3, 9, 27, 81, 243, 729, 2187, 6561, etc.

If you think of each number as a cycle, the series can be thought of as a cycle of cycles. This cycle of cycles doesn’t close, but gets bigger and bigger, according to a self-similar proportionality.

What may appear shocking to you, is that this open, growing, cycle of cycles generates a periodic, closed, cycle with respect to a modulus.

For example, take the geometric series formed from doubling squares and find the residues relative to modulus 3. This yields the period: 1, 2, 1, 2, 1, 2, 1, 2, etc. Now do the same for modulus 5. This yields the period: 1, 2, 4, 3, 1, 2, 4, 3, 1, 2, 4, 3, 1, etc. And for 7: 1, 2, 4, 1, 2, 4, 1, 2, 4, 1,etc. Try the same experiment with respect to the geometric series based on tripling. For modulus 5 it yields the period: 1, 3, 4, 2, 1, 3, 4, 2, 1, etc. Compare this with the period generated from the same modulus but the geometric series based on doubling. Modulus 7 for the same series yields the period: 1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, 1,etc.

This experiment is a simple example of what Gauss described as discovering certain truths by experiment. From where does this periodicity arise? What is its nature? What principles does it reflect?

Experiment with other geometric series and other moduli. Next week we will plunge ahead.