Itson Mechanics - The Levels of Infinity

By Lere O. Shakunle
Director of the Transfigural Math Laboratory at the
American Computer Scientists Association

Mailing and other addresses:

Lere O. Shakunle

The Matran School of Mathematics
Berlin, Germany
Tel.\Fax.: +49 30 423 52 91
email: leshak@berlin.snafu.de

* Abstract

1.0. The Making of Reality

1.1. Gateway to Itson Mechanics

1.2. Mechanics of Infinity

1.3. Lifeline, Itson and Infinity

1.4. Itson Mechanics

2.0. Postulates of Itson Mechanics

2.1. Creative Potential

2.2. Deep, Reach and Transit

2.3. Radiance and Expression

2.4. Gradient

2.5. Quotient

3.0. Itson Mechanics Laws of Equilibrium

4.0. Foundations of Itson Mechanics

5.0. Properties of Itson Mechanics

6.0. Unified Space and Creative Potential

6.1. Folding Space in Unified Space

6.2. A Zero Spiral

7.0. Levels of Infinity

7.1. Outer Infinity

7.2. Inner Infinity

7.3. Transit

7.4. Absolute Spectrum

7.5. Deep

8.0. Some Features of Itson Mechanics

9.0. Itson Cosmology

10. A Physics of Physics

* References

This paper sets out to introduce the physics of itson called itson mechanics or infinity mechanics. As a gateway to this introduction we shall visit some of established theories of physics. Generally itson mechanics is a physics of physics. What do we mean by this? On the one hand it is a physics while on the other it is a physics that can be used to broaden our knowledge of any physics there is or may be. Therefore it is as physics as well as a meta-physics. We know [Bunch, 187] that a line can have at most two dimensions with the second dimension a loop. Euclid gave us a line as one dimension. The geometry of itson mechanics is based on spiral geometry. Spiral geometry is a meta-geometry in that it combines Euclidean, non-Euclidean and other features that liberate geometry from Euclid as a frame of geometric imagination. This we call a Post-Euclidean Age. Spiral geometry is based on a genus of line called lifeline which derives from a philosophy of space called the unified space outlook [USO]. Unified Space Outlook is the source of Folding Space on which the zero spirals of spiral geometry are based. [For these works refer [Shakunle,....]. A lifeline is composed of three outer dimensions and a number of inner dimensions. Both inner and outer dimensions are contained in even as they are the source of the nuclear dimensions. Nuclear dimensions are called the nuclear domain. The nuclear domain is the seat of the Creative Potential which is the seat of Creativity, Imagination and the other equally higher faculties. Lifelines have inner, outer dimensions and nuclear dimensions. Generally itson mechanics provides answers to, among other things, such questions as how old a pebble is and if space allows, what differentiates it from a rose? By the way, can we determine how long a pebble has been by the riverside? And what is the connection between it and the river? Is it possible for somebody to shout in Hamburg and be heard in New York? How does the folding space make this possible? Yes, how does information travel in a folding space? Folding space unifies space, time and mass. In this second paper on Organic System Design [OSD] we shall present some of the central idea of itson as the basis for the formalisms that are to be found in other works on itson mechanics. This calls for the extension of the spectrum of some of the terminologies of science and the creation of a new spectrum that embodies the foundations even as it grapples with the challenge of a new science - itson mechanics.

1.0. The Making of Reality

There is something defiant about the world. It can overturn our knowledge of it with a single event. It sometimes does. Perhaps the world is not the problem. Perhaps the problem is in our stars. We don't know how to be humble. Successful with a law or a rule, we celebrate the glory of knowledge and ask the world to clear the tables. At the peak of this celebration, as our song of supremacy reaches its crescendo, we heard a voice, deep inside us, that we ignored. It says, "Don't forget the puzzle." Of course we know the puzzle: the world is the puzzle. We know too that this puzzle is enveloped in a larger puzzle: the puzzle of the being.

It is indeed a puzzle that what looks so simple could be so defiant. The laws of Nature are simple. And though they can become so complex to the extent that our predictions are reduced to the elements of the dew - ineffectual - all the same we are sure that the trouble can only be that some parameters are missing. Once we can get these and plug them in, Nature will play ball.

The puzzle remains. That the world is a puzzle was finally brought home by mathematics. Suppose, says mathematics, the world is natural numbers, what can this mean? "Yes", we said, "the natural numbers have been taken care of by Peano arithmetic and its axioms. There are a progression with 1n following zero and 2 following 1 and so on. And so where is the problem?" The world was simple until we remembered what Phythagoras did with the same natural numbers, simple until Fibonacci adds two neighboring numbers together to get his sequence, simple until Euler arrived, ever so simple until Conway was born unto the world, but then no longer simple until group theory and quaternions arrived and became complicated, indeed near impossible as other mysterious and metaphorically deep numbers arrived to enrich our intellectual heritage and rock our ship of certainty. So if the world is the same as the natural numbers of mathematics, we may begin to say, maybe it contains world systems we never ever can know. Reason enough to be humble!

We turn our gaze to the heavens. Space journey has contributed indirectly to the knowledge of ourselves. We have come to see that we are enveloped in a universe whose immensity we never can penetrate in its entirety. We know that we cannot reach everywhere with our rockets; that we can lose contact with them and so, even if they are manned, can disappear in the cosmos with no explanation as to their whereabouts. We have come to see that the world is big, terrifyingly big. We feel like a person alone in a big hall. It cannot be that this guy is just dropped into a hall so big, alone. There must be some else around the corner, we said. A common sense justification. There is however, a deeper justification provided by geometry. This we shall have opportunity to visit in this work.

We turn our gaze inward in contemplation. We would like to answer the deep question of being without the microscope nor the telescope. With dig deeper inward with the resolution to reach the depth of the being. At a stage, the self is lost. At this stage we are disembodied, lost to the world of cares. This stage is Ralph Waldo Emerson's In Tune with The Infinite, the Christians' Kingdom of God and the Hindus' Nirvana. If we can get there and remain there for a while, we may come back with illumination.

Everywhere we turn we are confronted with something we cannot fathom - Infinity. Sometimes as we look at these impenetrable wonders, we begin to say that any science fiction should be possible - somehow. Since imagination pastures where common sense fears to tread, we have every reason to believe as possible what we cannot dispute. It is not, and I repeat, it is not that a science fiction is true to life. What we say is that, based on the mysterious structure of the world, as a result of the fact that we never can know how many world systems exist and in what variations (which may be the justification for the Many-World Interpretation of Quantum Mechanics and the belief in other worlds in general), we never can say that the features of some science fiction may not be the fabric of a world system inaccessible to our ken - as of now at least. To take all this as a call to believe in science fiction is to misinterpret a metaphor. What I was trying to say is that the world, according to the geometry we shall explore together in this work, may be more terrifying than what we've heard from the Newtonian mechanics, Relativity Theory and even from Quantum Mechanics, to the extent that one is suddenly caught in fear at its mere contemplation. Another reason to be humble!

1.1. Gateway to Itson Mechanics

The quantum mechanics is impossible enough. Indeed there is no physics over which the adherents doubt their choice and the opponents question their stand as it is the case with quantum mechanics. The quantum reality is a metaphor whose meaning in shrouded in mystery. There is no need revisiting these matters here. A lot has been written on it to fill the old library of Alexandria.

But what is the power of quantum mechanics? I would say its power lies in its metaphysics. Here is a physics that let loose a duel between Mind and Matter. Before quantum mechanics, it was possible to read mind into some aspects of Newtonian mechanics and some puzzles of Relativity Theory. Such attempts were not intended by the makers of these world systems. It is only in quantum mechanics that the mind was part of the project from the beginning. Indeed, if anything, the quantum mechanics was the harbinger of the end of a purely mechanistic outlook with its glorification of matter.

The physics that is going to be presented in this work - belt up - is more impossible than quantum mechanics. In this work there is no confrontation between Mind and Nature at all. We settle the case for the Mind. But then, we need to accommodate those things in which Matter has shown to be effective. The Mind cannot do this without being reduced in the process to Matter. To solve the problem we need a common source for everything. This we call the Creative Potential.

Creative Potential is composed of different levels of infinity. The sum of these levels is called the Deep. And so itson mechanics, otherwise called infinity mechanics (mechanics of infinity) (see below) is based on the nature of the structure of systems that compose the Deep.

Let us not be deceived by the name. Itson is neither a mechanical nor a vitalist entity. Itson is a singular entity (refer Singular Concepts). A singular entity does not engage in the argument of the Other. It is an entity that is there, complete in itself. Infinity is also a singular concept. And so mechanism and\or vitalism have no place in them. They are entities that are there, like beauty and humanity.

1.2. Mechanics of Infinity

Georg Cantor was the first mathematician to figure how the natural numbers amy possibly look like at infinity. To the end of his life Srinvasa Ramanujan puzzled over how zero and infinity relate. Aristotle spoke of two types of infinity - actual and potential infinities. Actual infinity is a once-given entity. It is the kind of infinity in which the mind finds its pasture. It comes once and as a whole. It is spontaneous. Potential infinity is evolutionary. It comes one after the other. The Cantorian set is based on actual infinity. The mind can conceive the members of a set. Intuitionists take a potential approach to infinity. This infinity is constructive. It is forever becoming. Both infinities have their attractions. But is there a way to approach infinity to accommodate the two types of infinities? Can we integrate the two infinities in a way that they are they are not dichotomies in relation to each other? Can we make them to be two sides of the same coin whereby we have something in-between them that unifies them? Indeed can we get a geometry and an algebra that make it possible to speak of the two types of infinities as two aspects of an indivisible reality? To do all this may require a new foundation of knowledge. It is not possible - and this is supported by Gödel's Incompleteness Theorem - to make these things possible within the conventional setting of knowledge based as it were on the classical or/and modern foundations. A system must transform itself to understand its insoluble(s) at all. And it must transcend itself to solve them.

But while mathematicians have made and are still making efforts to grapple with the mystery - and challenge- of infinity, to the physicists, infinity wasn't a problem until one begins to wonder what happens at the edge of space. There are other areas where physics and infinity meet. The one that interests us - and it happens to be the most interesting - is the edge of space of cosmology.

Cosmologists wonder what happens at the edge of space. It is a very deep question. Assuming that the universe ends somewhere, the question is what next, that is, what follows after? Assuming ours is the only universe. And assume that this universe has an end. This we call the end of space. Now let's begin with a simple picture of infinity. Let a line begin somewhere and continues until it thins out of sight. The place where it thins out and vanishes, is infinity. We don't forget that we are talking geometry. In the common sense parlance, a place where a line thins out may be some kilometers away. The line we mean here reaches as far as the end of the universe. Let's try to make our picture a bit sharper. Let us say that gravity is effective everywhere in our universe. Let us say that our planet is enclosed in a universe whose shape is spherical. In other words, our planet and others are enclosed inside say a ball. This will then mean that if we catapult our spaceship anywhere inside this cosmos, it will reach the end and return to us describing a curve. Suppose this spaceship can be made to be coming and going, what we have is the earth at the center-point of a bundle of curves. But they are curves at the end of the universe. To us on the earth, they are folds. Now, all this is assuming that ours is the only universe with earth as the only planet. If there are other planets and the same thing is happening on them as on earth, what we have are different folds. It is for us to determine the nature of folds that ensures that the planets do not collide. Now, imagine somebody at the other end of our ball. Imagine gravity to be effective outside the ball. This is like saying that our universe is made of a magnetic stuff which draws to itself whatever is on it. Any line on the universe will return and describe a projectile. If it is possible to make this line behave outside as it does inside, what we get are folds.

All this leads to trying to know the nature of infinity. From the line to the spaceship, it is just easy to catch the mood of infinity in what we have said this long. A spaceship that reaches the end of the universe is lost to us - for a time. It will come back. How long this may take or what it will look like when it comes back may not be accessible to a definition. All we know is that since it takes so long to reach the end of our universe - which is our infinity - it may take thousands or billions of years for a spaceship that is lost in the cosmos to come back to us. There is every assurance why this should happen. The spaceship is made of earthly metals on which only our gravity has absolute effect. We know that to land on another planet we have to adjust to its kind of gravitational pull. Based on this we can conclude that just as no two planets are the same so are two universes. Every universe is unique. And so, it is not possible for what belongs to another universe to fall into our universe and remain there forever. Communication is possible. But this requires geometry. This is one of the areas of applications of spiral geometry [Shakunle, 1996]. Now comes something a bit disturbing. If every universe is unique and, based on this, if every planet is unique, is it possible for something from one of the planets to fall on the other planet? It is possible by the same law that makes it possible to land on the planet. If we take the planetary system as genetic pools, then we can say that by dint of planetary mutation, a chunk of a planet may break away from the mother planet and head for another planet anywhere. Whatever happens, the uniqueness of a planet lies in its own 'laws of Nature'. Physics works with the assumption that its laws are the same everywhere but with the gravity on the moon one-sixth that of the earth, one sees that this assumption is based on the earth dictating to the others.

Our concern here is the mechanics of itsons or infinity. We have tried to explore the behavior of bodies at infinity in relation to their reference. Our reference here embodies, among other things, the frame of reference. We say 'among other things' because reference is one of the features of Method in Transfigural Mathematics. Method is composed of (i) reference (ii) inference, and (iii) transference. For example natural numbers are a reference. Inference are the systems, their operations and rules from natural numbers. Fibonacci sequence is an inference. Transference is the use of a system to understand and solve problems in mathematics, the sciences and engineering. It is also transference if we use it for concepts and ideas in metaphysics, philosophy, literature and the arts. We see that reference here is broader than the frame of reference.

Simply put, itson mechanics is concerned with the behaviors of things at infinity. Itsons are based on lifelines. A lifeline is the life of infinity as a line. It is the nature of a lifeline that everywhere is a center. This is to be expected since every point on the line as infinity is infinity.

1.3. Lifeline, Itson and Infinity

We need, at this stage, to revisit the concept of lifeline in Part I of this work again. In that paper, I wrote:

"Lifelines are potential curves at every point. A lifeline line bends into a curve which grows into a fold at infinity. Every point on the lifeline is a curve. Every curve is a potential fold. Both sides of a point are minus and plus domains whose summation is an inexhaustible zero. The minus and plus depict positive and negative infinities whose addition is the absolute zero. This means every point on the line is the summation of infinities. There are three types of infinity in a lifeline. These are (i) inner infinity (ii) outer infinity (iii) absolute infinity. Point in a lifeline unifies the features of points in Euclidean and non-Euclidean geometries. The life history of a lifeline is concentrated on the point."

And, according to

"It is the nature of a lifeline that everywhere is a center."

This means the center is everywhere on a lifeline. Any reference to a line in this work where not otherwise specified, is a lifeline.

In its physics, itson is the indivisible unit of space. The shortest route to itson is to think in terms of two points on the line. We divide these two points into two and divide again into two and again and again. Physically we reach a place where this is no longer possible. We commit the whole thing to the mind and continue to divide. This dividing of a line in the mind is the inward infinity. At a stage we can no longer see with the eyes what is happening but can see the whole thing clearly in the mind. The inward infinity takes us to itson. Actually logic numbers [Shakunle, 1986 - ] make division unnecessary. A logic number does not permit of a space between two points on the line. Thus a logic number takes us into the innermost sector of things without the need for fractionalization. True we shall see lines in itson which are curls - made so by the infinity - all the same we should not forget that itsons are invisible entities. We look for their representations in Nature. But they are far beyond the conventional name of things in physics. They are neither fields nor particles. They can be used to unify both. What we know is that one itson is enough to create all the universes that are. On the one hand, itson is something that is there while on the other it is a deep metaphor.

1.4. Itson Mechanics

Itson mechanics is based on point, line and infinity. This is why it is either called Itson Mechanics or Infinity Mechanics. As we shall see the point and line in the geometry of itson mechanics called spiral geometry are neither Euclidean nor non-Euclidean. A point in spiral geometry is a point-event while a line is a lifeline. There are three types of infinities in spiral geometry. These are (i) inner infinity (ii) outer infinity, and (iii) absolute infinity. We never can know what is happening in absolute infinity. Inner infinity is a spur to creativity. Outer infinity is what we know but which contains building blocks for the construction of other systems. What readily comes to mind in this regard are the natural numbers. We shall have more to say on this later.

Each time we mention point, line and infinity, where not otherwise stated, we mean point-event, lifeline (which we have said before) and the three types of infinities. Both the inner and the outer infinities contain even as the transcend the features of actual infinity of the sets of Georg Cantor and potential infinity of Brouwer's intuitionism which is at the basis of the constructivist paradigm.

Point, line and infinity in itson mechanics are based on a unified space outlook, being the type of space on which spiral geometry is based.

2.0. Postulates of Itson Mechanics

Postulate I

Every thing originates from the Creative Potential

Postulate II

The Expression of a Creative Potential is its Radiance

Postulate III

All things are made of space

Postulate IV

Every space is a potential fold

Postulate V

Every fold is a specific form

Postulate IV

Every form is a substance

Postulate VI

A substance is an idealization of a concept

Postulate VII

The Radiance of a Concept is its Expression

2.1. Creative Potential

Each of the postulates of itson mechanics deserves a book to itself. In what follows we shall try to take a short excursion in the postulates. The Creative Potential is the source and maker of worlds. Everything in its world has its stamp. The Creative Potential can be likened to a universal pool of creative powers. Each set of things has a definite amount of these creative powers called their deep of creative potential. This amount of the creative potential of a set of things is called its deep. Thus all the flowers in the world have a specific deep of creative potential. Birds have their own deep which is not the same as the deep of flowers. All the deeps are inside the Creative Potential. Each member of a set partakes of this creative potential. One deep can be deeper than the others but this does not mean greater. Indeed the deeps can be likened to different rooms in the same building. The differences between one deep and the other is what I call their depth, intensity and focusity. These determine their radiance. The focus of a deep determines its closeness to the Absolute Infinity (which we shall come to later) while the greater the focus, the more intense and the closer to the innermost (the depth). All this have nothing to do with greater or less since all things partake of the same thing. The difference between all flowers and all birds determines the individual differences among their members. All birds are the same. They all have a common pool of the Creative Potential. The difference between an owl and a sparrow is to be found in their focus, depth and intensity. These are geometrical-algebraic entities which are so involved that we shall only show their geometry phenomenological in this work. Generally we expect everything to have the qualities of the deep of creative potential, henceforth referred to as "dcp" or simply as deep. All the deeps called the Deep constitute but do not exhaust the dcp since there are regions of absolute infinity whose contents are beyond our deep - the dcp of the Homo sapiens - to imagine.

2.2. Deep, Reach and Transit

There are certain levels of infinity which may not be accessible to us with absolute certainty. Levels of infinity are the deep of the systems of the world in the Creative Potential and the realization of these worlds which is called their reach. There are deeps as systems within a world and the deeps of worlds within the worlds. The same thing holds for the reaches. The transit is the boundary between the deep and the reach. This can be compared to a border between two aspects of a single thing. There is something remarkable about transit. It is not possible to fix its place.

2.3. Radiance and Expression

Every system expresses itself through its qualities. The totality of these qualities which constitute its Expression are called its Radiance. Fragrance is the radiance of a flower. We expect every itson of a flower to have this quality in different measure. The seat of this radiance of a flower which expresses itself as fragrance is the [amount of] creative potential of that particular flower. The same holds for a pebble or a bird. What we do is first get the different deeps of stones and birds in the Creative Potential and from there get the radiance of [particular] stones and [particular] birds. The problem is always a radiance since it determines the itson we are dealing with.

One of the advantages of the Deep is that it liberates the consciousness from appearances. From it we come to know that things are not what they seem. Indeed a rose may still be wearing the appearance of a rose when it is indeed a touch-me-not. This is one of the effects of cyclicality. In one cycle a rose may be a touch-me-not while at another cycle it is again a rose and at another something far beyond a rose. Some roses get to the touch-me-not and remain in the deep of touch-me-not while other roses are marching for the heavens. During this metamorphosis which are unknown to the external world, the rose remains a rose in appearance [to the external world].

2.4. Gradient

At the level of the deep the thing cannot make a choice. At the level of radiance it can determine how to use the radiance. This is determined by the coordinate [of the web] the curl of itson chooses. With gradient comes the freewill. Gradient releases the inner resources of a thing. It is what makes the flower blossom to release its radiance which expresses itself as fragrance. The line of the web which a curl has to cross is its gradient. No two flowers have the same number of lines and so no two itsons are the same. We see it in roses. One rose has one leaf more or less than the others. The branches are also not the same. The place where one rose occupies cannot be occupied by another rose. The same holds for two pebbles. No two pebbles are equally smooth. In itsonal world, everything is unique. The pebbles have their Deep. But this pebble which may have the same amount of radiance is different in the configuration of its itsons therefore not the same as every pebble.

2.5. Quotient

This is the amount of radiance at the disposal of gradient.

3.0. Itson Mechanics Laws of Equilibrium

In what follows we present a set of the laws of the mechanics of itsons. These are the laws of Equilibrium. The laws of equilibrium are derived from the Itson Mechanics Conservation of Equilibrium.

Conservation of Equilibrium

Every thing is in a state of balance

Laws of Equilibrium

Equilibrium law of Meaning

Every system seeks its balance

Equilibrium law of Necessity

Every system keeps its balance

Equilibrium Law of Cyclicality

Every system always returns to its balance

4.0. Foundations of Itson Mechanics

From what we have come to know about itson mechanics we can put certain things down as settled. The foundations of itson mechanics differ fundamentally from those of any classical and modern physics. Whether classical or modern, physics recognizes the existence of matter and mind. Where it does not have an answer for one it throws both inside the same pot. But this does not settle the matter either. Whether classical or modern there is only one identity for the collective. It does not matter whether this rock is round or this stone is smooth, this rock is the same as every other rock and same for the stone. There is nothing unique about matter. But there is something unique about the Homo sapiens. Yes, only the Homo sapiens are unique at least to some extent. A human being in this physics is made of atoms like the stone or the rock except that, well, he has a mind. Itson physics goes in a completely different direction. Its foundation is not built on the dichotomy of Matter and Mind which is the fundamental shortcoming of classical and modern physics. We see that quantum mechanics tries to break down the wall between Mind and Matter without really succeeding. The reason for this is clear. To do this you need a new, a completely new foundation. You cannot build a physics of unity on the foundation of disunity.

What we expect as itson unfolds is that we simply forget that there is something called Mind or Matter. Indeed itson mechanics shows us the nature of the unfolding of identities of things - unique. We cannot pin down what is happening to spontaneous creation or evolution. Itsons have the features of both. They are also have the inexplicable. For example, the nature of their creative potential and the changes they undergo within themselves even while their appearances remain the same. These things cannot be subsumed in neither spontaneous creative nor evolution. It is the nature of a creative potential to challenge the imagination. And so we are not surprised that things are like this in the itson scheme of things.

Based on its foundations, the itson is neither a field nor a particle but something beyond both. The wonder of it is that it incorporates both without being either one or the other.

On the other hand, itson can give us a deeper understanding of classical and modern physics. Indeed it gives their entities a sense of meaning. This means we can use some aspects of itson mechanics to tackle some of the normal as well as intricate and even paradoxical issues of modern and classical physics.

5.0. Properties of Itson Mechanics

Some of the properties of are as follows:

- Orientation

- Reflection

- Projection

- Rotation

6.0. Unified Space and Creative Potential

It is the nature of unified space that the doors are open between one discipline and the others, between one tongue and the others, between one type of space and other space types so that all the good things can speak with one another such that, in the process, one understands the others to the extent that one can speak of a common language. A unified space is based on the conviction that though the good things may differ all the same they all have a single source. We have mentioned the word 'good' often enough to give the reason for its presence. Creativity and the Good are two sides of the same coin. And so the Creative Potential is the source of the Good. By the Good we mean the Truth. If Falsehood is the opposite of the Truth, there is no Falsehood in the Creative Potential. This means we are dealing with an unconventional logic here. Here is a logic based on The Truth and the Truth alone. The Truth manifests itself in different ways and in different places. These different manifestations of the Truth which can be equated with the different expressions of the Creative Potential through its radiance, we call positive and negative truths. We represent positive truth as +T. This we call the region of positive infinity. Negative truth is represented as -T. This we call the region of negative infinity. These are the outer infinities. The nucleus which is the Creative Potential that is manifesting itself we represent as zero. It should be T but it is not. The reason for this is to be found in logic numbers. It is any other capital letter other that T. In the process of its manifestations the truth symbol is changed. It is numerically the zero. This is the basis of logic numbers [Shakunle, 1986 - ]. This is the region of inner and absolute infinities. Inner infinity is the source of The Zero Conjecture where the zeroids - the numbers of the zero - play an important role [Shakunle (in progress) ].

Also from what we have come to know of Creative Potential we see that it is not possible for polarities such as matter-mind, subject-object to exist since the Truth is indivisible. And so just as the unified space bring out the disciplines and principalities from their cocoon so does the Creative Potential happen to be the source of all the deeps around which are the resource pools of world systems.

6.1. Folding Space in Unified Space

Unlike Euclidean and non-Euclidean geometry's where the shape determines the nature of the line, there are no shapes other than space itself in folding space. In both Euclidean and non-Euclidean geometry when we talk of space we do so in relation to something else. And so for this reason points and lines are meaningful, the line especially, in relation to something else. We see this in non-Euclidean geometry's. Spheres and ellipses play important role as they show how the laws of the parallel lines breaks down in them. The curvature of space also derives from what happens to space in relation to something else. In folding space we are not dealing with something in relation to space. Indeed we are dealing with the transformation of space into those things to which it is later related. This is what the first postulate of itson mechanics means. All this results from different folds of space. In other words, everything is reducible to space. This means all that we see as solid are special kinds of space folding. There is nothing solid about anything! What makes all the difference is the nature of the packing of a web and the number of curls that spreads out to form the folds and its hooks. In fact when we talk of something in space what we are saying is a space that is folded inside a space that is still folding. Since things are hooked together we can show how these two things are intertwined. In such a scheme there is neither particle nor field to talk about.

But a folding space is even more mysterious than all this. There is motion in this space which in fact is non-motion. A paradox. In fact, motion in folding space is built on paradox. Since things move which in reality do not, the equation of motion proves to be an adventure in wonders of geometry. In other physics one hears of the speed of x and the speed of y and the time it takes x or y to cover t from which we get the differential equations that takes care of the distance covered in a specific time. There is no distance in folding space. Yet there is distance! This means something can be here yet be there. Like we said, all this is based on the wonders of a geometry, spiral geometry to be specific, based on a genus of spirals called the zero spirals.

Our shortest cut to what is happening is to imagine a line folding at infinity. We can compare this to thinking of an idea and following it until we reach a stage where we can no longer continue - for the moment - and come back to ourselves. In such a case the idea is the point which describes a line as we think over the idea which, as a result, grows in the process. When we come back to ourselves as we have to what really happens is that the point returns to us. It makes a point where we decide to come back. This place where we decide to stop to return to ourselves is infinity. It the infinity of the idea, a point, at this moment. This infinity is not static. It is always growing even when we are not working on the idea. When next we think of the same idea, we don't go in the same direction as we did the other time. And so by the time we return again to self, a spiral is made with the self as the nucleus. The self itself is transformed in the process and so it is no longer what it was before the idea occurred to it. There is a physical example which one can give even though its metaphor is not as rich as the one we give here. What happens really is this:

In folding space the Cartesian coordinates vanish. The reason is that the fold uses the basic lines of any space for its foldings. Where they remain, they are webs or webloids. Even as webs and webloids, they are still non-Cartesian coordinates. They don't behave as one! Indeed every line that forms a fold is a vanishing line. It vanishes in the transformation. Since every line is a fold and since every point on the line is a center this means there are no lines as we know them. Thus the Cartesian coordinates are imaginary lines in a folding space. We have ample examples of geometric figure that show the coordinates to be imaginary lines. With the imaginary line we have the last figure as a fold:

6.2. A Zero Spiral

Even if the Hindus had not (invented, discovered) the zero, the zero of itson mechanics (nuclear zero, potential zero) has been there all the time. Whether it is Nirvana or the Kingdom of God, whether it is prayer of the devout or the meditation of the mystic, whether in introspection as a mirror on the self or the interrogation of the images of identity, there has always been a state of intellectual and spiritual upliftment at which the being, in tune with The Infinite, rediscovers and, in the process, recreates itself. And so the traditional zero which is Nothingness which is not empty and the nuclear zero of itson mechanics which is the potential zero and nuclear domain of logic numbers which is the fountain of life in its various levels of manifestation, would have come to us somewhere, somehow.

Zero and infinity are two of the enigmas of mathematics. First, the zero. In mathematics there are two approaches to it. It is Nothing which is the same as none. In this case it is empty. Even then, as we shall see very soon, this not settled either. It is Nothingness, which in this case, is not empty. Even then Nothingness is not the same as the Vacuum which, as we know is not empty either [Bunch, 1989]:

"A different instance of electron-positron formation seems more mysterious. In that case, the electron-positron pair is formed from random fluctuations of the vacuum. A modern definition of a vacuum is the state of least energy for a region. We now know that this energy can be the lowest average energy. Even when the average energy is 0, which we would think of as a normal vacuum, there can be fluctuations above and below the average" [162-163]

This amounts to producing something out of nothingness. But now comes one of the mysteries of the zero which have eluded physics up to now and which is one of the sources of the thesis of itson mechanics which says: Everything is from the Creative Potential. This Creative Potential is the nucleus of the Zero." The Creative Potential, we do not forget is a universal metaphor. It is the source of all there is. If we confine it to the language of the physics we know, we can say that it the source of all energy. If we choose this partial interpretation then this would then mean that the zero is not the just average energy but the very source of all forms of energy. However, as we have seen, the Creative Potential is a mind at work. It is what makes all there is and so this explains the reason why its physics in beyond particle and waves, beyond the conventional terminology's of physics and all-encompassing in its range.

We dwell on the issue of energy a bit. For this we need Bunch again:

"When a particle seemed not to have an antiparticle, such as the photon, physicists decreed that it was its own antiiparticle. This is in analogy to the role of 0 among the signed numbers. Every number has its opposite or anti-number. The anti-number of +3 is -3. The anti-number of -5 is +5. But the anti-number for +0 and -0 is just 0, since all three symbols refer to the same number. Similarly, the photon is its own antiparticle in the same way as zero is its own anti-number"[167]

The +0 and -0 of Matran Logic [Shakunle, 1987 - ongoing ] derive from the nuclear (0) of Logic Numbers [Shakunle, 1986 - ]. Both +0 and -0 constitute the first step in the inward journey to the nuclear zero - in itson mechanics - called the potential zero or identity domain - in logic numbers. Since the 0 from which they derive is not empty, the +0 and -0 of matran logic take us in different directions of a zero which remains unattainable. According to matran logic, there is anti-photon! The argument of matran logic is this: To know what other particles are contained in the photon which is +0 we need the anti-photon which is -0. With the nuclear zero we are in for surprises. It is the domain of the Creative Potential - the Spectrum of Infinity - of Spiral Geometry [Shakunle, 1994] which is at the embodiment of the levels of infinity of Itson Mechanics.

All this takes us full-scale to infinity. Zero and infinity have something in common. They both have the stuff of which paradoxes are made. On the one hand they can be seen to be clear as a day while on the other it is not possible to pin them to a point and say, "Here is the end of zero. Here is the end of infinity." Both are spectra of possibilities. The zero is the spectrum of inward infinities while all negative numbers represented by -1 and all positive integers represented by +1 constitute the spectrum of outer infinity. We shall have more to say on these infinities in the next section.

We said above that an infinity is not static. What can this mean in practice? We take the natural numbers. There is a place where our mind regards - at a particular moment - the place where the symbol of infinity could be placed. The next moment we look at the natural numbers and see infinity again, we can be sure that this infinity is not the same as the infinity we saw the other time. What the transfinites of Georg Cantor do to settle the matter, something which the finitists find hard to swallow, is take infinity as something which the mind can see at once [Benardete, Rucker]. This is indeed a great achievement. However the spiral geometry with the lifeline on which it is based takes a different approach to infinity. It sees infinity as a project. Infinity as a project means infinity is dynamic. In this case, infinity is such that we can speak of it as we speak of dynamic systems. With a difference. There is no way to consume the potential of infinity. In other words there is no point out or in there which we call infinity. There are points which depict infinity. And there is a place where we can just put the point and call it the region of absolute infinity. This is what Cantor calls the Absolute. What we have below the Absolute is a spectrum of infinities. This spectrum of infinities is the same above as it is below. The spectrum of infinity is composed of different infinities. These are called the levels of infinity (see below).

Before now we said,

"This place where we decide to stop to return to ourselves is infinity. It the infinity of the idea, a point, at this moment. This infinity is not static. It is always growing even when we are not working on the idea. When next we think of the same idea, we don't go in the same direction as we did the other time. And so by the time we return again to self, a spiral is made with the self as the nucleus."

What results from all this is a spiral called the zero spiral. The idea of self is just a metaphor. It shows that this phenomena of infinity cover the idea of self. Applied globally, the point is where the self is the zero point of the Creative Potential.

What we have is the fold of a zero spiral below:

This follows from

At least two folds make a zero spiral, therefore:

7.0. Levels of Infinity

Infinity calls for freedom from the bondage of language. Our languages are such that condition us to see things as words and opposite. There is something mysterious happening between language and the mind. They influence each other. And so if language says that for every ying there is yang, the mind begins to live in a binary world. If it the mind sees the quantum logic as a reflection of itself, language creates the necessary terminologies to take care of the mind's invention. And so language makes the mind as the mind the language. To ask to know the source of dichotomy may need to explore the developments in the philosophy of mind and that of language. From the dichotomy's outlook follows that no word is spared its opposite. Since this outlook is based on either one or all, it therefore means that the axe must fall on all the words there is and those to come. This is what I call the bondage of language from which the infinity, like The Good and The Truth which have no opposites can free us.

We start by saying that some words have no negation. Such words are their own negation. Up to now I have found three words which have no negation. Generally in The Principles of Transfigural Mathematics we have three types of Concepts.

These are:

Singular Concepts

Words that have no opposites

Generally such words are very few.

Examples of Singular Concepts

Being; Truth; Goodness; Beauty; Love; God; Humanity; Whole; Infinity;

Holiness; Wisdom,......

Complementary Concepts

Complementary concepts are concepts that fit into each other to the

extent that where there is one there the other is expected to be there.

Mirror and complementary concepts are always in pairs.

The are not very many either. But they are more than singular concepts.

Examples of Complementary Concepts

mind, thought/ light, photon/,...

We don't put the complementary concepts in brackets because even though

to be in one is to be in the other, all the same only one is needed to be in the

other.

Mirror Concepts

Mirror concepts are symmetric-asymmetric concepts.

Examples of Mirror Concepts

(North, South), (Left handedness, Right handedness),.....

Dual Concepts

Concepts that have opposites

Most words belong to this group.

Examples of dual concepts

Growth-Decline; Upper-Lower;.....

In this work we are concerned with Infinity. Nowhere is infinity an intractable problem than in cosmology. The finitists quarrel with the aleph-null of Cantor and with Hilbert's Grand Hotel. But before we go, I hasten to say that a finitist is somebody who presents no counter argument to the infinite because finitism is not the opposite of the infinite. The finite is normally a subset of the infinite and we can show that even this subset is infinite since, if is a line, there is no way to fill it up with points. Between two points on a finite line it is always possible to put points. If we look inside this line we see two points which can also be joined by lines with infinite number of points between them.

It is believed that to the question, "Has the world an end?" there is only one answer. It is either 'yes" by which is meant that the world is finite or "no" which means the world is infinite. Which should not be the case. Benardete [1964] replaces the world with a wall whose end one cannot see and asks to know whether the wall is finite or infinite. Sound argument but for the confusion of language. The confusion lies in the juxtaposition of finite and infinite in which one is seen as negating the other. This, as we shall see in this work is not the case. Where infinity is the issue, the finite has nothing to say. This is because, infinity is a singular concept.

Before I show that infinity is its own negation, let's see the nature of the tussle between finite and infinite.

"Our language is thus seen to be systematically misleading in regard to the finite-infinite dichotomy, though altogether satisfactory in regard to perfect and imperfect" [105]

This leads to the postulate of infinity on which the itson mechanics is based.

I. Postulate of Infinity

Infinity is its own negation.

II. Postulate of the Deep

There are different levels of infinity

III. Postulate of Singularity

For every Transit, there is at least a Deep and a Reach

IV Postulate of Unispace

All infinities share a common thread

According to I. infinity has no opposite therefore it has nothing to do with finitude. A paradox, it has its own negation too. But this is not the same as the negation in dichotomy. From (II) we get to know that there are different infinities in the Deep. In other words the Deep is the summation of infinities. These infinities are not the same in terms of their contents. Thus even though each is an infinity all the same its depth and reach are not the same as those of others. This is what is meant by the infinity being its own negation. (III) is very mysterious. It says that it is possible to use one transit for more than one Deep and Reach. Is this a proof that other intelligences may use the transit of our universe to cross to other worlds? (IV) assures us that wherever we may land, we are in safe hands since one common thread runs through all infinities.

7.1. Outer Infinity

The world is a riddle. I hope nobody is trying to equate this with the universe. By the world I mean the totality of the human experience of being of which the universe is just a part. The relation between the world and the universe is that of a partwhole and wholepart in which case the universe is a partwhole. A partwhole is a whole that behaves like a part while a wholepart is a part that behaves like a whole [Shakunle, Matran Logic]. In reality what we have are wholes, wholeparts and partwholes. This also shows that the whole-part dichotomy doesn't hold. The whole is a thing with no parts if parts exist at all. What this means is that when I divide a cake into two, the whole still remains. Things which are whole by themselves cannot be divided. Each part of the cake that you get is your whole, though a different whole. It is therefore one of the manifestations of the whole. Call it your part of the truth if you like.

Being a partwhole, the universe shares some of the mysteries of the world. Most philosophers, excluding Immanuel Kant, made the mistake of equating the world with the universe. I think Kant was one of those few philosophers with a highly developed ontological sensibility. Kant knows that, ontologically, the world is larger than the universe but then the metaphysical accommodation of the world demands what is accessible to the vocabulary of philosophers. More than that, to tell his fellow philosophers that it is simply nonsensical lumping the world and the universe together as most of them did would be ungentlemanly. And Kant was indeed a gentleman. And he did what gentlemen did when they know that what others are saying lacks every philosophical depth and epistemological foundation and they themselves are not clear how best to order the situation: he denies that the world exists. Benardete [1964] was annoyed with Kant with reasons. He ventilates his ire in a sentence,

"Kant is guilty of an enormity only at the point when he draws the stupefying conclusion that the world simply does not exist! [110]

but when he gave us Kant's three reasons for his stand:

"(1) The world as a whole is no possible object of experience; (2) Only what is a possible object of experience may be rationally supposed to exist ; ergo (3) The world as a whole may not be rationally supposed to exist" [110]

we couldn't see the reason for his ire nor that of Kant dismissing the world for which our being stands as a proof of its existence. Of course you cannot see the world as you see the sparrows arriving home after a summer sojourn in other lands. If anything it belongs to the region of deep reality which the first part of Neils Bohr's Copenhagen Explanation of Quantum Reality denies [Herbert, 1985]:

"There is no deep reality.... The world we see around is real enough...but it floats on a world that is not real. Everyday phenomena are themselves built not out of phenomena but out of utterly different kind of being. Bohr's anti-realist stance is summed up in, 'There is no deep reality' ".

Though Part I distances itself from deep reality, the second part of The Copenhagen Explanation brings in reality through another door with a different name. According to this Explanation [Part I] the world in which we live is embedded in another world. We see that even if we don't agree with the Copenhagen Explanation, we see that it knows what it is saying: the world is not accessible to the laboratory. This, we say, is what makes it the deep reality! I think the problem is that Benardete interpreted Kant out of context. By reading further [Benardete, 111] one could see that he [Benardete] was fighting in the camp of philosophers who took the world for the universe. An ontological catastrophe. There is something fundamentally faulty about the premises of Kant. The world of feelings is not rational. This world exists all the same. Apart from this, reason can be faulty, as we have seen in Kant's conclusion, where the premises are wrong. Beyond all this, there are faculties which are higher than reason. Intuition and Revelation for example. When the prophets predict what will happen in thousand years, there is no way for reason to know what is happening nor can reason tell us what led to the Eureka of Archimedes. To say then that "only what is a possible object of experience may be rationally supposed to exist" is to delegate to reason what is beyond its terms of reference. The existentialist philosophers who have a richer language than Kant in this regard [Satre,1958] failed to achieve the synthesis of the world. The dichotomy of Being and non-Being throws the spanner in the works of their dialectics.

True, the existentialists contributed immensely to our knowledge of Nothingness in their phenomenology of the being. From them we gather that Nothingness is that which propels the being to its unfoldment - which is conditioned by its folds - all the same we do not know what this Nothingness contains nor do we know how it relates to the Nothingness of others. This Nothingness is part of the world. The universe also shares in this Nothingness. It is this Nothingness which is the Zero of the Zero Spirals. But how do the existentialists see Nothingness? Sartre has the word:

"Transcendence, which is "the pro-ject of self beyond" is far from being able to establish nothingness; on the contrary, it is nothingness which is at the very heart of transcendence and which conditions it" [18]

This attempt at synthesis was undone by what follows later:

"If I emerge in nothingness beyond the world, how can this extra-mundane nothingness furnish a foundation for those little pools of non-being which we encounter each instant in the depth of being" [19]

What Sartre takes as non-beings are the collection of negative statements. In the semantics of itson mechanics, these would not be negative integers. Both the positive and the negative integers are, in this scheme, two sides of the same coin whereby the coin can be likened to the being. At the center of this we have the being, the Nothingness, which is inexhaustible in its potentialities. In the transformation of the being to the beings, the being is itself transformed. This much we gather from the logic numbers. The zero spirals of spiral geometry on which itson mechanics is based tells us that even this being is contained in a larger being. Sartre spoke of transcendence as being contained in Nothingness. This means the transcendence of the being is contained in Nothingness. It is this transcendence we need for our contact with the world and the universe. Transcendence is the seat of Imagination. Thus we have some strands of existentialism in the philosophy of science of itson mechanics but, as we have seen above there are fundamental differences between it [existentialism] and the weltanschauung of itson mechanics. Because of these fundamental differences the two philosophies are not compatible. Ontologically Nothingness is itsonically the Inner Infinity and the region of the Creative Potential. The Creative Potential is the source of the Outer Infinity.

Mathematically the outer infinity is composed of integers - positive and negative natural numbers - with the positive integers ending with positive infinity +¥ and the negative integers with negative infinity -¥. In spiral geometry, the local domains which are composed of alpha and omega numbers are the outer infinities. Cosmologically, in itson mechanics, the outer infinity is the reach. In itson cosmology there are many reaches. One of these reaches is our universe. For every universe there is a reach, a transit and a deep. These reaches, transits and deeps are all in a unified space. In itson cosmology and the unified space the conventional belief expressed by [McDonough, 1989]:

"Most SETI [Search for Extraterrestrial Intelligence] projects such as META [Mega-channel Extraterrestrial Assay] have been designed to look for signals at special or 'magic' frequencies. These frequencies, like TV channels, would be known to other civilizations as well as ours because we have observed in studying galaxies and quasars that the laws of Nature are the same everywhere in the universe" [19]

doesn't hold. Based on the geometry of itson cosmology there are indeed other civilizations. According to the geometry of itson mechanics on which its cosmology is based, every universe is infinite including our own. When we say this we mean both in the reach and in the deep. The outer infinities are the Reach

7.2. Inner Infinity

The inner infinity is the Deep. The Deep is composed of infinities. The Deep which is the seat of the Creative Potential, the Reach which is the Expression of the Creative Potential and the Transit which is the Meeting-Melting Point, are an indivisible unit. This results from the Creative Potential which is the nucleus within the nuclei of the inner and outer infinities. The zero of the transit is a singular concept. As the Truth, it can manifest itself in different places and in different forms. It remains nothing but the Truth. This various manifestations of the Truth transforms the Truth itself. This transformation is a logico-mathematical event. The Truth glows in its transformation.

7.3. Transit

This is the nuclear domain in which the infinities - outer and inner - meet before one spirals outward to the Reach and the other spirals inward to the Deep.

7.4. Absolute Spectrum

This is the Highest and Deepest Region. It is accessible to the infinities of the Deep and the Reach which are different levels of infinities through the Creative Potential. From the Deep we know that things have to reach the depth of their Deep to get in tune with the Absolute Spectrum.

7.5. Deep

The Deep is the region of inner infinities. In this region, a thing might be in the deep of another thing even while its reach still remains the same. We bring everything together on the lifeline:

The above zero spiral is composed of two folds. Really, even then this two-fold zero spiral is more complicated than it looks. When we shall get into the physics of it, there we get to know that it is composed of four domains and three types of non-wave, non-particle entities whose primary building blocks are itsons.

8.0. Some Features of Itson Mechanics

In what follows we shall present the geometric representations of the features of itson without going into their mechanics.

Curl

The most indivisible segment of a fold at both ends of which are nuclei. A curl is the imprint of the lifeline on the web of experience.

Hook

The divergence of a fold

Folds

The summation of curls and hooks of a lifeline. There are three types of folds.

These are:

(i) Latents

These are shared by all itsons.

(ii) Determinants

These are folds that covers individuality on every itson.

(iii) Divergents

These folds are the sources of growth. They also generate hooks.

(iv) Cyclants

These are folds in cycles

Three behaviors of a fold are:

(i) Enfolding

This is a continuous exercise. The itson is always folding.

(ii) Defolding

In overcoming say a gradient, and itson defolds. What is defolded

is determined by what is enfolded.

(iii) Refolding

Since every itson is rooted in the laws of equilibrium, every defolding

is compensated for by refolding. Thus what is being defolded is being

enfolded at the same time in another direction. Thus let the gradient

by A on which defolding occurs, this is more compensated for by a simultaneous enfolding. In practical terms, if task X which is equal to

gradient k requires p defolds at time t1, this means that 'greater than p' enfolds has taken place in the itson such that at time t2 when it comes back to task X again, with the gradient still the same, a 'less than p' defolds is needed. This is applying some of the principles of Itson Mechanics on Newtonian and Modern physics. In itson mechanics itself, enfolding, defolding and refolding are even more interesting, though a little bit more involved. The reason is that instead of the classical time

where we have t2 - t1 whose result is non-vanishing, in itson, space

and time being unified influences each other to the extent that time

can vanish.

Others are:

- Bubbles

- Eddies

- Turbulence

- Divergence

- Cycles

- Periods

- Phases

- :

9.0. Itson Theory and Evolution

Evolution theory is based on something unfolding and spreading out therefore becoming. Itson mechanics is based on the inner folds which represent the share of the a set of things or species and its members in the resources of the Creative Potential. Thus itson theory would say that nothing is unfolding but that everything is folded and it is the nature of folding and the shape of the web [of experience] that makes things what they are. Unlike evolution theory, in itson theory, unfolding is internal to the thing and so has nothing to do with externalities. In scaling the problems of life the thing is folding, that is winding up so to speak, and once done, this fold represents its unfolding. This unfolding, which is also part of the lifeline, is also the potential curls which is the source of folds.

Let's see what we get from the example that follows. Imagine each winding on a bolt depicting the specific age of the machine in which the bolt is lodged. When the bolt slacks, the nut relaxes. This we can call depreciation. Imagine that the bolts are so connected that the out-winding of one leads to the in-winding of the other. The itson is the bolt and nut in this case while the windings are the folds that constitute the amount of power packed into the machine. The out-winding of a nut is compensated by the in-winding of another nut such that what we call the "conservation of equilibrium" in itson physics (infinity mechanics) is maintained. This is a very trivial example since the bolt and the nut, unlike real itson, do not grow, cannot take decisions, indeed are pawns on the chessboard of fate. And so, I am not sure whether the example does what it should do which is to fire the imagination on itson and its mechanics.

9.1. Itson Cosmology

Itsonal Universe

10.0. A Physics of Physics

The purpose of this work has been to represent a step that is a leap in itson mechanics. We did not go into formalization because the most important aspect of any project is the central idea.This work encapsulates some of the issues of the central idea of itson mechanics.

In itson mechanics, unlike in conventional setting where physical entities are translated into mathematical language and so complete their platonic existence in the equation, mathematical entities transform into those of physics to the extent that there is nothing to refer to as applied mathematics again.

Itson mechanics is not a physics created that looks round for a geometry. It is geometry and physics so interwoven that they are inseparable. This mechanics is a world built on the pillars of geometry. Indeed itson mechanics is a physics of physics.

By a physics of physics I mean, apart from itson mechanics being a physics, it is designed to be a source of inspiration for the invention of other physics.

Acknowledgment

Many Thanks to Professor Kenneth Hsü for his frank and friendly questions that led to my getting some clear pictures that constitute a lay person's inroad to itson mechanics. I must confess that I gained a lot from our discussions. The message of it all is that it is not enough for an invention or discovery to be superb, there is the need to bring it to a level that is accessible to conventional knowledge and where this is not possible as it is always the case with new ideas, a language should be invented that leads the imagination along the path that can lead to a spark of understanding.

References

Benardete, José A.: Infinity. An Essay in Metaphysics.

[Clarendon Press, Oxford 1964]

Bunch, Bryan : Reality's Mirror : Exploring the Mathematics of Symmetry.

[John Wiley & Sons, Inc. New York 1989]

Herbert, Nick: Quantum Reality. Beyond the New Physics.

[Doubleday, New York 1985] pp 16-17.

McDonough, Thomas R.: Continuing SETI : Scientists Share Findings at

Toronto Conference IN The Planetary Report, Vol IX. No. 1.

Jan./Feb. 1989, pp 18-19

Sartre, Jean-Paul: Being and Nothingness.

[Routledge, London 1958], pp 18-19

Shakunle, Lere O.: Spiral Geometry. The Principles (+ Discourse)

[Hitit Verlag, Berlin 1994]

- ibid - Foundations of Logic Numbers, Vols. I., II., III.;

[The Matran Schools Publications]

- ibid - Introduction to Logic Numbers, Vol. 1., Book 1.

[The Matran Schools Publications]

- ibid - Introduction to Matran Logic, Vols. I., II.

[The Matran Schools Publications]

@ Shakunle, Lere O. 1996