Identification with what is described

Dynamics of Symmetry Group Theorizing (Part #10)

Whilst mathematics may indeed offer a description of the sets of relationships obtained by abstraction (in the above table), the challenge in each case is the identification by individuals (or cultures) with what is described -- when they are implied by the description as in the psycho-social contexts of interest here. In this sense the "points" are points of identification -- which may well be psychologically highly charged. These may be clustered to form a set in which the relationships between the points may also be a focus of identification -- and of identity games (as typical of many social groups, including mathematicians). The distinctions between the points are in this sense one of complementarity essential to the operation of the set as a system -- with which larger framework an individual (as a "point") may identify.

In the right-hand portion of the table, the points find their identity challenged -- leading to competing sets ("them and us") which may well be reframed as distinct "points" in further description. In many situations the left and right-hand portions are to some degree interwoven as in competing ball games or (amicable) debate. This integration may however fail and lead to conflict that is not so contained.

The point to be emphasized is illustrated by the last row of the table. How are the "insights" of symmetry group theory to be understood in terms of a triangle, for example, where (rather than the conventional geometric or algebraic depiction) the "points" are associated cognitively with different "ways of knowing":

either three distinct smells (or tastes, or sounds, etc),
or three distinct senses (such as vision, hearing and smell)

As noted by Marcus du Sautoy, as a triangle, any such triangle has six "symmetries". From what understanding are such symmetries then recognized? Even when the more general term, "understanding", is used (rather than the vision-biased "perspective"), does the use of "from" then inappropriately imply a metaphorical "distance", as well-argued by George Lakoff and Mark Johnson (Metaphors We Live By, 1980)?

Marcus du Sautoy makes the point that a:

...triangle's symmetry was captured by things I could do to it that would leave it looking the same...the number of ways that I could pick up the triangle and put it down so that it fitted back exactly inside its outline on the paper. Each of these moves... was a 'symmetry' of the triangle. So a symmetry was something active, not passive.... symmetry as an action that I could perform on the triangle to replace it inside its outline, rather than some innate property of the triangle itself.... one could think of the total symmetry of an object as all the moves that the mathematicians could make to trick you into thinking that he hadn't touched it at all.

In geometric (and algebraic) terms these "moves" are described as including:

reflection symmetry (or mirror symmetry)
rotational symmetry
translational symmetry: leaving an object invariant under a discrete or continuous group of translations

Such symmetries, and others, may be combined such that for a given form (like a triangle) the set of combinations constitutes a "group" -- hence the term "symmetry group theory".

Clearly the question with respect to the example of "ways of knowing" is how to understand possible analogues to these "moves", especially when there is some form of psycho-social identification with "points" configured as a triangle, for example -- whether the identification is by a person, a group, a discipline, a "school of thought", a belief system, or otherwise. The challenge in these cases has much to do with the recognition of invariance despite a range of "moves" -- notably interesting in the case of ideological groups. How is the sameness of a behvioural pattern to be recognized underlying such "moves"?

The challenge to the conventional, and supposedly dominant, mode of understanding is evident in the range of such modalities as variously identified (Systems of Categories Distinguishing Cultural Biases, 1993). Of relevance here is how these variously understand "symmetry" and prefer to represent it in support of their own approach to psycho-social organization. The potential of such alternative perspectives has been elegantly argued, emphasizing Asian insights, by Susantha Goonatilake (Toward a Global Science: mining civilizational knowledge, 1999) and appropriately recognized, in traditional knowledge systems, by Darrell A. Posey (Cultural and Spiritual Values of Biodiversity, 1999).

Virtualization of identity through abstraction
Kenneth Boulding. Ecodynamics; a new theory of societal evolution, 1978.

Our consciousness of the unity of self in the middle of a vast complexity of images or material structures is at least a suitable metaphor for the unity of group, organization, department, discipline or science. If personification is a metaphor, let us not despise metaphors -- we might be one ourselves.

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Identification with what is described