Three generic transformations on patterns of containment are taken as axioms. An interpretation of round-brackets as powers and square-brackets as logarithms permits ease of mapping James Forms to conventional arithmetic. When nested, the two types of boundary maintain alternating exponential and logarithmic contexts which permit smooth transition between addition and multiplication.
Volume II and Volume III each contain a significant surprise. In Volume III it is a new imaginary unit, one that is more fundamental than √–1. In Volume II it is the disclosure that the James angle-bracket is only a convenient shorthand abbreviation that allows us to contrast negative and positive and thus remain in familiar cognitive territory where negative numbers are taken to exist. The numeric inverses too are configurations of only round- and square-brackets. In Volume II we’ll see that James algebra has only two independent boundaries that are bonded together by one void-based axiom and one rearrangement axiom. (p. 156)
⇒ 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.
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BRICKEN, William, 2019. Iconic Arithmetic Volume I: The Design of Mathematics for Human Understanding. Unary press. ISBN 978-1-73248-513-6, p. 152, 156.
To complete an operational description of round- and square-brackets, we will need an axiom that specifies how the two boundaries interact in configurations more complex than the double nesting of Inversion. Here is the James axiom of Arrangement. The axiom specifies an invariant structure across frames.