The diagram below appears to show a black circle against a white background. But we could equally well interpret it as a white circle with a black interior. The figure as such - the circle - is neither black nor white, foreground or background, but the boundary between the black field that it envelops and the enveloping white field around it. It is only a bounded black 'foreground' figure from an ordinary spatial point of view. From a counter-spatial perspective the figure is an internal configuration of its enveloping white field.
From a spatial perspective, the circle marks the circumference of a bounded inner field which it envelops, and is surrounded by an enveloping field that is essentially unbounded - the only boundary being the artificial one created by the diagram box itself. From a counter-spatial point of view, the circle is itself an internal boundary of the enveloping white field, and it is the enveloped black field which is boundless. This conforms with the general definition of counter-space as a type of space constituted not by an infinitely distant circumference but by an infinitely distant centre or 'counter-space infinitude'.
Once this topology of a two-dimensional figure such as a circle is grasped in this way it is clear that three dimensional forms too, cannot be identified with either their enveloped or enveloping fields or their constituent elements. Thus we can talk of bubbles of air in a fluid or of bubbles of fluid containing air. The bubble as such is neither. It is an air bubble in a fluid only from a spatial perspective. Counter-spatially, it is a fluid bubble containing air. But what then, of bubbles or balloons that are blown with air and also surrounded by air. These can indeed be said to be fluid bubbles, but they still constitute surface boundaries between enveloped and enveloping fields and spaces. And any such surface has itself spatial and counter-spatial dimensions. For the surface of a bubble or balloon can be viewed spatially, as the outer surface of the field it envelops, and counter-spatially, as the inner surface of an enveloping field. This picture is further complicated by the simple recognition that any surface, whether of two or three dimensions, has itself two sides. Thus a three-dimensional surface such as a balloon has an inner surface that constitutes a boundary with its interior or enveloped field and an outer surface that constitutes its boundary with an exterior or enveloping field. Any three-dimensional surface therefore, is not simply one surface but four surfaces in one:
the inner surface of an outer field
the outer surface of an inner field
the inner surface of an inner field
the outer surface of an outer field
From a spatial point of view there are only two surfaces to the balloon, an outer and an inner surface. Counter-spatially however, the outer surface of the balloon is the inner surface of an enveloping field, and the inner surface is an outer surface of this same field. The result is four surfaces, or rather five - for these four surfaces are distinct but inseparable aspects of that single surface, the balloon as such.
The field-relation between spatial and counter-spatial dimensions of three-dimensional surfaces is basic, not just to an understanding of hollow bodies like bubbles or balloons but so-called solid bodies, which we know are not solid in the way they appear to us to be but are largely composed of spaces between atoms. Above all it is a reminder that two-dimensional figures or three-dimensional forms as such are essentially massless, for like the circle figure in the diagram which is neither black nor white, three-dimensional forms are reducible neither to material bodies nor to the spatial or energetic fields enveloping those bodies.
The next step towards a Fundamental Topology is to consider both enveloping and enveloped fields not simply as pre-given spatial fields but as fields of spatial awareness. Here the human body or that of any organism provides us with a basic model. The human body as we perceive it from without, is the human body perceived within our own field of extensional, spatial awareness. As such its outer surface or skin is, for us, the inner surface of an outer field of awareness that is in no way bounded by our own skins. Similarly the walls of a room we are in are, and the surfaces of any objects or material bodies in that room are also inner surfaces of our outer field of awareness. As for our own inner field of awareness, the field of our inner bodily self-awareness, this is not itself visible to others from the outside, even if they were to open up our body and look inside it. From this external point of view, the interiority of the human body contains only organs, muscle, fat and bone tissue. It is composed largely of water. We ourselves are little more than skin-bags of water.
The outer surface of our skin is the outer surface of this water bag, but it is not the outer surface of our field of inner bodily self- awareness. For as a field of awareness, this can contract beneath the surface of our skins, expand beyond it, or expand within it. The inner field of our bodily self-awareness, unlike our field of outer spatial awareness, has a counter-spatial and non-physical character - it is an awareness that opens up within an enveloping field and within a skin or envelope of awareness. Counter-space is characterized by its centripetal character, in contrast to the centrifugal character of ordinary space. All physical skin sensations on the other hand - of warmth or contact, irritation or itching etc. are a counter-spatial awareness of things and people in our outer field - our clothes, the air around us, sound, light and other energies.
The physical skin itself forms no part of the human organism as a body of awareness with its own skin - its envelope of awareness or 'psychic envelope'. This psychic envelope's surface is a boundary not between the inside and outside of our body but between two fields of awareness - the field of our inner bodily self-awareness on the one hand, and the outer field of our sensory awareness of the world. The inner field enveloped by the psychic envelope is not bounded by it, for counter-spatially, it leads inward into a non-extensional or intensional space.
No Fundamental Topology is possible without a fundamental distinction, not only between space and counter-space, but between extensional and intensional space. Counter-space leads into intensional space. But intensional space, far from simply being enveloped by extensional figures and forms surrounded by their own enveloping extensional fields of awareness, actually envelope these enveloping fields. The extensional spaces of our dreams occupy no extensional physical space but open up and expand within a non-extensional or intensional field of awareness, and are in this sense enveloped by them. What is true of our dreams applies to extensional spaces and fields of awareness in general, all of which open up and expand, within intensional fields and intensional space and are in this sense enveloped by them. The relational topology between extensional and intensional space is difficult to picture, because this requires that we represent intensional space itself in an extensional way. Nevertheless, it is possible to use an extensional diagram to show the way in which an enveloped inner field can lead into an intensional space that envelops the enveloping field. Diagram 2 represents the human organism as an envelope of awareness in the form of a circle. The psychic envelope both envelops its own inner field or space of awareness and is enveloped by a seemingly boundless outer field of extensional spatial awareness.
Diagram 3, represents the way in which the inner field of the psychic envelope leads into an intensional space that actually envelops its own extensional field of physical space.
This 'keyhole' diagram is the key to understanding the relation of intensional and extensional space, showing how an enveloped field can lead inwardly - counter-spatially or centripetally - into an intensional space that envelops its own enveloping field. It also shows however, how the inner surface of an extensional figure or form, the inner surface of its inner field, is contiguous with that which lies 'behind' the outer surface of its outer field - in other words the inner surface of its enveloping field. If we think again of the walls of a room as the outer surface of our own outer field of awareness, then what lies 'beyond' or 'behind' these perceived wall surfaces is not just plasterboard or brick and mortar, and beyond them, the space around our planet. That is only their extensional behindness and beyondness. On a more fundamental level, what lies behind, beyond and within the extensional objects and surfaces we perceive around us is the intensional space within which extensional space opens up in the first place. Just as in a dream we can leave a room and go outdoors, even fly toward the sun or moon, without thinking for a moment that these expanses of extensional awareness are opening up within us, so in waking life we are never aware of the intensional space enveloping the extensional spaces around us.
Diagram 4 shows how intensional space constitutes a medium of direct inner relatedness between people - being a direct link between the inner fields of two psychic envelopes. From the point of view of Fundamental Topology, two people occupying the 'same' room actually occupy two distinct rooms, represented by the white outer fields of extensional, spatial awareness in diagram 4. Indeed there are as many rooms - or football stadiums - as there are people in them. That is because extensional spaces are essentially extensional spatial fields of awareness unique to each individual. They are perceived as common or shared spaces only because those individuals share common field-patterns of awareness and pattern what appears to be a common spatial field of awareness. It is not just that each of the people, by virtue of their unique position in a common space, perceives that space and the other people in it from a different angle. Instead, an individual's perception of another human body as a three-dimensional form is an outward perception of that person's inner form - the field-patterns of awareness that constitutes the organism or psychic envelope. The same is true of our perception of things. The outwardly perceived forms of objects in extensional space are our own patterned perceptions of their intensional reality or inwardness - their inner field-patterns or 'morphic fields' (Sheldrake). What appears to us as 'empty' space between physical objects, is filled, counter-spatially, with invisible pre-physical field-patterns of perceived forms, which achieve visibility as objects only as they reach a certain field density of intensities. Diagram 5 shows counter-space as a field-pattern converging on a perceived form in the shape of a rectangle.
Diagram 6 shows the overlapping of the space or field-patterns of two contrasting forms.
The light and other energies radiated or reflected spatially from these objects are the centrifugal energetic expression of these counter-spatial field-patterns of formative pre-physical energy or 'inergy'. It is because of the counter-spatial dimensions of both our inner and outer fields of awareness that things can get 'under our skin' - that we are inwardly sensitive to the perceived form and movements of everything and everybody in our outer spatial field of awareness. It is only through our own resonance with the inner field-pattern of an object or person that we can form a physical image of that object or person in the first place. Once formed however, that image affects us through its own resonant field, generated by self-resonance with its own inner field-pattern or morphic field.
From the point of view of Fundamental Topology therefore, field-patterns and spaces are one and the same, the essential distinction being between extensional and intensional fields of awareness, spatial and counter-spatial fields, morphic and resonant fields, inergetic and energetic fields. Whenever we travel in extensional space, encountering different places and people, we are also moving through intensional space and encountering different field-patterns and tonalities of awareness. What we sense as the 'atmospheres' of different places (whether different rooms in the same house, different streets or parts of the same town or village, or different countries and continents), or as the overall 'aura' of different people, is our felt sense of these field-patterns and tonalities, which though they have their source in intensional space, exert a resonant effect on our own field-patterns and tonalities of awareness.
Diagram 7 shows the field-pattern or morphic field of a body (circular) in dissonance with an already patterned enveloping or environmental field (rectangular).
The result of this dissonance can be that the morphic field of the environment tends to take on the field-pattern of bodies within it, that the latter take on the pattern of the environmental field or that a complex field topology takes shape producing morphic field-patterns intermediate between that of a body, its environmental field and other bodies within that field. What any body essentially is, is a field-boundary - a surface boundary or envelope between its own organizing field-patterns, on the one hand and those of its environment or other bodies within that environment.
The identification of reality with extension is rooted in the earliest eras of Western thought. To begin with extension was equated with discrete extensional bodies or with a continuous extensional medium - whether fluid or airy or aetherial. It was the Greek 'atomists' who first replaced the ontological duality of Being and Non-being with a cosmological duality of extensional bodies on the one hand and empty space on the other. But Aristotle denied even the possibility of a spatial vacuum free of extensional bodies or an extensional medium, as did Descartes centuries later. And Newton himself could never fully accept the idea that gravity represented action at a distance, unmediated by some form of subtle medium, material or immaterial. Like the Stoic philosophers, Newton's first picture of the cosmos was a stellar realm surrounded by an extra-cosmic void - a realm of pure chaos. To accept an intra-cosmic void or vacuum state seemed, as it did to countless previous generations intrinsically unnatural, in a way summed up in the maxim that "Nature abhors a vacuum". It also seemed irreligious, for through the identification of created reality with extension, the concept of a vacuum implied also an absence of God within the cosmos. Religious accounts of creation however, themselves posited 'ex nihilo', a nothingness from which the extensional world was created. Fundamental theological as well as philosophical and scientific issues all centred, therefore, on what sort of reality could be conceived beyond a created world or even an eternal and uncreated cosmos if the latter was essentially an extensional continuum. The only answer that seemed conceivable was some sort of extra-cosmic void or intra-cosmic vacuum, a nothingness out of which God created the world or, as in Eastern philosophies, a void which itself gave birth to the world.
Today we have no difficulty accepting the idea of an intra-cosmic void - a spatial vacuum possessing extension but not filled with a continuous extensional medium or aether. General Relativity seemed to make the idea of such an aether unnecessary to account for gravity, replacing it with the notion of curvatures of space itself. But that still left the question of what lies beyond extensional reality, now conceived as a shape of space in time. The lay inquirer still wonders, and rightly so, what sort of reality could have preceded the Big Bang, if the latter represented the birth of space-time. Or, if the entire extensional universe of space-time has itself an extensional shape - a torus or doughnut shape for example - what lies beyond it. The Big Bang gives us a how - provides an account of the cosmic evolution - that seems to fit with the facts but does not explain why the universe should have developed according to certain physical laws and not others. Indeed, Einstein's theory itself allows for universes obeying other laws and taking different forms according to the field-equations relating the geometry of space on the one hand with distributions of matter and energy on the other. Theoretically, a vacuum universe with no matter or energy is defined as flat rather than curved. But other solutions to the field equations allow for curved universes or 'spacetimes' even without the presence of matter or energy as we know it.
At the same time light and gravity have become more of a mystery than ever, the one defining the curvature of space, the other existing as waves filling the vacuum of space and not ultimately dependent on the existence of sources of gravity in the form of extensional bodies. The puzzle of 'dark matter' - the fact that 90% of the gravitational mass of the universe remains unaccounted for - indicates that physics still has a long way to go in attempting to fill the voids in current cosmologies, let alone explaining what they essentially are, and from whence they derive.
Historically, the universe has been conceived of as finite or infinite in extent, continuous or broken up into atoms and collections of atoms separated by empty space, surrounded by an extra-cosmic void or filled with intra-cosmic vacua, flat or curved, spherical or toroidal. But the recurring problem, and the one that cannot be gotten round, is that the origins of an extensional spacetime universe (or of multiple spacetime universes) cannot, in principle, be conceived or mathematically accounted for in purely extensional terms.
But what appear as abstruse or intractable problems of physics and philosophy, mathematics and theology are answered by everyday human experience. Aristotle defined a vacuum as a space in which the presence of a body is possible but not actual. And we all know of a non-extensional space which fits this definition - the space of our imagination. We all know too, of an extensional space with no actual extension in space - the spaces of our dreaming. To which we may add the space of our seeing and hearing, which forms no part of any space we see or any objects we hear in it. Perhaps it is not surprising therefore that it was not a physicist but a psychoanalyst - Donald Winnicott - who first ventured a description of non-intensional space not as an empty vacuum but as a 'potential space' in the Aristotelian sense, the space of our openness to possibilities that expands in play and creativity. Long before Aristotle however, it was the Greek thinker Heraclitus who first pointed to the non-intensional space of the psyche itself, and characterized the essential nature of intensional space as unbounded interiority. The term 'psychology' derives from the Greek words psyche and logos, and Heraclitus's saying, which unites these two words can be considered as the founding statement of any fundamental 'psycho-logy: "You shall not know the limits of the psyche, no matter how far you go about it, so deep is its logos". The term 'in-tension' means to tend inwards. The inwardness of the psyche is an inwardness of its word or logos. This is not an ordinary spatial insideness but an unbounded interior space of meaning or inner resonance. If a text is the flat two-dimensional surface of an intensional meaning space, then no degree of curvature of the sheet of paper on which it is printed, no alterations of its extensional geometries - will reveal anything of the fundamental, intensional space of meaning that constitutes the true inwardness of the text. It must be emphasized again, however, that this meaning 'space' is not space in a metaphorical sense. It is fundamental space, understood not as actual sensory space but as the space of potential meaning or sense.
© Peter Wilberg 2002