Dynamics of Symmetry Group Theorizing

Exploration of comprehension of symmetry and its psycho-social implications.

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Dynamics of Symmetry Group Theorizing
Significance of "explanation"
Progressive comprehension
Psychological recapitulation of historical development of mathematics
Knowledge and ignorance
Levels of abstraction
Defining progressive abstraction and the associated challenges of comprehension
Framing the mapping process
Configuration of differences
Identification with what is described
Symmetry as a strange attractor
Description, indication, depiction and form
Potential contribution of symmetry group theory
Self-reflexive non-conclusion
Possible questions for future symmetry-related exploration
Accommodation to constraints in symmetry comprehension: building your own "home"?
References

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your own "home"?

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Introduction

This is an exploration of partial or inhibited comprehension of insights that are said to be of the most profound significance. Unfortunately the language in which the insights are commonly expressed is one which I find essentially alienating. The formalization considered essential to articulation of those insights is precisely what inhibits my engagement -- despite a degree of intuitive sense of the potential meaning and its value for me.

To be clear, although I have studied mathematics through three years of university and although I have since written a set of papers on the potential significance of different branches of mathematics, the tantalizing set of insights (which are a continuing attractor) continues to be elusive.

A fundamental reason for my inhibited comprehension is that I am not satisfied by explication through formalization -- however much I respect such language, notably through decades of computer programming. My comprehension of the necessary formal operations, whether incomplete or adequate for a specific purpose, does not provide a psychological sense of completeness nor does it enhance my sense of what such completeness might be -- as I intuit that it might. From this perspective a "proof" is clearly formally adequate within mathematics as commonly understood but yet fails qualitatively to constitute the satisfier that seems possible.

None of this can be construed as a criticism of mathematics or of the explanatory power of mathematicians. It has much to do with my own intellectual inadequacy and the process of my mathematical education. Having attended some 10 schools in different countries it could be argued that this undermined a degree of continuity which might have brought the desired insights into focus on an appropriate foundation -- but then I would never have engaged in all the other activities for which I believe that mathematics has some as yet unrealized relevance.

The following is therefore an exploration of symmetry group theory as elegantly presented by Marcus du Sautoy (Finding Moonshine: a mathematician's journey through symmetry, 2008). This follows an earlier exploration of a related journey by Mark Ronan (Symmetry and the Monster: one of the greatest quests of mathematics, 2006) which I described -- according to my understanding -- in two complementary papers (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007; Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007).

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