## Saturday, 1 October 2011

### Expanding Consciousness in a Living Systems Universe

Ralph Metzner, Ph.D.

California Institute of Integral Studies

The word consciousness is derived from the Latin con-scire - “with-knowing”. We can ask “knowing-with” - what? A relational, or systems, view is implicit in this etymology: pointing to the relation between subject and object, between the knower and the known. Conscious knowing (con-scire) is knowing with knowing that you know. This can be contrasted to the unconscious knowing involved in my knowing how to grow hair, or skin cells over a wound; and my knowing how to tie my shoelaces, or ride a bicycle, which has become a kind of unconscious or automated knowing.

Historically there have been two main metaphors for consciousness, one spatial or topographical, and one temporal or developmental. The topographical metaphor is expressed in conceptions of consciousness as like a territory, a terrain, or a field, a “state” one can enter into or leave; or like empty space, as in Buddhist psychology. The spatial metaphor, can lead to a certain kind of fixity in one’s perception or worldview, a craving for stability and persistence, and anxiety about change. From this point of view, ordinary waking consciousness is the preferred state, and “altered states” are viewed with some anxiety and suspicion, -- as if an “altered” state is automatically abnormal. In many ways this is the attitude of mainstream Western thought toward alterations of consciousness—even the rich diversity of dreamlife and the changed awareness possible with introspection, psychotherapy or meditation is regarded with suspicion by the dominant extraverted worldview.

The temporal metaphor for consciousness is seen in conceptions such as William James’ “stream of thought”, or the stream of awareness, or the “flow experience”, as well as in developmental theories of consciousness going through various stages. Historically, we see the temporal metaphor emphasized in the thought of the pre-Socratic philosophers Thales and Heraclitus, in Buddhist teachings of impermanence (anicca) , and in the Taoist emphasis on the flows and eddies of water as the basic patterns of all life. From this point of view, wave-like fluctuations of consciousness are regarded as natural and inevitable, and health, well-being and creativity are linked to one’s ability to tune into and utilize the naturally occurring, and the “artificially” induced, modulations of consciousness.

According to Immanuel Kant, “space” and “time” are the a priori categories of all thinking. It seems appropriate that these are the two most common metaphors we have come up with in our reflections on consciousness. Perhaps the most balanced way to think about consciousness would be to keep both the spatial and the temporal metaphors in mind. We can recognize and identify the structural, persistent features of the perceived world we are “in” at any given moment, and we can be aware of the ever-changing, flowing stream of phenomena in which we are immersed. Heraclitus, in addition to the oft-quoted “you cannot step twice into the same river”, expanded on the metaphor by also saying that “when we step in the same river, the flowing water is always different and new.”

The definition of consciousness proposed by the Russian mathematician and physicist Victor Nalimov, in his book Realms of the Human Unconscious, is one that integrates both the topographical and the temporal aspect: he calls it a “semantic continuum”, i.e. a “continuum of meaning”. A continuum is defined as a “continuous extent, succession or whole”, which can be divided mentally into parts, but also considered as a whole. A continuum of meaning, like the sensorium of sense perception, conveys something of the elusive quality of consciousness, -- both stable and mobile, integrative and differentiated.

In contrast to cancient, Eastern and indigenous views, Western science in general and psychology in particular has never been comfortable with the study of the subjective side of life, with qualities of experience, purposes, intuitions, altered states or spiritual aspirations. Under the sway of the Newtonian-Cartesian mind-matter dichotomy, consciousness and experience were seen as belonging to the realm of religion, and science agreed to stay out of it. Later, as the ideological hold of the Church diminished and the materialist paradigm became paramount, consciousness and all subjective experience became even more firmly banished from scientific discourse.

In the 19th century, the German social philosopher Wilhelm Dilthey attempted to establish the “mental sciences” (Geisteswissenschaften), on an equivalent footing to the “natural sciences” (Naturwissenschaften). This idea never really took hold in the English-speaking world. Instead, the social sciences (psychology, sociology, anthropology, political science) adopted and imitated the empirical observational and quantitative analytical methods of the natural sciences. In psychology, the only observations that qualified as “scientific” were observations of behavior—to the extreme of B.F. Skinner’s behaviorism, in which mental states were said to be in an unknowable “black box”. Although the influence of strict behaviorism in psychology has waned in the latter half of the 20th century, the ideological commitment to a materialist worldview has not. In the leading paradigms of cognitive psychology or cognitive science (which includes brain sciences, computer modeling, information systems and the like) consciousenss is still treated as something to be explained (i.e. explained away) in the supposesdly more “real” terms of “neural nets”, “brain circuits” and the like.

In the latter half of the 19th century a European philosophical movement took a completely different and new approach to the study of consciousness. The German mathematician/philosopher Edmund Husserl (1859 - 1938) originally conceived of his phenomenology as an attempt to rescue philosophy and the quest for absolute knowledge from the “naturalism” and relativism of the newly arising experimental psychology. He criticized the psychophysical method of Wilhelm Wundt and G.T. Fechner as providing only correlations between subjective events and physical events, and ignoring the possibilities of “pre-understanding” of what consciousness was essentially. For Husserl, the abstract truths of mathematics are essences that are grasped by the mind directly, without relative or empirical observation. He proposed phenomenology as the method for directly arriving at essential and universal knowledge about the nature of consciousness and meaning, in part by clarifying the implicit pre-understandings that underlie other psychological approaches.

A core concept of Husserl’s phenomenology of consciousness is intentionality : consciousness is always intentional, always “of” or “about” something, always directed, like an arrow or a mathematical vector, toward some object of meaning. The objects that consciousness intends can be external, or they can be internal aspects of our own experience. Because intentional consciousness is always “constituting” the essential features of the various domains of existence, both external and internal, consciousness has a fundamental “ontological priority”—it is the “supporting ground of reality”. The focus on intention as the fundamental constituting attribute of consciousness is congruent with the emphasis on “set (and setting)” as the prime determinants of altered states, which I shall discuss further below. The ontological primacy of consciousness in Husserl’s phenomenology is consistent with the worldview of the mystics in Eastern and Western traditions as well as the insights coming from profound altered states.

A further innovative contribution to the phenomenology of consciousness was made by the French philosopher Maurice Merleau-Ponty (1908 - 1961). In his work, the focus of interest shifts from the subjective mind to the subjective body, or bodily experience (le corps propre). For Merleau-Ponty, perception is an inherently creative, participatory activity between the living body and its world. All subjectivity or consciousness presupposes our inherence in a corporeal world, a world that we perceive as the having depth, intimacy and horizon. The ecophilosopher David Abram has shown, in his work The Spell of the Sensuous, how in many ways Merleau-Ponty’s later thought anticipates the deep ecologists and others who are looking to develop a new conscious awareness of our embeddedness in the world of Nature.

The American philosopher William James (1842 - 1910) approached the psychology of consciousness in his characteristic multifarious manner. He may have been the first person to use the concept of “field” in talking about consciousness: human beings have “fields of consciousness”, which are always complex, -- containing body sensations, sense impressions, memories, thoughts, feelings, desires and “determinations of the will”, in fact “a teeming multiplicity of objects and relations.” He made it clear that his famous “stream of thought” image actually meant not just thoughts, but images, sensations, feelings, etc. He wrote that “the mind is, at every stage, a theater of simultaneous possibilities”. This idea is reminiscent of C. G. Jung’s conception of personality structure consisting of an aggregate of psychic entities or complexes: the persona, the shadow, the ego, the anima, the animus, with the Self forming the superordinate whole that includes all the parts, both conscious and unconscious.

William James explored the paranormal and mystical dimensions of consciousness, that usually lie outside the boundaries of scientific interest. He pursued a life-long interest in the phenomena of sub-liminal consciousness, what he called “exceptional mental states”, including those found in hypnotism, automatisms, such as sleep walking, hysteria, multiple personality, demoniacal possession, witchcraft, degeneration and genius. James’s interest in unusual states of consciousness led him to experiment with nitrous oxide, or “laughing gas” as it was then known, an experience that reinforced his understanding of transrational states of consciousness. He wrote that the conclusion he drew from these early “psychedelic” experiences was “that our normal waking consciousness, rational consciousness as we call it, is but one special type of consciousness, while all about it, parted from it by the filmiest of screens, there lie potential forms of consciousness entirely different.”

James wrote the above statement in his The Varieties of Religious Experience, probably his most influential book. In it he explored with great discernment and eloquence the nature and significance of mystical or “conversion” experiences, by which he meant not only a person’s change from one religion to another, but the process of attaining a sense of unity and the sacred dimension of life. In my book The Unfolding Self I adopted James’ empirical, comparative approach to the study of transformative experience — delineating the basic archetypal patterns of psychospiritual transformation.

It is only recently, in re-reading William James’ writings on his philosophy of radical empiricism that I came to realize that this philosophy actually provides the epistemology of choice for the study of altered states of consciousness. James started with the basic assumption of the empirical (which means “experience-based”) approach: all knowledge is derived from experience. Radical empiricism applies this principle inclusively, not exclusively: James writes:

To be radical an empiricism must neither admit into its construction any element that is not directly experienced, nor exclude from them any element that is directly experienced. For such a philosophy, the relations that connect experiences must themselves be experienced relations, and any kind of relation experienced must be accounted as ‘real’ as anything else in the system. (James, 1912/1996, p. 42)

Observations made in objective external reality perceived through our, aided and unaided senses, do not have an ontological priority and do not give “privileged access” to truth or validity. All knowledge must be based on observation, i.e. experience; so far this view coincides with the empiricism of the natural and social sciences. It’s the second statement that is truly radical and that explains why James included religious and paranormal experiences in his investigations. The experiences in modified states of consciousness are currently excluded from materialistic, reductionistic science, as are all kinds of anomalous experiences, such as UFO abductions, near-death experiences, and mystical or paranormal experiences. They need not and should not be excluded in a radical empiricism.

From the perspective of radical empiricism, it is not where or how observations are made that makes a field of study “scientific”, it is what is done with the observations afterwards. Repeated systematic observations from the same observer, and replicating observations from others, is what distinguishes the scientific method from casual or haphazard observations, or those made with intentions other than gathering knowledge. Whereas the ideology of fundamentalist scientism does not permit the objective investigation of subjective experience, the epistemology of radical empiricism posits that it is possible to be objective about subjective experience, using the accepted canons of the scientific method. The methodology of systematic introspection, and phenomenology, are the beginnings of such a more inclusive approach.

In Eastern systems of yoga and meditation, it has long been understood that reliable and replicable observations can be made in the interior landscapes of our experience. The meditative practice of “mindfulness” is exactly an attitude in which objectivity is added to the primary given (or “data”) of the experience. Religious texts describe the characteristic kinds of observations that a practicioner in a particular tradition may expect to make. This is, in essence, no different than a scientific text that describes the kinds of observations a student of biology can expect to make when looking through a microscope.

One of the most exciting consequences of adopting a worldview and epistemology in which no sources of new perceptions and observations will be ignored because of their provenance, leads to the opening up of vast new fields and possibilities of understanding, and a renewed bringing together of spiritual and scientific understandings into an integrated worldview, or sacred science.

In a systems view of humans and universe, a relational view of multi-level interconnectedness, things, objects and persons are temporary nodes, in ever-changing patterned relations with other nodes. All being is interbeing, in Thich Nhat Hanh’s felicitous phrase. Consciousness then is the experiential side, the knowing, feeling, sensing, imaging of relations, the “knowing-with”. It is the subjective, inner, concaveness of the transparent hollow sphere we call the space-time continuum; of which objective, outer, convexity is matter and energy, in their ceaseless transformations. According to Buddhist teachings, the inside of the sphere is empty, void, sunyata – hence no things, objects or persons actually exist. However, through the magic power of projection or maya, we seemingly perceive an objective universe, “out there”, of infinite differentiation and diversity.

The wise elder Thomas Berry, theologian turned evolutionary cosmologist, says that in the evolutionary transformation civilization is now going through, the transition from the cenozoic to the ecozoic era, our perspective on the world will change from seeing and measuring it as a “collection of objects”, to knowing and experiencing it as a “communion of subjects”. In other words, a living systems worldview, in which the conscious communion of living subjects is acknowledged as equally real and valid as the conscious perception of identifiable objects. We don’t need to stop utilizing the powerful methods science has developed for analyzing the complexities and details of our world; we add to them the inner, sensuous, feeling-based, aethestic appreciation of the essential oneness of the ever-transforming web of life.

Oneness and differentiation exist at every level of reality. The ecologists have made us aware of how the diversity of life, biodiversity and its preservation is the core requirement for sustainability. To this, the anthropologists and historians of culture have added the recognition of the importance of preserving cultural diversity – for the unique knowledge system or tradition that each culture has developed. As a psychologist, I would want to add to the celebration of diversity, an acknowledgement of the rich psychic multiplicity of our psychic inner life: we are all multiple personalities, each of us containing a “theater of simultaneous possibilities”. Learning from indigenous cosmologies as well as the spiritual traditions of ancient times, we may well want to return to a recognition of the animistic polytheism of our forebears, recognizing and celebrating the indwelling spiritual intelligences of all inorganic and organic life-forms, as well as of Earth and other planets, Sun and other stars, Milky Way and other galaxies, and Universe. As Buckminster Fuller said, wanting to include the metaphysical as well as physical in his definition, “Universe is everything that we can experience.”

The Centrality of Intention and Question

We have seen how in the phenomenological approach to the understanding of human consciousness, “intentionality” is the central organizing concept. Consciousness is said to always be “about” something; or, we are always “conscious of” something. The “mental status” examination in cases of shock or psychosis, asks whether the patient is oriented in time and space (do you know where you are? what day it is?). We have also noted how psychedelic and other altered states of consciousness can best be understood if one inquires into the “set” (= intention) that preceded or accompanied the catalyst that triggered the movement in consciousness. Carlos Castaneda, in his writings on the principles of Yaqui, or Toltec, sorcery, emphasizes repeatedly, that “intent” is the master key to sorcery; where sorcery may be defined as the intentional use of altered states, such as conscious dreaming, to acquire and exercise psychic power.

Following Michael Harner, I define shamanism as the intentional practice of altered states, called “shamanic journeys”, for healing, problem solving and guidance. In shamanic indigenous healing practices, as in ordinary medicine and psychotherapy, the intention is given by the problem or illness the patient brings to the practicioner. Indeed, it is perhaps too obvious to state, that in any situation, any state of consciousness, inquiring into the intention the person is holding, is a way of directing awareness to the experiencing subject or self, the starting point, the original orientation. In divination, which may be defined as the intentional use of the enhanced perception of altered states ( and sometimes divinatory tools such as cards or runes) to acquire knowledge and spiritual guidance from the divine of spirit world, conscious intention again provides the master key.

Intention, or “interest”, as William James phrased it, controls the selective function of attention, which in turn determines what it is we perceive or become aware of. We may express these relations as follows:

intention -> attention -> awareness

If our attention –perception is not guided by our conscious intention or interest, then it will be captured, or captivated, by whatever attractive, intense, insistent, prominent stimulus patterns present themselves to our sensorium.

Through my work with divination practices, both in the context of psychotherapy, and in the context of shamanic rituals using expanded states of consciousness, I have come to understand that the basic formula for divination is that of asking questions and receiving answers. Even when a mantic procedure, such as the I Ching, or casting runes, or laying out Tarot cards is used, inherent in the procedure is a question posed and an answer received (which may need to be interpreted). Divination in the context of healing aims at finding the origin or source of the illness – which in traditional societies may be attributed to sorcery in as many as 50% of illnesses. In Western medicine this process is called seeking a diagnosis, the origin and nature of the infection or injury. The primary difference between the indigenous shamanic approach in healing divination and the Western medical approach, is that traditional divination involves an induced altered state of consciousness, and asking questions of the practicioner’s tutelary spirits; whereas the Western approach involves highly focussed and detailed observations, supplemented by specialized instruments, in the ordinary waking state of conciousness.

The biologist Rupert Sheldrake has pointed out that the process of divination to obtain knowledge -- of the world, the environment, anything concealed or not understood, possible future outcomes – is a questioning process exactly analogous to the scientific experiment in Western science. The experiment is a precisely controlled situation, where detailed observations can be made to answer questions we are posing to Nature, the world, the environment, etc.

One can see then that intention and question are like two sides of the same coin, two equivalent ways of guiding our attention and perception, in both ordinary and non-ordinary states of consciousness. For example, if I am wanting to heal the effects of a childhood trauma, whether through shamanic soul retrieval work, or therapeutic regression, I can state my intention and say: “I want to (am interested, or aiming to) heal the effects of this trauma”; or, I can ask: “How can I heal (and integrate) the effects of this trauma?” There is a dynamic (yang) and receptive (yin) polarity here: the intention is focussed, directional, searching, seeking; the question (any question) is a basic gesture of receptivity and opening. If I ask a question of someone, I am placing myself in receptive mode to hear their answer. We can present the differences in these two ways of deploying attention by the following analogies:

Intention Question

psychic polarity dynamic receptive

process analogies hunting, tracking gathering, collecting

instrumental analogies arrow basket

mathematical analogies vector attractor

Levels of Consciousness in Shamanistic Cosmology and Living Systems Universe

Spiritual seekers and yogic meditators have described a universe of many levels of of reality and consciousness, to which we can gain access through various practices of consciousness expansion, though in ordinary life we are rarely, if ever, aware of these other “higher” dimensions. The Sufis say we are like a man who has a seven-story mansion, but who lives only on the ground floor, the most polluted, cluttered and unattractive parts of his mansion; to the point where he has completely forgotten that the other, grander rooms even exist, much less how to get there. The techniques of finding access again to these higher realms, is therefore a kind of memory exercise – often referred to in the literature as “recollection”, or “remembrance”.

We may formulate the relationship between states of consciousness and levels, planes, or worlds of consciousness as follows: in the ordinary waking state of consciousness, our awareness and identity is generally focussed on the level of ordinary, consensus reality. In meditative states, some dreams, expansive psychedelic states, mystical and visionary states, we may find ourselves seemingly in other kinds of reality altogether, other realms, other worlds. These other realms are then known to always exist, and we always to have potential access to them. Spiritual and shamanic practices are designed to intentionally allow us to move into and through these other realms, in which we can find sources of healing or spiritual knowledge, conveyed to us by spiritual teachers, elders, ancestors, angels, spirit animals or spirit guides. We call such practices of intentional access divination, as discussed above.

The level of “ordinary reality” is often called the physical or material plane, which implies the other planes are metaphysical or non-material (e.g. subtle, etheric). This ordinary, ground-floor level is also referred to as the time-space world, implying that in the other levels the laws of time and space as we know them don’t apply, or are different. This notion is readily verifiable when we recall that in dream states we can seemingly visit with, talk with, interact with, a person known to us, say a grandmother, who objectively lives thousands of miles away, or indeed may even be dead, and yet we traverse no space, and take no time, to get there. We also do not question the reality of a contact with a “dead” person. This malleability of the time-space framework is the reason why divination can reveal knowledge of probable futures, called “visions” (which are not “predictions”, as the dictionary definition of divination erroneously has it).

There have been numerous mappings of the different realms and levels of consciousness in the spiritual, meditative, and esoteric traditions of the world, and to compare and coordinate them all is a task far beyond my capability and the space limitations of this essay. I will briefly describe here, for illustrative purposes, (1) the traditional shamanistic cosmology; (2) a version of the seven-level map commonly found in the European and Asian esoteric, perennial and theosophical traditions; and (3) three of the many possible dimensional mappings that can be identified in a living systems worldview .

Those who have embarked on a serious psychospiritual practice of consciousness exploration using shamanic and yogic technologies, who are willing to trust their own experience more than the received views and concepts they have taken on faith, tend to find themselves gradually awakening to a vastly expanded and different worldview. I should point out however that many features of the traditional and newly revived shamanic-animistic worldview appear to be quite compatible with the most recent, growing edge theories of post-modern science. There is not the space here to enter into a discussion of these convergences in any detail. I will merely mention the particular relevance of ecology and living systems theory, as described for example in Fritjof Capra’s book The Web of Life; the Gaia theory of James Lovelock and Lynn Margulis; Rupert Sheldrake’s theories of morphogenesis; David Bohm’s “holomovement” interpretation of quantum theory; the integrative and synergetic cosmology of Buckminster Fuller; and the evolutionary cosmology articulated by Brian Swimme, Thomas Berry and others.

There is one fundamental commonality among the shamanic, Asian meditative, esoteric and living living systems worldview, and that is in their conception of the fundamental reality of the universe. The fundamental reality of the universe, according to the most ancient, and the most recent, post-modern formulations, is a continuum, a unitive field or fabric or process, of both energy and consciousness, that is beyond time, space and all forms, and yet somehow mysteriously within them; simultaneously transcendent and immanent. In traditional Asian religions, this unitive field is variously referred to as Tao, or Atman-Brahman, or Tantra (meaning “web” or “fabric”) or the “jewelled net of Indra.” Some Native North Americans refer to it as Wakan-Tanka, the all-pervading Creator Spirit. In the traditional Anglo-Saxon religion of the British Isles it was called the Wyrd, an invisible network of magical forces. In theistic religions like Christianity, this oneness corresponds to what is called the “Godhead”, a spiritual Beingness beyond the personal deity. In esoteric writings it is variously called The One, Absolute Beingness, the All That Is. In the systems language of post-modern science it is seen as an infinitely complex, multi-level system of interrelationships, or “web of life.”

(1) Shamanistic Cosmology. Worldwide, shamanic practicioners speak of a three-fold division of upper, middle and lower worlds. The names reflect the primary movement of the shamanic traveller, when engaged in divinatory practice. A journey to the Lower World, initiated by intention, and energized by psychoactive plants or rhythmic drumming, involves a downward movement in consciousness, into a cave, or opening, or tunnel – from which one emerges after a time to track the objects of one’s quest. Lower World journeys seem to be especially relevant for problems of healing, as if the traveller were actually “going down” into the material microcosm of the earth body. Upper World journeys involve ascent – through flying, soaring, climbing a World Tree or axis – and visiting light, airy worlds, with expansive vistas and luminous structures. These kinds of journeys may be involved when the shaman wishes to look ahead, “see” hidden realities, or envision future probabilities. In Middle World journeys, the movement is straight across, but may involve all kinds of strange, weird, magical, unusual places and beings – such as enchanted forests, magical castles, wastelands and wilderness. The characteristics of upper, middle and lower world journeys, as reflected in myths and fairy tales, as well as shamanic accounts, are described in more detail in the chapter on “Journeys to the Place of Vision and Power”, in my book The Unfolding Self.

Whereas the cosmography of three worlds is most often found, there are also mythic-shamanic traditions of five, or seven, nine or more worlds, often arrayed around a central World Tree or Axis Mundi. In the Nordic-Germanic mythic worldview, there are nine worlds arranged on the World Axis-Tree Yggdrasil. In the center is Midgard, the familiar ordinary world of humans, animals, and Earthly life; pathways go from this world to all the other worlds, which knowledge-seeking shamans can learn to find and travel. There are two worlds on the central axis above the Middle Earth realm: the World of Elves, who are light, airy spirits, and the World of the Aesir Sky Gods. Two worlds are vertically below Midgard: the World of Dwarves, or Dark Elves, the spirits of stones and minerals, and the world of Hel, the death goddess. Then there are four other worlds, in the same horizontal plane as Midgard, in the four directions: the World of Giants in the East and the World of Vanir Earth Deities in the West; in the North and South, two uninhabited worlds, of pure ice and pure fire respectively. I describe both ecological and mythic symbolic meanings associated with these nine worlds in my book on the Earth-Wisdom mythology of the Nordic-Germanic people, The Well of Remembrance. In myth and folklore of the European people, there is sometimes only one name for all the non-ordinary, nonmaterial realms that we humans may come across: they are called “Spirit World”, “Otherworld”, of “Faery World”.

(2) In esoteric and thesosophical traditions we usually hear of seven levels of consciousness, differing in vibratory rate or density, with the physical-material as the “lowest” or densest. We have a series of bodies, which are the bodies we occupy in the corresponding realm or world. Just as the physical body is the body we move around in while in the time-space physical world, the astral body is our body for functioning in the astral world or realm. Leaving aside for the moment the question of the different terms and names used for the different levels or realms, in the experiences of contemporary neo-shamanic practicioners, with or without mind-moving substances, experiences of visiting other worlds are quite common. Also, of course, they are accessible to us in dream states, and in meditative states. Alternatively, the person may feel that the veils, barriers or screens between worlds can become transparent or porous, so one can see and be in both the ordinary and the spirit world at the same time. In William Blake’s famous lines, “if the doors of perception were cleansed, everything would appear as it is – infinite”.

The mappings of different levels of consciousness or worlds, are based on accumulated observations by thousands of explorers and elders in the various spiritual traditions. In other words they are empirically based, just like maps of an unknown geographical area are based on accumulated observations; these maps then become available to be used by subsequent explorers. An individual explorer, using yogic, shamanic or meditative methods of journeying in consciousness, could, in theory, verify the accuracy of the mappings for him or her self. In practice, when we begin our explorations, we are not usually able to identify the particular worlds we are in. The dreamer, for example, may realize he is in a different world, an “otherworld”, but not be able to say whether it was the “astral” or some other realm. So, if we want to maintain a stance of radical empiricism, we can accept the maps given by the different traditions as provisional, to be verified by our others’ observations. The maps, as we know, are not the territory.

In that spirit, I present here a mapping of the seven major levels of consciousness, as I was taught them by my teachers, and as I continue to hold them, as a working hypothesis. The most useful analogy to help understand the differences between the levels, is, to my mind, the concept of frequency or vibratory rate, as in music theory. The different bodies or levels, differ in vibratory rate, like the tones of a scale or chord: that is why they can co-exist, co-incide, in the same place at the same time, without interference.

The seven/eight note structure of the octave, is also consistently found in mappings of the different levels of consciousness, including G. I. Gurdjieff’s formulation of the Law of Seven as one of the fundamental laws of the universe. It is not inconsistent with this principle that any given yogic or meditative practice may work primarily, or in the beginning stages, with two or three or five levels.

So, in ascending order of frequency rate, we have: the physical-material body and realm (in the Indian traditions this is called the “body/sheath of food” – annamayakosha); next, the perceptual or etheric body and realm; the emotional or astral body and realm (some combine the perceptual and emotional, calling it “psychic”); and the mental or noetic body and realm. These four together make up the personality-systems; they are subject to conditioning, and all the dualities and conflicts that we know at the physical body level. Above the mental, in vibratory rate, are the transpersonal, unobstructed dimensions, or “heavens”, in which there are individual differences, like different colors in a painting, or tones in a symphony, but no conflicts, dualities or antagonisms. The first transpersonal level is that of soul, called anandamayakosha, (“body/sheath of joy”), in the Indian tradition. In the writings of the esoteric teacher Rudolf Steiner, the intermediate personality levels are not called “bodies”, but levels of soul: in descending order, soul of understanding, soul of feeling, and soul of perception. If the physical is the DO of an ascending octave, the level of soul would be SOL, which is also the Latin word for “Sun”. Some alchemical texts describe the soul, when “seen” with clairvoyant vision, as a golden, sun-colored sphere.

Continuing with the upper part of the vibratory octave, we have, above the level of soul: the angelic or celestial realm, which corresponds in the Buddhist teachings to the mandala realm of the Dhyani Buddhas; the archetypal or cosmic, or realm of First Differentiation, or primordial Yin-Yang; and, as the upper DO of the octave, Formless Form, Spirit, Self or Atman, which is always in perfect union with the Macrocosmic Oneness (Atman=Brahman). The particular names used here for the inner and higher frequency dimensions, have no particular importance. Obviously, other languages have different names. What is important, is that once we accept the existence of these higher worlds, our participation in them, and our relationships with beings, usually called “spirits”, in these other worlds, a whole new focus on spiritual practices which provide us with access to these worlds, for greater knowledge, understanding, insight, creative inspiration, healing and problem-solving.

(3) Levels of consciousness in a living systems universe. Modern science is only slowly moving towards acceptance of an ecological or systems cosmology. Even in such a worldview, much less in conventional materialist science, the idea of subtle dimensions, differing in vibratory rate, has little credibility, mostly because there is no widely accepted way of measuring or quantifying such subtle levels of energy and matter. (Some mathematical-physical accounts of other dimensions have been written, for example in the work of William Tiller). In one important respect however, the sytems worldview is congruent with traditional esoteric or shamanistic traditions – and that is the notion of levels, or scales. A systems universe consists of a series of whole systems (also called holons) arranged in ordered series, in such a way that the parts of a whole at one level, become wholes at the scalar level “below” ; and are themselves parts of wholes at the scalar level “above”.

This layering of levels of complex ordering, has been called by some a hierarchical principle, since the holons at the higher levels include the holons at the lower level as parts. However, the word “hierarchy” refers to systems of social organization in human collectives, -- ecclesiastical, military and corporate – in which “command, control and communication” (the military’s favorite phraseology) flows from the top down, one way only. Systems of natural order, as found in nature, are quite different, and have for this reason been called “nested hierarchies” or “holarchies”. They are structural ordering principles of containment and interdependence. For example, the body or organism is a holarchy containing a dozen or so organ systems functioning as interdependent parts, each of which is itself a holarchy of cellular components. There is really no similarity, or even analogy, with a conventional hierarchy: the body does not issue commands to the organs, and does not control them; it contains them.

We may look then, to the holarchical levels of a living systems universe for means to map out inner journeys or explorations in consciousness. As stated above, consciousness is the experiential side, the knowing, feeling, sensing, imaging of relations, the “knowing-with”. It is the subjective, inner, concaveness of the space-time continuum; of which objective, outer, convexity is matter and energy, in their ceaseless transformations. Many different systemic ordering scales, in the physical universe, have been identified and named. I will select three for consideration, and use the octave principle here too to delineate the stages, although sometimes more and sometimes fewer levels have been identified by various writers. At each level, there is a characteristic aggregation of energy and matter; and consciousness is what Teilhard de Chardin called the “within” of things. Thus atoms, molecules, cells, stones, plants, animals, rivers, oceans, planets, galaxies… each have their “within”, their subjectivity, their spiritual essence, or even “gods”.

3.1 Humans-in-Universe. As the first example let us take the ascending holarchical octave series that goes from the individual human being to the whole Universe:

body-organism of the human animal;

the kingdom of all animal species;

the biosphere, which includes all five kingdoms of organic life;

the Earth, Gaia, which includes the biospher and all inorganic material systems, as well as oceans and athmosphere;

the solar system, including Sun and all the planets;

the Milky Way galaxy, with its ten billion stars;

the cluster of several thousand galaxies of which the Milky Way is one – a cluster quaintly referred to by astronomers as the “local group”; and

Universe – all galaxies and other cosmic formations.

One can note that within each step of the holarchy series, a further octave series of steps can be identified. For example, to get from the individual body-organism to all animals, evolutionary taxonomy tells us of the following seven stages: the species homo sapiens; the genus homo; the family hominidiae; the order primata; the class mammalia; the phylum chordata; and the kingdom animalia.

People in altered states using shamanic or entheogenic practices, have reported having thoughts, feelings and perceptions, i.e. consciousness contents, at any of those levels. Once the Gaia theory was put forward on purely scientific grounds, spokespersons from traditional societies as well as esotericists immediately pointed out the similarity to ancient notions of “Earth Mother” or Anima Mundi, a conscious, sentient, spiritual being that is the indwelling spirit of the Earth. Likewise, solar deities have been a part of many religious traditions, including early Christianity. “Cosmic Consciousness”, a field of consciousness expanded to the outer limits of the known universe, has often been identified and described in the writings of mystics. In their book The Universe Story, Brian Swimme and Thomas Berry have shown in detail how at each stage of the evolving universe, humans can not only know about that stage through external information, but can experience inwardly what it means to be a part of those evolving wholes; for example, what it means to be part of the story of star-formation.

3.2 The Microcosmic Descent. Here also, the physical and life sciences have identified and described in great detail the organization of matter and energy at each level. The primary difference of a systems view from the conventional physical science approach, is that in a systems view, there is no reductionism to the physics level as the privileged, “ultimate” science. So starting again with the body the descending octave series goes:

organism-body of the human animal;

organ systems of the body – a level to which it is relatively easy for anyone to expand awareness, as we do for example, in relaxation and healing meditation practices;

cells in their tissue clusters – cellular consciousness, perhaps because cells float in a fluid matrix, tends to be experienced as oceanic;

organelles, the sub-components of cells, which are probably the evolutionary descendants of monera (bacteria) that have been incorporated into cells by endosymbiosis, according to the microbiologist Lynn Margolis;

molecular level – an infinite web of interconnectedness, containing the DNA genetic code for all life forms, but extending into inorganic and elemental matter as well;

the atomic level;

the level of sub-atomic particles such as electrons, protons, quarks and the rest; and

the level of pure, photonic, mass-less energy or vibration.

Assuming that subjective, potentially conscious perception and knowledge is possible at each level, allows us to understand seemingly miraculous healings that have been reported as occuring without any physiological or pharmcological intervention: through healers laying on hands, transmitting energy through crystals and vibrational chants, through homeopathic dilutions in which no single molecule of the original substance is left, or through seeing, in expanded states of perception, into the interior energetic structure of the patient’s body. An example of how scientific understanding may be advanced and enhanced by conscious attunement to the cellular and molecular levels is given in the work of Jeremy Narby, author of The Cosmic Serpent, a book on serpent imagery in South American ayahuasca shamans. Narby brought geneticists and molecular biologists to take ayahuasca with traditional shamans; they would ask specific questions related to their field of research, and obtain useful answers (Narby, 2002). The ayahuasca apparently allowed these scientists to focus their perception and attention at the cellular and molecular levels, without resorting to electron microscope, or other instrumentation used in Western science.

3.3 Humans-in-Habitats. This is a holarchy series dealt with in the sciences of human ecology, anthropology and sociology. Starting here too, with the DO of the ascending octave :

the human body, in its immediate habitat of clothing; then,

the family, in its house/home habitat; then

the extended family or clan, in a cluster of houses; then

the community, perhaps of several hundred families, living in a village;

the tribe or population in a bioregion or urban environment;

the society, all speaking the same language, inhabiting a country or nation;

the multi-cultural, multi-lingual, multi-ethnic confederation or conglomeration inhabiting a whole continent ;

the whole of humanity on Earth, the “global village”, as the upper DO.

The notion that there exist collective forms of consciousness, beyond the individual self, is of course not new in psychology. In family systems therapy, for example, the focus of therapeutic attention is the relationship network of all the family members; the notion that certain beliefs, attitudes, characteristics may be shared by a whole clan or community, is also not new. C.G. Jung coined the concept of “collective unconscious”, meaning images, symbols and archetypes shared by all of humanity; later theoretical development by his students, identified a layer of “cultural unconscious” between the personal and the collective. This makes it clear that the “collective unconscious” is really the system of consciousness that includes all human beings – what Robert Lifton has called the “species self”. It also makes it clear that beyond the layer of collective human species consciousness (a term I prefer to “unconscious”, since many of these contents are not unconscious at all), are forms of consciousness shared by all primates, all mammals, all animals, all life on Earth, even the cosmos. Different series of holarchical systems branch off from every node point in the web of life.

Since we are part of the unified system of interdependence, just like every other being, we can never actually be outside of it, as a detached “objective” observer. But since the unified field is energy, we are energetically connected to every other form and being in the universe. And since the field is also consciousness, this enables us, as human beings, to attune with, identify with, and communicate with any and every other life-form, object or being in the universe, from the macrocosmic to the microscopic.

If, as shamanistic and spiritual teachings agree, we humans are multidimensional beings, living in a multidimensional universe, then there are profound implications for revolutionary changes in our perspectives on the world. The shamanistic animistic worldview, is although older historically, larger and more inclusive than mechanistic-materialistic worldview dominant in the modern world. In the modern worldview, only the material time-space world of measurable quantities is accorded reality status; intuitive, imagistic, spiritual impulses and needs are marginalized to lesser ontological status.

In the worldview of indigenous people everywhere, the worlds we know from dreams and visions, worlds inhabited by gods and spirits, including our own deceased ancestors, are accorded recognition of equal reality. Our disconnection from the recognition and honoring of the vastness of spiritual realms enclosing and permeating this physical world, amounts to a staggering gesture of self-crippling on the part of Western techno-industrial civilization. This opens up almost inconceivable possibilities of solving our problems and bringing about a truly sustainable civilization, infused with spiritual values.

References

Abram, David. The Spell of the Sensuous - Perception and Language in a More-than-Human World. New York: Pantheon Books, 1996.

Harner, Michael. The Way of the Shaman. San Francisco: Harper & Row, 1980.

James, William. 1890/1952. Principles of Psychology. Chicago: Great Books of the Western World. Encyclopedia Britannica.

James, William. 1901/1958. Varieties of Religious Experience. New York: New American Library.

James, William. 1912/1996. Essays in Radical Empiricism. Lincoln: University of Nebraska Press.

Metzner, Ralph. The Unfolding Self - Varieties of Transformative Experience. Novato, CA: Origin Press, 1998.

Metzner, Ralph . The Well of Remembrance – Rediscovering the Earth Wisdom Mythology of Northern Europe. Boston: Shambhala, 1994.

Metzner, Ralph. Green Psychology - Transforming Our Relationship to the Earth.

Rochester, VT: Park Street Press, 1999.

Narby, Jeremy. The Cosmic Serpent – DNA and the Origins of Knowledge. New York: Random House, 1998.

Narby, Jeremy. “Shamans and Scientists”. in Hallucinogens – A Reader. Ed. Charles Grob. New York: Jeremy P. Tarcher / Putnam. 2002.

Website: www.greenearthfound.org

A shorter version of this essay is to appear in a collection of essays on the “Primacy of Consciousness”, edited by Trish Pheiffer and John Mack.

http://greenearthfound.org/write/expanding.html

## Thursday, 2 June 2011

### NASA Announces Results of Epic Space-Time Experiment

May 4, 2011: Einstein was right again. There is a space-time vortex around Earth, and its shape precisely matches the predictions of Einstein's theory of gravity.

Researchers confirmed these points at a press conference today at NASA headquarters where they announced the long-awaited results of Gravity Probe B (GP-B).

"The space-time around Earth appears to be distorted just as general relativity predicts," says Stanford University physicist Francis Everitt, principal investigator of the Gravity Probe B mission.

An artist's concept of GP-B measuring the curved spacetime around Earth

"This is an epic result," adds Clifford Will of Washington University in St. Louis. An expert in Einstein's theories, Will chairs an independent panel of the National Research Council set up by NASA in 1998 to monitor and review the results of Gravity Probe B. "One day," he predicts, "this will be written up in textbooks as one of the classic experiments in the history of physics."

Time and space, according to Einstein's theories of relativity, are woven together, forming a four-dimensional fabric called "space-time." The mass of Earth dimples this fabric, much like a heavy person sitting in the middle of a trampoline. Gravity, says Einstein, is simply the motion of objects following the curvaceous lines of the dimple.

If Earth were stationary, that would be the end of the story. But Earth is not stationary. Our planet spins, and the spin should twist the dimple, slightly, pulling it around into a 4-dimensional swirl. This is what GP-B went to space in 2004 to check.

The idea behind the experiment is simple:

Put a spinning gyroscope into orbit around the Earth, with the spin axis pointed toward some distant star as a fixed reference point. Free from external forces, the gyroscope's axis should continue pointing at the star--forever. But if space is twisted, the direction of the gyroscope's axis should drift over time. By noting this change in direction relative to the star, the twists of space-time could be measured.

In practice, the experiment is tremendously difficult.

The four gyroscopes in GP-B are the most perfect spheres ever made by humans. These ping pong-sized balls of fused quartz and silicon are 1.5 inches across and never vary from a perfect sphere by more than 40 atomic layers. If the gyroscopes weren't so spherical, their spin axes would wobble even without the effects of relativity.

According to calculations, the twisted space-time around Earth should cause the axes of the gyros to drift merely 0.041 arcseconds over a year. An arcsecond is 1/3600th of a degree. To measure this angle reasonably well, GP-B needed a fantastic precision of 0.0005 arcseconds. It's like measuring the thickness of a sheet of paper held edge-on 100 miles away.

"GP-B researchers had to invent whole new technologies to make this possible," notes Will.

They developed a "drag free" satellite that could brush against the outer layers of Earth's atmosphere without disturbing the gyros. They figured out how to keep Earth's magnetic field from penetrating the spacecraft. And they created a device to measure the spin of a gyro--without touching the gyro. More information about these technologies may be found in the Science@NASA story "A Pocket of Near-Perfection."

Pulling off the experiment was an exceptional challenge. But after a year of data-taking and nearly five years of analysis, the GP-B scientists appear to have done it.

"We measured a geodetic precession of 6.600 plus or minus 0.017 arcseconds and a frame dragging effect of 0.039 plus or minus 0.007 arcseconds," says Everitt.

For readers who are not experts in relativity: Geodetic precession is the amount of wobble caused by the static mass of the Earth (the dimple in spacetime) and the frame dragging effect is the amount of wobble caused by the spin of the Earth (the twist in spacetime). Both values are in precise accord with Einstein's predictions.

"In the opinion of the committee that I chair, this effort was truly heroic. We were just blown away," says Will.

An artist's concept of twisted spacetime around a black hole. Credit: Joe Bergeron of Sky & Telescope magazine.

The results of Gravity Probe B give physicists renewed confidence that the strange predictions of Einstein's theory are indeed correct, and that these predictions may be applied elsewhere. The type of spacetime vortex that exists around Earth is duplicated and magnified elsewhere in the cosmos--around massive neutron stars, black holes, and active galactic nuclei.

"If you tried to spin a gyroscope in the severely twisted space-time around a black hole," says Will, "it wouldn't just gently precess by a fraction of a degree. It would wobble crazily and possibly even flip over."

In binary black hole systems--that is, where one black hole orbits another black hole--the black holes themselves are spinning and thus behave like gyroscopes. Imagine a system of orbiting, spinning, wobbling, flipping black holes! That's the sort of thing general relativity predicts and which GP-B tells us can really be true.

The scientific legacy of GP-B isn't limited to general relativity. The project also touched the lives of hundreds of young scientists:

"Because it was based at a university many students were able to work on the project," says Everitt. "More than 86 PhD theses at Stanford plus 14 more at other Universities were granted to students working on GP-B. Several hundred undergraduates and 55 high-school students also participated, including astronaut Sally Ride and eventual Nobel Laureate Eric Cornell."

NASA funding for Gravity Probe B began in the fall of 1963. That means Everitt and some colleagues have been planning, promoting, building, operating, and analyzing data from the experiment for more than 47 years—truly, an epic effort.

What's next?

Everitt recalls some advice given to him by his thesis advisor and Nobel Laureate Patrick M.S. Blackett: "If you can't think of what physics to do next, invent some new technology, and it will lead to new physics."

"Well," says Everitt, "we invented 13 new technologies for Gravity Probe B. Who knows where they will take us?"

This epic might just be getting started, after all….

Author: Dr. Tony Phillips Credit: Science@NASA

http://science.nasa.gov/science-news/science-at-nasa/2011/04may_epic/

## Saturday, 7 May 2011

### Pi=4!

*"...I’ll place a straight ruler next to it and measure it and so I'll make a square out of a circle and place a marketplace in its centre where all the roads will be straight… It’ll be like a star which, though round, all its rays go off in all directions in straight lines."*

~~Aristophanes, The Birds

We know Ï€ as the ratio of how many times the diameter of a circle fits in its circumference - somewhere in the region of 3.14159....ad infinitum. Because it is an infinite number, and therefore unknowable in its entirety, it means that in any situation where we use Ï€, the answer we get is also incomplete. Everytime we use Ï€, such as to determine the area of a circle, we only ever achieve an approximation - regardless of how accurate that approximation might be.

We imagine that in-order to increase the accuracy of Ï€, we have to add more and more numbers to the millions and millions of digits that we already have. This might increase the accuracy of Ï€, but it still only amounts to a more accurate approximation. If we want to know the exact answer, what we really need, is to KNOW pi.

Some argue that in certain situations, the infinite figure we have for pi is wrong, and that pi is really 4. This would mean that the number of times the diameter of a circle fits into its circumference is not three and a bit - but 4. Typically, we cannot draw a circle which has a circumference four times its diameter. We could try, but we would fail - and that failure would look a lot like a square.

If we were to roll-out the circumference of a circle, and compare it to the perimeter of a square with the same diameter, it would ably demonstrate that the length of the circle's circumference is roughly three-quarters that of the square. This being the case, how is it possible to produce a square from a circle which shares the same diameter?

In order to reach a length which is four times that of the diameter, it is necessary that the length of the circle's circumference is increased; that is, it is extended in some way. In effect, the line of the circumference has been stretched. Pi has mysteriously increased from the infinite figure of 3.14159... to the more finite figure of 4. What is more, if 4 can be produced by stretching Ï€, it would further suggest that Ï€ is in some way, a compressed form of 4.

The number 4 is nowhere near as slippery as Ï€ to work with - it is neither transcendental, nor irrational. It is not weighed down by an infinite amount of digits after the decimal point. 4 is a nice whole number. In using 4, we no longer get approximate answers, we get THE answer. But how has the number 4 come into existence?

The new number of 4 for pi is not quite as bizarre, nor as impossible, as it at first might appear. The theory goes that a square can be manipulated in such a way that it effectively takes on the semblance of a circle. This circular shape is known as an infinite-sided concave polygon. Concave, as in the sides of the polygon are made to quite literally "cave-in."

*"To remember what concave means it’s best to split the word up like this – “con” + “cave”. The important part is the “cave” part – the word concave is used to describe shapes that have something looking like a cave in them. When you talk about concave polygons, the cave is on the outside of the polygon. Another way of spotting concave polygons is to look out for any interior angles that are larger than 180°. Remember that angles larger than 180° are called reflex angles."*

*~~Image: The reflex angle of a concave polygon has an exterior angle which looks a bit like a "cave".*

If a polygon has a reflex angle, then it is said to be a concave polygon. A reflex angle is greater than 180 degrees and less than 360 degrees. An infinite-sided concave polygon will have an infinite number of sides which "cave-in", producing an infinite number of reflex angles.

A simple concave polygon gives the impression of a shape which is awkward in nature. Increasing the number of "caves" (exterior angles) seems to make the polygon appear only more complex, and irregular. However, just as increasing the number of sides of a regular polygon will see it assume more and more of a circular shape - so too does a concave polygon whose reflex angles are increasing in number, and in uniform fashion. In other words, if we keep increasing the number of sides, ensuring that each side is the same length, and that they occur at regular intervals on the perimeter, then the concave polygon will begin to adopt a shape which is more circular. If you were to try and picture how this might look so far, then it is perhaps easier to imagine the perimeter as the blade of a circular saw.

*~~Image: Circular saw blade on antique portable sawmill.*

In order to cram as many sides into the perimeter as possible, it is imperative that every side has the same length. The sides act as the legs of the reflex angle, and the space formed between them, the "cave" as it were, must be infinitesimally small so that an infinite number of "caves" might occupy the circumference. These exterior angles are fundamental to the shape, as they serve to prevent the legs of the reflex angle from ever touching each other. In the case of an infinite-sided concave polygon the distance between each side is as infinitesimally small as can be imagined. If these "caves" or spaces did not exist, it would be impossible for a perimeter to shake its length, and dissolve into a circumference.

An infinite-sided concave polygon will have an infinite number of reflex angles. The sum of the reflex angles must surely provide some dazzling figure for the number of degrees inside the shape. Funny enough mind, regardless of the infinite figure in the interior, it can be argued that the sum of a concave polygon's exterior angles, infinite-sided or not, shall always remain at 360 degrees. This shares a remarkable symmetry with convex polygons, whose exterior angles undergo the exact same phenomena - they too always add up to 360 degrees.

If we observe each indivual reflex angle in the infinite-sided concave polygon, we find that it is imperative that the number of degrees inside the angle only ever approach 360 degrees. The reflex angle must maintain an angle which is as close to 360 degrees as permissable, so that the exterior angle remains as acute as possible. It is important that the reflex angle is never allowed to fully complete 360 degrees, otherwise the exterior angle would come to equal zero degrees, and the "cave" would simply "pop" out of existence.

If we were to return to our previous analogy, then I suppose we'd be looking at a circular saw blade with an infinite number of tiny teeth - each tooth seperated from the other by an infinitesimal gap. If we ourselves were infinitesimally small, we would be able to see that the circumference of the blade followed a distinct zig-zag pattern. If we were to increase the distance between us and the saw, then it would challenge our previous conception, as the blade now appears perfectly smooth. It's interesting to think that our judgement is made so bias by perspective.

The entire process described above, is brought to dramatic effect in the illustration below. Known to some as "Ï€ = 4! Problem Archimedes?", it's already gained some notoriety on various maths and physics forums, where some others have come to call it - undeservedly perhaps - "Troll Pi":

*~~Image: "Ï€ = 4! Problem Archimedes?"*

What the illustration shows, and rather well, is how the perimeter of a square might be reduced to occupy the circumference of an inscribed circle. Ordinarily, this would be deemed impossible because a square's perimeter is four times its diameter, while a circle's circumference is just over three times its diameter. To get a square to squeeze into a circle, the length of the square's perimeter will need to be shortened by as much as a quarter.

It begins with the square's corners being "removed" or inverted, so that it adopts the shape of a concave polygon. Accordingly, more corners are inverted, meaning that more sides are added, and the perimeter pretty much resembles a ziggurat made up of "Lego" blocks. More and more of the perimeter is encroached upon as the number of sides increase. As their numbers swell, we see the size of the blocks decrease - so much so - that by the time we arrive at numbers which are infinite, the perimeter is now a curve which possess a row of barely imperceptible jagged little teeth. Under its new guise, the perimeter seems able to fit perfectly inside the circumference of the circle.

One might imagine the square as having been panel-beaten, then concertinaed into the shape of a circle. The circle has been created by nothing short of square-mangling. If this is true, then we might also assume that the circle is just as easily taken and bent back into the shape of the square. Flipping between these shapes - from circle to square, and from square to circle - we can see that nowhere in the entire process is anything either added, or taken away. At no-time is the line cut or dismantled. The only thing which is observed is MOTION.

For a square to transform into a circle, its sides are allowed to crumble and fall into rubble around the circle's circumference. Those once imposing corners of the square have somehow been lost - buried - in the defining line of the circumference. The square can be rebuilt however, and it begins by sweeping the rubble back into piles - piles which grow bigger and bigger - until they are at last gathered into four mountainous heaps.

To summarise then, "Ï€ = 4! Problem Archimedes?" is a revelation in how the circumference can act as a perimeter, and how the perimeter can change into a circumference. This simple illustration explains very well how this transformation takes place, clearly showing how it is possible for the length to remain unchanged. The length of the square's perimeter is ALWAYS the same - and remains so - even when it is asked to assume the role of a circle's circumference.

All this is very good in theory, but if we were to examine a circle on a page - how do we go about peeling the circumference of a circle from the paper, so that we might use it to make a square of the same diameter? Surely, everytime we unravelled the circumference and measured it, the answer will always be Ï€, not 4? To reach 4, something needs to be added to Ï€. From where has the required extra bit materialised?

What is required is a reliable mathematical method to show how the extra length is gained legitimately. In a landmark paper, "The Extinction of Ï€", Miles Mathis proposes that he has discovered one such method. Mathis appears to have resolved the problem of pi=4 by including time in his geometric analysis. It can be seen that his geometry is no longer describing some static abstraction, but something much more physical. Something real.

Much thanks:

http://en.wikipedia.org/wiki/Circle

http://betterexplained.com/articles/prehistoric-calculus-discovering-pi/

http://mathforum.org/~sanders/exploringandwritinggeometry/polygons.htm

http://paramanand.blogspot.com/2010/12/angle-sum-formula-for-polygons.html

http://www.jimloy.com/geometry/pi.htm

http://milesmathis.com/pi3.html

http://dyinglovegrape.wordpress.com/2010/11/17/homology-primer-2-triangulating-a-surface/

http://www.physicsinsights.org/pi_from_pythagoras-1.html

http://www.philosophynow.org/issue81/Mathematical_Knowledge_A_Dilemma

http://www.tutorvista.com/math/infinite-lines-of-symmetry

http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_number.php

http://www.mathopenref.com/polygoncentralangle.html

http://www.jimloy.com/geometry/pentagon.htm

http://forums.xkcd.com/viewtopic.php?f=3&t=67907

http://www.pi314.net/eng/aleatoire.php

http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Mylod/Math7200/Project/InscribedCircle.html

http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/integration/archimedes.html

http://www.themathpage.com/atrig/measure-angles.htm

## Thursday, 14 April 2011

### Circling The Square

*"Most people would say that a circle has no corners, but it is more accurate to say that it has an infinite number of corners."*

I like this fact because it forces us to examine the nature of a circle. If a circle has corners, then surely, it must be related to the square in a fundamental way. The circle is proof that mere appearances can be deceptive. In a mathematical sense, a square and circle are not two entirely different shapes as such. Those severe bends that we see forming the four corners of a square, can surprisingly, be found to also exist in those soft curves which form a circle. The best way to get from a square to a circle is by increasing its number of corners.

*~~Image: Red Square Painting (2009 Digital Remix) by Nigel Tomm*

A square has its four sides and its four corners, and if we add another side to it, to create a five-sided polygon, then it also generates another corner. If everytime we keep adding more sides, and everytime we make all the sides the same length - then we can go on to form a fantastic array of regular polygons.

If we keep on increasing the polygon's number of sides, and keep on increasing them, then theoretically, the sides of the polygon will eventually reach numbers which are infinite. It shall produce a polygon with an infinite number of sides - and for all intents and purposes - a shape that looks convincingly like a circle. It is by creating polygons that the early pioneers of geometry, such as Archimedes, were able to gain more and more accurate approximations of pi. The following extract below comes from this site, which also offers some excellent working demos of how pi can be approximated - one by unravelling circles, and another by inscribing polygons:

*"Ï€ is an irrational number, which means that we can never write the value of it completely accurately. So how do we calculate it? After all, it is difficult to measure round the edge of a circle. You could get an approximation by winding a piece of string round a tin, then measuring the string and across the tin, but this will not be very accurate. Another way is to fit a polygon (like a square or a hexagon) to the circle, either inside or outside. We can calculate the edge of a polygon. As we increase the number of sides in the polygon, it fits the circle better and better, so its edge becomes closer and closer to the circumference of the circle. What is more, the outer polygon will have a longer edge than the circle, and the inner one will be less. So we can get two approximations for for each polygon, one too big and one too small."*

As you increase the number of sides of a polygon, you increase the sum of the interior angles of the polygon. Each time we increase the number of sides by one, the sum of the angles increase by 180 degrees. For example, a square (quadrilateral) has the interior sum of 360 degrees, while a five-sided polygon (pentagon) has a sum of 540 degrees. Going further, we see that the sum of the interior angles of a ten-sided polygon (decagon) are 1440 degrees. The sum of the interior angles of a polygon are calculated by inscribing triangles (triangulating). If we know that the sum of the interior angles of a triangle are ALWAYS 180 degrees, and we can count the number of triangles being used to form the shape of the polygon - then we have the perfect formula for calculating the sum of the polygon's interior angles (n being the polygon's number of sides):

(n-2) × 180° = sum of interior angles

Basically, everytime we add a side to a polygon, we generate a new triangle inside the polygon, and increase the sum of the interior angles by 180 degrees. A square, for example, can be made up by two triangles (hence 2 × 180° = 360°), while a pentagon can be made up by three triangles (3 × 180° = 540°).

*~~Image: There are two triangles in a square.*

*~~Image: A pentagon has five sides, and can be made from three triangles.*

If a circle can be described as a polygon with an infinite number of small sides, then we must assume that the sum of the interior angles of such a circle will too approach figures somewhere in the infinite. If we were to try and triangulate such a polygon to try and reach the sum of the interior angles, we would have to deduct two from the number of sides to give us the number of triangles. This means we would be left trying to tackle the rather troublesome sum of infinity minus 2 (n-2) to achieve the answer.

Trying to add or subtract to infinity is always a little awkward. After all, infinity is considered to be a concept rather than a number - you can't just go around ripping bits off it, or for that matter, slapping things on it. In order to deduct 2 from infinity to get a number, it means that we would have to ask infinity to be a little less infinite, and be a bit more finite, which is probably asking the impossible. Or is it?

Assuming that the sum of the interior angles will reach amounts which end somewhere in the infinite, it remains that the sum of the exterior angles of a such a polygon, a polygon with an infinite number of sides, if measured, will still be found to equal 360 degrees. This is because the sum of the exterior angles of any convex polygon will ALWAYS add upto 360 degrees. Essentially, all the exterior angles amount to one full revolution (360°). In other words, adding all the exterior angles together is the mathematical equivalent of taking the shape and rotating it one complete turn.

*~~Image: In this diagram the exterior angles have been given different colours. You can see how they can be put together to make a full circle.*

If we add up the interior angle and the exterior angle of a regular polygon, we get a straight line - 180 degrees. The interior and exterior angles are distinctly related. We can increase the polygon's number of sides to figures which are infinite, and with it, we will also see an increase in the sum of the interior angles. In theory, the number of degrees should become infinite - infinitely big. However, each interior angle cannot be seen to be equal to, or exceed, the boundary of 180 degrees, otherwise we will encroach upon the space of the exterior angle.

*~~Image: Internal angle + external angle = 180°*

Increasing the number of sides will see the sum of the interior angles grow,and grow, but this growth is wholly reliant on each exterior angle, the one at each vertex, becoming smaller, and smaller - infinitely smaller. In other words, the growth of the sum of the interior angles is severely restricted. The infinite sum of the interior angles are by no means boundless.

*" The sum of the exterior angles of a polygon are 360 degrees regardless of the number of sides. That means that the measure of each exterior angle must get smaller as the number of sides increases. There is no "least possible measure" because even though the limiting value is 0 you can never achieve a 0 degree exterior angle and still have a polygon. You can get as close to zero as you like, but as close as you get, someone else can always come along and get closer. Another way to look at it is that a zero degree exterior angle measure implies that there are an infinite number of sides. And an infinite number of sides implies a circle, not a polygon." *

Personally, I would argue that a circle, or infinite sided polygon, does not possess a zero degree exterior angle. That's because an infinite amount of nothing will still give you nothing. Nevermind how much nothing you get, you'll still be left holding nothing. The sum of the exterior angles, regardless of the fact that they are infinite in number, shall always add upto 360 degrees. Therefore each exterior angle must be seen to amount to something, even if it is something infinitesimally tiny, in order to reach the sum of 360 degrees. Coincidentally, this exact same restriction we find outside the circumference of any polygon, or circle, is also at work in the shape's centre.

One of the defining properties of a circle, and indeed, any regular polygon, is that its entire central angle ALWAYS measures 360 degrees. If we were to add a central vertex, or central point to a pentagon for example, and inscribe triangles in the same way that we might slice up a pizza, then our pentagon would produce 5 triangles - all sides would have the same length, and all the interior angles would be the exact same size. The central angle of each triangle will also be the same, and the sum of these shall ALWAYS add upto 360 degrees.

*~~Image: The central angle of a regular pentagon (5 × 72° = 360°)*

Using this same method, we can imagine inscribing triangles to an infinitely sided polygon, to create an infinite number of infinitesimal triangles. But if we add the sum of these infinitesimally tiny central angles together, they produce the sum of 360 degrees. It doesn't matter how many triangles we have, infinite number or not, they shall always add up to 360 degrees. An infinite sided polygon does not have a central angle whose sum reaches an infinite number of degrees - it produces only an infinite number of ways to percieve the finite sum of 360 degrees.

I have always imagined infinity as an entity which fulfils the very definition of freedom. But where is all the freedom that I was hoping to embrace? No, infinity offers only the illusion of freedom. Infinity can never escape the confines of the finite. The term infinite is not actually describing the phenomena of ever-expanding space - it only pretends to.

We imagine that in order to behold the infinite we have to travel to some far-flung, incomprehensible horizon - but the reality is, for us to comprehend the infinite, we don't so much as have to leave the spot. Infinity is always describing the exact same space - a space chopped up into an infinite number of ways, an infinite number of ways in-which to percieve it - but it is the exact same space nonetheless.

If the state of infinity were truly free, then surely, there should be no restrictions to its growth whatsoever. Here however, we can see that infinity is shackled to enormous constraints. We can try to develop a sense of the infinitely big - building a polygon with infinite sides - but we find that that growth is constricted by a number of finite limits.

Infinite growth is restricted by constraints imposed both inside and outside the circle. The sum of the central angle, and the sum of the exterior angle can never exceed 360 degrees. Also, the linear pair of the exterior and interior angles can never exceed 180 degrees. Infinity is dependent upon how big we make each interior angle, and at the same time, how small we make each exterior angle. Each interior angle can never extend to, or beyond 180 degrees, and at the same time, the exterior angle can never be allowed to fall to zero. If any of these restrictions are breached, well, then you no longer have a perfect circle.

A circle may well be a polygon with infinite sides, but at its heart, it is still very much a square - a square bent infinitely out of shape - but a square nonetheless. A square of finite proportions.

The sum of the interior angles, the angles which exist inside the circumference of the circle as it were, can reach figures which are infinite - but this stands only as an expression of how limited our understanding is. The sum of the interior angles could never truly, unrelentlessly expand into infinite space - there is a limit in place. A limit so vast that it is unknowable - but a limit nonetheless. We may not be able to know the number of that limit, but we can see it. We see it all the time. That's because the limit itself is a construct of a remarkably simple shape - the circle.

*~~Image: The human eye - one of the most outstanding examples of a circle that we see everyday.*

The state of infinity perhaps, might best be described as a place that exists somewhere between a square and a circle. If this were true, what exactly does it mean for the supposedly infinite ratio of pi?

## Saturday, 9 April 2011

### Contemplating Infinity

*The word after "infinity" in my dictionary is "infirm," a definition of which is "weak of mind." This is how many of us who are not mathematically inclined feel upon contemplating infinity. (To see how mathematicians and similar thinkers regard infinity, see Working With Infinity: A Mathematical Perspective.)*

We feel weak because our finite minds can only go so far with the concept, and because every time we think we're on the verge of securing even a shadowy understanding, we're tripped up by something. A friend of mine once told me that trying to hold her hyperactive toddler was like trying to hold a live salmon. Infinity is like that for us "infirm ones": slippery as a salmon, forever eluding our grasp.

Becoming numb

This is true no matter how you approach the concept. Many of us might consider numbers the most sure-footed way to come within sight of infinity, even if the mathematical notion of infinity is something we'll never even remotely comprehend.

We may think, for starters, that we're well on our way to getting a sense of infinity with the notion of no biggest number. There's always an ever larger number, right? Well, no and yes. Mathematicians tell us that any infinite set—anything with an infinite number of things in it—is defined as something that we can add to without increasing its size. The same holds true for subtraction, multiplication, or division. Infinity minus 25 is still infinity; infinity times infinity is—you got it—infinity. And yet, there is always an even larger number: infinity plus 1 is not larger than infinity, but 2infinity is.

Try another tack: huge numbers. When we play with mind-boggling figures we non-math types may think we're playing in infinity's neighborhood, if not in the same playground. When we're told that the decimals in certain significant numbers, like pi and the square root of two, go on forever, we can somehow accept that, especially when we learn that computers have calculated the value of pi, for one, to over a trillion places, with no final value for pi in sight. (For more on pi, see Approximating Pi.) When we're told that there are 43,252,003,274,489,856,000 possible ways to arrange the squares on the Rubik Cube's six sides, we may feel intuitively (if not rationally) that we must be on our way to the base of that loftiest of all peaks, Mt. Infinity.

One reason we may feel this way is that such numbers are as intellectually unapproachable to the mathematically challenged as infinity itself. Take a Googol. A Googol is 10100, or 1 followed by 100 zeroes, and is the largest named number in the West. The Buddhists have an even more robust number, 10140, which they know as asankhyeya. Just for fun, I'll name a larger number yet, 101000. I'll call it the "Olivian," after my daughter. Now, doesn't an Olivian get me a little closer to infinity than the Googolians or even the Buddhists can get? Nope. Infinity is just as far from an Olivian as it is from a Googol—or, for that matter, from 1.

For many of us uncomfortable with infinity, the word number can be defined as “that which makes numb.”

Perhaps we infirm ones would be wise to take a leaf from the lingual book of Madagascar. The word there for a million is tapitrisa, which means "the finishing of counting." For some tribal groups in other parts of the world, counting stops at three, in fact; anything above that is "many." In some ways this makes sense. How many of us can keep more than a few things in our minds at once? I remember playing a game with myself as a child in which I would think "I'm thinking that I'm thinking that I'm thinking that I'm thinking...." After the third or fourth "I'm thinking," I could no longer retain in my head all the degrees it implies. Such infirmity holds for simple counting as well, as Lewis Carroll reveals so tellingly in Through the Looking Glass:

"Can you do Addition?" the White Queen asks. "What's one and one and one and one and one and one and one and one and one and one?"

"I don't know," said Alice. "I lost count."

"She can't do Addition," the Red Queen interrupted. "Can you do Subtraction?"

For many of us uncomfortable with infinity, the word number can be defined as "that which makes numb," as Rudy Rucker wryly notes in his book Infinity and the Mind (BirkhÃ¤user, 1982). This is especially true when a number is so outlandishly enormous that it smacks, however remotely, of the infinite. Galileo himself felt this way. "Infinities and indivisibles transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness," he wrote in his Dialogues of Two New Sciences of 1638. "Imagine what they are when combined." Rather not, thanks—makes me numb.

Incredible shrinking

Infinities do come in two sizes, of course—not only the infinitely large but also the infinitely small. As Jonathan Swift wrote, "So, naturalists observe, a flea/Has smaller fleas that on him prey/And these have smaller still to bite `em/And so proceed ad infinitum." We may not be able to conceive of Swift's infinitesimal fleas, because reason insists they don't exist, but we can imagine ever smaller numbers without much trouble. It's no hardship, for example, to grasp the notion of an infinity of numbers stretching between, say, the numerals 2 and 3. Take half of the 1 that separates them, we might tell ourselves, then half of that half, then half of that half, and so proceed ad infinitum.

Of course, just when we think we have infinity in the palm of our hands, we watch it evaporate in the harsh light of another of those confounding paradoxes: the numerals 2 and 3 are separated by both a finite number (1) and an infinity of numbers. This conundrum spawned one of the great paradoxes of history, known as Zeno's paradox. Zeno was a Greek philosopher of the fourth century B.C. who "proved" that motion was impossible. For a runner to move from one point to another, Zeno asserted, he must first cover half the distance, then half the remaining distance, then half the remaining distance again, and so on and so on. Since this would require an infinite number of strides, he could never reach his destination, even if it lay just a few strides away.

It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit. Thus, even though we can't count all the numbers between 2 and 3, we know they converge to 1.

No limits

As Zeno's paradox hints, considering infinity from the perspective of space has much correspondence with that of numbers. We can imagine, for instance, that space, like numbers, is infinitely divisible. We believe Hamlet when he says "I could be bounded in a nutshell/And count myself a king of infinite space." The shortest length physicists speak of is the Planck length, 10-33 centimeters. But might not there be an even shorter length, say, 10-333 centimeters, or 10-an infinite number of 3's centimeters?

Many of us are as queasy around eternity as we are around infinity.

As with numbers, we can also envision space as being infinitely large. After all, if the universe has a boundary, what's on the other side? We might flatter ourselves that we're somehow getting closer to infinity when we consider extremely large distances. On June 12, 1983, while traveling at over 30,000 mph, the Pioneer 10 spacecraft became the first human-made object to exit our solar system. Some 300,000 years from now, unless something interrupts its voyage, the craft is expected to pass near the star Ross 248, a red dwarf in the constellation Taurus. Ross 248 is about 10.1 light-years from Earth, or about 59,278,920,000,000 miles away. Pioneer 10 will still be in the early stages of its journey, though. When our sun bloats into a red giant about five billion years from now and incinerates our planet, our robotic ambassador will still be heading away, knocking off more than 250 million miles a year.

Are we making headway towards an infinite distance with such knowledge? Hardly. An infinite distance, as you've guessed, would be as far from where Pioneer 10 will be in five billion years as it is from the Earth now. If the universe is infinitely large, even the remotest stars we can detect, which are so far away that their light left them some 12 billion years ago, are as far from infinity as we are. (Things get tricky here: as one mathematician pointed out to me, infinity is an abstract concept, appearing only in our mental images of the universe. It is not actually in the universe.)

Forever and a day

Time is another way to contemplate infinity, though many of us are as queasy around eternity as we are around infinity. ("That's the trouble with eternity, there's no telling when it will end," Tom Stoppard writes in Rosencranz and Guildenstern Are Dead.) Yet isn't infinite time somehow easier to swallow than finite time? After all, what can stop time?

Many of us do indeed live our lives thinking that eternity is a given. And again, we may fool ourselves into thinking that we're on the way to eternity when we think of 12 billion years, or of any other frighteningly mind-bending length of time. One of the gamest attempts to define eternity appears in Hendrik Willem Van Loon's 1921 children's classic The Story of Mankind:

High up in the North in the land called Svithjod, there stands a rock. It is 100 miles high and 100 miles wide. Once every thousand years a little bird comes to the rock to sharpen its beak. When the rock has thus been worn away, then a single day of eternity will have gone by.

That passage gives you an inkling for just how gosh-darn long eternity is. But all the usual caveats apply: eternity doesn't have a length, that single "day" of eternity is as far in time from eternity itself as a normal day, etc., etc.

Fear of the infinite

If all this leaves you feeling numb, you're not alone. The Greeks, in fact, invented apeirophobia, fear of the infinite. (The term comes from the Greek word for infinity, apeiron, which means "without boundary.") Aristotle would only admit that the natural numbers (1, 2, 28, etc.) could be potentially infinite, because they have no greatest member. But they could not be actually infinite, because no one, he believed, could imagine the entire set of natural numbers as a finished thing. The Romans felt just as uncomfortable, with the emperor Marcus Aurelius dismissing infinity as "a fathomless gulf, into which all things vanish."

“Infinity is where things happen that don’t.”

The ancients' horror infiniti held sway through the Renaissance and right up to modern times. In 1600, the Inquisitors in Italy deemed the concept so heretical that when the philosopher Giordano Bruno insisted on promulgating his thoughts on infinity, they burned him at the stake for it. Later that century, the French mathematician Blaise Pascal deemed the concept truly disturbing: "When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there ... now instead of then." Martin Buber, an Israeli philosopher who died in 1965, felt so undone by the concept of infinity that he "seriously thought of avoiding it by suicide."

Most of us will never feel so put out by infinity that we'll resort to contemplating such extreme measures. We may feel weak of mind, like the anonymous schoolboy who once declared that "infinity is where things happen that don't." But our uneasiness will never get much greater than the schoolboy's delightfully dismissive attitude suggests his got. We can live with that level of discomfort, contenting ourselves with the knowledge that all we can reasonably expect in musing on infinity is to get a feeling for it, like that engendered by this gem from another anonymous sufferer of our common infirmity: "Infinity is a floorless room without walls or ceiling."

We feel weak because our finite minds can only go so far with the concept, and because every time we think we're on the verge of securing even a shadowy understanding, we're tripped up by something. A friend of mine once told me that trying to hold her hyperactive toddler was like trying to hold a live salmon. Infinity is like that for us "infirm ones": slippery as a salmon, forever eluding our grasp.

Becoming numb

This is true no matter how you approach the concept. Many of us might consider numbers the most sure-footed way to come within sight of infinity, even if the mathematical notion of infinity is something we'll never even remotely comprehend.

We may think, for starters, that we're well on our way to getting a sense of infinity with the notion of no biggest number. There's always an ever larger number, right? Well, no and yes. Mathematicians tell us that any infinite set—anything with an infinite number of things in it—is defined as something that we can add to without increasing its size. The same holds true for subtraction, multiplication, or division. Infinity minus 25 is still infinity; infinity times infinity is—you got it—infinity. And yet, there is always an even larger number: infinity plus 1 is not larger than infinity, but 2infinity is.

Try another tack: huge numbers. When we play with mind-boggling figures we non-math types may think we're playing in infinity's neighborhood, if not in the same playground. When we're told that the decimals in certain significant numbers, like pi and the square root of two, go on forever, we can somehow accept that, especially when we learn that computers have calculated the value of pi, for one, to over a trillion places, with no final value for pi in sight. (For more on pi, see Approximating Pi.) When we're told that there are 43,252,003,274,489,856,000 possible ways to arrange the squares on the Rubik Cube's six sides, we may feel intuitively (if not rationally) that we must be on our way to the base of that loftiest of all peaks, Mt. Infinity.

One reason we may feel this way is that such numbers are as intellectually unapproachable to the mathematically challenged as infinity itself. Take a Googol. A Googol is 10100, or 1 followed by 100 zeroes, and is the largest named number in the West. The Buddhists have an even more robust number, 10140, which they know as asankhyeya. Just for fun, I'll name a larger number yet, 101000. I'll call it the "Olivian," after my daughter. Now, doesn't an Olivian get me a little closer to infinity than the Googolians or even the Buddhists can get? Nope. Infinity is just as far from an Olivian as it is from a Googol—or, for that matter, from 1.

For many of us uncomfortable with infinity, the word number can be defined as “that which makes numb.”

Perhaps we infirm ones would be wise to take a leaf from the lingual book of Madagascar. The word there for a million is tapitrisa, which means "the finishing of counting." For some tribal groups in other parts of the world, counting stops at three, in fact; anything above that is "many." In some ways this makes sense. How many of us can keep more than a few things in our minds at once? I remember playing a game with myself as a child in which I would think "I'm thinking that I'm thinking that I'm thinking that I'm thinking...." After the third or fourth "I'm thinking," I could no longer retain in my head all the degrees it implies. Such infirmity holds for simple counting as well, as Lewis Carroll reveals so tellingly in Through the Looking Glass:

"Can you do Addition?" the White Queen asks. "What's one and one and one and one and one and one and one and one and one and one?"

"I don't know," said Alice. "I lost count."

"She can't do Addition," the Red Queen interrupted. "Can you do Subtraction?"

For many of us uncomfortable with infinity, the word number can be defined as "that which makes numb," as Rudy Rucker wryly notes in his book Infinity and the Mind (BirkhÃ¤user, 1982). This is especially true when a number is so outlandishly enormous that it smacks, however remotely, of the infinite. Galileo himself felt this way. "Infinities and indivisibles transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness," he wrote in his Dialogues of Two New Sciences of 1638. "Imagine what they are when combined." Rather not, thanks—makes me numb.

Incredible shrinking

Infinities do come in two sizes, of course—not only the infinitely large but also the infinitely small. As Jonathan Swift wrote, "So, naturalists observe, a flea/Has smaller fleas that on him prey/And these have smaller still to bite `em/And so proceed ad infinitum." We may not be able to conceive of Swift's infinitesimal fleas, because reason insists they don't exist, but we can imagine ever smaller numbers without much trouble. It's no hardship, for example, to grasp the notion of an infinity of numbers stretching between, say, the numerals 2 and 3. Take half of the 1 that separates them, we might tell ourselves, then half of that half, then half of that half, and so proceed ad infinitum.

Of course, just when we think we have infinity in the palm of our hands, we watch it evaporate in the harsh light of another of those confounding paradoxes: the numerals 2 and 3 are separated by both a finite number (1) and an infinity of numbers. This conundrum spawned one of the great paradoxes of history, known as Zeno's paradox. Zeno was a Greek philosopher of the fourth century B.C. who "proved" that motion was impossible. For a runner to move from one point to another, Zeno asserted, he must first cover half the distance, then half the remaining distance, then half the remaining distance again, and so on and so on. Since this would require an infinite number of strides, he could never reach his destination, even if it lay just a few strides away.

It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit. Thus, even though we can't count all the numbers between 2 and 3, we know they converge to 1.

No limits

As Zeno's paradox hints, considering infinity from the perspective of space has much correspondence with that of numbers. We can imagine, for instance, that space, like numbers, is infinitely divisible. We believe Hamlet when he says "I could be bounded in a nutshell/And count myself a king of infinite space." The shortest length physicists speak of is the Planck length, 10-33 centimeters. But might not there be an even shorter length, say, 10-333 centimeters, or 10-an infinite number of 3's centimeters?

Many of us are as queasy around eternity as we are around infinity.

As with numbers, we can also envision space as being infinitely large. After all, if the universe has a boundary, what's on the other side? We might flatter ourselves that we're somehow getting closer to infinity when we consider extremely large distances. On June 12, 1983, while traveling at over 30,000 mph, the Pioneer 10 spacecraft became the first human-made object to exit our solar system. Some 300,000 years from now, unless something interrupts its voyage, the craft is expected to pass near the star Ross 248, a red dwarf in the constellation Taurus. Ross 248 is about 10.1 light-years from Earth, or about 59,278,920,000,000 miles away. Pioneer 10 will still be in the early stages of its journey, though. When our sun bloats into a red giant about five billion years from now and incinerates our planet, our robotic ambassador will still be heading away, knocking off more than 250 million miles a year.

Are we making headway towards an infinite distance with such knowledge? Hardly. An infinite distance, as you've guessed, would be as far from where Pioneer 10 will be in five billion years as it is from the Earth now. If the universe is infinitely large, even the remotest stars we can detect, which are so far away that their light left them some 12 billion years ago, are as far from infinity as we are. (Things get tricky here: as one mathematician pointed out to me, infinity is an abstract concept, appearing only in our mental images of the universe. It is not actually in the universe.)

Forever and a day

Time is another way to contemplate infinity, though many of us are as queasy around eternity as we are around infinity. ("That's the trouble with eternity, there's no telling when it will end," Tom Stoppard writes in Rosencranz and Guildenstern Are Dead.) Yet isn't infinite time somehow easier to swallow than finite time? After all, what can stop time?

Many of us do indeed live our lives thinking that eternity is a given. And again, we may fool ourselves into thinking that we're on the way to eternity when we think of 12 billion years, or of any other frighteningly mind-bending length of time. One of the gamest attempts to define eternity appears in Hendrik Willem Van Loon's 1921 children's classic The Story of Mankind:

High up in the North in the land called Svithjod, there stands a rock. It is 100 miles high and 100 miles wide. Once every thousand years a little bird comes to the rock to sharpen its beak. When the rock has thus been worn away, then a single day of eternity will have gone by.

That passage gives you an inkling for just how gosh-darn long eternity is. But all the usual caveats apply: eternity doesn't have a length, that single "day" of eternity is as far in time from eternity itself as a normal day, etc., etc.

Fear of the infinite

If all this leaves you feeling numb, you're not alone. The Greeks, in fact, invented apeirophobia, fear of the infinite. (The term comes from the Greek word for infinity, apeiron, which means "without boundary.") Aristotle would only admit that the natural numbers (1, 2, 28, etc.) could be potentially infinite, because they have no greatest member. But they could not be actually infinite, because no one, he believed, could imagine the entire set of natural numbers as a finished thing. The Romans felt just as uncomfortable, with the emperor Marcus Aurelius dismissing infinity as "a fathomless gulf, into which all things vanish."

“Infinity is where things happen that don’t.”

The ancients' horror infiniti held sway through the Renaissance and right up to modern times. In 1600, the Inquisitors in Italy deemed the concept so heretical that when the philosopher Giordano Bruno insisted on promulgating his thoughts on infinity, they burned him at the stake for it. Later that century, the French mathematician Blaise Pascal deemed the concept truly disturbing: "When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there ... now instead of then." Martin Buber, an Israeli philosopher who died in 1965, felt so undone by the concept of infinity that he "seriously thought of avoiding it by suicide."

Most of us will never feel so put out by infinity that we'll resort to contemplating such extreme measures. We may feel weak of mind, like the anonymous schoolboy who once declared that "infinity is where things happen that don't." But our uneasiness will never get much greater than the schoolboy's delightfully dismissive attitude suggests his got. We can live with that level of discomfort, contenting ourselves with the knowledge that all we can reasonably expect in musing on infinity is to get a feeling for it, like that engendered by this gem from another anonymous sufferer of our common infirmity: "Infinity is a floorless room without walls or ceiling."

~~Contemplating Infinity:

A Philosophical Perspective

by Peter Tyson

http://www.pbs.org/wgbh/nova/archimedes/contemplating.html