Potential experiential implications in terms of concentration and meditation

Psycho-social Significance of the Mandelbrot Set: a sustainable boundary between chaos and order (Part #4)

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It is possible that the M-set usefully points to the challenges of concentration and meditation. Indeed it is possible that it was those with such skills that were instrumental in formulating myths to anchor their insights as an aid to wider understanding.

• "Oceanic awareness": This feeling, frequently reported by investigators of alternative states of consciousness, might well be associated with the cardioid portion of the M-set -- or quasi-similar sub-variants of it.
  
• Mindlessness: This condition (also termed "no mind"), to which meditators of some spiritual disciplines aspire, might be associated with the level of abstraction of the cardioid portion of the M-set. As expressed from a Kriya Yoga perspective, for example, in commenting on the Bhagavad Gita:
  

  Understanding of Gita happens in our bone marrow and blood-cells through repeated chanting and contemplation. Gita invites us to be available to the holistic consciousness of choiceless awareness, that is, to the 'ignorance' which knows -- not to the 'knowledge' which is ignorant! Gita is the unique world view that tolerates and requires holding together multiple positions simultaneously so that religion holds whole mankind and does not degenerate in bigotry and battle. The war in Gita is symbolic. It is the war between wickedness and wisdom, between mindlessness and 'no-mind' with mind as the bridge. Duryodhan is mindlessness, Arjuna is mind and Krishna is 'no-mind' i.e. pure intelligence (chaitanya). Gita liberates us from our prior self-preoccupied identities. Gita sets us free from disintegration to integration (yoga), from reaction to action (Kriya Yoga), from longing to living (swadhyay), from paradoxes to pure consciousness (Ishwara Pranidhan), from Prakriti (inherent traits and tendencies) to Purusha (transcendental truth of enlightened existence). Gita is the wisdom of sacrifice of the fruits of action, the distinction of Gunas, the emergence of equanimity, and the importance of non-doership. [more]

• Alchemy: Traditional representations of meditating practitioners of Taoist "internal alchemy" (nei-tan) bear a striking resemblance to the representation of the M-set. As discussed in an earlier paper (Cardioid Attractor Fundamental to Sustainability, 2005):

  Curiously it is Gregory Bateson in a section on Form, Substance and Difference [more] of his book Steps to an Ecology of Mind (1972) that relates the depth psychology work of C G Jung to the thermodynamics of Sadi Carnot. But it is in another book, translated by Jung's colleague Richard Wilhelm (1929), that Jung comments on a fundamental cycle identified in a Chinese text T'ai I Chin Hua Tsung Chih (The Secret of the Golden Flower) -- more recently translated by Thomas Cleary (1991) -- which showed Jung how to initiate this alchemy through the breathing cycle [more | more | more]. This focus has also been compared to the Nestorian Gospel of St Thomas [more].

  The much-cited Chinese work discusses the "circulation of the light" of awareness through various conditions during meditation [diagram] reminiscent (if only in the metaphors used to describe them) of stages of the Carnot heat cycle discussed above. The Taoist practitioners of "internal alchemy" (nei-tan) refer to a continual circulation of the ch'i (vitality principle) up the primary yang (positive), back or Governor Channel (tu mai or du mo) and down the primary yin (negative), front or Functional Channel (jen mai or ren mo) of the body. (see Lu K'uan Yü, Taoist Yoga, Alchemy and Immortality, 1970) [more].

  It is possible that the representation of the M-set is to be usefully related to the traditional concept of an alchemical "vessel" in which all "base matter" can be dissolved -- the container for the "universal solvent". Alchemy postulates the existence of such a universal solvent as being capable of transforming base metals into gold and bestowing eternal youth and therefore immortality on human beings. The universal solvent -- counterpart to the "philosopher's stone" -- is not ordinary water, but "philosophical" water, the water of life, aqua permanens, aqua mercurialis. For Robert Grinnell (Alchemy in Modern Woman, 1973) it is basic to the transformative alchemical process of solutio which facilitates the fluid, mobile basis of consciousness:
  

  For aqua permanens is a mode of the arcane substance; its symbol is water or sea-water, an all-pervading essence of anima mundi, the innermost and secret numinosum in man and the universe, that part of God which formed the quintessence and real substance of Physis, at once the highest supercelestial waters of wisdom and the spirit of life pervading inorganic matter.

  For Iona Miller (Chaos as the Universal Solvent, 1993):

  In reducing all to pure water, the prima materia and the ultima materia become synonymous. That primal consciousness state, that creative and chaotic consciousness is the beginning of the operation of "water", and its ultimate realization. It becomes easy to see why the operation of water is the "root of alchemy." Through consciousness journeys which liquify our rigid notions of self and world, we re-create the adventures of the hero or heroine. The theme is the loss and recovery of identity.

• Chakra system: In Hinduism, its spiritual systems of yoga, in some related eastern cultures, as well as in some segments of the New Age movement, a chakra (from the Sanskrit word for "wheel, circle") is considered to be an energy node in the human body [more]. The seven main chakras are described as being aligned in an ascending column from the base of the spine to the top of the head (see fractal representations). Each is associated with a certain color, multiple specific functions, an aspect of consciousness, a classical element, and other distinguishing characteristics [more]. Given the hypothesized organizing function of the M-set, it is interesting to consider where these chakras are each located in relation to the geometry of its graphical representation (in the "seated-Buddha" orientation):
  
  1. Muladhara (Root chakra) : Understood to be at the base of the human spine. On the M-set representation it is most closely associated with the junction of the inward-turned portion of the cardioid.
  2. Svadhisthana (Sacral / Hara chakra): Understood to be below the human navel. On the M-set representation it is most closely associated with the axial origin from which the cardioid is generated. Zazen as the study of the self, is a particular kind of meditation, unique to Zen Buddhism, that is at the "very heart" of the practice. Fundamental to that practice is the centering of attention in the hara as the physical and spiritual center of the body. In Hinduism, this is recognized as governing the function of the gonads and the reproductive system. It is associated with the emotional body, willingness to feel emotions and accept change. As the centre of gravity of the body, this awareness is also fundamental to some Eastern martial arts.
  3. Manipura (Solar Plexus chakra) : Understood to be at the level of the human solar plexus. On the M-set representation it is most closely associated with the intersection of the vertical axis with the horizontal line between the symmetrically positioned 3-period primary bulbs attached directly to the cardioid.
  4. Anahata (Heart/Lung chakra) : Understood to be at the level of the human heart. On the M-set representation it is most closely associated with the intersection of the vertical axis with the horizontal line between the symmetrically positioned 5-period primary bulbs attached dfirectly to the cardioid.
  5. Visuddha (Throat chakra) : Understood to be at human throat. On the M-set representation it is most closely associated with the junction of the cardioid with the circular period-2 primary bulb above the cardioid.
  6. Ajña (Third Eye chakra) : Understood to be between the human eyes. On the M-set representation it is most closely associated with the centre of the circular period-2 primary bulb above the cardioid.
  7. Sahasrara (Crown chakra) : Understood to be at the crown of the human head. On the M-set representation it is most closely associated with the crown-like features about the circular period-2 primary bulb above the cardioid.
     
  There is a case for investigating the extent to which the position of these chakras emerges from more complex mathematical features of the cardioid, its associated curves, and especially from its generation. These might be understood as significant associative pathways for those who embody this system.
  
  
• Types of concentration/meditation : In the terms, for example, of the Yoga Sutras of Patanjali (1.17-1.18):
  All objects are in one of four stages: Virtually all types, styles, methods, or objects of meditation are included in one of these four stages, levels, or categories (1.17):
  • Savitarka/Gross: relates to concentration on any gross object while still accompanied with other activities of the mind, including meditation on sensory awareness, visualized objects, the gross level of breath, attitudes, syllables of mantra, or streams of conscious thought.
  • Savichara/Subtle: relates to subtle objects, after the gross have been left behind; the subtleties of matter, energy, senses, and the mind are, themselves, the objects of meditation, inquiry, and non-attachment.
  • Sananda/Bliss: emphasizes the still subtler state of bliss in meditation. In this state, the concentration is free from the gross and subtle impressions that were at the previous levels.
  • Sasmita/I-ness: focuses on I-ness, which is even subtler, as it relates to the I that is behind, or witness to all of the other experiences.
  Objectless concentration: The four stages (above) all have an object to which attention is directed (samprajnata). Beyond these four is objectless concentration (1.18), where all four categories of objects have been released from attention (asamprajnata).
  
  
• Emptiness of mind: As with many spiritual disciplines (for which there are extensive web references [more | more]), Buddhism in its various forms (notably Mahayana Buddhism) places much emphasis on the emptiness of mind and the ways in which that understanding is obscured. Offering an intriguing association to the M-set cardioid, in The Heart Sutra the Buddha reportedly states, for example:

  ...form is emptiness and the very emptiness is form; emptiness does not differ from form, form does not differ from emptiness; whatever is form, that is emptiness, whatever is emptiness, that is form, the same is true of feelings, perceptions, impulses and consciousness.

  In another example, the Tibetan Kalu Rinpoche reportedly stated:

  You live in illusion and the appearance of things. There is a reality, and you are that reality, but you don't know it. If you should ever wake up to that reality you would realize that you are nothing, and being nothing, you are everything. That is all. [more]

  This could be understood in terms of the the human tendency to focus on, and identify with, the experience of the sensible dynamics which the M-set "embodies" -- but from a higher order of abstraction with which the individual may alternatively choose to identify.
  
  It is therefore interesting to note the Buddhist comparison between such understandings of emptiness and the origin of numbers from emptiness. For Buddhists, emptiness (sunyata) implies a sense of potential rather than nihilism. In Mathematics, mind, ontology and the origins of number the recognition by Geshe Kelsang Gyatso (1995) is noted that there are three levels of dependent relationship:
  
  • Gross dependent relationship - causality - the dependence of phenomena on their causes.
  • Subtle dependent relationship - structure - the dependence of phenomena on their perceived parts (including aspects, divisions and directions).
  • Very subtle dependent relationship - the dependence of phenomena on imputation by mind.
    
  
  For Buddhists, these ideas are noted [more] as being remarkably similar to the theory of the origins of mathematics, as proposed by the mathematician John von Neumann (1923) in the light of the theory of sets. He was the originator of the architecture used in most non-parallel-processing computers. Von Neumann suggested that all numbers could be bootstrapped out of the empty set by the operations of the mind. A set is a collection of things. An empty set is a collection of nothing at all. An empty set can be thought of as nothing with the potential to become something (that is to be become a set with at least one member). Again, the three orders of abstraction might be fruitfully related to the structure of the M-set.
  From a perspective of hermetic philosophy (About Spiritual Emptiness or the Void):
  

  Emptiness designates a state of mind, an inner level of consciousness based upon the renunciation of what one believes to be real, beyond all comprehension or lack of comprehension. Emptiness is, therefore, a higher level of the mind, an attunement with "nothing," i.e., with Pure Being that has no reflection at all. This is the revelation that the abstract or higher mind of a spiritual man receives. He is called "spiritual" because his mind is open and in harmony with the whole of creation, or the Unknown God. Hence, such a man is empty of a personal "interpretation" and is in touch with the world of spirit. As he manifests the world of spirit within him, a new one opens up before him, giving him a new understanding of life.

  Such references point to many possibilities of using the M-set as a means of holding the distinctions between various fundamental states of consciousness associated with "spiritual" insight. For example, given the various distinct modes of long term behaviour characteristic of the dynamical systems mapped by the M-set:

  • To what extent is the notion of iterations "escaping to infinity" to be usefully associated with the typical forms of distracted everyday thinking -- and in the most extreme form of attention deficit disorder? In fact are the "number of iterations" before thinking does escape to be understood as a measure of the degree of concentration?
  • To what extent is asymptotic attraction to a single value (an attracting fixed point) to be associated with "one-pointed" meditative states?
  • To what extent are bounded systems characterized by periodic or cyclic patterns of a certain frequency, to be considered as associated with forms of meditation (sustained by repetitive chants) or, otherwise, by thoughts going obsessively and habitually "round and round"?
  • To what extent are chaotically bound systems (neither attracted to a single value nor to a periodic cycle) to be associated with forms of concentration that are neither attracted to a particular focus nor to a periodic pattern of habitual reflection?
  
  The extensive discussions regarding Buddhist insights into emptiness offer many pointers to such a comparison [more].

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