Progressive comprehension

Dynamics of Symmetry Group Theorizing (Part #3)

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The 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.

Fig. 1: Progressive comprehension of profound symmetry as articulated and communicated by mathematicians
	Discovery / Innovation / Explanation (by mathematicians)		Learning / Education / Communication (from mathematicians)
	Individual (genius)	Collective (peers)	Individual	Collective
Single generation	A	C	a	c
Multiple generations	B	D	b	d

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 !

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