8. Parallel Numbers Postulate

— Eumetry of Paradox; Zerometry —  

The Number Line
The Unified Line

We think of a number line like this:

… +3 …+2  +1  ±0 1 … –2 … –3 …

Each number on this line an integer, in their midst Zero, central among numbers. We see every positive number has a negative number that parallels it, meaning Zero is his-her own parallel. (A geometrical analogue to zero could be the point and all forms considered analogue to it, each a parallel of any other; the point parallel the circle parallel the sphere.)

Parallel Numbers

Re-considering number as a set of parallels, we could think of the number line like parallel lines, in which case they would be given like this:

 … +1 … +2 … +3 …
 … –1 … –2 … –3 …

Re-presenting the number line in this way we need to find a non-numerical symbol for Zero as Ze — unity generalized — is always one and cannot be considered as either plus or minus, always and everywhere being exactly both. Just like Zero. It’s Platonic.

A-positive and A-negative are distinctly different, radically opposed entities, standing for uniquely non-same ideas, yet in this also being identical in the only way identity is possible in a non-trivial sense.

Re-thinking these integers as not generic integers — cardinal integers representing only themselves — but ordinal integers, each embodying an identity, the positive one corresponding to a negative one and vice versa, each denoting simultaneously a defined and limited area of definition and its precise, radical, opposite. An example might clarify: If a positive amount of apples equates ‘2 apples’, the corresponding negative ordinal integer gives exactly everything else everywhere, eternally. Identical universe minus two apples.

Re-formulating positive-negative in such terms retains the one-to-one correspondence, but provides us with a way in which we can view this correspondence as an identity and a radical opposition at the same time. It does not change the universe, but it might give a way to try out different scenarios.

We may also rethink the number line into a parallel consisting not only of Platonic Zero and Platonic One, but with every number upon it-them “Platonic”. Such a parallelism could be understood in several ways, below a duo of doubles.

Diophantic Parallels
  • Nominative. Positive numbers equal 1 times n.
  • Denominative. Negative numbers equal 1 through n.
  • Additive. Positive numbers equal 1 + n.
  • Subtractive. Negative numbers equal 1 - n.