Gestalt Mathematics


Gestalt means "organize"; the Greek root is stellein, to set in order, and the study of Gestalt is the study of how we organize our worlds, whether as figure against ground in Gestalt psychology, or as an individual in a community, as in Gestalt therapy; in either case as a discrete entity that stands firm against some sort of continuum. This sense is reflected in the Indo-European origin of the word, sta-, from which we derive modern stand, stage, stay, contrast, extant, state, system. The contribution of the Gestalt approach is  the recognition that figure and ground are formally the same and may be exchanged without violence to the whole, and in fact must be changed for a complete experience of the whole. When we understand something, we stand under it, experience it from without as well as within.

Recognizing that the experience of the whole depends on how how we organize the parts directs us to the boundary between them as the transcendental element. Fritz Perls called it the contact boundary between the organism and the environment, and located there the excitement which he identified with life. In Chinese metaphysics it is called the Tao. Christians call it the cross. The boundary is the place where opposites collapse into the same.  In mathematics, the process is called degeneration, meaning, literally, the undoing of what has been generated.

Attention to the actual names by which phenomena are called reflects the Gestalt process itself, although not all practitioners of Gestalt are willing to carry it that far. The etymologies are histories of consciousness, and show how the patterns we create have developed and intermingled. In discussing the relation of the discrete to the continuous, the words we use represent the boundaries we set, which we cannot see from both aides without considering the formal properties of language as well as the semantic. Transcendentalist Emerson saw a word as a "fossil poem." The sounds of the words we utter affect our experience directly, a fact once recognized in the sense of sacred speech and implicit in the notion of a mantram. The shapes of the letters we use to represent the words undergo iconic processing in the brain in parallel with the symbolic exercise of "reading." In the forms used to express Gestalt we may see a general theory.

To discuss Gestalt we must cell the names that set the boundaries to our experience. Adam's first task, was to name the plants and animals (Genesis 2, 19-20). And here we have a hint of our function as separate entities: the scribe records Jahweh's curiosity as to what Adam would call them.

Fritz looks at Gestalt as an experimental science, hampered by the lack of a unit of awareness by which data might be measured. Of course it could only be an experimental science if it had a unit, and in fact it does have such a unit available, although  the unit cannot be seen as such until it is defined and used.

Definition: the setting of a boundary. A definition distinguishes that which it defines by indicating its boundaries.

The boundary defines figure against ground. The Gestalt approach focuses on how the boundary is drawn rather than what it contains. Neither figure nor ground is valued higher than the other; they are created simultaneously, and neither could exist without the other. (This is how Gestalt therapy differs from other therapies, which focus on the states actually distinguished by the patient, and encourage him to enter other states.) So in the drawing of the boundary, the making of distinctions, we may see the unit of an experimental science of consciousness.

We may here distinguish consciousness from awareness; conscious and science have the same root, which means to cut through and separate. The mood sense is active.  The root of awareness means to watch out for and almost defines a state of receptivity. Awareness makes no distinctions and therefore we may not study it scientifically, that is, examine it from without; we may only experience it from within. We may however analyze how we make distinctions. The science for which Gestalt is an experimental process is the science of science itself, of how we make distinctions. It is self-referential.

The unit is the distinction. Name it in what is known as "English," i. e. about angles, something that bends (from swerve of shore to bend of bay), and so define it where English is generated: Latin distinguere, to separate, Indo-European steig-, to stick, pointed. What is separated, in Gestalt terms, is figure and ground. Any given figure/ground arrangement may change; what remains constant is not the boundary between them but the process of drawing a boundary.

Gestalt psychology studies this boundary process as an act of perception. Gestalt therapy treats it implicitly as an ego function; the nearest it comes to explicit recognition of the boundary is in the concept of closure, in which figure and ground are integrated and form the ground for the next gestalt. Both of these  disciplines treat the entire field of figure-­boundary-ground and in this respect derive the name of "holistic" theories. They do not however  analyze their own process. Fritz Perls stated a belief that consciousness is a property of matter, like extension and duration, but he did not see any unit with which to measure consciousness akin to feet and seconds for the others. Gestalt psychologists analyze the entire field of the gestalt in terms of relationships rather than distinctions.

At this point a pause for station identification. This is Sufi Central broadcasting in the 5550 angstrom band. Tonight's program is about Gestalt and how we may discuss the Gestalt process without reference to theories of perception or personality... or perhaps, if that seems an arbitrary deprivation, about how we may not discuss the basic nature of Gestalt, or anything else, without arriving at a place where the terms of any individual discipline break down. Why anyone should wish to discuss at this level may not be immediately clear...or it may be. However the basic pattern of Western science, which we are discussing, has been to create ever deeper and more general order for apparently unrelated events. We cannot understand the behavior of variables except from the standpoint of the constants which they represent. Otherwise we have no firm reference point.

Jung was aware of the importance of constant patterns, which he called archetypes. He did not say how they were generated. The pattern, or collection, is made up of constant units. We see the generation of the pattern in the combination of the units. So to find out what Gestalt is, we study how its units combine.

Formally, a distinction (our unit) is drawn by arranging a boundary so that we may not move from a point on one side to a point on the other side without crossing the boundary. In a plane space, a circle draws a distinction. We do not have a circle until the line that traces it closes on itself, so that the circle contains the space on its concave side. (Here we meet again the concept of closure.) So we may make the paradoxical definition: Distinction is perfect continence. A therapist may say that we do not have a  clear distinction without closure. This is half the picture; we may also ask how separate distinctions, simultaneous or successive are related to each other. What is the sense in which one distinction "contains" or does not contain, another? There are a lot of systems, hierarchical and heterarchical, in which we may see the patterns.

We describe the generation of the patterns mathematically.

Mathematics is a method of formal analysis in which we do not concern ourselves with the value which any form may take, but rather with the form in which we see the value. In the form of a word, for instance, whether as a pattern of sound or a written representation of the sound, we see the collapsed values of all the meanings which we may ascribe to the word. The word "group" has a different meaning to a mathematician than it does to a therapist, but in a formal sense there is no difference.

The way that we represent the group, in sound or in writing, is not the form itself but a name by which we indicate the form. The form itself is distinction, in which we no longer see the idea of a group. So we may see form as inner to indication, indication inner to the general pattern of the group, and the general pattern inner to any given state in which the group may exist.

To trace a phenomenon to its source, we abandon variable existence to unchanging archetypes, which we may take as always true; truth to indication, and indication to form. We may go deeper, but not in any form. At each step, what we gain in generality we lose in a sense of particular existence.

From a particular group meeting Thursdays at 2 o'clock, or a noncyclic simple group of order 25920 we go to the concept of a group, which is like a dictionary definition, and absorbs both of the specific examples. We next move to the name "group," which tells us almost nothing about the concepts it names, and then to the form itself, at which point we cannot even give it a name. The rest is silence. Each step toward generalization confuses something that had previously been distinguished. We may not bring these distinctions with us, although we may generate them again any time we wish to return from form to indication, truth, and existence. So from Gestalt therapy and Gestalt psychology we move inwards archetypal organization, the name "Gestalt" and the form of distinction. As a therapy, Gestalt is on the same level as any other therapy. Its exemplary value is in the clarity with which we may see its relation to the basic form underlying all therapies and indeed all conditions of life. That the journey inward must be made with awareness is the central motif of the Grail legends. The successful Grail knight was named Parsifal, meaning "pierce the middle," in which we may see a parallel to the definition of distinction as separation, and its root, a pointed stick.

At this point we can begin to see one way in which distinctions relate to each other, either on the same level, as with different members of the same class, or with one contained in another, as with the  member and the class.

We may only relate them precisely at the formal level because at any level shallower than, or outer to, the formal, (indicative, archetypal, existential), distinctions are seen as differing incommensurably in value. We may communicate, existentially, about what we imagine our values to be, we may discover archetypes, as Jung, and we may study the language by which we indicate the values. Each of these approaches eventually runs up against the boundary created by its own loss of contact with the deep patterns.

Our language for formal analysis is called mathematics. In order to uncover the deepest expressible nature of Gestalt as a discipline, we must learn its mathematics. This is not to say that we will then be able to practice Gestalt by formula. The deep structure of Gestalt confuses the existential distinctions made in therapy or psychology. It just don't make them. However, only in the mathematics can we see how the natural structure of the discipline creates itself.

This mathematics is not about number; it is pre-numerical. Mathematics is about space and relationships, and number comes into mathematics only as a measure of space and/or relationships. The branch of mathematics that is useful for analyzing two-valued systems, such as figure and ground, is called Boolean mathematics, after George Boole, the 19th century mathematician who developed it as a tool for logical computation.

Invoking once more the ghost of unser Fritz, we examine the process he called "the five layers of the neurosis." These are:

  • Cliche layer: Hello, how are you (chickenshit); the Giants will win it this year (bullshit); the neurosis has five layers (elephantshit).

  • Role playing layer: personality--doctor, lawyer, husband, saint.

  • Impasse: Where cliches and roles are abandoned but inner strength not yet felt. Environmental support not forthcoming, inner support not yet achieved.

  • Implosion: Retreat into self, inner focus.

  • Explosion: When all these inwardly directed energies burst outward into (a set of four here, rather than five) grief, anger, joy or orgasm.

Why five? To examine this structure formally, we take note first of its number. The link to other fives--fingers, basketball players, the Pentagon, the Chinese division of the elements (the number five is basic to ancient Chinese medicine; see The Yellow Emperor’s Classic of Internal Medicine, tr. Inza Veith, University of California Press, Berkeley, 1966) and the mysterious fifth power in numerical mathematics, a point at which "something overtakes something else," as in Zeno's paradoxes, and we are unable to find general solutions for equations  degree five and above, they do not exist, although they do exist, and have been worked out, for degree four and under. Some thing special about five, the geometry of it, some special way of indicating a boundary or limit. This is the form in which Fritz perceived the psyche, or chose to describe it. The number five seems to bear so little on the matter--the content of each layer--that it is hard to see how this relation may be expressed.

Suppose we draw it, though, as five concentric circles. It looks like an electron. In fact, in some circles it is an electron. Now we label each space with one of Fritz's layers. Or do we label each line? We have five of each, spaces and lines. If we label the lines, though, we are left with an inconvenient unlabeled inner space, so let us label the spaces and call the lines between each pair the boundaries. Notice that each of the circles draws a distinction that contains another distinction, or is contained by another distinction, or both.  We could not get from inside the inmost circle to outside the outmost without crossing five boundaries.

Had we drawn the circles side by side, each of the layers would be only one crossing from inside to outside. We could go from any layer to any other layer with only two crossings, and formally, the various layers would be indistinguishable from one another. That is, without labels, we could not tell one circle from another. Whereas each circle can be completely described in terms of the others, with no reference to external coordinates or directions, and each is clearly distinct from the others in the form. Therefore we may see in each a separate value, and give each a name to indicate that value. Fritz chose the names cliche, role-playing, impasse, implosion, explosion.

As we progress past each stage, we undo the distinction we made to create it. So we show at the impasse, the superficial layers have been undone, while the inner layers are behind a boundary. The way to reach the inner space is to cross the boundary. At this formal level the boundary is not any particular boundary, it is the idea of a boundary. In order to cross it, we must for a while become it. Here is the literal sense of the crucifixion and the cross in Christian iconography. Crossing the boundary entails a kind of death and resurrection. The Greek god of travelers was Hermes, whose shrines were at crossroads—messenger of the gods, god of information.

In the inmost state, explosion, we have left the form altogether. All of the circles have been broken and there is no structure at all, nothing, the Void you might say, as they do in the East, where the Void is worshipped as Christians worship the cross.  From unorganized energy comes the recurring Big Bang.

Following the more generalized terms introduced earlier, we can rename the outer two layers:

Layers of neurosis           Formal structure
Existential                        Existence
Archetypal                         Truth
Impasse                              Indication
Implosion                          Form
Explosion                          (Void)

What is existential is our individuality, that which stands out, ex plus stasis, the word has the same root as ecstasy, to shoplift a name from Frl, Lemarr. Archetypes do not exist, literally; they do not stand out, and so cannot be perceived, as figure against ground, but must be divined, dived for, since they are inner to, deeper than, existence. Archetypes are always true, whereas existentially a proposition may be true or false.

Abandoning the notion of truth and logical structure removes outside support. The way through the impasse leads toward form. This pilgrimage inward from existence to truth, indication, form and void is the route followed by the Grail knight, Parsifal, on his spiritual journey. Formally, the Grail legend, Fritz's model of personality, the Yellow Emperor's Classic and the structure of Laws of Form are the same, and this is presumably the same form revealed to Dante:

Substance and accidents, and their modes, became
As if together fused, all in such wise
That what I speak of is one single flame.
Verily I think I saw with my own eyes
The form that knits the whole world.
--Paradiso xxxiii 88-92.

The root of Grail means "to confuse." Presumably Dante was privileged to behold it, perceive directly what we are talking about here. Bertrand Russell had a dream in which he met Not, and he couldn't remember what it looked like. But Dante saw the First Distinction, and remembered.

Kurt von Meier
Clifford Barney
Circa 1975