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nullary

Nullary is a base-less numeric encoding. Unlike unary, which requires O(n) symbols to encode the number n, nullary encoding still requires just O(log(n)) symbols.

Nullary is the foundation of real numerology.

Nullary is a variant of the ‘xenotation’ described by Nick Land in his essays The Tic Xenotation and TX2.

why not use the xenotation

Tic Xenotation has an unnecessary base case.

Xenotation
----------

      : -> 2
     xy -> x * y
    (x) -> the x'th prime

Nullary fixes this with a natural base case.

Nullary
-------
        -> 1
     xy -> x * y
    (x) -> the x'th prime

canonical form

The expressions ()(()) and (())() encode the same value. The number of ways to encode a value is a key property of numbers. Nevertheless, it is useful to have a canonical encoding. The canonical encoding is defined to be the one which places the largest factors first. This means (())() is the canonical form of ()(()). Another way to express this is to say that the canonical form is the first item in a lexical sort if '(' < ')'.

forecast

Mr Land mentions an extension to identify 0 and 1:

((-P)): -> 0
  (-P): -> 1
      : -> 2

The idea is that ‘(-P)’ is an operator that unwraps an expression. In this case, it is applied to ‘:’. This formulation is unsatisfactory for a number of reasons…

For now, work some examples to get familiar with nullary. How do you encode your favorite number?