A primary reason for going to all the trouble to learn this new sensual language is to learn new ways of thinking.
It is not the concepts represented by the James language that are multi-dimensional, it is the language itself that has different dialects or “notations” expressed in different dimensions. One implication is that a series of transformations can be animated. Another implication is that many transformations can occur concurrently, all at the same time.
When you stop to consider the rationality of symbolic representation, it becomes clear that Symbols are highly discriminatory against our physical evolutionary heritage. The vast majority of the neurons in our brains are dedicated to managing the Interface of our physical Body with physical Reality. Everybody lives in a body, only a very few of us live in the conceptual fantasy of the Platonic reality associated with mathematics.
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L. Bunt, P. Jones & J. Bedient (1976) The Historical Roots of Elementary Mathematics p.122:
> Most mathematicians accept the modern philosophical ideas that their axioms are logically arbitrary and that their theorems are about mental concepts. These mental concepts cannot be actually observed in the physical world. This view of the nature of mathematics can be traced back to the Greek philosopher Plato.
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Abstraction is of interest to only a small portion of a brain; the skills of abstraction are exceedingly difficult to teach. The symbolic math currently taught in schools expects us to abandon both sensation and experience in favor of unnatural cognitive acts. No wonder students find it difficult to learn this disembodied language.
Boundary languages are visceral. Interpretation will remain constant as the boundary representation is transcribed across dimensions, from 1D strings to 2D icons to 3D architectures to 4D temporal experiences. There is no abstract/concrete dichotomy, so that boundary languages are much easier to understand. No mind/body split, so boundary forms are much easier to tolerate. In contrast, string encoding cannot be experienced, it must be learned via memorization. Consequently string languages remain necessarily cerebral. Mathematical nominalism holds that mathematics is about objects that exist. Container languages provide nominalistic consistency by requiring that concepts too have a manifest Form.
It is a distinct advantage to represent mathematical concepts across many different spatial formats, not only symbolically but also diagrammatically, physically and experientially. The printed page limits representation to symbolic and iconic forms, but by projecting volumetric forms onto paper, we can approximate concrete and experiential languages. The image of a box can elicit imagination of a box. Leibniz: “The best signs are images; and words, insofar as they are adequate, should represent images accurately.” (Leibniz to Tschirnhaus, end of 1679 (Math., IV, 481; Brief., I, 405), as quoted in L. Couturant The Logic of Leibniz, Ch. 4 footnote 93.)
⇒ Dialects
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Bricken, Iconic Arithmetic Volume I, p. 144–145.