There is a philosophy of mathematics and a Science on the theory of Numbers that has discussed this question and de-constructed our notions of Number, and lain bare the deficiencies of our Understanding on this matter better than we can hope to do so here. We remind the Student however, that in Mathematics itself, even a thing so simple as Number is defined according to its scope and needs. Thus in Geometry, a number represents a Length or a Ratio or a Quantity. In Arithmetic, it is always an identity; in Calculus, the limit of a function. In mathematical and philosophic theory, every number is infinite in itself and contains within it the possibility for engendering all other Identities by combining with other Numbers. Even the transcendental numbers which cannot be expressed finitely, can be written down as the operation between two numbers to some degree.1 Thus ∏, though incomprehensible as a string of numbers, is simply the circumference of the Circle divided by its diameter and expressible as a simple ratio of two quantities. Though the result is mathematically, a fiction; we allow it into our calculations, for we witness Circles in Nature and know this ratio to exist in Reality, though it does not in our current Field of integral Number.2
For our purposes, we are not so much concerned with Number as we are with the subset of positive integers and their prime factorizations. There are a number of reasons for this:
- First, all Number (except the irrational numbers) can be expressed as the operation of one integer upon another; and
- Every integer can be expressed as a finite product of primes.3
We note that, regardless of what number is or how we may define it, we represent these identities through a series of glyphs, and that every integer is in itself a Glyph. The set of glyphs is in fact the Arabic Numerals which we use here in the West, namely the set {0, 1, 2,..., 9}, consisting of 10 elements including the null element (0). Each of these elements is a glyph in itself, an irreducible identity which all other Integers are a composites of, both as mathematical expressions of quantities and as glyphs. Thus, though one-hundred twenty-three is a number in itself, as a glyph, we represent it as composite of three glyphs in precise sequence 1-2-3 or 123. This set of 10 glyphs {0, 1, . . . , 9} spans the integer space, is the basis for the modern decimal system and constitutes the Spheres of Manifestation in increasing degrees as exhibited in the Sephiras of the Tree of Life.
Before discussing the properties and definitions for the glyph set, we examine first its operation and modes of use.
1 This is not the case with irrational numbers such as √2 and the transcendental numbers (pi, e, and i).
2 There are an infinite number of transcendental numbers like pi. We know of only very few however. The discovery of these numbers correspond to the revolution of the Aeons and are the single most important discoveries of our entire History. Without pi, the physical world as we have shaped with architectural structures and machinery would not be possible. Physics and the sciences as they relate to change on the field of time would not be possible without e or the exponential number; and modern electronics and physics would not be possible without the discovery of i. There are an infinite variety of such transcendental numbers, though it remains today extremely difficult to prove that a number transcendental.
3 The proof is elementary: A number is either (a) prime or (b) composite.
(a) If it is prime, it’s only factors are itself and 1, both prime, and so QED.
(b) If it is not prime, then it is composite. We argue by contrapositive:
If not all numbers can be expressed as a finite product of primes then there exists a non-empty set S of all such numbers. We define:
Let S be the set of all number not expressible as a finite product of primes.
Furthermore, because the set S is non-empty, there exists at least one m such that m is an element of S. And of all the elements we can select the smallest element of S. (If S consists of one element only, we take this element as m). We define:
Let m be the least element in the set of S.
Now because m belongs to S, so it cannot be prime, otherwise it would be expressible as a product of 1 and itself, both primes. So m is composite. That is m = q x r, for some q, r. Now because, q and r divide m, they are both less than m (q, r < m). Because q and r are smaller than m, they cannot be in the set S, seeing as m is the least element (the smallest element) in S and there are no smaller numbers in it. Thus, q and r are both expressible as a finite product of primes, and r = r1 x r2 x... x rn and q=q1 x q2 x... x qs with pi and qi all prime, so that m= q x r = r1 x r2 x... x rn x q1 x q2 x... x qsall prime so that m itself is a finite product of primes, which contradicts our definition of m. Hence m cannot exist and the set S of all integers not expressible as a finite product of primes is empty. QED.
This argument actually constitutes the Euclidean proof for the prime factorization of integers. In fact, all modern cryptography and IT encryption algorithms depend upon this simple result.