Arthur M. Young

I have described experience as the simple sequential ordering of events (threefold), and mind as the integration of these events into forms or concepts (fourfold). Often, however, our initial concepts prove inadequate or false, and the only way we can test them is by encounter with objective fact at the level of the tangible world. This is the domain of determinism, the realm of law.

All such laws may be characterized as dealing with cause and effect: they link two events in casual relation. And we must heed this law, for it is basic. If we do not listen to it, we only repeat our mistakes. However, once we are aware of its operation, we can invert our relation to the law and make it work for us. I hope to show this in what follows.

The law of cause and effect is the basis of the deterministic world view, which denies the existence of free will. The determinist, observing that everything in the universe has a prior cause, concludes that since man is part of the universe there is no ground for the existence of free will.

The geometry we have proposed, however, enables us to see the law of cause and effect in perspective. While the law manifests at the level of objective fact, its existence as a concept is, of course, at the fourfold level, and as experience at the threefold. And the knowledge of cause and effect must initially and ultimately depend upon the threefold - the experience of the temporal ordering of before and after. There is no way to describe cause and effect conceptually. We require experience to know which comes first.

But the determinist's "scientific" point of view takes into account only the manifestation and the concept; it does not include the experiential aspect. The statement, the concept, of cause and effect is the mapping of a relationship. It is given in spatial terms, in simultaneity, and time is factored out. The deterministic view, or the logical view, or, in fact, any "mapping" in conceptual terms, omits temporal succession. It can relate two events but cannot say which is first.

On the other hand, the world view according to our geometry does include a temporal succession. In fact, as stated so far, it imposes a fixed temporal succession, which would support the determinist's view. But now consider this example:

In the old story, the barn burns down and the pigs are roasted.

(1) Fire -> (2) cooking. But then, the improved flavor of roast pig is discovered, and barns are burnt in order to cook pigs. Thus fires are made on purpose, and the purpose, to cook pigs, becomes the cause of the fire.

(I) Cooking (imagined) -> (2) fire -> (3) cooking (actual). This is where the fallacy of the determinist's view is revealed. The temporal ordering of cause and effect has been reversed. The cooking which was the effect of the fire has become its cause. A purposive intelligence has the power to reverse cause and effect.

In one sense, the reversal of cause and effect is a reversal of time. In another sense, it is not, for the envisioned result, the desired cooking, is imaginary, not actual cooking. So there is no "actual" or physical time reversal. The envisioned result or goal is nonphysical. Must we therefore dismiss it as nonexistent? No. For our cosmology postulates that experience, as threefold, is derived from some initial nonphysical unity. We can trace the envisioning of a goal, or purpose, back to the unity from which "physical" experience was derived. Purposive intelligence, taking the cause-and-effect relationship into account, realizes that a goal may be achieved by using this relationship. So it approaches experience from a different direction: it initiates action addressed toward a goal.

We have not yet found where this purposive intelligence arises. While the intelligence is at least partially dependent upon the faculty of comparison, and would thus be fourfold, purposiveness is neither in the fourfold nor in the threefold. It is certainly not in the objective world, and so there is only one place left for it: the initial unity from which both the fourfold and threefold are derived by division.

And this makes sense. Recall that we have referred to the initial unity as a projective potency. This concurs with the projectiveness of purpose and its power to initiate action leading to goals. Here, we might note in passing, we have confirmed teleology by uncovering purpose at this fundamental level of the universe.

There are good reasons for having said little about this initial unity itself. Since it includes all qualifications at once, no qualifying words can be used to describe it - let alone "define" it. For definition doesn't come into existence until the level of the fourfold division.

All we can say, then, of this initial and unitary totality is that it has dynamic and projective potential. This potential, by declension or division of itself, produces that which can be defined, but not without a residue of that which cannot be defined. In thus containing contraries, the initial unity is beyond knowing in the usual sense. At the same time, it is the source of knowing and of being, and, of course, all the qualifications we have discussed under threefold and fourfold divisions.

But what does the initial unity have to do with cause and effect? Simply this: it is the place of the first cause in any causal sequence. It is the purpose which initiates a chain of causal relationships.


Introduction of the twofold operator

Return now to our first discussion of the learning cycle. While we found that we could use the same diagram to describe both the learning cycle and the controlled motion of a body, recall that the order in which the steps were taken had to be reversed. While the analysis of motion used clockwise rotation on the diagram, learning was counterclockwise.

Having now distinguished two kinds of action (action toward unity, toward goals, and action away from unity, toward manifestation),* we can recognize their equivalence to the action which moves clockwise on the diagram and the action which moves counterclockwise.

The latter motion is toward means, toward manifestation; the former is toward goals. In the learning cycle, we have instinctive motion which causes encounter with the unexpected (the child burns his finger) and results in learning and self-control. In the application of the cycle of action, or the control of means to achieve ends, the rotation is in the opposite direction. Observation first, then reaction to produce what is wanted. Generalizing, we may say that clockwise rotation is informed or premeditated, whereas counterclockwise motion is natural or naive.**

*By manifestation I mean any step in the direction of making known to the senses, toward making tangible, visible, i.e., toward effects.

**Recall that, moving clockwise, each step in the cycle of action was the derivative of the one before. Since the derivative is equivalent to a ratio or slope, it is predictive. This is another sense in which the clockwise rotation is the informed direction.

As yet, we have no convention in our theoretical scheme for distinguishing naive action from informed action. But these considerations are so basic that for their formal expression we need a new operator - a twofold operator - to accommodate left and right-hand rotation in our system.

It is important to realize at the outset that this operator is not to be confused with "pseudo-dichotomies" such as positive and negative, true and false. Such apparent dichotomies, as I explained, actually involve four distinctions. In the case of positive and negative, the two hidden distinctions are positive and negative imaginaries; in the case of true and false, the distinctions are consistent and inconsistent, without which no use could be made of the true - false distinction.

The introduction of such an important idea as the twofold operator at this late stage challenges us to do some very basic thinking. Why have we not considered it before?

Let us first note that it is much easier to talk about the divided than the undivided. We began with the fourfold division, because it was the most accessible to rational treatment. We then took up the threefold only to explain that it deals with an area (experience) that is inaccessible to definition. Of the initial unity, we can say nothing, except to regard it as a dynamic potency (we cannot predict where such potency will lead).

It is natural, therefore, that the twofold should be inherently subtle and difficult to grasp. But we have found that it comes about through a change of sequential ordering. Let me use a concrete example:

An electric motor connected to a source of electricity may be used to pump water uphill. By reversing the sense, water flowing downhill drives a turbine (a pump in reverse) connected to a dynamo (a motor in reverse), producing electricity. A diagram of the two arrangements would be the same whether the electricity was consumed to pump the water or the water used to produce the electricity. There is no logical or analytical distinction between the two. To make the distinction, we have to talk about the order in which the elements are taken - which is input and which output?

This example shows that the distinguishing mark between the two situations - or between what we are calling right- and left-hand rotation - is the sequence of the three elements a, b, and c.* This distinguishing mark can be perceived only as a difference in temporal succession.

*Three elements are required to establish an order; ( . . . ababa . . . ) does not define an order, whereas ( . . . abcabca . . .) defines an order which is different from ( . . . acbacba . . . ).

Considering cause and effect, then, we can no longer say that these are twofold, because there is a third factor describing which comes first.


The twofold operator in nature

We may now ask an important question. Why does the twofold operator not come into play before the threefold and the fourfold? It would seem reasonable to have the twofold come in between unity and the threefold. Why does it not come into play until experience (the threefold) has been analyzed by intelligence (the fourfold)?

The answer is that it does. At each stage after unity, the twofold is operating, but of the two orders (or motions) only one manifests. This is the "natural" order, which invests in substance and form. It moves toward multiplicity and away from unity. The other order of the twofold moves away from substance and form toward unity (the goal). But if this latter order or motion occurs too early, before substance and form are attained, the movement turns back from manifestation toward the unmanifest, back toward the unity from which manifestation is derived by division. And we do have evidence for this turning back at the earlier stages.

There is a phenomenon in the production of nuclear particles which I interpret as an example of this clockwise rotation at the second stage of cosmic process. A very high-energy photon (pulse of light) has the capacity to turn into a material particle (an electron or a proton). When this occurs, there is simultaneously produced an antiparticle (a positron or an antiproton). The antiparticle almost immediately reverts into radiation or photons. Antiparticles are called antimatter, and we may interpret their production and subsequent disappearance as an example of the clockwise rotation at the level of nuclear particles, noting their non-productiveness at this stage.

I have recently done some speculation on antimatter. We know that the mass (i.e., energy) of antimatter cannot be negative, because when the photon produces a particle and an antiparticle, the positive energy of the photon is equally divided. The reversal must be one not of mass but of time. I prefer to think of the reversal of time not as negative time (-T) but as inverse time (1/T). This is, of course, conjectural, but it has some interesting implications. If one thinks of normal time as being very long (even if not infinite), then inverse time (1/T) would be very short - eternity in an instant. In the photon, it has long been known that the energy is inversely proportional to time (h = ET). This implies that in the "anti" world there might be an unlimited amount of energy in an instant of time, reversing our normal relationship between size and importance.

This compaction of time would give it the character of omnipresence - not going "backward" in time, away from the present, but instead going more deeply into the present. This interpretation has the merit of conforming to references in countless religions and mythologies to the super-sensible, nonphysical celestial world. "The Kingdom of Heaven is like a mustard seed..."

Returning to the operation of the twofold at subsequent stages of cosmic process, we find a corresponding duality at the third stage, the generation of atoms. There are certain kinds of atoms, helium, argon, neon, etc., known as noble gases, which have complete electron shells and do not enter into chemical combination with other atoms to form molecules. This "refusal to manifest" (by engagement with other atoms) can be interpreted as the clockwise rotation again, and, like antimatter, is nonproductive.

We have already, in Chapter VII, shown the progressive loss of freedom at successive levels.


I (Light)                   Unity                 Three dimensions free 

II (Nuclear)            Threefold          Two dimensions free

III (Atoms)             Fourfold             One dimension free

IV (Molecules)        Twelvefold         No dimensions free

This declension, however, depicts only the process of generation produced by the counterclockwise rotation, toward manifestation. The clockwise rotation, away from manifestation, is of course toward goals; but it is also toward greater freedom (more degrees of freedom, fewer degrees of limitation).

At the fourth, or molecular, level, a reversal can occur in the evolutionary process, and the appropriate direction moves now toward goals (what we have called clockwise rotation). Further counterclockwise rotation results only in molar aggregates, which do not participate in the process. The molar aggregates do not comprise another level because they are not fundamentally different from molecules in terms of the theory. As aggregates, they have lost that trace of the original potency which appeared in individuals on the molecular level as the phase dimension, or choice of timing. They constitute the inert "furniture" which is used by process to fulfill itself.

We can now realize the reason for the non productiveness of the clockwise rotation at levels II and III. When the clockwise rotation occurs at the nuclear and the atomic levels, before the complete loss of freedom at the molecular level, it cannot harness means (matter and law) to its purposes. The opportunity to use means comes at the fourth level, in the physical world. This is the world of commitment, where the initial freedoms have changed into restraints; the world of determinism.

This world is characterized by the increase of entropy. Entropy is the tendency of differences to average out, of stones to roll down from the mountains and fill valleys. In other words, it is the tendency for energy to become more uniformly distributed and hence unavailable.

In the molecular world, the reversal of direction permitted by the twofold operator does have significance, and a crucial one. It makes possible the decrease of entropy.* A small system can store order or energy which it draws from its environment, and build this order into itself. Such a system is the living plant.

*The decrease, that is, of local entropy. While the total entropy of a particular system always increases, it is possible in living systems to circumscribe a local region where entropy decreases; in inert systems no such region exists.


The emergence of life

Living things have the power to organize inert matter into complex and self-perpetuating forms. In human activity, this would correspond to the employment of law to produce an effect - using the fire to cause cooking. The demarcation between the living and the nonliving may be variously drawn, but the crucial distinction hinges on this capacity to oppose the increase of entropy, to store (and expend) order. The reversal of entropy does not imply a mere inversion of the former tendency toward disorder. Life process involves both conservation and expenditure of energy, but requires a net gain, just as a business must make more than it spends.

This view of the initiation of life provides a deductive and theoretical basis for what has hitherto remained a paradox, an apparent violation of the laws of matter. We mentioned earlier that the conclusion of determinism (there can be no free will) is shown to be erroneous when it is realized that the order of cause and effect can, in the human case, be reversed, and a desired effect can be the cause which is to produce itself.

We now extend this concept to apply generally. We observe in complex molecules the capacity to store order. There is no violation of the conservation of energy here; the order or energy is drawn from the environment. There is no creation of energy; the energy is made available by judicious sorting and storing in available form (as starch or sugar is stored in plants, for example).

Strictly speaking, we cannot, and do not have to, explain why this "judicious sorting" comes about. Since it involves the twofold operator, which carries back to the initial unity, we can appreciate that, as with all first causes, it cannot by definition be accounted for in terms of antecedents.

Nevertheless, the plausibility of our thesis is supported by relativity and quantum physics, both of which show that individual entities, in addition to physical volume, have an extra dimension which is equivalent to a free choice of timing. This is called the phase dimension and has the measure .* It

  supplies the necessary condition for reversal of entropy, making it possible for an individual entity to store order. These two different (but in the last analysis, equivalent) arguments run roughly as follows:

*Here means an angle of 360 degrees - it is called phase, and is a 

dimension because it is an additional unknown. For example, we say, "I don't know how this will turn out," or, "I don't know which way he will take this," "Which way" implies he may be for it, against it, or indifferent - this range of possibility can be covered by the angles possible in a circle.

1. The argument from relativity. The volume of the universe according to Einstein and Eddington is , not, as it would be for a physical sphere, The difference, as Eddington explains it,** is that the "ordinary" sphere, is multiplied by two added factors. One of these is , which Eddington calls the phase dimension. 

This phase dimension is an uncertainty of direction of , it being the character of uncertainty that its measure is angular. Thus we can describe the defining power (or certainty) of a lens as the angle subtended by two points that it can just discriminate. The smaller this angle, the greater the definition of the lens. It follows that the maximum uncertainty (lack of definition) is the largest possible angle. This is not infinity, but , the whole circle. (As the whole circle, this is also a unity.)

**Eddington, Arthur S, Fundamental Theory (p. 47). Cambridge University Press, 1946.

A second factor, 3/4, comes in as what Eddington calls "fixing the scale," which we may interpret as a reduction resulting from self-limitation. (See Chapter II: 3/4 point = control.) Eddington makes a provocative comment on the significance of this stabilization of scale: "But now that each particle, or small system has its own scale variate, a new field of phenomena is opened up to theoretical investigation, which is suppressed in the molar treatment of scale as an averaged characteristic."* The phenomena suppressed in the molar treatment are made possible by the phase dimension and the 3/4 stabilization of scale. These I interpret as:


a. Capacity to make choice of timing or .**

**There may seem to be a distinction between as direction (in  space) and "timing." But all time indicators in a finite universe have to be cyclic and as such show time as spatial angle.

       b. Self-limitation or stabilization of scale, 3/4.

The two factors, 3/4 and which multiply the ordinary sphere, 

thus render any small system, volume dimensionally equivalent to a universe (dimensionally equivalent means having the same essential nature, not the same size).

2. The argument from quantum theory. According to Planck's discovery, the photon, or pulse of electromagnetic radiation, is described as a quantum of action:

This quantum of action is also described as a quantum of uncertainty and contains the angle , a phase dimension, just as in the formula discussed above.

In other words, the light pulse is a "piece" of uncertainty. I cannot make this more palatable except to say that it can only be this way. Were it possible to predict or define or account for this uncertainty, we would simply transfer the ultimate reference, the first cause, to something else. "Where do babies come from?" "The stork brings them." "Where does the stork get them?" etc.

What is interesting and possible to account for is the chain of effects which this photon can produce. If it is very powerful (the smaller its wavelength, the more energy it packs), it can become a nuclear particle, either a proton or an electron. Some of the uncertainty has become mass (or certainty), but there is still retained a proportion of energy which is in the form of radiation (or uncertainty).

A further step combines the nuclear particle and other particles into an atom with further and very considerable loss of uncertainty, followed by still more at the molecular stage. At this molecular stage the remaining uncertainty, now much reduced in energy, persists as the bond which holds the molecule together. (We may think of a bond as equivalent to the energy necessary to break the bond.)

The measure of this uncertainty still contains , the choice of direction or phase, despite the fact that the quantity of energy is reduced. (As an example, think of a person with a much smaller pistol, but still having free choice of direction to shoot.) With the development of large molecules, this bond energy gets smaller and smaller until a point is reached where the energy of the bond and energy of a random molecule in the environment are of the same order.

At this point it would be possible in theory for the molecule to act as an energy sink, to draw energy from the environment and store it as order.

Now, the fact that life is very sensitive to temperature, that refrigerators preserve food because they prevent the chemical action of bacteria and molds, that fever and pasteurization destroy germs, and that harmful bacteria cannot exist at perpetual temperatures over 110 degrees Fahrenheit, confirms this speculation about bonds. The bonds of molecules involved in life processes must involve quite definite energies with rather narrow limits (plus or minus five percent in general, plus or minus one percent for warm-blooded animals). The energy of these bonds is of the same order as the residual uncertainty - about one-twentieth of an electron volt.

We see then that the theory of relativity and quantum theory agree. Both indicate that in large molecules there is a modicum of energy and choice of timing (phase dimension, ) due to which the molecule can extract energy

from its environment and build order or organization. Within a certain temperature range this emerges as life.


The evolution of life - increase of freedom

Up to this point, our account has described how the physical world of objects has come into existence. In contrast with the more "natural" explanation, which accepts physical objects as "being there in the first place," our interpretation holds that the originating and primary "stuff" is unitary, undifferentiated, devoid of properties,* without form or substance; it is known to physicists as radiation.

*Wavelength or frequency in our view is not so much a property as a potency of the photon. The velocity of light is not so much a property (i.e., limitation) of light as it is a limitation on matter.

Out of this primordial unity, there emerges, by division of itself in threefold fashion, the flow of time.** This division establishes the possibility of motion, of substance, and of relation. By a further division of itself, the fourfold, space (as a matrix of relationship and independent of time) comes into existence. In human terms, this fourfold division makes concepts possible; in terms of generation, it initiates form and interrelationship. From the combination of these two divisions, the threefold and the fourfold, comes the generation of the physical world, the world of inert molar objects, subject to the laws of determinism.

**Time does not exist for radiation, because at the speed of light clocks would stop.

According to this view, matter, or the world of inert objects subject to law, is not fundamental, but derived, and is three levels removed from the primordial and undifferentiated unity. Similarly, determinism, whose laws render inert objects predictable, is a derived principle, and owes its existence to the canceling out of the individual uncertainties of particular particles or subsystems.

However, the advent of determinism is, as we said, the last of a series of stages by which the initial freedom of the undifferentiated substratum (electromagnetic radiation) spent itself, first to acquire substance, then form, and finally concrete objective existence - a stake, as it were, a participation in the unique manifestation which we call the "physical world."

But the space-time of the physical universe is singular, unique. (As Kant pointed out, the remarkable property of the physical world is that there is no other. Heraclitus also noted the uniqueness of the physical world; he said that in waking life all men share one world, but in sleep they create their own and different worlds.) This fact, actually a corollary of the principle of determinism, makes it a truism that two bodies cannot be in the same space at the same time, that airplanes crash if their flight-paths cross. In short, it makes collisions possible, but by the same token provides laws which can ensure against them.

The progress of generation has carried us from the extreme of an initial and undifferentiated dynamic, to the contrary extreme of a derived and diversified objectivity (an unlimited number of kinds of molecule). The one is completely free, the other completely determined. The "descent" can go no further.

The ground is prepared for the next and only possible step, the reflection, the rebound, of what remains of the primary dynamic from the inflexible and unyielding barrier presented by the world. This possibility of a rebound is provided by the free phase dimension described above. The appropriate action, by foreseeing the result, can create the cause that will produce the effect. The result is the reversal of the tendency toward greater entropy, the possibility of drawing energy from the environment and storing it as order. This is the first step toward the regaining of the freedom lost, the fifth step from the beginning.

We are now ready to explore the nature of this first "upward" step and then the other possible steps. We have a choice whether or not to make use of empirical references to show how these steps manifest in nature. It would be easier from the point of view of clarity to do so, but the clarity gained would be at the expense of our deductive method. We are trying to layout the course of evolution from first principles by dead reckoning, "flying on instruments," and we would be going against the spirit of the endeavor if we "look out the window" to see where we are.

So I will ask the reader to bear with me and concentrate on deductive principles. These principles have already been established; we have only to draw out their implications.

First we recall how the threefold and then the fourfold operators brought about the loss of the freedom intrinsic to the original photon. The threefold created the limitation of movement along a time line which was one-dimensional, sequential, and non-ending. Then, with the fourfold operator, there was limitation in two dimensions. This produced forms and shapes which are necessarily finite, so that the first limitation, a commitment to one dimension with a freedom in two, was exchanged for a freedom in one dimension and a commitment in two.

If freedom is regained in the same way that it was lost, we may expect the first step toward freedom to conquer time, because time was the first constraint, and the second step to conquer space, because space was the constraint to which atoms are subject. Our first freedom will be infinite but one-dimensional, and the second finite but two-dimensional.

Finally, there will be the sum of these two; that is, unlimited goals in any direction.

Thus we may anticipate three distinct kinds of entity:

1. Entities moving toward an infinite goal with no choice of direction (one degree of freedom), conquering time.

2. Entities moving toward finite goals with choice of direction (two degrees of freedom), conquering space.

3. Entities moving toward a nonfinite goal with choice of direction (three degrees of freedom).


Categories of life exemplifying the three degrees of freedom

To the first class of entity we may assign plants (the vegetable kingdom), on the grounds that both growth and self-propagation comply with the definition "moving toward an infinite goal with no choice of direction." Heliotropism (growth toward light) is a specific behavior of this sort. Propagation through seeds conquers "the dimension of time."

To the second class of entity we may assign animals, since the mobility (voluntary motion) of animals, their pursuit of food and of each other, complies with the definition "moving toward finite goals with choice of direction" and conquers space.

The third class may be assigned to man, on the basis of his greater command of his environment, his greater freedom. This increased freedom is demonstrated by the pursuit of goals beyond his immediate needs and capabilities. Since the beginning of history, man has been motivated by principles, religious beliefs, or even by the pursuit of power for its own sake. He also exhibits this greater freedom in his manufacture and use of tools, his creation of language and culture, his pursuit of the arts, all activities which indicate a principle different from that underlying the voluntary motion of animals in pursuit of food.

There are reasons for reservation, however - not because man is a species of animal, for he may well also be something else, but because we should expect to find at this level a whole spectrum of development, a correspondence to the range between the lowest and highest forms of animals, between amoebae and elephants, in short a development that leads to supermen and to gods.

In this anticipated spectrum, man as we know him is by no means the highest development. But the principle here is quite independent of shape. Though we can recognize a great difference between one man and another, one is barely able to learn simple manual skills while another composes symphonies, both might look alike. Therefore we should not expect the principle underlying this third level to manifest in a variety of external shapes as did the animal principle.

We will not go into the development of plants, animals, and man here; it is the subject of another book by the author, The Reflexive Universe.* We need note only what is implied by the development in general: the freedoms which were lost in the descent into matter are regained in the ascent of evolution, but with a significant difference. What was random and blind on the descent becomes voluntary or controllable at will in the ascent. The trap of time and the trap of space become the power to organize forces and to animate forms. The means that brought the fall makes possible the ascent, and what is at first a constraint, when understood, becomes the agent for freedom.

*New York: Delacorte Press/Seymour Lawrence. 1976.


The Geometry of Meaning