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Configuration of differences
Dynamics of Symmetry Group Theorizing (Part #9)
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It is most intriguing that such a fundamental domain of mathematics, symmetry group theory, should be (proudly) held by mathematicians to be irrelevant to the psycho-social challenges of the mundane world -- whilst being upheld as the most competent in identifying and describing the beauty to be found in the universe. This is especially irritating in a world challenged by a need for greater harmony when symmetry group theory is recognized -- through music -- as indeed being fundamental to human understanding of harmony in all its variety. As a musician himself, Marcus du Sautoy stresses this relationship. But why is it that the forms of "harmony" that are the focus of strategic initiatives -- "harmonization" as advocated within the European Union -- are so simplistic in terms of the insights into harmony offered by music and symmetry group theory? Do understandings of "harmony" and "integration" need to be "liberated" as explored elsewhere (Liberation of Integration, Universality and Concord -- through pattern, oscillation, harmony and embodiment, 1980)?
One expression of this frustration is indicated elsewhere with respect to the territorial challenges of the Middle East and other conflicts (And When the Bombing Stops? Territorial conflict as a challenge to mathematicians, 2000). The account of Marcus du Sautoy is explicit in indicating his personal appreciation of the understanding of mathematics in the Jewish and Muslim cultures. But somehow those cultures have proven unable to understand the relevance of their insights into symmetry groups as they might relate to challenges of territorial apportionment -- considered mathematically "trivial" even though the simplistic "solutions" currently promoted give rise to bloody conflict.
Unfortunately mathematicians have primarily contributed their insights to the military applications associated with this lack of understanding and its reinforcement, rather than applying their insights into harmony to reconcile the differences of the parties -- on the assumption, for example, that this might require symmetry of a higher order. It would seem that the military applications are mathematically more interesting -- perhaps precisely because those regarding harmony are explored at too simplistic a level?
It is perhaps the most supreme form of irony that insights into the highest forms of symmetry -- fundamental to the structure of the universe -- emerge from a "group theory" of mathematics which is considered totally distinct from a "group theory" of the social sciences (as in Mary J. Fambrough, The Changing Epistemological Assumptions of Group Theory, The Journal of Applied Behavioral Science, 2006). It is the insights of the latter which are supposedly of relevance to the alleviation of the bloody lack of harmony between different perspectives in a global civilization. This confusion is only too evident in web searches. It matches that described with respect to the "correspondences" considered fundamental to the mathematical "moonshine" conjecture (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007).
In addition to the "social science" focus implied by issues of mathematics education (cf International Commission on Mathematical Instruction; International Group for the Psychology of Mathematics Education), one exception to this lack of connectivity between the two forms of group theory is perhaps to be found in the recognition by the International Society for Group Theory in Cognitive Science that:
Group Theory has emerged as a powerful tool for analyzing cognitive structure. The number of cognitive disciplines using group theory is now enormous. The power of group theory lies in its ability to identify organization, and to express organization in terms of generative actions that structure a space.
A series of relevant studies has been produced by the president of the group, Michael Leyton (Shape as Memory: a geometric theory of architecture, 2006; A Generative Theory of Shape, 2001; Symmetry, Causality, Mind, 1992).
The key question for the following exploration of symmetry group theory is associated with the manner in which mundane "differences" are transformed through the process of abstraction into mathematical objects. Studies of symmetry acquire their power and interest beyond binary relationships, however implicit these may continue to be in the mathematical objects that subsequently emerge. Interest might be said to start with the triangle, as it does in Marcus du Sautoy's account. But a triangle of what?
Abstraction, and the practice of mathematics, requires that any such triangle be a triangle of "points" defining the differences between three related "somethings" -- whose nature is quickly to be forgotten as being irrelevant to the efficacy of subsequent discovery of symmetry. It is assumed that the essence defining the significance of those points is appropriately retained by the process of abstraction. This assumption may be fruitfully challenged as illustrated by the following cases:
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