How Unity Relates to Number
In his De Docta Ignorantia, the German Cardinal Nicholas of Cusa carefully educed the relationship between unity and number as infinite in nature. The idea of an "infinite relationship" is paradoxical in that the two concepts of number and unity are irrelative. Since unity must necessarily exist prior to number, it is superlative and beyond relation to number, the nature of which is relation itself. Unity does not, in fact, relate to anything, since its nature is primary to that of relation.
For example, one could assert that the "relationship" between 2 and 1 is 2. 2 / 1 = 2. Similarly said, 1 * 2 = 2. All numbers describe relationships to other numbers and are bound by unity, but nothing new about them not contained in their identity is expressed or revealed when one compares them to unity. In that sense, one cannot in fact divide or multiply a number by 1. To say otherwise is a subtle sophistry that masks the important distinction between number and unity. For convenience, teachers and mathematicians tell their students that 1 divides 2 twice. But this is a meaningless statement, a vacuuous tautology, a short-circuit that avoids fascinating questions about the nature of multiplicative and divisive relation on an abstract level independent of measure.
For example, if you were to draw two circles of differing indeterminate measure and I were to ask you to "multiply the smaller of the circles by 3 and show me where on the perimeter of the larger circle that value lies, relative to any given starting point", you couldn't solve the problem. You cannot multiply by 3 without first defining a unit circle. Neither could you simply "multiply the larger circle by the smaller" if I asked you to do so, without first determining some unit of measurement to assign each circle.
However, the infinite relationship between unity and number can be expressed geometrically in a manner that reveals a singularity in one of the most elusive geometrical shapes in the history of mathematics--the straight line, which is an arc segment of an infinite circle.
The straight line lies at the heart of the controversy between Euclidean and non-Euclidean or anti-Euclidean geometries. How can one, for example create a cartesian manifold without them? How can one find parallels for curved lines? Euclid's parallel postulate falls short of being able to accurately describe and capture these relations in nature, since we've no basis upon which to physically construct straight lines. We take Euclid's nominalist a priori construction for granted, that a line is the shortest distance between two points, but it isn't necessarily so.
A number of nineteenth century geometricians Bolyai, Lobachevsky and especially Riemann contributed revolutionary advances to geometric science, opening the door for breakthroughs in physics and mechanics in the 20th century.1 They did this by abandoning the irrational portions of Euclid's "Elements", and drafting physical and rational constructions for geometric relationships independent of measure. Riemann's elliptical geometry adopted Gaussian cyclotomic number theory in order to draw the concept of the infinite into a finite contraction.
In this system, a straight line is a singularity or boundary condition in species of curvature. It conforms with an arc segment of an infinite circle. But how do we produce an arc segment of a circle of infinite diameter? Were we to draw it on your screen, wouldn't we need an infinite number of pixels?
That's a silly question. Infinity isn't a number, and cannot be contracted to number. Infinity, unity and even nullity are shared aspects of the same concept, as I've already shown. You can't "divide by infinity" because infinity exists prior to the concept of division itself. Division is wholly reliant on the existence of infinity-unity-nullity for even the possibility of its own existence. As the German Cardinal Nicolaus of Cusa was careful to show, infinity is beyond the scope of the concept of relation itself, and cannot relate to anything other than itself, which would normally be to say nothing, except that infinity, the absolute maximum, is three as well as one. And so the infinite can be shown to relate to unity, in a different manner than 2 relates to 3. It can likewise be shown to relate to nullity, with which it shares identity.
And so if number cannot relate to infinity, are we lost for showing the superlative curve? No. We can exploit the union of the maximally great (infinity) and the maximally small (unity), which are at once the same and yet relate. We can geometrically capture the infinite in a finite number of pixels by inverting the problem. A fascinating aspect of the line of maximal curvature is that it is the same for each species of curvature, whether it be a hypocycloid, a circle, a parabola, a catenary, or any other species of curvature.
Consider this: a curve of the second degree x2, and a curve of the third degree x3 are the same if x == infinity, just as they are the same should x == unity or x == nullity. Within the context of circular relation, then, we can capture the curve by establishing a maximum hypocycloidal curvature. I've done so here:
Here, within the boundaries of possible relation from the enclosing unit circle, we have established the infinitude of the relationship between the first number, which is 2, and the unity that gave it life. The resulting relationship is expressed in a hypocycloidal line of maximum curvature, the arc-segment of the infinite circle. This maximal curvature conveys the singular relation between its own infinite nature and the infinite nature of unity.
And so, the "difference" or "relation" between 1 and 2 is infinite. It is the difference between unity and number.
What are some philosophical implications of this geometric construction? What underlying assumptions about localized spacial curvature must be made for this evidence to hold? Can we invert those assumptions and learn something new about the nature of spatial relation itself? What truths can we educe from this simple and beautiful concept?
We shall investigate these paths on a later occasion.
Comments
The implications of the Gaussian system are quite interesting, given the cyclic nature of the ratio in general (c.f. Euclid, damnit) when working with factors, squares, etc. I'd like to hear your thoughts on the radius of the modular circle (wc?) as given by the radius of the unit circle, and how this initial relationship is necessary for constructing modular circles in general. My interest in the fundamental relationship between unity and the mode is, of course, great, but here particularly involves two questions:
First, the generality of the unit, such that one half in relationship to one is given by "re-configuring" the unit to be one half - that is granted. I suppose I may have been slightly and perhaps not rightly incensed by your emphasis on "one" as the numeric representation of the unit. Surely, one represents unity in interger relationships and perhaps, one could argue by using an inverse relationship, unity in all real numbers.
This is all kind of leading up to a larger question that I have been perusing outside of the constructs of your argument: Primes. The unique relationship between a prime and the unit is fucking fascinating, a ratio within which there is no mean! In order to multiply using ratios (again, c.f. the illustrious Greek, although he never said this -explicitly-) there is required a baseline relationship between unity and a prime number in order to re-set the prime as unity and therefore re-establish the same ratio again, creating a proportion such as 1:2::2:4, or 2:4::4:8. Essentially, the stuff we learned about fractions back in 4th grade or whatever about common denominators. To establish a smaller prime in the Gaussian system, we need only attempt to create a circle of, say, 13 and then find that no other mode is able to fit within the set points of the unit radius along the circumference of the modular circle without extending itself through the unit/mode into another rotation, leaving a remainder. Right? So how do we find bigger ones? Is there a similar way to construct prime modes without said prime already known as a factor of another, larger mode?
The second question is a bit more abstract. Pi, irrational, is the mean term in the ratio of the diameter of a circle to its circumference. Although the traction of the unit circle's radius within the circumference of the modular circle determines the points to which numbers are assigned, to put it coarsely, the relationship between the circumference of the modular circle and the radius of the unit circle -cannot- be determined by the specific relationship of the mode to the unit (1:5, as opposed to 1:7) nor by the relationship of unity to multiplicity in general, about which you have spoken quite well. How, then, do we understand pi as the necessary mean? How do we understand the relationship of the unit radius along the modular circumference -in general-?
Posted by: Margaret Flood | June 21, 2006 07:54 PM