Levels of Abstraction

Stepping Back from Unity

by Jonathan J. Dickau
©2006 – all rights reserved

        In order to fashion an accurate picture of anything, we need to employ various levels of abstraction.  This need arises from the basic relationship between an object and an observer, or a subject and its student.  There are qualitative changes, as one moves from one level of abstraction to the next, however, and seemingly a progression of form and/or formative principles.  Thus the process of abstraction itself becomes part of the picture, or one of the formative elements thereof.  Though this process enters into every observation we make, it is poorly understood, and not something that many people attempt to look at directly.  It would appear; however, that the process of abstraction, and of preserving observation and measurements, has many parallels with the formative process and this can give us insights into both.

        Abstraction and the realm of the abstract relate strongly to the world of symbols and the realm of archetypal forms, but they are not exactly the same.  A symbol is simply a representation of something else, whose nature is suggested by the form or content of that symbol.  A photograph is thus a symbol for the scene that is depicted thereby.  Likewise, a dot on a page, or a circle drawn on paper or stone, symbolizes an ideal or archetypal form, by being an approximation thereof.  The idea from Geometry of a point defining a location in space, and having zero size of its own, is a good example of a pure abstraction, as it is impossible to observe or achieve in the real world.  Strangely enough, the closest thing to a geometric point in the universe may be the entire universe at the moment of its birth.

        So; the process of abstraction can take us from the realm of the real to the realm of ideal and back again.  Abstraction is simply the process by which we come to have an understanding of any condition, object, or phenomenon.  It is the means by which we calibrate our observational strategy, in order to take in relevant details, and make sense of them.  It is also how we formulate a representational strategy, in order to preserve what we have learned somehow.  Thus, abstraction is a part of both perception and the creation of symbols to represent what we have come to know.  To develop an understanding ourselves, or to share our understanding with others, abstraction must be employed along the way.  How does this process proceed, however, and what does abstraction actually consist of?  Is it something that we can grasp, or do we need to employ it blindly, without regard for its fundamental nature.

        Thankfully, it appears that a complete understanding of this subject is possible, but this is a topic which philosophers have debated for centuries, and will probably continue to debate long after a definitive description is propounded.  In my view, the process of abstraction proceeds naturally through alternating phases of observation and exploration, and it’s especially evident in the initial stages.  That is; after observations are made, a step or steps are taken to acquire a new and unique viewpoint, then more observations are made and this process is repeated until all aspects of what is being examined are known, or can be described.  Abstraction itself is often the explorer’s role, in this scenario, as adopting a new point of view can consist of simply considering different possibilities, or entertaining alternative explanations which account for the same observed facts.

        If we start with a single viewpoint in an undivided expanse of space, the process of abstraction can be delineated in its entirety.  There is an implied connectedness in a truly empty space, because there is nothing to separate any one part from the whole, or from any other part.  So the zeroth (or 0th) level of abstraction is a state of ambiguous oneness, or indistinguishability, where one cannot tell oneself apart from the object or space under observation, until some actual observing takes place.  I believe that there is also a dynamical relationship of any potential observer with its surroundings, or with what will be its object of observation, even prior to taking in any information whatever.  The idea that we are surrounded by reality, and that it affects us even if we don’t open our eyes, should not raise any eyebrows.  However; I assert that this dynamical relationship exists even in the pure abstract, and in the absence of any form in the conventional sense.

        This idea has a long history, but has seen exciting proof recently.  In the Taoist tradition, mystics describe the primal state or Wu-Ji as a realm beyond and before distinctions, neither hot nor cold, neither light nor dark, and neither large nor small.  To some, a lukewarm space like that seems uninteresting, even boring.  But rather than being a place devoid of meaning and purpose, they describe a realm of pure process without form or limitations, and a state that is beyond all opposites or opposition.  This observation is borne out by the recent development (championed by French mathematician Alain Connes) of a new branch of Mathematics called noncommutative geometry.  This field studies the qualities and evolution of spaces which don’t follow normal laws of size and distance.  The term means simply that one can’t take 3 steps of 5 units length, turn around, then take 5 steps of 3 units length and end up in the same spot.  Nor is it merely a case of things not adding up.  It is often more like the challenge of navigating across a river or stream with changing eddies and currents.  This adds another twist.  Noncommutative spaces evolve with time!  And they appear to do so without anything to drive the process.  This makes otherwise ‘empty’ space pregnant with possibilities, and it makes the origin of our universe kind of magical.

        That is; the zero-point of reality is not totally featureless, as there is a way for the formless or emptiness to evolve toward form and substance just as there is a way for an open mind to evolve toward understanding and meaning-making.  Accordingly; moving from zero to the first level of abstraction requires observation, or taking in information.   We move from a sense of oneness into a kind of openness, or receptivity for input.  The process of opening ones eyes, the process of awakening from an unconscious state, and the idea of looking around and taking in the view of ones surroundings as a panorama, all entail ones being purely receptive.  To take it all in, without judgment or reasoning, is the first step toward abstract thinking, and the first part of the process of abstraction.  When we can open our minds to possibilities, without prejudice or bias, we have taken an important step in the direction of full and accurate cognition.  But to put oneself in a posture of being neither for nor against anything is difficult at times.  Rather; we find that most adults must really work hard to be purely objective.

        When we study how symbolic logic and abstract reasoning develop in children, we discover there are many mistakes and much confusion along the way, and that quite a lot of information must be taken in before we can develop accurate perceptions of the world around us.  If seems that infants make no assumptions about the world at all, beyond what they can actually see, hear, and touch.  One of the first landmarks is the development of object-constancy - the notion that objects remain intact and available, even when they are not in our immediate field of view.  Thus; a game like peek-a-boo, or hide and seek, can be endlessly fascinating to a small child, as they will be genuinely surprised each time you pop out from around a corner or behind a chair.  The fact is; making generalizations about things requires more than simple observation, as only a process of exploring and comparing can convince a small child that you are still there when you are in the next room, or that their toys still exist once they have been put away.

        When we begin to explore, we have begun a new phase, and entered the second level of abstraction.  To be fair, we should assume each new vantage point without making undue assumptions about what we will find once we examine the conditions there.  Younger children actually do this; they approach everything with a sense of awe and wonder.  The reality for most adults is quite different, as we go into the process of abstraction or cognition with many assumptions.  Once we start to see patterns, whether in arrangements of objects and spaces or rules and exceptions, we have entered yet another phase, and undertaken a third level of abstraction.  This all happens fairly quickly and the process moves on to the next stage automatically.  But the process of exploration itself, and the second level of abstraction, deserves a fair amount of attention and elaboration.  To explore can mean a literal movement from one place to another, or it can be a more subjective maneuver.  From pure observation, we already start to get a sense of what’s near and far, what’s part of me and what’s part of my surroundings, and so on.  But to verify that observation, we must explore and compare.

        Exploration is a process of stepping out, or of stepping back, from where we began to a new vantage point, where we will hopefully be able to distinguish new details of whatever we are trying to examine.  To explore may mean to actually move physically in space, or it may be the more subjective ‘movement’ from one way of looking at things to another view of things, a different understanding of how or why things appear as they do.  For our understanding of reality to grow, however, we need to be able to sustain the awareness of where we started, or how we got here, while acquiring new viewpoints, and accumulating new experiences.  So an important part of the exploration process is the ability to accept and incorporate what we have already taken in, while continuing to seek insight.  Part of this process involves asking questions, taking guesses, and making postulates.  This is the most familiar part of the process of abstraction, but it requires some faith in that process.  We must be prepared to consider ‘what-if,’ or proceed ‘as-if’ each time we consider new possibilities, in order to be true explorers.  However; we must also be prepared to find something different from what we expect, once we go to see.  And we need to acknowledge that we may change things, by going and observing.

        So; what does it mean to explore and compare, where does that get us, and how does this relate to the process of abstraction?  Again; observing the activities and developmental stages of children can provide insight into how abstract thinking evolves in humans, and presumably elsewhere in nature.  There is a testing process, at each step toward understanding, and an evolution of testing procedures.  Infants develop a sense of me and not-me, of here and not-here, then move on to perceive distinct objects and other people as persistent aspects of their reality.  For children to develop a sense of object-constancy requires multiple observations under varying conditions, because for the very young every new outlook is a brand new ballgame, where anything can happen.  This carries over, once object-constancy is learned, into the next phase of exploring.  When we introduce toddlers to models, photographs, floor-plans, maps, and other symbolic forms, there is often confusion as the child will try to sit in a model car, or on doll-house furniture, or perhaps attempt to put on a shoe that’s in a life-size photo.

        The idea that one form can represent another is an important abstraction, and a pivotal stage in the development of abstract thought.  A key concept here is that the representation (image, model, or map) is a distinct and separate object, while being in some way connected.  This concept includes and evolves from the idea of a projection.  A projection might be a shadow, a mirror image or reflection, or the actual projection of an image through a lens, and perhaps onto a photograph.  The idea that such a projection process turns a 3-dimensional object into a 2-dimensional form aptly conveys that the original object is represented by its projection.  To be clear, the object itself isn’t turned into a 2-d form, but rather that 3-dimensional form is represented in 2-d by its projection.  Likewise, a scale model, though it is 3-d, is still only a representation of the actual house, car, or train.  For children to play pretend with toy houses, cars, planes, trains, or even people, gives them opportunities to explore different possibilities and compare outcomes.  This gives them a mental vocabulary large enough to develop both abstract and symbolic thinking.

        Still; there is a stage in their learning where one can show children the exact location of an object in a scale model of a room, and then watch them look everywhere in the real room, for the life-sized object it represents.  For those who are too young, or too inexperienced and undeveloped, it remains a mystery to them, even if you show them exactly where to find something in the symbolic representation of that place.  So part of the process of abstraction is this sense of an object and its representation as both equivalent and independent simultaneously.  As Korzybski often reminded us, “The word is not the thing and the map is not the territory.”  However (as he also noted); a map, a photograph, or a painting, allows one to preserve the landscape as it is now, for future viewers to see how things were, once the landscape has changed.  Similarly; our internal abstractions capture the sense of what we have experienced, and make it available for comparison with what we are observing now, or will experience in the future.

        Our mental-image pictures of things, and our internal sense of feeling about what we perceive, are our own properties - to the extent that we can possess our own thoughts and feelings.  But any attempt to share our experience of life requires us to employ (create or adopt) symbols that can be interpreted consistently by others.  In other words; our personal abstractions can be open-ended and free of limitations, but we must use shared symbols and symbolic language to communicate with others.  There can be some communication through simple demonstration and direct imitation of actions and behaviors.  But this presents those who want to communicate with a very limited range of possibilities for conveying information.  To either express or learn more than this requires that we be savvy about projections, descriptions, ideas, names, and other symbols.  We must acknowledge that one thing can represent another, yet remain distinct as well as related.

        Adult human beings take the process of abstraction for granted, to some extent, because we are so familiar with it.  It has become such a part of our lives that it’s only barely conceivable how we might formulate detailed thoughts and ideas, without the method of abstract thinking we have come to know.  But there is quite a range of possible interpretations, and any number of ways to cognize the same facts.  It might be wiser to speak of how we learn to think abstractly, rather than trying to enumerate what the different levels of abstraction are, but in this case I believe we would arrive at the same end result.  Abstract thinking is the means by which we are able to digest and retain what input our senses bring us, and the means by which we engineer, motivate, and justify our choices and actions.  To some degree, abstract thinking is the only way that one can rise above simplicity, and the only way that more detailed descriptions of reality can be learned and understood.  The means by which the process proceeds beyond exploration and pure observation is through some form of classification and generalization.

        Once a certain amount of information is taken in, we begin to form opinions, categories, and rules for each object, group, or other sub-division of reality.  We sort what’s important from things that can be ignored.  This happens without much conscious thought, in higher life forms, because it is a part of our survival system and accordingly it is often fundamental to maintaining life.  But the kinds of value judgments we can make become more sophisticated as we develop.  The behavior of various creatures seen in nature can be related to their ability to cognize and adapt, as well as their other natural adaptations (of body architecture, etc.), given the challenges and pressures they face to remain alive.  The behavior of humans is on some level more sophisticated, but most of us adult human beings are remarkably socialized and conditioned, rather than having the full range of our cognitive capabilities and the true freedom of our choices.  I feel like I reach a painfully small percentage of my own potential, but I’m also aware that too many people aren’t even conscious of what glorious things they might be capable of.

        Unlocking that potential requires us to oversee the process of our own thoughts, and to examine how we form opinions and categories.  This is why any attempt to reveal the nature of the abstract itself gives us a better understanding of the world around us, and a significantly larger toolkit with which to analyze what we observe there.  Every attempt to cognize our world requires that we evolve a collection of measuring tools, sufficient to what we hope to understand.  It would appear that learning how to measure our world is a big part of what the process of exploration and comparison is all about.  Making sense of how things interrelate is a big piece of understanding what is real.  So we need symbols not only for objects and beings, but for the different ways in which various objects and beings can interact.  And once we have acquired a certain level of sophistication with that, we can begin to understand a little about how things come to be as they are.

        So; up to this point, it has been a journey from a kind of ambiguous oneness to a more sophisticated and detailed sense of order, and some fuzzy areas around the edges of our world-view.  However, in terms of a thorough understanding of things, we are only about halfway there.  But our culture seems content with this level of thinking.  The problem is that our understanding to this point has become quite sophisticated already, and it may afford us a considerable range of predictive capability, so it’s understandable why people might be reluctant to leave behind the comfortable illusion of a clockwork world.  The idea of natural law evolving the whole universe as a giant mechanism was quite popular until the end of the nineteenth century.  And even the great discoveries around the turn of the twentieth century have not created very many converts to a broader view.  Little by little, however, subjects like quantum mechanics, relativity, chaos theory, and fractal geometry, plus data from particle physics and observational cosmology, force us to accept that simple views must give way to a more accurate understanding of reality.

        The universe we live in is far more interesting, and far more magical than a purely mechanistic view can accommodate.  That is; what’s real lies beyond the sense of perfect order and orderly progressions of arrangements.  Sometimes, what allows for the next level of sophistication is not more of the same kind of orderly form-building that brought us to this point, but is instead a chaotic burst of growth, or a bold step in the direction of a totally different kind of order.  Or maybe it is more exploration and observation, but a change of phase is necessary to seek a totally new viewpoint to observe from.  Perhaps, in fact, it is chaos instead of order, or as an alternative to orderly progressions, that allows the next phase to be realized.  This is actually the next level of abstraction we encounter, or the next type of abstraction that must be considered.  That is; at this point in the process, we must step back from the orderly understanding we have developed and examine other possibilities.

        To step back from order, we must take an excursion into the realm of chaotic forms, or develop an understanding of chaos itself.  To grasp the world’s complexity in its fullness, we must learn to appreciate chaos as well as order, and to see the role it plays in evolving complex forms.  Without a touch of the chaotic, the universe would be a very boring place indeed.  However, chaos does not mean randomness.  The idea that complexity has no underlying order is erroneous, but some events do appear perfectly random.  The realm of the chaotic embodies many types of complexity, but perfect randomness is very hard to create.  Rather, collections of billions upon billions of individual particles (such as our bodies) display, or exhibit, an astonishingly large number of possible behaviors, and these take place in an even larger universe, so that our life events may sometimes appear random.  However, this is evidence that all systems which are sufficiently complex are inherently unpredictable.  We can’t always know how long something is going to take, or what the final outcome of events will be.

        Again; this does not imply that the progression of events, or the behavior of complex systems, is truly random at all.  What is essential to understand is that chaos actually arises from order, but with a certain level of order present, or with a large enough collection of orderly objects and processes, chaotic complexity is quite inevitable.  This doesn’t mean that chaos is meaningful, necessarily.  Nor does it imply that it is essential for us to allow ourselves to be the victims of chaotic events, just because we accept that such things really happen.  Sometimes; the only way to really make sense of the world is to acknowledge that the world doesn’t always make sense completely.  Nor are the events in our live always meaningful.  People who try too hard to force things into categories that don’t fit will eventually find themselves frustrated.  Sometimes we must be prepared to accept that our current perspective will not allow us to make sense of things (no matter how hard we try), and to know that sometimes the reality of life is neither orderly nor sensible.

        This is not to say there is no meaning there, nor do I mean to imply that there isn’t something to learn.  Einstein said that we should try to make things as simple as they really are, but no simpler.  Sometimes there are deep meanings awaiting only our acknowledgement of real depth.  It is reasonable to believe that there is a moral to every story, as well.  If we are not too quick to remove ourselves from the rest of the universe, or to consider ourselves separate from others and unique beyond comparison, the universal story will show itself to us in everyday events.  However, the tendency to consider ourselves special, and therefore different from everything and everybody else, keeps us from seeing beyond the superficial level, and it is not something that life abides for long.  The presence of chaos is a great leveler, in that chaotic events touch everybody, regardless of circumstance.  Chaos is also somewhat of a character builder, as the way we handle the unexpected determines our strength of character, or lack thereof.  People who do fine when all is well, but do not have the strength to handle adversity, may not be of much value in an emergency.

         Dealing with complex situations requires a more sophisticated way of thinking about things, in many cases, but it also requires us to be open-minded, and that allows us to assess how things are really developing.  To allow for complexity requires that we be aware there can be exceptions to any rule.  It also means we must be prepared to develop a new understanding, and new rules to fit the circumstances, when what we are dealing with no longer fits the parameters of what we have been taught, or what we have learned from past experience.  There is no guarantee things will be a particular way, even if that’s how it always has been, nor does it mean you’ll find the same things everywhere that you find here.  But there’s some sense to things nonetheless, and we now have plenty of evidence to support this notion.  It appears that there are limits to complexity, or borders at least, and order within chaos.  Though it had appeared, for a long time, as though there were no limits to how complex things can get, it seems that we have seen the opposite shore of chaos, finding new kinds of order there, and reason.

        In the last 10-25 years, there has been tremendous progress made toward creating definitive theories of chaos and complexity.  It has been found that even very simple systems can be made to exhibit quite complex behaviors.  Furthermore, complexity is a robust behavior of many systems.  Ergo; it is crucial that we include the dynamics of complexity and chaos in our study of the abstraction process.  The good news is that some very complex behaviors can be reduced to quite simple equations or computational systems.  In addition; the study of duality and complementarity principles has revealed how some very complex and difficult problems have a back door, so to speak.  Sometimes, one particularly thorny problem can be viewed as the complement of a different problem, which is easily solved.  Thus; an otherwise intractable question is seen to have a workable solution.  The process of solving a problem this way can be likened to a sculptor, who carves away stone in various places (often quite carefully mapped) to reveal the desired form underneath.

        A good example of the borders of complexity I’ve encountered is seen when people learn to play in different keys, on the Piano.  As sharps or flats are added, things get more and more complicated, but at some point you are playing almost entirely on the black keys, and things are once again quite simple.  Many phenomena have a similar nature, where enough exploration will reveal that there is a simplicity beyond the growing complexity we encounter at some point.  Although it may seem that things can become infinitely complex, the reality is that no physical system can encode an infinite amount of information, nor can any physical phenomenon be perfectly random, and order will be revealed when things are analyzed closely enough.  The degree of randomness which can be achieved (or observed in nature) is astounding, however.  So we must accept that things may become totally random, for all practical purposes, while knowing that we may find things resolving into simplicity, if we persevere, and continue our exploring.  Or we may find that there is a different kind of order, to what appears to be chaotic.

        It should be noted that the greatest complexity is found at the borders between dissimilar things.  The shore of the ocean is where the action is, and most of the danger to mariners.  Once the boat leaves the harbor, or once it gets far enough from shore, there is clear sailing.  The coastline of any landmass or body of water is a place of tremendous complexity, where possibilities abound and change is almost continuous.  The exact boundary is actually quite difficult to define, as the size of our measuring stick will determine the level of detail we can delineate, and the measure of the perimeter will increase as our measuring rod gets smaller.  The reason for this is that the details present do not come in even intervals, and no measuring rod can precisely follow a border.  But even if we were to substitute a rope, which will bend around promontories and coves, we have a similar situation as a string will yield (or reveal) a larger measurement for the perimeter and a thread will give a still larger measurement for a given boundary.

        This fact is one of the reasons Benoit Mandelbrot discovered Fractals, and developed the subject of Fractal Geometry.  Form in nature is not so simple that ordinary geometry does it justice.  Mountains, coastlines, clouds, trees, and a host of other things found in the natural world, are quite difficult to model using conventional geometry, but can be studied or re-created with great ease, using fractals.  By viewing these things as objects having a fractional dimension, rather than a whole number, a very accurate representation can be made.  Michael Barnsley discovered that some ferns can be accurately modeled using iterated function systems, with only a handful of numbers as ‘seeds’ for the process.  To see that a small array of numbers can be turned into something that looks almost precisely like a spleenwort fern is miraculous to behold, and makes one wonder if this is the method by which nature ‘devises’ the form for such things.  But exploring the vast array of form found in an object like the Mandelbrot Set can make ones head spin, as the detail available to view seems inexhaustible (and may well be).

        However; we must remember that complexity does have limits, or is somehow contained.  Though it may display infinite complexity, the far edges of the Mandelbrot Set are all contained in a circle with a radius of two.  And like the coastline I spoke of above, most of the detail to be found in the Mandelbrot Set is located at, or near, the boundary.  In a discussion of how abstraction arises, and how the process of abstraction proceeds through its stages of development, we must acknowledge that finding the borders of complexity indicates a new phase, or a new stage in the development of abstract thinking.  More properly; finding out that there is a border to chaos and complexity, or that there is order within and behind what appears chaotic, marks the beginning of a new stage in our development, and a new phase in the evolution of abstract thought itself.  Learning to understand what kind of order there is to find is how we navigate that stage.  Finding out the other side of the story is the fruition of that part of the journey, as we come to know how to comprehend what has been unmanageably complex, and learn how to deal with it.

        So now; I can discuss the final stage of the process, the end of all exploring, and the reunion with what has always been real.  This is a difficult part of the story to describe in plain language, though many have spoken about this final stage of the journey, and some comments have been quite profound.  T.S. Eliot shared this wonderful insight, in a poem called “Little Gidding”.
“We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time.”
Once we have taken the journey, arriving home is a whole new story.   Earlier in the same poem he states that “What we call the beginning is often the end.  And to make an end is to make a beginning.  The end is where we start from.”  These thoughts characterize what I describe as the final phase in the process of abstraction.  The fact is that every end is a new beginning, and the end of every phase does begin a new phase of things.  However; in terms of different qualities or styles of thinking that have to be evolved, there is a culmination of the learning process, and even an endpoint to learning what can be known.  This involves both having a detailed knowledge, and having the wisdom to appreciate what is not knowable (yet).

        Beyond this, the most important aspect of this phase of things is realizing that all of what you have learned and everything else that is, is a part of you.  That is; what we perceive is not separate from us, and we are not actually separate from the rest of reality.  Being the observer does not make us any less a part of what is real.  We are participants in what goes on here, it is true, but there is an even deeper level to this stage or phase.  On some level, the distinction we make between ourselves and our surroundings is artificial.  We are a living part of life.  We are an aspect of the universe we inhabit, or an extension thereof, expressing intelligence and awareness.  In a very real way, we are the Cosmos itself.  We are the universe looking back at itself, after a long period of exploration and comparison.  We are the product of the very same process by which the universe came to be.  The fact is; those elements in our bodies which make us solid, like the calcium in our bones or the carbon which is the basis for living tissue, all come from the heart of long dead stars that originated deep in space.

        The Earth we inhabit is currently on a journey through space, which makes us all space travelers too.  But all of the heavier elements that make up our bodies come from the remnants of supernova explosions, from dying stars.  The lighter elements are older still, and yet more cosmic, as they originated from the Big Bang itself.  This means that we are clothed in the stuff of the cosmos.  Our bodies are made of earthly elements, it is true, but the Earth is made of star-stuff, and so are we.  To know all as ourself is therefore an acknowledgement of where we are now, or who we are already, rather than some misguided attempt to become something we are not.  The idea of enlarging the ego to fit the universe is foreign to most westerners, but the Hindu tradition teaches that we can reach enlightenment in just this manner.  It has been suggested that once one has learned to expand the ego to identify with the whole universe, it becomes transparent and disappears entirely.

        That is; once we can get out of our own way, we can see reality for what it is.  When we see ourselves as integral to the world around us and the world as an integral part of the larger universe, we obtain a perspective which was impossible to achieve at other stages of the journey.  We learn to identify with the totality of being, rather than just a limited piece of the puzzle.  Thus; we have come full circle.  At the beginning of this essay, I spoke of the unified state as the starting place for our journey, the zeroth stage of the abstraction process.  I believe that the final stage is much the same, as what must be acknowledged is that our separateness is an illusion and that the unified state has persisted despite the appearance of form and the development of abstract thinking, our detailed knowledge, and other aspects of the learning process.  We remain a part of the universe we inhabit and an aspect of the reality of which we are a part.  But now we know more about who we are, as well, and we have acquired a set of tools with which to measure the universe.  Once we have enough knowledge and sufficient understanding, we can develop wisdom.  We can “arrive where we started, and know the place for the first time.”  This is the fulfillment of the promise held out by the process of abstraction.

©2006 Jonathan J. Dickau – all rights reserved

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this page was first posted
July 14, 2006,
and revised on
July 30, 2006