Mirroring within the Monster

Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks (Part #13)

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The factorized representation of the Monster, with high order exponents, suggests that in terms of cognition there is a high degree of mirroring -- mirrors within mirrors -- perhaps to be understood as an effect of symmetries. Consideration of mirror symmetry was one approach used to the detection of sporadic groups, as Ronan notes with respect to those of Fischer:

Fischer's groups are very large, but the way to understand them is to look at the complex arrangement of mirrors.... The largest group, Fi₂₄, has more than a million million million million symmetries, but the number of mirrors is less than a third of a million... (p. 163)

It is therefore relevant to note a collection of discourses from the Sufi Inayat Khan (The Palace of Mirrors, 1935-1976) which introduces comments on mirroring and reflection as follows:

The book deals with the "mirroring" faculty, a specific quality of mind, which often passes unobserved under the "rust" of the usual, emotional and logical, processes. It is only recently that this faculty has come to be recognized on a wider scale under different technical terms, such as synchronicity, gestalt, non-sequential thinking or self-remembering

Each discourse starts with reference to aspects of this understanding, where the Palace of Mirrors could be a way of understanding the factoral mirroring within the Monster:

• The mind-world in the terms of the Sufi poets is called 'Aina Khana, which means the Palace of Mirrors. One knows very little of the phenomena that this Palace of Mirrors has in it [more]
• The phenomenon of reflection differs in its nature and character, especially by reason of the nature of different personalities.
• A thought may be compared with a moving picture projected upon a curtain. It is not one picture, but it is the several parts of the picture that, changing every moment, complete the picture.
• The impression that is made upon the mind has quite a different character from the impression that is made upon objects. Man is living, therefore creative. Whatever impression his mind takes it not only holds as a stone holds an impression, but it produces the same several times in a moment, thus keeping it a living impression.
• The phenomenon of reflection is is such that every action, every thought, is reflected in oneself, and there starts a production. Something is produced, something which gives direction to one's life and which becomes a battery behind everything one does, a battery of power and thought.
• The heart, which is called a mirror in Sufi terms, has two different actions which it performs. Whatever is reflected in the heart does not only remain a reflection but becomes a creative power, productive of a phenomenon of a similar nature.
• A clear vision depends upon a clear heart, open to reflection.
• All that one learns and expresses in one's everyday life has learnt by the way of reflection...

Mirroring has long been used to reflect on the insights of self-reflexiveness through which internal and external realities mirror each other (see Mark Pendergrast, Mirror, Mirror: a history of the human love affair with reflection, 2004; Paul Demiéville, The Mirror of the Mind, In: Peter N Gregory, Sudden and Gradual; approaches to enlightenment in Chinese Thought, 1991) -- as discussed in Mirrors of My World (2002). In Buddhism, for example, the mirror is one of a group of eight auspicious symbols relating to right thought on the eightfold path, reflecting things as they really are. In a famous passage Nichiren Daishonin (On Attaining Buddhahood, 1999) states

Even a tarnished mirror will shine like a jewel if it is polished. A mind which presently is closed by illusions originating from the innate darkness of life is like a tarnished mirror, but once it is polished it will become clear, reflecting the enlightenment of immutable truth.

A quite different line of reflection is associated with uncertainty and speculation (Louis Bachelier, Theory of Speculation: the origins of modern finance, 2006). Mirror divination (catoptromancy), or mirror scrying, has long been common to many cultures. The Tibetan art of mirror divination has even given rise to an official Divination Mirror of the State Oracle of Tibet.

Ironically, and potentially of relevance to understanding how mathematicians formulate conjectures in the knowledge creation process, a study of mirror divination by Katherine Swancutt (Representational vs conjectural divination: innovating out of nothing in Mongolia, Journal of the Royal Anthropological Institute 12, 2006, 2, pp. 331-353) distinguishes between "representational" and "conjectural" forms of divination. Swancutt demonstrates how conjectural divinations initiate processes of innovation wherein repeated questioning leads to combinatory thought which imposes novel combinations on people, who perceive the need for innovation, access an innovation, and finally recursively posit that innovation's conceptual origins.

The theme of what may be hidden in the mirror of higher dimensions, as an attractor for popular interest, has been the subject of a valuable exploration by physicist Lawrence M. Krauss (Hiding in the Mirror: the mysterious allure of extra dimensions, from Plato to string theory and beyond, 2005).

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