Xenotation (#1)¶
From Euclid’s Fundamental Theorem of Arithmetic (FTA), or unique prime factorization theorem, we know that any natural number greater than one that is not itself prime can be uniquely identified as a product of primes. The decomposition of a number into (one or more) primes is its canonical representation or standard form.
Through the FTA, arithmetic attains the cultural absolute. Number is comprehended beyond all traditional contingency, as it exists for any competent intelligence whatsoever, human, alien, technological, or yet unimagined. We encounter the basic semantics of the Outside (comprehending all possible codes).
Insofar as numerical notation is constructed in a way that is extraneous to the FTA, we remain Greek. Our number signs fall lamentably short of our arithmetical insight, stammering deep patterns in a rough, ill-formed tongue. Stubbornly and inflexibly, we translate Number into terms that we know deform it, as if its true language was of no interest to us.
Yet, given only the FTA, the code of the Outside — or Xenotation — is readily accessible. Nothing is required except compliance with abstract reality.
A single operation suffices to count. In words, it matters little what we call it — implexion, envelopment, wrapping, or bracketing describe it with increasing vulgarity. For convenience, parenthesis — ‘( )’ — provides a sign. The semiotic (or purely formal) equation ‘( ) = 0’ offers additional economy. Xenotation needs nothing more.
One is redundant to the FTA. It begins with two, the first prime. This introduces our sole notational principle, and operation.
Every number has an ordinality and a cardinality (an index and a
magnitude). Crudely represented, through a mixture of barbarous signs,
we can see these twin aspects as they are relevant here:
First (Prime =) 2
Second (P =) 3
Third (P =) 5
Fourth (P =) 7
By wrapping an ordinate (or index), itself a number, the Xenotation marks a magnitude. So ‘(first)’ or ‘(1)’ = 2. One, we know, is superfluous, and thus economized: (1) = ( ) = 0. Remembering that ‘0’ is henceforth the sign for the initial implexion, and not the familiar (though cryptic) numeral, we can now depart from all notational tradition. [The further usage of decimal numerals, in hard brackets, will be strictly explanatory, and dispensable.]
An implexion signifies the number designated by the enclosed index. Once this rule is understood, Xenotation unfolds automatically.
0 [= 2]
(0) [= 3, the second prime]
((0)) [= 5, the third prime]
(((0))) [= 11, the fifth prime]
Compound numbers are signified in accordance with the FTA:
00 [= 2 x 2 = 4]
000 [= 2 x 2 x 2 = 8]
(0)0 [= 3 x 2 = 6]
((0))(0) [= 5 x 3 = 15]
For primes with compound indices, the procedure is unchanged:
(00) [= 7, the fourth (2 x 2) prime]
((0)0) [= 13, the sixth (3 x 2) prime]
((0)(0)) [= 23, the ninth (3 x 3) prime]
So the xenotated Naturals [from 2-31] proceed:
0, (0), 00, ((0)), (0)0, (00), 000, (0)(0), ((0))0, (((0))), (0)00, ((0)0), (00)0, ((0))(0), 0000, ((00)), (0)(0)0, (000), ((0))00, (00)(0), (((0)))0, ((0)(0)), (0)000, ((0))((0)), ((0)0)0, (0)(0)(0), (00)00, (((0))0), ((0))(0)0, ((((0)))) …
[That’s probably more than enough for now]