# Books

In his first book, Geometric Wholeness of the Self, author Donald Harms explores the meaningful relationships between mathematics, sacred geometry, philosophy of The Enlightenment and the spiritual life of psychologist Carl Jung.

In bringing this to light, the author examines sacred geometry within the Golden Section as it relates to analytical geometry expressed by little known German geometer Jacob Steiner. Geometric Wholeness of the Self delves into the limits and failings of modern mathematics while revisiting geometry of the ancient past, including classic Euclidian theory and geometry that preceded it.

Philosophical and psychological ideas are explored through symbolism in The Geometric Wholeness of the Self. Donald Harms examines history of mandalas, culminating with an analysis of Carl Jung’s monumental Systema Munditotius mandala.

### Geometric Wholeness of the Self

# Geometric Wholeness of the Self

## The Mandala as a Psychological and Spiritual Representation of the Self

### Donald Harms

*With* *With this book I acknowledge the support and sharing of ideas of my wife Patricia Damery, Jungian analyst;*

*the extensive editing by Leah Shelleda, prof. emeritus of humanities; and the thoughtful reading of my friend Jimalee Plank, writer.*

Copyright © 2016 by Donald Harms All rights reserved.

isbn 978-0-99113098-9-4

3185 Dry Creek Road, Napa California, 94558 usa Telephone: 707.257.2683 dharms@napanet.net

Cover: The Golden Rectangle

of the geometry of The Golden Section (Divine Proportion).

### Contents

**I. The Geometry of the Self:**

The Christian Era through the Age of Enlightenment 1

The Christian Era through the Age of Enlightenment 1

Continuity of Gnostic Beliefs 4

Enter the Age of Enlightenment and the Changing Psychic Center 5 Kant’s Philosophy and the Self 8

Implied Philosophic Meaning of the Mandala 12

The Mandala as a Spontaneous Psychic Occurrence 13

**II. Enter the Modern Era of Mathematics and Philosophy 15**

Wholeness as a Term of Contemporary Understanding 15

Kant on Geometry and the Grounding of Sensible Intuition 16

A Philosophic Principle: Geometry Precedes Number 18

For a Common Understanding of the Philosophic Language Expressions Used Here 19

Geometry as a Synthetic Object of Wholeness 23

The Unitary Circle and the Mathematical Roots of Unity 26

The Philosophical/Psychological Keeping Body and Soul Together 31

Geometry of the Mandala 36

Representation of the Self Contained in the Geometric of the Mandala 37

Kant’s Critique of the Mathematical Proof, The Excluded Middle 39

Scientific Intervention in Natural Systems 47

Kant’s Progression of Perception vs. the Scientific Concept of Progress 47

Incompleteness and the Concept of Wholeness 50

Kant’s Philosophical Inner and Outer Spheres and His Copernican Consciousness 51

**III. The Necessity for Wholeness in the Contemporary World 55**

Carl Jung’s Recognition of the Unconscious 55

The Carl Gustav Carus (1789–1869) Understanding of the Unconscious 61

Mathematician David Hilbert on the Subject of Intuition 62

The Unconscious Holding of Completion of the Philosophical and Psychological Whole 63

Geometry as a Representation of Wholeness 66

An Analog of Perceptions –Apperceptions / Apperceptions – Perceptions 69

A Critical Commentary on the Mathematics of the Nineteenth Century 76

Dreams, Visions, and Visitations Arising as Complete Conceptions 78

Inner and Outer Geometric Representation of the Self 79

**IV. A Geometric Representation of the Relation of the Self to the External Universe 83**

Inversive Geometry: The Relation of the Philosophical/Psychological Internal to the External 83

Re-emergence of Intuition as Represented by Synthetic Projective Geometry 85

The Polar Line through the Center Placing the Self in the Cosmos 90

Kant’s Transcendental Aesthetic and the Basis of Perceptive Understanding 96

The Fate of Icarus Flying too High 98

Perception in Relation to the Monad as It Was Understood by Leibniz 101

Kant’s Perception of the Infinite 102

Philosophical Reflections of a Principal Twentieth-Century Mathematician 103

The Order of Classifications within the Self 108

Jung’s Inner Spheres and the Personal and Collective Unconscious 113

Jung’s Definitions of the Endopsychic Sphere(s) 117

The Psychic Affects and Invasions 118

Invasions of the Self in the Modern World 121

Jung’s Ectopsychic Sphere(s) and the Intuited Creative Process 122

**V. Carl Jung’s World Systems Mandala – Systema Munditotius 127**

Jung Embraces Gnosticism 127

Jung Gives a Sermon on the Infinite 129

Jung’s Anthropomorphic Mandala: Systema Munditotius 136

Jung Lays Christian Theology on Its Side 137

The Synthetic Geometry of Jung’s Systema Munditotius Mandala 142

Philosophical Meaning in Jung’s Systema Munditotius 145

Reflections on the Status of the Self in the Modern Era 145

Sustaining Life and Living in the Modern World 148

The Philosophical and Psychological Meaning of the Mandala 150

The Divine Proportion Between the Objective and Subjective Self 150

## Preface

** **When the object is *wholeness, *one can take license to reflect across the differentiation’s that make a whole. In the following I have traversed philosophical, psychological, geometric, as well as other mathematical domains to explore the geometry of the mandala in the present-day context.

As an introduction to what follows, these thoughts of Plato from twenty-five hundred years ago, retold by Kant some two hundred fifty years ago, are set out as a beginning. This passage, from both Plato and Kant, is cited to focus not only on Kant’s *Critique of Pure Reason*, a tract that demands of us extraordinary mental concentration, but is introduced here to shine light on Kant’s giving importance to *intuition *and preceding that, the mystical *a-priori. *These are presented here as parts of a psychological process by which we raise our consciousness of not only being but that which, as Kant refers to below, the “soul” and “a faculty for divinity.”

Kant referring to Plato:

His (Plato’s) hypothesis was mystical. For he took as the basis a pure intu- ition of the non-sensible or = (equal to) a supersensible intuition and assumed that the soul, before it was delivered into bodily condition, had a faculty for divinity, and even if it no longer participates in it in this life, never the less a consciousness of those ideas of pure understanding could be awakened in human beings, and that this consciousness is the source of *a priori *cognition (Kant 1997, 422).

I begin here with the notion that the mystical, as referred to above by Plato and Kant, has lost meaning in our age of science and technology, but that the mystical is found in the realm of ideas in number, geometry, and in the philosophical principles of Immanuel Kant. That is why we begin here, to define a transcendent understanding through geometry and philosophy.

In my vocational experience, the practice of architecture, there is an unwrit- ten understanding that persists in which the architect will take command and creatively apply the several modes of understanding: engineering, physics, environ- mental science, health, history, psychology; and if done successfully, there will be a geometric synthesis of the forms we call architecture. As in the practice of archi- tecture, in the following I have accounted for the *mathematical*, naturally the *geomet- ric*, and not the least, the *philosophical/psychological, *as the latter is where the architect, the occupant, and society meet to construct forms that will house both body and soul. This is what I seek to do in the *Geometric Wholeness of the Self*, where you will find both the philosophical and mathematical approaches leading to not just the *psycho- logical *but also explores the meaning of the form of consciousness in the geometric of the mandala itself, linking the philosophy of Kant and, in turn, his understand- ing of Plato as well.

As is the tradition in philosophic writing, presented here is what are under- stood to be universal truths, including that which in recent philosophic discourse is the logic of mathematics called *symbolic logic*.1 Some may find it inappropriate that I rely on my personal psychological experience to present a current symbolic logic form of discourse; but that is precisely the perspective I offer: the experience of the human psyche within the Self. Psychological experience is understood to hold all meaning; therefore the forms of human consciousness become the principal focus that I believe is so lacking in recent philosophy. It is my belief that the recent philo- sophical discourse has been unnecessarily limited by preoccupation with a form of logic that is principally defined by mathematics.

I have acknowledged mathematics as having a prominent role in what follows here because that is the reality of the modern era we live in. However, my treatment of mathematics in this book is not entirely positive. I have treated some of mathematics with negative criticism in order to restore geometric proportion to this post-modern perspective. Neither do I take what is common in mathematical analysis—the A or B, plus or minus alternatives—as being the optimal principle available to us. Nor do I propose that mathematics should not be a part of solutions to the modern technological dilemmas that as a total world humanity we are just beginning to face up to. I believe that in our modern culture we have a collective appreciation of the wonders that mathematics has produced; but we do not see its inadequacies. More importantly, the uncritical use of mathematics in both philosophic and psychologic ways has had adverse effects in the modern world. Nor do I wish to represent mathematics as being in complete opposition to the philosophical ideas described here; the philosophical ideas are rather an expanded and more inclusive of a more comprehensive view of the universe we live in every day.

The alternatives proposed here are found in the mandala, more specifically in the geometry of the mandala. What is implied by its geometry and the phenomenon by which the mandala is psychically and spiritually manifested in us is the basis for what follows.

## Introduction

In any quest to find wholeness in personal experience and in contemporary ter- minology in the psychological, we are confronted with ever more complexity. In a natural desire to limit or even dispose of complexity, we tend, as a culture, to grasp ideological life jackets that can keep us afloat, particularly in a focus on our com- mon ideology of science and technology. But that ideology alone does not bring us to a place where we can move about each day with a feeling that with this modern day ideology, that contemporary life, with all of the enhancements of science and its technology, has acquired through it any truly transcendent philosophic, psychic, or spiritual meaning.

Progress, that is to say the common notion that technological change that is the direct result of progress in science, is the path to cultural fulfillment. But it continues to be a motivating force. As a culture we have haphazardly acquired a collective experience around our technology that, when viewed and analyzed from the opposite, that is to say, the originating point of a linear progression mentioned above, we can greatly expand our perspective of what our modern belief system means to us. The reader is advised that I will briefly revisit basic formal knowledge of number, geometry, and indeed some forms of consciousness as they have histor- ically been understood and used.

You are also advised here, in the beginning, of a principle that all of these ideas are formed around. That principle and its form is that the *Self *is the

vehicle, the propagator, and the holder of not just the psychological but the philosophical, the geometric, and, not the least of all, the mathematical. This may seem to be an obtuse idea to some, for we are informed, particularly by mathematicians, that these subjects—philosophy, geometry, and number—are universal if they are any- thing of real value to us and are not the property of any individual. But notion of

the Self is also not just the physical body, which is the reducted notion that is systematically implied by modern science. The simple anecdotal evidence that I raise in promoting the *centrality *of the Self is the fact that virtually all of the innovative mathematical, geometric, philosophic, and psychological findings are given the name of the individual who conceived them: the Pythagorean theorem, Euclidean geometry, the Cartesian coordinates, the Galois numbers, Platonic thinking, Freudian psychology, and so forth. These individuals and their concepts are a matter of historical record. They are not labeled as such simply for the sake of convenience, but because these individuals found the intuited originality, thinking, and the history behind their conceptual processes that became formal collective knowledge. We continue to acknowledge these personalities because the Self made these discrete findings conscious, aside from and in addition to any *universality *as may exist in their discoveries. If the Self is the holder of everything known *and *the *con- tainer *of the form of understanding of that which is not known, or the *irrational*, then of course the grand question that follows is what is the relation between the inner and the outer of the Self? And how is it maintained? How the inner of the Self exists in concert with the ‘outer’ will be found near the end of this not-so-humble Self-asserting dissertation.

Beginning in the early twentieth century, there was the notion that mathematicians could define the *infinite *wholly within number theory; but I believe it became evident to some mathematicians that without limit this approach was precarious and, I believe, not supported by reason. Mathematicians, of course, were not in any significant way working with the irrational but had, over a period of time in the recent centuries, constructed a system of axioms and proofs that was strictly ‘rational.’ 2 In practice, by a constructed rationalizing of the irrational when it was naturally encountered, it was thought, that through this method, mathematics had the capability to explore the state of the infinite.3 It would be convenient here to describe parallels to the religious ideals about the infinite: infinite insight, power, grace, and aspirations of the Christian era of Europe, but that would need to be under the aegis of God. In the contemporary context, I propose here those powers are, indeed, not of mathematics and of mathematicians alone but also belong in the realm of philosophical reason that has been more recently expanded by our under- standing of the psychology of the unconscious.

In addition to the Self as the center of understanding I want to make some distinctions about the *form *of understanding, or more precisely, the *forms *of what is discussed here, namely: language, geometry and number. Other than pure music and physical art forms, we commonly think that language is the sole means of expression; and as a means of communication, that is nearly so. In what follows, the geometric is given what I believe is its due as a distinct and differentiated form of understanding in addition to language. There is, of course, a third form of under- standing—that of mathematics, specifically counting (the number continuum) and arithmetic—which we rely upon to be true no matter what. In what follows, I have differentiated between the *geometric *and the *arithmetic*, even though the two are not commonly distinct in our culture in the contemporary collectively held under- standing of mathematics.

I have sought here to reintroduce geometry as a synthesis of principles underlying knowledge.4 To do this, it is necessary to use geometry not just as a grounding for number, which is the prevailing view of mathematicians in general, but as a distinct mode of understanding what we call space, and more to the point, as a tool of how we conceptualize both philosophically and psychologically within the Self. Just as mathematics is thought to have a logic, so does geometry and, for that matter, so does language. But, as I propose here, *logic*, which was so rigorously sought after in the last century in mathematics and specifically in the number continuum, is not the primary basis of human understanding. Of course, in the Western world, we have long had a formal set of logical findings we know as Euclidean geometry, and they served as a standard of logical truths until mathematicians, beginning in the sixteenth century, sought ways to override and, in some critical instances, to denigrate the truths that were formalized in Euclidean geometry…

Would you like to read more? You can purchase this book from Amazon.com