The [ ] unit in combination with angle-brackets creates a cacophony of non-numeric structures including the exotic and indeterminate expressions of conventional math such as negative infinity, infinitesimal, divide-by-zero, square-root-of-negative-one and logarithm-of-negative-one. Volume III retreats to the uninterpreted James arithmetic to examine the forms that correspond to these exotic expressions, viewing the development of theorems as design choices that come with both strong and weak points. This quasi-mathematical approach is thus a hybrid of rigor and realism. Volume III also focuses on one particular form, [<( )>], with very unusual properties.
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Bricken, Iconic Arithmetic Volume I.
The three challenges to which a modern doctrine of number must address itself are those of the infinite, of Zero, and of the absence of any grounding by the One.
The ellipsis, ..., is finite, there is no intention to introduce infinite collections. However, at least with regard to countable infinities, Whitehead and Russell say that
> Cardinal arithmetic is usually conceived in connection with finite numbers, but its general laws hold equally for infinite numbers, and are most easily proved without any mention of the distinction between finite and infinite.
A. Whitehead & B. Russell (1910) Principia Mathematica Preface p.vi.
[…] When sets became philosophically popular about 1910–1920, fusions were abandoned. The motivation, perhaps, was that putting-replicas-together worked well for whole numbers, but not for everything in mathematics. Here, fortunately, we are only trying to understand common numbers, so we’ll avoid (at least until later) the bewildering consequences of using infinite sets to describe counting, or for that matter, any cognitive act.
[…] Here we have used forms that are themselves not rational (i.e. [2]) to describe the transcendental number π. This makes our interpretation of container-based forms inherently more expressive than algebraic expressions from the start. However, the physical semantics of James forms is not commensurable with real numbers defined solely by infinite series.
⇒ The Cut