Defining progressive abstraction and the associated challenges of comprehension

Dynamics of Symmetry Group Theorizing (Part #7)

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• education itself might be understood as the progressive understanding of abstraction, itself framed in an academic context in terms of "degrees"
• in other contexts, levels and degrees of comprehension and insight are commonly recognized, with creative breakthrough notably described metaphorically as being "born again", as reviewed elsewhere (Varieties of Rebirth: distinguishing ways of being "born again", 2004) -- abstraction as "rebirth" into a new space?

It would appear that neither mathematics, nor computing, nor education has as yet clarified the shifts in understanding associated with shifts in levels of abstraction. Even the tantalizing title of the study by J. Mason (Mathematical Abstraction as a Result of a Delicate Shift of Attention. For the Learning of Mathematics, 1989) seems to imply only a single shift in attention -- rather than either a linear series of such shifts, or possibly a non-linear progressive comprehension of a system (as hypothesized in Authentic Grokking: Emergence of Homo conjugens, 2003).

And yet the history of mathematical discovery with regard to symmetry suggests that a succession of assumptions is made that successively challenges:

• constraints on the known levels of "abstraction"
• unquestioned conventional "perspectives"
• misplaced "concreteness" and the nature of "objects"
• the nature of invariance

Despite the reservations regarding psychological recapitulation, that history of discovery might therefore be understood as itself mapping or encoding the successive levels of abstraction -- and the shifts in perspective necessary to appreciate the underlying and higher orders of symmetry to be discovered. In that sense, as argued above, the ontogeny of the student's comprehension may indeed recapitulate the phylogeny of the discipline of mathematics -- whether the beauty to be discovered "exists" or is a "taste" acquired through education of the suitably predisposed student's "palette".

The "levels of abstraction" have effectively been explored to a greater degree through the term "levels of reality" on which there is an extensive literature, notably the work of physicist Basarab Nicolescu (Transdisciplinarity and Complexity: Levels of Reality as Source of Indeterminacy, 2000; Levels of Reality and the Sacred, 2002) and John van Breda (Exploring Non-Reductionism and Levels of Reality: on the importance of the non-separability of discontinuity and continuity of the different levels of reality, 2008). 

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