Riemann for Anti-Dummies: Part 27 : Gauss’ Declaration of Independence

Riemann for Anti-Dummies Part 27


In September 1798, after three years of self-directed study, C.F. Gauss, then 21 years old, left Goettingen University without a diploma. He returned to his native city of Brunswick to begin the composition of his “Disquisitiones Arithmeticae.” lacking any prospect of employment, he hoped to continue receiving his student stipend. After several months of living on credit, word came from the Duke that the stipend would continue, provided Gauss obtained his doctor of philosophy degree, a task Gauss thought a distraction and wished to postpone.

Nevertheless, he took the opportunity to produce a virtual declaration of independence from the stifling world of deductive mathematics, in the form of a written thesis submitted to the faculty of the University of Helmstedt, on a new proof of the fundamental theorem of algebra. Within months, he was granted his doctorate without even being required to appear for oral examination.

Describing his intention to his former classmate, Wolfgang Bolyai, Gauss wrote, “The title indicates quite definitely the purpose of the essay; only about a third of the whole, nevertheless, is used for this purpose, the remainder contains chiefly the history and a critique of works on the same subject by other mathematicians (viz. d’Alembert, Bougainville, Euler, de Foncenex, Lagrange, and the encyclopedists … which latter, however, will probably not be much pleased) besides many and varied comments on the shallowness which is so dominant in our present-day mathematics.”

In essence, Gauss was defending, and extending, a principle, that goes back to Plato, in which only physical action, not arbitrary assumptions, defines our notion of magnitude. Like Plato, Gauss also recognized it were not sufficient to simply state his discovery, without a polemical attack on the Aristotelean falsehoods that had become so popular among his contemporaries.

Looking back on his dissertation 50 years later, Gauss said, “The demonstration is presented using expressions borrowed from the geometry of position, for in this way, the greatest acuity and simplicity is obtained. Fundamentally, the essential content of the entire argument belongs to a higher domain, independent from space, (i.e., anti-Euclidean) in which abstract general concepts of magnitudes, are investigated as combinations of magnitudes connected by continuity, a domain, which, at present, is poorly developed, and in which one cannot move without the use of language borrowed from spatial images.”

It is the intention of this installment to provide a summary sketch of the history of this conception, and Gauss’ development of it. Because of the difficulties of this medium, it can not be exhaustive. Rather, it seeks to outline the steps which should form the basis for extended oral pedagogical dialogues, such as is already underway in various locations.

Multiply-Extended Magnitude

A physical concept of magnitude was already fully developed by those circles associated with Plato, expressed most explicitly in the Meno, Theatetus, and Timaeus dialogues. Plato and his circle demonstrated this concept, pedagogically, through the paradoxes that arise when considering the uniqueness of the five regular solids, and the related problems of doubling a line, square, and cube. As Plato emphasized, each species of action, generated a different species of magnitude. He denoted such magnitudes by the Greek term, “dunamais”, a term akin to Leibniz’ use of the word “kraft”, translated into English as “power”. That is, a linear magnitude has the “power” to double a line, while only a magnitude of a different species has the “power” to double the square, and a still different species has the “power” to double a cube. (See figures 1a, 1b and 1c). In Riemann’s language, these magnitudes are called, respectively, simply, doubly, and triply extended. Plato’s circle emphasized that magnitudes of lesser extension lacked the capacity to generate magnitudes of higher extension, creating, conceptually, a succession of “higher powers”.