Knowledge and ignorance

Dynamics of Symmetry Group Theorizing (Part #5)

Whilst the table highlights the progressive increase in comprehension of symmetry over time, it also implies a corresponding ignorance of such symmetry by those who have not matched the insights acquired by others. In each case there is a sense of struggling with what is known (correctly or incorrectly) and what is not known -- and remains to be discovered (whether from others or by future generations). Ignorance may also be intimately related to information accessibility, overload and imagined priorities regarding what it is worth endeavouring to know -- notably as constrained by cost and confidentiality.

It is also presumably the case that some may have an understanding of profound symmetry -- in resonance with their own sense of the "inner structure" of their intellect -- without being able to communicate it through conventional mathematical formalism. This is clearly the case with music and architectural patterns, for example, which may offer the possibility of reinterpretation in mathematical form -- as discussed by Marcus du Sautoy. Appreciation of pattern may also be cultivated -- quite independently of mathematics -- as it is expressed through nature, as notably highlighted by Christopher Alexander (The Nature of Order: an essay on the art of building and the nature of the universe, 2003-2004). It is of course also the case that some, later acclaimed to be mathematical geniuses, may express their insights through formalisms foreign to conventional mathematics, as was the case with Srinivasa Ramanujan.

Given this struggle between knowledge and ignorance, life-long experience may be understood as being on the boundary between comprehension of profound symmetry (in resonance with inner structure of the intellect) and profound ignorance or misunderstanding of any such relationship -- namely on the boundary between order and chaos (cf Psycho-social Significance of the Mandelbrot Set: a sustainable boundary between chaos and order, 2005). The implication is that appropriate insight will be forthcoming when appropriate knowledge is acquired through learning. For mathematicians this is dependent on the "proof" of such a higher order of symmetry -- understood as the end of a kind of journey to that level of comprehension.

The linearity of such a progression -- especially if one has to live with one's own ignorance in failing to comprehend some intuited higher order of symmetry (possibly illusory) -- suggests that the understanding or realization of such "completion" may itself (always) be incomplete, as discussed from a different perspective (Happiness and Unhappiness through Naysign and Nescience: comprehending the essence of sustainability? 2008).

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Knowledge and ignorance