The Identity of Unity, Nullity and Infinity

The basis of Gaussian number theory rests on his use of "relative manifolds", also sometimes referred to as "cyclotomic fields" and "rings". The use of these relative manifolds is ever-present in his Disquisitiones Arithmeticae.

Relative manifolds replace the number line as the representative environment upon which we study numerical relationships. On a number line, each end of the line extends to "infinity", or "negative infinity", and each "number" is assigned to a "point" on the line. The series of whole positive and negative integers is assigned to each point extending to the right and to the left in equally spaced intervals. All relations which exist between these points are included in the line. That is to say, if one were to draw a line to represent the difference between 0 and -3, that line would be a section of the number line in its totality.

A relative manifold, by contrast, is represented by a circle. The circle is divided into multiple equal parts. The number of parts is determined by / determines the "mode" of the manifold. This is also called "modulus". So, a relative manifold of mode 5 has 5 possible locations to which numbers may be assigned. The series of whole integers, when assigned to these locations, "wraps around" the manifold until it reaches its origin and repeats its steps. So, in a manifold of 5, the "number" 7 and the "number" 2 are equivalent. They are identical.

To understand the implications of this arrangement, one must shed one's notion of "number" as "quantity", and replace it with a notion of "number" as a relationship to unity, and nothing more. In our example above, 7 and 2 are identical because they express the exact same relationship to the completion of the circle, which is an aspect of unity. In a relative manifold of 5, there exist but 5 possible relationships to the unity which is the completion of the circle of mode 5. The illustration below shows an example of these 5 possible relations:

The larger circle (radius 5) is evenly divided into 5 parts by the rotations of the smaller unit circle (radius 1). It is possible to divide a circle into n parts by rotating a unit circle inside of a circle of radius n. So, 7, 17, 87, 887 divisions are possible. Below is an example of a relative manifold of 17; a circle divided into 17 parts by a unit circle (scaled to the size of the preceding illustration).

Carl Friedrich Gauss determined the method for accomplishing this feat when he was 18 years old. His discovery of the 17-gon is intrinsic to his entire numeric theory. One simply cannot understand "number" as Gauss did without understanding his concept of the relative manifold and relative circular rotation.

Returning to our manifold of 5, we find that every numerical relation in existence, from 0 to infinity, must necessarily be represented as one of the 5 possible points on the circumference of the manifold. Notably, 0 and 5 are identical. Gauss would call this identity "congruence" throughout his entire Disquisitiones Arithmeticae. The relative manifold serves to bind and contain the scope of possibility for expressions of numerical relation. One might say accurately that a relative manifold is a "geometric contraction of infinite possibility to finite relation". This contraction makes it possible for a human analyst to perceive patterns in relation which were invisible on the number line.

For instance, 4, 9 and -1 are necessarily the same "number" relative to a manifold of 5. Because this is true, the "square root of -1", 2, -3, 3 and 2 are all different ways to express the exact same numerical relationship within this manifold. Gauss sufficiently proves and explains this truth in the first two chapters of his Disquisitiones Arithmeticae, and makes use of a trigonometrized version of this truth in his 1799 dissertation on the fundamental theorem of algebra, wherein which he produced the world's first complete proof of said theorem, without resorting to the use of "imaginary" numbers.

No such relationship can be represented on the number line. In fact, there is no distinction between a number line, and an arc segment of a relative manifold of modulus infinity. This statement is a paradox, because relationships to the infinite are not possible. The infinite exists prior to the existence of relationships, and is a necessary precursor to the possibility of relationship itself, so to call something an arc segment of a relative manifold of modulus infinity is to say nothing.

You may liken the statement to the concept of looking through a magnifying glass which is infinitely large. An infinitely large magnifying glass could not in fact magnify whatever one was looking through it at.

This concept of infinity as beyond relation is drawn directly from Cardinal Nicholaus of Cusa's "De Docta Ignorantia", which has been translated and published by the University of Minnesota's own Jasper Hopkins, with whom I've recently had the privilege of sharing an excellent discussion on the subject. Gauss' cyclotomic number theory is a worthy illustration of Cusa's conception of the infinite, for the following reasons:

The "absolute minimum" and the "absolute maximum" in a given relative manifold share an identity. Relative to a manifold of 5, the "absolute minimum" is 0, and the "absolute maximum" is 5, both of which are identical. In addition to these truths, the completion of the perimeter circle is the completion of 1 rotation, which is an expression of unity. Furthermore, the nature of "modulus 5" itself springs forth only as a relation between the circle of radius 5 and a unit circle, and so the numbers 1, 2, 3, 4 and 5 on the perimeter are themselves each a record of the completion of a unit circle in its rotation on the perimeter of the larger circle of modulus 5, so they each also share in unity. On the unit circle itself, 0 and 1 share an identity.

It is not possibly the case that infinity relative to modulus 5 could be anything other than 5, or 0, each of which record the completion of the circle of 5, and so are each identical with the unity of that circle. I say this because relative to a numeric relational mode of 5, the "numbers" 4, 3, 2 and 1 each admit degrees of greater and lesser completion. The notion of the Infinite is precisely that notion which does not admit a greater or a lesser, for it is superlative. It is therefore necessarily the case that the relations to unity conferred by 0 and 5 share their identity with the infinite.

5 then, relative to a numerical mode of 5, can be accurately said to represent the infinite.

However, the question of unity persists and must be addressed, for relative to a modulus of 5 we have defined two different instances of unity--that represented by each of the 5 units along the perimeter of the circle of 5, and the completion of that perimeter in 0, 5 or Infinite. How is it possible that unity may coexist on relative manifolds of differing magnitude?

That question is a paradox, because we are not studying conceptions of magnitude, but only those of relation. Magnitude is a secondary concept which itself may not exist prior to conceptions of relation, and it is possible to study relation independent of magnitude. The concept of unity, however, exists prior to concepts of relation. No relation is possible prior to unity.

To illustrate this point, I ask that the reader perform the following exercise:

Draw a circle of two.

The clever reader will immediately ask "two of what?" Herein lies my point. One may not create a circle of a given number except as a relation to a unit circle. Quantitative concepts, such as the assertion that 2 is "more" than 1 and so must of course require the lesser quantity to create the greater, have no bearing on this exercise, either. To illustrate this, I ask that the reader draw a circle of one-half.

Quantitatively, "one-half" is "less" than what might be mistaken for the "number one", and one may assert that "one is not possible without one-half". This is not a meaninful concept of number, however. Relationally, "one-half" may not exist prior to a concept of unity. One may not draw a circle of "one-half" until one has a firm notion of unity. Because both "two" and "one-half" may only exist as a relation to unity, we can state reasonably that unity precedes all numerical conceptions.

And so, the "unity" of a manifold of modulus "five" exists before it becomes a manifold of "five". The concept of "five" is superimposed upon the unity of the manifold by establishing a differential numerical relationship with a second manifold, to which we assign the role of the "unit". The relation of this "unit" circle to the circle of "five" is expressed by the rotation of the unit circle along the interior circumference of the circle of "five".

Similarly, were we to rotate the circle of "five" on the interior of a circle with a radius of "twenty-five", it would divide the latter into five equal parts.

The circle of "twenty-five", then, shares the relationship to the circle of "five" that the circle of "five" has with the "unit" circle. In this case, the "number five" on the circle of "five" marks "one-fifth" of the circle of twenty five. The completion of the perimeter of "five" is in this sense a type of "one" relative to the manifold of modulus twenty five, even though the unit circle is also a type of "one".

One exists, then, as the completion of the unit circle, the unit of the circle of five, as the completion of the circle of five, and as the unit of the circle of twenty-five. In all of these cases it represents unity, not as a fixed quantity, but as a geometric relationship between manifolds.

In each case, unity is not a mere number, but unity creates the potential for meaningful numerical relationship. In each case, unity shares its identity with 0 or nullity, which is an aspect of unity that describes a polarity of relationship with a unity of a lower degree (such as the relationship between the circle of 5 and the unit circle, or the relationship between the circle of 25 and the circle of 5). In each case, unity shares its identity with infinity, which expresses the completion of all relationships and the fulfillment of the mode of relation in each numerical manifold.

In each possible relative manifold, zero or nullity does not represent "nothing", but an absolute minimum bounds of possible relation with the completion of a unit manifold of lower degree, indeed the promise of such a relationship. In each possible relative manifold, nullity expresses the absolute maximum bounds of possible relation relative to a unit manifold of lower degree, indeed the completion of that numerical relationship. In each possible relative manifold, nullity expresses the completion of possibility and so shares its identity with infinity.

In each possible relative manifold, infinity expresses the exhaustion of possible relationships, and indeed the possibility for all possible numerical relationships. It is, as absolute minimum and absolute maximum, bound by the circle and by the types of numerical relation conferred to the circular manifold by its predetermined relationship to unity. It shares its possibility and its identity with nullity and unity.

Through the lens of a relative manifold, nothing does not exist, infinity is both relative and beyond relation, and unity can be found both in the whole and in each of the subordinate parts. Through the lens of a relative manifold, the three concepts of unity, nullity and infinity are aspects of a single greater truth.

What are the implications of these truths on philosophy, theology, economy, mathematics, physics, music, or art? The answer to this question is the subject of my continuous pursuit in each of the above fields of research.

Special thanks to Jasper Hopkins and Arthur A. Clarke for contracting the Cusa and Gauss texts into English, so that future generations may tease out the implications of the ideas of these two brilliant minds.