Introduction to Gauss's Work on Higher Arithmetic
By Peter Martinson
In an environment caked with the residue of that classic British operation called the French Revolution, which led into the brutal dictatorship of amateur mathematician and Lagrange worshiper, British agent Napoleon Bonaparte, Carl F. Gauss was afraid to publicly lead the Leibnizian revolution in science which had been thrown into his lap by his teacher Abraham Kästner. Instead, Gauss had to go under cover, and although he did present the results of his work, the revolutionary discoveries themselves were cloaked behind a smokescreen of mathematics. Written into that mathematics, however, as a form of code, were hints that he hoped future creative thinkers would be able to decipher.
For example, though his discovery of the orbit of Ceres shocked the scientific world, his reported method of discovery appeared to be little more than a restatement of mathematical techniques that were already in use by various scientists around Europe. When pursued by other astronomers to reveal his method, Gauss’s response was to publish a full textbook on the theory of the motions of planetary bodies, which held little in common with his actual discovery of the orbit of Ceres, as he would himself admit.
From the above, along with several other similar accounts, we know that Gauss made crucial discoveries, which were underlain by a scientific method that differed from that of his contemporaries. How did he make these discoveries, and what was his method? In order to come at this question from the top, it is necessary to get inside Gauss's mind during the period leading up to his discovery of the orbit of Ceres. A glance at all of Gauss's work, previous to his discovery of the asteroid orbits, reveals his first love in science: Arithmetic.
The Pythagoreans recognized that no science exists that is not based on the notion of Number. One conceptualization of the Pythagorean Quadrivium, founded each of the four studies upon the notion of Number. Arithmetic is the study of Number, Geometry is the study of Number in space, Music is the study of number in time, and Astronomy is the study of Number in both space and time.
What, then, is Arithmetic exactly? Here is what Gauss says in his first book, published in 1801, the "Disquisitiones Arithmeticae":
“Just as we include under the heading Analysis all discussion that involves quantity, so integers (and fractions in so far as they are determined by integers) constitute the proper object of Arithmetic. However what is commonly called Arithmetic hardly extends beyond the art of enumerating and calculating (i.e. expressing numbers by suitable symbols, for example by a decimal representation, and carrying out arithmetic operations). It often includes some subjects which certainly do not pertain to Arithmetic (like the theory of logarithms) and others which are common to all quantities. As a result it seems proper to call this subject Elementary Arithmetic and to distinguish from it Higher Arithmetic, which includes all general inquiries about properties special to integers. We consider only Higher Arithmetic in the present volume.”
What Gauss means by Higher Arithmetic, is not what modern mathematicians call Number Theory. The assumptions of this so-called "Pure Mathematics" are founded on the axioms of logical deduction. Each new theorem must be able to be derived, logically, from a previous set of theorems, going all the way back to a foundation network of axioms, postulates, and definitions. Each new theorem, therefore, must also be logically consistent with the others. The mathematicians of today feel that their job amounts to deducing theorems, all day long, from those axioms and related theorems.1
The validity of the entire system thus depends upon the validity of the initial set of axioms, postulates, and definitions, and also upon the consistency of the rest of the theorems premised on those assumptions. A single real event or phenomenon that contradicts any part of the network of theorems and assumptions, renders the entire complex flawed. Hence to avoid such potential calamities of their precious theorem lattice, modern mathematicians avoid any situations where paradoxes from experimental evidence could enter into their considerations. Any new discovery which contradicts the old theory will only be accepted if it can be stated in the language of/ shown to be logically consistent with the latter!
This is contrary to the method of Gauss, who was constantly changing his hypotheses about the fundamental nature of not only number, but of the Universe itself. For Gauss, mathematics was not a collection of theorems derived logically from a set of assumptions, but was an experimental domain. In his investigations of Higher Arithmetic, Gauss constantly invented new forms of mathematics. He did not develop logically consistent theorems, and then discard the physical phenomena that contradicted them, but subjected Number itself to the standards of physical experiment.
This world of Gauss, however, was quite secret after his discovery of the orbit of Ceres. A peek inside his personal journal during his student days, reveals that his attention was focused on Higher Arithmetic. Every entry notes a new personal discovery in Higher Arithmetic, including his division of the circle into 17 parts, his four first proofs of the law of Quadratic Reciprocity, his initial investigations of elliptical functions and the arithmetic-geometric mean, and his development of the Complex Domain. Out of nowhere, the entries abruptly end in 1801, with a seemingly out-of-place note of Gauss's discovery of a new method of investigating the elements of a planetary orbit. At that point, Gauss had made the decision to go underground, and only publish work scrubbed clean of all remnants of that method of discovery which rendered him dangerous.
How is the young, defiant scientist of today supposed to decipher the revolutionary discoveries of Gauss, which laid the foundations for the work done by Dirichlet, Riemann, Vernadsky, Einstein, and LaRouche? In the following pages, we have prepared a broad sweep through the early (and not so early) arithmetic work of Gauss. With a happily insolent attitude toward modern dogmas, and some patience, the young fighting scientist will come to recognize a kindred mind in Gauss, who did not hold arithmetic to be "pure" mathematics, but a core domain of physics. Therefore, prepare to throw some axioms overboard!