Unification

As a type of unit, [ ] must have different additive properties than ( ), otherwise the two would be indistinguishable.

Round-brackets provide a Tally via Accumulation, which explicitly forbids deleting replicated units. In contrast, [ ] dominates rather than accumulates. We can express this as an iconic equation, one that reduces apparent multiplicity to unity. The Unification axiom defines the Behavior of Square-Brackets.

Square-Brackets are blind to multiplicity, making them clearly non-numeric. Strictly speaking, [ ] does not absorb the forms within its environment (no forms in the same space interact). More accurately, [ ] completely fills its container, so that the outer container can support no other contents. From the perspective of the outer container, Unification asserts that there is only one empty square-bracket whereas Accumulation asserts that every round-bracket makes a difference.

Unification provides an interpretation of conventionally extreme numeric cases, particularly when 0 is implicated. It is the interpretation of [ ] that may be cause for concern, because we immediately encounter forms that cannot be interpreted as numbers. Our current interpretation reads square-brackets as taking a logarithm. An empty square-bracket represents the logarithm of zero, but what is log 0?

BRICKEN, William, 2019. Iconic Arithmetic Volume I: The Design of Mathematics for Human Understanding. Unary press. ISBN 978-1-73248-513-6, p. 175–176.