Riemann for Anti-Dummies: Part 19 : The Known is Only a Special Case of the Unknown

The Know is Only a Special Case of the Unknown

On June 10, 1854 Bernhard Riemann presented his now famous Habilitation Lecture, “On the Hypotheses that lie at the Foundation of Geometry”, to the faculty of Gottingen University. To begin to comprehend Riemann’s revolutionary address, imagine yourself in the audience, looking over the shoulder of Carl Gauss who had chosen the topic from among three proposals submitted by his student. Think, if you can, of what Gauss was thinking as the 28 year old Riemann stated his intention to lift the darkness concerning the basic assumptions of geometry. “One sees neither whether and in how far their connection is necessary; nor a priori whether it is possible.”

“From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor by the philosophers who have labored upon it. The reason of this lay, perhaps, in the fact that the general concept of multiply extended magnitudes, in which spatial magnitudes are comprehended, has not been elaborated at all. Accordingly, I have proposed to myself at first the problem of constructing the concept of a multiply extended magnitude out of a general notions of quantity. From this it will result that a multiply extended magnitude is susceptible of various metric relations and that space, accordingly, constitutes only a particular case of a multiply extended magnitude. A necessary sequel of this is that the propositions of geometry are not derivable from general concepts of quantity, but that those properties by which space is distinguished from other conceivable triply extended magnitudes can be gathered only from experience. There arises from this the problem of searching out the simplest facts by which the metric relations of space can be determined, a problem which in the nature of things is not quite definite; for several systems of simple facts can be stated which would suffice for determining the metric relations of space; the most important for present purposes is that laid down for foundations by Euclid. These facts are, like all facts, not necessary but of a merely empirical certainty; they are hypotheses; one may therefore inquire into their probability, which is a truly very great within the bounds of observation, and thereafter decide concerning the admissibility of protracting them outside the limits of observation, not only toward the immeasurable large, but also toward the immeasurably small.”

Riemann’s words echoed the thoughts of Gauss, who, more than 60 years earlier, had been prompted by his teacher, A.G. Kaestner, to reject the assumption that the characteristics of Euclidean geometry were true. Gauss responded to Kaestner’s provocation by embarking on a life-long effort to free science from the Aristotelean straight-jacket of a priori assumptions about physical space. While, it formed the core of his thinking, and the basis for his discoveries in physical science, Gauss only published a few “hints” on the subject, to which Riemann referred in the opening of his lecture.

Nevertheless, Gauss’ efforts to develop a “general concept of multiply-extended magnitudes” were already evident in his earliest discovery of the division of the circle.(fn.1) There, Gauss showed that the divisions of the circle that were “constructable”, that is formed by magnitudes commensurable with the diameter of the circle, or its square, and therefore, “knowable”, were only a special case of those divisions that were based magnitudes that were “unknowable”. These “unknowable” magnitudes were only “unknowable” from the standpoint of the circle itself . They were not generated by the circle, but by a higher principle, of which the circle was only a reflection. But, since those higher principles were not perceivable by the senses, they could only be “known” by magnitudes that were “unknowable”. The “unknown” was no less real than the “known”, but could not be measured by the “known”. Instead of thinking of the “known” as real, and the “unknown” as imaginary, Gauss, considered the “unknown” primary, and the “known”, as only a special case.

This required the creation of a new set of metaphors, by which we can represent to the mind, a precise concept of the “unknown”, or , as Riemann would call it, “a general concept of multiply-extended magnitude”, the first hints of which were supplied by Gauss.

The first hint to which Riemann referred was Gauss’ “Second Treatise on Bi-quadratic Residues”. While it was not published until 1832, Gauss had worked out its essential concepts during the period of the writing of his “Disquisitiones Arithmeticae” and during the early phases of his investigations in astronomy and geodesy. It may at first seem odd that concepts concerning the relationships among whole numbers could somehow be related to discoveries in physics, but that is only because the accustomed way of thinking these days is so infected with Aristotelean-Bogomilism. For Gauss, the paradoxes that arise in the mind when it contemplates itself are necessarily congruent to the paradoxes that arise in the mind when it investigates the world outside itself. Take the case of whole numbers. It is obvious that whole numbers do not arise in the mind from counting things. Rather, as Cusa says in “On Conjectures”:

“The essence of number is therefore the prime exemplar of the mind…. In that we conjecture symbolically from the rational numbers of our mind in respect to the real ineffable numbers of the Divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world.”

Thus, investigations of the relations among numbers will, as Plato says in the Republic, “Lead the mind from becoming to being.” It is from this standpoint that Gauss developed the means to investigate the “metaphysics of the theory of space” through the relationships of whole numbers, a sort of experimental approach to numbers.

In order to grasp this, the reader will have to work through some basic principles from Gauss’ “Disquisitiones Arithmeticae” which will lead into the relevant sections of “Bi-Quadratic Residues.” This will be the basis for some fun work that will unfold in the next succeeding installments.

1. See Gauss’ work on the division of the circle, “Riemann for Anti-Dummies” Parts 11 and 12.