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Dynamics of Symmetry Group Theorizing
Exploration of comprehension of symmetry and its psycho-social implications.
Dynamics of Symmetry Group Theorizing
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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).
1Significance of "explanation"The focus here is necessarily on the challenge of the explanation of symmetry group theory to my comprehension of its implication for my understanding. Marcus du Sautoy introduces his own exploration with a very meaningful quote from
The universe is built on a plan the profound symmetry of which is somehow present in the inner structure of our intellect
That theme could be understood as having been explored by the cognitive linguist George Lakoff and the psychologist Rafael E. Núñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).
The challenge would therefore seem to be the personal (necessarily subjective) comprehension of the relationship, or resonance, between that "inner structure" and what is offered objectively as a formal explication of that "profound symmetry".
There have been centuries of struggle by mathematicians of the highest ability to understand and give meaning to that profound symmetry. Given their many partial steps on the way (with their associated partial comprehension), it is appropriate to question whether a modern "explanation" is necessarily immediately meaningful -- whatever one's capacity or degree of application. This is clearly unfortunate if such symmetry is of such profound significance and implicated to such a degree in the inner structure of our intellect, whether individually or in terms of any understanding of collective intelligence and its application to the challenges of the times..
There is also the question of the profound significance now of the symmetry that only future generations of mathematicians will come to comprehend and explicate.
1Progressive comprehensionThe explanation described above can be usefully reframed in terms of an experience over time implying a process of communication. This might be expressed as follows.
Legend:
- Mathematical community
- A: Progressive discoveries of a genius (typically isolated or challenged by communication with peers)
- B: Succession of discoveries by isolated individual (pioneers) over generations
- C: Development of mathematics within a community of peers in a given generation
- D: Development of mathematics by the community of mathematicians over generations
- Dependent on insights of mathematicians
- a: Progressive learning by an individual from explanations by (other) mathematicians
- b: Progressive development in mathematical education of an individual over generations
- c: Progressive improvements in comprehension of mathematical explanations by (other) mathematicians
- d: Progressive development in mathematical education over generations
It is appropriate to note that a mathematical "proof" -- confirming the existence of the Monster group -- which takes the form of 10,000 pages spread across 500 journals (as also indicated by Marcus du Sautoy) raises important issues regarding the nature of the connectivity constituting such objects, and their credibility and communicability with respect to human comprehension. This theme is discussed at greater length in the earlier comment on the Monster group (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007). Related factors might be implied by the cost (say $20) for anyone seeking to access each such copyrighted journal article over the web, and the extent to which such knowledge then remains essentially confidential to a particular community, to be affirmed as a credo by those unable to confirm the proof, especially if they lack the capacity to comprehend it !
1Psychological recapitulation of historical development of mathematicsWith respect to the above table, given the assumptions regarding the fundamental relationship between symmetry and the inner structure of the intellect, it might be assumed that the theory of recapitulation -- expressed as ontogeny recapitulates to progressive comprehension of such symmetry. In evolutionary biology the theory holds that embryonal development of an individual organism (its ontogeny) follows the same path as the evolutionary history of its species (its phylogeny). The theory has been variously refuted although it continues to be held to offer interesting insights as a first approximation. It may be equally suggestive in the case of progressive comprehension and learning.
Giorgio T. Bagni, et al. (History and Epistemology in Mathematics Education, 2003) indicate that this possibility of mathematics education "had its apotheosis" in a famous book by Benchara Branford (1921). Fulvia Furinghetti and Luis Radford (Historical Conceptual Developments and the Teaching of Mathematics: from phylogenesis and ontogenesis theory to classroom practice, 2000) review current renewed interest in this approach f
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