A Fractal Epistemology for a Scientific Psychology: Bridging the Personal with the Transpersonal [1 ed.] 1527540235, 9781527540231

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A Fractal Epistemology for a Scientific Psychology

A Fractal Epistemology for a Scientific Psychology: Bridging thePersonal with the Transpersonal

Edited by

Terry Marks—Tarlow,Yakov Shapiro, Katthe P.Wolf and Harris L. Friedman

Cambridge Scholars Publishing

A Fractal Epistemology for a Scientific Psychology: Bridging the Personal with the Transpersonal Edited by Terry Marks—Tarlow, Yakov Shapiro, Katthe P. Wolf and Harris Friedman This book first published 20 20 C ambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, N E 6 EPA, UK British Library Cataloguing in Public ation Data A catalogue rec ord for this book is available from the-British Library Copyright © 2020 by Terry Marks—T arlow, Yakov Shapiro, Katthe P. Wolf, Harris L. Friedman and contributors

All rights for this book reserved. N o part of this book may be reprod uc ed, stored in a retrieval system, or transmitted, in any form or by any means, electronic , mechanical, photocopying, rec ording or otherwise, witho ut the prior permission of the copyright owner. ISBN (10):1—5275—4023—5 ISBN (13): 978-1-52 75-4023—1

NO ONE WILL BE CONSIDERED SCIENTIFICALLY LITERATE TOMORROW WHO IS NOT FAMILIAR WITH FRACTAL S.

—JOHN ARCHIBALD WHEELER, PHYSICIST

TABLE OF CONTENTS

Advance Praise for the Book...................................................................... X Contributors ............................................................................................. xvi Foreword ................................................................................................. xxi Richard Taylor Introduction ........................................................................................... XXiv Towards Fractal Foundations for Transpersonal Science Terry Marks-Tarlow,Ha1ris L. Friedman, Yakov Shapiro, and Katthe P. Wolf PART 1 : TRANSPERSONAL EPISTEMOLOGY

Chapter One ................................................................................................ 2 A Fractal Epistemology for Transpersonal Psychology Terry Marks-Tarlow Chapter Two ............................................................................................. 33 More than Merely a Model or Metaphor? The Contributions a Fractal Epistemology Might Make to Transpersonal Psychology Katthe P. Wolf and Harris L. Friedman Chapter Three ........................................................................................... 65

Towards a Naturalistic Science of Transpersonal Experience: Fractal Evolution and Nonlocal Neurodynamics Yakov Shapiro Chapter Four ........................................................................................... 104 Transpersonal Psychology and Fractal Evolution I. Rowan Scott

Chapter Five ........................................................................................... 144 Fractal Epistemology and the Biology of Emotion Katherine Peil Kauffm an

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Table of Contents

l 86 Chapter Six ............................................................................................. 186 Epistemology of the Neurodynamics of Mind Frederick David Abraham Chapter Seven......................................................................................... 208 Fractals Transcendent: Bridging the Transpersonal Chasm William Sulis Chapter Eight Eight .......................................................................................... 243 Chapter Foundations of Fractal On the Mathematical and Transpersonal Foundations Geometry and Dynamics Jonathan Root FRACTAL APPLICATIONS PART PART 22:: FRACTAL

Chapter Nine........................................................................................... 274 Dreams, Synchrony, and Synchronicity Terry Marks-Tarlow Chapter Ten ............................................................................................ 303 Transcendent Experiences A Fractal Topology of Transcendent Experiences A Allan Combs Sally Wilcox and Allan Chapter Eleven ....................................................................................... 324 Boundaries: A Subjective Approach. Fractal Boundaries: Galatzer—Levy Robert M. Galatzer-Levy Chapter Twelve ...................................................................................... 361 How Fractals Help Us See and Understand the World Larry Larry S. Liebovitch Thirteen ..................................................................................... 372 Chapter Thirteen How the Cerebellum and Cerebral Cortex Collaborate to Compose Fractal Patterns Underlying Transcendent Experience Larry Vandervert Chapter Fourteen .................................................................................... 391 Ancient Hidden in Plain Sight as the Sky Holds a Cloud: Fractals in Ancient Chinese Philosophy Anthony S. Wright

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Chapter Fifteen ....................................................................................... 417 All the Inbent Fractals of Connection William J. Jackson

Chapter Sixteen ...................................................................................... 452 The Fractal Qualities of Hallucinatory Phenomena: On Form Constants and Their Implication for the Psyche Jesus-Mario Serna

Chapter Seventeen .................................................................................. 494 Cracked Orlando: Dramma per Musica e Fractals, Dimensions of a Fractal Baroque Opera Jonathan Dawe Afterword ............................................................................................... 524 Yakov Shapiro, Terry Marks-Tarlow, Harris L. Friedman and Katthe P. Wolf Index ....................................................................................................... 530

ADVANCE PRAISE FOR THE BOOK

“Fractals are the essence of being human, not just in the building of our lungs, our nerves and our bloodstreams, but in our individual and collective

behaviors. This is the brave new world for fractal researchers. A Fractal Epistemology for a Scientific Psychology belongs firmly to this exciting world and its quest to bridge the personal with the transpersonal will broaden the scope of fractal thinking. In m y discussions with Mandelbrot,

he was delighted to see fractals venture from their mathematical shell and shake the world. He would have been delighted to read this book.” —Richard Taylor, PhD, Professor of Physics, Psychology and Art, Head, Department of Physics, University of Oregon; author of 315 publications, including 12 in Nature and 4 in Science; 52 awards for research and teaching spanning the arts and sciences, including an InnoCentive Prize, Cottrell Scholarship, Pollock-Krasner Residency,

Nobel Foundation Travel Award, and a British Royal Society Award. “Intellectually engaging and provocative, A Fractal Epistemology for a Scientific Psychology provides the reader with exciting perspectives on the promise of fractal mathematics and geometry for illuminating mind, behavior, and consciousness. Its potential applications to transpersonal psychology are particularly noteworthy and are likely to serve as the basis for new avenues of research and theory development. This book will be a challenging but delightful read for scientists and erudite laypeople with an interest in fractals and consciousness.” —Doug1as A. MacDonald, PhD, Associate Professor of Psychology, University of Detroit Mercy; Associated Distinguished Professor, California Institute of Integral Studies; Associate Editor (Research) of

Journal of Transpersonal Psychology; Senior Research Editor of Journal ofHamanistic Psychology. "Fractal Epistemology, as presented in this timely and inspiring collection, is a breath of fresh air in the closed room of traditional epistemology. This book offers a truly expansive freedom of thought that can liberate us from the imprisoning dominance of epistemologies of patriarchy. Understanding fractals can help us to develop an inclusive model of thought that is more

reflective of feminine, and feminist perspectives. This important book is

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vital to the expansion of our understanding of thought that sheds light on

the hidden dynamics of all the forgotten and ragged edges of thinking and feeling that are neglected or misunderstood

through more standard

psychological models. A Fractal Epistemology for a Scientific Psychology is the first academic approach I have found to provide me with truly nuanced methods of understanding experiences of trance state awareness and insights. These essays help to explore with subtlety the experiences of insight in liminal and meditative practices like Yoga Nidra and allied states of being, such as dream, recollection, and intuitive knowing invoked

through ritual trance induction in indigenous, earth wisdom practices of ceremony and story." —Uma Dinsm ore-Tuli PhD, Author of Yoni Shakti: A Woman ’5 Guide

to Power and Freedom through Yoga (2014) and Nidra Shakti: A De— Colonizing Encyclopaedia of Yoga Nidra (2020), co-founder of Intemational Yoga Nidra Network. “After more than a century of trying to explain “what is personality?” psychological theory and research brought us much closer to an understanding of a psychologically healthy human being that integrates emotion, experiences, cognition, behaviors, and traits into a more holistic

view of the person. A Fractal Epistemology for a Scientific Psychology opens a new vista that deploys the mathematical properties of fractal geometry—non—integer dimensions, self-similarity, scalability, and selforganization—to the understanding of how minds move and change over time and co-evolve with a changing environment. The contributions to this volume use fractal geometry to identify qualitative patterns and explain how

emergent dynamics lead to holistic phenomena. They also present some intriguing speculations as to how synchronicity, telepathy, psychokinesis, clairvoyance, and pre-cognition could be identified and understood if they were investigated from this new vantage point.” —Stephen Guastello, PhD, Editor-in—Chief of Nonlinear Dynamics,

Psychology, and Life Sciences, Professor of Psychology, Marquette University. “A rich, wide—ranging collection of creative chapters on the fundamental roles of fractal patterns in nature and the human experience, especially in transpersonal contexts, including psychotherapy. Destined to become a classic in the field.” —Allan N. Schore, PhD, Author of The Science of the Art ofPsychotherapy and Right Brain Psychotherapy.

Advance Praise for the Book

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“A Fractal Epistemology for a Scientific Psychology is a prodigious and much-needed exploration into not just what are quite literally the building blocks of our world, but also into the underpinnings of our unique experiential worlds and interpersonal relationships. Marks-Tarlow and her combined authors, each from their own unique perspective, offer a strongly integrative tour de force into the realms of personal, transpersonal, and scientific phenomena while mounting a successful argument for the integration and synthesis of historically incommensurable schools of thought. Along with investigating vital, contemporary conceptualizations of

our fractal world, A Fractal Epistemology persuasively and artfully reunifies the objective and the subjective, the scientific and the transpersonal, providing essential avenues for disparate disciplines to join hands in exploring how our world, including our emotional lives, actually lives and breathes.” —William

.I. Cobum, PhD, PsyD, Founding Editor Emeritus of

Psychoanalysis, Seiy and Context (formerly the International Journal of Psychoanalytic SelfPsychology)‘, author of Psychoanalytic Complexity: Clinical Attitudes For Therapeutic Change (2014, Routledge)‘, Associate Editor of Psychoanalytic Dialogues; Faculty Member and Training and Supervising Analyst of Institute of Contemporary Psychoanalysis, Los Angeles. “This is exactly the book I have been waiting for! It establishes transpersonal psychology as a full-fledged member of scientific psychology.” —Stanley Krippner, PhD, Associated Distinguished Professor, California Institute of Integral Studies, Co—Editor of Varieties of Anomalous

Experience: Examining the Scientific Evidence. “In an era mesm erized by binary technology, where our society is in danger of losing its human sensitivities, the field of transpersonal psychology attempts to embrace all that is the mystery of individuality and relatedness. A Fractal Epistemology for a Scientific Psychology promises the reader a bridge back to our unique selves, while at the same time offering a visual path that returns us to the inseverable bond that ties us to each other and to our natural surroundings. This beautifully balanced compendium fills the reader with hope to retum us to a humbling sense of that which may join us together, rather than what separates both hearts and minds. How extraordinary to fill pages with such a mix of both precision and poetry. If “big bang” theory doomed us to social fragmentation, the authors’ imagination about fractals may hold the promise of a psychology that

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returns us to a harmony of human experience. Like modern day Kabbalists, the authors remind us of the economy of the universe of hum an experience.” —Rabbi Peter M. Rosenzweig,

PhD, Faculty, Division of Clinical

Psychology, Northwestern University; Machon Shiluv Institute for Marital and Family Therapy, Jerusalem. Author of Married & Alone: The Way Back; Introspection; Teshuva and Personality; What did the

Prophets Say? and numerous articles on the art and science of psychotherapy. “Surprise! Our lives don’t manifest in straight lines or fixed dualities but in a dynamic matrix of intertwined possibilities and permeable boundaries, limned by the fractal, self-similar, “fingerprints of chaos,” across evolving infinities, present and potential. New understandings emerge for empathy, altruism, expanded self, exceptional experience, creativity, culture, and

more. This book is a “must” as we enter a new era.” —Ruth Richards, MD, PhD, author of Everyday Creativity and the Healthy Mind: Dynamic New Paths for Self and Society (a 2018 Silver Nautilus Award winner) and co-editor of the forthcoming book, Nonlinear Psychology: Keys to Chaos and Creativity in Mind and Nature. “This remarkable, well edited collection provides a broad, thorough study of fractals with applications to psychology. Working to span subjective and objective aspects of reality, important bridges are built between personal and transpersonal. The archetypal nature of fractals shines forth throughout the text, granting greater access to a new scientific holism, offering much needed renewal for psychology at this time. The editors have provided a

valuable gift to interested readers.” —Joseph Cambray, PhD, President/CEO Pacifica Graduate Institute;

Past President of International Association for Analytical Psychology; U.S. Editor of Journal ofAnalytical Psychology, Author of Synchronicity: Nature and Psyche in an Interconnected Universe. “Transpersonal psychology rs at a crossroads, at the centre of which is the meaning of a seemingly simple term, naturalism. One path holds that nonm aterial beings and transcendent experiences cannot be squeezed into the container of naturalism', the other expands the container, emphasizing the mysterious and multidimensional basis of our natural order. As this book so richly illustrates, A Fractal Epistemology. for a Scientific Psychology has the potential to place a much-needed signpost at the crossroads. The authors collected in this ground-breaking volume demonstrate the ways in which

transpersonal phenomena follow the logic of fractals and open an

Advance Praise for the Book

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epistemological approach that aligns transpersonal psychology with naturalistic

science. Essential

reading for a 215t-century transpersonal

psychology ! ” —Brian Les Lancaster, PhD, Professor Emeritus of Transpersonal Psychology, Liverpool John Moores University, UK; and Founding Director and Academic Dean of the Alef Trust.

“A Fractal Epistemology for a Scientific Psychology explores the role of fractals in illuminating the knowledge of our interconnectedness with all that is. As an indigenous Maya familiar with ancient sacred geometry such as the Mayan calendar, I found rich possibilities explaining a very complex

subject in relationship with our psyche. I endorse the book as important contribution to this field.” —Yoland Trevino, International Ambassador of the Maya Confederacy,

Guatemala. “Fractal thinking transfers perspective from one knowledge space to another, from one scale to an entirely other one—that’s the key to finding

ways to understand people at the scale of new human interconnections. That’s the fascinating perspective that this book outlines for our futures of hum an change.” —Franco Orsucci, MD, Professor, Niccolo Cusano University, London;

Visiting Professor, University College, London; Founding Editor of Chaos & Complexity Letters, author of Neuroscience in the Age of Complexity and Human Dynamics. “This book is a revolutionary and evolutionary landmark in publishing! Vast in its wide-ranging capacity to unify science and mathematics and the personal and transpersonal dimensions of psychology with a scientific rigor that has never been so clearly presented. The expansive and universalist

perspective this book embodies fits perfectly into the ancient Taoist worldview and organic sensibility as well. The book is a joy to read. Using styles ranging from scientific to transcendent, the volume perfectly captures the voices of nearly 20 major theorists—scientists, neuroscientists, psychologists, psychiatrists, scholars, philosophers, and visionary thinkers—to express a common passion for never settling for reductionist arguments and always respecting the ways in which we, as part of the intricate web of nature, reflect and contribute to a wholistic vision that both includes yet transcends us all. Reading A Fractal

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Epistemology for a Scientific Psychology is delving into a sumptuous feast for the mind and senses.” —Carl Totton,

PsyD, Director

of Taoist Institute, Los Angeles,

California; Host of regular podcast: What’s this Tao All About? Psychologist in private practice and consultant; Former Professor and Department Chair, School Psychology, Phillips Graduate Institute. “This is a mind-expanding volume for those interested in understanding how complex psychological phenomena can be modeled by the rich and

robust mathematics of fractals. As just one lucid example, the author contrasts the stepwise linear logic used by the conscious mind with the dreaming mind, illuminating its fractal nature that sees symmetrical equivalence of wholes with their parts.” —Shan Guisinger, PhD, clinical psychologist and evolutionary biologist; co-editor of the forthcoming book, Nonlinear Psychology: Keys to Chaos and Creativity in Mind and Life. “This is a wonderful book, with contributions from an all-star cast! Whatever you already know or don’t know about fractals, whatever your interests in the wonders and puzzles of hum an consciousness: This volume provides new entry points for grasping these instantly appreciated—as well as notoriously difficult—phenomena, and then offers path after fascinating path toward deeper knowledge.”

—David Schulberg, PhD, Professor of Psychology, University of Montana; Director of evaluation, National Native Children’s Trauma Center; co-editor of the forthcoming book, Nonlinear Psychology: Keys

to Chaos and Creativity in Mind and Life.

CONTRIBUTORS

Terry Marks-Tarlow, PhD, is a Clinical Psychologist in private practice in Santa Monica, California. She is also an Adjunct Professor at Pacifica Graduate Institute, Santa Barbara, and Core Faculty at the Insight Center, Los Angeles. She has authored and edited several books, including Play & Creativity in Psychotherapy, Clinical Intuition in Psychotherapy, Awakening Clinical Intuition, and Psyche ’s Veil, all of which she illustrated herself. Along with clinical practice, writing, and family life, she happily

immerses herself in the arts, including dance, yoga, and writing opera librettos, one of which premiered at Lincoln Center.

Yakov Shapiro, MD, is a Clinical Professor in the Department of Psychiatry at the University of Alberta, Faculty of Medicine and Dentistry, psychotherapy supervisor, and Director of the Integrated Psychotherapy/ Psychopharmacology Service. He specializes in the psychobiological systems approach to the treatment of trauma, mood and personality disorders, with a special interest in a dynamical systems approach to the neurobiology of psychotherapy and integrative health. Katthe P. Wolf, MA, has returned to academia after a 30-year-hiatus spent building a successful non-profit career in family support, social justice, and child abuse prevention, currently serving as CEO of Be Strong Families. Wolf is currently a doctoral student in Integral and Transpersonal Psychology at the California Institute of Integral Studies with a research interests in fractals, fractal epistemology and the nature of self. She recently served as guest-editor on the special focus issue of the International Journal

of Transpersonal Studies on the potential role of fractals for modeling transpersonal phenomena and synthesized the commentators’ perspectives in her own: “The Nature of Nature is Fractal.”

Harris L.

Friedman, PhD, is retired Research Professor of

Psychology at University of Florida, Distinguished Consulting Professor at the California Institute of Integral Studies and Visiting Scholar (2019-2020) at Harvard University. He also practices clinical and organizational

psychology, and has written extensively on transpersonal psychology,

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cultural change, and research methods. He is Associate Editor of both The Humanistic Psychologist and the Journal ofHumanistic Psychology, as well as Senior Editor of the International Journal of T ranspersonal Studies.

J. Rowan Scott, MD, is a Clinical Professor of Psychiatry at the University of Alberta, Canada. He has an interest in Complex Systems as well as fundamental questions regarding the reductive scientific paradigm. These interests inform his research and writing on the subject of consciousness. He teaches a seminar on consciousness as well as family therapy in the Department of Psychiatry at the University of Alberta. He has a private practice

in general

Psychiatry and analytically

informed

psychotherapy. Katherine Peil Kauffman, MA, is an affiliate of Northeastern University and the Harvard Divinity School. As Founding Director of nonprofit EFS International, whose mission is to foster global emotional wisdom, she is especially interested in an evolutionary perspective on how the new biology of ”emotion" can shed light on various mind-body conundrums. Frederick David Abraham, PhD, was a cognitive neuroscientist at UCLA’s Brain Research Institute. Abraham is a pioneer in the application of chaos and dynamical systems to the field of psychology who co-authored A Visual Introduction to Dynamical Systems Theory for Psychology (Ariel Press, 1990), as well as co—edited Chaos Theory in Psychology (Praeger, 1995). William Sulis, MD (Psychiatry), PhD (Math), PhD (Physics), is Associate Clinical Professor in the department of psychiatry of McMaster

University, where he is also director of the Collective

Intelligence Lab and co-director of psychological services. He researches collective dynamics, synchronization in complex systems, cellular autom ata, random graphical dynamical systems, and mathematical psychology among other diverse topics. Jonathan Root, PhD, received his doctorate in mathematics from Boston University in 2016 under the supervision of Mark Kon. He taught English in rural China during the 2016-2017 school year and is an instructor at the Hunan Institute of Science and Technology.

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Contributors

Sally Wilcox, PhD, studied at the California Institute or Integral Studies (CIIS) in San Francisco and completed her PhD in 2012. Her dissertation topic was A Fractal Topology of the Transcendent Experience with Allan Combs as Chair of her committee. Wilcox lives in British Columbia,

Canada and continues researching fractal nonlinear dynamics as they relate to various states of consciousness. Allan Combs, PhD, is a transpersonal psychologist, consciousness researcher, neuropsychologist, systems theorist, and President of The Society for Consciousness Studies. He holds appointments at The California Institute of Integral Studies (CHS), where he is Director of the Center for Consciousness Studies; the Saybrook Graduate School; and the Graduate Institute of Connecticut, where he is the Director of the MA program in Conscious Evolution. Professor Combs is author of over 200 articles, chapters, and books on consciousness and the brain, including, Consciousness Explained Better: Towards an Integral Understanding of MultifacetedNature of Consciousness, and the award—winning, Radiance of Being: Understanding the Grand Integral Vision. Professor Combs is also co-founder of The Society for Chaos Theory in Psychology and the Life Sciences; Co-Editor of the Journal of Conscious Evolution, and Editor of CONSCIOUSNESS: Ideas and Research. for the 21 st Century.

Robert Galatzer—Levy, MD, is a psychoanalyst of children, adolescents and adults who teaches at the University of Chicago and the Chicago Psychoanalytic Institute. Besides his psychiatric background he did graduate work in mathematics at N.Y.U. Courant Institute of Mathematical Sciences. In addition to clinical psychoanalysis his interests include

nonlinear dynamic systems theory, complexity, forensic psychiatry, and life course development.

Larry S. Liebovitch, PhD, is Professor of Physics and Psychology at Queens College of the City University of New York and serves as Adjunct Senior Research Scientist for AC4. At Florida Atlantic University, he served as the interim director of the Center for Complex Systems and Brain Sciences and has used nonlinear methods to analyze and understand molecular, cellular, psychological, and social systems, including as author of Fractals and Chaos: Simplifiedfor the Life Sciences (Oxford University Press, 1998) and coauthor of Fractal Analysis in the Social Sciences, Quantitative Applications in the Social Sciences, Volume 165 (SAGE Publications, 2010).

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Larry Vandervert, PhD, is a retired college professor who has published and edited works in the neurosciences, creativity, innovation, child prodigy giftedness, and science in general. His major research interest is in how, through practice, and in collaboration with the cerebral cortex,

the cognitive functions of the brain’s cerebellum constantly optimize both mental and behavioral performance. In his publications he has applied the findings of recent brain-imaging studies of the cerebellum to creativity (2003, 2007, 2015), the evolution of language (2011), of culture (2016), of child prodigies (2016), of play (2017), and the cerebellum’s prominent role in the rise of Homo sapiens (2018). The cerebellum’s prominent role in creativity will appear in Encyclopedia of Creativity (3rd ed). Dr. Vandervert is a Fellow of the American Psychological Association since

1992, and now writes under the egis of American Nonlinear Systems. He presently lives in Spokane, WA, USA.

Anthony S. Wright, PhD, completed his doctorate at Califomia Institute of Integral Studies in San Francisco on ”Principle and Pattern: Zhu Xi and Complexity Theory—Completing Wisdom through Fathoming Organic Pattern," which sought parallels in fractal organic patterns in Chinese philosophy and complexity science. Previously a Lecturer in the Philosophy Department at Sonoma State University in Rohnert Park, California, he is currently an Assistant Professor in the English Taught Program in Intemational Business, in the College of Management at Shih Chien University in Taipei, Taiwan. His fields of interest continue to be parallels of Chinese Philosophy and natural organic patterns found in complexity science. He has been a piano technician since 1970 and presently lives in Taipei, Taiwan. William J. Jackson, PhD, is a Professor Emeritus who taught courses in comparative religion and Asian traditions in the Department of Religious Studies at Indiana University-Purdue University at Indianapolis. He earned his doctorate in the Study of Comparative Religion at the Graduate School

of Arts and Sciences, Harvard University. He is the author of books and articles about archetypes in the lives and works of South Indian singersaints, and books such as Heaven ’s Fractal Net: Retrieving Lost Visions in the Humanities (Indiana University Press, 2004); The Wisdom of Generosity (Baylor University Press, 2008); andAmerican Tricksters: The Shadow Side of a Culture ’s Psyche (Cascade Books, 2014). Jackson expands upon the relevance of fractal geometry to transpersonal psychology by illuminating

spiritual archetypes, inspiring wonder and awe, as well as providing a Visual

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Contributors

lexicon for paradoxical concepts, such as existence as multiple, yet “all one” simultaneously. Jesus—Mario Sema, is a Licensed Clinical Psychologist, PhD candidate in Research in Psychoanalysis and Psychopathology (Paris Diderot), and Research and Teaching assistant at the Institute of Psychology, Université Sorbonne Paris-Cite (Descartes), laboratory of Clinical Psychology, Psychopathology, Psychoanalysis (PCPP). His ongoing research at the Center for Research in Psychoanalysis, Medicine and Society (CRPMS) on fractals and the psyche focuses on the self-similar aspects of repetition. A certified attendee of the Santa Fe Institute's CSSS program, he collaborates with several interdisciplinary complex systems research teams around the world. Jonathan Dawe is a composer based inNew York City. His music often involves a synthesis of compositional workings based upon fractal geometry applied to fragments and sounds of Baroque music. He is also a member of

the Doctoral, Graduate Studies, and Music Theory and Analysis at The Iuilliard School.

FOREWORD THE PROFOUND NATURE OF FRACTALS RICHARD TAYLOR

What do we see in the wispy edges of clouds, in the intricate branches of trees, and in the jagged peaks of a snowy mountain range? For many years, it was assumed that these images were a haphazard mess devoid of any pattern. However, the past fifty years have seen a remarkable revolution in the way we study nature’s scenery, which has brought scientific inquiry and artistic views of nature closer together. At the heart of this revolution lies the discovery of intricate patterns called fractals. Dramatically referred to by many as the fingerprint of life, fractals have been shown to be the basic building block of many of nature’s patterns, ranging from clouds, trees, and mountains through to our brains, blood vessels, and lungs. No one should be surprised that nature uses fractals so prevalently. The fractal geometry of nature is profound, both in the simplicity of its construction and in the favorable properties that emerge. Fractals repeat patterns at increasingly fine magnifications. Yet, with this simple act, they build a rich and intricate shape possessing a level of complexity that Euclidean shapes such as triangles, squares and circles cannot match. Mathematicians have studied the exotic consequences of this complexity since the 1860s. However, a century passed before Benoit Mandelbrot realized that nature was using this same pattern repetition to build the world within and around us. Upon his discovery, he struggled to find an umbrella term to unite the earlier mathematical work with that of nature. Marveling at the jaggedness of fractal lighting, he focused on its fractured character and, on a whim, morphed the Latin translationfractus into the now familiar term fractal. Armed With this quirky name, a new era of understanding nature was welcomed in. Many subsequent studies were fueled by bio-inspiration—the

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principle of learning from nature’s repetition and applying it to artificial systems. For example, we now have fractal storm barriers based on coastlines, fractal solar panels based on trees, even my own fractal electronics based on neurons. Clearly, the future shines brightly for scientific applications of this building block of nature. Even more exciting, fractals have the potential to build bridges from the sciences to the arts. Surely, artists and scientists have a shared interest in understanding fractals? For me, the most staggering factor in the story of

fractals is that artists have been creating fractal patterns in their artworks long before these recent scientific breakthroughs. Examples include Leonardo da Vinci’s drawings of turbulent rivers, Jackson Pollock’s epic organic paintings, and M. C. Escher” s mind—bending prints. Pollock’s fractals have evenbeen shown to reduce people’s stress—levels, perhaps explaining that magic feeling of awe that many people experience when facing one of his creations (Taylor, 2006). This deep resonance between the observer and their fractal world is not a new discovery. Experiments from the 1980s show that hospital patients recover far more quickly from major surgery when given a room with a view overlooking nature (Ulrich, 1984). This effect is called fractal fluency—our eyes have become fluent in the visual language of nature’s fractals. In a sense, we are “hardwired” to appreciate fractals. One theory for fractal fluency pictures fractals as being embedded deep in our psyche, perhaps forming the basic structure of the

Jungian collective unconscious. Another theory builds on the fact that our eyes trace out fractal motions when searching for visual information. Sim ilar to the eye hunting through im ages, many animals undergo fractal searches through their terrains when foraging for food. Ongoing research looks to see if people’s daily journeys similarly follow fractal patterns. This prevalence of fractal searches triggers a flood of more profound questions related to our human behavior. In terms of creativity, perhaps our minds exploit fractal searches when exploring the landscapes of our imaginations? If so, perhaps our minds use fractals to drive many emotional, cognitive, and spiritual aspects of our lives? Such hum an questions might surprise those who associate fractals with their mathematical origins. However, as Galileo is often quoted, “the book of nature is written in the language of mathematics.” In fact, a number of

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defining studies on the road to Mandelbrot’s discovery foreshadowed the potential of fractals for exploring questions of humanity. Mandelbrot’s work evolved from Lewis Richardson’s 19505 work attempting to develop models of why nations go to war. Even earlier, Ralph Elliott’s research from the 1930s pictured the stock market as following fractal up and downs, a phenomenon latter proposed to indicate that society exhibits a collective fractal mood. In the future, we might well conclude that fractals are the essence of being human, not just in the building of our lungs, our nerves and our bloodstreams, but in our individual and collective behaviors. This is the

brave new world for fractal researchers. “A Fractal Epistemology for a Scientific Psychology” belongs firmly to this exciting world and its quest to bridge the personal with the transpersonal will broaden the scope of fractal thinking. In my discussions with Mandelbrot, he was delighted to see fractals venture from their mathematical shell and shake the world. He would have been delighted to read this book. References

Taylor, RP. (2006). Reduction of physiological stress using fractal art and architecture. Leonardo, 39, 245-251.

Ulrich, RS. (1984). View through a window may influence recovery from surgery. Science, 224(4647), 420-421.

INTRODUCTION TOWARDS FRACTAL FOUNDATIONS FOR TRANSPERSONAL SCIENCE

TERRY MARKS-TARLOW HARRIS L. FRIEDMAN YAKOV SHAPIRO KATTHE P. WOLF

This volume represents an expansion of a special issue of the International Journal of Transpersonal Studies that proposed establishing a rigorous epistemological foundation for transpersonal science based on the applications of fractal geometry (Marks-Tarlow & Friedman, in—press). We want to extend a heartfelt thanks to the broad range of colleagues who felt inspired to participate in this project. That such an esteemed group of physicists, biologists, mathematicians, psychiatrists, psychoanalysts, religious scholars, and neuroscientists were moved enough to weigh in on the subject indicates the wide-ranging potential of applying fractal mathematics across the spectrum of physical and social sciences. Whether as physical objects, spatial or temporal patterns, or mathematical attractors underlying the processes of emergence and self-organization, fractal dynamics are ubiquitous in nature. Fractals’ presence on all sizes and scales of spatial, temporal and psychological complexity is precisely what elevates its epistemological candidacy. Traditionally, the subject of transpersonal psychology has been primarily confined to humanist and postmodern thinkers, who often dismiss mathematics and the hard sciences as crude reductionist tools that do not apply in the transpersonal domain. A similar attitude is evident among many psychoanalytic thinkers who eschew the recent developments in neuroscience and neural network dynamics in favor of subjective and intersubjective exploration. However, just as unconstrained reductionist attitudes have served to marginalize consciousness and transpersonal

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studies, they can also impede psychological and social theorists. Whether one focuses on objective reality at the expense of subjective experience or privileges psychological reality at the expense of its physical foundations— both approaches ignore systemic connections at different levels of complexity, while perpetuating the reductionist paradigm. Fractal properties of self-similarity, scale invariance, and trans-dimensionality offer a unique potential to build a conceptual bridge between materialist and psychological perspectives, helping us to expand the reductionist paradigm towards a rigorously scientific systemic-holistic perspective that has the potential to

re—unify brain/mind, objective/subjective, and personal/transpersonal do— mains (Shapiro & Scott, 2018). In putting forth a fractal epistemology, we do not wish to make a “one

size fits all” claim. We are not asserting that fractal geometry is the only branch of mathematics worthy of providing metaphors and models for transpersonal phenomena. Swiss psychiatrist Carl Jung came to view numbers as the basic quality of existence. In crafting an archetypal theory, his theory of number doubled over as a theory of mind. Jung (1955/1973) attributed to number the power to bring inherent order into the chaos of appearances, referring to material existence less as objectively unfolding andmore as subjectively perceived by an observer. For Jung, numbers serve as the most fundamental foundation of perceived reality, the place where observers and observed merge at the level of symbol, synchronicity, and meaning. In building a bridge between mind and matter, Jung and his dedicated follower, Marie-Louise von Franz (von Franz & Verlag, 1986), were interested primarily in the concept of numbers as founts of inexhaustible metaphor for conscious experience. Whether in dream, mythology or art, the number one tends to symbolize undifferentiated unity; two signifies the first distinction or duality; three indicates dynamic change and movement away from the static opposition, while four suggests stable manifestation. Jung viewed number as the realm where mind and matter

meet, sometimes referred to as the psychoid level of existence and at other times the Units Mundus. Similarly, Spencer-Brown (1969) referred to mathematics as the cradle of creation, both abstractly in the domain of mind and concretely in the domain of matter. Within this psychophysical cradle of creation, the realm of mathematical abstraction is said to be discovered in so far as it is rulebound and capable of uncovering quantitative facts about the workings of the external world. At the same time, it is also invented as an abstraction, indicating something qualitative about the subjective realm of mind and

meaning. A seminal paper by Robin Robertson (1989), Jungian psychologist

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and mathematician, expanded the concept of number as an archetype of order and traces a history of the qualitative development of human consciousness based on the evolution of quantitative, mathematical discovery. Robertson traced four major stages of human collective awareness. The first stage began with counting numbers, where products of mind and products of matter are symbolically merged. The second stage involved the purely abstract discovery of zero, an absence that becomes a presence, allowing for the modern numbering system and negative numbers necessary for the debt/credit system of economic exchange. The third stage involved the discovery of infinity, which allowed for calculus through the discovery/invention of limits and enabled measurement of complex and moving objects that served as the foundation for the modern scientific/technological society. Robertson’s fourth stage began with the recursive mathematics of Godel, who proved that no system of logic can be simultaneously consistent and complete. Godel’s method models recursive loops of consciousness necessary for self-reflection, as well as the nascent study of self—awareness, which uses the mind recursively to study the mind. To provide a geometrical illustration, consider the Mobius band (see Figure v-l), which is made by cutting out a long strip of paper, giving it a half twist and then taping or gluing the ends together. The result is the topological oddity of a 2-dimensional object that occupies 3-dimensional space with only one side and one edge. The Mobius band functions like an Uroboros, or snake eating its own tail, prototypical symbol of self-creation, based on the workings of recursive feedback loops, where each cycle ending becomes a new cycle beginning (Marks-Tarlow, 2008', Robertson & Combs, 2002).

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Figure v-l. Mobius Band. (From Marks-Tarlow, 2008)

A 3-dimensional equivalent is the Klein bottle (see Figure v-2), which starts with moving a Mobius band up one dimension by enclosing all the edges and stretching out its other aspects. What was the half twist at lower dimensions becomes a self-intersecting feature in higher dimensions. From our limited human perspective that is restricted to 3-dimensional space, the Klein bottle appears to contain both an inside and an outside. Yet, it is actually the 3-dimensional shadow of a 4-dimensional object, which, much like the psyche, has porous boundaries that interpenetrate its insides with its outsides.

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w ' ‘ITMarks-Tar l 090?

Figure 11-2. Klein Bottle. (From Marks-Tarlow, 2008)

Both the Mobius band and the Klein bottle relate to fractals, in that they share the quality of being interdimensional. It is precisely this quality of betweenness that is so relevant to transpersonalists who love to explore interdimensional phenomena, such as mind travel through physical space or the mind’s capacity to influence matter. The psychologist Steven Rosen (1994) has written a fascinating book, Science, Paradox and the Moebius

Principle: The Evolution of a “Transcultural ” Approach to Wholeness, which launches off these topological oddities to explore boundary crossings and paradoxes, such as “the two as one” within a philosophical position he dubs “nondualist dualism.” Complementary positions are those of Roger Sperry’s (1977) monistic interactionism and David Bohm’s active information (1980/1997), which postulated a common informational substrate to all

reality that differentiates into physical (brain) and mental (mind) domains in the ongoing fractal unfolding of evolutionary emergence.

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The Mobius band and Klein bottle can be seen as precursors to fractal geometry. They are paradoxical, interdimensional objects with the concept of infinity implicitly tucked into their infinitely stretchable, topological spaces. By contrast, fractal geometry utilizes infinity more explicitly within the new concept of fractal dimensionality. The infinite stretch between ordinary dimensions is what renders fractal objects ideal for incorporation into religious architecture and art. To behold a progression of self-similar steeples as they unfold upwards from a Buddhist temple (see Figure v—3) is to get an embodied feel for fractals as grounded in the finite realm of matter,

while stretching towards the infinite realm of spirit.

Figure v—3. Rajbana Vihara, Rangamati, Chittagong. (Public domain)

In a paper entitled, “Semiotic seams: Fractal dynamics of re-entry,” Marks-Tarlow (2004) extended Robertson’s history of human consciousness beyond the mathematics of Godel, stating: I argue for the importance of fractal dynamics to model entangled relations between observer and observed. To recognize the broad foundation of fractal geometry within infinite recursion on the imaginary plane can enhance our understanding of reality as finitely perceived in nature. Conversely, to comprehend how fractals manifest ubiquitously at the joints in nature, in turn, self-referentially expands our understanding of mind, especially the deep relativity that exists between observer and observed at all scales of

Introduction

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observation. I introduce self-similarity as a new symmetry in nature that represents the sign of identity. Explored semiotically, self-similarity can be seen as a distinction that leads to no distinction. I relate this paradoxical

equivalence of change and no-change to the operation of cancellation within Spencer-Brown’s arithmetic of first distinctions, as well as to Varela’s reentry dynamics characteristic of autonomous systems. My thesis is that

fractal separatrices between inside/outside, self/other, subjective/objective levels, as well as conscious/ unconscious underpinnings of experience, represent an imaginary/real foundation for the entangled co—creation of

world and psyche. (pp. 49-50) The concept of infinity that bridges mind and matter also arises in the work of Ignacio Matte Blanco (1980), a Chilean psychiatrist and psychoanalyst. Matte Blanco developed a rule-based structure using the

mathematics of infinite sets in order to make sense of a-logical aspects of primary process thought typical of the unconscious. According to Matte

Blanco, the ordinary logic of the conscious mind conforms to additive, reductionist, asymmetrical properties of finite sets. For example, the conscious mind follows stepwise, Aristotelean, tautological, if/then logic: “If I do not do m y laundry, then m y clothes will not be clean.” By contrast, the a-rational logic of the unconscious conforms to the symmetrical equivalence of wholes with their parts that is found within the mathematics of transfinite numbers, where for example, the set of infinite whole numbers is equivalent in size to any subsets, such as the set of all even numbers. Psychologically, Matte Blanco equated this property to children who generalize from parts to whole by calling all dogs “Fido,” or adults who espouse racist ideology,

equating each individual member of a group with properties attributed to the group as a whole. hi sum, we believe that the mathematics of fractals is rich and robust

enough to model the most complex psychological phenomena, which corresponds to the Mandelbrot set as the most complicated mathematical object known to humankind (see Figure v—4). The chapters in this volume apply the unique properties of fractal mathematics within their respective fields.

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Figure v-4. Detail of the Mandelbrot set. (Courtesy of Wolfgang Beyer with the program Ultra Fractal 3)

In part I, Transpersonal Epistemology, Terry Marks-Tarlow offers a fractal epistemology for psychological and consciousness studies broadly. Katthe P. Wolf and Harris L. Friedman explore some of the signs and practical ramifications of embedding a fractal epistemology into transpersonal psychology, the difference it could make. Yakov Shapiro relates fractal dynamics to informational processes underlying neurobiology of consciousness, extending to transpersonal and psi phenomena that span the quantum-classical divide. Rowan Scott applies Gddel’s incompleteness theory to the reductive paradigm in natural sciences, illustrating the holistic potential for a fractal epistemology by adding bottom-up and top-down causal loops. Katherine Peil Kaufl'man looks at fractals through the evolutionary lens related to emotion and universal sentience that transcends the human brain. Frederick David Abraham examines the fractal dirnens ions of mind and its neural network substrates. William Sulis addresses the use of mathematical metaphors in subjective and objective experience. Jonathan Root gives a technical look at fractal geometry juxtaposed with poetic and metaphysical properties of the mathematics involved.

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L1 part II, Fractal Applications, Terry Marks-Tarlow uses fractal properties to examine the intersubjective landscape of repetitive dreams, physiological synchrony, and synchronistic events, where mind and material reality interpenetrate in meaningful ways. Sally Wilcox and Allan Combs equate far-from-equilibrium emergence to rare and unpredictable subjective transpersonal experiences. Robert M. Galatzer-Levy guides the reader through a subjective exploration of the fractal aspects of individual experience. Larry Liebovitch affirms the utility of fractal metaphor in physical, biological, psychological, and social systems. Larry Vandervert explores the interrelationship between form and function within the fractal neurophysiology of the cerebellum. Anthony S. Wright draws parallels between contemporary fractal mathematics and ancient Chinese philosophy. William S. Jackson illustrates the widespread appearance of fractal images and concepts across world religions. Jesus-Mario Serna examines fractals in visual hallucinatory patterns. Finally, Jonathan Dawe describes the fractal structure of his baroque opera “Cracked Orlando: Dramma per musica e fractals,” which premiered in New York City in 2010. The Wide range of different perspectives will appeal to readers from all backgrounds and serves to stimulate further conceptual and empirical research Within the fields of transpersonal science, psychology and consciousness studies as a Whole. Fractal epistemology may help us to expand the reductive paradigm in natural sciences to incorporate subjective, intersubjective, and transpersonal phenomena in all their complexity. We sincerely hope that some of the ideas expressed in these pages can be extended towards new scientific horizons, currently invisible to us all. References Bohm, D. (1997). Woleness and the implicate ordeic New York, NY: Routledge. (Original work published 1980) Jung, C. (1973). Synchronicity: An acausal connecting principle. Princeton, NJ: Princeton University Press. (Original work published 1 955) Marks-Tarlow, T. (2004). Semiotic seams: Fractal dynamics of reentry. Cybemelics and Human Knowing, 11(1), 49-62. —. (2008). Psyche ’s veil: Psychotherapy, fractals and complexity. New York, NY: Routledge. Marks-Tarlow,

T., Robertson, R., & Combs, A. (2002). Varela and the

uroboros: The psychological significance of reentry. Cybernetics and Human Knowing, 9(2), 31-47.

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Marks-Tarlow, T ., & Friedman, H. (Eds.)(in press). Potential role of fractals for modeling transpersonal phenomena [special issue]. International Journal of Transpersonal Studies, 38(2). Matte Blanco, I. (1980). The unconscious as infinite sets: An essay in bilogic. London, England: Kamac. Robertson, R. (1989). The evolution of number: Self-reflection and the archetype of order. Psychological Perspectives, 20(1), 128-141. Rosen, S. (1994). Science, paradox and the Moebius principle: The

evolution of a “transcultural” approach to wholeness. Albany, NY: State University of New York Press. Shapiro, Y., & Scott, J. R. (2018). Dynamical Systems Therapy (DST): Complex adaptive systems in psychiatry and psychotherapy. In E. Mitleton—Kelly, A. Paraskevas, & C. Day (Eds), Handbook of research methods in complexity science: Theory and application (pp. 567-589). London, England: Edward Elgar. Spencer—Brown, G. (1969). Laws of form. London, England: Allen and Unwin. Sperry, R. W. (1977). Bridging science and values: A unifying view of mind and brain. American Psychologist, 32(4), 237-245. von Franz, M. L. & Verlag, E. K. (1986). On number and time: Reflections leading toward a unification of depth psychology and physics (Dykes, A., trans). Evanston, IL: Northwestern University Press.

PART 1: TRANSPERSONAL EPISTEMOLOGY

CHAPTER ONE A FRACTAL EPISTEMOLOGY FOR TRANSPERSONAL PSYCHOLOGY1

TERRY MARKS-TARLOW2

Since the inception of transpersonal psychology in the late 1960s, controversy has surrounded the scope and definition of the field (see Laj oie & Shapiro, 1992). One early aim of this incipient discipline was to transcend limitations of research and methods available to conventional psychologists. Spawned within the humanistic movement, Maslow’s 1971 book title

expresses his desire to explore The Farther Reaches of Human Nature. Much of his early work involved documenting peak experiences, altered states of consciousness, and spiritual dimensions of life. This was in line with the field’s early goal to reject a narrow focus on psychopathology and bring into scope the whole person. From the beginning, highly subjective phenomena of interest to transpersonal psychologists possessed the characteristics of being ineffable and ambiguous. They often involved rare and unreproducible states of mind, fringe rather than normative experiences, and aspects of awareness that are highly personal and culturally specific. Beyond difficulties in quantifying such mental states, an identity confusion has pervaded the subfield of transpersonal psychology, within academic psychology at large. Are these social or “soft sciences” primarily qualitative, descriptive endeavors akin to the humanities, or are they quantitative, empirical undertakings more like chemistry and other “hard sciences”?

1A version of this chapter was published in the International Journal of Transpersonal Studies, 38(2). 2 Core Faculty, Insight Center, Los Angeles; Visiting Professor, Italian Universita Niccolo Cusano London; Research Associate, Institute for Fractal Research, Kassel, Germany; Adjunct Faculty, Pacifica Graduate Institute. E-mail: [email protected]

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Theorists, such as Walsh (1992) suggested an inclusive, integrative strategy, whereby all approaches to transpersonal phenomena are recognized as valuable. Likewise, Wilber (1983) argued for an expanded epistemology that includes sensory, mental-phenomenological, and contemplative data. To these folks, any single approach, no matter how objective-seeming, is nonetheless only partial, limited, and unable to capture the whole truth. On the other side of the transpersonal divide, Friedman (2002, 2013) argued for greater scientific rigor. From his point of view, “theories of everything,” such as that of Wilber, which arise out of a fully inclusive attitude, water everything down so fully as to explain nothing. Friedman proposed that the designation of “transpersonal psychology” be reserved for only that which can be empirically studied, while purely subjective phenomena and fringe disciplines, like astrology, be gathered under the looser designation of “transpersonal studies.” As the debate rages on, Jorge Ferrer (2014) counter— argued that Friedman’s supposedly “objective” lens of strict empiricism is guilty of its own charge: since no perfectly objective stance exists, every perspective is fraught with its own set of assumptions and biases. This chapter and book as a whole aims to present an epistemology for the field of transpersonal psychology that helps to heal an ever-widening schism between these two positions. To honor the call for objective rigor, I offer up the mathematics of fractal geometry as model, method, and metaphor for otherwise ambiguous and inaccessible transpersonal phenomena. To preserve the breadth and richness of personal and cultural phenomena called for by more inclusionary approaches, I suggest that this nascent mathematical field provides a wider framework than conventional empirical approaches from which to consider even the most unique and subjective of mental states, as well as to tackle the complex interrelationship between subjective and objective realms. To choose a branch of mathematics as an epistemological framework could be powerful, because there are clear underlying assumptions, plus unambiguous “right” and “wrong” answers for many, if not most, mathematical problems. For multiple reasons, mathematics is often considered the most rigorous discipline of all. What is more, quantitative experiments within any subfield of psychology (or any social or physical science for that matter) rely upon mathematics at their foundation, often in the form of statistics. Yet, despite this reputation for rigor, Lakoff and Nunez (2000) asserted that even math has no objective origins. In their book, Where Mathematics Comes F rom, these researchers argued that mathematics is instead a fully embodied discipline emerging from the movement of our bodies as they

4

Chapter One

interact in a physical world. Lakoff and Nfifiez pointed out metaphorical origins for even as basic a concept as “number,” which can be conceived in multiple ways, depends upon which metaphor is chosen. Whether considered a collection of objects, a member of a set, or a point on a line, this has important implications, including entaihnents that lead not only to wholly different branches of mathematics, but also at times, to contradictory assumptions among these various branches. Since his discovery/invention of fractal geometry during the 19705, Benoit Mandelbrot considered this new branch to be the mathematics best suited to understanding features of the natural world. In fact, in The Fractal Geometry of Nature, his manifesto published in 1977, Mandelbrot offered fractals as a fi'amework for modeling aspects of nature previously considered too ambiguous, irregular, unique, discontinuous, or complicated for traditional mathematical methods. Over the past 50 years, tens of thousands of researchers have used fractal geometry to model every facet of nature, from microscopic patterning within the quantum realm to the cosmic patterning of galaxy clusters, as well as everything in between, at the mesoscopic level. By assigning quantitative number (in the form of fractal dimension) to qualitative aspects, fi'actal geometry is ideal for understanding natural

features like the fiufi‘iness of clouds, the jaggedness of a shoreline, or the ruggedness of a mountain range. This mathematical power to model complicated patterns extends from outside to inside the human body. Pioneer nonlinear researchers such as West (2013) and Liebovitch (1998) documented how fractal patterns pervade the complicated physiology of our lungs, circulatory system, and neural structures. Other examples of its utility include fractal measurement to differentiate tumor from normal cells (Baish & Jain, 2000), as well as differences in the visual productions of famous artists who suffered from degenerative brain conditions versus those who did not (Forsythe, Williams, & Reilly, 2017). In my own work as a clinical psychologist, I have written extensively

about the fractal geometry of human nature (Marks-Tarlow, 1999, 2004, 2008, 2010, 2011, 2012, 2015). I believe that nonlinear science broadly, and fractal geometry specifically, provide a holistic, flexible meta-framework for understanding even the most complex psychological, social, cultural, and historical systems. Because fiactal patterns extend across space, time,

as well as symbolic realms (Schroeder, 1991), fractals can illuminate complex interrelationships, such as the interpenetration between brain and mind, self and other, and inner versus outer realms.

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In sections to follow, I begin with a brief description of the history of fractal geometry, including its uncanny parallels with the early history of transpersonal psychology. I then describe specific features and properties of fractal geometry that are useful for conceptualizing otherwise inaccessible qualities of transpersonal phenomena. This chapter ends with a list of principles derived from fi'actal geometry in hopes of providing a novel epistemology for transpersonal psychology.

What is a fractal? Everywhere we look, fractals surround us—in the branching patterns of a tree, the spots of a leopard, or the wrinkles of an elderly human face. Although each of us understands fractal patterns intuitively, in an embodied way, very few of us tend to “see” them consciously. ~Why is this? An important reason may be because the field of fractal geometry is too new. Few of us have grown up with fractal objects as part of our visual or mathematical lexicon. Instead, traditional Western education has privileged linear lenses by highlighting straight lines and regular forms, such as Platonic solids and Euclidean dimensions. It is easy to remember elementary school activities of playing with such shapes—for example, cutting out and pasting a larger triangle onto a smaller rectangular base and calling it a pine tree. Yet, all the while, we could sense our productions as mere approximations of the real thing. What constitutes the “real” thing? In other words, how do natural shapes differ from human-made ones? Is there an archetypal meta-pattern—that is, a pattern of pattems—that Nature draws upon again and again? The answer appears to be “yes.” Nature loves recursively enfolded shapes (i.e., patterns that are repeated over and over on multiple size and/or time scales). When in elementary school, we could have just as easily played with fractals. Had we cut out multiple triangles, each the same shape, but slightly different in size, placing the smallest one atop of a layered series, all laid upon the smaller rectangular base, we would have played with self-similar, fractal objects while producing a more realistic pine tree. It is ironic that so few of us have developed a conscious awareness of fractals despite our implicit awareness of them, given what may be a fractal stage of most children’s art (Marks-Tarlow, 2008), akin to Gardner’s (1982) tadpole figure (a circle atop of a stick) to represent the human figure. Figure 1-1 represents an example of fractal art, spontaneously created by my then 5-year-old daughter. Whether the shape consists of a heart, diamond, or oval, there appears to exist a universal desire in children to play with the

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Chapter One

same shape on different size scales. Meanwhile, just about every parent

recognizes some variation of this drawing within their own children’s early art productions.

Figure 1-1. Fractal stage of children’s art. (Courtesy of Darby Tarlow) An important reason that fractals may play a stage in children’s art involves the dynamics of the visual field (see Marks-Tarlow, 2010). As people approach or retreat from babies, similar shapes on different—size scales appear and re—appear successively upon the flat surface of our retinas. Objects or people appear larger as they move towards us (or we move

towards them) and smaller as they (or we) move away. In this respect, our eyes intuitively understand the multi—scaled quality of fractal dynamics, which works as an algorithm to make sense of our own position relative to people and things in our Visual landscapes. This hallmark property of a fractal, as stated more formally, is called

“self-similarity.” Within fractal geometry, self-similarity means that the large-scale pattern of the whole gets repeated on multiple size or time scales

within its parts. Self—similarity involves recursive, that is self-reflexive, symmetry. A related fractal property is called scale-invariance, which means that the same pattern repeats itself either identically or approximately

across multiple size or time scales. Many growth processes are self-similar

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7

as well as scale—invariant. Consider the successive growth of a nautilus shell, as illustrated in Figure 1-2. The mathematical qualities of the algorithmic spiral reveal how the shell’s basic shape, or identity, gets

maintained by preserving part/whole relations, despite successive changes in size. The numbers inside the boxes—1,1, 2, 3, 5, 8—are named the Fibonacci sequence, first defined by Euclid and written about in the 15‘h

century by Luca Pacioli, an Italian monk reputedly “drunk on beauty” (Olson, 2006). To get the next number in the series, simply add together the previous two numbers. By dividing each pair of successive numbers, one arrives ever closer to “the golden ratio” (1.61803...) For millennia, the golden ratio number has been capitalized upon in art and architecture, romanticized in literature, and spiritualized under the name of “sacred geometry” (Lawlor, 1982). Because of the self-similar preservation of

part/whole relations, the Fibonacci series represents an early recognition of this special quality of fractals that describes many common aspects of natureifrom the reproductive rate of rabbits, to the spirals of a sunflower, to the helical form of a pinecone.

Figure 1-2. Self—similar construction of a nautilus shell

Beyond the Fibonacci series, there exist multiple ways to construct a fractal. One involves applying the same algorithm, or procedure, over and over to a seed shape. Consider the Koch snowflake (see Mandelbrot, 1977) in Figure 1-3. The seed shape consists of a triangle; the algorithm involves removing the middle third of each side and replacing it with two thirds of a smaller triangle. The figure below reveals the first three stages, or iterations,

of this process, which can extend indefinitely, at least in theory, even though at a certain point our eyes fail to see the tiniest iterations.

it 1“

Figure 1-3. Seed shape plus first 3 iterations of a Koch snowflake

Fractals like the Koch snowflake are linear, because the identical pattern is repeated on each size scale. Fractals can also be nonlinear, by tossing a bit of chance or randomness into each iteration. Herein lies a critical difference between the regularity of human-made objects and the irregularity of natural ones. Consider for example the genetic code: despite a single underlying growth algorithm, intensely variable conditions within the environment tweak the resulting epigenetic manifestations, from ever so slightly to quite dramatically. Figure 1-4 helps to visualize the difference between linear fractals and nonlinear ones.

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Figure 1-4. Linear versus nonlinear fractal branching algorithms

History of fractal geometry Within the history of mathematics, linear objects and additive methods have prevailed. Consider the invention of calculus in the late 17th century simultaneously by Newton and Leibnitz, designed to capture continuously

evolving dynamics of motion (see Figure 1-5). By chopping the space under a curve into smaller and smaller units, each subsection could be added together to reveal the total area. Using the device of calculus, very complex curves could be broken down and measured. The concept of infinity creeps into the methodology as follows: it lurks implicitly as an idealized point at the limit of the measurement as the size of units become infinitesimally

small, shrinking towards zero.

Chapter One

10

Ay

f(X)

/

a

b’x

Figure 1-5. Calculus

The concept of infinity plays a more explicit role in the invention of fractal geometry. Consider the mathematician Georg Cantor in the late 19th century, whose work was an important precursor to fractal geometry (Mandelbrot, 1977). Up until Cantor’ s time, it was assumed that infinity was absolute, that is that infinity comes in one and only one size. This idealized and definitive view was why some mathematicians equated mathematical

productions with the hand of God, whose infinite power was conceived as equally absolute. By innovating a method of one-to—one correspondences, Cantor discovered that there are endless “flavors” or sizes of infinity, such

as the difference between the set of rational numbers versus the set of irrational numbers. Even bigger is the size of the set of all sets of numbers. By introducing an “infinity of infinities,” Cantor gave birth to a new field called “transfinite” mathematics. At the inception of transfinite

math, many mathematicians

were

shocked. Poincare referred to Cantor’s ideas as a “grave disease” that was “infecting” the field of mathematics (Dauben, 1979). Whereas Cantor

believed his discoveries had been handed to him by God, some Christian theologians feared Cantor’s work challenged the uniqueness of God’s absolute infinity. One way to understand Cantor’s brand of infinity is to contemplate his fractal contribution, called Cantor dust (see Figure 1-6). Whereas the Koch snowflake (Figure 1-2) uses each successive iteration to add structure, Cantor dust uses each successive iteration to remove structure, in this case, the middle third of each line.

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Figure 1-6. Cantor dust

Another 19fh century iconoclast and precursor to fractal geometry was Guiseppe Peano (Mandelbrot, 1977), whose space filling curve, much like both the Koch snowflake and Cantor dust, is notable because it eludes conventional methods of calculus (see Figure 1-7). Because the Peano curve possesses no tangents, it is considered “undifferentiable” or outside the scope of calculus.

Figure 1-7. Four iterations of the Peano curve

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Mathematical forms produced by Koch, Cantor, and Peano represented new and unconventional objects that were met with high suspicion. At times, mathematical colleagues dismissed these objects as irrelevant. At other times, they rejected them as outliers or deemed them “pathological” and “a gallery of monsters” (Mandelbrot, 1977). Looking back at the revolution of ideas that separated the classical mathematics of the 19tll century from the modern mathematics of the 20tll century, it is ironic that the very shapes dismissed as irrelevant and rejected as monstrous have proven over time to conform most highly to Nature’s recursive patterning. Within the history of transpersonal psychology, Stanislav Grof (200 8) documented the predilection of mainstream psychologists and psychiatrists to similarly reject and psychopathologize transpersonal phenomena of interest. As a trend, Western materialistic scientists easily dismiss the realm of spirituality as reflections of mere ignorance, gullibility, superstition, selfdeception or primitive magical thinking. Meanwhile, the experiences of visionaries, prophets, or saints at the root of the world’s major religions are frequently seen to be indicative of serious mental illnesses. In the words of

Grof: St. Anthony has been called schizophrenic, St. John of the Cross labeled a “hereditary degenerate,” St. Teresa of Avila has been dismissed as a severe hysterical psychotic, and Mohammed’s mystical experiences have been

attributed to epilepsy... Franz Alexander (1931), known as one of the founders of psychosomatic medicine, wrote a paper in which even Buddhist meditation is described in psychopathological terms and referred to as “artificial catatonia.” (pp. 47- 48)

Perhaps these parallels between the early days of fractal geometry and those of transpersonal psychology are less coincidental than they might seem. While mainstream mathematics was busy addressing conventional issues, Cantor, Peano, and Koch were examining fringe ideas. In parallel fashion, while mainstream psychology was following its own set of normative trends, transpersonal psychologists were also drawn towards the fringes. Grof asserted ontological realism for transpersonal experiences of the interconnection between all beings and levels of existence, an idea dismissed primarily by reductionist scientists. Perhaps pioneers in fractal geometry and transpersonal psychology were rejected as unconventional, even heretical, largely fiom the perspective of reductionist science. Perhaps the two fields share a similar history because, through more holistic, integrative lenses, they both model the same thing—what is unique, irregular, and rare in nature, including human subjective experience.

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There is a famous paper by the mathematician Wigner (1960) entitled, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Wigner’s focus was primarily on amazing correspondences between mathematical formulae and outer physical realms of the material level. Perhaps we are on the cusp of a transformation by perceiving the unreasonable effectiveness of mathematics within the social sciences. In the spirit of Wigner, I suggest transpersonal psychology needs a more holistic scientific/mathematical fractal framework to embrace the full breadth and depth of its psychological and experiential scope.

Fractional dimensionality: The endless space between dimensions To understand how fractals model identity in nature and bridge various realms of space, time, and imagination (Marks-Tarlow, 2004, 2012), it is important to examine how fractals illuminate aspects of subjective experience. Where does consciousness begin? Where does it end? What are its bounds, especially given that the subj ectivefeel of conscious awareness seems to extend across boundaries (from inside our heads to outside our bodies)? How does the invisible substance of consciousness relate to the materiality of our brains and bodies? What is the difference between an objectively measurable event and a subjectively held experience? All remain disputable issues often relegated to the realm of philosophy. A complete answer to these questions is beyond the scope of this chapter. However, before highlighting a couple of issues relevant to this discussion, I begin with a disclaimer. In sections that follow, including the list of epistemological principles at the end of this chapter, I do not claim to have solved what is labeled the “hard problem” of consciousness, as formulated by David Chalmers (1995) in the following: It is undeniable that some organisms are subjects of experience. But the question of how it is that these systems are subjects of experience is perplexing. Why is it that when Our cognitive systems engage in visual and auditory information-processing, we have visual or auditory experience: the quality of deep blue, the sensation of middle C? How can we explain why there is something it is like to entertain a mental image, or to experience an emotion? It is widely agreed that experience arises from a physical basis, but we have no good explanation of why and h0w it so arises. Why shOuld physical processing give rise to a rich inner life at all? It seems objectively

unreasonable that it should, and yet it does. (p. 201)

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Chapter One

Chalmers formulated the hard problem as difficulty explaining the contents, or qualia, of conscious experience. He outlined and then dismissed the success of various case studies proclaiming to explain consciousness, including Crick and Koch’s (1990) suggestion that gamma oscillations in the cerebral cortex provide the neurobiological correlates of consciousness or Penrose’s (1994) suggestion fi'om within a nonlinear dynamics perspective that nonalgorithmic processing explains mathematical reasoning. I want to clearly state that I am not presenting a theory of conscious awareness or an explanation of how we come to experience its various qualia. Instead I offer fractal geometry as a means of modeling some features that pertain to the structure of subjective experience, including the possibility of open boundaries between conscious awareness and physical, material levels of brain, body, and surrounding environment. Keeping these limitations in mind, Iturn next to an important distinction between objectively measurable events and subjective experience. Objectively measurable events are discrete and observable. To measure something, clear boundaries must be present, plus a clear value within those finite bounds. Subjective experience, by contrast, carries the feeling of being immeasurable and infinitely deep, with borders that feel fuzzy and ambiguous. Perhaps there is a unified field of consciousness—a truly transpersonal extension of invisible subjective dimensions into objective realms. Shamans who claim to transport themselves through their astral bodies could possibly be such a case. The true potential of consciousness remains unknown. But again, most relevant to this discussion is the subjective feeling of fuzzy boundaries and infinite extension, both during contemplation of inner worlds, as well as during perception of external worlds. This very sense of boundary-less interconnection and complete interpenetration of inside and outside realms corresponds to mystical experiences and peak states like “nondua ” awareness, whether facilitated by psychedelic substances or occurring naturalistically. How does a mystical sense of infinite extension relate to fractals? When I first came across this new branch of geometry in the early 1980s, I immediately had the intuition of something profound about fractals. At the time, I was attending a weekly drawing group that included the physicist Richard (Dick) Feynman, and we had become quite good friends. Because Feynman was deemed the smartest man in the world (after Einstein), I rushed to him with the question, “Don’t you think fractals are profound?” Someone standing nearby asked what a fractal is. Dick took several minutes to give a state of the art explanation of fractal geometry—its hallmark features of self-similarity, scale invariance, and more. I waited patiently,

A Fractal Epistemology for Transpersonal Psychology

15

and once Dick had finished, I asked him again, “Don’t you think fractals are profound?” His response—“I don’t understand them”—absolutely shocked me. How could this be, when Feynman had just explained fi'actals so eloquently? Crestfallen, I was left utterly alone to find my way forward. It has taken me decades to flesh out my understanding of fractals, including 15 years to

write Psyche ’s Veil (Marks-Tarlow, 2008), which applies chaos theory, complexity theory, and fractal geometry to clinical practice. To this day, I still do not understand fractals fully, nor do I believe I ever will. But one thing I am quite certain of—my initial feeling of profundity relates to the role infinity plays within fractal construction. Let me explain. Ordinary Euclidean dimensions are finite, that is, they consist of whole numbers such as integers. Points are 0 dimensional (0-D); lines are 1 dimensional (1-D); planes are 2 dimensional (2-D); solids are 3 dimensional (3-D). Einstein offered time as the 4th dimension, while others view imagination as the 4‘5h dimension (see Marks—Tarlow, 2008). Human made objects, such as the top of a table, have clear boundaries within the confines of finite Euclidean dimensions. The measurement of a table’s circumference is resolvablegwe always arrive at the same approximate answer, no matter how large or small our measuring device. Whether our ruler is 6 inches long or 6 feet long, the measurement of a table’s circumference remains essentially the same. None of these conditions apply to fractals, which are multi-scaled objects that are not finite, but infinitely deep, at least in theory. Because of the properties of being multi-scaled and infinitely deep, fractals do not have clear boundaries. Their measurement is not fixed but fuzzy and dynamical instead. To illustrate this, consider the Mandelbrot set (Figure 1-8), the granddaddy of all fractals and the most complex mathematical obj ect known to humankind (Dewdney, 1985). To construct the Mandelbrot set, the same formula, f(z)—> :2 + c, is iterated for every point on the complex number plane. Iteration means that the end product of an equation is fed back into the beginning over and over, that is, recursively, until the equation resolves itself (or does not).

16

Chapter One

Figure 1-8. The Mandelbrot set, f(z)—>z2 + 0. (Courtesy of Nicolas Desprez)

In Figure 1-8, the solid black areas represent the finite zone where the

formula resolves to a fixed number. The white areas represent the infinite zone where the formula goes on and on, extending towards infinity. The complex border between these two zones represents the dance of the Mandelbrot’s intricate, multi-scaled pattern. This edge of complexity is

infinitely deep. This means that, when the computer is used as a “microscope” to zoom in on a particular area, ever new pattern emerges

dynamically and unpredictably. Figure 1-8 reveals four scales of zoom on the Mandelbrot set’s complex edges. Notice the self-similarity that re— appears in the fourth square, such that the very similar shape of the whole

reappears, making it quite difficult to tell what is inside and what is outside its borders. From this example, we can see that fractal geometry is a very visual form of mathematics that is intimately dependent upon the prodigious calculating power of the computer. This fact helps explain why fractal geometry was not discovered until the 1970s. Fractal zooms abound on YouTube, and it is

highly illustrative to watch a few of these short videos to get a feel for the endless beauty and depth of fractal geometry.

.9. clsl Epdsumelemr' fear Tmpfi'sfldfil Psychchy

17

'To more fully understand the infinite aspect of fractal geometry, I now present the concept of fractal dimensionality. Contiary to ordinary assumptions, fractals grow in the endless space betw een finite Emlidean dimensions. Mathematically, the discovery of fractals required expansion of flte 1irery notion of dimensionality, such that each mathematical fractal not only has a discrete Euclidean dimension, but also has a fractal dimension,

consisting of a fractional number that carries the potential of being infinitely T.Iariable. Subfigures I-Qa and I-Qb illustiate how ordinary scribbles as well as more fiarmal Sierpinski carpets occupy the space betweena I—D straight line and a 3-D plane. Subfigures I—filc and Lid illustrate how the Sierpinski triangle and a fractal mountain generated from triangles occupy the space between ail-D plane and 3-D solid.

18

Chapter One

A Fractal Epistemology for Transpersonal Psychology

19

60 Figure 1-9. Fractal dimensionality a) Scribble, b) Sierpinski carpet, c) Sierpinski triangle, (1) Fractal mountain In general, no matter what the Euclidean dimension, the higher the fractional dimension, the more complex the fractal object. Figure 1-10 shows the same fractal mountain scape rendered in lower versus higher fractal dimensionality. We can now begin to see how fractals help to quantify qualitative features of Nature, like the ruggedness of a mountain scape, the jaggedness of a coastline, or the fluffiness of a cloud. In an interesting recent application, the fractal dimension of Rorschach test figures was quantified (Abbott, 2017). Despite initial speculations that

Rorschach dimensional complexity would mimic that of Nature (e.g., cloud patterns that resemble Mickey Mouse or a submarine), the Rorschach figures are relatively lower dimensional than Mother Nature, revealing a slightly different “fractal sweet spot” that is best suited to the projection of visual imagery from imagination.

20

Chapter One

Figure 1-10. The same fiactal mountain scape Lower dimensionality (top), higher fractal dimensionality (bottom). (Courtesy of Nicolas Desprez)

21

A Fractal Epistemology for Transpersonal Psychology

Fractalparadoxes Mandelbrot (1967) posed a now famous question, “How long is the coastline of England?” At first blush, the answer might seem straight

forward. Yet because of the multi-scaled quality of a coastline’s fractal shape, paradox lurks Within, connected to the construct of fractal dimensionality. Mandelbrot claimed that the length of the coastline of England is infinitely long, and What is more, every other natural coastline is also infinitely long, along with any arbitrarily short subsection of coastline!

Mandelbrot’s assertion emerges from fractals as multi-scaled objects. The property of infinite depth renders a single, definitive measurement impossible. Instead, the number one arrives at depends intimately on the size of our measuring device. Counterintuitively, the smaller the ruler, the larger the number. This resulting quality of observer dependent measurement (Marks-Tarlow, 2008) is illustrated in Figure 1-1 1.

F.1.l.|.l.|.l.|.l.|.|.|Ll 0

1

2

0

1

2

0

1/2

3

3

9

4

5

6

7

6

8

12

Figure 1-11. Fractal relativity of measurement. (Courtesy of Terry Marks-Tarlow)

22

Chapter One

Notice that when the ruler is 6 inches long, it is too crude to capture any detail of the Koch curve. When the ruler is 2 inches long, it is short enough to capture more detail, and the length measured extends to 8 inches. As the ruler shrinks to half an inch, the measurement captures yet more detail and extends to 12 inches. Two important additional observations: 1) Even at half an inch long, the measuring stick still does not capture all the detail of the Koch curve; 2) Generally, the shorter the measuring stick, the longer the measurement, such that at the mathematical limit of an infinitely small measuring stick, we obtain an infinitely long measurement. Hence, this supports Mandelbrot’s claim regarding the infinite length of any section of coastline. In both the Koch curve example and Mandelbrot set, we see how infinity quite literally exists at the edges of fi'actal objects. This helps us to grasp how fi'actals can model irresolvable seeming subjective boundaries. To get a fuller feel for fractal boundaries, such as I claim represent the psychological edges of the self, consider Figure 1-12, which illustrates Newton’s method of approximation. Each shaded area converges towards one of four correct solutions to the simple equation, X4 — 1 = 0. Each solution consists of a black circle within the center of each of four quadrants, more technically known as basins of attraction. Whereas each solution is finite, the boundary zone separating the four basins of attraction is infinitely deep. What is more, this mathematical rendering also reveals the infinite interpenetration between the parts and the whole. This is because each of the fractal boundary zones contains all of the other basins recursively, an infinite number of times. The notion of interpenetrating boundaries, such as exists interpersonally, that is, between one person and another, is a subject of great interest to me as a clinical psychologist. I have written about the relational unconscious (Marks-Tarlow, 2008), as shared between therapist and patient, beneath the level of conscious awareness. For example, in Psyche ’S Veil, I cite the case

of apatient who one day brought into our session my own childhood dream. This tidal wave dream was very different from anything she had ever remembered dreaming, as most of my patient’s dreams involved scary chases and attacks. Especially given the flood of change that happened next, both of us experienced this dream as an unconscious bid to break the enactment stalemating our psychotherapy for several months.

A Fractal Epistemology for Transpersonal Psychology

23

Figure l-lZ. Newton’s method of approximation. (Adapted from Gleick, 1987) The notions of fuzzy, interpenetrating boundaries between self and other, mind and brain, and brain and body, is consistent with the work of Scott Kelso (e.g., 1997, Kelso & Engstmm, 2006). Kelso, a nonlinear researcher also interested in how science and philosophy intermingle, has studied and written extensively about coordination dynamics, that is, how

patterns of coordination form, dissolve, adapt, and change through processes of self—organization. When examining implications of coordination dynamics for the brain~mind and brain~behavior barriers, Kelso uses the tilde to

symbolize the dynamic nature of complementary pairs, whose polar ends are not only of significance, but everything in between. Kelso has also

24

Chapter One

studied how dynamical patterns of muscular motion existing within one person extend to others, such as when people fall into lockstep or when musicians coordinate so precisely as to anticipate each other’s next moves. Kelso’s recent work on hyperscanning (Kelso, Dumas, & Tognoli, 2013) extends these examinations even further. Hyperscanning involves the simultaneous brain scanning of two individuals’ as they interact in real time. This fascinating line of contemporary neuro-research reveals very little difference between intrapersonal and intelpersonal communication. In other words, how messages are sent fi'om one part of the brain to another share similar coordination dynamics to how messages are sent between brains. Such research points towards fluid, dynamic boundaries between self and 7 other, inner and outer realms. Figure 1—12 provides a visual representation of fluidboundaries between inner and outer realms in the case of mathematical intuition—subjective guesses at objective answers. Here is how Newton’s method of approximation works. To solve the equation, X4 — 1 = 0, begin with a random guess at a solution, then calculate the formula using your guess as the starting point. How close your guess is to one of four actual solutions determines what happens next. The closer your beginning guess is to an actual solution, the quicker you arrive at the solution; if your initial guess diverges too far, it will land within the chaotic boundary zone between 7 solutions, from which there is no exit. This visualization is particularly interesting given paranormal intuition as an important subject of interest within the field of transpersonal psychology (Daniels, 1998). I have also used the diagram to model the chaotic boundaries that so often surround people diagnosed with Borderline Personality Disorder (BPD). As a clinician who works often with people suffering from this diagnosis, I can attest to the frequent feeling of loose boundaries that trigger my fall into dangerous, double-bind territory, from which there is no escape. I am “damned if I do, and damned if I don’t,” from the perspective of the other—and utterly helpless to assert my own independent perspective. Finally, I suggest another excellent use of Figure 1- 12 for re— conceiving Ken Wilber’s integral grid. To add fractal boundaries in place of straight lines between his four quadrants increases the power of his model for understanding interpenetrating subjective, intersubjective,

objective, and inter-objective realms.

A Fractal Epistemology for Transpersonal Psychology

25

Power laws: A new kind of statistic As mentioned earlier, most applied sciences depend upon mathematics to supply the necessary rigor for their foundations. “Hard” sciences, such as physics, fiequently rest upon predictions supplied by pure mathematical formulas. For example, Einstein needed to delve into the strange, nonlinear world of non-Euclidean mathematics to prove his theories of relativity. “Softer” sciences, such as biology or economics, often use statistical methods to test between competing hypotheses. Across all subfields of psychology and other social sciences, normative statistics are traditionally used. This type of statistics, sometimes called a Bell curve, seeks the central tendency, that is, the mean or norm of a population or sample. Unfortunately, normative statistics contain underlying assumptions that often prove to be false within most complex systems, as is described in detail in West’s (2016) book on the topic. One underlying assumption that frequently proves false is the requirement that all underlying variables operate independently (or orthogonally) from one another. Having had the honor of writing the foreword to West’s book, I relayed a mathematical tale from my youth. At the time, a common statistic was floating around that the average American family had 2 ‘/2 children. What the heck does that mean, I pondered, given that no family has 2 ‘/2 children? Because transpersonal psychologists are so often interested in idiosyncratic states and non-repeatable circumstances that have nothing to do with central tendencies, the poor fit between normative statistics and phenomena of interest may be particularly exaggerated within this subfield of psychology. Fortunately, another mathematic distribution exists, called a power law, which excels for modeling rare, unpredictable, and unique events. Power laws are temporal fractals, where statistical self-similarity manifests as scale invariant patterns across multiple time scales. As an example, Mandelbrot and Hudson (2010) applied temporal fractals to model stock market fluctuations. Whether examined over the period of a day, year, or decade, the ups and downs of the market reveal statistically self—similar patterns. With chance and randomness part of natural fractal fluctuations, we begin to understand how fractals help us to model transpersonal phenomena that

are fundamentally unpredictable, yet simultaneously ordered. A good example of a power law distribution in Nature is the frequency of earthquakes of various magnitudes, as measured on the Richter scale, a logarithmic metric. It turns out that the chances of a very large earthquake

26

Chapter One

are very small (sometimes called a Black Swan event); the chances of atiny earthquake are quite large; and the chances of a medium level quake are medium sized. Much like patterns on the stock market, we cannot predict the specifics at any given point, yet we can determine the coarse-grain picture. With normative statistics organized around a mean score, their power lies in the center, such that all variability collapses into a single number at the peak of the Bell curve, hence the conclusion that the average American family contains 2 1/2 children. With larger and larger sample sizes, normative statistics gain both in certainty, as well as in predictive power. By contrast, the power of a power-law distribution is not in the center, but in the tails, where rare events exist. This type of statistic allows for unpredictability while preserving variability. What this means is that the larger the sample size, the greater the variability one finds. Simply put, the more people you sample, the greater the differences you will find among them. Psychologically, this trend certamly corresponds with my professional experience as a clinician. Although depression is ubiquitous as a symptom, to me no two cases look alike and, if they did, I am probably in the wrong profession. The ability of power law distributions to predict the occurrence of highly rare occurrences, but not their precise timing, seems invaluable for validating, if not tracking, transpersonal phenomena. Here is an empirical

example, related to my 1999 paper, “The Self as a Dynamical System.” In this paper, [predicted that changes relevant to the self would follow a power law distribution. Much like earthquakes, this would mean that people rarely experience huge changes but often only experience tiny shifts. Delignieres and his French colleagues (2004) decided to test this hypothesis. Twice daily, for 512 days, a small group of subjects rated six subjective dimensions: global self-esteem, physical self-worth, physical condition, sport competence, attractive body, and physical strength. Results indicated that changes in self-esteem, as well as changes in perception of physical self, did indeed reveal a fractal distribution. Each subject demonstrated an array of self-similar fluctuations that possessed a unique fractal—dimension exponent. Results confirmed my conceptualization of the self as a hierarchically nested, self-organizing, dynamical system. Subjective research such as this fulfills Friedman’s (2013) call for rigor, while mirroring his own (1983) research on self-expansiveness as a transpersonal construct.

A Fractal Epistemology for Transpersonal Psychology

27

The computer as aid to the human eye In previous sections, I demonstrated how fractals offer a way to visualize otherwise invisible dimensions, as well as how fractals can model interpenetrating boundaries within highly complex, open systems. We saw how invaluable the computer is in the process, since the entire branch of fractal mathematics depended upon its invention for a complete Visualization. Generally researchers interested in nonlinear dynamics, including the complexity sciences, utilize computer simulations to model highly complicated systems that contain unpredictably emergent or highly idiosyncratic elements. Computer-aided methods, such as agent—based modeling, allow researchers to simulate complex systems by tweaking underlying parameters (values) and then running the system again to see what happens. In their book The Philosophical Computer, Grim and his colleagues (1998) described their use of the computer to model paradoxical philosophical issues too complex to otherwise visualize. Consider, for example, the self-referential assertion, “This statement is false,” known since antiquity as the Liar’s paradox. The statement is paradoxical because it is true only if it is false, and false only if it is true. Translated into a mathematical equation iterated by computer, Figure 1-13 shows a way to visualize the paradox as a periodic attractor bouncing back and forth between 2 values: 1 (true) and 0 (false). Logic is ordinarily considered to exist outside of time; yet by adding time into their equation, Grim and his colleagues found a way to solve the age—old paradox. Their solution functions much like a light switch that contains two contradictory states (on and off), which cannot co-exist but can oscillate over time (Marks-Tarlow,

2008).

Figure 1—13. The simple Iiar’s paradox. (Courtesy of Patrick Grim, Group for Logic and Formal Semantics, Department of Philosophy, SUNY at Stony Brook)

, 28

t

Chapter One

assertions:

3

s

A more complicated, interpersonal variation of the Liar’s paradox exists, also known since antiquity: “Socrates asserts, “Plato speaks falsely,” while Plato counters, “Socrates speaks truly.” To visualize the interpersonal variation of the Liar’s paradox, Grim’s group used fuzzy logic to supply an infinite—valued scale between truth and falsity applied to the following x: xisastrue asy; y: y is as true as x is false

When converted into mathematical equations that were iterated by computer, Figure 1—14 reveals the resulting fractal escape diagram. Computer modeling of interpersonal dynamics demonstrates yet again how

fi‘actal boundaries arise out of complex feedback loops between inner and outer processes, such as self and other. Grim and his colleagues, in fact, offer up fractal geometry as a means for modeling not just paradoxical logic,

saw

,, '17

but in fact, all formal systems.

Figure 1-14. Interpersonal variation of the Liar’s paradox. (Courtesy of Patrick Grim, Group for Logic and Formal Semantics, Department of Philosophy, SUNY at Stony Brook)

A Fractal Epistemology for Transpersonal Psychology

29

Epistemological principles for transpersonal psychology Having explained fiactals and given examples of how they are constructed and how they have been used at the edges of psychological research, this final section offers fractal geometry as an epistemological framework for transpersonal phenomena. I propose the following principles: Fractal geometry models and bridges recursive patterns in space, time, and the imagination; Fractal geometry offers quantitative methods for revealing qualitative patterns in nature previously deemed too complex, irregular, or discontinuous from the perspective of linear lenses and reductionist techniques; Fractal geometry models hidden, as well as higher, dimensional phenomena that exist inthe infinite expanses between ordinary, finite (Euclidean) dimensions; Fractal dimensionality captures key features of the structure of subjective experience, such as the endless feeling of contemplation, the boundary-crossing experience of consciousness as it leaps from inner to outer worlds, and the paradox of full engagement, such that closer one looks at something, whether inside or outside the imagination, the more there is to see; Fractal geometry highlights idiosyncratic, non-repeatable, and rare events, by offering power law statistical distributions over time; Power law distributions present an alternative to normative statistics in which variability is preserved, while unpredictable, chance events are factored in, such that order is conserved in the form of an underlying growth or decay algorithm; Fractal measurement illuminates observer dependence, whereby what we see depends upon how we look, including our scale of observation plus other qualities of ourselves as measuring devices; Fractal geometry presents a way to conceptualize fuzzy, irresolvably complex borders between various realms, levels, and dimensions of existence, including full interpenetration as it exists at flactal boundary zones; and Fractal edges model paradoxical insights related to traditional mystical experiences and nondual states of awareness, including how the whole of things can be enfolded within the parts of existence, plus Buddhist notions of emptiness and interbeing.

Chapter One

30

I conclude this chapter with a plea not only for transpersonal psychology, but also for all of psychology to adopt a fractal epistemology. As a result, researchers and clinicians will be better equipped to model idiosyncratic, rare, and unpredictable phenomena. Meanwhile, both qualitative and quantitative aspects of nature can be simulated within a single, mathematically rigorous umbrella.

References Abbott, A. (2017). Fractal secrets of Rorschach’s famed ink blots revealed. Nature Magazine, February 15. Retrieved from https:l/www.scientificamerican.com/article/fi‘actal-secrets-of— rorschach—rsquo-s-famed-ink-blots-revealed/ Baish, J., & Jain, R. (2000). Fractals and cancer. Cancer Research, 60(14),

3683-3688. Chalmers, D. (1995). Facing up to the problem of consciousness. Journal of Consciousness Studies, 2(3), 200—219. Crick, F ., & Koch, C. (1990). Toward a neurobiological theory of consciousness. Seminars in the Neurosciences, 2, 263-75.

Daniels, M.

(1998).

Paranormal psychology

and

the

paranormal.

Transpersonal Psychology Review, 2(3), 17-31. Dauben, J. (1979), Georg Cantor: His mathematics and philosophy of the infinite. Boston, MA: Harvard University Press. Delignieres, D., Fortes, M., & Ninot, G. (2004). Fractal dynamics of selfesteem and physical self. Nonlinear Dynamics in Psychology and Life Sciences, 8, 479-510.

Dewdney, AK. (1985). A computer microscope zooms in for a look at the most complex object; in mathematics. Scientific American, 253(2), 16-

24. Ferrer, J. (2014). Transpersonal psychology, science, and the supernatural. Journal ofTranspersonal Psychology, 46(2), 152-186. Forsythe, A., Williams, T., & Reilly, R. (2017). What paint can tell us: A fi'actal analysis of neurological changes in seven artists, Neuropsychology,

31(1), 1-10. Friedman, H. (1983). The Self-Expansiveness Level Form: A conceptualization and measurement of a transpersonal construct. Journal of Transpersonal

Psychology, 15(1), 37- 50. —. (2002). Transpersonal psychology as a scientific field. International

Journal of Transpersonal Studies, 21(1), 175-181. —. (2013). The role of science intranspersonal psychology: The advantages of middle-range theory. In H.L. Friedman & G. Hartelius (Eds), The

A Fractal Epistemology for Transpersonal Psychology

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Wiley-Blackwell hancflaook of transpersonal psychology (pp. 203-222). Malden, MA: Wiley Blackwell.

Gardner, H. (1982). Art, mind and brain: A cognitive approach to creativity. New York, NY: Basic Books. Grim, P., Mar, G., & St. Denis, P. (1998). The philosophical computer:

Exploratory essays in philosophical computer modeling. Cambridge, M A : MIT Press. Grof, S. (2008). Brief history of transpersonal psychology. International Journal ofTranspersonal Studies, 27(1), 46-54. Kelso, S. (1997). Dynamic patterns. Cambridge, MA: MIT Press.

Kelso, S. & Engstrom, D. (2006). The complementary nature. Cambridge, M A : MIT Press. Kelso, S., Dumas, G., & Tognoli, G. (2013). Outline of a general theory of behavior and brain coordination. Neural Networks, 37, 120—131. 7 Kestenberg, J. (1985). The flow of empathy and trust between mother and child. I n B. Anthony & G. Pollack ( E d s ) , Parental influences in health and disease (pp. 137—163). Boston MA: Little, Brown. Lajoie, D., & Shapiro, S. (1992). Definitions of transpersonal psychology:

The first twenty-three years. Journal of Transpersonal Psychology, 24(1), 79-98. Lakoff, G., & Nufiez, R. (2000). Where mathematics comesfrom: How the

embodied mind brings mathematics into being. New York, NY: Basic Books. Lawlor, R. (1982). Sacred geometry: Philosophy d’c practice (Art (1’: imagination). New York, NY: Crossroad.

Liebovitch, L. (1998) Fractals and chaos simplified for the life sciences. Oxford, UK: Oxford University Press. Mandelbrot, B. (1967). How long is the coast of Britain? Statistical selfsimilarity and fractional dimension. Science, I 5 6, 636-638.

—. (1977). The fractal geometry of nature. New York, NY: W. H. Freeman. Mandelbrot, B., & Hudson, R. (2010). The (mis)behaviour of markets: A fractal view ofrish rain and reward. London, England: Profile. Marks-Tarlow, T. (1999). The self as a dynamical system. Nonlinear

Dynamics; Psychology, &Life Sciences, 3(4), 311-345. — (2004). Semiotic seams: Fractal dynamics of reentry. Cybernetics and Human Knowing, I 1(1), 49-62.

—. (2008). Psyche ’s veil: Psychotherapy, fiactals and complexity. New York, NY: Routledge.

—. (2010). Fractal self at play. American Journal ofPlay, 3(1), 31-62. —. (2011). Merging and emerging: A nonlinear portrait of intersubjectivity

during psychotherapy. Psychoanalytic Dialogues, 21(1), 110-127.

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—. (201 2). Fractal geometry as a bridge between realms. In F. Orsucci & N.

Sala (Eds), Complexity science, living systems, and reflexing inteifaces: New models and perspectives. Hershey, PA: IGI Global.

—. (2015). The nonlinear dynamics of clinical intuition. Chaos & Complexity Letters, 8(2-3), 1-24.

Maslow, A. (1971). The farther reaches of human nature. New York, NY: Viking. Olson, S. (2006). The golden section: Nature ’s greatest secret. New York, NY: Walker.

Penrose, R. (1994). Shadows of the mind Oxford, UK: Oxford University Press.

Schroeder, M. (1991). Fractals, chaos, power laws. New York, NY: Freeman. Walsh, R. (1992). The search for synthesis: Transpersonal psychology and the meeting of East and West, psychology and religion, personal and

transpersonal. Journal of Humanistic Psychology, 32(1), 19—45. West, B. (2013). Fractal physiology and chaos in medicine: Vol. I 6. Studies of nonlinear phenomena in life sciences (2nd ed.). Hackensack, NJ: World Scientific.

—. (2016). Simplifying complexity: Life is uncertain, unfair and unequal. Sharj ah, UAE: Bentham. Wigner, E. (1960). The unreasonable effectiveness of mathematics in the

natural sciences. Communications on Pure and Applied Mathematics, 13(1), 1-14. Wilber, K. (1983). Eye to eye: The questfor the new paradigm. Garden City,

NY: Anchor/Doubleday.

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CHAPTER TWO

MORE THAN MERELY A MODEL OR METAPHOR? THE CONTRIBUTIONS A FRACTAL EPISTEMOLOGY MIGHT MAKE TO TRANSPERSONAL PSYCHOLOGY KATTHE P. WOLF & HARRIS L. FRIEDMAN1

Transpersonal psychology can be understood as a science concerned with exceptional human experiences, multiple states of consciousness, and the areas between spirituality and scientific studies of mind and behavior. It can also be understood as a subdiscipline of psychology seeking to radically transform the mainstream discipline, including reconciling methodologies for understanding mind—body relationships, traditional Eastern and Western, as well as indigenous, worldviews, and numerous other perplexing divides. Hartelius, Roth, and Roy (2013) offered the following definition: Transpersonal psychology is a transformative psychology of the whole person in intimate relationship with a diverse, interconnected and evolving world; it pays special attention to self-expansive states as well as to spiritual, mystical, and other exceptional human experiences that gain meaning in

such a context. (p. 14) As a discrete approach to psychology, it is generally agreed to have arisen in the 1960s in the San Francisco Bay Area (Hartelius et al., 2013), although its name traces its earlier roots from William James through Carl

Jung until Abraham Maslow and others later adopted it (Daniels, 2013). The 19605 genesis of the conversation between transpersonal and mainstream 1 Katthe P. Wolf: Doctoral student, lntegral and Transpersonal Psychology PhD Program, California Institute of Integral Studies Harris L. Friedman: Professor of Psychology (retired), University of Florida Email: [email protected]; [email protected]

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Chapter Two

psychology was like a collective epiphany addressing phenomena that real people actually experience all the time, but that are considered anomalous and suspect by the mainstream because they do not comport with current scientific understanding of ontology (that is, what is real), and epistemology (how we know this). Intuition, synchronicity, psi (including ESP, precognition, mystical states, out-of-body experiences, the so-called “ineffable” knowledge that often comes from psychedelics, etc.) beg for explanations in a rigorous, scientific manner. After all, there are millions of people throughout the world who incorporate these phenomena into their understanding of reality seamlessly. Could it be that such traditional Eastern and indigenous understandings as well as esoteric and mystical Western traditions, hold not just a legitimate, but possibly even better, approximation to the totality of reality? Could aspects of these cosmologies be scientifically approached, allowing for a more holistic understanding that does not ignore these important sources of data? Transpersonal psychology historically has, however, had an uneasy relationship with science. Friedman (e.g., 2002, 2013, 2018) has written extensively about the importance of pursuing this relationship to enable mainstream psychology to become more holistic, as well as to allow career pathways for transpersonal scholars and practitioners to flourish for the sustainability of the subdiscipline. Accordingly, Friedman (2013) wrote: Transpersonal psychology can be seen as an attempt to replace traditional spiritual and folk psychological worldviews with perspectives congruent with those of modern science that can develop scientifically through

empirical research. Specifically, this means making these perspectives amenable for empirical exploration. (p. 301) Friedman’s (1983) model of self-expansiveness offers the field one such scientific approach, and he has strongly argued over the years that the subdiscipline should find a balance between the limiting scientism of mainstream psychology and the overly speculative romanticism of New Age thinking (e.g., Friedman, 2002, 2013, 2018). This balance could both expand responsible science within transpersonal psychology, while also expanding the boundaries of mainstream psychology to the benefit of both.

He described the context for his work as follows: My contributions have landed in an interesting limbo, namely between those who experientially appreciate the transpersonal but who tend to reject the worth of science for furthering it on one hand, while on the other hand are more conventional scientists who reject the worth of the transpersonal and are often resistant to even scientific evidence supporting it. I am interested

More than Merely a Model or Metaphor?

35

in the proverbial third hand, developing the interstices between these two oppositional stances. (Fracasso, Franco, MacDonald, & Friedman, 2011, p.

513) Friedman’s commitment to a balanced empiricism in psychological investigations does not endorse apurely reductionist approach, such as one that views consciousness as only being an epiphenomenon of brain activity. Instead, he has vigorously argued against both a narrow materialism that unduly restricts scientific investigation (e. g., Friedman & Pappas, 2006) and romanticism, including what he called “romantic scientism,” which blends science with romanticism (Friedman, 2018; Friedman & Brown, 2018). In charting a path with relation to Cartesian dualism, Friedman has argued for both/and approaches in forming a holistic integration. In addition to focus on transpersonal psychology fi'om a scientific perspective, Friedman has also focused on its role in enhancing health (e. g., Elmer, MacDonald, & Friedman, 2003, as have many others (e. g., Walsh & Vaughn, 1993). As a so-called “fourth force” in psychology, growing out of prior behavioristic, psychoanalytic, and humanistic movements, it has been “always grounded in a strong therapeutic, and soteriological impulse and has been particularly concerned with alleviating suffering on the individual, social, and ecological dimensions” (Lahood, 2007, p. 2). Consequently, in this chapter we explore the contributions a fractal epistemology might make to transpersonal psychology as a science and as a healing approach based on Marks-Tarlow’s work (2008; this volume, chapters one and nine), and tentatively explore the compatibility of Friedman’s self-expansiveness construct with a fractal epistemology. Marks-Tarlow (this volume, chapters one and nine) offered a way to undermine both the romanticism and scientism that plagues contemporary psychology generally and transpersonal psychology specifically, while being grounded in a solid-scientific approach to understanding the realm of the transpersonal. She offered a non-reductive approach that explores non— linear, complex informational dynamics that underlie and structure the material world. In so doing, she offered a bridge to a possible new paradigm within not only transpersonal psychology but also science and philosophy in general. She illuminated the terrain of the current paradigm shift (Kuhn,

1962)—as the paradigm is shifting, without offering a theory of everything (such as attempted by Wilber, 1995). Instead, she provided a window into

what the future could be, based on Mandelbrot’s insights about the nature, of nature.

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36

Mandelbrot’s relevance for ontology, epistemology, and a scientific transpersonal psychology Fred Abraham (this volume, chapter six) described the relationship between epistemology and ontology as a yin/yang type of entanglement: You can’t fabricate knowledge about reality unless you have some concept or commitment to the nature of reality; and your concepts about the nature of reality are under constant revision as you continue to investigate it. There is an ongoing dialogue between epistemology and ontology; thus, they are

parts of an organic, holistic process no longer to be considered separately.

(p. 187) Abraham’s subject is the neurodynamics of mind. However, what he says applies equally to the way that understandings about the material world

condition scientific inquiry. Ontologically speaking, Mandelbrot’s (1977) primary insight was that the nature of much of nature is fractal. Marks-Tarlow (this volume, chapter one) wrote that Mandelbrot “offered fractals as a framework for modeling aspects of nature previously considered too ambiguous, irregular, unique, discontinuous, or complicated for traditional mathematical methods” (p. 4). She further stated that, “Although each of us understands fractal patterns intuitively, in an embodied way, very few of us tend to ‘see’ them consciously” (p. 5). In fact, people are immersed in fractals, and see fractals all the time. If looking out a window at trees, which are fractal in structure, the issue is not that people do not see fractals or that fractals are simply a framework that overlays onto reality to model certain aspects of nature. The deeper reality is that much of nature in all of its reality is fractal, and that modern Western people have been taught to “un—see” fractals, from kindergarten onwards. Contrariwise, man-made reality has traditionally been non-fractal, bounded by Euclidean geometry contained in Platonic solids, which are artificial and constructed. Just about everything in nature has a rough edge. There are very few, if any, smooth lines. This is known not only intuitively but, also, experientially. In art classes, modern Western students learn how to represent reality the way it is commonly seen, using what Filippo Brunelleschi (1377-1446) taught the world about three-dimensional, linear perspective, and mixing colors to create depth. However, Euclidean geometry has been and remains the predominant way employed to map or draw in Western mathematical classes, using straight lines and curves to approximate or model reality. Without adequate mathematical language to

More than Merely a Model or Metaphor?

37

express a more complex reality, science is trapped in a hegemonic intellectual framework and limiting paradigm. Notwithstanding the political implications of education that supports existing power relations (e. g., Friere, 1970), or feminist and other critical theories of the way that reality is described (e.g., Plumwood, 1993), this limitation has had implications for science, philosophy, and technology in terms of the ability to advance their agendas. The point is twofold: 1) somewhere along the way, in all the sciences—including psychology, academics started confusing man-made models of reality, which were crude and over-simplified approximations of it, with what actually is; and 2) this tendency has limited the ability to both understand and manipulate reality. Marks—Tarlow (this volume, chapter one) explained this: We stern education has privileged linear lenses by highlighting straight lines and regular forms, such as Platonic solids and Euclidean dimensions. It is easy to remember elementary school activities of playing with such shapes for example, cutting out and pasting a larger triangle onto a smaller rectangle and calling it a pine tree. Yet, all the while, we could sense our productions as mere approximations of the real thing. (p. 5)

This insight is directly aligned with Mandelbrot’s (1977) introduction to The Fractal Geometry of Nature: “Clouds are not spheres, mountains are not cones, coastlines are not circles, bark is not smooth, nor does light travel in

a straight line” (p. 3). In chapter one of this book, Marks-Tarlow limned out how fractals (as structures and processes with rough edges, permeable boundaries, and holographic self-similar dynamic replication) are redolent in nature. It is not that fractals are a subset of the natural world: it is that pretty much everything in nature has more in common with fractals than with Euclidean shapes. Therefore, understanding how fractals work produces a better understanding of body/mind, as well as subjective experience in relationship to the objective world. With a better understanding of the dynamic functioning of the natural world, there is a greater possibility of developing technologies that preserve the environment instead of destroying it (Benyus, 1 997). It is important to emphasize that Mandelbrot’s contributions were not just theoretical mathematical abstractions: they were grounded in observable and observed reality in the physical world, such as understanding coastlines and turbulence (Mandelbrot, 1967, 1974, 1977). Mandelbrot’s genius was to get closer to the reality of the physical world, to know it so that it can be understood and manipulated, arguably job one of science. Mandelbrot’s

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(1977) creation of fractal geometry was a game-changer for science precisely because he discovered ways of understanding how the natural world works that had not theretofore been understood, surfaced, or articulated. Solving the equation for infinity, mathematically expressing the dynamics of infinite, self-similar complexity in a simple equation as the “DNA” of the natural world had practical technological implications. Without Mandelbrot, there would almost certainly be no Pixar animation or sophisticated bio-imaging technology because human circulatory, nervous, and respiratory systems all are more fi'actal than Euclidean in both structure and function. So, without ways to accurately and adequately represent them, namely without a deep and specific understanding of their fractal nature, such technologies would not exist. This leads to whether or not such a revolution in thinking has applicability for the types of problems faced by transpersonal psychology, many of which have thus far been intractable. ”What current scientific trends towards non-reduction and advances in areas such as epigenetics, chaos and complexity theory, fractal geometry, social neurobiology, and many more fiontiers have to teach transpersonal psychology is that natural does not need to be equated with materialist or mechanistic conceptions in any narrow way. Within transpersonal psychology, Ferrer (2017) approached this stance by advocating for a “more liberal open naturalism—one that is receptive to both the ontological integrity of spiritual referents and the plausibility of subtle worlds or dimensions of reality” (p. 2). Friedman and Pappas (2006) advocated a similar perspective in the context of advancing a transpersonal cartography of self-expansiveness as a “a unified field of the self/Self as based on both non-materialistic transcendent consciousness and materialistic existence” (p. 50). They wrote: Our relationship with the material world is thus conceptualized as interactive and overlapping systems within systems such that the world and the individual are in no defensible way distinct, but, rather, part of a unified system. (p. 45)

Central to Friedman’s self-expansiveness construct (Friedman, 1981, 1983, 2018; Friedman & Pappas, 2006, and his other works generally) is a dichotomy between materialist and non—materialist, transcendent and material, sacred and profane—all of which may be included in so-called reality. However, while the ontological existence of the non-dual, nonmaterial, spiritual, supernatural, transcendent, and so forth may be maintained as possible, even if to some implausible, it remains, according to Friedman, inaccessible to scientific, empirical inquiry—and even

More than Merely a Model or Metaphor?

39

phenomenological methods can only capture its traces. However, this does not foreclose on the possibility that advances, such as in technology, can possibly bring what is now seen as ineffable into a perspective amenable to scientific inquiry in the future. With Friedman, many adherents of the scientific approach in transpersonal psychology are mapping routes for the discipline to become more scientific that do not necessitate shifting epistemes. For example, Adam Rock and Charles Laughlin (2018) envisioned that path as adopting a neurophenomenological approach involving more qualitative, quantitative, and mixed-methods research studies on transpersonal topics, noting that 78% of papers published in the IntemationalJoumal of Transpersonal Studies were essays between 2008-2012. The call, a la Friedman (2002, 2013, 2018), is for more application of scientific methods to understand and elucidate the dynamics of transpersonal phenomena. Rock and Laughlin cite Friedman and Hartelius (2007) as champions of expanding quantitative research methods in transpersonal psychology. They also quote Leahy (1992) regarding why it’s difficult for transpersonal psychology to advance as a

solence: For scientific research to be progressive, the scientific community in a particular research area must agree on certain basic issues. Its members must agree on the goals of their science, on the basic characteristic of the real world relevant to their subject, and on permissible research methods and mathematical techniques. (Rock & Laughlin, 2018, p. 17)

It is in the area of agreeing on the basic characteristics of the real world that Mandelbrot’s work is instructive, and the area of permissible research methods and mathematical techniques that Marks-Tarlow (this volume, chapter one) offered a radical suggestion—to adopt a fractal epistemology in order to advance transpersonal-psychological science. Proposing a shift in episteme is not a trivial suggestion and would not be a minor addition to the conversation. This is not just looking at and learning from fractals— which is another avenue of research that can be pursued using existing scientific methods and without shifting epistemologies. It is not just identifying fractal dynamics in psychological phenomena. Rather, a fi'actal epistemology could substantively shift the focus in transpersonal psychology. Adopting a fractal epistemology may change the categories and the way they are treated in scientific analysis, as well as some of the major analytical tools and methods. In Marks-Tarlow’s view (this volume, chapter one), fractal geometry is key not just as a discrete subfield relevant to mathematicians, physicists,

40

Chapter Two

and neuroscientists that can be borrowed from or used as poetic inspiration. The power of Marks-Tarlow’s suggestion is that fractal geometry has the potential to offer transpersonal psychology the possibility of new epistemes to close the ontological gap between “metaphysics (what exists) and [by illuminating] the submerged epistemology (knowing what exists) interface between the unconscious and the nonlocal reality that defines the field of transpersonal science” (Shapiro, in press). Exploring and immersing in this new epistemology would put those who identify as transpersonal psychologists in the same conversation with other psychologists who are currently integrating physics, math, and complexity theory into their work (e. g., Guastello, Koopmans, & Pincus, 2009). A fractal epistemology may also allow for better mapping of the dynamics of transpersonal phenomena and processes. Refreshingly, Marks-Tarlow’s approach possibly could rescue the field from rehashing the same old debates constrained by Cartesian epistemology and mechanistic metaphors, and allow for new, truly generative and transfonnative, explorations.

Studying fractals in transpersonal psychology Before going to how shifting epistemology may transform transpersonal psychology, it is important to describe the lines of existing psychological research—transpersonal and mainstream—that study how fractals and fractal processes show-up and how they are involved in human brain structure, function, and information processing (e.g., Vandervert, this volume, chapter thirteen).This is a promising direction for mixed-methods research in transpersonal psychology, such as by employing neurophenomenology (Varela & Shear, 1999) to understand transpersonal phenomena (Laughlin & Rock, 2013) without adopting substantially new methods for analysis. At the forefront of this avenue of research related to fractals is Taylor’s work at the University of Oregon. Taylor, trained as a physicist and also an abstract expressionist artist, leads an interdisciplinary research network that investigates the positive physiological changes that occur when people look at fractals—specifically the fractals in Jackson Pollack paintings, which the group verified mimic fractals in nature (Taylor, Micholich, & Jonas, 1999, 2002). Their experiments over time have used eye-tracking equipment, as well as qBEG and fMRI imaging techniques, to measure brain activity when viewing fractals (Hagerhall, Laike, Kuller, Marcheschi, Boydston, & Taylor, 2015; Hagerhall, Laike, Taylor, Kuller, Kuller, & Martin, 2008; Spehar & Taylor, 2013; Taylor, 2006; Taylor, Spehar, Van Donkelaar, &

More than Merely a Model or Metaphor?

41

Hagerhall, 201 1). They have found that when people look at a specific form of fiactal found in nature and reproduced in art, their stress levels go down by up to 60% (Taylor, 2016). Taylor (2016) explained: This stress-reduction is triggered by a physiological resonance that occurs when the fractal structure of the visual system matches that of the fractal image being viewed. Our discovery that exposure to fractals automatically relaxes people holds crucial implications for society: the U.S. spends ever $300 billion annually on stress-induced illnesses... (para. 1)

The implications for this line of research are also huge vis a vis transpersonal psychology’s agenda of “human survival and betterment” (Friedman, 2002). For example, one of the Taylor group’s long-term goals is to collaborate with artists and architects to incorporate stress-reducing fractals into novel indoor and outdoor environments. Taylor (2016) explained: “These ‘biophilic’ fractals could be used in many applications, ranging from keeping astronauts calm on their long journeys into space to soothing anxious patients in dentist waiting rooms” (para. 2). Transpersonal psychologists might be interested in researching human responses to dynamic fractals, such as the Mandelbrot Zoom, which is available on YouTube in multiple forms (e.g., https://www.youtube.com/ watch?v=pCpLWbHVNhk). The Mandelbrot Zoom is a visual representation of the solutions to the equation,f(z)—> 22 + c, which Mandelbrot discovered, thereby solving one of the mathematical “monsters” of his day, the Julia Set (see Root, chapter eight, this book, for a thorough explanation of the relationship between the Cantor Set, the Julia Set, and the Mandelbrot Set). Wilcox and Combs (this volume, chapter ten) quoted James Gleick (1987), a science writer who makes chaos theory accessible to non-physicists, describing the Mandelbrot Set as follows: “An eternity would not be enough time to see it all, its disks studded with prickly thorns, its spirals and filaments curling outward and around, bearing bulbous molecules that hang, infinitely variegated, like grapes on God’s personal vine (p. 221).” For many in popular culture and parts of the psychedelic community, the Mandelbrot Zoom depicts an “eternally existing, self-reproducing, chaotic, and

inflationary universe” (Linde, 1986). For this reason, its nickname on the intemet is commonly the “Thumbprint of God.” A popular takeaway is that infinity is real and exists in a world that appears finite. As Jackson (this volume, chapter fifteen) wrote: “...knowledge of the scale of the universe and the depth of our psyches, fractal research is promising, still at the beginning of its career, continuing to uncover life-changing perspectives

and applications” (p. 451—452).

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Systems biology theorist, Kauffman (this volume, chapter five) described Mandelbrot’s equation and its implications. She wrote that the equation, (z)—> 22 + c shows the “forging (of) the exquisite fractal structures by feeding back into itself, squaring its own output solutions, adding with each iteration a new factor (C) that contains both a real and an imaginary componen ” (p. 171). She further speculated: What if the complex plane somehow captures the still mysterious process, force or mechanism that unites quantum and classical worlds? Might this suggest thatthe ubiquitous dance of complementary opposites might all flow fi'om a deeper interactive dance between quantum and classical realms themselves? Might it be that the infinity in fractals connotes a realm of ontologically real “quantum possibles” (Kastner, Kauffman, & Epperson, 2017), a fiilly interpenetrating, unifying, nonlocal realm, tucked everywhere within the fractal boundaries of the classical realm like the glorious fractal image of Newton’s method? (pp. 171-172)

Clearly this is topic for philosophical debate. It is a point of view that seems aligned with Ferret’s (2017) “soft naturalism,” for example. Could it also be a subject for scientific transpersonal inquiry? Could it auger a wholesale cultural shift in perception of what constitutes agreed-upon ontological reality? There might also be salutary applications of a fiactal epistemology to studying dynamic fiactals. For example, when people are exposed to ordered, infinite complexity via the Mandelbrot Zoom, they often describe what they experience and the effect of the experience in positive, perhaps even healing, ways. How is visually immersing oneself in the Mandelbrot Zoom similar to or different from other experiences that involve relaxing psychological resistance and surrendering to the moment, such as observing the ocean, ingesting psychotropic drugs, or doing holotropic breathwork (Grof & Grof, 2010)? Does viewing the Mandelbrot Zoom have the same calming effects as static fractals that mimic those found in nature? If so, what, if anything, does the experience of viewing the Mandelbrot Zoom have in common with so-called “transcendent experiences” (Wilcox & Combs, this volume, chapter 10) or similar experiences, such as “awe” (Bonner & Friedman, 2011)? These questions illustrate pragmatic possibilities for bettering the human condition by studying fractals’ impact on human psychology.

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Epistemological themes for a scientific transpersonal psychology of the future science will be limited to a naturalistic Transpersonal-psychological worldview, with naturalistic understood broadly and deeply in the sense of ontologically presupposing the “mind-body” existence of phenomena in a way that is “real.” Supernatural terms that are now all too often misused (and misunderstood when used), such as spirituality, soul, and even the nondual, can be bracketed as outside of the direct purview of science, while many of the phenomena to which they refer may be understood and explored as the result of natural processes. A fractal epistemology may provide a viable way to incorporate these scientifically—and enable them to be accessible via both theory and empirical research. We assume that many such phenomena may eventually be described, mapped, analyzed, understood, and manipulated. Transpersonal—psychological science proceeding this way would thus be empirical, inclusive, and integrative. It would explore questions and hypotheses that are testable, in accord with Sulis (in press): “[T]ake notice, and build theories with these ideas and then develop tools to test them out in the real world” (n.p.). These emerging fractal epistemes of transpersonal psychological science have four distinct characteristics or themes, which are also simultaneously and currently emerging from and within sciences, social sciences, and popular culture. They would: 1) be meta/trans-binary; 2) utilize a dynamic-systems approach; 3) have a decentralized focus, such as on borders and boundaries; and 4) reposition and revalue transpersonal phenomena in the emerging epistemological fi‘ame.

Meta/Trans-binary Following Shapiro (in press) in affnming the possibility of “reintegrating transpersonal phenomena into the realm of the natural sciences, and expanding the reductionist paradigm to incorporate multi-level emergent complexity in Nature” (up), one step (not to say necessarily the first step) would be to avoid focus on binaries as opposite and distinct polarities. Transpersonal psychological science could track with phenomena in the natural world, which can be viewed through a binary lens, but which do not need to be understood in dualistic ways, as opposites. This meta/trans—binary stance would be consistent with the way many young people now are experiencing the world into which they are born. In

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Western popular culture, binaries seem to be eroding and giving way to more complex understandings of phenomena that have traditionally been understood in dualistic (and hierarchical) terms. Sex and gender are one example of this trajectory — but arguably, supernatural and natural, nature and culture, science and religion are all equally implicated. We describe the relationship of sex and gender, society/ culture, and science at some length as an example of the emergence of a meta/trans-binary landscape. Traditionally, newborns have been assigned a sex at birth —either male or female. Gender, the social category based on biological sexual markers, has flowed from that assignment and has been understood as relatively immutable. Gender has circumscribed social roles, within the patriarchal context. Psychological theory and praxis have tracked with these and assumed a role of assisting people with adapting to and reconciling their selves with this reality. For example, theories like Freud’s penis envy were directly related to the binary-hegemonic episteme and the logical possibilities that inhered from it. Over the last 50 years, sex and gender have become increasingly complex: feminist discourses and activism have been unraveling and severing the tethers between gender and socio-cultural destiny; the dualisms of male/female and feminine/masculine have been questioned and deconstructed; queer and trans individuals and communities are challenging both the immutability of sex assignment at birth and the assumed correlation between biology and gender while exploring how to shift their gender and sexual identity within and outside of the binaries. Examples of other cultures that have dealt with gender differently have resurfaced. For example, the two-spirit reality of Native American cultures, where, at point of colonial contact, five genders were recognized: male, female, two-spirit male, twospirit female and transgender (Brayboy, 2017). Science too, has been questioning the binary of sex and gender since at least 1968, when biologist Moore (1968), in the context of rules for Olympic sports, identified nine different components of sexual identity. The biological complexity is being reaffirmed with acknowledgement of the existence of millions of intersex individuals (Ainsworth, 2015), which problematizes the binary by showing that anatomy, hormones, cells, and chromosomes are all involved in “sexing the body” (Fausto-Sterling, 2000). In short, there has been growing awareness that Western culture has reinforced its binary sex and gender taxonomy, despite scientifically knowing that it is insufficient for describing the reality that exists, and that there are better descriptions of what exists (i.e., more accurate, and therefore

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also more helpful and less traumatizing). As the socio-cultural landscape changes, psychology must adapt as well, such as in Hyde, Bigler, Joel, Tate, and Anders’ (2019) article “The Future of Sex and Gender in Psychology: Five Challenges to the Gender Binary” in the American Psychologist, in which they argued that, while binary gender has shaped the history of psychology as a science, scientific evidence undermines the gender binary as physiological reality. Rather, gender is culturally determined and malleable and, within the current sociocultural landscape, relying on the gender binary has significant costs. With respect to duality, transpersonal psychological science should not “throw the baby out with the bathwater.” Since much Western philosophical and scientific knowledge has been produced within a Cartesian episteme, and, perhaps therefore, binaries are foundational to human experience (e. g’., Kauffman, this volume, chapter five), it is important to consider, experiment, and construct with duality. The first logical step in moving beyond a two-handed binary worldview as part of afractal epistemology is embracing a third hand, as in the dialectic approach of thesis/anti-thesis/synthesis, recognizing that whenever there is two, there is the possibility of three and from there, fractal replication into infinity. As Root (this volume, chapter eight) quotes the Tao Te Ching: The Tao beget one. One beget two. Two beget three. And three beget the ten thousand things. (Lao Tsu, Tao Te Ching, translated by Feng & English, 1972, Chapter 42)

An example of this meta/trans—binary stance with relation to transpersonal psychology and gender would be Sell’s (2001) article entitled, “Not Man, Not Woman: Psychospiritual Characteristics of a Western Third Gender.” Another example from the transpersonal psychology literature might be research utilizing Ferrer’s (2018) schema for understanding the existing range of intimate relationships. Ferrer wrote: While the traditional understanding of monogamy and nenmonegamy as polar opposites has been deconstructed, the conceptual and experiential territory beyond the non/monogamy system (and attendant mono/poly binary) has not been systematically discussed. This article considers three that plural relational modes fluidity, hybridity, and transcendence disrupt and arguably outdo the non/monogarny system. Moving beyond the mono/poly binary opens up a fuzzy, liminal, and multivocal semanticexistential space this article terms nougamy. After describing several

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transbinary pathways, the article concludes by sketching a critical pluralist approach that eschews universal sequences or hierarchies among monogamy, nonmonogamy, and nougamy, as well as provides tools for the making of qualitative distinctions both within and among relational style. (p.

3) A fractal epistemology deals differently with dualism, attempting not to reduce the complexity of a phenomena to simplified binary descriptions. It recognizes the continuum between polar opposites. Within a fractal epistemology, dualisms are contextualized and situated in the larger context of anon-linear dynamic system with multiple levels and the boundaries and borders between what are understood to be artificially, oversimplified categories—including dualisms. These can then be investigated, explored, traced, mapped, plumbed, and so forth.

Dynamic systems approach There is a larger context that contains transpersonal phenomena and everything else (including the natural world and perhaps some or all of what may be currently understood as supernatural). Within a fractal epistemology, this larger context may be understood as complex, non-linear, dynamic, multi—level, and fractal—that is self-similar, iterative and recursive. Logically, if natural laws represent the qualities of dynamic/non—linear systems, then they can be grasped at any level of any natural system. Applying the science of how systems work to transpersonal psychology, Shapiro (in press) wrote: From a non-dualist vantage point, brain/mind is seen as a unified psychobiological system nested mid-way within a hierarchy of selforganizing complexity extending from quantum to atomic, molecular, cellular and neural networks on the lower levels to individual, group, cultural, ecological and technological processes. Each level is manifested by the emergence of qualitatively novel processes absent at lower levels of organization such as temperature, rigidity, or super conductivity as a function of collective behavior of a large quantity of atoms that cannot be reduced to phenomena at atomic or subatomic scales (Anderson, 1972;

Laughlin, 2005). In a similar way, it would be meaningless to discuss higherlevel interpersonal or cultural phenomena,

such as attachment

or

mythology, in terms of quantum or neural network interactions, even though no known cultural process can arise without them [italics added]. The ‘horizontal’ causal loops operating on each level have to be complemented with ‘vertical’ between-level

causal interactions,

which involve both

reductive bottom-up and emergent top-down causation. . .. (we can understand)

47

More than Merely a Model or Metaphor? living organisms as systems of systems what happens or is expressed in one level has a fractal correspondence with what happens

may happen or

is expressed in a different level or scale. Such a fiamework may help us expand the reductive paradigm in the natural sciences in order to incorporate “holistic” phenomena characterized by emergent properties at higher orders of complexity, such as living, conscious, and entangled systems that cannot be understood by analyzing their constituent components alone. ( u p )

For Shapiro (in press), reality is organized in levels, and different processes occur at different levels. The processes contained within each level operate in the same basic ways—they show fractal properties of selfsimilarity and scale invariance. However, it is not fruitful (or possibly meaningless and even dangerous) to try to reduce phenomena at one level to those of another. What may be extraordinarily fruitful, and even therapeutic, is applying what is known about how non-linear, dynamic systems operate to understanding dynamics of the psychotherapeutic process (Marks-Tarlow, 2008; this volume, chapter nine) and to understanding psychological concepts, such as “self’ (Marks-Tarlow, 1999, 2010). Marks-Tarlow (1999), in her article entitled “The Self as a Dynamical System,” wrote: Ln the model proposed in this paper, psychological health resides at the edge a healthy organization of self displays optimal environmental of chaos sensitivity and behavioral flexibility. The ability of healthy individuals to adapt to changing situational contexts is consistent with situationalist thinking, such as Mischel (1968, 1984) or Bandura (1969), as well as with social role theory (for a summary, see Biddle & Thomas, 1966).... Perhaps

a dynamical systems model can help resolve the old dispute between dispositionalists, who focus on internal stability, and situationalists, who focus on cross-situation inconsistency. (p. 335)

What can be seen in this quote is also how a dynamic-systems approach emanating from a fractal epistemology can resolve debates within a discipline when those debates rely on a binary-dualistic opposition. A parallel in the world of neuroscience is in accounting for free will and creativity. While it is beautiful to be able to understand that human improvisation and creativity might occur through the communication between the cerebellum and the cerebral cortex, maybe via a fractally structured sequence generator (Vandervert, this volume, chapter thirteen) and that the fractal nature of hallucinatory phenomena in psychedelic experiences may be mirrors of neural dynamics related to the structure and function of the brain (S ema, this volume, chapter sixteen), it is not necessary or desirable to jump levels and reduce creativity or transcendent insight to

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an epiphenomenon of neural functioning. This type of epistemological move happens when the dynamics of causality between levels of a dynamic system are confused or reduced. Shapiro explained this as follows: The problem in reductive science is that only bottom-up causal loops are

considered to be valid/efficacious, while top-down or emergent causation is considered to be an epiphenomenon (which is why there is no such thing as “free will” in reductive neuroscience). So, biology can’t impact on physical

laws, psychology can’t impact on biology, social processes can’t affect physics or biology

all should be reduced to quantum/physical phenomena

and analyzed from that perspective. (personal communication, September 18, 2018) In constructing a rigorously scientific framework for psychological science based on a fractal epistemology, it is important to accurately depict the complexity of top-down and bottom-up impact across levels of the system. Both top-down movement and bottom-up movement and causality can be emergent; both can also be reductive (Scott, this volume, chapter four). Moreover, the action may occur in both vertical directions (across levels) and horizontal directions (within levels) simultaneously and at all times. That is why it is called complex, and the good news is that these types of dynamics can be modeled without being reduced or oversimplified. A practical concrete example of how a fractal epistemology and nonlinear dynamic systems approach in transpersonal psychology can potentially integrate and resolve perspectives is Dale’s (2013, 2014) work on Piaget and transpersonal development culminating in his 2014 book,

Completing Piaget’s Project: Transpersonal Philosophy and the Future of Psychology. In his review of Dale’s book, Ferrer (2015) wrote: One important upshot of Dale’s nonlinear transpersonal paradigm is that it arguably resolves important disputes in the transpersonal literature, in

particular those around the competing developmental models of Washburn (1988, 1990, 1998, 2003) and Wilber (1990, 1995, 1999, 2001). If transpersonal development is both nonlinear and pluralistic, then there is no needto choose between these supposedly conflicting developmental models; instead, both can be recognized as (at least potentially) equally valid accounts of different individuals’ developmental pathways. To illustrate this

diversity, Dale described five transpersonal developmental patterns associated with different transpersonal theories such as Wilber (1996), Assagioli {1988/2007), Hunt (1995), Washburn (1988), and Maslow (1971), among others as well as various triggers of transpersonal growth such as meditation practices and psychodynamic maturation. (p. 125-126)

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A fractal epistemology is grounded in understanding of non-linear dynamic systems. This worldview is compatible with the genesis and underlying assumptions of transpersonal psychology and will assist the discipline going forward with accurately describing and modeling, and ultimately understanding, its subjects of interest. New methods may be needed to study the complexity, variation and

fractal dynamics of transpersonal experiences. Marks—Tarlow (1999; 2008; 2011; this volume, chapters one and nine) suggested the applicability of power laws in physics for modeling fractal dynamics. This adds to the existing methodological toolbox by allowing expansion beyond normative statistics to interpret data, because normative statistics are inappropriate for and ill-suited to understanding complex systems. A similar insight has been previously elaborated at length with respect to the human body and the medical field (W est, 2006). This has an extremely relevant impact when it comes to understanding transpersonal phenomena, as Marks-Tarlow (this volume, chapter one) wrote: Unfortunately, normative statistics contain underlying assumptions that often prove to be false within most complex systems... One underlying

assumption that frequently proves false is the requirement that all underlying variables operate independently (or orthogonally) from each other. Because transpersonal psychologists are so often interested in idiosyncratic

states and non—repeatable circumstances that have nothing to do with central tendencies, the poor fit between normative statistics and phenomena of interest may be particularly exaggerated within this subfield of psychology.

With chance and randomness part of natural fractal fluctuations, we begin to understand how fractals help us to model transpersonal phenomena that are fundamentally unpredictable, yet simultaneously ordered. (p. 25)

The insight that, in the power law distributions common to complex systems, the power is not in the center but in the tails (Schroeder, 1991), assists with describing empirical regularities in seemingly random phenomena. Power laws have been used successfully in other social sciences, such as economics, to find regularities and predict behavior in diverse areas, such as wealth distribution, stock market returns, and executive pay—even when the regularities lack an empirical explanation (Gabaix, 2009). This allows transpersonal psychological science to admit unpredictability while not only allowing, but investigating, exploring, and honoring variability. Pincus, Cadsky, Berardi, Asuncion, and Wann’s (2019) article entitled “Fractal Self-Structure and Psychological Resilience” is an example of the type of research that can be done using power laws to investigate psychological phenomena. The study measured reaction time of

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subjects answering questions about themselves using MMPI—Z to answer the question of whether or not the personality appears to be fractal in structure, as defined by an inverse power law (IPL) and whether the rigidity of the [PL was related to psychopathology. Pincus et al. found that their results supported recent theories and empirical evidence that the personality is a self-organizing system and that the structure of the self is a complex, nonlinear, dynamic network producing fi'actal outputs. Further, higher levels of rigidity, which is more linear and less fractal, “self-organizing at the edge of chaos” (Marks-Tarlow, 1999) did seem to predict psychopathology. Besides power laws, there are other methods to identify fractal dynamics in psychological phenomena. A technical example, Delignieres, Fortes, and Ninot (2004) used Rescaled Range Analysis, Dispersional Analysis, and Scaled Windowed Variance Analysis to identify fractal patterns in selfesteem and relation of individuals to their physical selves. Recognizing fractal patterns and knowing how fractals behave may allow transpersonal psychology to better identify, focus on, and map them in transpersonal experiences. Sulis (this volume, chapter seven) wrote, “whenever fractal structure appears, one can rule out a linear deterministic process as the generator of such behavior” (p. 240), while Wilcox and Combs (this volume, chapter ten) explored boundary conditions of individual consciousness during non-drug-induced so-called transcendent experiences, comparing them with other natural boundary conditions, and identifying patterns that can be found at far-from-equilibrium conditions. Wilcox and Combs proposed that: ...(C)onsciousness seems to behave in a nonlinear fashion, as characterized by the fact that small changes in inputs often create disproportionately large and unpredictable changes in outputs. Considering the common feature among physical dissipative structures, namely that they spontaneously reorder themselves at far-from—equilibrium conditions, it is reasonable to speculate that the boundaries of individual consciousness can also reorder under the seemingly far-from-equilibrium conditions of transcendent experiences. Thus, the threshold between ordinary individual experience on the one hand, and nondua] consciousness on the other, can be understood as dynamic and responsive, potentially allowing for, and participating in the sense of relaxation and unfiirling that characteristically merges the

individual with the transpersonal. (p. 306)

They concluded that, in mystical and other so—called transcendent experiences, the boundaries of the self may exhibit non-linear dynamics,

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and the perception of dissolution of ego or self may be better understood as a fi'actal unfurling into an infinite and porous boundary. Focus on boundaries and borders One of the most important ways a fractal epistemology encourages interacting with dualism and separate, man-made, artificial categories is by investigating the nature and function of boundaries. Instead of focusing on the centers and contrasting and comparing them, assuming their ontological separateness—the focus can be equally on the similarities dwelling in their permeable, interpenetrating boundaries and how the opposites replicate in fractal patterns. Think of the yin/yang symbol as a proxy for any of the constraining dualities: objective/subjective, mind/matter, feminine/masculine. Instead of exploring yin versus yang, and classifying things into those boxes—and documenting the similarities and differences—focus instead on the boundary/border between yin and yang, and the way the yin/yang as a dynamic complex system reproduces itself as being self—similar and scale invariant. It is likely that the border between yin and yang is actually permeable, that yin and yang interpenetrate and that the boundary between the two seeming opposites approaches infinity in its depth. It is possible to study how yin and yang hold together perhaps through a process akin to quantum entanglement (Abraham, this volume, chapter six). Using a fractal epistemology results in asking different questions and being interested in different answers: it could drive focus on and attention to connections, complexity, and overlaps rather than develop models to test hypotheses that document means and norms and central tendencies as the defining characteristics of what is real and what can be empirically studied by rigorous scientific methods. For example, returning to the discussion of gender binaries, Galatzer— Levy (this volume, chapter eleven) wrote: A huge literature describes both the inadequacy of the idea and the

oppressive consequence of its adoption. In addition to its oppressive nature, the gender binary asserts the existence and normality of a sharp distinction between masculinity and femininity. This conceptualization interferes with

the intellectual and aesthetic possibilities that result when the boundary between masculinity and femininity is conceptualized as fractal. (p. 350)

is When the boundary between masculinity and femininity conceptualized as fi‘actal, it is treated as porous, permeable, complex, and

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dynamic or continually emerging. Confusion, difficulty, and psychological distress result fi'om oversimplifying and reifying the distinction. Another example relevant to transpersonal psychology is MarksTarlow’s (this volume, chapter nine) exploration of the fiactal boundaries and dynamics present in dreams, in interpersonal synchrony, and in synchronicity—and how understanding these as such assists with clinical intuition and psychotherapy. She attended to two key-fractal features: selfsimilarity” and “paradoxical boundaries between self and other and self and world that are simultaneously open and closed and observer dependent” (p. 292). Marks-Tarlow explored the notion of the relational unconscious that emerges between patient and therapist with synchronic, overlapping dreams and illuminated the possible dynamics of this phenomena by referencing recent advances in social neurobiology that use hyperscanning to demonstrate how “interpersonal synergies span organisms by extending beyond boundaries of skin” (p. 292). With relation to self-world boundaries and synchronicity, she posited “acausal” connection as an underlying driver: When the pattern of the whole is present in its parts, we must embrace more complex models of causality and relatedness, including the notion of acausal

connection. With acausal connection, the “glue” between parts is not based on a linear or temporal chain of events. Because fractal patterns exist outside of any particular time or size scale, fractal elements connect with one another instead through self-similar symmetry. In the case of fiactals, acausal connection preserves fundamental identity of the underlying wholeness by

permeating fractal parts. When acausal connection takes the form of synchronicity, the identity of the whole serves to unify inner and outer worlds, spirit and matter. When two seemingly unconnected things happen simultaneously, they can be acausally connected to one another through hidden self-similar channels of meaning. (p. 294-295)

Acausal connection is an example of a way of theoretically conceptualizing and describing the relationship between transpersonal and interpersonal which may be able to be verified scientifically.

Reposition and revalue so—called transpersonal phenomena within the emerging epistemological frame Marks-Tarlow (this volume, chapter one), along with many other and perhaps most, psychologists—including some transpersonal psychologists, seems to accept as axiomatic the dogma that transpersonal phenomena are anomalous, rare, unpredictable, idiosyncratic, unrepeatable, unreproducible, as well as amorphous, ineffable, and ambiguous. She added the insight

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that, notwithstanding these characteristics, they may also be ordered and understandable, as are chaos and fractal dynamics. But, what if all of these adjectives describing transpersonal phenomena are not accurate in the ordinary lives of ordinary people, including scientists, but only anomalous within the context and fiamework of reductive, materialist science? Adopting a fractal epistemology allows transpersonal psychologists to ask this question, to consider the possibility that transpersonal phenomena are actually the canary in the coal mine of mainstream psychology. These phenomena could be similar to what are called mathematical “monsters” that threaten to expose the current epistemology underlying orthodox science as unsatisfactory, partial, incomplete, and fundamentally misguided, therefore unhelpful for creating definitive knowledge about the human psyche. If bringing transpersonal phenomena into the mainstream of psychology requires reconceptualizing the whole, that seems a significant contribution to the enterprise. As Sulis (this volume, chapter seven) wrote: “Transpersonal psychology could focus on transpersonal experience as utilize such experiences as a normative, creative, and adaptive, and resource...” (p. 231). After all, there is ample evidence to support the assertion that so-called anomalous, transpersonal experiences are prevalent in cultures worldwide, and it is only a Western psychological bias that pretends otherwise. FlorHenry, Shapiro, and Sombrun (2017) wrote: ...(A)nthropological studies show that of the 488 societies studied worldwide, over 90% found to have an institutionalized form of ASCs [altered states of consciousness], and 57% of these were possession trances (Oohashi et al., 2002). A recent cross-sectional study of normal British population (Pechey & Halligan, 2012) estimated the prevalence of anomalous experiences at 48%. Culture-bound religious and/or spiritual beliefs are a major factor determining their specific context. Within a

particular social group, altered states can be experienced as either pathological (such as possession trance) or beneficial (such as meditative and shamanic practices). In fact, from the perspective of some meditative

practices such as Zen Buddhism, it is the normative “dual” state of consciousness that fragments our perception into inner and outer reality and contributes to suffering and psychopathology. The expressed goal of many meditative practices therefore lies in achieving a non-dual state of unitary awareness (Berman & Stevens, 2015). Conversely, the Western rational tradition tends to pathologize ASC phenomena based on reductionist and behaviorist medical models, although there has been an increasing number of anthropological and neurobiological studies in recent years attempting to

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clarify their normative phenomenology and neurobiological mechanisms (Iilek, 2005; Vaitl e t al., 2005). (p. 4)

This means that the transpersonal psychological science of the future may no longer need to advocate for the existence of transpersonal phenomena. It may also need to decouple itself from the concept of “supernatural.” As Shapiro (in press) wrote: “There are no ‘supernatural’ phenomena in the scientific domain; both subjective and transpersonal experiences are natural processes [italics his], which must work in compliance with natural laws, even though our current understanding of these laws is necessarily limited” (n.p.). We choose, however, to adopt an agnostic stance regarding the existence of phenomena beyond nature and therefore beyond science, understanding that much once considered supernatural has later come to be accessible to scientific investigation (see

Radin, 1997, 2006, 2013, 2018). Since a fractal epistemology is potentially nested within this ontology, it can greatly assist the advancement of a scientific transpersonal psychology that better approximates reality by allowing for the elaboration of theories that can be deployed to interpret such data, such as non-reductive firstperson neuroscience. For example, Flor-Henry et al. (2017) conducted the first neurophysiological study of a self-induced trance—by a female subject, the first Westemer to achieve the status of shaman in the Mongolian tradition. Using EEG and LORETA (low resolution electromagnetic tomography) to study brain changes during her trance, they found that the shamanic state of consciousness (SSC) involves a “shift from the normally dominant left analytical to the right experiential mode of self-experience and from the normally dominant anterior prefrontal to the posterior somatosensory mode,” and one of their motivations was to better understand how changes in consciousness that accompany trance can shed light on the neural networks contributing to the “experience of autobiographical self; the subjective demarcation of ‘self’ from others and reality at large; and normative vs. pathological domains of self—experience” (pp. 2-3).

While identifying neural pathways is one direction for situating and elaborating the mechanisms of transpersonal experiences, another is theorizing along pan-psychic and coo-psychological lines. For example, Kauffinan’s (this volume, chapter five) embrace of Theise and Kafatos’ pan-psychic model of S/self: In their model everything occurs within a fundamental monistic (“nondual”) awareness; what {might call the Self with a capital S and others may call God. They describe a mathematical symmetry-breaking dynamic wherein

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the unbounded Self, can parcel itself into infinitely many local subject/obj ect subdivisions, but forging local and relative self/not-self boundaries. Best of

all, their model notes three universal components that occur on all levels of scale: 1) Interactivity, between parts and wholes; 2) Complementarity, (our dance of Yiaang opposites), and 3) ongoing, Recursion the iterative, self-reflexive, cyclic nature of feedback the engine driving the creation of fractal structures. In such a scenario the deepest fundamental comparison in

the loop of mind, might actually be the Self / Not—Yet— Self possibilities giving quite literal meaning to the imperative of Self-actualization. Indeed, beneath the level of the living system, the imperative for stable selfpreservation is meaningless, as form itself emerges from the deeper creative dance of change. All that remains is the developmental regime and the positive emotional spectrum, a possible source of ecstatic bliss of “nonbeing” or of the God as Love metaphor. (p. 177)

There is excitement in the possibility of dispensing with the necessity of proving the existence of transpersonal experiences or treating them as beyond the reach of empirical scientific inquiry. Within the emerging fractal episteme, the boundary between transpersonal psychology and other scientific sub-disciplines—as well as between transpersonal and mainstream psychology—may be seen as fuzzy, permeable, open, and overlapping, and that the border approaches infinity in its depth. Ubiquitous fi'actal processes found in nature unite disparate fields of academic inquiry as can be seen in this book, as well as in the concurrent (Marks-Tarlow & Friedman, in press)

special issue of the International Journal of Transpersonal Studies where Marks-Tarlow simultaneously advances her radical proposition.

Self-expansiveness As has been evident in this chapter, the nature and boundaries of self and the relation of that self to “not self” (e. g., others and the rest of existence) is a topic of interest to many psychologists and theorists. A fractal epistemology has been demonstrated to be of potential value for explaining how the self may function. Transpersonal psychologists are particularly interested in accounting for experiences where the sense of self is not experienced as limited to being within the physical body—such as in some alternate states of consciousness, including those induced by psychedelics or holotropic breathwork, trances, near-death experiences, mystical experiences, and psi. In this regard, Friedman (1981, 1983, 2018) has worked extensively over many years with developing the construct of “selfexpansiveness,” and a related approach to measuring that construct, notably the Self-Expansiveness Level Form (SELF). Although this construct and measure were not explicitly developed from a fractal perspective, they are

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consistent in many ways with a fractal epistemology, and they align with much of how transpersonal science has been conceptualized from a fractal vantage in this chapter. The construct currently relies on a distinction between aspects of the natural world with which one can identify as being part of, or even the same as, one’s self as depicted on a cartography or map of all that exists in time and space (see Figure 2-1). The relationship between what is depicted within the boundaries of the map and what is outside of these boundaries portrays an opposition between what is natural and what might possibly be beyond nature (or “supernatura ”). Seen through such a unifying way, namely by being depicted on the same map—although the natural is explicit and the possible supernatural is only implicit, self-concept in its furthest levels of transpersonal self-expansiveness can be seen as non-binary and integral (Friedman & Pappas, 2006). The self—expansiveness model identifies three levels of self—concept, namely personal, middle, and transpersonal, depicted across both spatial and temporal dimensions. This model is compatible with an ontology of a complex, non-linear, dynamic system, as it can be seen as both top-down and bottom-up. Although the word fractal is not used in the description of the boundary between self and non-self, the construct is based on an assumption of a relationship between self and non-self as “inherently unlimited to such an extent that all absolute distinctions between the two are untenable. The self is thus seen as inextricably embedded in the universe .. .”

(Friedman, 1983,p. 38). Further, the boundary between transpersonal and personal concepts of self is mediated by the middle level—a border that is permeable, open, and allows for interpenetration between the transpersonal and personal levels of self-concept, in other words, a fractal boundary. In fact, the very definition of the transpersonal self-concept in this model asserts that there is a dissolution of persons’ perception of themselves as isolated individuals existing only in the present time and space. Thus, it tacitly assumes that all levels of self-concept and all modes of relating to the world are potentially contained within every individual at all times, even ontologically, such that no individuals actually exist separate from the whole, despite their seeming separateness outside of rare occasions of self-realization. The premise of the SELF is that, while the metaphysical question of ultimate reality may be

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inaccessible to scientific inquiry, the subjective perception of the selfccncept is. This is in synchrony with Wilcox and Combs’ [this volume, chapter ten} description of the topology of self-dissolution in far-fictitequilibrium conditions and with Marles-Tarlow’s [1999} description of the self as a non-linear dynamic system. Using a trayel metaphor,they focus on describing the process of self-expulsion, perhaps seen as a journey through Various alternate states of consciousness,

while Friedman’s construct

measures the trait of self—concept in terms of a stable structure present at each destination and even multiple destinations. Cln the final characteristic of a scientific transpetsonal psychology emerging from a fractal epistemology, the self-expansiyeness construct neither requires supernatural, such as spiritual explanations,nor does it deny these as possibilities. Instead, it works with what exists in nature while avoiding arrogantly proclaiming

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that there is no possibility of anything more or different, sidestepping potential quandaries that have stymied generations of scholars. There are a myriad of ways that future research could utilize the SELF model in a fi'actal way to examine the borders between transpersonal and other levels of self-concept. While the SELF is currently a trait rather than a state measure, it could be adapted so that neural correlates of individuals’ travel between personal, middle, and transpersonal levels could be explored similar to the way F lor-Henry et al. (2017) mapped the progression into a Shamanic trance state. Mental health/illness, parapsychology, and religious experience are additional domains for which the SELF model could be useful. For example, researchers could study the relationship between trans— personal self-expansiveness and other variables, such as positive well— being, psychotic patterns, parapsychological phenomena, meditation, or other intentional practices to alter consciousness. Finally, in accord with the theme of this chapter—and recursively in this book in which it is contained, se]f—expansiveness itself could be explored as a fractal process.

Conclusion Transpersonal psychology as a science does not need to posit anything other than the natural world as the context for transpersonal phenomena, yet it does not have to deny other possibilities. It could embrace complexity, dynamism, and non-linear non-reductionist scientific explanations, including those aided by fractal insights and methods. It could allow its own borders and boundaries to be penetrated and permeated, as well as seek connection with the larger community of scientists and psychologists. Then, it can become a self-described “transformative psychology of the whole person in intimate relationship with a diverse, interconnected and evolving world” (Hartelius et al., 2013, p. 14). In this regard, fractals can offer immense potential for the future of a scientific psychology that bridges the

personal with the transpersonal. This does not necessarily imply, however, that fractals or a fractal epistemology—or even non-linear dynamic-systems theory—can ever fully explain all that is transpersonal. Psychological science, including its transpersonal sub-discipline, strives for so-called objectivity while sometimes incorporating so-called subjective accounts of what is, the nature of nature. This is not meant to imply any firm demarcation between subjective and objective accounts in science, because all experience, including scientific data derived in what might be called an objective manner, are still

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accessed through a subjective lens of human experience. The value of such subjectivity is found in many areas of psychology, such as within cognitive neuroscience where, through using subjective reporting combined with objective technology, scientists have been able to better map neural circuitry. However, there are still aspects of subjective experience that appear unknowable except by their reported impact on a participant or, in the case of work in areas like social neurobiology, by interviewing all involved participants. Fractal epistemology will likely allow for some aspects of transpersonal experiences to become more accessible and, perhaps more predictable as the mechanisms underlying their production are exposed. At the same time, there will likely remain an ineffable terrain, mysteries that cannot be completely solved through empirical-scientific investigation. Whether the so-called supernatural is conceptualized as existing in a separate realm or at a level of reality nested in a larger-dynamic system impacts its theoretical accessibility to scientific exploration. Fractals, both as static forms and as dynamic processes, clearly provide a powerful-mathematical model to understand and explore the so-called objective exteriors related to the transpersonal arena, and possibly to bring more so-called subjective interiors into this domain as well. Fractals also provide a compelling metaphor to comprehend how the inner and outer might contain each other, and together constitute a more integrated whole. However, are fractals more than merely a model or metaphor, insofar as they might be able to contribute to the interiors, not just exteriors, that are essential to much of transpersonal psychology? In this regard, we are hopeful for the many contributions a fractal epistemology might add to transpersonal psychology, but we are also guarded in our prognosis for fractals explaining the very qualia of experience that lie at the heart of the so-called hard problem of consciousness (Chahners, 1996), including the many transpersonal phenomena that might be called mystical, non-dual, transcendent, and unitive. 7

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References Ainsworth, C. (2015). Sex redefined. Nature, 518, 238-291. Beny'us, J.M. (1997). Biomimicry: Innovation inspired by nature. New York, NY: HarperCollins. Bonner, E., & Friedman, H. (2011). A conceptual clarification of the

experience of awe: An interpretative phenomenological analysis. The Humanistic Psychologist 39, 222—235. Brayboy, D. (2017). Two spirits, five genders, one heart. Indian Country

Today. Retrieved from: https:l/newsmaven.io/indiancountrytoday/archive/two-spirits-one— heart-five—genders-9UH_xnbfVEWQHiNn0rQQ/’

Chalmers, DJ. (1996). The conscious mind: In search of a fundamental theory. Oxford, England: Oxford University Press. Dale, E.J. (2013). Neo-Piagetian transpersonal psychology:

A new

perspective. Journal of Transpersonal Psychology, 45(2), 118—138. —. (2014). Completing Piaget’s project: Transpersonal philosophy and the future of psychology. St. Paul, MN: Paragon House. Daniels, M. (2013). Traditional roots, history and evolution of the

transpersonal perspective. In H. L. Friedman & G. Hartelius, (Eds), The Wiley-Blackwell hancflaook of transpersonal psychology (pp. 23-43). West Sussex, England: John Wiley. Delignieres, D., Fortes, M., & Ninot, G. (2004). The fractal dynamics of

self-esteem and physical self. NonlinearDynamics, Psychology and Life Sciences, 8(4), 49-510. Elmer, L., MacDonald, D., & Friedman, H. (2003). Transpersonal psychology, physical health, and mental health: Theory, research, and

practice. The Humanistic Psychologist, 31, 159- 181. Fausto-Sterling, A. (2000). Sexing the body: Gender politics and the construction of sexuality. New York, NY: Basic Books. Ferrer, TN. (2015). Neo-Piagetian transpersonal psychology: Dancing inbetween pluralism and perennialism. An essay review. Journal of

Transpersonal Psychology, 4(7), 124-138. —. (2017). Participation and the mystery: Transpersonal essays in psychology, education, and religion. Albany, NY: State University of New York Press. —. (2018). Beyond the non/monogamy system: Fluidity, hybridity, and

transcendence in intimate relationships. Psychology & Sexuality, 9(1), 3-20.

More than Merely a Model or Metaphor?

61

Flor-Henry, P., Shapiro, Y., & Sombrun, C. (2017). Brain changes during a Shamanic trance: Altered modes of consciousness, hemispheric laterality, and systemic psychobiology. Cogent Psychology, 4(1), 1-25. Fracasso, C., Franco, Z., MacDonald, M., & Friedman, H. (2011). Measuring the transpersonal: The research and mentoring contributions

of Harris Friedman. NeuroQuantology, 9(3), 499-514. Friedman, H. L. (1981). The Selxpansiveness Level Form.'A conceptualization and measurement of a transpersonal construct. (Unpublished doctoral dissertation). Georgia State University, Atlanta, Georgia. —. (1983). The Self-Expansiveness Leve1 Form: A conceptualization and measurement of a transpersonal construct. Journal of Transpersonal Psychology, 15, 37—50. —. (2002). Transpersonal psychology as a scientific field. International

Journal of Transpersonal Studies, 21, 175-187. —. (2013). The role of science in transpersonal psychology: The advantages of a middle-range theory. In H.L. Friedman & G. Hartelius, (Eds), The

Wiley-Blackwell handbook of transpersonal psychology (pp. 300-312). West Sussex, England: John Wiley & Sons, Ltd. —. (2018). Transpersonal psychology as a heterodox approach to psychological science: Focus on the construct of self-expansiveness and

its measure. Archives of Scientific Psychology, 6(1), 230-242. http://dx.doi.org/IO.1037/arc0000057 Friedman, H.L., & Brown, N.J.L. (2018). Implications of debunking the “critical positivity ratio” for humanistic psychology: Introduction to special issue. Journal of Humanistic Psychology, 58(3), 239-261. Friedman, H.L., & Hartelius, G., (2007). Applying quantitative research methods in transpersonal psychology: Introduction to special topic

section. International Journal of Transpersonal Studies, 26, 56-57. and selfFriedman, H.L., & Pappas, J. (2006). Self-expansiveness of transcendence and contraetion: Complementary processes

immanence. Journal of Transpersonal Psychology, 38(1), 41-53. Friere, P. (1970). Pedagogy of the oppressed. London, England: Penguin Classics. Gabaix, X. (2009). Power laws in economics and finance. Annual Review of Economics, 1(1), 255-294. Gleick, J. (2015). Chaos: Making a new science. London, England: The Folio Society. (Original work published 1987)

Grof, C., & Grof, S. (2010). Holotropic breathwork: A new approach to self-exploration and therapy (SUNY Series in T ranspersonal and

Humanistic Psychology). Albany, NY: State University of New York Press.

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62

Guastello, S.J., Koopmans M., & Pincus, D. (Eds.) (2009). Chaos anaT complexity in psychology. Cambridge, England: Cambridge University Press. Hagerhall, C.M., Laike, T., Kuller, M., Marcheschi, E., Boydston, C., & Taylor, R.P. (2015). Human physiological benefits of viewmg nature: EEG responses to exact and statistical fractal patterns. Nonlinear Dynamics Psychology and Life Sciences, 19, 1-12. Hagerhall, C.M., Laike, T., Taylor, R.P., Kfiller, M., Kfiller, R., & Martin, T.P. (2008). Investigations of human EEG response to viewing fractal patterns. Perception, 37(10), 1488-1494. Hartelius, G., Roth, G., & Roy, P.J. (2013). A brand from the burning: Defining transpersonal psychology. In H. L. Friedman & G. Hartelius,

(Eds), The Wiley-Blackwell handbook of transpersonal psychology (pp. 3-22). West Sussex, England: John Wiley. Hyde, J.S., Bigler, R.S., Joel, D., Tate, C.C., & van Anders, S.M. (2019). The future of sex and gender in psychology: Five challenges to the

gender binary. American Psychologist, 74(2), 171—193. Kastner, R.E., Kauffman, S. & Epperson, M. (2017). Taking Heisenberg’s potential seriously. Retrieved from: https://arxiv.org/pdf/1709.03595.pdf

Kuhn, TS. (1962). The structure of scientific revolutions. Cambridge, MA: Harvard University Press. Lahood, G. (2007). The participatory turn and the transpersonal movement: A brief introduction. Re Vision, 29(3), 2-6. Laughlin, C., & Rock, A. (2013). Neurophenomenology: Enhancing the experimental and cross-cultural study of brain and experience. In H. L.

Friedman & G. Hartelius (Eds) The Wiley Blackwell handbook of transpersonal psychology (pp. 261-280). West Sussex, England: John Wiley.

Leahy, TH. (1992). A history of psychology: Main currents in psychological thought (2nd ed.). Englewood Cliffs, NJ: Prentice Hall. Linde, A. D. (1986). Eternally existing self-reproducing chaotic inflationary universe. Physics Letters B, I 75(4), 395-400. Mandelbrot, BB. (1967). How long is the coast of Britain? Statistical selfsimilarity and fractional dimension. Science, 156(3775), 636-638. —. (1974). Intermittent turbulence in self-similar cascades: divergence of

high moments and dimension of the carrier. Journal ofFluidMechanics, 62(2), 331-358. —. (1977). Thefractal geometry of nature. New York, NY: W.H. Freeman. Marks-Tarlow, T. (1999). The self as a dynamical system. Nonlinear

Dynamics; Psychology, and Life Sciences, 3(4), 311-345.

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—. (2008). Psyche ’s veil: Psychotherapy, fractals, and complexity. New York, NY: Routledge.

—. (2010). The fractal self at play. American Journal of Play, 3(1), 31-62. —. (2011). Merging and emerging: A nonlinear portrait of intersubjectivity

during psychotherapy. Psychoanalytic Dialogues, 2] (1), 110-127. Marks-Tarlow, T., & Friedman, H. (Eds), (in press). Potential role of fi'actals for modeling transpersonal phenomena [special issue]. International Journal ofTranspersonal Studies, 38(2). Moore, K.L. (1968). The sexual identity of athletes. Journal of the American

MedicalAssociation, 205(11), 787-788. Pincus, D., Cadsky, 0., Berardi, V., Asuncion, C.M., & Wann, K. (2019). Fractal self-structure and psychological resilience. Non-Linear

Dynamics Psychology and Life Sciences, 23(1), 57—78. Plumwood, V. (1993). Feminism and the mastery of nature. London, England: Routledge.

Radin, D. (1997). The conscious universe: The scientific truth of psychic phenomena. New York, NY: HarperCollins.

—. (2006). Entangled minds: Extrasensory experiences in quantum reality. New York, NY: Simon & Schuster.

—. (2013). Supernormal: Science, yoga, and the evidencefor extraordinary psychic abilities. New York, NY: Random House. —. (2018). Real magic: Ancient wisdom, modern science, and a guide to the secret power of the universe. New York, NY: Penguin/Random House. Rock, A.J., & Laughlin, C.D. (2018—unpublished manuscript). The advancement of transpersonal psychological science: A neurophenom— enological traj ectory.

Schroeder, M. (1991). Fractals, chaos, power laws. New York, NY: Freeman. Sell, I. (2001). Not man, not woman: Psychospiritual aspects of a western

third gender. Journal of Transpersonal Psychology, 33(1), 16-36. Shapiro, Y. (in press). Toward a science of transpersonal phenomena: Commentary on “A fractal epistemology for transpersonal psychology.”

International Journal of Transpersonal Studies, 38(2), n.p. Spehar, B., & Taylor, R.P. (2013, March). Fractals in art and nature: Why do we like them? Retrieved from: https:l/www.researchgate.net/profile/Branka_Spehar/publication/258 8 1 3746_Fracta1s_in_Art_and_Nature_Why_do_we_like_fl1em/links/54C9

a72e00t2f0b560243148pdf Sulis, W. (in press). Toward a science of transpersonal phenomena: Commentary on “A fractal epistemology for transpersonal psychology.”

International Journal of Transpersonal Studies, 38(2), n.p

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Taylor, RP. (2006). Reduction of physiological stress using fi'actal art and architecture. Leonardo, 39, 245. —. (2016, February 3). Fractals in psychology and art. [Web log] Retrieved fiom: https:l/blogs.uoregon.edu/richardtaylor/ZO16/02/03/human— physiological-responses—to-fi'actals-in-nature-and-art/ Taylor, R.P., Micolich, A.P., and Jonas, D, (1999), Fractal expressionism, Physics World, 12, 25-28. —. (2002). The construction of Pollock's fractal drip paintings, Leonardo,

35, 203. Taylor, R.P., Spehar, B., Wise, J.A., Clifford, C. W. G., Newell, B. R. and Martin, T. P. (2003). Perceptual and physiological responses to the visual complexity of Pollock’s dripped fractal patterns Journal ofNon—

linearDynamics, Psychology and Life Sciences, 9(1), 88—114. Taylor, R.P., Spehar, B., Van Donkelaar, P., & Hagerhall, CM. (2011). Perceptual and physiological responses to Jackson Pollock’s fiactals. Frontiers in Human Neuroscience, 5(1). doi:

10.3389/fiihmn.2011.00060 Varela, F. & Shear, J. (Eds) (1999). The view from within: First person approaches to the study of consciousness. Thorverton, England: Imprint Academic. Walsh, R., & Vaughan, F. (1993). On transpersonal definitions. The Journal

of Transpersonal Psychology, 25(2), 199-207. West, B]. (2006). Where medicine went wrong: Rediscovering the path to complexity. Hackensack, NJ: World Scientific. Wilber, K. (1996). Sex, ecology, spirituality: The spirit of evolution. Boulder, CO: Shambala.

CHAPTER THREE

TOWARDS A NATURALISTIC SCIENCE OF TRANSPERSONAL EXPERIENCE: FRACTAL EVOLUTION AND NONLOCAL NEURODYNAMICS1 YAKOV SHAPIRO2

The fractal dimensions of conscious experience Introduction: Homo supiens’ cognitive revolution Science is neither more nor less than patient and detailed attention to the world, and is integral to our understanding of it and of ourselves. Ian McGilchrist, T he Master and His Emissary, 2009, p. 7

Scientific exploration since the Age of Enlightenment has opened vast vistas of knowledge unprecedented in the history of life as we know it In the intervening four centuries, mankind has moved from the religious straitjacket of the Middle Ages to an explosion of disciplines in natural sciences and humanities that re—defined our view of ourselves and our place in the Universe. The sphere of knowledge has been increasing exponentially in all areas of human endeavor, and virtually every society has been transformed by its impact Our technological breakthroughs are making it possible to map the structure of the Universe, from cosmic microwave background radiation, an afterglow of the Big Bang some 13.7 billion years ago on the cosmological macroscale, to the Higgs field, which underlies the existence

1 This is an expanded version of a commentary published in the special issue of the International Journal of Transpersonal Studies, 38(2). 2 Clinical professor of psychiatry and psychotherapy, supervisor at the University of Alberta, Edmonton, Canada. E-mail: [email protected]

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of matter and mass on the quantum microscale (Randall, 2013). We can finally tie together disjointed observations about the history of life into a biological, and cultural comprehensive theory of physicochemical, evolution on the “pale blue dot” of our planet and beyond. For the first time in history we can contemplate universal human rights irrespective of race or religion, global environmental awareness, and transcending our tribal heritage of war, plague and famine (Harari, 2014). The scientific process is a powerful tool that allows us to formulate hypotheses about the nature of reality, design ways to test our 61 prion beliefs, refining them into scientific theories, and use theory-based epistemology to discover scientific facts about the world that separate wider reality from our imagination. Wilson (1998) refelred to this complex pattern of spreading and deepening scientific knowledge that coalesces into a unified tapestry of the comprehensible world around us as “consilience.” Yet as the sphere of knowledge expands, we face an ever-increasing frontier between what is known and what is “as yet unknown” or unknowable; the more answers we get — the more questions they entail. The outer boundary of knowledge that focuses on the world around us is complemented by the elusive inner boundary of understanding the conscious mind itself. What is consciousness as distinct from physiological processes in the brain? How does our sense of “self” come to be, and what happens to it after we die? Do we possess “free will” that can impact the world around us, or are our choices pre—determined by our biology and physical laws? The inner boundary of knowledge brings to the forefront perhaps the most fundamental question in all of science: What is it that enables the process of knowledge itself?

Consciousness and evolved neurobiology How does complicated, analytical thought relate to intuitive certainty?

Which of the two should we trust more? Pascal Mercier, Night train to Lisbon, 2009 While primitive intentional behavior, such as directed locomotion, is present in all living organisms, and underlies the evolution of life and environmental adaptation, the capacity for abstract thought and creative design is a very recent evolutionary acquisition. The latest evidence (Dodd et al., 2017) suggests that primitive life may have emerged within only 300 million years of the formation of Earth some 4.5 billion years ago (Bya); yet anthropological studies place the first use of stone tools by Australopithecines to c. 3.3 million years ago (Mya), with the use of fire

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practiced by Homo erectus c. 1.5 Mya (Gowlett 2016). The earliest evidence of burial ceremonies goes back to the middle Paleolithic period between 300,000 and 50,000 BCE in both human and Neanderthal social groups, with cave and representational art only emerging between 40-35,000 BCE (Lewis-Williams & Pearce, 2004). The evolution of higher cortical networks responsible for cognitive, emotional and social information processing parallels the evolution of symbolic and artistic capacity (Deacon, 1998). In his far-reaching book, The Master and his Emissary, Ian McGilchrist (2009) suggests that cognitive evolution proceeds along two axes defined by complementary ways of relating to our environment by the two cerebral hemispheres. The evolutionarily older “master” right hemispheric networks allow for experiential readout of the organism’s here-and—now landscape and define the horizontal axis of Gestalt stimulus-response (S-R) adaptedness. By contrast, the more recent “emissary” left prefrontal systems specialize in the serial analysis of events in linear time, its vertical axis allowing for the capacity to lift above the primary consciousness of S-R landscape and map

a wider territory of reflective awareness in linear time, including hindsight of what has unfolded, insight into what is, andforesight of what may come (Edelman, 2004). The evolved left hemisphere interpreter (LHI) functions are thus tied to symbolic language capacity, which is lateralized to left prefrontal networks in over 95% of the Western population. While the right brain experiential networks allow for parallel processing of multiple channels of information but remain in the perpetual present of the organism’s external and internal landscapes, the LHI reflective consciousness networks allow us to perceive unfolding events in the subjective space of a story about the world. In turn, our ability to share subjective stories and plan mutual strategies with others forms the basis of cooperative dynamics that lie at the core of our species’ success in dominating the planet since the cognitive, revolution some 70,000 BCE (Harari, 2014). Neurobiological research supports the hemispheric asymmetry argument. Newberg and d’Aquili (1998, 2000) described right-hemispheric parietooccipital holistic operator, which occupies a posterior domain and “allows for the abstraction from particulars or individuals into a larger contextual involved in generating gestalt understanding about both framework sensory input and various abstract concepts” (2000, p. 254). By contrast, the left-hemispheric fronto-parietal causal operator networks lie in the anterior domain and “handle abstraction of generals from particulars, the perception of abstract causality in external reality, the perception of spatial or temporal sequences in external reality, and the ordering of elements of reality into

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causal chains giving rise to explanatory models of the external world, whether scientific or mythical” (1998, p. 190). Dennett (2006) described this hardwired tendency to attribute intentional causes to natural events as an intentional stance instrumental not only to all known animistic and theistic religions, but also to widespread beliefs in transcendental souls and disembodied consciousness. Vadim Deglin’s group (1996) in St. Petersburg, Russia carried out one of the most creative studies of right versus left modes of information processing using unilateral electroconvulsive therapy (ECT), which functionally deactivates the affected hemisphere for a brief period of time following the treatment. They employed a false syllogism paradigm such as the following: Major premise (false):

Northern lights are ofien seen in Africa

Minor premise (true):

Uganda is in Africa

Question:

Are northern lights seen in Uganda, or not?

Patients in the control (pre-ECT) condition would easily identify the major premise as false and conclude that northern lights are not seen in Uganda. The same patients under left hemisphere suppression (in the right

holistic operator mode) would emotionally reject the argument and provide an empirical answer based on their a priort knowledge: “This is a lie! Northern lights never occur in Africa!” By contrast, under right hemisphere suppression (in the left causal operator mode), the same patients would ignore their knowledge of reality and calmly conclude that one can indeed see northern light in Uganda because “it says so here.” They respond with an analytical answer based on “theoretical, deductive reasoning even when factual answer was known a priori, the premises were obviously false, and the conclusions were absurd” (Deglin & Kinsbourne 1996, p. 285). The conclusions derived from laterality research are stark: our ordinary states of consciousness (OSC) are based on a schizophrenic (“schizophrenia” = split mind) mix of experiential and symbolic representations of reality. The empirical mode of experience, which is biased towards the nondominant hemisphere (right in the right-handed population), is grounded in one’s physical, psychological, and cultural landscape and responds with righteous indignation to any challenge of its a pn'on‘ beliefs. The analytical mode of thought, which is biased towards the dominant hemisphere (left in the right-handed population), displays dogmatic adherence to symbolic

rules and responds with la belle indzference when its deductions bear little

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correspondence to external reality. We are faced with two simultaneously

conflicting and complementary modes of relating to the world (Figure 3— 1), any capacity for functional adaptation depending on integrated functioning between the two.

Figure 3-1. Artistic representation of the right holistic and left analytical modes of experiencing reality. (Courtesy of Dr. Dharini Maheswaran, University of Alberta)

Fractal geometry was developed by Benoit Mandelbrot in the 19703. It

displays the principle of evolving symmetry that results in unique properties of self-similarity and scale invariance that lead to one definition of fractals as “a shape made of parts similar to the whole” (F eder, 2013). Mandelbrot described fractal mathematics as “the geometry of nature” because fractal patterns universally recur within physical and biological systems, from coastlines to clouds, snowflakes to tree leaves, river deltas to circulatory and neural networks. Living systems display fractal self—similarity at different scales of complexity, such as semi-isolated living cells coalescing into semiautonomous multicellular organisms, which aggregate into semiindependent sociocultural groups, and so forth. Fractals tend to arise naturally in dynamical systems, which include the complex adaptive system (CAS) of brain/mind and illustrate the principle that extraordinary evolutionary complexity can arise out of very simple rules if they are iterated repeatedly. Mandelbrot (1989) envisioned the language of fractals as a bridge between scientific and artistic dimensions of human experience,

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stating that “mathematics gains by not attempting to destroy the ‘organic’ unity that appears to exist between seemingly disparate but equally worthy activities of Man, the abstract and the intuitive” (p. 9).

Altered states: The ultimate scientific challenge We are all patchwork, and so shapeless and diverse in composition that each hit, each moment, plays its own game. And there is as much difference between us and ourselves as between us and others. Michel de Montaigne, Essays, II, 1

The contraposition of abstract]analytical versus experiential/intuitive modes of knowing is a running thread in the history of natural sciences. While religious and mystical philosophies have traditionally relied on an introspective apprehension of reality, Western empirical science since Galileo Galilei, Sir Isaac Newton, and Rene Descartes increasingly adopted a reductive method heavily weighted towards an analytical mode of thought (Capra, 1984). Descartes’ famous formulation of Cogz'to ergo sum (“I think, therefore I am”) laid a radically dualistic foundation for the budding empirical sciences, with material res extensa, the domain of substance, separated from the ineffable res cogi tans, the domain of thought, which was considered to be immaterial and thus outside of the natural science domain Consequently, just as rationalism was persecuted and marginalized by the religious establishment in pre-Renaissance Europe, the study of consciousness and its impact on the material world (free will) has been increasingly marginalized within the reductive scientific paradigm. In view of their

inability to resolve the hard problem of consciousness—how physical processes in the brain may result in “immateria ” subjective experience of the mind (Chalmers, 1995)—extreme reductive views have dismissed conscious experience and intentionality as an epiphenomenal illusion, arriving at the reductive epiphenomenalism of consciousness paradox: I think; therefore I am not

Even a cursory review of consciousness research shows a perpetual tug of war between analytical and experiential modes of thought, with dichotomies of brain vs mind, analytical vs intuitive, reductive vs holistic, rational vs mystical, secular vs sectarian, and so forth. Within the transpersonal field, there is an ongoing split between empiricists, who attempt to fit anomalous phenomena within the scientific paradigm (Friedman, 2002), and subj ectivists, who focus on experiential exploration as an alternative to empirical science (Wilber, 2011). This dichotomy brings into focus the need to expand the limits of reductive reasoning and construct a meta-reductive

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paradigm for natural sciences in order to incorporate causally-efficacious conscious phenomena that allow for “organic unity” between the abstract and the intuitive envisioned by Mandelbrot. It also begs a more specific question: Can transpersonal science itself serve as a bridge between empirical and intuitive modes of knowing? The spotlight of conscious awareness is not limited to ordinary consciousness alternating between familiar waking and dreaming states. There is a spectrum of conscious phenomena beyond the fringes of normative awareness often referred to as alternate states of consciousness (ASCs). In the normative domain, they include mystical states (trance, nondual awareness, out-of-body and near-death experiences, divination, mediumship), peak experiences (creativity, flow, spiritual epiphanies, lucid dreaming), extra-sensory perception (ESP—telepathy, clairvoyance, precognition), and mind-matter interactions (psychokinesis, distant healing).

The latter two categories fall within the domain of psi phenomena investigated by the British and American Societies of Psychic Research since the 18805 and established as an experimental discipline of parapsychology by J. B. Rhine at Duke University in the 1930s. It took until 1980 to establish the Association for Transpersonal Anthropology for the specific purpose of studying psi-related practices in indigenous settings. A Wide definition of transpersonal science, therefore, includes ASCs in all their manifestations, from mystical to peak to psi phenomena, and it intersects with psychiatry and psychoanalysis, where pathological manifestations such as dissociative and possession-trance disorders, lycanthropy, some forms of psychosis, and temporal lobe epilepsy are addressed. Advanced neuroscience tools in the past two decades introduce the potential not only to map physiological changes accompanying transpersonal experiences with the help of spectral electroencephalography (EEG) and neuroimaging techniques, but also to induce them by means of trans-cranial magnetic stimulation (Persinger, 2010). Alternate states by their very definition extend beyond symbolic ego boundaries, displaying essential subjectivity, limited accessibility to verbal descriptions, sporadic and often spontaneous nature with sensitivity to initial conditions, and Gestalt apprehension of experiential reality poorly amenable to linear analysis. Notwithstanding ongoing controversy about psi research, the cumulative weight of evidence strongly suggests that ASC and psi phenomena are commonplace and operate under both naturalistic and experimental conditions (Bem et al., 2015; Puthoff, 1996; Radin, 2006). Of the 488 societies studied worldwide, over 90% are found to have an institutionalized form of ASCs (Lewis-Williams & Pearce, 2004), and a

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recent cross-sectional study estimated the prevalence of psi-related experiences among the normal British population at 48% (Pechey & Halligan, 2012). Unfortunately, the task of constructing naturalistic transpersonal science remains elusive, hampered by the failure of reductive scientific accounts and ongoing resistance to integrating mystical and psi states into the mainstream of natural science (Mayer, 2007; Cardena, 2014). The purpose of this review is to propose a meta-reductive paradigm based on the languages of information, emergent complexity and quantum neurobiology, which can incorporate consciousness in its ordinary and alternate domains (Shapiro & Scott, 2019). A critical point needs to be addressed at this point. Intuitive and mystical knowing is often associated with the domain of the “supernatural” or “nonphysica ” promoted by religious and New Age gurus, often in opposition to empirical science. Consequently, the study of ASC and psi states is often dismissed as pseudoscientific. One example involves Shamanic states of consciousness (SSCs), which predate all known religions but have been largely dismissed by Western science as magico-religious practices or psychotic experiences (Flor-Henry, Shapiro, & Sombrun, 2017). Yet, there is emerging evidence that Shamanic trance is a crosscultural phenomenon associated with consistent neurophenomenology, positive psychosocial outcomes, and patterns of physiological changes in the brain that may account for its subjective and transpersonal qualities (Hove et a1., 2015). As in all scientific endeavors, transpersonal science has to be based on empirical criteria of scientific epistemology (Popper, 1959),

which state that our hypotheses have to be verifiable and falsy‘iabie, not merely descriptive in nature, and carry predictive validity in order to separate scientific facts from popular imagination. By these standards, both mystical and psi phenomena are natural processes, which work in compliance with natural laws, even though our current understanding of the,

specific mechanisms involved may be limited.

The fiactui dimensions of evolved complexity in nature: Evolution, emergence and complex adaptive system Is there an archetypal meta-pattern that is, a pattern of patterns that Nature draws upon again and again? (Marks-Tarlow, 2018)

Fractal dynamics are not limited to the static geometry of leaves and coastlines or colorful computer-generated diagrams; they are based on

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iterative processes of self-organization and pattern formation that are ubiquitous throughout natural evolution, fi'om quantum to physicochemical to biological, psychological, and sociocultural domains. In generating conscious experience, neural network processes in the brain follow similar iterative dynamics by utilizing signal re-entry and redundant hierarchical loops of information processing that allow for the progression fi'om rudimentary responsiveness in an organism’s environment (basic sentience) to the primary consciousness in the “perpetual presen ” of stimulusresponse behavior, to the self-reflective awareness of awareness (Edelman, 2004). In the words of Mandelbrot (1989): (T)he process of iteration effectively builds up an increasingly complicated transform, whose effects the mind can follow less and less easily. Eventually, one reaches something that is ‘qualitatively’ different from the original building block. One can say that the situation is a fiilfilment of what in general is nothing but a dream: the hope of describing and explaining

‘chaotic’ nature as the cumulation of many simple steps. (p. 6) Mandelbrot’s dream can now be extended to fractal informational dynamics that may help to re—integrate conscious processes with the natural world—not as immaterial res cogz'tans, but as embodied, iterative patterns of information flow that underlie recurrent patterns of feeling, thinking, and relating in objective, subjective, and intersubjective domains. The very concept of subjective qualia can now be re—formulated as “geometry of

integrated information” (Balduzzi & Tononi, 2009). The authors stated that:

(S)pecific qualities, such as the “redness” of red, while generated by a local mechanism, cannot be reduced to it, butrequire considering the entire quale. Ultimately, the present framework may offer a principled way for translating qualitative properties of experience into mathematics. (p. 1)

Instead of being reduced to their neurophysiological substrates, the quality of subjective experience can be specified by infannational relationships between its active components (whether patterns of neural network activation or associative connections). These informational connections are impartial to the brain/mind dichotomy and show the property of entanglement, forming a unified informational whole across spatio-temporal domains involved. Reductionist and materialist conceptions of reality are limited in their capacity to integrate the full complexity of emergent phenomena (Nagel, 2012). Consequently, a comprehensive framework for transpersonal processes would require a shift to nonlinear

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complexity models operating with informational language. One example is the rapidly growing science of coordination chinamics that postulates reciprocal causality between individual and collective processes in selforganizing systems. They are based on context-dependent functional information that arises out of a system’s coupling to its external environment (Kelso & Engstrom, 2006). These authors proposed that: [J]ust as new qualities of matter arise when a group of atoms behave as a single particle—the so—called Bose-Einstein effect—new functional states of coordination arise on a given level of organization when different components such as molecules, genes, neurons and muscles aggregate together to form a single coherent entity (p. 93). In keeping with this model, there is functional neuroimaging evidence that changes implemented in the course of cognitive therapy can directly affect physiological changes in the brain (Beauregard, 2009). Marks- Tarlow (2015) made a similar point in her recent review of the non-linear dynamics of clinical intuition: “Nonlinear approaches preserve natural complexity, partly by incorporating circular models of causality that permit bidirectional loops of interaction [where] minds can alter brains (through topdown mental dynamics), at the very same time that brains can alter min ”

(13- 3)In more general terms, Nobel Laureate Robert Laughlin (2005) suggested that emergent phenomena of all kinds, from superconductivity to collective dynamics, are not in fact predetermined by lower level physical or mathematical laws, but spontaneously unfold as a consequence of systemic self-organization, creating novel laws which are absent at lower level domains. In his words: “The myth of collective behavior following from the law is, as a practical matter, exactly backward. Law instead follows from

collective behavior, as do things that flow from it, such as logic and mathematics” (p. 209). From this perspective, consciousness and introspective knowledge are only reducible to lower level processes, whether brain or quantum interactions, at the expense of losing a set of emergent laws that define the very nature of these phenomena. To illustrate, just as the Earth prior to emergence of life was a physicochemical system that lacked emergent principles of biological adaptation and diversity, Earth’s ecosystem prior to emergence of reflective consciousness lacked novel principles of artistic creativity and scientific endeavor — the very

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factors that define the emergence of Anthropocene as a qualitatively new

geophysical stage. From a non—dualist vantage point, brain/mind is seen as a unified

psychobiological system nested mid-way within a hierarchy of selforganizing processes extending from neural network down to cellular, molecular, and subatomic levels at the lower limit versus up to personal, interpersonal, group, and cultural levels of organization (Figure 3—2).

Emergence of consclousncss

Quantum-classical limit A

C

ACTIVE INFORMATION Figure 3-2. Multilevel causal loops between different levels of evolutionary complexity PPP!‘

Implicate level of active information below the quantmn—classical limit Explicate level of physicochcmical processes with level-specific causal loops Explicate level of biological processes with level-specific causal loops Explicate level of psychological processes with level-specific causal loops B — reductive bottom-up causation T — emergent top—down causation

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Each level is manifested by the emergence of qualitatively novel properties, such as directional movement in living systems, which are absent at lower levels of organization. The behavior of atomic/molecular constituents is vastly different depending on whether they comprise inorganic or complex living systems. It would be meaningless to discuss higher-level interpersonal or cultural phenomena, such as patriotism or mythology, in terms of quantum or neural network interactions, even though no known cultural process can arise without them. Therefore, reductive bottom-up causal loops (molecules to neurons; neurons to brain/mind) have to be complemented with emergent top-down causation (brain/mind to neurons; neurons to molecules). In addition, level-specific causal interactions need to be considered (e.g. brain/mind to brain/mind — figure 3— 2, 2—4), where information exchange through words, pictures or music can result in lasting physiological changes in the brain (Beauregard, 2009).

Ongoing processes of emergence and self-organization define perhaps the most fundamental view of evolution as a fractal process. David Bohm (1990) made a distinction between “implicate” (quantum) and “explicate” (classical) levels for all reality, where the implicate domain operates according to maniacal-participatory dynamics that continually unfolds into explicate local-interactive processes across the quantum-classical (Q-C)

limit (Figure 3-2, 1). Both implicate and explicate levels can be described in as dynamic modes of organization conceptualized informational rather than matter/energy terms. The recursive processes of self-organization in the explicate domain are manifested by increasing diversity and complexity of physicochemical and biological evolution, with novel emergent laws ultimately leading to self-awareness and sociocultural phenomena. However, what remains unappreciated is that at its quantum foundations evolutionary process proceeds in a qualitatively different way. Instead of the negentropic arrow of increasing order and complexity inherent in macro-evolution, evolution in the implicate realm down to Planck scales3 operates with “being” rather than “becoming”; the very notions of space and time are quantized and the definition of “evolving” loses its meaning. This difference can be conceptually illustrated by

3 In the late 1890s Max Planck calculated a set of fundamental scales that define the lower limits at which all known physical laws can operate. With the elaboration of quantum physics, the smallest quantum of space was calculated to be ~10'33 cm, and

the lowest quantum of time ~1043 sec. Below these scales space-time breaks down and all four fundamental forces (electromagnetic, weak, strong, and gravitational) merge into one. Both quantum mechanics and relativity theory break down at the Planck scale, and a new theory of quantum gravity is required.

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contrasting two views of the evolution on Earth since its formation out of a prom-stellar dust cloud shortly afier the birth of our sun some 4.6 billion

years ago (Bya) (see Figure 3—3). rodmologlal

“MW”

. * evolution

:

_ Cultural evoluuon

Biological

Expllale

evolution

> Physical evolution 5m! Ilmll > lulplcn:

Formation of Earth

" 4.5 By:

Amhmpocene

Origin of life

" 3.7 9V!

0.13;“ of sell-awarenESS "' 3.5 Mya

Figure 3-3. Implicate vs. explicate evolution of complexity in the Earth ecosystem.

While physicochemical and biological evolution have since resulted in the bewildering variety of self—organized complexity that comprise Earth’s geo- and ecosphere, the atomic composition of the Earth system has remained relatively stable (disregarding collisions with interplanetary bodies, ejection of the moon, solar wind, and radioactive decay). In fact, the entire spectrum of evolved structures on Earth has been nothing more than an ongoing process permutations and re—combinations of the same pri-

mordial assortment of atomic ingredients that catalyze macro—evolution while remaining largely unchanged themselves. All explicate processes share active bi—directional connections with the implicate domain, potentially allowing for “macroscopic entanglement,” where spatially and/or temporally separated systems are correlated non— locally, or beyond classical space—time causal links. While the existence of quantum—level entanglement, where relevant properties of the particles involved show correlated behavior irrespective of the distance between them, is well-documented, and biological utilization of quantum processes is becoming increasingly evident (Maldonado & Gomez—Cruz 2014), macroscopic entanglement in biological systems remains controversial.

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Nevertheless, EEG correlations between isolated human subjects were documented as far back as 1960s (Duane 1965). The first EEG/fMRI observations of macroscopic entanglement between two isolated human subjects were reported by Leanna Standish’s team in Washington (2003) and independently corroborated by Radin (2004), Wackerman (2004) and others. Michael Persinger’s team in Canada recently showed evidence of EEG correlations between physically isolated human subjects who had been “entangled” with a complex electromagnetic field and separated by over 300 km (Burke, Gauthier, Rouleau, & Persinger, 2013). It is important to qualify that even though macroscopic and biological systems may share information non-locally, quantum mechanisms do not support “information transfer” by nonlocal means; only nonlocal correlations between entangled components are allowed (Gisin 2014). If any information does become available through nonlocal channels, it would have to be shared outside the explicate space-time rather than being “transmitted” through it, possibly by tapping into the underlying implicate domain. It may be study significant that both mystical and remote perception/telepathy participants similarly report a sense of “reaching in” and “sharing” psi-related information rather than “transmitting” or “receiving” it (Jahn & Dunne 2011).

Active information and quantum neurobiology It from bit. Otherwise put, every it every particle, every field of force, even the spacetime continuum itself derives its function, its meaning, its very existence entirely... from binary choices, bits... in short, that all things physical are information-theoretic in origin and this is a participatory universe. John Wheeler, 1990, p. 312

The reductive epiphenomenalism paradox is rooted in Cartesian dualism, which postulates a fundamental dichotomy between configurations of matter (physical, chemical, and biological systems) versus configurations of the mind (psychological and sociocultural constructs). It is tempting to accept this distinction as factual given all too obvious differences between physical objects (rocks, dolphins, watches, and the like) and seemingly insubstantial perceptions, feelings, thoughts, and desires. A similar dichotomy is apparent between measurable physiological processes in the brain and elusive contents of subjective experience, which is referred to as

the “hard problem of consciousness” (Chalmers, 1995).

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Yet, the validity of Cartesian dualism within a meta-reductive framework is illusory. The lower one descends on the size scale of matter, the more elusive the definition of “matter” becomes. Einstein’s relativistic framework already makes it clear that matter and energy are interchangeable (E=mc2), and at subatomic scales the very distinction between particle and wave descriptions of reality is erased. Quantum phenomena only respond in particle-like or wave-like manner to our measurements depending on the experimental setup. Therefore, clearly definable classical concepts, such as position and velocity of an object or matter/energy distinctions, merge into a unified informational whole in the quantum domain. In addition, quantum mechanics demonstrates an irreducibly nonlocal foundation of material reality, where a classical conception of material objects localized in space— time no longer applies. The Schrodinger equation describes quantum processes as wave-like phenomena with elementary particles behaving as spread-out stationary waves rather than as Newtonian miniature billiard balls. Experimental confirmations of Bell’s inequality theorem4 unequivocally demonstrate that no combination of local—interactive mechanisms can account for quantum entanglement (Gisin, 2014). Consequently, quantum nonlocality is now accepted as the basic foundation of physical reality, while spatiotemporal locality is seen as an emergent property of macroscopic space-time. There is increasing evidence that our classical local-interactive reality is an emergent phenomenon valid in the intermediate range of scales; it coalesces out of the quantum nonlocalparticipatory domain just as the property of fluidity manifests out of the interaction of a vast number of water molecules, but loses its meaning at the sub-atomic level (Anderson, 1972). Space and time themselves are seen as emergent properties above the Planck scale, beyond which they become quantized and lose their meaning. The fabric of material reality is thus built on nonlocal informational processes that transcend spatiotemporal characteristics of matter, rather than on Newtonian clockwork pieces that interact across space in a unit of time. The eminent British physicist Iohn Wheeler (1990) epitomized this unified vision of reality by pointing out that all matter ultimately rests on informational foundation: “It from bit.”

4 The Irish physicist John Stewart Bell formulated his inequality Theorem in 1964, which defines experimentally

verifiable domains where local versus nonlocal

variables can account for quantum entanglement effects. It has since been tested in multiple experiments, which demonstrate that our Universe is fundamentally nonlocal, and local realism only applies as an approximation in the classical macrodomain (Gisin, 2014).

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Conversely, processes of the mind are embodied, a fact increasingly appreciated by clinicians and neuroscientists dealing with the brainfmind interface (Varela et al., 2017). While matter, such as rocks, can exist without consciousness as an attribute, mind and consciousness cannot be understood in isolation fi'om physiological processes of body/brain and their wider environment (Penrose, 1997). Carl Sagan famously stated: “Extraordinary claims require extraordinary evidence.” Disembodied consciousness may exist in the human imagination, much as elves and unicorns do, but we are far from possessing extraordinary evidence for its ontological reality. Any purportedly “disembodied” experience, such as near-death states, still requires access to a functioning brain in order to recall and communicate it (Craffert, 2015), and may be more parsimoniously explained by psi-related phenomena. Rather than postulating an immaterial mind, David Bohm’s (1990) “ontological interpretation” of QM suggests that implicate active infomaiion may underlie both mind and matter processes. Both physical and psychological processes tap into the quantum domain, being guided by a nonlocal-participatory informational “pilot wave,” just as a ship may be guided by GPS data its electronic systems tap into. What we see as dualistically opposed matter and mind represent the proverbial tips of the implicate iceberg, just as seemingly isolated ocean islands are really just interconnected mountain peaks of the underwater terrain. Bohm states: One may then ask: what is the relationship of these two processes? The answer that I want to propose here is that there are not two processes. Rather, I would suggest that both are essentially the same. This means that that will, in a natural way ultimately move the which we experience as mind body by reaching the level of the quantum potential and of the ‘dance' of the particles. There is no unbridgeable gap or barrier between any of these levels. Rather, at each stage some kind of information is the bridge. (13. 283)

Theise and Kafatos (2013) made a similar argument, defining sentience as “sensing of the surrounding environment, complex processing of information that has been sensed... and generation of a response” (p. 378). Starting fiom Maturana and Varela’s autopoietic theory (V arela, Thompson, & Rosch, 2017) they suggested that primitive sentience exists all the way down to living cells (“where there is life, there is mind” , and further argued that if complexity principles of self-organization are applied at even lower scales down to the Planck limit, there is no apparent lower limit for sentience: From the subatomic level downward... all entities are defined by wave functions that extend infinitely in all directions, overlapping with all others [so that] there is no ‘extemal’ to be sensed and no ‘internal’ processing to

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create a response to the external; rather, the component activities that define sentience are inherent and pervasive, to be currently described, in part, by the concepts of quantum entanglement and non-locality.” (Varela, Thompson & Rosch, p. 385)

The common informational foundation for mind and matter is directly relevant to transpersonal science. Both physical and conscious dynamics

can be conceptualized as quasi—stable vortices of informational flow, the illusion of localized stability maintained by the recursive unfolding/re— enfolding process between implicate and explicate domains. Just like our experience of stable autobiographical Self rests on a vast foundation of implicit perceptions, affects and reflexive responsiveness that coalesce into

explicit thoughts, the solidity of material objects dissolves into nonlocal informational dynamics at lower scales of organization. What is seen as dichotomous matter/mind processes thus represents a collective emergent phenomenon, with the language of information equally applicable to neural network processes in the brain and to patterns of emergent conscious experience in the mind (Figure 3—4).

Causal operator (explicit)

Classical (local) macro-dynamics

f \ \ Holistic operator (implicit)

J

Explicare/umued Implicate/Enfolded

;

Quantum (nonlocal) micro-dynamics

Figure 3-4. The realms of empirical and transpersonal sciences 1. Natural science: objective knowledge of the classical macro-world (local— interactive) 2. Psychological science: objective/subjective knowledge of mind/brain processes

(local-interactive) 3. Quantum science: objective knowledge of the micro-world (local-interactive + microscopic nonlocality)

4. Transpersonal science: subjective knowledge of the world and other brain/mind systems (nonlocal—participatory)

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In keeping with the above, the outer and inner boundaries of the sphere of knowledge enabled by reflective consciousness display self-similarity in physical and psychological domains. Just as psychologists and psychoanalysts may study unconscious processes that operate according to “primary process logic”5 but give rise to explicit consciousness and intentionality, quantum physicists study subatomic phenomena that operate with distinctly different logic and causality, yet give rise to the everyday “macro-logic” by which classical reality operates. But while conventional empirical sciences, whether psychological (Figure 3-4-1), natural (Figure 3-4-2) or quantum (Figure 3-4-3) rely on the participation of a conscious observer, it is the “submerged” interface between unconscious and nonlocal-participatory implicate reality that defines the field of transpersonal science (Figure 3—4— 4). The process of knowing, therefore, can proceed along two separate

channels: an explicit, local-interactive channel that underlies the outer boundary of studying the world and physical processes in it (including

ourselves), and an implicit, nonlocal—participatory channel that defines the inner boundary of mystical and extraordinary knowing. The fractal epistemology

approach helps to shed light on another

question vital for transpersonal science: Can verifiable infomation about external reality be obtained by purely introspective means? The validity of mystical inner knowing or psi—derived extraordinary knowing (EK) has haunted transpersonal and parapsychological fields for decades; both tend to be viewed with skepticism by the mainstream scientific community. What is overlooked in this controversy is that all “hard sciences” rely on inner knowing in the form of mathematics. Nobel Laureate Eugene Wigner (1960) posed the question of the “unreasonable effectiveness of mathematics in natural sciences,” stating:

5 According to Freud’s elaboration of pSychoanalytic theory, the rational process of thought is based on the reality principle, which allows us to assign cause-effect relationships and separate subjective from objective events. By contrast, in the primary process domain, such as dreams or psychotic states, the distinction between internally and externally derived information is lost. Psychoanalytic techniques of free association and dream analysis are specifically designed to bring primary process symbolism to rational awareness, thereby providing a rational understanding for why we feel, think, and react the way we do. In psychobiologica] terms, our reflective dual consciousness (experience of “inner self” observing the world and our interactions with it) is based on symbolic capacity mediated by the Left Hemisphere Interpreter networks, while right hemispheric “experiential self” underlying ASC experiences tends to blur the boundaries between self/not self and subj ective/ objective reality.

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The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither

understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse,

to our pleasure even though perhaps also to our bafflement, to wide branches of learning. (p. 14) However, the common informational foundation between physical and conscious processes would in fact predict a fractal correspondence between abstract thought responsible for mathematical constructs and larger physical reality. Nobel Laureate Brian Josephson (1987, p. 18) argued for a similar perspective in asking: “Why should the thinking involved in doing mathematics (which is as introspective in character, as is meditating) serve as a legitimate component of science, while mystical experience does not?” While the interplay between abstract/analytical and experiential/intimive modes of knowledge forms the foundation of scientific process, of primary importance to transpersonal science is the interplay between macro-level nonlinear complexity models and micro-level quantum neurobiologyfi (Tarlaci, 2011). Just as the processes of life are positioned at the edge of chaotic dynamics where linear models are inadequate to model evolutionary adaptation, the brain/mind system spans the interface between quantum and classical levels of reality, where classical local—interactive models only address a subset of mind/brain interactions, and a more general neuroquantology model is required. Consequently, classical neurobiology only serves as an approximation of integrated brain/mind dynamics, in a similar way that classical Newtonian physics is an approximation of a wider relativity fi'amework. Stapp (2011) comments that “the classical principle does not generally hold in quantum of causal closure of the physical

5 Quantum neurobiology extends classical neurophysiological mechanisms of neural network dynamics to quantum effects at the level of cellular sub-components, such as microtubules and intercellular communication in synaptic junctions and dendritic

fields. Relevant quantum effects in biological systems, which include quantum tunneling, superposition of multiple co-existing states, biological entanglement, and nonlocality depend on the capacity to maintain quantum coherence in room

temperature “warm and wet” environments. It has now been experimentally demonstrated that multiple biological processes, such as avian compass, visual and olfactory receptor fiinction, photosynthesis, and enzymatic catalysis reliably utilize

quantum effects (Maldonado & Gomez-Cruz, 2014). Consequently, neural network information processing does not involve classical computation alone, but utilizes quantum computing mechanisms by means of controlled rather than random

decoherence.

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would effectively inject a theory” (p. 48), and “a quantum Zeno effect7 rapid sequence of mental intentions into the course of brain activity” (p. 42). Consequently, he suggests that “according to quantum physics all causal effects of consciousness act within the latitude provided by the uncertainty principle, and this latitude shrinks to zero in the classical approximation, eliminating the causal effects of consciousness” (p. 39). In their recent review of quantum biology, Maldonado and Gomez-Cruz (2014) describe living organisms as “systems of systems” where “what happens or is expressed in one level, has a fractal correspondence with what happens—may happen—or is expressed in a different level or scale” (p. 180). A “quantum biology” framework may help us expand the current reductive paradigm in the natural sciences to incorporate “holistic” phenomena characterized by emergent properties at higher orders of complexity, such as living, conscious, and entangled systems that span the

quantum-classical divide (Pylkkanen, 2014).

Transpersonal experience Within the natural science domain Transpersonal science as a bridge between analytical, intuitive and exiraordinary knowing The finest emotion of which we are capable of is the mystic emotion. Herein lies the germ of all art and all true science. Albert Einstein, letter to Hoffman and Dukas, 1946

To summarize the argument so far:

7 Quantum Zeno effect describes the fact that quantum measurements repeated in

quick succession inhibit transitions between quantum states. It derives from the ancient Greek philosopher Zeno of Elea’s “arrow paradox”: if at any given instant a flying arrow is motionless, how is it that it can move? Quantum Zeno effect, first postulated as a thought experiment by Alan Turing, has now been experimentally verified and provides one of the best frameworks for genuine biological autonomy

and free will. Repeated re adouts of neuronal state superposition in the brain have the potential to preference a specific neural pattern without violating energy conservation laws, thus providing a quantum-classical mechanism for causally

efficacious free will.

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Both reflective consciousness and the spotlight of knowledge are enabled by continuous interplay between experiential/intuitive and abstract/analytical modes of knowing, which enables human capacity to lift over the “perpetual presen ” of the stimulus-response landscape and access a wider terrain of cause-effect relationships in fourdimensional space-time. Classical physical and psychological processes take place in the

explicate local-interactive domain, which continually unfolds across the quantum-classical (Q-C) limit from the implicate nonlocal— participatory domain, where material and mind processes share common informational foundation. The matter/mind distinction is only relevant in the classical macro-world, while informational processes span the (Q—C) limit and underlie both objective neural network dynamics and subjective “qualia,” such as experiential phenomenology of perceptions, feelings, beliefs and desires. Brain/mind can be conceptualized as an emergent Complex Adaptive System that operates in far—from—equilibrium conditions at the border of chaotic dynamics and spans the Q-C limit. In addition to the abstract;’ local-interactive channels of analytical and experiential/intuitive information exchange (ALI and ELI) accessible in ordinary conscious states, brain/mind may be capable of operating with a nonlocalparticipatory channel (NPC) of mystical and extraordinary (psi-based) knowing, largely accessible in alternate states of consciousness. The fractal epistemology approach allows us to model self-similar patterns at different scales of complexity across the Q-C limit, from implicate to physicochemical, biological, and psychological levels of organization. It may provide means of constructing a naturalistic transpersonal science at the intersection of analytical, intuitive, and extraordinary knowing domains where nonlocal-palticipatory sources of knowledge stand on an equal footing with the abstract/analytical and experiential/intuitive modes that comprise empirical natural sciences (Figure 3-5).

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ANALYTICAL KNOWING

EXPERIENTIAL KNOWING

(Au mode)

(Eu mode)

TRANSPERSONAL SCIENCE

BITRAORDINAR Y KNOWING (NPC mode)

Figure 3-5 . Transpersonal science at the intersectional of analytical, experiential and extraordinary knowing: ALI — Analytical local-interactive mode of knowledge

ELI — Experiential local-mteractive mode of knowledge NPC — Extraordinary nonlocal-participatory mode of knowledge

A critical task for transpersonal science is to construct a naturalistic

model of mystical and psi experiences. Charles Tart (1972) introduced a seminal paradigm of state-speczfic sciences, where he argued against both excessive subjectivism (blind faith in one’s “feeling of knowing”) and

exclusive reliance on reductionist models with their illusion of a “detached observer.” Here, a self-similar parallel between quantum—level observations and subjective experience becomes apparent: the very process of observation and measurement inevitably affects the processes under study. An observer has to be incorporated into a larger unified system with the processes observed, which incorporate objective, subjective, intersubj ective and interobjective (collective and cultural) perspectives. This participant observer paradigm is also evident in the psychotherapy setting, where both abstract/analytical and experiential/intuitive modes of knowing comprise

the patient/therapist intersuly'ective field defined as a complex matrix of nonlinear interplay between two interacting and interconnected participants

(Stolorow, 1997). The phenomenon of clinical intuition may be a good case in point here.

Patient/therapist interaction typically proceeds along three interrelated channels:

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1. Right-to-fight brain communication, where patient and therapist’s holistic operator networks interface on an implicit (non-verbal) level based in part on mirror neuron network information processing (Iacoboni, 2007). This process manifests a complex matrix of somatic, affective, and associative interactions, coordinated autonomic reactivity and body postures, and non-verbal interplay that parallel mother-infant communication (Stern, 2007). The therapist can then utilize his or her countefiransference reactions8 to empathically resonate with the patient’s implicit content “beyond the words” (Shapiro, Marks—Tarlow, & Friedman, 2017). Note that, in this mode of communication, there is no meaningful separation between the observer and observed, the information becoming available on an intuitive level in the joint intersubjective space co-determined by both participants. In the natural science domain, such interaction parallels quantum-observer effects where an observer inevitably becomes an extension of the system observed.

2. Lefi-to-right brain communication, where a therapist explicitly focuses on the patient’s nonverbal communication, including facial expression and eye contact, body posture, prosody and tone of voice, the timing and rhythms of the patient’s responses, and silences in the therapeutic communication as a counterpart to the words. In this case, the therapist works to bring into awareness the patient’s holistic operator processes rather than their left hemisphere interpreter-based story and content, an approach common in process-oriented therapies, such as relational and psychodynamic treatments. In the natural science domain, such process of Gestalt perception parallels intuitive “flashes of insight” that may provide a qualitatively new understanding of the phenomena under study, such as Einstein’s thought experiment of riding a beam of light that led to formulation of special relativity, or Kekule’s dream of an

s COuntertransference is usually defined as a therapist’s emotional response to a patient. Psychoanalytic affect theory separates unconscious aflects at the level of the mirror neuron system, which are processed to conscious feelings that we can name, and to propositional emotions attached to an object. For instance, in the course of

psychosocial development an infant matures from crying with distress in response to loud noise (an affect of fear), to being able to verbalize “I’m scared” (a feeling of fear), to becoming aware of being “scared of thunder” (an emotion of fear).

Similarly, countertransference reactions as a readout of a patient’s affective state are processed from a therapist’s somatic response (such as unconscious body tension), to empathic reactivity (“I’m in pain”), to an associative awareness of the here-andnow intersubjective context (“I am feeling sad listening to your story”).

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3. Left-to-left brain communication, where a patient tells his or her story, and a therapist formulates an appropriate treatment plan based on his or her theoretical orientation. Many forms of therapy are based on this model including cognitive-behavioral, interpersonal, and solutionfocused modalities. The treatment process mainly occurs at the level of explicit interaction between the patient’s and therapist’s causal operators, the therapist utilizing his or her abstract/analytical capacity to help patients understand and change some of their dysfunctional cycles. In the natural science domain, this process parallels sequential analysis of classically-derived data, where a scientist can be reasonably separated from the processes under study. The above 3-channel model describes conventional local-interactive routes of communication, both intuitive and symbolic. Daniel Kahneman (2014), a Nobel Laureate in economics, made a similar distinction between fast and slow systems of cognition, the evolutionarily older “fast” system relying on implicit Gestalt heuristics, and the more recently evolved “slow” system corresponding to explicit symbolic reasoning. However, both intuitive knowing (channel 1) and explicit information exchange (channels 2 and 3) have to be distinguished from mystical/extraordinary knowing, which may be not just nonverbal but nonlocal in origin. Specifically, while channel relies on Gestalt apprehension of the experiential/intuitive information obtained by local-interactive sensory means (such as parallel processing heuristics of sensory-derived information in limbic and mirror neuron systems), mystical knowing may operate with nonlocalparticipatory dynamics that go beyond the sensory data. These experiences have been labelled as “anomalous” or “uncanny” in the psychoanalytic literature, going back to Freud’s “thought transference” (1921/1975), Carl Jung’s “synchronicity” (1959), telepathic dreams (Marks-Tarlow, this volume), and apparent access to private information without sensory sources (Tennes, 2007). In all these cases, accurate information may be shared within a patient/therapist dyad, apparently without access to either verbal or nonverbal communication means. Many experienced therapists have encountered “uncanny” phenomena in clinical practice, which can now be studied with functional neuroimaging tools, such as hyperscanning and real-time multi-channel electroencephalography, to construct a meta-reductive science of transpersonal phenomena in health and psychopathology. For instance, Acunzo’s team (2013) reviewed methodological issues in using functional neuro-

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imaging to identify neurobiological substrates of ESP phenomena, noting that five out of six published studies demonstrated positive results. By their very definition, transpersonal and parapsychological experiences cannot be reduced to either objective or subjective data alone, and must incorporate fuzzy boundaries between analytical, intuitive, and extraordinary knowing domains (Figure 3-5). Classical reductive neuroscience is insufficient for this task, and more integrative models are needed. One example is “first-person neuroscience” (Northoff & Heinzel, 2006), the authors commenting that “in order to reveal the true neuronal correlates of mental states, first- and third-person perspective must be linked to each other” (p. 3). In a recent study of brain changes during a Shamanic trance (F lor-Henry et a1., 2017), which utilized quantitative electroencephalography mapping and low-resolution electromagnetic tomography (LORETA) source imaging, we demonstrated trance—specific disruptions in the interhemispheric integration most prevalent in the central temporal regions. In addition, there was a prominent shift from normative left hemispheric dominance in the “default self” ordinary consciousness (OSC) mode to pronounced right hemispheric dominance in the “trance self’ mode of Shamanic state of consciousness (SSC). The transition from OSC to SSC is also accompanied by the transition from the normative left anterior prefrontal to right posterior temporo-parietal networks, where a sensorirnotor/experiential rather than abstract/analytical perception of reality predominates (Figure 3-6). The transition from OSC to SSC is thus accompanied by the transition from left anterior prefrontal to right posterior temporo-parietal networks, where a rather than abstract/analytical perception of sensorimotor/experiential reality predominates. This shift was accomplished at will by our subject, a trained Mongolian shaman, enabled by a release from normative contralateral and prefrontal inhibition, where temporary right holistic operator networks dominance resulted in a qualitatively different experience of inner and outer reality. This was evidencedby qualitative changes in our subject’s normative sense of self with the emergence of a wolf alter- ego, inner visions and music, and synesthetic experiences (such as “a smell of discord”). Similar findings of “inhibition of inhibition” in critical hubs of default consciousness network were recently reported by Carhart—Harris’ team in psychedelic-induced states (2012).

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Len Analytical Self

Right Experiential Self

Causal Operator {Local-Interactive)

Holistic Operator (NonlacaI-Parfid » , a .

Figure 3-6. Dual left-to-right and anterior-to-posterior network shifts in transition

from ordinary waking consciousness to Shamanic trance. In their discussion of Shamanic states, Frecska and others (2016)

suggested that they may tap into a “nonlocal—intuitive” mode of knowledge, where “nonlocal information about the physical universe offers the missing link between scientific objectivism and subjective experience, including consciousness and spiritual experiences” (p. 160). The authors described SSCs as “a form of focused and expanded consciousness, closer to meditative states, in which the participant intentionally shifts his or her awareness from ordinary perception toward a different ‘input,’ which seems to originate from ‘within’” (p. 157). Such inner knowing may complement ordinary “perceptual-cognitive” information processing by allowing us to shift between local and nonlocal channels of experiencing unified mind/matter reality in a similar way to shifiing at will between mutually exclusive face/candlestick perceptions of a Visual illusion, which one can never hold simultaneously (Figure 3-7). What is lacking to date is a naturalistic model of mystical and EK phenomena that would be compatible with both quantum/classical physics and the nonlinear dynamics that define

mind/brain as a quantum—classical Complex Adaptive System.

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Figure 3-7. The two-facesfcandlestick illusion: note that you can shift between the

black facesfwhite candlestick perceptions of the picture at ml], but neverhold both simultaneously. (Biblie domain)

Nominal! mar-admonitionA M d MEI qfarfi'aorflinmy knomhg The filndamental reality is ‘infinity,’ the unknown, the situation for which

there is n o language—not even one borrowed by the artist or the religious— which gets anywhere near to describing it-

—'Wilfred Bion (quoted in Eigen, 2012'.)

One of the most promising integrative models for transpersonal science to date was developed by J ahn and Dunne (2001), who called it a Modular Model of Mind-Matter Manifestations (MS). Based on their comprehensive database of remote viewing and mind-machine interactions (random event.

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generators, or REGs) they suggested a parallel between classical/quantum levels of physical reality and explicit/implicit levels of psychological experience (Figure 3-2). Similar to Bohm’s theoretical model, the boundaries between implicate/implicit and explicate/explicit domains of mind/matter reality cannot be defined in binary terms but, rather, represent fuzzy transitions between qualitatively emergent phenomena at new levels of informational complexity. In their later works, Jahn and Dunne (2011) underscored the fact that the deeper we look into implicit and quantum domains, the more self-similarity they show, including progressive lack of localizability, lack of subj ect/obj ect differentiation, spatio-temporal breakdown of linear causality, fundamental nonlocality and uncertainty, and co-participatory rather than interactive dynamics. Consequently, they postulated “emergence of both cognitive experience and physical events from a common underlying substrate where the unconscious mind and the undifferentiated world of physical potentiality merge, and where the distinction between mind and matter blurs into uncertainty” (2011, p. 249). They similarly suggested that “the primary currency of reality is information, which may flow in either direction; i.e., consciousness may insert information into the environment as well as extract information from i t ” (p. 264). Unfortunately, the critical question of a physical mechanism for such nonlocal information sharing, which may include mind—to-mind (telepathy), mind-to-explicate systems (psychokinesis), explicate systems--to-mind (clairvoyance), and retro-causality (precognition), has remained unexplained in View of the fact that any information transfer beyond simple correlations between entangled-system components would violate special relativity, and has not been observed in quantum systems (Gisin, 2014). Another problem is that, in spite of the fact that biological processes undoubtedly tap into quantum-level phenomena at subcellular levels of electrochemical reactions, biological “warm and we ” environments were thought to be too noisy to maintain sustained quantum coherence necessary for entanglement

correlations. To address the second issue first, the elaboration of quantum neurobiology in the past decade unequivocally demonstrates that quantum coherence is reliably utilized in the nervous system, and viable theories of such mechanisms have been proposed. Hameroff and Penrose (2003) postulated an Orchestrated Reduction theory in microtubules (cellular components universally present in all living cells including neurons and neuroglia), identifying potential interface between quantum- and classicallevel processes in all biological systems. For instance, a universal property

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of living systems is their capacity for directed movement, which is present all the way down to unicellular organisms without specialized nervous systems. It may be significant that in paramecia, membrane-bound cilia responsible for coordinated movement are similarly rich in microtubules (Penrose, 1997). In addition, recent work by Vattay and others (2014) provided one of the first models for sustained quantum coherence in living tissues, suggesting that “at the critical edge of quantum chaos, coherence and transport can coexist for several orders of magnitude longer than in simple quantum systems ” (p. 1). Sherlock Holmes, a legendary character of Sir Arthur Conan Doyle’s novels, once famously remarked to his friend Dr. Watson that “once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.” Paradoxes occur not in nature but in the minds of those who contemplate it—the fact noted by St. Augustine: "Miracles occur in contradiction not to nature, but to what is known to us of nature" (quoted in Rao & Palmer, 1987). Given extensive evidence for nonlocal-information sharing reviewed previously, transpersonal science is faced with the seemingly impossible proposition that such processes are forbidden by both special relativity and quantum entanglement rules. However, this would only apply to macroscopic systems within explicate space-lime, where no superluminal information transmission is allowed. In reviewing existing data for entanglement correlations, Gisin (2009) states: In modern quantum physics, entanglement is fundamental; furthermore, space is irrelevant at least in quantum information science, space plays no central role and time is a mere discrete clock parameter.... No story in space-

time can tell us how nonlocal correlations happen; hence, nonlocal quantum correlations seem to emerge, somehow, from outside space-time. (p. 1358)

If sentient dynamics that underlie explicit self-awareness do in fact share common informational roots with physical matter in the implicate domain, exchange is no longer limited to local-interactive information communication in space-time but also includes nonlocal-participatory “information sharing” within a fundamentally unitary implicate system that underlies both matter and mind processes. In the implicate realm, the very definitions of “information exchange” and mind/matter differentiation dissolve into probabilistic uncertainty of the nonlocal wave function. It is only in the process of unfolding into explicate reality that classical systems become delineated from each other and conscious subjects become aware of objective or subjective events in space—time. This transition is an inevitable consequence of crossing the Q—C limit between micro— and macro—reality, above which emergent classical effects predominate, in order

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to make a measurement by amplifying quantum-level processes to macroscopic domain. More recent theories quantify Bohm’s unfolding process by postulating an ongoing process of continuous spontaneous localization (Smolin, 2019) that re-establishes ontological realism and avoids the dualistic implication of conscious observers being responsible for the “collapse of the wave function” in quantum measurements. Nonlocal information sharing that potentially underlies psi-related phenomena would literally operate “underneath space-time,” without violating special relativity or quantum rules. From this perspective, there is no meaningful separation between neural and conscious states, and the “hard problem” of consciousness becomes an artefact of imposing classical assumptions on a fundamentally quantum/ classical system (Stapp, 1997). Specific mechanisms that enable extraordinary and mystical knowing can now be formulated based on the proposed nonlocal neuroafynamics model. On a classical level, a functioning human brain operates as a nonlinear Complex Adaptive System (CAS) that incorporates multi—level information processing that underlies both “objective” synaptic network physiology and “subjective” self-awareness and intentionality (Shapiro, 2015). Classical interactions between the CAS of brain/mind and its wider environment involve local-interactive analytical (ALI) and experiential (ELI) modes of verbal/nonverbal communication. Note that all localinteractive channels incorporate bi-directional information exchange, which involves sensing information from either external or internal environment and responding to it by means of verbal, nonverbal, motor, or autonomic modalities. However, mind/brain information processing cannot be described by classical processes alone. Just as the CAS of mind/brain operates atthe edge of chaotic dynamics in the classical domain, it spans the quantum-classical limit and incorporates quantum processes of superposition and

entanglement at its foundations. It is the implicate-to-explicate flow of active information across the Q-C limit that forms the substrate for both mind and brain processes, with quantum Zeno effect potentially allowing for “free will” in selecting a single neural/subjective state out of a superposition of all potential ones. The resultant brain/mind states are unitary and causally efficacious, allowing for both local and nonlocal knowing in full accordance with quantum and relativity principles.

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The neurodynamics of Q-C transition defines a bi—directional nonlocalparticipatory channel (N PC) of nonlocal awareness that may underlie mystical and psi-based modes of information processing. In keeping with the meta-reductive fiamework, both bottom-up and top- down causal chains come into play, with NFC-related information shared nonlocally between systems and their wider environment, such as in brain/mind telepathy/clairvoyance, or inserted nonlocally, such as in PK (Table 3-1). The top-down causal chains thus operate along both local-interactive and nonlocal-participatory channels, allowing not only for conventional choice of action (as in picking up a cup of tea), but also for nonlocal “information insertion,” as in psychokinetic REG or distant healing effects. While the precise nature of neural mechanisms underlying NPC capacity is presently unclear, there is preliminary data that it involves a shift from the normative left prefrontal “default self” mode to the right temporo-parietal “trance self ’ and becomes available in somatosensory rather than symbolic modalities. Mode of knowledge

Proposed model

Abstract/analytical local-interactive knowing (ALI channel): 1.

Inner explicit (e. g., mathematics)

2.

External explicit (e. g., empirical sciences)

Dominant left-prefrontal causal

operator networks based on localinteractive symbolic information exchange

Experiential/inmiiive loc Lil—interactive

knowing (ELI channel): Implicit (Gestalt/heuristic, flow states,

Non-dominant right-posterior holistic

flashes of creativity); no inner/extemal distinction

operaior networks based on the localinteractive Gestalt information exchange mode

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Mode of knowledge

Proposed model

Mystical and extraordinary nonlocal-

participatory knowing (NPC channel): 1.

ASC-based (Shamanic, meditative, mystical states)

Systematic or spontaneous ability to “tune in” to the NPC mode of awareness by shifting away from

sensory channels/causal operator towards nonlocal holistic operator 2.

Psi-based:

i.

Clairvoyance/remote viewing

Nonlocal spatial connection between a brain/mind system and its environment

ii.

Telepathy

Nonlocal spatial connection between two (or more) brain/mind systems

iii. Recognition/premonition

Nonlocal temporal connection between a brain/mind system and other brain/mind systems or their

environment iv. Mind-matter interactions

Nonlocal spatiotemporal information insertion from a brain/mind system into its environment

v.

Distant healing

Nonlocal spatiotemporal information insertion from one brain/mind system

into another

Table 3-1. Modes of knowledge and their proposedneurodynamic substrates.

Further delineation of ASC— versus psi—based modes of extraordinary knowing can be accomplished with systematic studies using currently available neuroimaging tools, such as real—time qEEG, MRI, and transcranial stimulation that utilize ESP/psychokinesis tasks under the conditions of complete electromagnetic shielding (e.g., Faraday-cage paradigm). The ultimate challenge in assessing the scientific validity of transpersonal experience would lie in demonstrating that the information obtained during mystical or psi experiences is not limited to subjective or objective changes in a single brain/mind system, but bears direct correlation

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to independently verifiable data, whether in extemal reality or other brain/minds. One prediction of the proposed nonlocal neurodynamics model is that the ability to reliably access ASC and psi phenomena would be inhibited by LHI analytical functions. These paradoxical intentionality effects have in fact been described, such as the experimenter effect in parapsychology and series position eflect in remote viewing experiments, which require the participants to “bypass conscious intention” (Meier, 2007). Similar observations have been made in the psychoanalytic field, where explicit cognitive processing inhibits the therapist’s capacity to access implicit/primary process data. Therefore, in studying anomalous experiences and sharing knowledge of these states, transpersonal science may need to utilize the well-known psychoanalytic concept of “evenly hovering attention,” monitoring the nonlinear flow of implicit/experiential content unconstrained by explicit expectations, verbal formulations, or pro-existing theoretical

frameworks.

Conclusion: A fractal epistemology for transpersonal psychology All scientific thought is ultimately an interplay between analytical and intuitive modes of knowing, where creative insight is elaborated through symbolic means and verified empirically in order to achieve scientific synthesis. Fractal epistemology extends the recursive dynamics of self— similarity and scale invariance not only to the physical world around us but also to the processes of global evolution and mechanisms of knowledge itself. Coupled with the informational framework, it allows us to construct

a rigorously scientific holistic epistemology that transcends the dualistic language of conscious-observer states and unifies dichotomies of physical versus mental, classical versus quantum, analytical versus intuitive, and ordinary versus extraordinary modes of knowing. The classical local-interactive perspective is inadequate to account for quantum-classical transitions and thus for the process of thought; the Universe is fundamentally nonlocal in nature, and a nonlocal-participatory model is required to complement classical logic and incorporate conscious observers into anaturalistic—scientific framework. Just as analytical capacity lifted humans over the 3-dimensional landscape of primary stimulusresponse consciousness, and allowed us to map higher-dimensional spacetime terrains, nonlocal knowing can help us transcend the categorical

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limitations of symbolically constructed dual self and tap into the common informational substrate of all possible terrains. Causally-efficacious consciousness in general and transpersonal/psi experiences in particular do not contradict any physical laws but, rather, point at inherent limitations of the reductive paradigm in the natural sciences, suggesting the need for an extended meta-reductive paradigm, in a similar way that relativity and quantum theories extended the Newtonian vision of mechanistic “clockwork universe.” Herein lies the power of transpersonal science to bring the full spectrum of consciousness into the unified naturalistic picture of mind/matter reality. Everything that exists is rooted in active information of nonlocal implicate reality, just as seemingly separate trees form a unified eco-system of a mature forest connected below the visible soil. The “deep ecology” model (Zimmerman, 1994) can be seen as another example of a fractal selfsimilarity pattern extending to the Universe at large: we are the proverbial trees, and our roots are touching (whether brain/mind—to-brain/mind or Transpersonal knowing is not about sending brain/mind-to-world). superluminal messages from one tree top to another; it is a form of sharing the nonlocal connections below the surface of classical macro-reality. Ludwig Wittgenstein (1922/2016) is often quoted to have said: “Whereof one cannot speak, thereof one must be silent.” The emerging meta-reductive paradigm powerfully argues for expanding this view: Whereof we cannot speak, thereof we must share in silence.

References Acunzo, D. J ., Evrard, R., & Rabeyron, T. (2013). Anomalous experiences, psi and functional neuroimaging. Frontiers in Human Neuroscience, 7,

133-135. Anderson, P. W. (1972). More is different. Science, I 77, 393-396. Balduzzi, D., & Tononi, G. (2009). Qualia: The geometry of integrated

information. PLoS Computational Biology, 5(8), e1000462, 1-24. Beauregard, M. (2009). Effect of mind on brain activity: Evidence from neuroimaging studies of psychotherapy and placebo effect. Nordic

Journal of Psychiatry, 63, 5-16. Bern, D., Tressoldi, P., Rabeyron, T., & Duggan, M. (2015). Feeling the future: A meta-analysis of 90 experiments on the anomalous anticipation of random future events. F I 000Research, 4, 1-33. Bohm, D. (1990) A new theory of the relationship of mind and matter.

Philosophical Psychology, 3(2-3), 271-286.

Towards a Naturalistic Science of Transpersonal Experience

99

Burke, R. C., Gauthier, M. Y., Rouleau, N., & Persinger, M. A. (2013). Experimental demonstration of potential entanglement of brain activity over 300 Km for pairs of subjects sharing the same circular rotating, angular accelerating magnetic fields: Verification by sLORETA, QEEG

measurements. Journal of Consciousness Exploration and Research, 4(1), 35-44. Capra, F. (1984). The Tao of physics: An exploration of parallels between modern physics and Eastern mysticism. London, England: Flamingo. Cardena, E. (2014). A call for an open, informed study of consciousness. Frontiers in Human Neuroscience, 8, article 17, 1—4.

Carhart—Harris, R. L., Erritzoe, D., Williams, T., Stone, J. M., Reed, L. J ., Hobden, P. (2012). Neural correlates of the Colasanti, A., psychedelic state as determined by MRI studies with psilocybin.

Proceedings of the National Academy of Sciences, 109, 2138—2143. Chalmers, D . J. (1995). Facing up to the problem of consciousness. Journal

of Consciousness Studies, 2(3), 200—219. Craffert, P. F. (2015). Do out-of—body and near-death experiences point towards the reality of nonlocal consciousness? A critical evaluation. The

Journal for Transdisciplinary Research in Southern Africa, 11(1), 1-20. D’Aquili E. G., and Newberg, A. B. (1998). The neuropsychological basis of religions, or why God won’t go away. Zygon, 33(2), 187-201. Deacon, T. (1998). The symbolic species: The co-evolution of language and the brain. New York, NY: W.W. Norton.

Dennett, D. C. (2006). Breaking the spell: Religion as a natural phenomenon. New York, NY: Viking. Deglin, V. I., & Kinsbourne, M. (1996). Divergent thinking styles of the hemispheres: How syllogisms are solved during transitory hemisphere suppression. Brain and Cognition, 31, 285-307. Dodd, M. S., Papineau, D., Grenne, T., Slack, J. F ., Rittner, M., Pirajno, F ., Little, C. T. (2017). Evidence for early life in Earth’s oldest & hydrothermal vent precipitates. Nature, 543(7643), 60—64. Duane T. D., & Behrendt T. (1965). Extrasensory electroencephalographic induction between identical twins. Science, I 5 0(3694), 367.

Edelman, G. M. (2004). Wider than the sky: The phenomenal gift of consciousness. New Haven, CT: Yale University Press. Eigen, M. (2012). Kabbalah and psychoanalysis. London, England: Karnac. Peder, J. (2013). Fractals. New York, NY: Springer Science+Business Media. Flor-Henry, P., Shapiro, Y., & Sombrun, C. (2017). Brain changes during a Shamanic trance: Altered modes of consciousness, hemispheric

Chapter Three

100

laterality, and systemic psychobiology. CogentPsychology, 4:1313522, 1-25. Frecska, E., Hoppal, M., & Luna, L. E. (2016). Nonlocality and the Shamanic state of consciousness. NeuroQuantology, I 4, 155—165. Freud, S. (1975). Psychoanalysis and telepathy. In J. Stratchey (Ed.) The

standard edition of the complete psychological works ofSigmundFreud, (Volume. XVIII) (pp. 177-193). London: Hogarth. (Original work published 1921). Friedman, H. (2002). Transpersonal psychology as a scientific field.

International Journal ofTranspersonal Studies, 21, 175—187 Gisin, N. (2009). Quantum nonlocality: How does nature do it? Science

326(5958), 1357-58. —. (2014). Quantum chance: Nonlocality, teleportation and other quantum marvels. Geneva, CH: Springer. Gowlett, J. A. J. (2016). The discovery of fire by humans: A long and

convoluted process. Philosophical Transactions of the Royal Society, B 371, 1-12. Hameroff, S., & Penrose, R. (2003). Conscious events as orchestrated space-time selections. NeuroQuantology, 1(1), 10-35.

Harari, Y. (2014). Sapiens.‘ A brief history of humankind. New York, NY: Random House. Hove, M. J ., Stelzer, J ., Nierhaus, T., Thiel, S. D., Gundlach, C., Margulies, Merker, B. (2015). Brain network reconfiguration and D. S., perceptual decoupling during an absorptive state of consciousness. Cerebral Cortex, 26(7), 3116—3124. Iaeoboni, M. (2007). Face to face: The neural basis of social mirroring and

empathy. Psychiatric Annals, 37(4), 236-241. Jahn, R. G., & Dunne, B.J. (2001). A modular model of mind/matter

manifestations (MS). Journal of Scientific Exploration, 15(3), 299—329. —. (2011). Consciousness and the source of reality. Princeton, NJ: ICRL. Josephson, B.D., and Pallikari-Viras, F. (1991). Biological utilization of

quantum nonlocality. Foundations ofPhysics, 21 (2), 197-207. Jung, C. G. (2011). Synchronicity: An acausal connecting principle. Princeton, NJ: Princeton University Press. (Original work published

1959) Kahneman, D. (2011). Thinking, fast and slow. Toronto, ON: Doubleday Canada.

Kelso J. A., & Engstrom, D. A. (2006). The complementary nature. Cambridge, MA: MIT Press.

Laughlin, RB. (2005). A difierent universe. New York, NY: Basic.

Towards a Naturalistic Science of Transpersonal Experience

101

Lewis-Williams, D., & Pearce, D. (2004). Inside the Neolithic mind:

Consciousness, cosmos and the realm of the gods. London, England: Thames & Hudson. Maldonado, C. E., & Gomez-Cruz, N. A. (2014). Synchronicity among biological and computational levels of an organism: Quantum biology

and complexity. Procedia of Computer Science, 36, 177-184. Mandelbrot, B. (1989). Fractal geometry: What is it, and what does it do?

Proceedings of the Royal Society of London A, 423, 3-16. Marks-Tarlow, T. (2015). The nonlinear dynamics of clinical intuition.

Chaos and Complexity Letters, 8(2-3), 1-25. —..(2019).

A

fractal

epistemology

for

transpersonal

psychology.

7 International Journal ofTranspersonal Studies 38(2), n.p. Mayer, E. L. (2007). Extraordinary knowing: Science, skepticism, and the inexplicable powers of the human mind New York, NY: Bantam. McGilchrist M. (2009). The master and his emissary: The divided brain and the making of the Western world. New Haven, CT: Yale University Press.

Mercier P. (2008). Night train to Lisbon. New York, NY: Grove. Montaigne, M. (1958). The complete essays of Montaigne. Redwood City, CA: Stanford University Press.

7

Nagel, T. (2012). Mind & cosmos: Why the materialist neo-Darwinian conception ofnaiure is almost certainly false. Oxford, England: Oxford University Press. Newberg, A. B & d’Aquili, E. G., (2000). The neuropsychology of religious and spiritual experience. Journal of Consciousness Studies, 7(11—12),

251—66. Northoff, G & Heinzel, A. (2006). First-person neuroscience: A new methodological approach for linking mental and neuronal states.

Philosophy, Ethics and Humanities in Medicine, I (1), 3-10. Pechey, R., & Halligan, P. (2012). Prevalence and correlates of anomalous

experiences in a large non-clinical sample. Psychology and Psychotherapy: Theory, Research and Practice, 85, 150—162. Penrose R. (1997). The large, the small and the human mind Cambridge, England: Cambridge University Press. Persinger, M, A., Saroka, K. S., Koren, S. A., & St—Pierre, L. S. (2010). The electromagnetic induction of mystical and altered states within the

laboratory. Journal of Consciousness Exploration and Research, I (7), 808-830. Popper, K. (1959). The logic of scientific discovery. London, England: Hutchinson.

102

Chapter Three

Puthoff, H. E. (1996). OLA-initiated remote-viewing program at Stanford

Research Institute. Journal of Scientific Exploration, I 0(1), 63-76. Pylkkanen, P. (2014). Can quantum analogies help us to understand the process of thought? Mind & Matter, 12(1), 61-91. correlations Radin, D. (2004). Event-related electroencephalographic

between isolated human subjects. Journal

of Alternative and

Complementary Medicine, 1 0(2), 315—323. —. (2006). Entangled Minds: Extrasensory experiences in a quantum reality. New York, NY: Paraview Pocket.

Randall, L. (2013). H iggs discovery: The power of empty space. New York, NY: HarperCollins. Rao, K. R, & Palmer, J. (1987). The anomaly called psi: Recent research and criticism. Behavioral and Brain Sciences, 10, 539-643. Shapiro, Y. (2015). Dynamical Systems Therapy (DST): Theory and

practical applications. Psychoanalytic Dialogues, 25(1), 83-107 —. (in press). Toward a science of transpersonal phenomena: Commentary on “A fractal epistemology for transpersonal psychology” by Terry

Marks—Tarlow, International Journal of Transpersonal Studies, 38(2), n.p. Shapiro, Y., Marks-Tarlow, T., & Fridman, J. (2017). Listening beneath the words: Parallel processes in music and psychotherapy. American

Journal of Play, 9(1), 228-251. Shapiro, Y., & Scott, J. R. (2018). Dynamical Systems Therapy (DST): Complex adaptive systems in psychiatry and psychotherapy. In E.

Mitleton-Kelly, A. Paraskevas, & C. Day (Eds.) Handbook of research methods in complexity science: Theory and application (pp. 567-589). London, England: Edward Elgar.

Smolin, L. (2019). Einstein ’s unfinished revolution: The search for what lies beyond the quantum. Toronto: Alfred A. Knopf Canada. Standish, L. J ., Johnson, L. C., Kozak, L., & Richards, T. (2003). Evidence of correlated functional magnetic resonance imaging signals between

distant human brains. Alternative Therapies, 9(1), 122-128. Stapp H. P. (1997). Science of consciousness and the hard problem. The

Journal ofMind and Behavior, 18(203), 171- 194. —. (2011). Mincb‘ul universe: Quantum mechanics and the participating observer (211d ed.). Berlin, Germany: Springer-Verlag. Stern, D. (2007). Applying developmental and neuroscience findings on other-centered participation to the process of change in psychotherapy.

In S. Braten, S (Ed.), On being moved: From mirrorneurons to empathy (pp. 35-47). Amsterdam, Netherlands: John Benj amins.

Towards a Naturalistic Science of Transpersonal Experience

103

Stolorow, R. D. (1997). Dynamic, dyadic, intersubjective systems: An evolving paradigm for psychoanalysis. Psychoanalytic Psychology,

14(3), 337-346. Tarlaci S. (2011). From quantum biology to quantum neurobiology.

NeuroQuantology, 9(4), 692-701. Tart, C. (1972). States of consciousness and state-specific sciences. Science,

176, 1203-1210. Tennes, M. (2007). Beyond intersubjectivity: The transpersonal dimension of the psychoanalytic encounter. Contemporary Psychoanalysis; 43(4),

505—525. Theise, N. D ., & Kafatos, M. C. (2013). Sentience everywhere: Complexity theory, panpsychism & the role of sentience in self—organization of the

universe. Journal of ConsciousnessExploration &Research, 4(4), 378— 390. Varela F. J., Thompson, E., & Rosch, E. (2017). The embodied mind: Cognitive science and human experience (2“‘1 ed.). Cambridge, MA: MIT Press. Vattay, G., Kauffman, S., & Niiranen, S. (2014). Quantum biology on the 7 edge of quantum chaos. PLoS ONE 9(3): e89017, 1-6. Wackermann, J. (2004). Dyadic correlations between brain functional states: Present facts and future perspectives. Mind and Matter, 2(1),

105—122. Wheeler, J.A. (2018). Information, physics, quantum: The search for links.

In W. H. Zurek (Ed), Santa Fe Institute studies in the science of complexity: Vol. VII. Complexity, entropy and the physics of information: The proceedings of the workshop on complexity, entropy,

and the physics of information heldMay - June, 1989 in Santa Fe, New Mexico (pp. 3-28). Boca Raton, FL: CRC / Taylor & Francis. Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959. Communications on

Pure andApplied Mathematics, I 3(1), 1-14. Wilber, K. (2001). Eye to eye: The quest for the new paradigm (3rd Ed). Boston, MA: Shambhala.

Wilson, E. O. (1998). Consilience: The unity of knowledge. New York, NY: Vintage. Wittgenstein, L. (2016). Tractatus logico-philosophicus. New York, NY: Chiron Academic. (Original work published 1922). Zimmerman M. E. (1994). Contesting earth ’sfitture: Radical ecology and postmodernity. Berkeley, CA: University of California Press.

CHAPTER FOUR TRANSPERSONAL PSYCHOLOGY AND FRACTAL EVOLUTION

J. ROWAN SCOTT1

Transpersonal psychology and reductive science Transpersonal psychology is a relatively recent subfield of the study of psychology focused on assimilating spiritual and transcendent human experiences within the framework of modern scientific psychology. The anomalous status of transpersonal phenomena sitting at the edge of the reductive sciences creates a unique opportunity to explore the limits of reductive scientific logic and to confront the shortcomings of the Reductive Scientific Paradigm. A fractal epistemology for transpersonal psychology can be developed within a meta-reductive perspective to confront the limits and shortcomings of the reductive paradigm. Transpersonal psychology is, therefore, well positioned to guide the reductive-scientific paradigm through a necessary adaptation or transformation. The transformation described brings transpersonal phenomena more fully within the natural sciences, but also creates a meta-reductive and fractal framework capable of addressing other unsolved reductive-scientific problems. Transpersonal experiences involve states of consciousness in which the sense of identity or the sense of self extend beyond the individual psyche or the personal to include the interpersonal, a sense of humankind, the greater ecosystem of life, or the cosmos (Walsh and Vaughn, 1993). Transpersonal psychology has also been defined as psychological development beyond what is considered “conventional” personal or individual levels of organization (Scotton, 1996). The phenomena studied by transpersonal psychology includes development of self beyond the ego, peak experiences,

1 Clinical Professor, University of Alberta, Department of Psychiatry Email: j .rowan.scott@ gmail. com

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altered states of consciousness, trance states, mystical experience, spiritual self-development, spiritual practices and crises, religious conversion, and numinous or sublime kinds of unusually expanded states of consciousness and experiences of being. Transpersonal psychology has been a domain of scientific research since

the early 1960’s (Lajoie & Shapiro, 1992). Since its inception, the transpersonal research community has been embroiled in a debate regarding the scope and definition of transpersonal research (Hartelius, Caplan, & Rardin, 2007). The criticisms faced by transpersonal psychology include a critique that the field introduces a subtle but deeper form of Cartesian dualism (F errer, 2001), particularly when it attempts to employ anaturalistic metaphysical worldview that is considered by many to be inappropriate for the exploration of spiritual experience (F errer, 2014). Others suggest the field in many instances lacks conceptual, evidentiary, and reductive scientific rigor, and is underdeveloped as a natural science (Friedman, 2000, 2002). Reliance on qualitative, anecdotal, and introspective clinical evidence (Adams, 2006), and undervaluing quantitative research methods (Matthews, 1999) is suggested as limiting the scientific value of transpersonal research. Cunningham (2011) summarized a wide range of criticisms, which include the suggestion that transpersonal psychology employs a naive metaphysics and underdeveloped epistemology, incomplete operationalization of many of its concepts, lack of attention to biological and evolutionary underpinnings of behavior and experience, and inappropriate application of fundamental physics concepts as explanations of transpersonal states and altered states of consciousness. There is a tendency in the field to consider spiritual and transpersonal experience to be unique kinds of natural phenomena, and data that resist conceptualization as quantifiable states and processes, which may not be well suited for conventional reductive scientific inquiry (Elmer, MacDonald, & Friedman, 2003). There is a split between more integrative research approaches and strong reductive research approaches in transpersonal research. The integrative side of the split accepts as appropriate diverse research methods, including qualitative reports of subjective experience, eschewing narrowly defined reductive and objective scientific and experimental approaches. In so doing, it accepts the ineffable, ambiguous, and uncertain boundaries of subjective experience associated with a wide range of transpersonal phenomena. On the other side of the divide, a strong reductive approach guides the research community exploring the same transpersonal phenomena but in a more immediate and rigorously reductive manner, as a local experience reducible to particular brain states. The more reductive perspective often dismisses

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subjective exploration and reports as a valid research method in favor of rigorous reductive logic and objective empirical study (Marks-Tarlow, this volume). The neurosciences carry on a similar debate, divided on acceptance of subjective reports as evidence versus accepting only objective—rigorousreductive measures in research protocols. Phenomena, such as attachment in a parent and child relationship, are described in interpersonal neuroscience (Siegel, 2012) as involving an extended and interactive interpersonal social mind, transcending brain states (Siegel, 2016). In the alternative, more reductive approach in the neurosciences, the same interpersonal attachment can be described with a more reductive methodology focusing on reducing this state of mind to states of brain. Researchers supporting the use of strong reductive logic often make the assumption that advances in brain imaging techniques will ultimately close the gap between subjective reporting and the objective images created in neuroirnaging studies (Boleyn-Fitzgerald, 2010). However, there are conceptual, theoretical and philosophical reasons indicating the remaining gap between subjective report and objective measure cannot be closed by functional neuroirnaging alone (Seager, 2012). These parallel debates in transpersonal psychology and the neurosciences are related to the residual Cartesian gap held open by the enigmatic split between Descartes’ res cogitcms (the immaterial mind) and res extenso (the material brain) (Grim, 2008). The debates are also associated with an interaction between Descartes’ contributions to the philosophy of science and his contribution to the philosophy of mind. The former includes scientific doubt and an early formulation of bottom-up reductive logic and

bottom-up reductive thought (Descartes, 1637/2009). The latter refers to a fundamental separation of dual, non-commensurate substances in the form of brain from mind. The link between Descartes’ contribution to the philosophy of mind and the philosophy of science is immortalized in his

famous phrase: “Cogito, ergo sum”—“I think: therefore I am ” (Cotting— ham, 1988). Descartes excluded mind from the early formulation of accessible phenomena studied by reductive science and from the reductive hierarchy of natural sciences, ostensibly in order to avoid persecution by the Catholic Inquisition (Stevens, 1995), but also because he formulated brain as scientifically accessible material substance and mind as a fundamental scientifically inaccessible immaterial substance. The interactive consequence of Descartes’ interrelated decisions in philosophy of science and mind intersect in a contradiction. “I think,

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therefore I am” validates the existence of mind, but dismisses mind from the domain of scientifically accessible phenomena. This statement collides with the modern manifestation of Descartes’ rigorous reductive logic. In the contemporary scientific era, rigorous sustained “bottom-up” reductive logic

arrives at the statement of the reductive epiphenomenalism of consciousness (REC) proposition, which efi'ectively reduces mind to quantum epiphenomena, dismissing consciousness, conscious will, intention and causal agency fi'om the scientific description of natural evolving phenomena in the Universe. Reductive epiphenomenalism CRobinson, 2015) derived by employing Descartes’ rigorous “bottom—up” reductive logic and his philosophy for natural science contradicts Descartes’ philosophy of mind, by finally

arriving at a condensed statement in the form: I think reductively, therefore I am not. The REC statement is a say-referencing paradox (Bolander, 2017). An intentional, conscious, “I am,” rigorously applying bottom-up reductive logic, paradoxically dismisses its own existence, rejecting conscious agency and causally impactful conscious “will” as well as reducing subjectivity to quantum epiphenomena—ultimately becoming “I am not.” The paradoxical statement leaves its reader oscillating between a conscious phenomenon, “I am,” caught in the act of “thinking reductively,” and the conclusion that consciousness is only a “quantum-based epiphenomenon,” thus “I am not.” Is it the subjective “I am” or the epiphenomenal “I am not” that is doing the

thinking? Is there any thinking at all? REC cannot be resolved without precipitating contradiction or paradox—a problem that modern science attempts to avoid in theory and practice (Yanofsky, 2016). The paradox can catastrophically collapse reductive logic into inconsistency, producing a trivial logic capable of proving anything and nothing (Scott, 2018). Some proposed resolutions of REC (Seager, 2012) choose to desert reductive logic, restricting its use to limited contexts in models of natural systems or abandoning entirely the only logic available to natural science (Dennett, 1995). Another proposed “solution” involves simply concluding, on the basis of no logic whatever, that REC is absurd and can be ignored. Although occasionally effective as workarounds, none of these “solutions” succeed in unraveling Descartes’ contradiction or the self-referencing paradox of REC, and all leave the value of reductive logic in doubt (Seager, 2012). Analysis of the structure and impact of Descartes’ self-referencing paradox, by using an analogy derived from the work of Kurt Godel on formal incompleteness, reveals an unrecognized limit on the application of

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reductive logic. In conformity

with Godel's proof of the two formal

incompleteness theorems, it is possible to define REC as an undecidable reductive proposition, thus protecting reductive logic from inconsistency, at the cost of declaring rigorous reductive logic susceptible to formal reductive incompleteness. One significant implication of reductive incompleteness is the realization that the scientific narrative, composed by the modern reductive science paradigm and bottom-up reductive logic, is fundamentally and formalbz incomplete (Scott, 2018). A further implication of formal incompleteness, as defined by Godel, is the possibility that alternative-formal systems using slightly different premises and axioms, may be able to decide what was undecidable in the initial-formal system in which incompleteness is first manifest. An analogy, when transferred across into reductive logic and reductive science, predicts that an undecidable reductive proposition found in the reductive paradigm, such as the REC proposition, could ultimately be resolved and decided, without inconsistency, within a more general meta-reductive scientific paradigm (MRP) employing slightly different premises, assumptions, and methods (Scott, 2018). There is great significance for transpersonal psychology in finding a limit associated with formal reductive incompleteness influencing “bottomup” reductive logic and reductive science. Transpersonal psychology remains a science at the edge of the reductive paradigm, and transpersonal phenomena are still considered anomalous fi'om within the “normal” natural sciences. The analogy with Godel’s work suggests a MRP may be capable of bringing both conscious agency and transpersonal phenomena with their ineffable, ambiguous, and uncertain boundaries into the mainstream of scientific thought.

Transpersonal psychology and the implications of incompleteness Transpersonal psychology began as investigations of anomalous phenomena, which are difficult to predict or replicate and not easily open to reductive analysis. No matter their orientation, all transpersonal researchers attempt to effectively address transpersonal phenomena using the diverse theoretical and experimental resources found across the natural sciences. In order to become a more-mature science, transpersonal psychology requires a strong explanatory—scientific theory supported by a robust body of experimental evidence, whether objective, subjective, or both. This goal appears to be

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within sight, particularly through the development of a fractal epistemology explored in this volume, by incorporating a more detailed understanding of “self-similar thinking” in natural science (see the other chapters in this volume). The discovery that the formal-reductive logic underpinning the reductive sciences has an unrecognized limit, in the form of reductive incompleteness (Scott, 2018), suggests the reductive sciences may be on the edge of a paradigm transformation similar to the crisis precipitated by Godel's original publication of the two incompleteness theorems (Casti and dePauli, 2000; Godel, 1931/1992; Kennedy, 2016; Raatikainen, 2015). Transpersonal psychology, as a science at the forefront of exploring reductive anomalies, may therefore be well positioned to offer insight into the nature of the crisis and the structure the transformation reductive science needs to undergo in order to subtly step past the limit set by reductive

incompleteness. It may well be that reductive science needs to mature in order to create a novel-scientific paradigm capable of holding transpersonal phenomena and theory. The historical foundational pillars of reductive science have been

reduction, replication, and prediction (Rosen, 2008). These three pillars were originally associated with a conception of laws of Nature (Feynman, 1965; Penrose, 2004). In the modern era, replication and prediction have become more narrowly defined and limited to very special contexts. Particularly influential in this regard was the appearance in the 1970s of chaos theory and nonlinear science (Scott, 2006, 2010), the development of alternative ways of viewing evolving rules in natural systems (Wolfram, 2002), the increased understanding of emergence and complex systems (Holland, 2014), as well as the development of fractal mathematics, and self-similar thinking, which now augments symmetry thinking in the physical sciences (Stauffer, Stanley, & Lesne, 2017). Modern understanding in fundamental physics further limits replication and prediction. There is increased awareness of entangled non-local probabilistic forms of causation occurring in quantum systems where nothing can exceed the speed of light (Gisin, 2014) and quantum participatory—conscious observers who are integrally involved in quantum—experimental set—ups (Stapp, 2007, 2009). The third foundational pillar of natural science, reduction and reductive thought, is limited by extremes where the three-step process of reductive

science, including reduction, theoretical formulation, and synthesis (Scott, 2010), breaks down or cannot successfully be applied. Reduction works within the scientifically accessible universe, but not everywhere within the

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accessible domain. If the earliest scientifically accessible state related to the unique origin in the Big Bang, or the later scientifically inaccessible states in the depths of black holes (Blundell, 2015), are homogeneous high-energy states (Pielou, 2001), then reduction is limited by there being nothing further to reduce. As well, at these extremes, the obj ective—scientific observer is precluded from existence and cannot observe (Rosen, 2008). Reductive logic may also be limited by being domain specific within the accessible universe (Shapiro, personal communication). For instance, if fundamental physical laws are found to be natural, but radically emergent phenomena enabled by prior states in the evolutionary sequence of the universe, they cannot be conservatively or reductively fully described or predicted by those prior states (Seager, 2012); then bottom-up reductive logic is insufficient when used alone to describe the radical emergence of fundamental physical laws in the sequence of natural evolution. In these circumstances, reductive thought may need to be supplemented by some form of complementary “emergent though ” (Laughlin, 2006). There are suggestions that some natural phenomena have characteristics

that cannot be reduced (Kauffman, 2008, 2016). Kauffinan’s interpretation often involves emergent phenomena that disappear when reduced, or phenomena that entail complex forms of emergent meaning or function that simply are not included in a bottom—up reductive description. Wolfram

(2002) has similarly suggested there are computationally irreducible phenomena that can only be understood by watching what they do. Reduction eliminates these phenomena and makes observation of their behavior impossible (Wolfram, 2002). Reductive thought aims to prove it could reduce and then synthesize a reintegration of the whole phenomenon. In reducing, theoretically formulating, and then synthesizing, the behavior of computationally irreducible phenomena can be better understood and again observed, even if the phenomena cannot be exactly replicated or predicted. Suggesting these are non-reducible phenomena may however be premature and ignores the adaptive power of reductive logic. Systems of wide degrees of complexity can be reduced (Seager, 2012), and selecting a narrowly defined system that excludes the complexity of the phenomena does not mean reductive logic could not be used successfully within an

extended definition of the selected system. Reductive incompleteness places a limit on reductive logic in a very precisely defined way (Scott, 2018). Bottom-up reductive logic works but it must be protected. When an unresolvable reductive proposition is located, it must be left unresolved; it must be relabeled as an undecidable reductive

proposition in order to avoid contradiction, self-referencing paradox, or

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reductive inconsistency. An undecidable-reductive proposition can then only be resolved and decided in an MRP. Reductive incompleteness therefore brings into focus a profound and precise logical boundary beyond which the bottom-up logic of the reductive-natural sciences contends with uncertainty regarding the relationship reductive-scientific epistemology has with meta-reductive epistemology and a meta-reductive interpretation of natural ontology. In exploring the epistemological implications of reductive incomplete— ness, it is reasonable to ask whether Nature may manifest natural— ontological forms of evolutionary compositional incompleteness, which is the hypothetical natural complement of reductive incompleteness (Scott, 2018)—the evolutionary logic of Nature could encompass a natural evolutionary form of compositional incompleteness, particularly in contexts involving mixtures of conservative and radical emergence (Seager, 2012). Epistemological forms of reductive incompleteness theoretically could parallel ontological forms of evolutionary compositional incompleteness. Isaac Newton was the first scientific mind to mention Nature’s conformity to herself (Gell-Mann, 2007). The modern interpretation of Nature’s conformity to herself may require inter-locked conceptions of ontologicalevolutionary incompleteness and epistemological-reductive incompleteness in a profound self-similar evolutionary pattern of incompleteness and fractal dynamics. In order to be considered scientific, theory and experimental investigation of transpersonal phenomena must be open to reductive logical analysis, prediction, and replication, or must clearly demonstrate how these scientific goals are limited. Even more importantly, according to Popper’s (1959) criteria, transpersonal phenomena and their explanation must be open to experimental verification or falsification. Therefore, an immediate goal of transpersonal research is to define a clear-scientific understanding of why transpersonal phenomena are hard to predict or replicate and difficult to analyze using bottom-up reductive logic. Transpersonal phenomena frequently theoretically or experimentally disappear when reduction is attempted, and thus, they may fit still poorly defined criteria for nonreducibility and computational irreducibility. The phenomena also offer particular challenges because of their fi'equent unconscious character, or a necessary suppression of conscious intention in order to allow the phenomena to occur. As well, they are difficult to study because of their non-linear character and their close relationship with emergence. The fuzzy boundary between ordinary states of consciousness and anomaloustranspersonal states of consciousness suggests there could be a pattern of

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systemic complementarity associated with transpersonal phenomena that may sometimes be associated with quantum entanglement (Shapiro & Scott, in press). Exploring the precise limit on reductive logic set by reductive incompleteness will make it possible to more clearly spell—out why bottomup reductive logic employed in the modern reductive science paradigm creates an incomplete picture and is insufficient to the task of effectively describing many natural phenomena, including transpersonal phenomena. Potential MRP’s could soon provide a necessary framework for scientific explanation shedding light on reductive incompleteness and the nature of the challenge transpersonal phenomena offer modern reductive science and the RP.

Historical examples consistent with reductive incompleteness Historical examples reveal the structure of what we might now call a Gédel Incompleteness Transformation. Kurt Godel’s work on formal incompleteness in logic and formal mathematical systems provides the initial framework for the specific scientific analogy defining reductive incompleteness, thereby justifying the pursuit of MRP3 (Scott, 2018). The scientific analogy can be generalized across any domain of human thought based on a definable logic. The structure and sequence of a Godel incompleteness transformation in any domain of human thought involves a transition from one logical frame of reference to another more encompassing logical frame of reference, precipitated by the discovery of an unresolvable contradiction, self-referencing paradox or potential logical inconsistency. In order to protect the underpinning logic, an undecidable proposition must be declared, so that novel frames of reference can be explored in search of one in which the unresolved and undecidable becomes resolvable and decidable. When an alternative framework is imagined into existence, the original logic is preserved and adapted, the inconsistency is resolved, the undecidable becomes decidable, and the original “problem” and its solution become more understandable. The novel, encompassing framework, will open onto an unexplored vista of logic, conceptual, scientific, or mathematical possibility. An historical example is found in the history of mathematics where the frame of the Gfidel Incompleteness Transformation appears in relation to

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the story of Pythagoras and his followers. In pursuing the application of whole numbers and whole-number fractions, they were confronted by: a2 + b2 = 02. A crisis was created in the Pythagorean “closed” universe and framework of whole numbers and whole number fractions by running into the unresolvable and undecidable number, V2. The crisis led from a presumed closed mathematical universe to the discovery of an open mathematical universe. The resolution of the unresolvable and undecidable V2 in the “whole” number domain appears in the form of the irrational numbers and the discovery of a vista of modern mathematics (Yanofsky, 2016). The frame of the Godel Incompleteness Transformation appears again in fundamental physics, when Einstein turned Newton’s unresolvable gravitational problem of instantaneous action at a distance (Gleick, 2003) into a resolvable proposition involving adapted premises defining spacetime and the invariant limit set by the speed of light in Relativity theory. This transition turned Newtonian mechanics into an approximation and a special case within Relativity theory (Scott, 2018', Yanofsky, 2016). Locating further examples of paradoxical scientific concepts that are unresolvable in the frame of the reductive scientific paradigm will reveal how prevalent undecidable reductive statements are and how significant reductive incompleteness may be for the future of reductive science.

Reductive incompleteness and its manifestations in modern science Three examples come to mind. The first example involves the unresolved conflict associated with the non-corresponding boundary between quantum mechanics and relativity theory (Laughlin, 2006). Quantum mechanics describes a universe in which phenomena are quantized. The discontinuity, non-locality, and indetenninism of quantum physics and quantum states create an unresolved non-correspondence with the smoothness, continuity, locality and determinism of relativity theory and relativistic states (Pylkkanen, 2007). These two inconsistent viewpoints arise from two contradictory but vastly successful theories that remain manifestly incommensurate. The fundamental incompatibility of the two perspectives suggests that a novel theoretical structure or paradigm might be required to resolve the problem and reach consilience. The resolution of the conflict might exist in a novel framework including an understanding of reductive incompleteness and an adapted understanding

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of reductive logic. An adaptation involving dual, bidirectional, “bottom-up” and “top-down” reductive logic may be required (Ellis, 2016). This transformation could lead to a novel, dual interpretation of system evolution, self-organization, emergence and complementarity. Thus, complementarity (Kelso & Engstrom, 2006) with dual, bidirectional evolutionary “compositiona ” incompleteness existing in Nature may require the development of a dual, bidirectional meta-reductive framework and paradigm folr science. Quantum mechanics and Relativity theory thus may operate in a complementary relationship inhabiting a bidirectional transition zone, existing between the two theories and the two natural domains, such that a mutual pattern of complementarity and emergence occurs across natural and theoretical boundaries (Humphreys, 2016; Laughlin, 2006). The second example involves the modern science of complexity which, like reductive science (Dennett, 1995), has no fixed definition and no agreed definition across its multiple interdisciplinary relations. Instead there are many shifting definitions for the concept “complexity” (Mitchell, 2011). Each definition highlights particular properties pursued by scientists seeking an understanding of evolution, self-organization, emergence, complex-adaptive systems (Holland, 2014), or even more complicated networks involving hierarchies of complex systems (Caldarelli & Catanzaro, 2012). There is uncertainty regarding when it is sufficient in the long sequence of evolution and emergence of the hierarchy of complexity to employ the “old” scientific metaphor describing reductive “bottom-up” causal interactions (Morowitz, 2002). When does it become necessary to apply a new metaphor in which reductive or complexity science must describe both “bottom-up” and “top-down” causal influence, as well as circular positive and negative feedback, “upward” and “downward” interactions in complex systems? It appears there is an expanding need to describe complex evolving systems of different degrees of complexity using dual, bidirectional, “bottom-up” and “top-down” causal influence (Ellis, 2016). The expansion of the metaphorical conception of “upward” and “downward” causation across many levels of complexity involves the concept of emergence, which originally was confined to more complex phenomena. With increased awareness plus theoretical and experimental exploration, emergence is now finding use in the fundamental sciences (Humphreys, 2016; Laughlin, 2006). The spreading web of application of basic concepts, such as evolution, self-organization, complexity, emergence, and complementarity, points

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again toward an impending shift in the historical “bottom-up” reductive premise of natural science. Nature, by conforming to herself, and natural science, by attempting to conform to Nature, may soon need to shift from the central historical “bottom-up” reductive premise in order to begin using a dual, bidirectional “bottom-up” and “top-down” model of cause-effect relationships throughout the hierarchy of sciences. A third example involves the anomalous and difficult to study phenomena of transpersonal psychology. The unresolved problems, the challenges to reductive logic as well as the limited range of “acceptable” permitted in conventional-research reductive-causal understandings protocols tend to push transpersonal phenomena into the “bin of anomalies” not easily explored by traditional “bottom-up” reductive logic and the MRP. Transpersonal phenomena range across many categories, including shamanic and meditative experiences, clairvoyance and remote viewing,

telepathy, precognition and premonition, as well as various mind-to-mind and mind-to-matter interactions (Shapiro & Scott, in press; Shapiro, this volume, chapter two). Each kind of transpersonal phenomena challenges simple “bottom-up” reductive interpretation. Specific challenges include diffuse boundaries, atypical non-local causal relationships, and hidden mechanisms that appear to contradict conventional bottom-up causality. The limited frame of a selected experimental reductive semi-isolated system may not contain a wide enough interpretation of potential-causal relationships to be capable of containing many transpersonal phenomena, which can be non-local, non-linear, theoretically non-reducible and computationally irreducible in nature. As well, epistemological reductive incompleteness and ontological evolutionary compositional incompleteness introduce the possibility of MR? more fit to incorporate transpersonal

scrence. Scientific causal explanation involving defined mechanisms is not abandoned in the face of reductive incompleteness. However, incompleteness does suggest that the list of kinds of causal relationships so far explored in natural science may still be unfinished. Three local kinds of causal interaction are well studied in natural science, including one thing influencing another, two things influencing each other, and a third thing influencing two other things. Since the discovery of quantum mechanics, other forms of causal influence have been studied. Acausal, non-local quantum entanglement of micro-scale systems defines a novel category of interaction in which a causal connection is sustained between two quantum states, but no information can be transferred faster than light between the two distant-entangled points (Gisin, 2014). Macro-state non-local entangle-

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ment has now been demonstrated between two human subjects at distances up to 300 km (Burke, Gauthier, Rouleau, & Persinger, 2013). Quantumparticipatory consciousness is another causal relationship having quantum character involving the micro-state of the quantum experiment and the macro-state of the experimenter’s conscious intention (Stapp, 2007, 2009). The proposed meta—reductive hypothesis discussed later in this paper predicts the existence of possible kinds of “downward” or “top—down” causal interaction involving emergence and complementarity in both fundamental and complex systems. These three examples demonstrate that the universe of causal interactions known by science may be growing and may be more complicated and complex than science has yet conceived (Dawkins, 2005). Reductive incompleteness and transpersonal phenomena may at first stretch human credulity, but they also have the potential to

expand the range of natural kinds of causal interactions still to be understood. MRPs may ultimately provide a framework in which transpersonal phenomena are encompassed, allowing for an easier “under-standing” of the kinds of interactions transpersonal research pursues. For instance, multiple reductive premises may need to be adapted before “reduc-tive” incompleteness associated with reductive epiphenomenalism can be resolved and transpersonal phenomena can become “acceptable” and more easily understood in a scientific meta—reductive frame.

011 the persistence of transpersonal phenomena, transpersonal psychology, and reductive science The attempt to achieve scientific validation for transpersonal psychology is a 60 years old project (Grof, 2008; Laj oie & Shapiro, 1992). The phenomena of transpersonal psychology and parapsychology, including Shamanic and meditative states, clairvoyance and the assumption of telepathy, premonition, divination, precognition and accounts of mind matter interaction, go as far back as recorded human history. Cultural artifacts in the archeological record, including complex-burial ceremonies suggesting “magical thinking” and Nature spirits, imply a very much older history, even predating Homo sapiens and possibly involving competing species such as the Neanderthals (Wong, 2017; Wood, 2005). Trephining, probably to remove evil spirits rather than based in a clear understanding of neuroscience and the use of burr holes to relieve pressure caused by trauma and hemorrhage is found in skulls going back at least 10,000 years to the Neolithic period at the end of

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the Stone Age (Irving, 2013; Stout, 2017). Therefore, transpersonal phenomena likely go back millions of years, particularly if they are not limited to our species. If some transpersonal experiences are based in fundamental-physical processes, they may preexist and outlive our species. There is accumulating evidence that precursors of “scientific thought”

in human consciousness are at least 4000 years old (Fara, 2009). “Precursors” of a “scientific sense” might be as simple as cognitively formulating an intention to try something to see what happens—a behavior observed in many species. The 400-year history of success of the reductive scientific paradigm partially depends on the adaptability and shifting definitions of the reductive paradigm. This makes it difficult to tell the difference between less dramatic adaptations and more serious transformations, which might revolutionize a single science or change the whole reductive paradigm (Nickles, 2017). While specific reductive sciences have undergone much adaptation, the paradigm as a whole has never gone through a scientific revolution of the kind predicted by Kuhn

(Kuhn, 2012). The basic “mathematical sense” appears in monkeys and orangutans as well as in early hominid predecessors (Dehaene, 2011; Holt, 2018). The embodied experience of mathematics (Lakoff & Nunez, 2000) appears to emerge in even deeper eras of natural evolution, suggesting a basic mathematical sense has been around for millions, if not billions, of years and involves a multitude of species. Homo sapiens has so far lasted about

200,000 years and Gott (1993) estimated we may survive up to 8 million years at a 95% confidence level (Holt, 2018). Therefore, if life survives on Earth, the “number sense” and the “scientific sense” are likely to survive our species and many other species like ours. The study of irregular mathematical shapes is just 50 years old. Benoit Mandelbrot coined the word “fractal” around 1975 when he began writing the book, The Fractal Geometry ofNaiure (Mandelbrot, 1977). Constructing a scientific and mathematical epistemology based on fractal conceptions is therefore a very recent conception. Nature, however, seems to have ontologically been manifesting self-similar, fractal structures and processes (Mandelbrot, 1977; Schroeder, 1991), as well as manifesting structures and processes best described by symmetry and group theory, for close to 14.3 billion years (Lederman & Hill, 2004; Rosen, 2008). The fractal epistemology suggested by Marks-Tarlow (this volume) could provide an important adaptation and fi'amework for the development

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of future transpersonal research. For it to be most effective, however, the epistemology may need to be encompassed in a novel, bidirectional MRP bringing to an end the era of the monist, “bottom-up” Reductive Paradigm. Transpersonal psychology and transpersonal phenomena may find an easier scientific home within a transformed MRP.

The fractal boundary between epistemology and ontology Benoit Mandelbrot coined the word “fractal” to capture a particular group of irregular mathematical forms that characteristically contain self-similar and scale-invariant patterns within irregular geometric objects. The word “fractal” became the name for a class of mathematical structures Mandelbrot invented, with the Mandelbrot Set becoming the cultural symbol purported to be the most complex mathematical object known. In

his book, The Fractal Geometry of Nature, Mandelbrot suggested “fractals” are first invented in a human mind, but then also discovered and observed in a wide range of contexts in the vast complexity of Nature and her evolution. The mixed origin of the fiaetal concept, both invented by Mandelbrot and discovered in Nature, creates some of the sense of wonder and awe at the vast library of natural examples and beautiful computational demonstrations. Another property of many fractals inviting a sense of profundity is their relationship with infinity (Marks—Tarlow, this volume; Stewart, 2017). Although computational examples are practically limited by how much time one possesses to compute, in principle a fractal can be iterated infinitely, creating self-similar patterns on an infinite range of scales. Nature is also limited by the sequence of available time involving the natural evolution of our universe. On the largest scales, Nature is limited by the speed of light and the space-time dimensions of our universe and, on the smallest scales, limited by the Planck constant associated with quantum mechanics (Schroeder, 1991). Nevertheless, despite these practical and evolutionary limits, the fractal concept remains useful for engaging in “self— similar scientific thought.” Fractals reveal something profound about how human insight, imagination and observation can plumb the mysteries of the Universe, returning with what might be one inter-related element of the foundational design and pattern organizing the Cosmos (Barrow, 1992). Terry Marks-Tarlow, inspired by the fractal conception, ponders whether fiactals may be enough of a fundamental property of Nature to hope that they could form a natural basis for a scientific epistemology capable of

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providing guidance in healing the polarized divide that has developed in the transpersonal community. She states: As a clinician and theoretician, it is my hope and vision that the adoption of

a fractal epistemology might help to heal the divide between transpersonal psychologists, by eliminating any need to divide the field into more versus less rigorous subfields. (Marks-Tarlow, this volume, chapter one, p. 30).

The similarity between transpersonal theoretical polarizations, polarized theoretical divides in the neurosciences, and the deep relationship with the residual-reductive schism in the Reductive Paradigm, suggests MarksTarlow’s vision for a scientific-fractal epistemology may also have important implications for the future neurosciences, consciousness studies, and all the reductive natural sciences. How far can a formulation of a fractalscientific epistemology go in the task of resolving reductive science problems or bringing anomalous phenomena into mainstream-natural science? Could a fractal-scientific epistemology precipitate an adaptation of the reductive-science paradigm such that the universe could more easily be seen to be prone to evolve life, mind, and consciousness (N agel, 2012)? For an expansive transformation of scientific and mathematical understanding leading to such wide-ranging consequences, the fractal conception of an epistemology would need to tap into something very profound and fundamental in Nature. Marks-Tarlow asks: What constitutes the “real” thing? In other words, how do natural shapes

differ from human-made ones? Is there an archetypal meta-pattern, i.e., a pattern of patterns that Nature draws upon again and again? The answer appears to be “yes.” Nature leves recursively enfolded shapes, that is, patterns that are repeated again and again on multiple size and/ or time scales. (Marks-Tarlow, this volume, chapter one, p. 5)

Marks-Tarlow’s hope is that a fundamental ontological fractal pattern in Nature can be discovered, and that it might offer necessary insight and inspiration to formulate a fractal epistemology, which by closely approximating the natural pattern in evolution and the emergent hierarchy of complexity, could then offer means to heal the multiple rifts and chasms in many academic communities including transpersonal psychology. This is a far-reaching goal. However, the similarity between multiple academic debates cannot be ignored, and once seen, it seems inevitable that someone should suggest that resolving the polarization in one scientific domain might lead to resolution of many. As well, germane to the subject matter of fractals, the similarity between the various academic debates

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reveals itself to be self-similar, scaled and iterated in the evolution of the involved sciences: thus, inherently fractal. For these reasons, it is worthwhile stepping into the challenge in order to see if a fractal epistemology might illuminate a path toward a unification of the sciences (Cat, 2017), resolution of multiple polarized debates, and movement toward theoretical consilience (Wilson, 1999) of non-corresponding theoretical positions (Pylkkanen, 2007). Logicians, scientists, and applied mathematicians (Higham, 2015) employ scientific epistemology (Anderson, 2016; Steup, 2018) in an attempt to theoretically mimic and experimentally spell-out the inner workings of Nature, and thus to understand more fully the natural ontological logic (Hofweber, 2017) of evolutionary complexity. Nature’s conformity to herself, noted by Isaac Newton (Gell—Mann, 2007) is a kind of selfsimilarity. Self-similarity is a kind of symmetry (Schroeder, 1991). These ways of “thinking” seem to be closely associated with Nature’s inherent ontological evolutionary logic. Scientific models of Nature interweave symmetry and self-similarity, revealing iterative, algorithmic process in scaled relationships, reaching across the entire sequence of natural evolution and throughout the layers of the hierarchy of complexity. Conformity to self, self-similarity, and symmetry are human observations of Nature, as well as scientific and mathematical languages underpinning our modern scientific epistemology. A fractal epistemology could cross-link these significant conceptual and mathematical relationships, and could provide a novel approach for natural science to model fractal pattern within epistemology and ontology. How could a fractal epistemology heal a polarized academic split? The template of the imagined epistemological Fractal might iterate in a scaled, self-similar fashion in the conceptual space between the two polarized camps, making it clear a significant fractal relationship can be woven in between the two positions in a large number of contexts. In the transpersonal case, an epistemological Fractal could fill the space and dimension between the two polarized positions in the debate. It might iterate as a self-similar, scalable conceptual frame, cross—linking the rigorous reductive scientific intention on one side of the debate with the integrative, subjective position on the other side of the debate. In hopes of offering something useful to the ongoing conversation, I turn my attention to creating a meta—reductive complex-adaptive system model (CAS). With a meta—reductive CAS in place, it will then be possible to resolve and decide the reductive epiphenomenalism of consciousness

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proposition (REC). Once the paradox of the REC is dealt with, it becomes possible to conceptualize a fiactal epistemology attempting to closely model itself on Nature, ontology, and the inherent self-similarity, symmetry, and conformity found in our evolving universe.

Imagining a meta-reductive complex adaptive system The vast history of success and adaptation of reductive natural science (Dennett, 1995; Fara, 2009) and the unfailing effectiveness of reductive logic within domain-specific regions of the scientifically accessible universe (Seager, 2012) suggest any imagined MRP must preserve and encompass “bottom-up” reductive logic within the structure of any proposed meta-reductive logic or MRP. An adaptation of “bottom-up” reductive logic is already in use in Complex System Science, in models of Complex Adaptive Systems (CAS) (Mitleton—Kelly et a1, 2018; Morales, Gershenson, Braha, Minai, & Barr-Yam, 2018), complex social systems (Miller & Page, 2007), and in models of brain, emergent mind, and consciousness (Shapiro & Scott, 2018). The experimentally verified presence of bidirectional “bottom-up” brain changing the mind, and “topdown” mind changing the brain, leads directly to the appearance of dual “bottom-up” and “top-down” reductive description in CAS models describing brain/emergent mind and consciousness (Beauregard, 2007; Miller & Page, 2007; Schwartz and Begley, 2002; Shapiro & Scott, 2018). A Meta-Reductive Complex Adaptive System model, using dual bidirectional reductive logic, can be composed as a first approximation for an encompassing MRP (See Figure 4-1).

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Figure 4-1. Meta-Reductive Complex Adaptive System Model: The “bottom-up” brain changes the mind; the “top-down” mind changes the brain. Figure 4-] depicts a Meta-Reductive CAS of brain, emergent mind and consciousness. An “upwar ” (or “bottom-up”) and “downward” (or “topdown”) metaphor is employed, associated with descriptions of causal relationships between brain and emergent mind. There is also an initial representation of a “horizontal” or “lateral” metaphor, which intends to capture extended interactions and relationships of the brain, mind, and consciousness reaching beyond the boundary of the brain and beyond

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personal-subjective experience within the mind. Such extended interactions may entail quantum or fundamental physics properties such as entanglement but will also appear in the sequence of individual development, in interpersonal and inter-subjective interactions, in complex social and cultural contexts, in environmental engagements, and in various kinds of transpersonal experiences of the cosmos. A number of features of the Meta-Reductive Complex Adaptive System model depicted in Figure 4-1 are worth highlighting: - The historical and modem logic of modern “bottom-up” reductive science is encompassed as a special case and component of the model, embedded in the dual, bidirectional reductive logic of the MetaReductive CAS framework. 0 A modern “bottom-up” reductive CAS model contains “anomalous” but scientifically verified “downward” influences, which are now encompassed and embedded in the Meta-Reductive CAS frame as scientifically verified—causal influences. 0 Theoretical and experimentally accessible predictions of a Meta— Reductive CAS model are included. The Meta-Reductive

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Resolving and deciding “bottom-up” reductive epiphenomenalism A meta-reductive CAS and related Meta-Reductive Paradigm (MRP) using an adapted premise involving dual, bidirectional “bottom-up” and “topdown” reductive logic appears sufficient to collapse the contradiction and self-referencing paradox of reductive epiphenomenalism of consciousness (REC) (Scott, 2018). In an adapted MRP, the REC proposition becomes resolvable, and Descartes’ dualism of brain/mind will find alternative

solutions. The brain and em ergent mind can be described as a bidirectionalcomplementary form of complex organization composed fiom the neural

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components of the brain mutually interacting with the emergent mind with its significant developmental interactions and input from the interpersonal world and greater environment. From the initial moment of individual hum an development, brain/m ind are mutually engaged with “brain changing the mind” and “mind changing the brain.” The developmental process involves many causally impactful inter-level hierarchical exchanges and inter-system interactions occurring within local and extended brainemergent mind networks (Siegel, 1999, 2012, 2016). The complex evolving composition, mutual complementary interactions and systemic relationships defining the emergent brainfmind are all “reducible,” but they are only very narrowly “reducible” to just the brain. The mind relates to the interpersonal world and to the universe as a whole. Complex system modeling of brain and emergent mind must be expanded to include “being” and “experiencing” the universe as a “whole”—the

system being reduced is therefore significantly expanded. In a dual, bidirectional-reductive paradigm, “top-down” patterns of interaction must be included in the list of possibilities. The proposed MRP in which the m etareductive CAS is contained further differentiates reductive analysis into four categories, listed as follows. Beyond this, other possible forms of analysis may be created in the future supplementing dual “bottom-up” and “top-down” reductive analysis. It is possible that the reductive method itself will fail in certain domains within complexity—possibly, in irreducibly complex/non—computational systems where some form of complementary holistic/non—reductive analysis may have to be employed (Shapiro, personal communication). “Bottom-up” reductive epiphenomenalim of consciousness cannot be

logically, scientifically or philosophically sustained in a bidirectional CAS or MRP. The absurd implications of REC, including reduction to quantum epiphenomena, can be rejected, resolved, and decided in a dual Meta— Reductive Paradigm, where REC is transformed into a description of dual, “bottom-up” and “top—down” bidirectional reductive phenomenalism, with causally complex, causally impactful brain and emergent mind, and intentional consciousness. Although adapting the singular reductive premise into a dual, bidirectional premise appears sufficient to confront and resolve reductive epiphenomenalism, the adapted dual-reductive premise is not sufficient to create an MRP or to address the complex and anomalous phenomena of transpersonal psychology. For this, further adaptations to the reductive paradigm are required.

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Moving step—by—step toward a meta-reductive paradigm The dual, bidirectional, “bottom -up” and “top-down” reductive logic of the Meta-Reductive CAS model is a good jumping off point for the proposed structure of a Meta—Reductive Paradigm (MRP). The MRP employs multiple altered and adapted premises, assumptions, and methods, differentiating it from the Reductive Paradigm (RP). In each step, the RP is encompassed as a special case Within the MRP. The MRP addresses reductive incomplete-ness and attempts to fill in the unfinished narrative of the Reductive Paradigm.

1. Adaptive semi-isolated system The Meta-Reductive frame encompasses and then expands the usual reductive scientific concept of a semi-isolated system (Rosen, 2008). As employed in reductive science, a selected semi—isolated system of interest is a reduction intended to simplify the task of theoretical and experimental scientific investigation. The reductive semi-isolated system is limited to a selected phenomenon of interest (for example, brain and emergent mind), with a presumptive lower “bottom-up” subsystem of the cause (brain) and a presumptive higher “bottom-up” subsystem of the eflect (emergent mind). The selected reductive semi-isolated system of interest is presumed to fully contain the important causal relationships involving brain and emergent mind. By contrast, the Meta-Reductive definition encompasses and preserves the usual “bottom-up” reductive semi-isolated frame but then extends its conceptual reach. The Meta-Reductive semiisolated system also includes mappings of possible “bottom-up” and “top— down” influences, as well as extended “lateral” or “horizontal” influences.

This begins the process of stepping out of the RP and stepping into a space of multiple adjacent possible meta-reductive paradigms.

2. A special meta-reductive paradigm The abstract, structure and composition of the proposed MRP are most easily understood through a specific example (see Figure 4-2).

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model the brain, emergent mind, and consciousness. The limited special Meta-Reductive Paradigm of brain/mind can be applied in the neurosciences, the study of consciousness, and m transpersonal psychology. It employs dual bidirectional-reductive logic and the expanded definition of a semiisolated system.

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The layers and domains of natural systems, their composition, interaction, and relationship, require synthesis and integration in the proposed metareductive model of brain, mind, and consciousness—these are represented

by the short/lighter and long/darker horizontal bars. In the quantum realm, both holistic and reductive perspectives are developed The theory addressing these two alternative viewpoints is represented by David Bohm’s theoretical formulation of implicate and explicate evolution in quantum systems (Pylkkanen, 2007), and by his theory of the relationship between mind and matter (Bohm, 1990). Holistic and reductive perspectives are then developed throughout the layers of MRP theoretical formulation. The darker and longer bars represent systemic

relationships, across and between domains. The lighter and shorter bars represent more local descriptions and consequences related to brain, emergent mind and consciousness. The scientific process of reductive

analysis, theoretical formulation and synthesis is dual, with bidirectional “bottom -up” and “top-down” reductive models of causal influence. The dual causal description leads to a further division of the reductive analysis, theoretical formulation and synthesis into 4 fields represented in the following way: 1. The bottom leftfield of the diagram models the encompassed special case of “bottom—up” logic describing “bottom—up,” enabling influences leading to organization and composition of brain, as well as emergent interactions and relationships of mind. 2. The upper left field of the diagram depicts the “bottom-up” composition, interaction and extended “horizontal” engagement of brain and mind in the environment, interpersonal world, and cultural contexts. 3. The upper right field of the diagram depicts extended “horizontal” interactions and relationships, as well as “top-down” influences arising from the environment, social and interpersonal world.

4. The lower fight-hand field of the diagram depicts “top-down” regulation and influence of the mind on the brain. 5. The center field of the diagram depicts consciousness and selfreferencing, self-reflective participatory consciousness. The special MRP can be transformed into a general MRP intended for application across all the natural sciences. In order to achieve this, I first formulate a hypothetical fractal template.

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3. A fluctal unit of organization The most powerful fractal epistemology will be one that not only fills the space and dimension between the polarized positions in the transpersonal debate, but one that does so throughout the hierarchy of natural sciences, from fundamental quantum physics, through neuroscience and interpersonal neuroscience, and into the anomalous domains of consciousness and mind,

finally to integrate the fractured-conceptual dimensions of transpersonal psychology. Such a fractal-epistemological framework should closely approximate ontological-fractal templates discovered in Nature. The templates may be found to iterate on all scales in Nature, and also to iterate in a scientific evolutionary narrative that models Nature. Such a self-similar

pattern, both invented in scientific imagination and discovered in the sequence of evolution, would reveal a profound fractal “design” in Nature that could be used to cross-link a fractal—scientific epistemology with Nature’s fractal-evolutionary ontology. One intriguing property of fractals is how a simple template, composed of a simple geometric form, simple iterative rule, or a simple mathematical formula, can encapsulate everything necessary in order to produce a vastly complicated mathematical or geometric structure. Many abstract mathematical and geometric fractals, including the Mandelbrot set, produce very complicated geometric forms, but they are forms with little logical depth and very little effective complexity (Scott, 1997). A fractal epistemology intending to model the universe must conform to already well-validated reductive scientific facts. The accessible point of origin of our universe is believed to be an undifferentiated, very high-energy state localized in a very small and unique initial condition (Pielou, 2001). Some 14 billion years later, in the present moment of natural evolution, our own brains and emergent minds are likely the most complex evolved

phenomena hum ans have access to. If Nature employs a template, an archetypal meta-pattem, linking these two extremes of evolution, the fractal template in ontology studied and mirrored by natural science, must be capable of starting from an initial undifferentiated state and evolving toward vast complexity, ultimately to embody an immense degree of diversity, much logical depth, and a very high degree of effective complexity in the context of the brain and emergent mind. An ontological fractal template, mimicked by an epistemological fractal template, should therefore conform to Einstein’s advice—it should be as simple as possible, but not simpler. A proposal fitting these essential

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conditions creates a self-similar epistemological template mapping a parallel self-similar ontological template (see Figure 4-3). Initiate a self-similar, iterative, scaled, theoretical and experimental process of meta-reductive scientific exploration and approximation aimed at determining whether a fractal Epistemology can model a fiactal Ontology in an MRP.

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The Fractal Complex Natural History defines a self-similar template on the epistemological side of the fractal formulation (Fodor & PiatelliPalmarr'ni, 2010; Unger & Smolin, 2015). The parallel-ontological template hypothesized to be the simplest, iterative, scalable, self-similar fractal organizational pattern in natural evolution can then be defined as a Fractal

Complex Cycle of Existence. These two inter-related abstract conceptions and hypothetically realized templates are then used to create a fractal framework for describing everything, from the cycle of existence of the

universe as a whole, the cycle of existence of a single photon, or the multitude of forms and cycles of existence involving life, the evolution and

complexity of brain and emergent mind, and ordinary or altered states of consciousness.

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The dual, bidirectional investigative structure of the MRP can then be used to analyze segments in the sequence of evolutionary change in selected Fractal Complex Natural Histories and related Fractal Complex Cycles of Existence. It is now possible to suggest a framework for a general MRP.

4. A general meta-reductive paradigm

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The special and general MRPs encompass the RP 4-dimensional space-

time frame of reference, and then employ an extended MRP eight—foldframe of reference. A selected phenomenon of interest (P) is placed within the modern RP

four-fold frame of reference, which utilizes three dimensions of space and a fourth dimension of time. In modern science space and time are complex and unresolved scientific conceptions (DiChristina & Gawrylewski, 2018; Greene, 1999, 2004, 2011; Le Poidevan, 2015; Markosian, 2016; Zeh, 2010). The exact nature of space and time continue to be debated, dividing quantum physics and relativity theory, leaving fundamental questions about quantum physics unanswered (Pylkkanen, 2007; Schlosshauer, 2011; Smolin, 2013; Unger & Smolin, 2015). Despite the controversies and unsolved fundamental issues, these concepts can still be used as a useful reference frame in scientific work (Penrose, 2004).

The phenomenon of interest (P) is then placed in the MRP eight-fiJld frame, which includes four RP dimensions of space and time as well as the four-fold reductive analyses of the MRP. The extended vertical and horizontal metaphor, the four reductive analyses in the eight-fold frame, and the extended semi-isolated system of the general MRP encompass the RP. Brain, emergent mind and consciousness, as previously modeled in the MRP CAS and special MRP, provide a complex example of what the

general MRP can do in modeling sequences involving Fractal Complex Natural Histories and related Fractal Complex Cycles of Existence.

5. From reductive to meta-reductive scientific observers The historical Reductive Paradigm (RP) and its objective scientific approach is based on the assumption that consciousness should not and does not influence the essential experimental set up (Fara, 2009; Rosen, 2008).

This leads to the assumption of an objective-scientific observer. Over the last four centuries, a range of further “anomalous” scientific observers have appeared in the RP and have become increasingly relevant as natural science pursues diverse subjects of scientific, mathematical, and computational investigation. 0 0

The reductive objective observer is still maintained in many contexts. An anomalous reductive quantum participatory observer is

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postulated (Stapp, 2009). consciousness and its A participatory observer—participatory influence on the experimental set-up can limit the selected reduction and alter the experimental interpretation. between two observation Inter-subjective observer—mutual participants leads to the shared construction of meaning in a shared two-person context.

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Interpersonal three-person observer—a two-person context is

observed by an observing third person. Transpersonal observer—various transpersonal phenomena require complicated experimental arrangements of one, two, three, or more, participants and observers (Shapiro & Scott, in press).

The MRP firmly places consciousness and mind back in science, Nature, and the evolving Universe, and therefore requires multiple observer

positions in scientific work. The MRP then entails multiple kinds of scientific thought (see below).

6.

Kinds of meta-reductive scientific thought

The modern reductive science paradigm (RP) employs only one kind of scientific thought composed around “bottom-up” reductive logic. The altered premises so far composed under the MRP encompass the singular RP “reductive kind of scientific thought” and integrate it with an additional “three kinds of scientific thought” (See Figure 4-5). The four MRP ‘kinds of scientific thought’ are composed in a cyclic pattern (a self-similar, iterative and scaled structure).

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‘Halisllc Though? addresses the

of complexity. ‘Mela-Reducn‘w Emergent ‘Reduclive Emergent Though!’ c‘ ‘ a

Meta-Reductive Thought

Figure 4-5. “Reductive thought” can be encompassed and synthesized in a composite of four kinds of “meta-reductive thought.”

In combination, the four kinds of scientific thought depicted in the MetaReductive frame predict multiple kinds of “top-down” emergence may be

possible and should be looked for in experimental contexts. 7. Systems thinking in applied mathematics and science In order to function effectively, the MRP must encompass and integrate the

co-evolutionary relationship shared by science and mathematics (Higham, 2015; Stauffer et al., 2017;). The MRP highlights three specific inter-related forms of “mathematical thinking,” as well as their translation into each other: systems thinking, symmetry thinking, and self-similar thinking.

Historically, abstract formal system thinking and piecemeal conceptual scientific-system models have been used to describe particular aspects of natural-evolving systems (Berlinski, 2000; Higham, 2015; Reeves, 1993;

Schlosshauer, 2011). Beginning With Ludwig von Bertalanffy’s Work on

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General System Theory (von Bertalanffy, 1968), “systems thinking” became more widely applied throughout the natural sciences. In recent decades, the conceptual but not mathematical approach used in General System Theory, has been surpassed by many inter-related applied mathematical-system models (W olfi'am, 2002). Within the web of interrelated modern-system models there are many properties best described by

fractal conceptions (Flake, 1998). An important equivalence of system models reveals how formal incompleteness and reductive incompleteness should appear in applied mathematical contexts. Formal system models, dynamical system models, and information system models are all used in scientific contexts and these three models can be reliably and exactly translated into each other (Casti, 1994). Within reach of these equivalent and translatable system models are fractals (F eldman, 2013), non-linear chaos, power laws (Schroeder, 1991), and emergence (Scott, 2010). There should be within the conceptual and mathematical frame of relationships, three-equivalent translations of reductive incompleteness (Scott, 2018)—f0rmal system incompleteness, dynamical system incompleteness, and information system incompleteness.

8. Symmetry and self-similar thinking in science Symmetry is a unifying two-part mathematical concept underlying the scientific understanding of cause and effect relationships (Rosen, 2008). Symmetry and the related field of Group Theory provide an extremely powerful mathematical method underpinning scientific observation and the study of correlation, equivalence relationships, and symmetry relationships (Rosen, 2008). Through this powerful mathematical fi'amework, science can achieve its best approximation of causal relationships in natural systems. In modern reductive science, self-similarity, another two-part concept, is considered a special kind of symmetry, one in which symmetry presents

as self-similarity on logarithmic scales (Schroeder, 1991). Self-similarity and “self-similar thinking” can be used to infer and study self-similar causal relationships, which can be found linking divergent and convergent iterative phenomena on numerous scales of organization within natural evolution.

9. Causal relationships in meta-reductive science In understanding the four kinds of causal relationship previously discussed, the RP has focused its attention on determining the presence of simple correlation (two things keep happening together), then equivalence

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relationships (the subsystem of the presumed cause is the same as the subsystem of the presumed effect), and finally symmetry relationships (in which the symmetry of the subsystem of the presumed cause is preserved in one form or another in the subsystem of the presumed effect) (Rosen, 2008). The relationship between symmetry and self-similarity allows reductive and meta-reductive science to seek out causal relationships revealing symmetry or self-similarity across the entire sequence of natural evolution or within the details of the evolved and emergent hierarchy of complexity. The future understanding of causal relationships in an MRP encompasses the understanding derived in the RP but widens the application of mathematical methods and deepens the appreciation of possible relationships mind and consciousness can have within the understanding of causality. The MRP also deepens the appreciation of definable limits on the three historical pillars of science: redaction, replication, and prediction (Rosen, 2008), and clarifies the limit on reductive logic set by reductive incompleteness. A fractal MRP epistemology incorporating an appreciation of reductive incompleteness will a110w natural science to explore causal relationships associated with ontological evolutionary incompleteness and to more deeply understand incompleteness within the natural logic of Nature.

Conclusion: Ordinary to altered states of consciousness in a meta-reductive paradigm The Reductive Paradigm (RP), “bottom-up” reductive logic, and the RP concept of a semi-isolated system, shrink the evolving system of interest down to something “manageable,” but thereby exclude from the domain of scientific interest both Ordinary States of Consciousness (OSC) and Altered or Alternate States of Consciousness (ASC). By contrast, the MetaReductive Paradigm (MRP), the extended MRP semi-isolated system, the bidirectional frame of reductive logic in the MRP, as well as the epistemological fractal structure of the MRP, can hold any selected phenomena of interest within the scientifically accessible universe including

OSC and ASC. The epistemologz'cal-ontological fractal template of the MRP can be used to define an epistemological-fractal complex natural history and a related ontological-fractal complex cycle of existence. These MRP conceptual tools allow for description of OSC or ASC. OSC or ASC can be placed in the special or general frame of the MRP where selected states within the sequence of OSC or ASC evolution can be analyzed using the MRP fiamework.

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The “Fractal Template,” the cyclic definition of MRP “kinds of scientific thought,” as well as defined “kinds” of MRP mathematical thought, bring the reductive perspective of the RP into integration and consilience with a

bidirectional conception of emergence and an integrated holistic MRP perspective. The encompassed and preserved RP can continue to narrowly focus on particular reductive aspects of OSC or ASC. However, the MRP frame encompasses the narrow RP focus within an expansive emergent and holistic awareness. Ultimately, in a complete causal account any state of consciousness has a relationship with the “whole” universe, being enabled by the “whole” and potentially engaged with the “whole.” The expansive frame of the MRP is better positioned to provide a structure for investigating these complex relationships. Shapiro, in his contribution to this volume, supplies a table in which he summarizes the major kinds of OSC and ASC, and their probable neural substrates. In a previous paper, we (Shapiro & Scott, in press) created a functional taxonomy of transpersonal states, which were placed within an earlier iteration of a Meta-Reductive Paradigm. The present iteration of the MRP model proposed in this paper can incorporate within the rubric of “normal meta-reductive science” all OSC, as well as many, but not all, unusual and hard to replicate ASC pursued though the study of transpersonal psychology. Still, there remain some transpersonal phenomena that the expansive MRP framework will not be able to bring into the scientific fold. The MRP focuses on those transpersonal phenomena that can be contained within a framework underpinned by a definition of natural science, employing bidirectional-reductive logic, theoretical and experimental accessibility, and Popper’s criteria of falsifiability. Those transpersonal phenomena, including, for example, some aspects of subj ective—spiritual experience that cannot be experimentally operationalized will remain anomalous in relation to the fractal epistemology of the proposed MRP. A fractal epistemology and the Meta-Reductive Paradigm are capable of healing many of the polarizations and splits within different academic communities, including the field of transpersonal research. However, the discovery of reductive incompleteness introduces a logical and theoretical collision with theorists assuming complete and comprehensive scientific models are possible (Lincoln, 2017). The future understanding of reductive incompleteness will ultimately reveal the limited contexts in which science

can create models that are closed resolved decided and complete. Reductive incompleteness means scientific models, when sufficiently

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complex, must be open, unresolved undecided and incomplete. As well, future meta-reductive paradigms (MRP’s) will inevitably reveal their own “kinds” of reductive incompleteness (Scott, 2018). It is in the nature of the evolution of the universe, science, and human consciousness that natural evolution and the scientific work modeling it should be ever changing and unfinished. In conformity with Nature, the frames, models and theories of natural science must forever be incomplete.

References Adams, W. (2006). Transpersonal heterophenomenology? Journal of Consciousness Studies, 13(4), 89-95. Andersen, H., & Hepburn, B. (20 16). Scientific method. In E. N. Zalta (Ed), Stanford encyclopedia of philosophy Retrieved from: URL = 16/entries/scientific.

Barrow, ID. (1992). Theories of everything: The quest for ultimate explanation. London, England: Vintage. Beauregard, M. (2007). Mind does really matter: Evidence from neuroimaging studies of self-regulation, psychotherapy and the placebo effect

Progress in Neurobiology, 81(4), 218-236. Berlinski, D. (2000). The advent of the algorithm: The 300-year journey from an idea to the computer. New York, NY: Harcourt. Blundell, K. (2015). Black holes: A very short introduction. Oxford, England: Oxford University Press. Bohm, D. (1990). A new theory of the relationship between mind and matter. Philosophical Psychology, 3(2—3), 271-286. Bolander, T. (2017). Self-reference. In E.N. Zalta (Ed), Stanford encyclopedia of philosophy. Retrieved from: URL = .

Boleyn-Fitzgerald, M. (2010). Pictures of the mind: What the new neuroscience tells us about who we are. Upper Saddle River, NJ: Pearson. Burke, R.C., Gauthier, M.Y., Rouleau, N., & Persinger, MA. (2013). Experimental demonstration of potential entanglement of brain activity over 300 km for pairs of subjects sharing the same circular rotating, angular accelerating magnetic fields: Verification by sLORETA, QEEG measurements. Journal of Consciousness Exploration and Research,

4(1), 35-44. Calderelli, G., & Catanzaro, M. (2012). Networks: A very short introducti on. Oxford, England: Oxford University Press.

Chapter Four

13 8

Casti, I. (1994). Compleication: Explaining a paradoxical world through the science of surprise. New York, NY: Harper Collins. Casti, J., & DePauli, W. (2000). Godel: A life of logic. Cambridge, MA: Perseus.

Cat, J. (2017). The unity of science. In E.N. Zalta (Ed.), Stanford encyclopedia of philosophy. Retrieved from: URL = .

Cottingham, J.G. (1988). Descartes. In R.L. Gregory (Ed.), Oxford companion to the mind (pp. 189). Oxford, England: Oxford University Press.

Cunningham, PF. (2011). A primer on transpersonal psychology. Nashua, NH: Rivier College. Dawkins, R. (2005). W h y the universe seems so strange [Video file]. Retrieved from: https:l/wwwled.com/talks/richard_dawkins_on_our_queer_universe/u p-next?language=en

Dehaene, S. (2011). The number sense: How the mind creates mathematics (Revised Edition). Oxford, England: Oxford University Press.

Dennett, DC. (1995). Darwin’s dangerous idea: Evolution and the meanings of life. New York, NY: Simon and Schuster. Descartes, R. (2009) Discourse on the method of rightly conducting one ’s reason, and seeking truth in the sciences: Portland, OR: The Floating Press. (Original work published 1637) DiChristina, M . & Gawrylewski, A. ( E d s ) . (2018). A matter of time (Special collectors edition). Scientific American, 2 7(2). Ellis, G. (2016). How can physics underlie the mind: Top-down causation in the human context. Berlin, Germany: Springer-Verlag. Elmer, L.D., MacDonald, D.A., & Friedman, H.L. (2003). Transpersonal psychology, physical health, and mental health: Theory, research, and

practice. Humanistic Psychologist, 3I (2-3), 159-181. Fara, P. (2009). Science: A four thousand year history. Oxford, England: Oxford University Press.

Feldman, D P . (2013). Chaos and fractals: An elementary introduction. Oxford, England: Oxford University Press. Ferrer, I N . (2001) Revisioning transpersonal theory. A participatory vision

of human spirituality. Albany, NY: State University of New York Press. —. Transpersonal psychology, science, and the supernatural. Journal of

Transpersonal Psychology, 46(2), 152-186. Feynman, R. (1965). The character of physical law. Cambridge, MA: The MIT Press.

Transpersonal Psychology and Fractal Evolution

139

Flake, G.W. (1998). The computational beauty of nature: Computer explorations of fractals, chaos, complex systems and adaptation. Cambridge, MA: The MIT Press. Fodor, J ., & Piattelli-Palmarini, M . (2010). What Darwin got wrong. New York, NY: F arrar, Straus and Giroux. Friedman, H.L. (2000, May). Toward developing transpersonal psychology as a scientific field. Paper presented at Old Saybrook II conference, University of West Georgia, Carrolton, Georgia. —. (2002) Transpersonal psychology as a scientific field. International

Journal of Transpersonal Studies, 21(1), 175—187. Gell-Mann, M . (2007). Beauty, truth and physics? [Video file] Retrieved from: https:l/www.ted.com/talks/murray_gell_mann_on_beauty_and_truth_i n_physics/up-next?language=en 7

Gisin, N. (2014). Quantum chance: Nonlocality, teleportation and other quantum marvels. New York, NY: Springer. Gleick, J. (2003). Isaac Newton. New York, NY: Vintage.

Godel, K. (1992). On formally undecidable propositions 0 f Principia Mathematica and related systems (B. Meltzer, Trans.). New York, NY: Dover. (Original work published 1931) Gott, J .R. (1993). Implications of the Copernican principle for our future prospects. Nature, 363(6427), 315-319.

—. (2001). Time travel in Einstein’s universe: The physical possibilities of travel through time. New York, NY: Houghton—Mifflin.

Greene, B . (1999). The elegant universe: Superstrings, hidden dimensions, and the quest for the ultimate theory. New York, NY: Vintage.

—. (2004). The fabric of the cosmos: Space, time and the texture of reality. New York: Vintage.

—. (2011). The hidden reality: Parallel universes and the deep laws of the cosmos. New York, NY: Alfred A. Knopf.

Grim, P. (2008). Philosophy of mind: Brains, consciousness and thinking machines. Chantilly, VA: The Great Courses, The Teaching Company. Grof, S. (2008). A brief history of transpersonal psychology. International

Journal of Transpersonal Studies, 27(1), 46-54. Hartelius, G., Caplan, M . , & Rardin, M A . (2007). Transpersonal psychology: Defining the past, divining the future. The Humanistic

Psychologist, 35(2), 1—26. Higham, N J . (Ed.) (2015). The Princeton companion to applied mathematics. Princeton, NJ: Princeton University Press.

Hofweber, T . (2017). Logic and ontology. In E.N. Zalta (Ed), Stanford encyclopedia of philosophy. Retrieved from: URL =

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140

.

Holland, J.H. (2014). Complexity: A very short introduction. Oxford, England: Oxford University Press.

Holt, J. (2018). When Einstein walked with Godel: Excursions to the edge of thought. New York, NY: Farrar, Strauss and Giroux. Humphreys, P. (2016). Emergence: A philosophical account. Oxford, England: Oxford University Press. Irving, J. (2013). Trephination. Ancient history encyclopedia. Retrieved from: https:l/www.ancient.eu/Trephination/

Kauffman, S. (2008). Reinventing the sacred: A new view of science, reason and religion. New York, NY: Basic. —. (2016). Humanity in a creative universe. Oxford, England: Oxford University Press.

Kelso, J.A.S., & Engstrom, DA. (2006). The complementary nature. Cambridge, MA: The MIT Press.

Kennedy, J. (2016). Kurt Godel. In E.N. Zalta (Ed), Stanford encyclopedia of philosophy. Retreived from: URL = .

Kuhn, T. (2012). The structure of scientific revolutions (50th anniversary edition). Chicago, IL: University of Chicago Press. Lajoie, D.H., & Shapiro, S I . (1992) Definitions

of transpersonal

psychology: The first twenty-three years. Journal of Transpersonal Psychology, 24(10, 79. Lakoff, G., & Nunez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York NY: Basic. Laughlin, R B . (2006). A difiarent universe (reinventing physics from the

bottom down). New York, NY: Basic. Lederman, L.M., & Hill, C.T. (2004). Symmetry and the beautiful universe. New York, NY: Prometheus. Le Poidevin, R. (2015). The experience and perception of time. I n E.N.

Zalta (Editor). Stanford Encyclopedia of Philosophy (Summer edition), Retrieved from URL = .

Lincoln, D. (2017). The theory of everything: The quest to explain all reality [Video file and course guidebook]. Retrieved from: https://www.thegreatcourses.com/courses/the-theory-of-everythingthe-quest-to-explain-all-reality.html.

Mandelbrot, B. (1977). The fractal geometry of nature. New York, NY: WH. Freeman.

Markosian, N. (2016). Time. In E N . Zalta (Ed.). Stanford encyclopedia of

Transpersonal Psychology and Fractal Evolution

141

Philosophy. Retrieved from URL = . Matthews, CO. (1999). Psychotherapy and spirit (book). Counseling &

Values, 44(1), 75. Miller, J.H., & Page, SE. (2007). Complex adaptive systems: An introduction to computational models of social life. Princeton, NJ: Princeton University Press.

Mitchell, M. (2011). Complexity: A guided tour. Oxford, England: Oxford University Press. Mitleton—Kelly, E., Paraskevas, A., & Day, C. (2018). Handbook of

research methods in complexity science. Cheltenham, England: Edgar Elgar. Morales, A.J., Gershenson, C., Braha, D., Minai, A.A., & Barr-Yam, Y.

(Eds). Unifying themes in complex systems IX: Proceedings of the Ninth International Conference on Complex Systems. Cambridge, MA: Springer.

Morowitz, HJ. (2002). The emergence of everything: How the world became complex. Oxford, England: Oxford University Press.

Nagel, T. (2012). Mind and cosmos: Why the materialist neo-Darwinian conception of nature is almost certainly false. Oxford, England: Oxford University Press.

Nickles, T. (2017). Scientific revolutions. In E.N. Zalta (Ed.), Stanford encyclopedia of philosophy. Retrieved from: URL = .

Penrose, R. (2004). The road to reality: A complete guide to the laws of the universe. New York, NY: Vintage.

Pielou, E.C. (2001). The energy of nature. Chicago, IL: University of Chicago Press.

Popper, K. (1959). The logic of scientific discovery. London, England: Hutchinson.

Pylkkanen, P. (2007). Mind, matter and the implicate order. New York, NY: Springer-Verlag. Raatikainen, P. (2015). Godel's incompleteness theorems. In E.N. Zalta

(Ed.), Stanford

encyclopedia

of philosophy.

Retrieved from:

URL=.

Reeves, H. (1993). Malicorne: Earthly reflections of an astrophysicist. Toronto, CA: Stoddart.

Robinson, W. (2015). Epiphenomenalism. In E.N. Zalta (Ed), Stanford encyclopedia of philosophy. Retrieved from URL =

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D. Thus, D is in some sense optimal and unique. (This is closely related to the concept of Hausdog?rDimension—see Falconer, 2003, p. 31.) I will now formalize the above conceptualization. In general, when determining the “area” of a complex set (such as a fractal), one first approximates or covers the set by a sequence of simplified sets which have a well-defined area; this was done above via the use of polygonal line approximations of length 6. The most basic of fractals, however, such as the Cantor set or the Sierpinski Triangle, may be realized as a union of (covered, approximated by) self-similar, scaled subsets. The relationship between the number of such self-similar, scaled subsets required to cover the fractal set and the scaling factor will determine the dimension. Let us first see how this relationship works in such standard geometric objects as the unit square and unit cube. Fix n > 1. We may realize the unit square as a union of 112 smaller squares, each of side length 1/n—the scaling factor is l / n . We may realize the unit cube as a union of 113 smaller cubes each of side length 1/n—the scaling factor is again 1 [11. Thus, in

both cases we derive the analogue of (1): f = (1 /m)d, where f =number of self—similar pieces, m =scaling factor, and d =dimension. Said another way, the dimension satisfies the formula,

d = —log£’/logm

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The dimension of the square is 2 = —109n2/109(1/n); the dimension

ofthe cube is 3 = —109n3/109(1/n). Let us apply this method to the Cantor set and the Sierpinski triangle. The Cantor set may be regarded as comprising four copies of itself scaled by a factor of 1/9. Thus, we might say that the dimension of the Cantor set

is —1094/109(1/9) = 1092/1093. In general, the Cantor set may be regarded as comprising 2" copies of itself scaled by a factor of 1 / 3 " ; in all cases, however, we arrive at the same dimension: —109 2" / 109 (1 / 3") = 1092/1093. The Sierpinski triangle may be regarded as comprising three copies of itself scaled by a factor of 1/2; or 3“ copies of itself scaled by a

factor of 1 / 2 " . In any case we arrive at a dimension of 1093/1092 (see figure 4). This construction is known as the similarity dimension. In the general case, when self-similarity is not preserved in a constant way, we proceed as follows. Denote by R" n—dimensional Euclidean space, which is the collection of all vectors (x1, x2, , x n ) where each xi is a real number. Let U be any non-empty subset of R". We define the diameter of U as simply the maximum distance between any two points in U: | U |: = sup{|x — y | : x, y E U }. Now let F be any bounded, non-empty subset of R". Denote by N€(F) the smallest number of sets of diameter at most 6 which can cover

F. As in equation (1) above, a dimension of F is determined by the power law (if any) obeyed by NE(F) as E —> 0 (Falconer, 2003, p. 39). That is, NAF) ~ 26‘” , for constants A, D, with D representing the dimension of F. Take the Cantor set, for example. For e = 1/3", we have NE(C) = 2".

—1092/1093 1 = 6—1092/1093' NeCC) = 2 n = ( fl )

Then,

logNe (F) dimB(F): = 15:33 —1096

I

With this in mind, we define the box dimension of F by assuming the limit exists. For example, we have shown that the box dimension of the Cantor

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set is 1092 [1093, which agrees with its similarity dimension calculated above.

(e) V

(9) Figure 8-4. Self-similar coverings

Fractals and dynamics Ihave introduced the notion of a fiactal in isolation. It is just as important, if not more, to consider how fractals arise within dynamical systems. Dynamical systems is a mathematical study of processes in motion. Planetary motion, the dynamics of the weather and the stock market, or the

motion of a simple pendulum are all examples. Processes can be viewed in discrete time, continuous time, and through the lens of complex variables.

In the discrete case, this is mathematically represented by a fimction f: D —>» D (i.e. y = f(x), where 36,); are in D, D a subset of Euclidean space of

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dimension n, R", or D a subset of the complex numbers C), and its iterates or orbits,

{fk(x)}?=o = {Xif(x)if(f(x))lf(f(f(x)))i ...} where f°(x) =x,f1(x) = f ( x ) , f 2 ( x ) = f ( f ( x ) ) , and so on. In the continuous case, this is mathematically represented

by a differential

equation, x = 2% = f (x), along with its family of solution curves x(t), each of which is defined uniquely for a given initial condition x(0) (assuming some technical conditions on the function f , such as “smoothness”). In both cases, we can let time approach infinity and inquire as to the end behavior of the system. This is indeed an interesting problem because one is often interested in predicting the future of the system (will it rain tomorrow?).

Roughly speaking, we define an attractor (repeller) of the system to be a set to which all nearby orbits/solution curves converge (or diverge) in time

(defined in both the discrete and continuous case). This set can manifest itself as a fiactal; moreover, when it does so the dynamics of the system is often chaotic}. What is perhaps most interesting, however, is that very simple systems even systems depending on only one variable, may behave just as unpredictably as the stock market, just as wildly as a turbulent wateifall, and just as violently as a hurricane (Devaney, 1990, p2). This is, in essence, the notion of chaos.

Fractals and discrete dynamics Consider first a discrete system, given by a continuous function f : D —> D

and its orbits {f k(x)},‘f=D. For example, if f (x) = cosx then f k(x) —> 0.739 as k —>_ 00 for every initial value x in R. We consider three cases in general:

1. f k ( x ) converges to a fixed point w, that is, a point w in D such that f ( w ) = w. Convergence means that |fk(x) — w| —> O as k —> 00. A

fixed point w is stable or unstable according to whether or not the derivative of the function is smaller or larger than 1 in magnitude:

|f ’(w)| < 1 or > 1. Stable fixed points attract nearby orbits, unstable fixed points repel them.

2. f"(x) converges to a periodic orbit, that is a set of the form {12, f (v), f 2(1)), ..., f p‘1(v)}, where p is the minimal integer such that f 3’ (v) = 1:. Convergence inthis case means I f k(x) —- f k(v)| ->

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0 as k —> 00. A periodic point 17 of period p is stable or unstable according to whether or not the derivative of the pth iterate is smaller

or larger than 1 in magnitude: |(fp)’(v)| < 1 or > 1. Stable periodic points attract nearby orbits, unstable periodic points repel them.

. f k (x) may appear to move about at random, but always staying close to certain set, which may be a fractal. A common example of a discrete dynamical system exhibiting chaotic dynamics is given by the logistic map fi : R —) R , defined by 15106) = Ax(1 — x ) , where )l is a positive constant (see Figure 8-5). I outline the

dynamics of f ) for various 1. 0 < )1 S 1. Then f1 maps the unit interval to itself, and so restricting our attention here we find that 0 is an attractive fixed point: flk (x) —>

0 — i.e. the kth iterate of f1 approaches 0 as kapproaches infinity— for all x in [0,1]. 1 < 11 < 3. fit has an unstable fixed point at 0 and a stable fixed point

at 1 — 1/1. We have that ff(x) —> 1 — 1/1for all x in (0,1). As 1 increases through the value 11 = 3, the fixed point at 1 — 1/1 becomes unstable and almost every point in (0,1) attracts to a new stable orbit of period 2. When A increases through 12 = 1 + Vb the period—2 orbit becomes unstable and a new stable orbit of period 4 is born. This behavior continues as )1 increases fiirther, with a stable orbit of period 2‘1 appearing at A = )lq; this orbit attracts all but countably many points of (0,1). As q —> on, A —> Am z 3.570, the period doubling occurs more frequently, and we obtain a sequence of attracting (periodic) orbits approximating a Cantor set When )1 = )1m the attractor is a Cantor-

like set. The Hausdorff dimension of this set has been approximated to be 0.538 100 < A < 4 . Several types of behavior occur. There exist a set K of

positive measure such that for l in K , f ) has chaotic behavior. The behavior of the logistic map is universal in the sense that it is qualitatively the same as any family of transformations of the form

f (x) = Af(x), provided that f has a single maximum at a point c with f ”(c) < 0. The values of A which mark a change in the qualitative behavior of the

dynamics (iterates) of 131 are known as bifurcations. Thus, above for the

On the Mathematical and Transpersonal Foundations of Fractal Geometry and Dynamics

logistic map, such values include 11 = 3,12 = 1 + V3,

257

This sequence

lists a series of period-doubling bifurcations leading to chaos. I summarize this behavior in figure 8-5, known as the bifurcation diagram for fl. Each

value along the horizontal axis represents a value of /1, above which is plotted the long-term behavior of the iterates f f (x) for suitable x (as described).

1

.

.

0.9 0.0 .

0.7 0.6 -

/

0.5 0.4 0.3 02 0.1 o

2.4

I

I

I

I

I

I

I

2.6

2.8

3

32

3.4

3.8

3.8

4

Figure 8-5. The bifurcation diagram of the logistic map f [1(x) = xlx(1 — x)

Fractals and complex dynamics Perhaps the most intriguing of fractals occur within the realm of complex dynamics. By complex I am referring to both the complex number system

and the complex plane, the two being related by the identification x + iy I—> (x, y). That is, each complex number, x + iy, where x and y are real

numbers, is associated with the vector (x, y) in the Euclidean plane. The set of all complex numbers is denoted by C. For any complex number 2 = x + iy, we say that x is the real part of z, denoted Re(z), and y is the imaginary

part of z, denoted Im(z). Multiplication in this number system is most significant because i2 = —1; if i is the vector (0,1), then i2 = —1 is the

vector (— 1,0). Note that these two vectors are related by a simple 90-degree

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counterclockwise rotation! Thus, complex multiplication becomes inherently creative in the sense that embedded within the concept of complex multiplication is the geometric operation of rotation. Thus the inherent symmetry contained within the various complex fractals (Julia

sets), as for example displayed in Figure 8-6.

Figure 8—6. The Julia set off(z) = 22 — 1 Let f : C —> C be a polynomial

of degree n 2 2 with complex

coefficients. We are again interested in orbits {f k (2)},‘2‘;0 of various complex Z. In particular, we are interested in the boundary zone between orbits which diverge to 00 and orbits which do not. This “boundary” is known as the Julia set of f . Specifically, we define the filled in Julia set of

f as the set of all points whose orbits do not diverge to infinity: K (f): = {z E C: f k(z) +> 00}. (The symbol E means “in” in the sense of “is an element o f ” ) The Julia set o f f , denoted ] (f), is then the boundary of K (f).

Rigorously, said boundary is defined by the property that for every point z E ] (f), there exists a small disk DZ (1”) of radius r > 0, and points w and

17 in D20”) withfk(w) -/-> 00 (as k —> 00)andfk(17)—> 00 (as k —> 00). The Julia set of the function f (z) = z2 is the unit circle. Points inside

this circle iterate to 0, while points outside the circle iterate to 00. However, if f is perturbed ever-so slightly to fc(z) = z2 + c, then the possible structures of the Julia set become diverse, often fractal. The pictures that arise in these examples are majestic to say the least. See, for example, Devaney (1990) for an introduction to this topic.

As for the Mandelbrot set, which contains within its definition the concept of the Julia set, it is, in words, the collection of all complex c values

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Geometry and Dynamics

for which the Julia set of 12(2) = z2 + c does not break into pieces (Le. is

connected). It turns out that the Mandelbrot set M may be defined as the collection of all complex numbers c E C for which the orbit of 0, {3‘6" (0)}§=0, does not diverge to infinity:

M = {c E C: fi‘m) +> no}. The Mandelbrot. set may be viewed as a picture book of the various Julia sets that arise as c varies in fc. This is because each 6 value corresponds. to

a specific connected Julia set. This is displayed in figure 8-7.

fr)

Figure 8-7. Julia sets for c at different points in the Mandelbrot set

Curiously enough, not only does the definition of the Mandelbrot set embed

Within it various fractal Julia sets, it is itself a highly complicated fractal. It has a main cardioid region to which a series of circular ‘buds’ are attached In addition, fine ‘hairs’ grow out from the buds carrying within miniature

copies of the entire Mandelbrot set. The Mandelbrot set is connected

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(topologically) and its boundary has Hausdorff dimension of 2. The Mandelbrot set is pictured in figures 8-8, 8-9, 8-10, and 8-11.

Part II: An introduction to the poetic and metaphysical aspects of self-similar iterative schemes The poetry of fiactal geometry and dynamics We began our overview of fractal geometry with the unit line segment [0,1] and a simple iterative procedure. This iterative procedure quite literally

dissolved the unit line segment until all that was left was both surprisingly imaginative and endlessly detailed in nature. This was the Cantor set. This set resided between the ordinary zero-dimensional realm of discrete points and the one-dimensional realm of the real line. In fact, we attached to this set a dimension of 1092/1093 — a number which quantifies how much space the set fills, precisely defined as the exponent in certain power law relationship. This set was highly elusive in that the more we searched within the set, the more details we found — details which brought us right back where we started. Indeed, the Cantor set is self-similar on infinitely many scales. In reflecting on my own life journey, I see that both the concept of and the journey into the Cantor set can be minored through a transpersonal and

inner perspective: The farther] reached upward the tighter] grasped; the deeper] dove, the less structure I found yet the more detail there was to get lost in. It was all so confusing, to find myseK my struggles, my highs and lows mirrored back to me, forcing me to question to what extent within me is you, within you is me, and whether or not we are connected without by a web of profundity. And I realized that lyre was happening to me, dissolving my isolated conceptualizations, my isolating boundaries, forcing me ever deeper into myseU— an ever-elusive space, a say-similar dream which was simultaneously manifesting within and without. At the same time, we saw how fractals arose in the fate of a dynamical system. In a one—dimensional system, such as the logistic map, we saw that for certain values of )l the long-term behavior of the system attracts to a

fractal, Cantor-like set. That is, in trying to predict thefuture we onlyfound chaos. And yet within this chaos exists an inherent order; the fractal attractor is highly self—similar and quantifiable in the sense of its fiactal dimension. We may consider fractals in nature as well. A coastline exists at the boundary between land and sea. It is rather curious to remind ourselves that

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261

the sea is the great symbol of the divine. Thus, the coastline—a natural

fractal — existing at the boundary between land and sea, symbolically exists at the boundary between the physical and spiritual planes, between the concrete and the transpersonal. In complex dynamics, fractals exist as Julia

sets, defined as the (topological) boundary between convergent and divergent orbits. Between the finite and the infinite. We thus have a physical

boundary between land and sea, or a mathematical boundary between the finite and the infinite, both emanating from the same archetypal pool. We are thus reminded of the Pythagorean ideal of a mathematical order, magnanimous in its beauty, embedded in the fabric of the cosmos itself; a divine nous, the logos of a supreme Kosmos (which in its original Greek meaning describes the world as intelligently ordered and interconnected). It is this topic which we take up now. Im[c] ‘. 1.

.41‘ Figure 8-8. The Mandelbrot set in the complex plane with coordinates c = x + iy,

x = Re(c), y = Im(c)

0n the archetypalforms

The Platonic Forms, Ideas, or archetypes are posited as pervading and dynamically ordering the flux and form of the sense-perceived world. They are considered eternal in that they exist in a transcendent metaphysical

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framework which is out of time and place. As such, they exist in a state of being behind that which they order, which is in a state of becoming. Thus, one must awaken to them in the form of an intuitive, poetic knowing. Beauty comes and goes, but its essence, one may realize, is eternal; virtue may be described in this way by one person, and that way by another, but its

fimdarnental nature—what is it?

Figure 8-9. Seeing the ‘hairs’ of the Mandelbrot set Mathematical concepts exist in their own right. A circle is an eternal form which pervades all that is perceived as circular. Such a direct, intuitive knowing was considered the only way to realize the essence of things, and the only avenue through which true knowledge could be ascertained. Seek it in the grace of a breeze, the fall of rain, or in the texture of a voice; what it is is never here, never there, yet always present. For beyond time an archetype rests, and in time an archetype takes shape.

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Figure 8-10. Glimpsing the Mandelbrot set within the Mandelbrot set The Ideas (which in Greek denoted the form, pattern, essential quality, or nature of something), were thus divine. The circumference of the unit circle, namely H, is thus numinous, and if the ancient Greek’s knew of the celebrated Euler’s formula — e m = —1 — they would no doubt consider it

as a message from the sacred. And indeed, Plato’s philosophy grew out of the ancient Greek mythic perspective, which viewed reality as permeated with a sacred essence, as exemplified in the gods and goddesses of the great Homeric epics (see Tarnas, 1991/1993). A mathematician was not simply driven by an “aesthetic function,” but was a follower of Eros, the god of love. The philosopher is not just after wisdom but is a follower of Zeus. That which lends an archetype (or archetypal complex) its shape and form, or its essence—not its material form—was given a name by the

ancient Chineswfif. (Ch’ien/Qian)—translated as “The Creative” in Wilhelm and Baynes (1967), and imaged as rain by Confucius: “The clouds pass and the rain does its work, and all individual beings flow into their forms” (p. 4 and p. 370). One must be carefitl, however, to not take this image out of context, for the rain does not act alone; it is also received—it

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is received by the earth. And it is the combination of the two which bring

the Ideas into their material forms (the ten thousand things). The Earth is thus archetypal, representing the divine feminine. Moreover, these two archetypes—Heaven (the Creative) and Earth (the Receptive)—can never be spoken of in separate contexts, for they were always envisioned in dynamic union. This union manifests as a constant, continuous flux between the two. And this flux, this motion—it is called Tao. Tao is the one in the many, the unseen yet always present, it is the nourisher of the “ten thousand things.” The archetype of the Creative was given a four-character Judgement (seen to be its significance upon divination) by King Wen (circa 1150 B.C.): 7—13 . $- . Tl] , E. (yuan, hEng, 1i, zhén). This has been translated by Wilhelm as, “The Creative works sublime success, Furthering through perseverance.” (Wilhelm & Baynes, 1967, p. 4). The character if: denotes the beginning of all things and is a reference

to the archetypal forms. It is translated by Wilhelm as sublime or sublimity: “Great indeed is the sublimity of the Creative, to which all beings owe their beginning and which permeates all heaven.” (This quote has been attributed to Confucius, see Wilhelm & Baynes, 1967, p. 4, p. 370.) It is the character g!- which expresses the ability of man to commune with the archetypal realm and has been imaged as rain above; this character has the implied meaning of sacrifice or communication and is translated by Wilhelm as success. The third character W has the meaning of harmony, justice, or benefit, and has been translated by Wilhelm as furthering. This character embodies a natural order inherent in nature which operates by “creating that which accords with the nature of a given being” (Wilhelm & Baynes, 1967, p. 5). The last character E connotes the proper way of the Creative, which is “correct and firm,” and has been translated by Wilhelm as perseverance; the paradigmatic example of perseverance as given in the text and its commentaries is the consistent, daily motion of the heavens (and it is thus embodied in linear time and the changing seasons). This is to be contrasted with the Judgment given for the Receptive (i513, Kun), the essence of which is contained in the first two lines of the Wilhe1m translation: “The Receptive brings about sublime success, Furthering through the perseverance of the mare” (Wilhe1m & Baynes, 1967, p.11). Thus, the character E, that is perseverance, has been modified to suit the nature of the divine feminine, which is embodied in the characteristics of

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the mare. Moreover, the context for the character it —sublimity or great beginning—has changed and is to be envisioned as abirth, resulting from a union with the Creative: Perfect indeed is the sublimity of the Receptive. All beings owe their birth to it, because it receives the heavenly with devotion. (This quote has been attributed to Confucius, see Wilhelm & Baynes, 1967, p. 3 86) A mare belongs to the creatures of the earth; she roams the earth without bound. Yielding, devoted, furthering through perseverance: thus the superior man has a direction for his way of life. (This quote has been attributed to Confiicius, see Wilhelm & Haynes, 1967, p. 387)

The changes: A metaphysical selflsanilar iterative scheme The philosophy of the I Ching embraces the archetypal cycle as its

fundamental starting point, implicit within which is change — the Tao. The three paradigmatic examples (or embodiments) being the day, the year, and the waxing and waning of the moon (the lunation cycle). As the sun reaches its peak at high noon, so too does it reach its low in the dark of the night. These two points, diametrically opposed to one another in space, are also diametrically opposed to one another at an archetypal, metaphysical level: they represent the primal masculine and feminine poles — yang and yin — which are seen to be the basis of change. This is represented to a possibly greater extent in the lunation cycle, which is fundamentally a cycle of relationship. This interplay between masculine and feminine, light and dark, day and night, full and new moon, acts as aripple which spreads throughout

the entirety of the I Ching, underlying all its formulation. Such archetypal iteration carries with it, or leaves room for the emergence of the notion of fractal in the broad sense of the term. The foundation of this may be seen in the following two-stage seed pattern: 1. This stage posits the existence of a state of unity, imaged by a straight

line (—), or a circle (0). 2. With the assumption of a fixed frame of reference, that is, the straight line, or circle, we now introduce the concept of change by defining the broken line (— —) as the complement to the straight line (—). The two are to be seen as a pair, and positioned at two fixed (diametrically opposed) poles of the circle, with the straight line positioned on the top of the circle and the broken line on the bottom. The straight line is to be called “the light,” “the firm,” or yang. The broken line is to be called “the dark,” “the yielding,” or yin. Thus the

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Chapter Eight top of the circle is to represent the south, where the sun reaches its highest point in the day and in the year; it is also to represent the full moon, when the moon reflects the light of the sun fully and lies in direct opposition to the sun on the zodiacal belt. The bottom of the circle is thus to represent the north, where the sun reaches its low; or it is to represent the new moon, when the moon has turned completely dark and merged with the sun on the zodiacal belt. We

thus posit a union of opposites, inherent within which is a dynamic motion, which we call Tao, and which acts by opening the straight

line into the broken line and closing the broken line into the straight line. This is to be imaged as a continuous-clockwise motion between the two poles of the circie. Tao operates at each and every level of the construction and exists as a continuous movement between opposites. With the introduction of cyclic motion, we obtain the four seasons or the four phases of the moon, represented in the four lines: solid (yang/light), broken (yin/dark), and two which are in a state of change (old yin and old yang). This is sometimes represented by two lines stacked on top of each other, each of which is either broken or solid, and thus once again four in total (see Wilhehn & Baynes, 1967, p. 319). A changing yang line (old yang) is simultaneously imaged as the time when the light energy is at its peak (and thus beginning to contract), and the dark energy is at its low (and thus beginning to expand); this is summer or high noon. A changing yin line (old yin) is simultaneously imaged as the time when the dark energy is at its peak (and thus is beginning to contract), and the light energy is at its low (and thus beginning to expand); this is winter or midnight. Thus, we have a dual movement of light and dark: “Counting that which is going into the past depends on the forward movement. Knowing that which is to come depends on the backward

movement.” (Shuo Kua: Discussion of the Trigrams, Wilhelm & Baynes, 1967, p. 265). This dual movement between yin and yang is most commonly exemplified in the (apparent) course of the sun throughout the day and the year. I would suggest that it is also most elegantly embodied in the lunation cycle, which is displayed in figure 8-11. It is to be noted that the phases of the moon do not describe the position of the moon, nor do they describe the moon herself, but rather the relationship between the sun and the moon.

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first quarter

new moon

‘\

I,

full moon l

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0.' l

l

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waxing crescent

waxing glbbous

waning crescent

waning gibbous

last quarter Figure 8-11. The lunation cycle. Note the natural parallel between, for example, the image of first quarter moon and the little (changing) yang line symbolism, which may be represented as a broken line stacked on top of a solid line.

Thus, as I fix the line symbolism below, each item is to be seen in

relationship to the whole: '

Full moon = old (changing) yang line; peak of the yang cycle;

'

New moon = old (changing) yin line; peak of the yin cycle;

'

First quarter moon = young (not changing) yang line; middle of the

yang cycle; and °

Last quarter moon = young (not changing) yin line; middle of the yin cycle.

The Chinese sages of old thus discretized the cycle by considering the seeds of change, as embodied in the lines and their “changes.” By

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considering the infinitesimal, they were able to affect the “easy” and the “simple”—the two adjectives characterizing the movement of the Creative (yang movement) and the Receptive (yin movement) respectively; in effect, they invented a form of differential calculus (see pp. 286-287 of Wilhehn &

Baynes, 1967; the title I Ching is written 5 % in Chinese (another title of

the book being E5), and 5 means not only “change” but “easy”). Thus, it is said: “Only through what is deep can one penetrate all wills on earth. Only through the seeds can one complete all affairs on earth. Only through the divine can one hurry without haste and reach the goal without walking.” (Wilhelm & Baynes, 1967, pp. 315-316). Adding a third line to this construction we arrive at the trigram symbolism. This structure is emergent in the sense that as a whole it is greater than the sum of its parts. Each trigram is either feminine or masculine, yin or yang, light or dark, two of which are primal, the rest being “derived.” To each trigram was associated an archetypal Image. Eight in total, two trigrams represent the primal active and receptive forces of nature, the father and mother archetypes, the Image of which is heaven and earth, respectively. Three more trigrams represent the threefold unfoldment of the principle of movement, which is a yang cycle, and is often associated with the three sons in order of age. These three trigrams were assigned the respective Images, thunder, (moving) water, and mountain (rest). And lastly three more trigrams represented the threefold unfoldment of the principle of devotion, which is a yin cycle, and is often associated with the three daughters in order of age. These were assigned the Images, wind fire, and lake (see Wilhelm & Baynes, 1967, pp. 266, 269, 284 for two different sequential arrangements of the trigrams around the circle and their respective commentaries; see also pp. 318-319 and Wilhelm’s Introduction to the book in general). Inherent within the symbolism of the number three is what has been called the three primal powers: man, heaven, and earth. Embracing the Tao of each, that is, the union of opposites, thus gives six: the Tao of the earth gives two lines, the Tao of man gives two more, and the Tao of heaven gives the final two. Thus marks the proper context for the I Ching in general, and the hexagram symbolism specifically. For a hexagram was first developed as something which was to be divined; such a divinatory foundation was founded on the three-fold relationship between man, heaven, and earth, each of which were seen to embody one another. When the Changes are “stimulated, they penetrate all situations under heaven. If they were not the

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most divine thing on earth, how could they do this?” (Wilhelm & Baynes, p. 315). Says Wilhelm on this topic: The way in which the Book of Changes works can best be compared to an electrical circuit reaching into all situations. The circuit only affords the potentiality of lighting; it does not give light. But when contact with a definite situation is established through the questioner, the “current” is activated, and the given situation is illumined. Although this analogy is not used in any of the commentaries, it serves to explain in a few words the entire meaning of the text. (Wilhelm & Baynes, 1967, p. 315)

Both the trigrams and their composite hexagrams are thus dynamic processes which follow the natural rhythms of nature. Along with their associated Images, they were seen as ordering the empirical world in what was considered a very complete fashion: The holy sages were able to survey all the movements under heaven. They

observed forms and phenomena, and made representations of things and their attributes. These were called the Images. (Wilhelm & Baynes, 1967, p.

304) The Master said: Whoever knows the tao of the changes and transformations, knows the action of the gods. (Wilhelm & Baynes, 1967, p.

313) In fact, this entire process has been summarized before, in thirteen Chinese characters and four lines of English verse:

EE — _ HI- :

:E E

EEE M The Tao begot one. One begot two. Two begot three. And three begot the ten thousand things.

(Lao Tsu, Tao Te Ching, translated by Feng & English, 1972, Chapter 42) Thus, to the extent that the Changes do model the phenomenological world, it too would have to be fractal in the self-similar sense. Of course, a fractal or dynamic model implies a notion of time, and time in the objective sense

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is clearly insufficient in dealing with a book such as The Book of Changes. There does exist, however, a very individual form of time—what one might

call subjective time or subjective duration. This is based on thefelt sense of each moment. It would seem that the ancients knew a great deal about this form of timthey charted it according to the waxing and waning of the moon, the lunation cycle. Subjective time is thus conceptualized as inherently lunar, while objective time is inherently solar.

The average length of a lunation cycle is 29.53 days. A lunar year comprises 13 months, and thus 29.53 x 13 = 383.89 days. This becomes significant when one realizes that the basic numerical unit of a hexagram is 6 (lines), the basic numerical unit of a complete sequence of hexagrams is 64, and 6 x 64 = 384 z 383.89. (This numerical fact or coincidence was also noted in McKenna & McKenna, 1993, p. 126.) Says Wilhelm on the six-fold unfoldment of change: All movements are accomplished in six stages, andthe seventh brings return. Thus the winter so]stice...comes in the seventh month after the summer solstice; so too sunrise comes in the seventh double hour after sunset. Therefore seven is the number of the young light, and arises when six, the number of the great darkness, is increased by one. In this way the state of rest gives place to movement. (Wilhelm &Baynes, 1967, p. 98)

It is also significant to note that lunation cycle is essentially never mentioned in the Wilhelm translation of the I Ching and its commentaries. (The moon in isolation is mentioned in passing on different occasions, in part because it is embodied in the t r i n for moving water; the lunation cycle is vaguely mentioned in the Confucian commentaries on p. 338 of Wilhelm & Baynes, 1967, and mentioned in the commentary on the fourth line of the hexagram Inner Truth, p. 238 -- see also pp. 277, 302, 319.) We must keep in mind, however, that not only does the human follow the earth, but the moon as well, and the cyclic relationship between the sun and the

moon is the archetypal exemplar of the relationship between Man and Woman. Conclusion Fractal geometry, or mathematics proper, is not just a field of study where problems are to be solved and numbers “crunched,” nor is the emotional, aesthetic experience detached from its epistemology. Rather it transcends the logical rigor of its conscious foundation. To the extent that mathematics is a symphony in which one participates, one does not do mathematics, but

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dreams mathematics. To the extent that the cosmos is a kosmos, in the true sense of the word, one does not so much do anything, but participates in everything, at the grandest of scales. It is thus equally interesting, if not more so, to consider the larger concepts to which the mathematics points, the self-similar ripples that exist across the mathematical, physical, and inner worlds. It seems that it is not just that mathematics is excellent at modeling the

physical world, but that the two worlds are archetypally self-similar, and further this self-similar ripple extends beyond a mere physical manifestation. What this implies is a return to the simple, for formulations and symbols which mathematicians call “trivial” would be allowed to be so, not thought so, and thus would assume true profundity. Thus, the metaphysical truths of reality would not be so much sought within the depths of a Fermat’s Last Theorem (the proof of which eludes the majority of mathematicians), but rather in the depths of the symbols themselves; for if given a chance, perhaps such symbols would tell a tale far grander than any conscious plunge into the mathematical labyrinth erected by shear logic alone. And this is precisely the view one must take when considering such works as the I Ching. We saw how the Chinese sages of old felt very strongly that the sixty-four hexagrams of the I Ching “penetrate all situations under heaven,” thereby serving as a complete model of the phenomenological world. As elaborated on already, it seems that the I Ching is, in a rather complete sense, a variation on a theme of light and dark: a

creative iteration of an archetypal seeaT shape, an archetypal fiactai which cuts across inner and outer realities. Finally, inherent within a fractal paradigm is a dynamic synergy, where, for example, iteration (which is linear) of a non-linear process (which may be viewed as “creative”) converges into a dynamic synthesis. It is a synthesis of the rational with the intuitive, the limit of which is often glimpsed as both boundless and inherently ordered. But moreover, within the fi‘actal concept itself lie simple yet profound philosophical truths. They are embodied in the Cantor set: If I were to imagine myself manifesting in the deepest of depths of a random Cantor set, [would be overwhelmed by a seemingly orderless detail. But if by chance I was one day able to step outside of my separate existence, and perceive the whole from a higher dimensional perspective, the maj esty of a pattern would emerge. I was never separate to begin with, but a self—similar realization, caught in an illusion of low—dimensional separateness, unable to glimpse the forest through the trees; thoughts run astray, lost from the moon.

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References Bohm, D. (1980). Wholeness and the implicate order. New York, NY: Routledge and Kegan Paul.

Devaney, R.L. (1990). Chaos, fractals, and dynamics. Boston, MA: Addison—Wesley.

Falconer, K. (2003). Fractal geometry: Mathematical foundations and applications Hoboken, NJ: Wiley. Kingsley, P. (2003). Reality. Point Reyes, CA: The Golden Sufi Center. Lao Tsu (1972). Tao te citing (G. Feng & J. English, Trans). New York, NY: Vintage.

Mandelbrot, BB. (1977). The fractal geometry of nature. New York, NY: W.H. Freeman. McKenna, T., & McKenna, D. (1993). The invisible landscape. New York, NY: Harper Collins. Richardson, LP. (1961). The problem of contiguity: An appendix of

statistics of deadly quarrels. General Systems Yearbook, 6, 139-187. Schroeder, M. (1991). Fractals, chaos, power laws. New York, NY: Freeman.

Tamas, R. (1993), The passion of the Western mind New York: Ballantine. (Original work published 1991).

Wilhelm, R., & Baynes, CF. (1967). The I Ching (3rd ed.). Princeton, NJ: Princeton University Press.

PART 2:

FRACTAL APPLICATIONS

CHAPTER NINE DREAMS, SYNCHRONY, AND SYNCHRONICITY

TERRY MARKS-TARLOW1

Dreams serve as powerful mirrors to the psyche through their holistic potential. Dreams emerge at the edges between conscious and unconscious processes to display fractal properties. A fractal epistemology posits boundary conditions as dynamic zones of transaction across different states and scales of existence. Rather than smooth and fixed like a cup cleanly separating inside from outside, fractal boundaries are semi—permeable and infinitely deep, at least in theory and on an endlessly iterating computer (Mandelbrot, 1977; Peitgen, 1986; Schroeder, 1991). When examining a dream, we can use powers of conscious observation to illuminate self—similar patterns reflected within dream structure. Each time we revisit a dream, the opportunity exists to find another angle of meaning. Whenever working with dreams, the observer is inextricably and recursively linked to the observed, as dream structure is both discovered and created through the very act of looking. The process is infinitely deep in that endless opportunities exist to revisit any given dream. As with fractals, the closer we look at a dream, the more there is to discover. In past writings (Marks-Tarlow, 2008, 2012, 2014), I suggested fractal consciousness, by which we perceive the whole in the parts of experience, represents the essence of therapeutic intuition. Through clinical intuition we implicitly sense the fine texture of experience during clinical interactions partly by linking patterns in those tiny “now” moments to self-similar patterns surrounding larger events and conceptualizations. Fractal consciousness is also useful as an explicit tool for self-exploration, allowing us

1 Core Faculty, Insight Center, Los Angeles; Visiting Professor, Italian Universita Niccolo Cusano London; Research Associate, Institute for Fractal Research, Kassel, Germany; Adjunct Faculty, Pacifica Graduate Institute. E-mail: [email protected]

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to gain and retain perspective, again by appropriately linking small- and large-scale self-similar events into meaningful self-narratives. Fractal consciousness is especially useful for approaching dream interpretation, where we often capture the essence of a dream by looking at

any dream fiagment, no matter how tiny, as if it reflects the whole of the psyche. Alongside illuminating the geography of attachment dynamics, this holistic perspective also applies to temporal dimensions. Especially because of their highly visual nature and spatial extension, dreams more than verbal narratives exist in a timeless realm. Here they can simultaneously capture past as well as the present dynamics, while simultaneously pointing towards the future. Over history, dreams have enjoyed a full range of interpretations (Van de Castle, 1994). Ancients honored them for their power to prognosticate disease and foreshadow important events in history. One biblical example was Daniel's interpretation of King Nebuchadnezzar's dream. The Persian Magi plus Indian yogis and fakirs established elaborate rituals to cultivate nonlocal awareness within dreams, while the Sumerians and Egyptians formalized “sleep temples” towards the same end; by the 4th century BCE, the formalize linkage between dreams and nonlocal awareness was central to Greek religious life (Schwartz, 2018). Freud declared dreams the “royal road to the unconscious.” In the history of science numerous discoveries have been attributed to dreams. One famous example is the benzene ring in chemistry, which emerged out of Fredrich Kekulé’s dream of a ring of snakes, each biting another’s tail. A second example of discovery through dreams was Carl Jung’s notion of the collective unconscious, emerging from his nighttime reverie of descending to the subterranean floor of a house filled with the bones of ancient ancestors (see Jung, 1961/1989). The purpose of this chapter is to use a fractal lens to approach the subject of dreams. If a fi'actal epistemology is to be useful for transpersonal psychology, and consciousness studies more broadly, it should effectively model and integrate multiple levels of observation. In the pages ahead, I present an ongoing psychotherapy case to highlight a patient’s dream sequence. My purpose is to attend to two key fractal features: 1) Selfsimilarity as a connecting principle, whereby the pattern of the whole is reflected in the parts at multiple scales of observation; and 2) Paradoxical boundaries between self and other, as well as self and world, that are simultaneously open and closed, as well as observer dependent.

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First, I briefly describe the clinical case of Sabina, including a progresssion of self-similar dreams over the course of psychotherapy. I examine how even the tiniest dream snippets can reflect the whole of her developmental history and the evolution of our transference dynamics. I explore the open quality of intersubj ective borders existing between patient and therapist. On the one hand, the therapist enters the inner life of the patient by way of traditional transference as revealed through the patient’s dreams. On the other hand, the patient in this particular case entered the inner life of the therapist in less traditional, more transpersonal ways. I next present research suggesting that fractal physiology plays a key role in linking physiological micro-levels with psychological macro-levels during REM stages of sleep. I also offer research on physiological and brain synchrony suggestive of open borders between self and other. A nonlinear perspective on coupled dynamics equates intrapersonal communication

between one part of the brain and another with interpersonal communication between two brains, especially during intimate moments. The chapter ends by addressing the topic of synchronicity, or meaningful coincidence, whereby trajectories of fate imply open borders between self and world that include self-similar patterning and “acausal” connection.

Meet Sabina Sabina, who graciously gave me permission to write about our work together, is an East Indian woman who entered treatment in her mid— twenties, approximately 18 years ago. She continues to be my patient to this day. As a teenager in India, Sabina was sexually molested by her father while being blamed for the incest by her mother, who in addition, was physically and emotionally abusive. During her late teenage years Sabina’s emotional pain grew so intolerable that she attempted suicide by drinking pesticide. Fortunately, she lived through the episode, although her kidneys remain vulnerable. Several years later, Sabina fled India in search of a better life in the United States. Soon after arriving in the United States with no money or social support, Sabina met and married an East Indian man. At the point of seeking psychotherapy, Sabina did not drive or work, though she was taking classes at a local college. Her impetus to seek treatment came from fear of a male teacher; yet the true source of her fears was physical abuse by her husband. Caught between current and past traumas, Sabina was so haunted by her feelings and nightmares that she regularly cowered in bed, literally afraid to set foot on the floor in the dark.

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Through decades of intensive psychotherapy, Sabina became a US. citizen, got a driver’s license, completed a master’s degree, and now teaches at the community-college level. Over the past two years, she felt sufficiently healed and financially independent enough first to separate from her husband and then to request a divorce. As I write this chapter, Sabina is in the middle of nasty court proceedings, including a legal battle over custody of their 9-year-old daughter. In the beginning of our work together, Sabina was highly emotionally dysregulated and so overwhelmed by a host of physical symptoms that she regularly wound up in one emergency room after another, sometimes multiple times in a week. Especially during the early years, Sabina brought more dreams into psychotherapy than any other patient I have seen in over 30 years of clinical practice. During early stages of treatment, most of Sabina’s dreams were nightmares, from which she would awaken in a

scream with her heart pounding. The content was horrifying. Sabina was either being raped, forced to witness lurid sexual acts, or chased by figures carrying knives or guns who would stab or shoot her, as often to the point

of death as not. Many chase dreams early in therapy contained monsters who were either fantastical creatures or mythological figures attacking her in scary environments filled with creepy-crawly spiders or maggots. As our work progressed, Sabina’s assailants morphed more regularly into human beings. At first, those who chased and hurt her tended to be male strangers, often dressed in white, the color of death in India, but also a symbol of spiritual purity. As therapy progressed, the villains transformed again, now into members of her own family, whether past or present. Here is an example from the present: My husband is a murderer actively hacking bodies to pieces and hanging the parts on the ceiling. I am watching with great discomfort. He receives money for his eflorts. I feel scared about what is happening and urge him to

return the money. But he refirses. I grow afraid people will come after us and suggest we run away. No sooner do we start running than people begin

to chase us. Eventually, they discover me in a closet. I’m half naked. One of them puts a gun to my head. I wake up. As we continued to do trauma work, Sabina’s father, who had been dead for over a decade, frequently appeared in her dreams. In one nightmare, her father hit her again and again with the blade of an axe, blood Spurting everywhere. In another, Sabina’s father raped her. In a third, her father lay

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naked on a bed, the top of his penis exposed, as two Arabic women entered to make love to him. But most of the nightmares with recognizable relatives involved Sabina’s mother, the action often taking place in her childhood school where her uncle (father’s brother) was the priest. In real life, this uncle was all too publicly having an affair with Sabina’s mother, headmaster and a teacher employed at the same school. In one nightmare, Sabina’s mother stoned her. In another, her mother nearly cut off her head. In a third, her mother used a pressure hose to shoot water so hard that Sabina died. Here is a particularly gruesome variation: My mother and I enter a coflee shop. A blender is sitting on a countertop.

My mother shoves my head into the blender. She turns it on, and my head gets fillly minced She doesn ’t stop there. She smfis my whole body into the blender. When she turns it on again, I see my body in pieces. There is blood

everywhere; it is splattered all over the walls.

Self-similar themes A common way to interpret recurrent dreams is to look for external meaning. From this perspective, the need for repetition reflects identical or similar situations existing in real life. A different way to understand recurrent dreams is to look for internal meaning, an approach that informs my own clinical style. I have speculated (Marks-Tarlow, 2012, 2014) that repetitive dreams often reflect the earliest relational landscapes, constituting a kind of preverbal topography at the foundation of attachment dynamics. In the case of Sabina, a prolonged history of emotional, physical, and sexual abuse reflected personal space and a sense of self repeatedly violated by the most significant people in her life. From the start of life, those who were supposed to protect her were the very ones who endangered her, a recipe for a disorganized attachment. Child-researcher Beatrice Beebe and colleagues (2010) conducted micro-analyses on babies at 4 months in search of the roots of disorganized mother-child attachment dynamics evident at 12 months. In these instances of intrusive, mis-attuned mothers, Beebe observed paradoxical binds in action—children felt drawn to approach their caregivers while simultaneously attempting to avoid them. At times the resulting behavior appeared downright bizarre. One baby approached its mother backwards. Another arched away while braying toward its mother.

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Much like Beebe’s baby subjects, Sabina was a young child dependent upon abusive caregivers for her very survival. Equally caught in the approach-withdrawal dilemma of disorganized attachment dynamics, Sabina’s best option was to run away, first in fantasy, then in real life. Yet, without the horrors of her youth fully processed, Sabina was forced to revisit horrific scenes at night. Through bringing this self-similar sequence of dreams into psychotherapy, we could begin to make meaning out of them by re—processing traumatic memories in an atmosphere of safety and trust. The theme of being chased in service of being harmed and even killed reflected past traumas that reached deep into Sabrina’s body, threatening her health and even her life through a past suicide attempt and current somatic symptoms. Simultaneously, the progression of self-similar themes became a present-centered gauge of her ongoing transformation within psychotherapy towards ever greater safety and trust. Not only did Sabina’s

nightmares grow steadily more infrequent, but as Sabina began to calm down, the horrific monsters no longer chased, caught, or killed her as often, indicating that Sabina was feeling less haunted. As the months and years went on, and Sabina’s assailants took more human form, her defenses were growing less primitive and somatically lodged. The eventual direct appearance of Sabina’s family members in her nightmares suggested an enhanced capacity for internal regulation, while

being reminiscent of Mitchell’s (1998) poignant notion of the “transformation of ghosts into ancestors.” As Sabina slowly stopped running in response to inner horrors, steadily she was less caught by amygdala—driven, fight/flight impulses of trauma-based physiology. This change was apparent within the following dream (reported in Marks-Tarlow, 2008, p.13): There is a werewolf in the middle of the road. It is dead, and itsfitr is ripped open to reveal lots of blood and guts. I feel disgusted. Yet, I feel drawn toward the beast anyway, stmck by the sawtooth shape ofits wound.

When Sabina and I reflected on the meaning of this dream, she grew animated. She expressed feeling inspired by the jagged pattern of the wound. She could imagine herself artistically emblazoning a similar design onto a canvas to be painted in bold colors. To me, it was clear at this point that Sabina felt safe enough to be fully captivated by the path of selfreflection. Not just at conscious levels, but also unconsciously, Sabina now embraced the beauty of her own experience, no matter how scary or painful.

ses

{Slept-er Nine Draam interpretation as pro] action

Sabina’s self-similar progression of dream themes fmm monsters to stlangers to the very humans who have hurt her indicate increasing organizaiiocn within her psyche, a pmgression that closely resembles Eamefs {19159) classic system of Rorschach interpretation. For readers unfamiliar with the Rorschach, this projeciiue style of psychological testing consists o f a series ofinlcblots presented to a viewer who is asked to flee associate. y-erbalize, and elaborate on what they “see“ in the cards. Dreams resemble Rorschach cards in that both consist of ambiguous foJms onto which meaning is projected. Interestingly, the very Rorschach card inldJlo’Is themselves taJ-re fmctal form. According to a report {Abbot 2131?] based on the research otaylor and colleagues {Eill’i'}, these is a fractal “sweet spot“ within each Roischach card {see Figure 9-1] displaying a characteristic fractional dimensionality. Taylor‘s group hypothesized that the fractal dimension otfthe cards would mimic that of naiural objects, such as clouds. However, the

fiactal

dimensionality o f a typical card turned out slightly less than that of most natural features. Apparently, this may be what maximises the potential for projection ofinner forms and meaning onto the outer shapes ofthe cards.

Figure 9-1. Comet to The real iiilrblet {left}, a fem with no fractal features. {Jigs}, or with no rrmch fractal cenpledy, remces on: abiJity to p-eroeive and. pnject hitkienimages. {Rep-Med from RPTaybi er a1, 20-11, m m pandas-en} The validity of the Rorschach as a projeciiue device. where inner contents of consciousness get projected onto out-er objects. may result partly from its fnctal edges that provide open portals at the threshold between conscious and unconscious processes. Tl'tisis eracfly where dreams lie. The next section eramines the neurobiology of dreamsI where fractal neuro-

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dynamics amplify self-similar pattern, fiom the micro- levels of brain cells to the macro-levels of behavior and conscious awareness.

To dream, perchance to learn Although dreams have been revered by some, they have also been maligned by others. The contemporary scientist Crick, co-discoverer of the DNA helix, is one such maligner who dismisses the significance of dreams,

declaring their content meaningless. Crick and Mitchison (1995) viewed REM states as a response to overload in the brain, mere epiphenomena of random neural firings that help bring the brain back to an open state by “unleaming” unnecessary patterns. Yet chaos theory informs us that even seemingly random events at the surface can carry deep fractal order within the structure of underlying strange attractors (see Figure 9-2). Kahn, Combs, and Krippner (1997) attempted to reconcile Crick’s claims about random firings with the perspective of nonlinear science to integrate a theory of selforganization with the neurophysiology of dreams.

Figure 9-2. An example of the first strange attractor discovered, called the Lorenz attractor after its discoverer. The Lorenz attractor models surface unpredictability of the weather (no wonder people make fun of weathermen!) While revealing underlying fractal order. (Public domain)

At Harvard University, neuroscientist Anderson and colleagues (1998) studied neuronal firing patterns during REM sleep in fetal sheep. Anderson detected fractal pattern at the neurophysiological level, in the form of l/f power law temporal distributions. Much like the dynamics of the stock

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market, when power laws operate, self-similar fluctuations become evident on multiple time scales, here detected within the random surface pattern of neural firings. From this data Anderson and colleagues speculated that dreams serve to integrate various levels of functioning, ranging from the neurophysiological underpinnings right up to the level of consciousness and

behavior (Anderson & Mandell, 1996). It is no wonder that fetuses and newborn babies spend most of their time in REM sleep. Nonlinear—researcher Buzsaki (2006) has also made important links between micro- and macro-levels by studying single place cells of rats. These cells form a kind of grid in the hippocampus to fire whenever the rat enters a particular geographical area. Buzsaki compared the pattern of place-cell firings when rats ran through a maze during the daytime with place-cell firings during REM states at night. The result was remarkably identical patterns. This provided evidence that learning and memory get consolidated at night, especially during the dream phase of sleep. When Buzsaki exposed rats to two different mazes during the daytime, the subsequent pattern of place-cell firings at night revealed a creative synthesis of the two maze patterns, which suggests an integrative function within the creative power of dreams. In the rat hippocampus, place cells help the animal orient and navigate through the physical environment. In humans, the hippocampus has acquired an additional function. Because the human hippocampus is the seat of learning and memory, it helps us to orient and navigate through the social environment (Marks-Tarlow, 2012). Buzsaki’s research reinforces Anderson’s conclusions that processes at the micro-level cascade into self-similar processes at the macro-level, winding up in cognitive and behavioral changes.

Transference dreams and the relational unconscious From the beginning of our work together, Sabina had a series of transference dreams in which I, as her psychotherapist, am included. With interactive regulation, the major tool for developing secure attachment during early development or within psychotherapy, transference dreams become especially important indicators of therapeutic status and progress. In an early transference dream, Sabina entered my office, also my home in her dreamscape, where she caught a glimpse of my daughter as well as of my “dirty laundry.” This dream snippet emerged during initial stages when Sabina seemed to be testing me unconsciously. Although she desperately

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wanted and needed safety with me, this young woman had never known deep trust with anyone and had no implicit foundation for the experience of it. No wonder that she needed to check out my dirty laundry to ascertain whether my personal issues would prove trustworthy or too dangerous for self-revelation and exploration. From the perspective of fractal consciousness, where even the smallest dream snippet potentially sheds light on the whole of things, many of Sabina’s subsequent transference dreams shared the central feature of this initial one: either she entered my personal space or I entered hers. This held in dream after dream to follow, as in this slightly later version. You enter my house (not really where I ’77: living now). There is a boxed black and white picture of my mother that is standing at a low angle shot flom beneath in an unusual way. You comment on it, finding it interesting,

especially because it has an iridescent shine to it. Together, we take it apart to reveal a picture behind the picture. This solves the mystery of the

iridescence. Then the scene changes. I am 5 years old. You are my mother. You pat my hair very afiectionately and play with my cheeks. In contrast to the scary figures that haunt and continue to traumatize Sabina, this dream indicates I have gained enough trust to help her

understand her own “bad” mother from a safe distance (the picture is black and white, not color, and boxed in). In this role, I become the longed for, long-lost “good mother” with whom she may regress to earlier emotional stages, which could then allow us access to primitive defenses surrounding dissociated and walled off, highly vulnerable, aspects of self. The last scene, where I expressed affection and kindness by patting Sabina’s hair is especially poignant, given the contrast with her own biological mother. Many of Sabina’s most painful memories around age 5 surround her mother pulling her hair in anger when brushing it, or at other times later in life, criticizing the looks of her hair. Here is a particularly interesting iteration of this transference series: You and I are at ywr house. We are in the kitchen. You are showingme how to make soap using a method that involves taking used bars and recycling

them, by putting them together in a big pot. You say to me, "You ’re going to stay, aren’t you? ” I happily reply yes. We both know without speaking any words what this means—I will live with you forever. Sabina here appears to have reached full-object constancy, such that I have become fully internalized and will live inside her forever. She also seems to feel “contained” enough by me emotionally to work under

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conditions of heat (high arousal) with old memories and feelings. Recall that Sabina’s suicide attempt involved swallowing poisonous pesticide. In her dream, we work together to make soap, as if to wash clean any trace of previous toxins. We are in my kitchen, where I act as her guide in this hearth of the home, place of nourishment, both literally and figuratively. Sabina feels safe enough to melt old defenses and address core vulnerabilities. Symbolically, I help her “clean up her ac ” by teaching her how to recycle the old by integrating painful memories into something new. This view of the dream fits nicely with Lane, Ryan, Nadel, and Greenberg’s (2015) proposal that the universal indicator of change in effective psychotherapy is memory reconsolidation, by which previously unprocessed, traumatic memories are brought back up to consciousness where they may be re— consolidated in a more regulated, integrative fashion.

While cautious of universal symbolism, I do believe that the regular appearance of houses is often associated with foundational aspects of self. Consider Jung’s dream, previously mentioned. He descended through a house until he reached a hidden chamber in the basement where ancient bones were buried. This imagery inspired one of Jung’s most important contributions to psychology, the notion of a collective unconscious, arealrn of images and symbolism believed universally present in the deepest levels of the psyche and archetypally shared. Whether arising at the level of culture or of personal dreams, Jung’s concept of the collective unconscious is a transpersonal notion. Some contemporary scholars, such as Eenwyk (1991) and Harle (2010), conceived of Jung’s archetypes in fractal terms as “strange” attractors that take self-similar form across various cultures and historical eras.

The relational unconscious in the space between self and other Jung’s notion of the collective unconscious bears an important relationship to what psychoanalysts Gerson (2004) and Sands (2010) referred to as the “relational unconscious.” Here, open underground channels of communication exist beneath conscious levels of awareness, especially between people in close relationship. Much like the notion of the collective unconscious as being transpersonal, so too is that of the relational unconscious, although the latter is narrower in scope, being restricted to the intersubjective field shared between people with close ties.

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To me, Sabina’s transference dreams described above reflect a strong wish to be very close to me. They are the inverse of the boundary violations reflected by her horrific chase nightmares. Here, open boundaries lead to intimacy and enhanced life, rather than to torture and death. The open boundaries whereby Sabina enters my house, or I enter hers, seem to indicate ready access to each other’s inner sanctums. Indeed, at this point in our work together, Sabina and I mostly felt close. I had earned the right to stand in as the “good” mother, and we were in an ideal position to address early relational damage by dredging up previously unprocessed trauma. As I write this chapter, two more transference dreams came, back to back, during a very difficult period as Sabina goes through a very painful and protracted divorce. In the first dream, I was again in Sabina’s house, along with Sabina daughter, being very emotionally supportive. In the second dream, I was literally holding Sabina as a baby, cradling her in a way that felt really good. Important themes are reflected in the following transference dream (reported in Marks-Tarlow, 2008, p. 21): You were underwater, waving ymir arms as if holding paintbrushes in both

hands you used to create swirling shapes on pieces of paper, one for each hand. As each painting was completed the papers=floated upwards towards

the surface. Suddenly among all the papers, the dead body of an adolescent boy also floated upwards. You kissed the back of his neck. The boy ’s eyes popped open and suddenly he came alive. I woke up. The dream was so scary. The images stayed with me for hours. I couldn ’t get back to sleep.

Notice the opposite-energetic pattern evident in this dream—rather than someone being killed, someone dead is being brought back to life. I have come to understand this dream as a concrete illustration of what Kestenberg (1985) called “dead spots” in the infant’s subjective experience. What emerged from the dream, plus Sabina’s associations to its images, was a new stirring inside that “brought alive” a realm of dissociated emotion connected to a sexual trauma that took place during Sabrina’s adolescence. The incident, involving an adolescent boy, had not been repressed (it remained accessible within the recesses of Sabina’s memory); rather, it had remained emotionally unprocessed until now. Sabina’s transference dreams appear to serve as an underground form of communication. Such communication, from unconscious to unconscious, has been especially important in this case because of the highly relational nature and focus of Sabina’s psychotherapy with me. Together, we have undertaken the difficult challenge of shifting Sabina’s attachment status

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from disorganized towards organized and earned-secure, which occurs when someone rises above harmful effects of childhood abuse and trauma to feel securely attached to another human being later in life. I would like to bring home the porous unconscious boundaries of our relationship by sharing one more transference dream that occurred years later and extends even further into transpersonal realms. First, however, let me provide some background context. As opposed to the rich texture of previous stages in treatment, at this point the sessions felt stale. Compared with the highly dramatic, emotionally charged quality previously, we now seemed to be locked in a mutual stance of defensive disengagement. On Sabina’s end, she talked repetitively about jealousy concerning insignificant and tangential women in her husband’s life, in whom, from my perspective, her husband seemed to have little real interest. From my end, a low-level frustration was slowly building at the repetitive nature of the material. Clearly, we had reached an impasse. Then one day everything changed with the appearance of a dream unlike any other (Reported in Marks-Tarlow, 2008, p. 43): I am back in my old house in India. The roof is knocked ofi’. My mother is there. So is my daughter Maya (a two-year old toddler) who wanders ofin

search of a shoe. Suddenly I notice a tidal wave heading towards me. It’s absolutely huge. The water is clear. I feel desperate to protect myself I back up against the wall. From here I watch the wave approach. But while looking at this wave I don’t realize there is another one coming from the opposite direction. The water is black, and this second wave weeps over the house. Water fills the room and rises and rises. I’m struggling for my l i e . But somehow, I manage to make it through. Slowly the water recedes. Then I see my mother coming towards me. She ’s got Maya in her arms, holding her high in the air. Maya appears dead. Her body is limp. My mother thrusts my daughter towards me. She is snarling with an almost satisfied look of contempt on her face. I look on in horror.

This nightmare terrified Sabina and was unlike any other she had dreamed. It certainly deviated from her typical theme of being chased by monsters, relatives, or other scary people. What I found remarkable about this dream is that the tidal wave imagery perfectly matches the recurrent

theme that I used to dream as a child. To encounter huge tidal waves was by far the most frequent dream I used to have. I now understand this to be a classic anxiety dream. Within my own nighttime topography, sometimes a giant tsunami would approach too slowly and wash over me, occasionally occurring when I was backed against a wall. Sometimes the dream would start when I was already in the water, whether alone or with others. I never

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died in these dreams, and they were not always unpleasant. At times they were even comical, as when I dreamed of furniture floating all around me, alongside other people in the water, as if all were sharing a communal bath in the living room. These dreams peaked during middle childhood, when my parents were often out at night or traveling abroad for weeks at a time, making little or no contact with me while away. The moment Sabina recounted her tidal—wave dream, it touched and triggered my own memories in a deep and raw interior place that previously had been asleep for years. As I listened to Sabina’s dream, sensory-based, full-body memories of feeling alone, abandoned, and scared suddenly swept over me. I had vivid images of how terrified I used to feel as a little girl, especially at night. I remembered sucking my thumb and rocking. I recalled my fear every time my parents went out that they would never come back. I was terrified at their advanced age (my mother was 40 when she had me,

unusual for my generation). Over and over, I used to obsessively calculate how old I would be if I lost them at particular ages, plus imagining at what age I might feel safe, if only I could make it there. This highly charged, embodied, emotional experience triggered a sudden recognition in me regarding this case. Over these past several months I had been subtly aloof. I had disconnected fi'om Sabina as she expressed jealousy towards other women in her husband’s life. Meanwhile, my own father had had frequent affairs, including a 15-year one with the mother of my best childhood friend. During our impasse, I had noticed my own mild irritation, but had brushed off its significance. I had missed recognizing my own counter transference and the meaning of my dampened experience. I suddenly understood that Sabina was hitting upon a dissociated pocket of pain, betrayal, and terror within my own psyche. I then recognized how my own insensitivities had been playing into the picture. By avoiding intense feelings and brushing off my own discomforts, I was also brushing off Sabina’s. My empathy had become blocked, which only exacerbated Sabina’s experience of isolation and desperation.

Synchrony through the relational unconscious Psychoanalyst Susan Sands (2010) proposed that, within psychotherapy, dreams sometimes “activate powerful forms of unconscious affective communication between patient and analyst, which crucially facilitate the transformation of dissociative mental structure” (p. 357). Sabina’s dream appeared to emerge from what Gerson (2004) dubbed the relational unconscious to bring me into greater synchrony with her by seeking-and-

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finding deeper emotional congruence. We both came to view this dream as a transpersonal attempt to open wider channels of empathy. Indeed, Sabina’5 dream helped me attune to my role in our impasse.

Now I couldfeel where our interior worlds overlapped and where I had erected defenses against painful memories of betrayal and separation. Rather than the attuned behavior of the “good” mother, I had become the emotionally withholding, insensitive “bad” mother who was ignoring, shaming, and blaming her daughter for her failures. After all, I had recently called Sabina’s attention to the fact that she had hardly mentioned her daughter in months. Yet, simultaneously I had been enacting the child, who was upset at her own mother’s neglect. I now understood my mild irritation with Sabina as a defense against rousing my own underlying abandonment fears. Sands (2010) suggested that implicit—level dream communication is particularly likely to occur when overwhelming experience is dominating treatment. Indeed, Sabina’s dream came just as major crisis was erupting involving her daughter who, at age three, was recently diagnosed with severe autism following my suggestion she be evaluated. In the weeks that followed, a torrent of ruptures and re-solutions broke loose. This was a scary time. Extreme shame and suicidal feelings arose. Sabina even stopped eating at one point. She had grown utterly hopeless. Her life hung on a frayed line; bonds of trust stretched thin between us. This transpersonal transmission functioned in dual fashion as a life threatener and life saver. By communicating primitive levels of panic and separation anxiety directly from one unconscious to another, Sabina’s dream helped to restore my fuller experience of compassion. Although this period was excruciatingly painful for us both, we proved able to withstand the massive waves of high arousal that followed. Once the torrent of intense emotion passed, we were brought to a new level of trust. In the process, a greater level of complexity emerged in Sabina’s internal organization. She felt less wounded and more compassionate towards the short-comings of herself and others. A newfound sense of resilience eventually enabled Sabina to reconnect with her own very flawed mother, still in India, after

many years of estrangement.

A fractal model of biobehavioral synchrony One of the most valuable contributions a fractal epistemology could make, to psychology broadly, and to transpersonal psychology specifically, is a

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mathematically rigorous way to model open, semi-permeable boundaries between self and other. Many of us unconsciously hold a model of interpersonal relationships that presumes a Cartesian split, as if people are thoroughly encased within their bodies and cleanly separated from others by smooth boundaries. Especially in Western individualistic culture, the ideal of healthy adulthood promotes independence, such that it becomes easy to take for granted autonomous functioning. Yet, cross-cultural studies reveal this as only one perspective among many. Indigenous and collectivist cultures often envision porous-psychological boundaries, not only between self and other but also between self and world. Open channels are apparent in phenomena such as spirit possession, Shamanic travel, and shapeshifting. Individualistic lenses obscure the degree to which human beings, especially those in intimate relationships, operate in sync. We are deeply “wired” to be interconnected to each other via body, brain, and psyche. On the one hand, this is little wonder, since most mammals, including human beings, are herd animals with social minds, bodies, and brains that have evolved to care for our young and live in groups (Cozolino, 2017). On the other hand, new fiontiers are opening in science, which allow us to visualize and measure physiological coupling and neural synchrony with ever greater precision. Researchers, such as Aftanas and colleagues (1998), used nonlinear methods to capture overlap in brain and body processes otherwise invisible when purely linear methods are used. New scientific fields are emerging, such as interpersonal neurobiology that reveal neural correlates of how relationships shape the minds, bodies, and brains of others. Ruth Feldman and colleagues’ (2011) study on physiological synchrony is illustrative. Mothers and their 3-month old infants were observed during face-to-face interactions. Cardiac output was collected in both mother and baby, while the behavior of both was videotaped and micro-analyzed to mark episodes of mutual gaze, affect, and vocal synchrony. Results revealed a concordance of mother and infant biological rhythms that increased significantly during episodes of mother-child affect and vocal synchrony, compared to non— synchronous moments. Feldman’s group’s research points to the openness of the autonomic nervous system to maternal social influences. Feldman et a1. ’5 (2011) approach builds on the work of others who have identified three main channels of nonverbal synchrony. Kaye and Fogel (1980) studied how gaze synchrony enhances social relatedness and cognitive

growth. Cohn and Tronick (1988) studied the role affect synchrony plays in children’s development of self—regulatory capacities. Jaffe and colleagues

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(2001) identified how vocal synchrony during “proto-conversations” (e. g., Trevarthan, 1989) between parent and child builds language skills and attachment security. These studies point to the importance of biobehavioral synchrony during development. But what about adults? After the formative years, does the development of agency and autonomy signal the end of sync? The answer appears a resounding no. Structural coupling of affective, cognitive, and motor systems is as natural to adults as it is to children. We know this partly because brain imaging has advanced far enough to approach minute-tominute embodied processes as they occur between people, within context. A particularly exciting line of research is anew form of brain imaging called hyperscanning (Babiloni & Astolfi, 2014; Dumas et al., 2010; Liu & Pelowski, 2014). Hyperscanning, which consists of simultaneous measurements of brains, was innovated to assist in parapsychology experiments and only later extended to ordinary social functioning (e.g., Bouten, Pantecouteau, & Debruille, 2015). Riley, Richardson, Shockly and Ramenzoni (2011) examined neural substrates of cooperative tasks between people. Whether involving turntaking or simultaneous action, they concluded that movement systems in different actors become coupled to form low-dimensional reciprocally compensating synergies. Similar results extend beyond motor systems. Anders, Heinzle, Weiskopf, Ethofer, and Haynes (2011) identified a “mirror” representation of another person’s affect in the perceiver’s brain. Interbrain synchronization is especially strong in people who are emotionally engaged with one another versus detached observers (Dumas et al., 2010). Schilback et a1. (2013) likened the remarkable progress made in social neuroscience, marking the beginnings of a two-person neuroscience, as comparable to the discovery of dark matter in physics. It may seem controversial that interpersonal synergies span organisms by extending beyond boundaries of skin. Yet, from a neural perspective, there appears to be little difference between transmission of information between two areas within a single brain and transmission of information between two individuals (Hasson, Ghazanfar, Galantucci, Garrod, & Keysers, 2012). This makes sense given how important neural synchrony is in the development of cortical networks within a single brain in the first

place (Uhlhaas, Roux, Rodriquez, & Rotarska—Jagiela, 2009). The science of hyperscanning is still in its infancy, yet results are remarkable in the degree of neural sync that crosses over the threshold of

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both physical and psychological boundaries. Consider a ground-breaking paper by the lab of Princeton University neuroscientist Uri Hasson (Stephens, Silbert, & Hasson, 2010). The group solved previous difficulties with the noise and physical clumsiness of flVJRI machinery to simultaneously measure two brains during real-time communication. They tracked the electrical activity in the brain of a speaker telling a story along with the brain of a listener hearing it. What the scientists discovered was widespread neural resonance between the two brains that extended far beyond the parietal and premotor areas that contain mirror neurons (specialized to fire whether a person makes an action or watches someone else making the same action), as well as beyond cortical areas related to speech production and reception. What is more, the lab’s results also suggested that the greater the understanding displayed by the listener, the greater the brain synchrony with the speaker. Most remarkably, listeners who displayed the greatest understanding of the story revealed areas of neural resonance that anticipated the brains of speakers. Apparently, during excellent communication, not only do we follow the words of another person, but we also hang on to every nuance such that we can forecast what is to come. Hasson et a1. (2012) believed brain-to-brain coupling is the primary mechanism for creating and sharing social worlds.

Synchronicity and acausal connection Whereas synchrony requires open emotional and physiological borders between self and other, synchronicity requires open emotional and physiological borders between self and world. Synchronicity is a term coined by Carl Jung to describe meaningful coincidence. Where we might expect only chance or a random sequence of events, synchronicity instead indicates meaningful patterning. Carl Jung (1973) believed synchronicity occurs when an emotionally charged complex is activated and material in the unconscious is blocked from conscious awareness. Because of this block, rather than to experience something internally within our own psyches, material is forced out into the world and then backwards into consciousness through physical rather than emotional channels. To Jung, synchronicity was evidence for the Unus Mundus—the one world connecting mind and matter—with meaning serving as the thread between. Several contemporary theorists (e.g., Colman, 2011; Hogenson, 2005, 2009; Main, 2007) have expanded upon Jung’s ideas.

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Modern day spiritual guru Deepak Chopra (1995) presented The Way of the Wizard as an opposing but related stance. Rather than the result of blocked consciousness, Chopra celebrated synchronicity as a kind of hyperconsciousness. He views synchronistic events as the manifestation of our spiritual intentions in the Universe. Synchronicity occurs when we both seek and catch glimpses of the fundamental interconnectedness between mind and matter during higher states of open consciousness rather than lower states of blocked consciousness. I follow in the footsteps of Allan Combs (Combs & Holland, 1990) and Joseph Cambray (2009) to examine synchronicity through the lenses of contemporary science. Whereas intersubj ectivity is borne of open boundaries between self and other, what I have called interobjectivity (Marks-Tarlow, 2008) is home of open boundaries between self and world. In the latter case, fractal boundaries arise from structural coupling between self and environment when self-similar patterns become evident at the interface between subjective and objective levels. The fractal idea of the whole in the part is one scientific approach to synchronicity at the “joints” between mind and matter. Whether a branch in the road, a node of destiny, or a chance encounter, many encounters at the crossroads reveal self— similar pattern underneath. Fractal boundaries tend to occur when multiple basins of attraction are governed by the same underlying attractor (Schroeder, 1991). This notion of one underlying attractor governing the whole in each part of the universe fits well with concepts about indeterminate quantum waves and ideas about equipotentiality. Out of decades of deep and quiet contemplation of nature, physicist Bohm anticipated chaos theory partly by positing a single “attractor” in the form of a unified underlying order of the cosmos (Peat, 1997). Meanwhile, solid-state physics was the first arena for discovering self-similar patterns in time in the form of long-range intercorrelations during chaotic phase transitions from one state to another. Bak (2013) brought this insight from microscopic to macroscopic levels when he perceived the unity in nature between physical and temporal fractals. Ideas such as these from physics wreak havoc on linear conceptions of cause and effect, which dictate instead that one event will lead to the next, which leads to another, and so forth. When the pattern of the whole is present in its parts, we must embrace more complex models of causality and relatedness, including the notion of acausal connection. With acausal connection, the “glue” between parts is not based on a linear or temporal chain of events. Because fractal patterns exist outside of any particular time

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or size scale, fractal elements connect with one another instead through selfsimilar symmetry. In the case of fractals, acausal connection preserves fundamental identity of the underlying wholeness by permeating fractal parts. When acausal connection takes the form of synchronicity, the identity of the whole serves to unify inner and outer worlds, spirit, and matter. When two seemingly un-connected things happen simultaneously, they can be acausally connected to one another through hidden self-similar channels of meaning. At the most global-spiritual level, the very largest scale pattern for each one of us is our pattern of fate. Especially, when we look back at our life trajectories, I believe that patterns of fate tend to take fractal form. In my 2008 book, Psyche ’s Veil, which applies chaos and complexity theories and fractal geometry to clinical practice, I defined fate as the fi'actal residue of chance, using the Chaos Game invented by mathematician Michael Barnsley

(1993) to illustrate this concept. The Chaos Game is simple to play. Create a 3-sided die to throw (impossible in real life). Get a sheet of paper and draw a triangle. Label the three corners, or vertices, A, B, and C. Pick a starting point (Zo) anywhere within the triangle. Roll the die. Go to the vertex indicated and then halfway back. Mark this point (Zl) as your new starting point. Repeat the process again and again, marking each new starting position with a dot. After repeating the process over and over, what do you imagine the resulting pattern of dots to look like? One naturally might guess that the randomness of the die throws would result in a chaotic looking mess of dots. Instead the result is a well-known fi'actal, the Sierpinski triangle (see Figure

9-3).

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The Chaos Game ,

Sierpinski Triangle

1,2

5,6 Figure 9-3. To play the Chaos Game, start at Z0. Roll and plot Z1, Roll and plot 22, Roll and plot Z3. Keep going until underlying pattern emerges. The result is the Sierpinski triangle. (Courtesy of Terry Marks-Tarlow)

Fractal order appears under the randomness o f the Chaos Game due to the consistent underlying seed algorithm. To extend this principle to people, imagine that each one o f us has an underlying seed algorithm that is partly determined by genetics and partly determined by epigenetics (how the genes get turned on and off by environmental context, including patterns o f nurture). Much like the chaos game, it would then take a certain amount of time for the underlying fractal attractor to reveal itself. Perhaps the Chaos Game helps us to understand Why self-similar patterns o f fate become most

evident over the full course of our lives. Synchronicity is evidence for acausality in nature, for the natural tendency to self-organize using self-similar threads, Where wholes

continually fill in the holes of our experience. Just like the Mandelbrot set, Where self-similar representations o f the Whole keep popping up in unexpected places, in a synchronistic universe, islands of local order are interconnected beneath broad seas of disorder. As human experience stretches out into the world only to fold in again, events enfold recursively

upon themselves to reveal self-similar pattern at multiple levels of organization.

My own belief in synchronicity represents for me the ideal blend of scientific with spiritual and professional sides. I see synchronicity as evidence for invisible channels of spiritual connection deep under the life’s material surface, with self-similar flows governed by hidden attractors in a

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sea of quantum potentiality. By my account, synchronicity represents selfreferential symmetry occurring when spirit manifests physically such that outer events become animated with inner meaning. I suggest synchronicity is a form of temporal fiactal most likely to occur during highly emotionally charged and chaotic phase transitions in life. If so, no wonder that full crisis plus surrender to the unknown can move us into fi'actal boundary zones, allowing new portals to open us into and out of old dilemmas. Within psychotherapy I believe the presence of synchronicity often signals the melding of two psyches or the impending emergence of change. Even though we can never interpret the future meaning of a present event with certainty, we still should try to understand the symbolic significance of outer, in addition to inner, levels, albeit with doubt and humility. As a therapist, I find it invaluable to stay open to synchronistic happenings,

especially when people ripe for change find themselves hovering at edges of chaos. A final dream of Sabina’s On the very day I had planned to ask Sabina permission to use her dreams in this chapter, she brought in the following dream: I am with my mother on a deck (reminds me of the Titanic, which was one of my daughter ’sfavorite movies and the last movie I watched with my mother the day before I left India). My mother (her current age of 70) stands with a railing behind her when she makes the announcement that she is pregnant. My father (aged 47, his age ofdeath) then makes an appearance. He stands to the left of my mother, who rests her hand on his shoulder. I feel disgusted at the whole scene and walk up the stairs to a higher deck. I look at the two of them for a moment and then walk away.

When asked what stood out to Sabina, she was struck by her parents being so “together” in this dream. This contrasted sharply from memories of their relationship as highly strained, conflictual, and distant. The dream seemed to resonate with Sabina’s feelings of greater inner coherence and organization, that is, her more regulated internal landscape where her mother’s “railing/raling” stands behind her. Sabina observed that instead of feeling hate towards her parents in the dream, she felt disgust. Furthermore, no one was chasing her. She experienced enough fi'eedom to climb to a higher level and then walk away of her own accord. These observations seem consistent with Sabina’s “higher” level defenses (less somatization, enhanced ability to tolerate and process emotions using her conscious mind instead of her body) plus her greater sense of agency in the world. We both

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had the sense of this dream completing the previous chase dream series presented earlier. On my end, I was rather stunned by the synchronicity of Sabina bringing in this dream on the very dayI intended to ask permission to use her dreams in my writing. This feeling was enhanced by the fact that she had not brought in a dream in months and months, given so much external drama surrounding her divorce proceedings. The sense of resolution surrounding this dream not only felt like a gift to Sabina for all her hard work in psychotherapy, but it also felt like a gift to me to complete this chapter.

A final dream of mine A fractal vision of life filled with chance events, random occurrences, and fearful unpredictability is also a vision of life meaningfully ordered. Although chaos theory dictates that the specific events in our lives remain fundamentally unpredictable, the possibility of fractal borders between inner and outer worlds suggests that nothing is truly random. Especially when looking back, despite the occurrence of so many chance events, we can usually detect underlying attractors in the form of self-similar fractal pattern from the start. I end this chapter with my own childhood dream, which similarly has structural qualities of being fi'actal, and in hindsight appears synchronistically prescient (Reported in Marks-Tarlow, 2008, p. 210). I ’m sitting in the dark, on my bed in South Orange, New Jersey, surveying the sea of lights and treetops outside my window, as I love to do before going to sleep. Suddenly above all else towers the silhouette of the Statue of Liberty, appearing as a giant figure in the distance, as she rapidly pursues me. Terrified, I place all my stuffed animals along my windowsill in a =frantic efi'ort to protect myself After lining all the animals up, I dive under the covers. In the morning] awaken to the sound of the front doorbell. Alone in the house I go downstairs to answer the door. But when I look outside, n o one is there. Instead, sitting atop the stone bench stoop to the left, is a

miniature replica of the Statue ofLiberty. Filled with delight I scoop her up and bring her into the house, shutting the door behind me. This is one of the few dreams I remember from childhood and the only chase dream I ever recall having. I am currently aware of a lovely symmetry between mine and Sabina’s dreamscapes, with us each having had one major dream akin to the internal landscape of the other. At various stages of life, I have returned to my own dream’s scary but beloved images. Within

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my personal psychotherapy and during clinical training, I enlisted various therapists and supervisors to approach the dream fi'om multiple perspectives and theoretical orientations. I have understood my dream in various ways: in terms of conflicts with my mother; fear of my femininity; narcissistic wounds, including anxiety about being crushed under the enormity of important others; and terror surrounding my personal liberty and creative freedom. With each new iteration has come a new shade of meaning. No one interpretation seems any more “correc ” than any other. Each is observer-dependent, as seen through lenses of current relationships and states of being. The dream has reflected the whole of my psyche in different ways at different stages of my life. Whereas Sabina’s dream felt transpersonal through open self-other portals, this dream feels transpersonal in reflecting, as well as presaging, core themes yet to emerge. Symbolically, within the very structure of this dream, a fractal appeared as a gift from my unconscious, long before my conscious mind knew what fractals were. In fact, I had this dream in the 1960s, long before fractals were discovered/invented by Mandelbrot (1977). The appearance of the Statue of Liberty on several size scales and multiple levels is a fractal image that presaged my current intellectual and therapeutic interests. In my dream the central symbol of the Statue of Liberty appeared explicitly at two size scales—giant and miniature. She also appeared implicitly on the third scale of her real-life manifestation in New York City, where my father worked. The City was an exciting but scary place for me to visit from my sheltered—suburban New Jersey home. In my dream, scale reveals much about my internal conflicts as a child. The initial large-scale appearance of the Statue of Liberty suggested anxiety-provoking internal struggles that felt larger than life and too much to handle. Lining up my stuffed animals and then plunging myself under the covers, where I could no longer see what was happening, was a concrete enactment of my tendencies to use lines of reason (intellectualization) plus ducking and sleeping (denial) as central defenses. In the morning Liberty’s reappearance in static and miniature form is suggested eventual mastery of this central conflict, by my ability to titrate my fear and happily assimilate her female form on a manageable scale into my psyche at ground/grounded level. When I first wrote about this dream for my 2008 book, I drew a fractal image to accompany the dream, which I dubbed “Liberty in Hand” (see

Figure 9-4). Along with the central fractal image that presaged my own

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interest in fractals, this drawing also contains an eerie foreshadowing. Note the date that accompanies my signature: July, 2001—exactly two months

before 9/11. Never had I drawn a city skyline before. At the time, I felt shaken to have captured the twin towers so soon before their destruction. As I gaze upon my drawing today, I see something I had not noticed before. The main figure looks quite angry, as if she understands the upcoming threat

to America’s democracy and freedoms. Her mouth seems to scowl, while her eyebrows are furled.

Figure 9-4. Liberty in hand. (Courtesy of Terry Marks-Tarlow)

Perhaps the seeming magic of synchronicities such as this occurs less out of foresight and more out of archetypal resonance with the whole of

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things, that transcendent place where past, present, and future converge. Dreams emerge in the space between conscious and unconscious aspects, reflecting not just who we are in a present moment, but also what we might become in the future, in places where being shades into becoming, the individual into the collectivity. Successful dream work carries the fractal feeling of detecting the whole of ourselves in the world in the pieces of nightly reverie. Thorough dream work can demonstrate how creativepattern formation during successive iterations of consciousness constitutes the essence of effective introspection.

References Abbott, A. (2017). Fractal secrets of Rorschach’s famed inkblots revealed. Nature: International Weekly Journal of Science. Retrieved from: https:flwwnature.com/news/fractal-secrets-of-rorschach—s-famed— ink—blots—revealed—l .2 1473. Aftanas, L.I., Lotova, N.V., Koshkarov, V.I., Makhnev, V.P., Mordvintsev, Y.N., & Popov, SA. (1998). Non-linear dynamic complexity of the

human EEG during evoked emotions. International Journal of Psychophysiology, 28(1), 63-76. Anders, S., Heinzle, J., Weiskopf, N., Ethofer, T., & Haynes, JD. (2011). Flow of affective information between communicating brains. Neurolmage, 54, 439-446. Anderson, C., & Mandell, A. (1996). Fractal time and the foundations of consciousness: Vertical convergence of 1/f phenomena from ion channels to behavioral states. In E. Mac Cormac & M. Stamenov (Eds), Fractals of brain, fractals of mind (pp. 75-126). Amsterdam, Netherlands: John Benjamin. Anderson, C., Mandell, A., Selz, K., Terry, L., Robinson, 8., Wong, C., Robertson, 8., & Smothennan, W. (1998). The development of nuchal atonia associated with active (REM) sleep in fetal sheep: Presence of recurrent fractal organization. Brain Research, 787(2), 351-357.

Bak, P. (2013). H ow nature works: The science of self-organized criticality. Berlin, Germany: Springer Science & Business Media. Babiloni, F ., & Astolfi, L. (2014). Social neuroscience and hyperscanning

techniques: Past, present and future. Neuroscience & Biobehavioral Reviews, 44, 76-93.

Barnsley, M. (1993). Fractals everywhere. Burlington, MA: Morgan Kaufmann. & Beebe, B., Jaffe, 1., Markese, S., Buck, K., Chen, H., Cohen, R , Feldstein, S. (2010). The origins of 12-month attachment: A

Chapter Nine

300

microanalysis of 4-month mother—infant interaction. Attachment & Human Development, 12(1-2), 3-141. Bouten, S., Pantecouteau, H., & Debruille, J .B. (2015). Finding indexes of spontaneous brain-to-brain communications when looking for a cause of the similarity of qualia assumed across individuals. F1000Research, 3:316. Retrieved from: https:l/wwwncbi.nlm.njh.gov/pmc/articles/TMCS992535.2/

Buzséki, G. (2006). Rhythms of the brain. Oxford, England: Oxford University Press. Carnbray, J. (2009). Synchronicity: Nature and psyche in an interconnected universe. College Station, T X : Texas A&M University Press.

Chopra, D. (1995). The way of the wizard New York, NY: Harmony. Cohn, J.F., & Tronick, E.Z. (1988). Mother-infant face-to-face interaction: Influence i s bidirectional and unrelated to periodic cycles in either

partner's behavior. Developmental Psychology, 24(3), 386-392. Colman, W. (2011).

Synchronicity

and the meaning—making psyche.

Journal ofAnalytical Psychology, 56(4), 471—491. Combs, A., & Holland, M. (1990). Synchronicity: Science, myth, and the trickster. St. Paul, M N : Paragon House. Cozolino, L. (2017). The neuroscience of psychotherapy: Healing the social brain. New York, NY: W.W. Norton. Crick, F., & Mitchison, G. (1995). REM sleep and neural nets. Behavioural Brain Research, 69(1-2), 147-155. Dumas, G., Lachat, F., Martinerie, J., Nadel, J., & George, N . (2011). From social behavior to brain synchronization: Review and perspectives in hyperscanning. IRBM 32, 48-53. Dumas, G., Nadel, J., Soussignan, R., Martinerie, J ., Gainero, L. (2010). Interbrain synchronization during social interaction. PLoS ONE, 5(8): e12166. Eenwyk, JR. (1991). Archetypes: The strange attractors of the psyche.

Journal ofAnalytical Psychology, 36(1), 1-25. Exner, J.E. (1969). The Rorschach systems. New York, NY: Grune & Stratton. Feldman, R., Magori—Cohen, R., Galili, G., Singer, M . , & Louzoun, Y . (2011). Mother and infant coordinate heart rhythms through episodes of

interaction synchrony. Infant Behavior and Development, 34(4), 569577. Gerson, S. (2004). The relational unconscious: A core element of intersubjectivity, thirdness, and clinical process. Psychoanalytic

Quarterly, 73(1), 63-98.

Dreams, synchrony, and Synchronicity

301

Harle, R. (2010). Chaos, archetypes, and the all-integrating field. Nonlinear Dynamics, Psychology, and Life Sciences, I 4(1), 101-115. Hasson, U., Ghazanfar, A.A., Galantucci, B., Garrod, S., & Keysers, C . (2012). Brain-to-brain coupling: a mechanism for creating and sharing a social world. Trends in cognitive sciences, I 6(2), 114-121. Hogenson, G B . (2005). The self, the symbolic and synchronicity: Virtual

realities and the emergence of the psyche. Journal of Analytical Psychology, 50(3), 271-284. —. (2009). Synchronicity and moments of meeting. Joumal ofAnalytical

Psychology, 54(2), 183—197. Jaffe, F., Beebe, B., Feldstein, 8., Crown, C.L., & Jasnow, M D . (2001).

Rhythms of dialogue in infancy. Monographs of the Society for Research in Child Development; 66 (2, Serial No. 265), i-149. Jung, CG. (1989). Memories, dreams, reflections. New York, NY: Vintage. (Original work published 1961)

Jung, C.G., & Jacobi, J. (1973). CG Jung: Psychological reflections. A new anthology ofhis writings, 1905—1961 (V 01. 31). Princeton, NJ: Princeton University Press. Kahn, D., Combs, A., & Krippner, S. (2002). Dreaming as a function of chaos-like stochastic processes in the self-organizing brain. Nonlinear Dynamics, Psychology, and Life Sciences, 6(4), 311-322. Kaye, K., & Fogel, A. (1980). The temporal structure of face-to—face communication between mothers and infants. Developmental

Psychology, 16(5), 454-464. Kesterman, J. (1985). The flow of empathy and trust between mother and

child. In B. Anthony & G. Pollack (Eds), Parental influences in health and disease (pp. 137-163). Boston, MA: Little Brown. Lane, R.D., Ryan, L., Nadel, L., & Greenberg, L. (2015). Memory reconsolidation, emotional arousal, and the process of change in

psychotherapy: New insights from brain science. Behavioral and Brain Sciences, 38, 1-64. Liu, T. & Pelowski, M . , (2014). Clarifying the interaction types in twoperson neuroscience research. Frontiers in Human Neuroscience, 8(276). Retrieved from: https://www.fi'ontiersin.org/articles/ 10.3389/fi3hum.2014.00276/full. Main, R. (2007). Synchronicity and analysis: Jung and after. European Journal o f Psychotherapy and Counselling, 9(4), 359-371. Mitchell, S A . (1998). From ghosts to ancestors the psychoanalytic Vision

of Hans Loewald. Psychoanalytic Dialogues, 8(6), 825-855. Mandelbrot, B. (1977). The fractal geometry of nature. San Francisco, CA: W.H. Freeman.

Chapter Nine

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Marks-Tarlow, T . (2008) Psyche ’s veil: Psychotherapy, fiactals and complexity. New York, N Y : Routledge. —. (2012). Clinical intuition in psychotherapy: The neurobiology of embodied response. New York, NY: W.W. Norton. —. (2014). Awakening clinical intuition: An experiential workbook for clinicians. New York, N Y : W.W. Norton.

Peat, D. (1997). Infinite potential: The life and times of David Bohm. Reading, MA: Helix.

Peitgen, H 0 . (1986). The beauty of fractals: Images of complex afvnamical systems. New York, NY: Springer. Riley, M.A., Richardson, M.J. Shockley, K . & Ramenzoni, V.C. (2011). Interpersonal synergies. Frontiers in Psychology, 2(38), 1—7. Sands, S H . (2010). On the royal road together: The analytic function of dreams in activating dissociative unconscious communication.

Psychoanalytic Dialogues, 20(4), 357—373. Schilback, L., Timmerrnans, B., Reddy, V . , Costall, A., Bente, G., Schlicht, T . , Vogeley, K . (2013). Toward a second-person neuroscience. Behavioral and Brain Sciences, 36, 393-462. Schroeder, M . (1991). Fractals, chaos, power laws: Minutes from an

infinite paradise. San Francisco: W.H. Freeman. Schwartz, S A . (2018). Nonlocal consciousness and the anthropology of

dreams. EXPLORE, 14(21), 107-110. Stevens, G., Silbert, L., & Hasson, U . (2010). Speaker-listener neural

coupling underlies successful communication. Proceedings of the National Academy ofScience USA, 107, 14425—14430. Taylor, R R , Martin, T.P., Montgomery, R.D., Smith, J.H., Micolich, A.P., Spehar, B . (2017). Seeing shapes in seemingly random Boydston, spatial patterns: Fractal analysis of Rorschach mkblots. PLoS One. Retrieved from: https:// j oumals.plos.org/ plosone/ article?id=1 0.1371/joumal.pone.0 171

289. Trevarthen, C. (1989). Development of early social interactions and the affective regulation of brain growth. In C. Trevarthen, C. von Euler, H.

Forssberg, & H. Lagercrantz (Eds), Neurobiology of early infant behavior: Proceedings o f an international Wallenberg symposium at the Wenner—Gren Center, Stockhohn, August 28 — September 1, 1988 (pp. 191-216). London, England: Macmillan. Uhlhaas, P., Roux, F., Rodriguez, E., & Rotarska—Jagiela, A. (2009). Neural synchrony and the development of cortical networks. Trends in Cognitive Sciences, 14(2), 72 — 80.

Van de Castle, R. (1994). Our dreaming mind. New York, NY: Ballentine.

CHAPTER TEN A FRACTAL TOPOLOGY OF TRANSCENDENT EXPERIENCES] SALLY WILCOX AND ALLAN COMBS2

Most problems of real interest combine fractal and topological features in increasingly subtle fashion. Benoit Mandelbrot (1977, p. 17)

In this collection Terry Marks-Tarlow gives us the first epistemology of transpersonal psychology aimed at healing the longstanding rift between what William James termed tender-minded approaches that are rationalistic, idealistic, optimistic, and often spiritual and humanistic, and tough-minded approaches that are often empiricist, reductionistic, and skeptical. The present essay is a contribution to this purpose. To be specific, this paper explores boundary conditions of individual consciousness during non-druginduced transcendent experiences. Unique characteristics of transcendent experiences are compared to those found in other natural boundary conditions. Analogous patterns are identified that suggest far-fromequilibrium conditions may be present during such experiences that spontaneously reorder the boundaries of individual consciousness. Boundaries in nature are so common they are often unnoticed and imperceptible. In the physical realm we need only turn our attention to our own bodies to appreciate that each organ is bounded by form and function despite interacting with other corporeal systems. Each cell and organ is differentiated by its structure and specialization, yet needs to function as an integral part of the whole. For example, each neuron in the brain coordinates 1 A version of this chapter was published in the International Journal of T rampersonal

Studies, 38(2). 2 California Institute of Integral Studies Email: Sally Wilcox, [email protected] Allan Combs, [email protected]

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and communicates a vast amount of information through interactions with other neurons. Biological central-nervous systems, our vast vascular system, as well as our respiratory system show fi'actal characteristics in structural and functional parameters (Babloyantz & Louren, 1994). Likewise, every individual’s body is separate fi'om other bodies, and yet research in the realm of psychicphenomena discloses many apparently valid cases of cognitive and emotional experiences shared at a distance (e.g., Baruss & Mossbridge, 2017; Radin, 1997). Such shared experiences suggest the possibility that individual consciousness is less constrained in space and time than conventionally thought. These shared experiences point to the possibility that under certain special circumstances consciousness unfurls into an unbounded, or nearly unbounded, nonlocal presence. Indeed, there is much evidence that nonlocality is a fundamental feature of consciousness and the natural world itself (Baruss & Mossbridge, 2017; Schwartz, 2007). In the pages that follow, we explore how the apparent boundaries of individual experience may be fractal-like, exhibiting self-sinnlarity, temporal iteration, and scale independence (Bieberich, 2012; Crick & Koch, 2003). Our attempt in this chapter is to overlay scientific findings onto the rather diaphanous topic of consciousness. Nobel biologist Gerald Edelman and neurobiologist Tononi (2000) comment, “[w]e suggest that consciousness can be considered a scientific subject and that it is not the sole province of philosophers” (p. 3). As science advances in understanding the laws of the universe, there is a tremendous opportunity for cross-pollination of laws and theories that may advance our understanding of consciousness. It is our intention to consciously and carefully appropriate some of these scientific findings and apply them to our hypothesis of the transcendent experience. Many questions and challenges will surface until more is known or proven about consciousness. We posit that consciousness seems to behave in a nonlinear fashion, as characterized by the fact that small changes in inputs often create disproportionately large and unpredictable changes in outputs. Considering the common feature among physical dissipative structures, namely that they spontaneously reorder themselves at far—from—equilibrium conditions, it is reasonable to speculate that the boundaries of individual consciousness can also reorder under the seemingly far-fi‘om-equilibrium conditions of transcendent experiences. Thus, the threshold between ordinary individual experience on the one hand, and nondual consciousness on the other, can be understood as dynamic and responsive, potentially allowing for, and participating in the sense of relaxation and unfurling that characteristically merges the individual with the transpersonal.

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Clarifying what is meant by input and output as it relates to our premise of the nonlinearity of transcendent experiences is necessary at this point. It is generally assumed that an individual’s normal waking consciousness is in a state of equilibrium, with minor fluctuations and is the ground upon which an individual’s experiences are recognized (Gupta, 2003). Normal waking consciousness is generally assumed to be the awareness we have of ourselves and the world around us. Input would be those sensory and thought perceptions that arise in our awareness, including our awareness of ourselves as separate from the world around us. According to the fundamentals of nonlinearity, the behavior of a system is highly sensitive to initial conditions where small perturbations in input can produce disproportionately large and unpredictable changes in output (Lorenz, 1963). If we apply this nonlinearity principle to the transcendent experience, we understand through a significant number of reported subjective accounts, that an expansion, change, or perturbation in the sense of self as separate and agentic (input) can in the transcendent experience produce a profound sense of interconnectivity and nondual consciousness. A multitude of subjective accounts describe a loss of sense of self, the dissolution of boundaries and sense of time and space in the transcendent experience (d’Aquili & Newberg, 2000; Underhill, 1911). Boundaries, in general are multifunctional. They can be gatekeepers when differentiation and specialization is required, and they can act as porous fractal-like tendrils when driven far-from—equilibrium, as we propose happens in transcendent experiences. The present exploration into experiential-boundary conditions is undertaken first in terms of Carl Jung’s transcendent function, positioned at the boundary between the conscious and unconscious aspects of the psyche (e.g., Miller, 2004), then more broadly in terms of boundary conditions widely found in the physical world, which can be pushed into far-from-equilibriurn states that exhibit fractal characteristics at gradient interfaces. Boundaries as a gradient interface offer rich surfaces for information exchange or processing. An example close to home is our skin which absorbs and releases moisture as information about the surrounding environment is processed. Conclusions suggest that such fractal considerations may apply to transcendent experiences as well.

Topology In the purest mathematical sense, topology is the study of geometric properties that are preserved under transformation, like stretching but not

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tearing as exemplified by the Mobius strip and the Klein bottle. In this paper, topology is used to describe the texture of the boundaries of individual consciousness in the transcendent experience. This approach facilitates the visualization of ideas. As Gleick (1987) writes: Linking to help system, system,

topology and dynamical systems is the possibility of using a shape visualize the whole range of behaviors of a system. For a simple the shape might be some kind of curved surface; for a complicated a manifold of many dimensions. (p. 47)

Important to the study of transcendent experience is the topological notion of isomorphism. It suggests that a form can be stretched or transformed radically without losing its identity. This is important for the individual who has undergone a transcendent experience in that they remain that same individual after the transcendent experience but with an expanded sense unfurling into an unbounded state of at-one—with the universal (d’Aquili & Newberg, 2000; Underhill, 1911). Visual similes include the Mobius strip and the Klein bottle, where the insides and outsides of the objects themselves are continuous with each other and therefore reconciled. It is this topological feature that is helpful in describing the involution and evolution of the self and the universal, individual and collective, in transcendent experience. As with the Klein bottle visually, we sense that there is an inside and an outside to our being as it relates to our environment and experiences, but topologically there is not; they are one and the same. As a model, the fractal topology of the transcendent experience approximates the dynamic boundaries that an individual may sense as a reconciliation of inside and outside, across the boundary that previously served to delineate experience as I versus not I, and in the extreme reaching infmite dimensions to become a vanishing interface with an unbounded universe, known by a thousand names yet ineffable.

Transcendent function Carl Jung’s transcendent function (Miller, 2004) emerges at the boundary between conscious and unconscious aspects of the psyche. If the role of the psyche is to encounter, synthesize, and integrate aspects of the conscious and the unconscious, then the role of the transcendent function is to reconcile this duality, and is the process by which the psyche draws a person forward toward wholeness. The transcendent function is the key psychic mechanism through which purposeful guidance takes place (Miller, 2004). The discovery of the true-whole self is impossible without discovery of the

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unconscious, and Jung suggests the transcendent function is the vehicle of reconciliation between the conscious ego and the mysterious unconscious (Jung, 1961/1989). Consciousness remains mysterious and its relationship to neurocorrelates in the brain has yet to be determined; however in this examination, we are exploring what we see as characteristics of consciousness in the transcendent experience. Consciousness does not appear to be static and seems to have many states such as normal waking consciousness, dream state, and an expanded or transcendent state. Peak, mystical, or transcendent experiences incorporate significant neurobiological dimensions according to d’Aquili and Newberg (2000), who probed the epistemology of consciousness along a continuum that at one end is grounded in everyday perceptions and experiences, while at the other end represents a state of what is sometimes referred to as cosmic consciousness (Bucke, 1905) or mystical union (Underhill, 1911). These states exhibit common characteristics documented throughout history and are referred to with respect to Jung’s transcendent function as individual consciousness interfacing and unfurling into collective unconscious (Jung, 1961/1989; Franz, 1995). The transformative potential of the transcendent function suggests a dynamic process with porous tensegrity, or dynamic tensions that support compressed and hidden psychic elements in fluid suspension (e.g., see Fuller, 1961). Pushing this idea further, we suggest that during a transcendent experience, individual consciousness is driven to a far-fromequilibrium state where the boundary does not actually rupture, as formally presumed, but unfurls and spontaneously reorders to increase its fractal dimensionality. For the sake of visualizing this boundary as it relates to consciousness and its chenille-like quality in the transcendent experience, we need only to examine the many similar features in the body. The tongue and the small intestine are covered in chenille-like villi that absorb nutrients and interact and exchange information with the surrounding environment. Their stretchy surfaces are crumpled to increase surface area while conserving space, exhibiting an economy of scale. This is not to say that consciousness necessarily has physical qualities but that it may behave in such a stretchy unfurling in the transcendent experience. Boundaries, as stated earlier, have many functions. They separate for specialization and they unite for the purpose of connecting or exchanging information. We will expand on this concept of boundary as information in a later section under the heading of boundary physics. For now, let us explore boundary as gate—keeper that can

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transform under certain conditions.

A rupture assumes separateness, whereas infinite-fractal unfurling with a porous- gradient interface suggests expanded or non-dual characteristics. This change fiom a bounded consciousness that is rigid, dualistic, and excluding to one that is richly textured, porous, having a nearly infinite surface area alters the fundamental role of the boundary from that of dissociating to that of unifying and absorbing the individual self within the universal. The ubiquitous, non-linear systems of the physical world (Mandelbrot, 1977) suggest a model to understand the topology of the boundaries of consciousness itself. In this way, the fractal topology of the transcendent experience provides aframe in which individual consciousness transcends itself in a state of non-duality, opening to a broader or even dimensionless experience.

Kant and Hegel Immanuel Kant wrote, “All our knowledge begins with experience . . . but . . . it does not follow that it arises from experience” (Wood, 2001, p. 24). That is, Kant argued we must distinguish between objects as perceived and the experience that perceives them. This distinction is between knowing

about something as cognition, contrasted to knowing it directly, which for Kant was intuitive or transcendent knowing, beyond our capacity to know directly (Wood, 2001). Hegel (1807 /1997), on the other hand, argued that to know a boundary is to know that which it bounds, and knowing that which it bounds is also to know that which lies beyond. This, he wrote, is an inherent state of transcendence. In phenomenology, the transcendent is that which lies beyond our own consciousness, thus creating a dualistic boundary. Alternatively, to have a direct experience is to sense becoming one with the object. Underhill (1911) frequently refers to a felt sense of being “at-one-with ” in the context of religious “conversion” and transcendent experiences. “This awakening, from the psychological point of view [and limited sense of reality], appears to be an intense form of phenomenon of ‘conversion’” (Underhill, 1911, p. 163).

Transcendent experience Normal waking consciousness is the standard by which altered states, including the transcendent experiences, are most frequently compared (Baruss & Mossbridge, 2017). Transcendent experiences represent an alteration of consciousness in that they appear to interact supervem'ently;

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that is to say, transcendent experiences could not be experienced or known without a kind of elemental or ground level consciousness (Gupta, 2003). Transcendent experiences manifest in many variations, known by many names and meanings. Maslow’s (1964) psychological approach, for instance, refers to peak experiences. Whitehead (1969) refers “the universal throughout actuality” (p. 190) and the “inevitable continuity” (p. 191). Hollick (2006) and many others describe transcendent experiences as uniiy consciousness, and C. Grof and S. Grof (1992) refer to cosmic consciousness (also see Bucke, 1905) as the unity or transcendent experience. They are known as mystical experiences by Underhill (191 1), who famously explored them as unfolding spiritual consciousness. Briefly and as mentioned earlier, the commonly described characteristics of the transcendent experience

significant to this paper include the following (James, 1911; Underhill, 191 l): Sudden spontaneous onset; An expanded sense of self or loss of sense of self and agency; Space—time distortion; A sense of at one with the Divine or the Void; Certitude of the experience itself; Profound effect on the individual after the experience; and Ineffability in describing the experience. Transcendent experiences are not a geographically localized, novel, or recent phenomenon; they are historically familiar to Eastern and indigenous cultures around the world (Underhill, 1911). What is new and interesting is a consilience across formally disparate disciplines such as the “hard” sciences and psychology, allowing for fresh and rich dialogue. Examining new sciences of far-from-equilibrium conditions and boundary characteristics in fractal dynamics can render fresh insights into consciousness in the transcendent experience. Both fractal dynamics and far-from-equilibrium conditions address dynamic distortion of system tensegrity and the potential richness of the boundary interface.

Far from equilibrium The branch of physics concerned with condensed matters and material physics (Committee on CMMP, 2010, 2007)) has recently begun to recognize the language of far-from-equilibrium (FFE) systems is applicable to numerous problems in other fields. Consistent with our thinking, patterns

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of behavior in far-fi'om-equilibrium systems echo the characteristics experienced in transcendent experiences. Significant characteristics of F FE systems include the following (Jaeger & Lui, 2010): c

a I

There is spontaneous reordering at the boundary of an FFE system; Rich structures and complex behaviors arise from simple parts and few variables; They are typically non—linear; Systems driven far-from—equilibrium produce unanticipated phenomena; and FFE systems can be driven to find unique solutions in very rapid fashion.

Equilibrium is a static or unchanged state over time where all competing forces are balanced. Far-from—equilibrium systems, on the other hand, are in fact ubiquitous, and are discovered not only in materials, but also in a

wide variety of systems and processes. For example, the Committee on CMMP (2010, 2007) report states, Far-from—equilibrium behavior is not confined to special conditions or certain types of materials. Instead, it arises across the entire spectrum of

condensed-matter and materials physics in a host of problems of fundamental interest. Far-from—equilibrium phenomena also benefit and plague us in technology and in everyday life. Indeed, some of the most

complex outcomes of behavior far—from-equilibrium emerge in situations familiar in everyday experience. (p. 91) Examples of far-from—equilibrium phenomena in nature include, turbulence, hurricanes, earthquakes, avalanches, galaxy formation, swarming fish, the behavior of birds flocking, and consciousness (Committee on CCMP, 2007; Jaeger & Lui, 2010). Systems and structures are driven to FFE behavior when their energy levels increase. Subjecting physical materials to FFE conditions results in properties that are otherwise unattainable; and subjecting systems to FF E conditions results in emergent properties that do not reside in the constituent components themselves (Committee on CMlVIP, 2010). In some cases when an energy source is removed, a transformed system will relax, returning to its previous state, while in other cases it undergoes an irreversible phase transition. For example, heating a pot of water drives it through a series of non-equilibrium transitions involving an increasingly complex network of convection currents before it comes to a boil. These

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dissipate when the water is allowed to cool again. But if, for instance, the pot also contains ingredients for making candy, the cooling process does not return the system to its original composition but moves it to a new (and better tasting) composition. In the latter instance, the changes caused by driving the system FFE are permanent. Such is the case with rapid heating and cooling processes that are instrumental in the creation of many fabricated materials, including plastics, alloys, polymers, the crystalline structure of silicone chips, and the annealed edges of blades (Committee on CMMP, 2007, 2010). In the above examples, we see that in FFE states, nonlinear systems can be driven to spontaneously reconfigure themselves into unique and rich patterns of activity. The latter can also emulate edge of chaos processes associated with dramatically increased expressions of creativity and even intelligence (e. g., Lewin, 2001; Peak & Frame, 1994). Such phenomena are conceptually analogous to events at the edge of individual consciousness in the FFE condition of a transcendent experience. All of this suggests that the boundary of individual consciousness does not so much dissolve into nothingness, but rather its fractal dimensionality relaxes and increases towards infinite expansion. Fractals and consciousness Fractal is a term coined by French mathematician Benoit Mandelbrot (1989) to depict the geometry arising from nonlinear dynamics. In nature, fractals are ubiquitous and can be seen in structures such as mountains, clouds, and coastlines, as well in the dynamics of flowing rivers, avalanches, and weather systems, as Figure 10-1 illustrates. From a sociological point of view, fractal dynamics are useful for understanding patterns of human interactions, through an economic lens they can describe market behavior, and can even be used to model other phenomena, such as traffic jams. Anticipating fractal self-similarity dynamics, Kant himself wrote: We see the first members of a progressive relationship of worlds and systems; and the first part of this infinite progression enables us already to recognize what must be conjectured of the whole. There is no end but an

abyss... without bound. (Kant, 1910, p. 65)

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Figure 10—1. Naturally Occurring Fractals. These naturally occurring fractals demonstrate fractal development in ice crystals on the top (Helen Filatova, public domain) and Romanesco broccoli on the bottom (Jon Sullivan, public domain)

In the last 40 years, the discovery of fractals has facilitated accurate descriptions of irregular shapes and processes that had previously only been depicted through linear approximations. Fractal geometry has replaced the

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former linear Euclidian standard as the mathematical tool used to describe nonlinear systems. “It is based on a form of symmetry that had previously been underutilized, namely self-similarity, or some more general form of invariance under contraction or dilation” (Mandelbrot, 1989, p. 3). The possibilities of fractal computation emerged with the dawn of the digital age. With the help of computers, Mandelbrot and others were able to find solutions to nonlinear equations, a laborious process that was beyond manual computation, given the sheer number and the complexity of calculations required. In the early days of fractal geometry Mandelbrot programmed the Julia set equation into his computer, left it to work for the weekend, and was astonished at the graphics that were produced (Mandelbrot, 1989). The shapes that emerged continuously depicted selfsimilarity at different orders of magnitude. It was the scale of the observation—the number of iterative calculations where output was fed back as input—that provided a large enough perspective for the emergence of the fractal. Laszlo (1972) later wrote of the iterative nature of nonlinear systems that give rise to fiactals that “From within and without [they] call forth innovations, and the innovative system produces new kinds of inputs on all systems with which it communicates... thus a change in one triggers changes in others” (p. 65). Fractals are important examples of scaling, or tiling, that mathematically demonstrate invariance and self-similarity under displacement. They describe the border between order and disorder (Mandelbrot, 1977). This border between order and disorder bears conceptual self-similarity to Jung ’3 transcendent function that seeks to unite the conscious with the unconscious

(Jung, 1961/ 1989). The fractal border both differentiates and unites the two. The most common feature of transcendent experiences is the loss of the sense of self when the individual experiences a dissolution of the boundary that separates his or her sense of self fi'om a more cosmic or unbounded sense of reality. The boundary at the edge of individual consciousness is

that which is transcended in the transcendent experience. As such the nature of that boundary is central to this inquiry. It is in boundary conditions—between states of order and disorder, at the edges of phase transitions, and at the threshold of physical transformations, that fractal dynamics are indispensable as a tool for understanding. As Scaruffl (2013) comments, referencing the pioneering biomathematician, Stewart Kauffman:

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Kauffman showed that the best strategy for reaching the peaks occurs at the phase transition between order and disorder, or, again, at the edge of chaos. The same model applies to other biological phenomena and even nonbiological phenomena, and may therefore represent a universal law of

nature. (para 3) Key to the present inquiry is the question of whether the boundary of individual consciousness, like many physical boundaries and boundary conditions, is indeed fractal in nature. As we have shown, consciousness is self-similar, iterative and exhibits self-same characteristics regardless of scale. It is also nonlinear in that small changes in input can create large and unpredictable changes in outputs. The disproportionately evolution of experience from moment to moment, as well as over large spans of time, is iterative, in which each new experience feeds back into the hopper of the ongoing flow, generating new iterations, that are the very kinds of process that produce fractal dynamics mathematically, in physical systems, and apparently in experiential reality as well (e.g., Combs, 1996; James, 1890).

Fractal characteristics of gradient interfaces in nature There are numerous examples of fractal-gradient boarders in nature. One only needs to sit at the ocean’s edge to appreciate the dynamic interplay of the waves at the shoreline. If the shore is rocky, the nutrient-rich intertidal zone is a fractal at many scales. From an airplane, meandering rivers and streams dot the landscape while fractal mountain peaks seem to touch the floating fractal-like clouds. Fractals are as abundant in our natural surroundings as they are in the vascular and neurological networks of our own bodies. Gell—Mann (1996/2010) posited that nature is conformable to itself and is self-similar. Following this, it is no stretch of the imagination to think that, if individual consciousness itself is fractal in nature, then boundaries at the edge of individual consciousness exhibit fractal characteristics as well. This might seem a dramatic shift from physical to cognitive or informational structures or processes, but iterative cognitive and informational processes can be seen in many ways to be parallel. It is beyond the scope of this essay to expand this point in detail, but this is examined elsewhere both in terms of information flow and dynamical

feedback systems (Combs, 1996; Kampis, 1991; Kauffman, 1993).

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One specific fiactal, the Koch snowflake, is brieflyr examined here to

illustrate how their characteristics are analogous to the boundary characteristics of individual consciousness in a transcendent experience.

Koch Snowflake As illustrated in figure 10-2, the construction o f a Koch snowflake is an accessible example o f fractals as a mathematical process. Constructing a Koch snowflake begim with a triangle with sides of length 1. At the middle of each side, add a new triangle one-third the size, and continue doing this. As this process continues the actual enclosed area remains less than the area of a circle drawn around the original triangle. Thus, an infinitely long line

surrounds a finite area. In other words, the border can be considered infinite depending on the scale upon which it is examined.

figure .1 lit-2. The Koch mow‘flake. (Public domain}

'The unique characteristic o f the snowflake is that, asthe boundary gets more and more detailed, increasing in its fractal “dimensionality,” a

paradoxical situation is generated in which the finite area inside the snowflake is enclosed by an infinite boundary. As the boundary is increasingly refined from a few simple triangles to aninfinitely crystalline surface, the ratios between, and therefore the rehtionship between the container, the contained, and the excluded space, changes. Much the same can be said o f the boundaries o f individual consciousness during a

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Where a simple boundary once sufficed merely to identify the Koch snowflake and conserve its identity, differentiating the inside from the outside of the shape, an infinite boundary provides infinite potential for interaction between the contents of the snowflake and its boundary. The fractal nature of the boundary can embody a shift from merely conserving information, to providing potentially infinite opportunities for interaction with both the contained and that which is considered excluded. In this way, the characteristics of the Koch snowflake can be said to be analogous to the boundaries of individual consciousness during a transcendent experience. As individuals enter into a transcendent experience, they sense the boundaries of self and reality dissolving, and they experience an immersion in an expanded nonlocal experience (Loy, 1997). While empirically we know that individuals do not dissolve during transcendent experiences, even as their sense of self does, the paradox can be addressed by suggesting a Koch-like fractal boundary to individual consciousness. Here it might be worth noting that, though technically speaking the fractal snowflake does not actually change form, it expands indefinitely. Perhaps more importantly, however, even in the expanded geometric form, like the expanded state of consciousness, the elemental phenomenological ground of experience does not actually change (Gupta, 2003).

Fractal dimensionality and enfoldment If, as suggested in this paper, the boundaries at the edge of individual consciousness are fi'actal, it follows that reports of transcendent experiences would share descriptions of consciousness that resemble fractal dynamics. Fractal dimensionality appears to be just such a corollary dynamic. All fractals have a fractal dimensionality, that is, their degree of complex enfoldedness. Crumpled paper, folding of proteins, and vortices that arise as back eddies in laminar flow, are visually accessible examples of flactal dimensionality. As fractal dimensions increase, so too does complexity. The Mandelbrot set, with its high fractal dimensionality, is one of the most complex mathematical objects known. “An eternity would not be enough time to see it all, its disks studded with prickly thorns, its spirals and filaments curling outward and around, bearing bulbous molecules that hang, infinitely variegated, like grapes on God’s personal vine” (Gleick, 1987, p.

221).

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Boundaries of natural systems are universally nonlinear, are often in flux, and they often exhibit fi'actal characteristics. This also seems true of the boundaries of consciousness, we posit. What fractal dynamic accounts for the sense of dissolution of self and immersion in nondual experience prevalent in many transcendent experiences? Descriptive accounts over a wide range of such experiences suggests that consciousness changes during transcendent experiences as an individual’s consciousness unfurls or increases its fractal dimensionality at the boundary separating individual consciousness from nondual consciousness. This unfurling is experienced as spontaneous, due perhaps to the far-from—equilibrium state of transcendent experience. Consistent with far-from—equilibrium phenomena in general, we suggest the boundary of individual consciousness can undergo rapid reordering, where the border between the individual and the universal increases with

fractal dimensionality. This increase in the boundary’s fractal dimensionality increases the richness of its chenille-like texture. The experience of encountering the Absolute, the mystical, nondual, or Samadhi reported in transcendent experiences could be understood through the almost infinite uptake of information made possible via the higher fractal- dimensional ratio at the interface of individual to universal rendering a holographic perspective. This uptake of information is addressed in the next section. As the boundary of one’s consciousness increases, its fractal dimensionality and its porous-fractal tendrils of perception reach towards infinity, and the boundary undergoes a phase transition from a demarcation of inner and outer experience (Sheets-Johnstone, 2016), to a zone of rich interaction between self and not self. The resultant exchange of information described in transcendent experiences is not experienced in normal waking consciousness.

Boundary physics Physicist Maldacena has convincingly presented an argument “that everything taking place within the specified universe is a reflection of laws and processes acting themselves out on the boundary” (Greene, 2011, p. 263). In a conversation between two physicists, Brian Greene asked John Wheeler what the dominant theme in physics would be in the next while and Wheeler’s eventual answer was “Information” (Greene, 2011, p. 239). This theory of information uptake at a boundary surface is based on the intense work studying black holes. “Black holes inform us about information storage in any contex ” (Greene, 2011, p. 263).

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If, as we are positing, consciousness exhibits FFE characteristics as it unfurls in the transcendent experience revealing an infinite surface area, its boundary surface may simply embody information about a holographic universe. This would explain the subjective interpretation of experiencing the Absolute, the mystical, nondual or Samadhi. As Greene (2011) comments, “ .. information storage capacity [is] determined by the area of a bounding surface .. [where] there is good reason to believe that information is indeed stored at the horizon” (pp. 255-256).

Conclusion In this chapter we have brought forth a hypothesis that in the transcendent experience, individual consciousness displays fractal characteristics and, when driven far-from— equilibrium, it unfurls to reveal an infinite and porous boundary. We have also suggested that this boundary may be an information gradient interface between individual consciousness and a holographic universe. These are no doubt bold suggestions, however, as research continues to reveal laws of the universe, our understanding of consciousness will expand. With few exceptions, boundaries tend to be thought of as passive, rupturing in individual consciousness when the person is flooded with mystical or transcendent experiences (McDermott, 2001). On closer examination it appears that the function of the boundary in transcendent experiences may not be limited to that of a gatekeeper. Given the nearly universal fact that the boundaries of dissipative structures can spontaneously reorder at far-from-equilibrium conditions, it is reasonable to conclude that the boundaries of discrete consciousness can also reorder under the far-from—equilibriurn condition of transcendent experience. It is worth noting: Far-from—equilibrium behavior is ubiquitous... [it] underlies a wide range of phenomena outside the traditional boundaries of condensed matter physics, including earthquakes, hurricanes, galaxy formation, and consciousness. As a result, breakthroughs in the area have the potential for far—reaching impact across many scientific disciplines . . .. Far-fiom-equilibrium behavior is not a simple extension of equilibrium or near-equilibrium physics. Instead, it corresponds to qualitatively different types of behavior and response, typically associated with crossing some threshold into a new regime. (Jaeger & Lui, 2010, p. 2)

In this way, thresholds between ordinary and transcendent experiences can be considered dynamic and responsive, potentially allowing for, or

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participating in, the sense of rupture that merges the individual and the cosmic. As condensed matter and materials physics research suggests, “farfrom-equilibrium processes can achieve structural and dynamical richness even with the simplest of ingredients, such as the intricate dendritic growth realized in snowflakes” (Committee on CMMP, 2007, p. 95). This is not to ascribe agency to the boundary, but rather to suggest that, as with many nonconscious systems, the boundary at the edge of individual consciousness appears to exhibit nonlinear dynamics. One way to move a system from a state of rest and to push it into a nonlinear FFE regime is by the introduction of disruptive or patterning forces. This disrupted pattern and change of state, as in water crystallizing into ice or ice melting, demonstrates a phase transition (Laughlin, 2005). With respect to states of consciousness, such driving influences are reviewed in detail, for example, by Tart (1975) and Combs (1996). Such

driven systems give rise to rich unanticipated phenomena (Committee on CMMP, 2007). This is the case as well for conscious experience itself (Baruss & Mossbridge, 2017; Combs, 2015; & Metzner, 2017). Anecdotally, subjects who describe their own returns from transcendent experiences report that their consciousness has been forever expanded, and their previous sense of discreteness has changed to an irreversible openness and sense of interconnectivity. This would suggest that in addition to the learning brought about by any experience, the transcendent experience may also result in a structural change in the boundary of the self. The authors believe that the fractal dimensionality of the boundary of individual consciousness may maintain a residual degree of openness after many transcendent experiences. This is a common occurrence found in other FFE conditions that do not relax back into equilibrium, even after all driving forces have been removed. (Committee on CMMP, 2007).

As the previously enfolded fractal tendrils of consciousness unfurl in the transcendent experience, the rich, chenille-like texture of awareness is increased, allowing the emergence of a nearly infinite interaction between individual consciousness and information at its boundary with a holographic universe causing a nondual or mystical experience. Fractal topology can assist the understanding of the transcendent experience of seamlessness and the subjective dissolution of the sense of self. Further research into fractal characteristics of individual consciousness may reveal laws of the universe concerning information at that boundary. This may be a gradient interface between individual consciousness and a holographic universe.

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References Babloyantz, A., & Louren, C. (1994). Computation with chaos: A paradigm

for cortical activity. Proceedings of the National Academy of Science USA (91st ed., pp. 9027-9031). Baruss, I., & Mossbridge, J. (2017). Transcendent mind: Rethinking the science of consciousness. Washington, DC: American Psychological Association. Bieberich, E. (2012). Introduction to the Fractality Principle of Consciousness and the Sentyon Postulate. Cognitive Computation 4(1),

13-28. Bucke, RM. (1905). Cosmic consciousness: A study in the evolution of the human mind. Philadelphia, PA: Innes & Sons. Combs, AL. (1996). The radiance of being: Complexity, chaos and the evolution of consciousness. St. Paul, MN: Paragon House. —. (2015) The nature of consciousness. J.A. Davis & D. Pitchford (Eds)

Stanley Krippner: A life of dreams, myths, and visions; Essays on his contributions and influence (pp. 21-40). Colorado Springs, CO: University Professors. Committee on CMMP, Solid State Sciences Committee, National Research

Council (2007). Condensed-matter and material physics; The science of the world around us (Electronic version]. Retrieved from: https://catalog.loc.gov/vwebv/holdingsInfo?searchId=17024&recCount =25&recPointer=10&bibId=15032427

—. (2010). Condensed-matter and material physics; The science of the world around us (Electronic version]. Retrieved from: https://catalog.loc. gov/webv/holdingsInfo?searchId=16952&recPoint er=7&recCount=25&bibId=1778458 1 Crick, F., & Koch, C. (2003). A framework for consciousness. Nature Neuroscience, 6(2), 119-126. D’Aquili, E.G., & Newberg, AB. (2000). The neuropsychology of aesthetic, spiritual, and mystical states. Zygon, 35(1), 39-51. Edehnan, G., & Tononi, G. (2000). A universe of consciousness: How matter becomes imagination. New York, NY: Basic.

Franz, M.V. (1995). Projection and re-collection in Jungian psychology: Reflections of the soul. La Salle, IL: Open Court. Fuller, B. (1961). Tensegrity. Retrieved from: http://Www.rwgrayprojects.com/rbfiiotes/fpapers/tensegrity/tenseg01.h

trnl Gell-Mann, M. (2010). Nature conformable to herself. In H. Fritzsch (Ed)

Murray Gell—Mann: Selected papers (pp. 378-382). Hackensack, NJ:

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World Scientific. (Original work published 1996) Gleick, J. (2015). Chaos: Making a new science. London, England: The Folio Society.

Greene, B. (2011). The hidden reality: Parallel universes and the deep laws of the cosmos. New York, NY: Alfi'ed A. Knopf. Grof, C., & Grof, S. (1992). The stormy search for the self: A guide to personal growth through transformational crisis. New York, NY: 1P. Tarcher / Perigee. Gupta, B. (2003). CIT consciousness. Oxford, England: Oxford University Press.

Hegel, G.W., Miller, A.V., & Findlay, J. (1997). Phenomenology of spirit. Oxford, England: Oxford University Press. (Original work published

1807) Hollick, M. (2006). The science of oneness: A worldviewfor the twenty—first century. Winchester, United Kingdom: 0. Ingber, DE. (1993). Cellular tensegrity: Defining new rules of biologica1

design that govern the cytoskeleton. Journal of Cell Science, 104, 613— 627. —. (1997). Tensegrity:

The architectural basis of cellular mechano-

transduction. Annual Review of Physiology, 59(1), 575-599. —. (2003). Tensegrity II. How structural networks influence cellular

information processing networks. Journal of Cell Science, 1 1 6(8), 13971408. Jaeger, H., & Lui, A. (2010). Far from equilibrium physics: An overview. Unpublished manuscript, Chicago, IL. Retrieved from: https:l/j f1.uchicago.edul~j ae ger/ group/ J an gerGroupPapers/ other/FarFro mEquilPhys.pdf

James, W. (1890). The principles of psychology. New York, NY: Henry Holt.

—. (1911). The varieties of religious experience.'A study in human nature. New York, NY: Longmans and Green.

Jung, C.G. (1989).Memories, dreams, reflections (A. Jaffé, Ed.; R. Winston & C. Winston, Trans). New York, NY: Random House. (Original work published 1961) Kampis, G. (1991). Selfimodifying systems in biology and cognitive science. New York, NY: Pergamon. Kant, I. (1910) Kant’s cosmogony.‘ As in his essay on the retardation of the rotation of the earth and his natural history and theory of the heavens: with introduction, appendices, and a portrait of Thomas Wright of Durham. (A. Hastie, Trans.) Glasgow, Scotland: I. Maclehose.

Chapter Ten

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Kauffinan, S . (1993). The origins of order: Self-organization and selection in evolution. Oxford, England: Oxford University Press. Laszlo, E. (1972). The systems view o f the world: The natural philosophy of the new developments in the sciences. New York, NY: G. Braziller. Laughlin, R B . (2005). A different universe: Reinventing physicsfrom the bottom down. New York, NY: Basic. Lewin, R. (2001). Complexity: Life at the edge of chaos. London, England: Phoenix. (Original work published 1992) Lorenz, E N . (1963). Deterministic non-periodic flow. Journal of the

Atmospheric Sciences, 20(2), 130-141. Loy, D. (1997). Nonduality: A stuafv in comparative philosophy. Amherst,

'NY: Humanity. Mandelbrot, B B . (1977). The fractal geometry of nature. New York, NY: W . H . Freeman. —. (1989). Fractal geometry: What i s it, and what does it do? Proceedings

ofthe Royal Society A, 423, 3—16. Maslow, A H . (1964). Religions, values, and peak-expefiences. Columbus, OH: Ohio State University Press. McDermott, R. (Ed.) (2001). The essential Aurobindo: Writings o f Sri Aurobindo. Hudson, NY: Lindisfarne. Metzner, R. (2017). Ecology of consciousness: The alchemy of personal, collective, and planetary transformation. Oakland, CA: Reveal / New Harbinger. Miller, J.C., & Jung, CG. (2004). The transcendent. function: Jung ’s model of psychological growth through dialogue with the unconscious. Albany, NY: State University of New York Press.

Peak, D., & Frame, M. (1998). Chaos under control: The art and science of complexity. New York, NY: Freeman.

Penrose, R. (2005). Shadows of the mind: A search for the missing science of consciousness. London, England: Vintage.

Radin, D.I. (1997). The conscious universe: The scientific truth of psychic phenomena. New York, NY: HarperOne.

Scaruffi, P. (2013). The nature of consciousness: The structure of life and the meaning o f matter. Berkeley, CA: P . Scaruffi. Retrieved from http://www.scaruffi.com/nature/emergenchtml Schwartz, S A . (2007). Opening to the infinite: The art and science of nonlocal awareness. Buda, T X : Nemoseen. Sheets-Johnstone, M. (2016). Insides and outsides. Exeter, England: Imprint Academic. Tart, C T . (1975). Transpersonal psychologies. New York, NY: Harper & Row.

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Underhill, E. (1911). Mysticism: The nature and development of spiritual consciousness. New York, NY: E.P. Dutton. Whitehead, A.N. (1969). Process and Realiiy. Riverside, CA: Free Press. (Original work published 1929). Wood, A.W. (Ed.). (2001). Basic writings of Kant. New York, NY: Modern Library.

CHAPTER ELEVEN

FRACTAL BOUNDARIES: A SUBJECTIVE APPROACH ROBERT M. GALATZER-LEVYI

In this chapter, I invite you to think more complexly about boundaries of all kinds, but especially to think about psychological boundaries. The ideas for this chapter historically arose from the study of fiactals and complexity theory. But in this chapter, I hope to introduce and elaborate those ideas based on experience or experience near ideasz, and so invite readers to explore boundaries as they are experienced. Fractals are beautiful! Fractals are interesting! And fi'actals can enrich thinking about boundaries. But this only happens if there is a good enough match between personal experience and the mathematical world of fractals. This chapter explores the world of fractal boundaries from a subjective viewpoint.

Psychologists, especially those of a humanistic bent, are easily put off by attempts to introduce mathematical ideas into our field because they picture mathematics as narrowing the range of experience and limiting it with rigid rules and incomprehensible formulae. My aim here to show that an understanding of fractal boundaries has exactly the opposite effect, that it expands our vision of what is possible and what can be explored.

1 Clinical Professor of Psychiatry and Behavioral Neuroscience, University of Chicago; Faculty, Training, Supervising and Child & Adolescent Supervising Analyst, Chicago Institute for Psychoanalysis.

E-mail: [email protected] 2 For those interested in a more traditional exposition of these ideas aimed at the critical scholar, please see Galatzer-Levy (in press).

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A walk Each summer I return to Woods Hole, a Cape Cod community centered on a group of laboratories where scientists, mostly biologists and oceanographers from around the world, come to study. Each morning, as I have done for more than 40 years, I go out before breakfast. At some point

these outings changed from jogs to a more sedate pace. They also became longer. Please join me for a few minutes on one of these walks. I am walking on a road with a tangle of green foliage on either side (see Figure 11-1).

Figure 11-1. A walk on a road. (Courtesy of the author)

Glancing at a plant, I see recurring patterns of leaves (see Figure 11-2). One broad-leaf plant subdivides into six-oblong leaves at its pinnacle (the whole thing is a leaf I recall from the botany class I took with Mr. Abbot

when I was ten years old at the Children’s School of Science; Mr. Abbot was such a nice man). Each leaflet is somewhat thick and leathery, with a spine down the middle of the leaf, somewhat indented from the rest of the leaf From the spine radiate symmetrical veins, and I imagine the

interconnected nature of the cells.

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Figure 11-2. Close-up of plant leaves. (Courtesy of the author)

In fact, as I look around, the fractal nature of the growth of the plants glares at me, at which point my thoughts bifurcate to two books, one being a black-covered handbook that I used in Science School class when I was about 11 years old with a title something like “Key to the Plants of New

England.” If I had it with me, I would try to discover the name of the plant, and this brings me back to Mr. Abbott, who my parents had invited to lunch, where we sat at the same table where my fractalized worldview began in my mid 20’s. It is a Danish modern table my parents brought back from Denmark. At it I read René Thom’s Catastrophe and Development3, whose red cover appears to me quite vividly, though its exact title does not. Now I

am thinking about how perhaps eight feet away from where I was reading, though almost a half century later, my 4—year—old grandson Ezra is doing a “science experiment” which, to his grandfather, appears to be mixing water with the ashes of the barbecue of the night before. I recall my own

experience of doing similar “experiments,” which were dominated by ideas 3 Actually, I have misremembered this title though I was quite sure of it. Thom’s book is titled Structural Stability and Morphogenesis: An Outline of a General Theory ofModels.

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of what it was my father did in his laboratory, and a delight in making a mess which, as a proper psychoanalyst, I recognize is related to anal or

rather fecal pleasures. My thoughts are then interrupted by a cyclist who Cheerily calls out, “on your left,” abruptly shifting my attention to the manmade environment that includes a paved road on which I am walking with a yellow line down its center. I attempt to suppress my annoyance at “these people” who interrupt my walk with their bicycles and cars, and then think

of some of the signage along the bike path (See Figure 11—3).

mums msnoum ‘ m truss mm

Figure 11-3. Signage along the bike path. (Courtesy of the author)

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I attempted the day before to look at the signage with “beginner’s eyes,” to see it without my knowledge of how to read -- and discovered once again that it is quite impossible for me to see the letters of the alphabet without interpreting them as words. This paradoxically seems easier to do for the more complex notations than for the simpler ones like the stop sign which, when I attend to it, I start rearranging the letters—TOPS, POTS. SPOT, and so forth. But then I remember another sign I had seen the day before describing how Cape Cod was laid down by receding glaciers approximately 25,000 years ago, “only 25,000” years I think, whereas Martha’s Vineyard, an island approximately three miles away and clearly visible across the water, itself had been laid down 10,000 years later, long after there were human beings capable of noticing and appreciating the scene, but I doubted there were any people near the glacier to appreciate the view I was now looking at, much less to make anything of it. Looking across the water, I notice no sharp separation of land from sky and water but rather an apparently fuzzy area beyond which is neither clearly land, nor water, nor sky. I will stop here, first because Proust, Woolf, and Joyce have done this exercise so much better, and also to point out to inevitable incompleteness of my description of these few minutes of the mental life, that incomepleteness resulting flom a range of factors flom simple inattention, to the interstices that open as one moves through a mental landscape, to the active largely unconscious censorship of some lines of thought. Maj or aspects of the mental process have been actively bracketed in the sense of being present and influential but left undescribed. Inparticular, my fantasies about you, the reader, and your reactions to what I have written were ever present as I observed myself. Looking back on the process I have just described, anyone with a passing knowledge of fractals and nonlinear dynamics system thinking will quickly recognize old friends. But for those less familiar with these ideas, let me point them out. Fractals, that is structures built up of simple, recurring patterns which together emerge as an overall pattern, joining together into still larger patterns, are clearly visible in a description of physical objects (e. g., segments of leaflets, leaflets forming a compound leaf, the leaf as part of a plant with an overall structure, the plant part of a wooded area, the wooded area part of the landscape of the Woods Hole community, etcetera), and also fractal structures in time (my father’s laboratory, my own experiments, my grandson’s experiments); the minutes it took to have these thoughts embedded within the years of thinking and presumptive brain development on the one hand and social development on the other, in which

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they occurred embedded within a vision of human history which is, in turn, embedded in the context of recent geological events. I suspect that if the reader carries out a similar experiment with their own stream of consciousness or free associations, a very similar picture is likely to emerge with regard to the fiactal nature of consciousness. Various factors will influence the process such as the question of what is important to you at the moment, what you have been educated to think about, and how, within the various communities of which you are a part, thought processes are conceptualized to be valid and permissible. If you have been educated in various meditative processes (including the psychoanalysts’ free association), the extent and breath of your thoughts are likely to be wider than if you have not been so educated. Thoroughly non-fractal associative patterns, which are often referred to as rationality, is in fact a sign of psychopathology. People who experience themselves as moving logically from thought to thought or who censor thoughts which cannot be fit into a logical sequence have profound limits in their capacity to find anything new in their experience or to creatively adapt to new situations. As West (2006) wrote, it is a characteristic of pathological systems that they are unduly rigid and over-regular in their functioning, which makes them incapable of developing and/or adapting to changing situations. The example I have provided of something like a stream of consciousness has traditionally been understood as a demonstration that the boundaries that we ordinarily believe to exist, whether as a function of the internal or external world, are in fact less real or less necessary (if imposed by our own mental processes), than is commonly believed. The same example could be used to demonstrate the ubiquity of boundaries and differentiations in thought. The imposition of boundaries in thinking about situations leads to easier investigations, albeit at the expense of accuracy. The most glaring of these combined successes and failures is the description of individual psyches (or even brains) as though they existed in any significant sense independent of the world of which they are a part. Although there are important exceptions, 20th century psychoanalysis focused strongly on the activities occurring in individual minds, even going so far as to attribute almost all of the experience of the external world to projection of elements of an isolated “mental apparatus.” In this, psychoanalysts and other psychologists followed a tradition of reductionism that has been inordinately successful in enriching our picture of the physical world by

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isolating small portions or elements of the world in a test tube in which controlled experiments were possible. In fact, massive efforts to disrupt the illusion of boundaries on various levels and using various techniques to this end have become central to psychological practice and methods ranging from the “relational” or “field theory” points of view associated with psychoanalysis to the importation of segments of Buddhist meditative practice as a psychotherapeutic methodology (Wright, 2017). These attacks on the illusion of boundaries, while certainly freeing aspects of a description of the world, nonetheless tend to omit central and important aspects of the experience-near description from any discussion of our example. The example obviously includes many elements that could rightly be called boundaries but, at the same time, illustrates that the premature characterization of those boundaries as

absolute separators of the elements of the story leads to a confused and inaccurate understanding of the teller’s experience.4

Fractal boundaries Geometry is the mathematical study of relationships in spaces of all kinds. The spaces may be the descriptions of the ordinary visible world, which is the basis of the etymology of the word “geometry” (earth measurement) and forms the content of the subject matters of elementary coursework in the subject. But the spaces may also be abstractions from many other sources, such as the space of musical sounds, the networks of interconnected thoughts, and the relationships represented in genealogical trees and networks of various kinds. The single most important advance in geometry in the past century was Benoit Mandelbrot’s investigation of rough (as opposed to smooth) surfaces and his rediscovery of fractals. While mathematicians like Julia and Kock

recognized some of the properties of fractals in the late nineteenth and early twentieth centuries, their explorations

were severely limited by their

4 It is only recently that a group of physicists and cosmologists have come to recognize many of the central laws of physics, such as the value of the gravitational constant, reflect events which occurred in the very early universe, the accidents of which historical moments are reflected in the physical constants of the present universe. Similarly, certain elements of the biological world such as the I e s cycle appear to be universal to all things we recognize as living on Earth, and yet might have been different before the accident of their occurrence at the origin of life

(Rovelli, 2018).

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inability to visualize the objects they were studying. The fortuitous accident that Mandelbrot was working at IBM as computers with the capacity to produce graphic representations of fractals was emerging allowed him to literally see the richness of the geometric forms he was studying. Fractals are geometric configurations with the property of self—similarity (if you take some appropriate portion of the fractal and magnify it, the magnified portion maintains the statistical features of the whole). As a result, any appropriate part of a fractal when magnified further will exhibit the same properties as the original fractal. Mandelbrot discovered that rough surfaces of infinite complexity could be generated by very simple rules. He studied a particular such configuration which has been given his name, the Mandelbrot set (see Figure 11-4).

Figure 11-4. Mandelbrot Zoom (courtesy of Alberto Zambrano)

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In this figure we see the property of self-similarity as it appears in the Mandelbrot set, each panel representing a magnification of the rectangle demarcated by the outline rectangle in the previous panel. Although most easily illustrated as diagrams of two-dimensional Euclidean surfaces, fractals may be of any dimension and exist in spaces very different from Euclidean space, such as the space of musical sounds or the structure of stock prices. Although no physical object can be fractal in the sense that magnification of an arbitrarily great degree continues to reveal a fractal structure, real world configurations such as the form of cauliflowers, coo-systems, and the water waves have fractal structure across five to seven orders of magnitude of magnification as is evident since, at a molecular level, the fractal structure is no longer present. A surface with a fractal structure is easily made selectively porous by removing structures below a specified magnification. The resulting holes in the surface have various shapes imposed by the fractal structure so that they may serve not only as a simple sieve but even select for the shape of the objects that can pass through the fractal. The obvious application of this feature to biological systems — gases passing between blood and air in lungs, ions passing across cell membranes — can be thought of as occurring with regard to the flows of mental contents across fractal boundaries. Fractals are beautiful and interesting (Peitgen & Richter, 1986). This is important to clinical work because clinical endeavors of all kinds must hold the clients’ and clinicians’ interest if they are to be explored and worked on. Fractal configurations, not necessarily under that name, invite this desirable engagement.

Flow Throughout this chapter, I will attempt to describe mental entities as patterns of flow that are sufficiently stable that the human observer attributes to them a thing-like quality. Thus, for example, a rock can be conceptualized as a stable, for practical purposes, unchanging thing that remains pretty much as it is unless considerable force is applied to it. It can also be thought of as a very large number of somehow connected molecules flowing through space in relation to one another so slowly that, to a human observer, it appears unchanging. Alternatively, one might think of the rock as an aspect of a geological fluid that is sufficiently coherent to appear to have immobilesharp boundaries. The relationship of flow to things is most obvious when the things are changing rapidly. Water waves are a good example. We speak

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of a wave as a thing, but a moment of thought informs us that it is a perturbation in a fluid. The mental states that are referred to as “things” are usually better conceptualized as flows or movements or perhaps as mental processes or actions (Schafer, 1976)? An idea constitutes a mental state which, in addition to having a formal structure that we associate with ideas, is sufficiently stable in the flow of consciousness to be recognizable as a distinct entity of the type we call an idea. The Heraclitian aphorism concerning never stepping in the same river twice points to the fact that a river is characterized not by the stuff of which it is composed, nor by the configuration of that stuff at any particular moment, but rather by a pattern of motion of that stuff. Anything that can be described as an object can also be described, albeit more richly, as a pattern of flow, and of course any pattern of flow upon being named could be thought of as an object. A thorough discussion of this topic, which goes back to the pre—Socratic philosophers Parmenides and Heraclitus, is beyond the scope of this chapter. However, examining mental “things” as flows is not only closer to reality but also simplifies the discussion of the interaction of entities, and particularly their boundaries. Certainly, in thinking about fluids in motion it is easier to picture complex boundaries between flowing liquids than it is to imagine complex and dynamic interdigitations between solid, stone-like objects. Returning to the example of my train of thought during a walk, notice that the individual ideas into which these thoughts may be divided are connected on multiple levels. Given my own propensity to think in words, the individual ideas are embedded in English vocabulary and grammar. English is composed of sentences, so separations are introduced into the flow of ideas by its grammatical structure. The structure of sentences provides/imposes boundaries in verbal thinking. In thought that is less verbal than mine, and for non-verbal aspects of thought, different boundaries would be induced resulting in a different

5 Recognizing that the language of psychoanalysis, which commonly treats psychological entities as concrete objects, needs “action language” that requires reformulating psychoanalytic ideas in terms of acts. While this exercise succeeded in convincing many analysts that terms like ego or superego that implied the existence of psychological “structures” confused psychoanalytic thinking, the awkwardness of the proposed new language made it unattractive as a replacement and it never caught on.

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understanding. Suppose that instead of thinking in English sentences I organized thoughts by the color of the things. The experience would be a sort of spectrum to which the shape of obj ects and their functions, much less the words assigned to them, would be secondary. Einstein (1945) said that “words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be ‘voluntarily’ reproduced and combined” (pp. 142-143). (He also included a “muscular elemen ” as an element of his thought). The boundaries of the elements of thought are highly contingent on the mode of thought and are in most instances fluid and flowing, though with varying degrees of rapidity. The demarcation of boundaries can constitute a creative act. The text of Genesis 1 describes creation as repeated acts of division—the heavens from the earth, the waters from dry land, and so forth. Boundaries are implicit in each of these acts of division. Aristotle, especially in the Categories, separates the mental world into component parts though he would probably have characterized his activity as recognizing natural divisions in the world. The placement of boundaries in this mental world creates the entities demarcated by those boundaries.

Beach boundary Continuing my walk, I came to a beach on Vineyard Sound, which separates Cape Cod from Martha’s Vineyard. As Iwalked toward the beach, it seemed from a distance that there was a sharp, if moving, demarcation between the water and the sand. As I moved closer, looking at the receding waves, I saw that each wave left behind a thin layer of water which quickly turned into wet sand. Even closer it became clear that the edges of the wave were not in a straight line, or even smooth curves, but jagged configurations. Looking closer, the roughness of this border continued, down to individual grains of sand, the closest I was able to see. Looking at the water itself, I saw waves coming towards me, but noticed that on top of the waves were smaller waves which themselves were composed of yet smaller wavelets in the manner of the Japanese print, the Great Wave of Kanagawa, but on a much smaller scale.

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The moving border between the sand and the ocean is not imposed by any rule or barrier. Rather, it is an emergent property of the configuration of the water and the elevation of the sand beneath it. Other emergent configurations are everywhere visible. The tangle of dried seaweed that marks high tide, which results from the seaweed’s no longer floating in the receding water is an emergent configuration on one time and size scale. On another scale of time and size is Vineyard Sound, the product of a ten thousand year pause in the melting of the glaciers as they retreated northward leaving behind the debris (moraine) called Martha’s Vineyard and Cape Cod to be further shaped by the flow of sea and rain water over it. The sandy beach itself is the product of a process by which the flow of water moves the sand which is left in place as the water recedes. When the water returns, it shapes the water’s flow and hence its reshaping of sand. The sand beaches on either side of Vineyard Sound being the product of the tidal flow of water. In psychology, the search for clarity produces a strong urge to erect clear conceptual boundaries between various states, for example, between masculine and feminine, between various stages of cognitive and emotional development, between pathology and health, between various forms of pathology, as well as between right and wrong behaviors. Exploring the motives for such categorizations often initially results in statements to the effect that sharp clear demarcations improve thinking and thereby promote rational discourse about difficult topics. Some psychoanalysts have gone so far as to assert that the capacity to differentiate and recognize the border between say masculinity and femininity is equivalent to the ability to know reality, that it is madness to deny such a sharp border (Chiland, 2009). However, since the insistence on such sharp demarcations so often flies in the face of reality, there must be more at work than the simple love of knowledge. The need for sharp demarcations seems to arise most flequently when significant decisions must be reached. Physicians, for example, are daily faced with the reality epitomized in the Hippocratic saying, “Art is long and life is short, decision difficult, and the occasion fleeting” (i.e., that medicine is complex, physicians have inadequate time to learn it, but they must make momentous decisions quickly). Across the entire history of Western medicine this has led to a preoccupation with the diagnosis, because, in theory, precise diagnosis entails clear prognosis (the main thing physicians had to offer until the mid—twentieth century), and to an ever—increasing

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extent following the Second World War, with the invention of antibiotics, diagnosis supported specific therapeutics for a range of conditions. But the emphasis on diagnostic categorizations and the development of ever more precise boundaries between disease states has come at aprice. Disease states are but an aspect of the complex systems that are usually thought of as the patient’s and the community of which the patient is a part. For example, diagnosing a particular form of malaria is useless and confusing outside the context of the particular person with the illness and knowledge of the community in which the patient lives. For all the power that diagnostic categorization has had, the exclusive focus on it has led to a failure to attend to the complex systems of which the illness is but a part and to a falsely linear view of illness (West 2006). This medical model to believe that we know what we are doing has been carried over into psychotherapy, where supposedly well— defined disorders are treated with supposedly well-defined interventions and referred to as evidence based despite the ever-mounting evidence that neither the disorders nor their treatments represent natural types in any sense. A similar urgency for clear categorization results in overly clarified boundaries concerning gender. In fact, the entire concept of gender, including the belief that there are two types of people corresponding roughly to the biological sex but including a vast array of non-biological features such as social position, appropriate clothing, purported personality features, among others appears to arise fiom pressure to categorize. This urgency of categorization may be closely related to primate social structures. Clear gender boundaries are actively constructed in the sense that the supposed differences between the sexes, even at the level of biology, far exceed anything for which there is empirical evidence (e.g., brain differences between the sexes—Fine, 2014).) Whatever the sources of the urgency to maintain the gender binary, the sharp artificial distinctions designed to police this boundary are evident from the very beginning of infant’s lives in the manner in which they are handled and continue in the sex specific, clothing that even the youngest children in our society wear. Perhaps the most obvious group of artificial boundaries are the geopolitical boundaries whose creation and maintenance require enormous resources when they do not correspond to natural geographic structures such as mountain ranges or large bodies of water. This example provides a particularly interesting instance of the consequences of the difference between natural and artificial boundaries. The few places where modern democracies arose early—the United States, England and Switzerland—all had formidable natural boundaries—the Atlantic Ocean, the English

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Channel, and the Alps. These natural boundaries effectively protected-them

from invasions so that these countries were able to exist without the standing armies that were needed in most European countries to protect their borders. Inlconsequence, no powerful and democratic military was needed.

Leonardo, a student of boundaries Leonardo’s preoccupation with complex boundaries may have originated in his. fascination with his own hair and later his beard, both of which took overall form from the concatenation of individual hairs (see Figure 11-5}.

Figure 11—5.'The Turin self-portrait, Leonardo d a Vinci, circa 1510-1515

His fascination with curls was evident also in his erotic choices, most

obviously Salai, his “little devil” who was his companion and presumably lover during the latter half of Leonar do’s life. Drawing Salai’s hair occupied Leonardo’s notebooks for 2? years (see Figure 11-6).

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Figure 11-6. Leonardo da Vinci’s drawing of Salai, c. 1508 Fascination with various forms of what we would now recognize as

fractals continued throughout his life as evidenced for example by his drawings of turbulence (see Figure 11-7).

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And in the so-called “Deluge drawings” (see Figure 11-8).

Figure 11-8, A deluge drawing, Leonardo Da Vinci’s, c. 1517. Other works, especially his studies of anatomy and natural objects point to an appreciation of the fractal nature of biological reality. For example, he commented that branching systems of arteries parallel branching trees. Leonardo da Vinci is probably the greatest student of the Visual

perception of boundaries. As Vercocchio’s apprentice, he painted portions of Baptism of Christ (see Figure 11-9).

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Figure 11-9. Baptism of Christ, by Andrea del Vercocchio with Leonardo da Vinci,

c. 1474—1475 It provides vivid contrasts between the sharp clear lines of Verrocchio in outlinjng the figures of John the Baptist and Christ and the subtler portrayal by Leonardo of Christ’s feet, his pubic hair and the shadow under his belly. Compare the angels on the lower left of the painting (see Figure

11-10).

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Figure 11—10. Two angels, detail of Vercocchio’s Baptism of Christ

The angel on the left is painted by Leonardo. Note the subtle blurring or roughness in the outline of the left angel’s face contrasted with the sharp features of the one on the right and a similar quality of the borders of the angels’ clothing.

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In this painting we have the first clear example of the technique Ernst Gombrich (1950) called sfixmafo, a deliberate blurring and roughening of contours and edges. It is a way for artists to render objects as they appear to our eye rather than with sharp contours. He states: “Leonardo’s famous invention, the blurred outline and mellowed colors that allow one form to merge with another and always leave something to our imagination” (Gombrich, 1950, p. 187). The term sfumato derives from the Italian word for “smoke,” or more precisely the dissipation and gradual vanishing of smoke into the air. Leonardo wrote, “Your shadows and lights should be blended without lines or borders in the manner of smoke losing itself in the air” (Isaacson 2017, p. 41). Isaacson writes, “From the eyes of his angel in the Baptism of Christ to the smile of the Mona Lisa, the blurred and smoke— veiled edges allow a role for our own imagination. With no sharp lines, enigmatic glances and smiles can flicker mysteriously” (Isaacson, 2017, p. 41). But one should add that the sfumato is simply truer to what we actually see. Hence, it is the ambiguity of our actual vision, rather than the specialness of Leonardo ’s technique that precedes an ambiguous perceptual space with its room for creativity. Isaacson describes these effects in detail (Isaacson, 2017): Like most artists, Verrocchio drew lines to delineate the contours of his angel’s head and face and eyes. But in Leonardo’s angel, there are no clear edges that delineate the features. The curls dissolve gently into each other and the face, rather than creating a hairline. Look at the shadow underneath the jaw of Verrocchio’s angel, done with visible brushstrokes of tempera paint that create a sharp jaw line. Then look at Leonardo’s; the shadow is more translucent and blends more smoothly, something that’s easier done with oil. The almost imperceptible strokes are fluid, thinly layered, and occasionally smoothed by hand. The contours of the angel’s face are soft.

There are no perceptible edges. (p. 54) We can also see this beauty in the body of Jesus. Compare his legs, painted by Leonardo, with those of Verrocchio’s John the Baptist. The latter have sharper lines, unlike what a careful observer would see in reality. Leonardo even minutely blurs the curls of Jesus’ exposed pubic hair. This use of sfumato, the smokiness that blurs sharp contours, was by now a

hallmark of Leonardo’s art. Alberti (2013), in his treatise on painting, had advised that lines should be drawn to delineate edges, and Verrocchio did just that. Leonardo took care to observe the real world, and he noticed the opposite: when we look at three-dimensional objects, we don’t see sharp lines. “Paint so that a smoky finish can be seen, rather than contours and profiles that are distinct and crude,” he wrote. “When you paint shadows

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and their edges, which cannot be perceived except indistinctly, do not make them sharp or clearly defined, otherwise your work will have a wooden appearance” (Isaacson, 2017, p. 55). Leonardo understood that borders, in the form of lines delineating the boundaries of various objects, should never be drawn because they are never seen by the eye. We do not ordinarily see edges but instead recognize a transition where one thing turns into another. To be realistic the artist must, Leonardo tells us, represent the shape and volume of objects using light and shadow rather than directly representing aboundary, which is not visible as such to the eye. Leonardo’s sfumato was no romantic idea equating vagueness and imprecision with emotionality as the impressionists were to do centuries later. Nor was it an attempt to educate the viewers’ capacity to see in richer ways than they did naturally. It was, instead, an application of the scientific study of the optics of vision. His amazing investigations of perspective are incorporated into the Last Supper, such that the fresco’s figures appear in perspective from many positions, and his discovery that images of objects appear on the retina in multiple locations resulted from empirical investigation. So too sfumato. Leonardo realized that shadows rather than contour lines define the shapes of objects. He observed that natural objects have no visible outlines or borders and that this was a quality of natural objects themselves. Nature does not have precise lines! “Lines are not part of any quantity of an object’s surface, nor are they part of the air which surrounds this surface. The line has in itself neither matter nor substance and may rather be called an imaginary idea than a real object; and this being its nature it occupies no space” (Isaacson, 2017. 269).

Natural and artificial boundaries Leonardo’s study of the representation of boundaries and his realization that representing natural boundaries through lines of demarcation inevitably points to an almost always reliable distinction between natural and artificial boundaries. Artificial boundaries reflect someone’s intent to create a boundary and hence the boundary itself is directly represented. In contrast, most natural boundaries do not reflect an intention to create a boundary, so that they do not have a reality apart from that which they separate and have, no representation aside from this separation.

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Boundaries in biological systems commonly combine properties of artificial and natural boundaries. Consider the cell membrane, a structure that actively serves to separate the interior and exterior of cells, and which evolution has designed for this purpose”. The cell membrane is itself a distinct structure. Yet this structure reflects the emergence of membranelike structures from which cell membranes evolved. Molecules of lipids (fats) tend to join together in the presence of water. They do nothave electric charges and are said to be hydrophobic (water avoiding.) Charged molecules interact with water and can form solutions. Some molecules have both hydrophilic and hydrophobic portions, so they automatically line up with the hydrophobic portions joined and the hydrophilic portions sticking out into available water, spontaneously forming structures like those shown in Figure 11-11.

Figure ”-1 l . Naturally Occurring structures emerging from molecules containinglipid and ionicpoles.

5 Here I am using the terms design and purpose to refer not to'conscious intent but rather to the fact that natural selection favors structures that carry out fimctions needed for the organism’s survival.

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Cell membranes are essentially bilayer sheets shaped into abag. In actual cell membranes these sheets are interrupted by numerous structures that

allow material to pass into and out of the cell and also by structures that permit signals to pass in and out of the cell. For our purpose, the most important fact is that these boundaries are sharper and clearer than purely natural boundaries, but at the same time reflect a naturally occurring process of evolution acting on a spontaneously emerging boundary, the bilayer

sheet. Natural boundaries are often the result of emergent processes characterized by similar forces operating on natural substrates at various

levels of magnification and at various locations. Consider the natural course of a river which emerges from the combination of the presence of a source of water, gravitational forces, slight variations in the density of the earth in which the water is flowing, and the past history of the river. As the river digs its way through the landscape, similar forces are simultaneously in operation at every level of magnification (see Figure 11-12). The resulting

river has a fractal structure.

Figure 11-12. Aerial river photo. (Courtesy of Drew Coffinan, Unsplash) Natural boundaries arising, as they frequently do, from emergent processes commonly have a fractal structure because those emergent processes involving the repeated application of a simple rule result in fractal

or fractal-like structures. These structures have the characteristics of other fractal structures that despite being generated by simple rules, they are

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like a segment of a smooth curve nor is it two dimensional like a triangle or circle but, in a meaningful sense, it has a dimension between one and two. Fractal structures as they occur in the real world do not have infinite length or similar features because at some level of granularity the fiactal structure breaks down—all material objects are composed of molecules. Nonetheless, fractals are excellent models for a very wide range of real world phenomena from the form of lungs, to stock market prices, to the shapes of broccoli and trees, to coastlines (Mandelbrot, 1977, 2001/2019). Real-world structures that are modeled well as fractals have very large and very complex rough surfaces. They are often of great beauty. For the sake of brevity, these real-world structures are referred to as fractal, though it is wise to keep in mind that they do not have all of the properties of a mathematical fractal, just as the triangular objects of everyday life do not have all of the properties of a mathematical triangle. Conversely, a feature of real-world fractals not present in mathematic ones is that, since at some point the fractal structure breaks down, spaces between the materials that make up the fractal are not filled in. Thus, there are open spaces in the boundary formed by fiactals through which things can pass. For example, lungs have structures that repeatedly branch (a total of about 6 or 7 times) eventually becoming sufficiently small so the gasses can pass between the remaining openings but larger structures like dust particles, bacteria and blood cells cannot. Thus, fractal boundaries can make sophisticated sieves. Many have the property of allowing transport of material in one direction but not the other. One of the most important lessons from the observation that boundaries may be, and often are, fi'actal is that it makes the study of boundaries more interesting and complex by introducing richer possibilities for boundaries than seem to have come into psychological and other disciplines’ thinking from other sources. Consider the gender binary. A huge literature describes both the inadequacy of the idea and the oppressive consequence of its adoption. In addition to its oppressive nature, the gender binary asserts the existence and normality of a sharp distinction between masculinity and femininity. This conceptualization interferes with the intellectual and aesthetic possibilities that result when the boundary between masculinity and femininity is conceptualized as firactal. Adrienne Harris’ brilliant Gender as Soft

Assembly makes essentially this point in a somewhat different manner. She

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shows that the individuals’ gender experiences are emergent properties of the very many cultural, biological, and individual-psychological factors whose interaction leads to some kind of gendered (or anti-gendered) experience of self (Harris, 2011). It is the complexity and changing nature of this boundary combined with a felt urgent need to oversimplify it that leads to confusion and difficulty. Consider the relationship of same sex desire to other elements of gender. Well into the current millennium, it was often asserted that men who desired sex with other men were likely to have a large group of gendered psychological features in common with other such men, such as dressing more flamboyantly than “straight” men, or having a supposedly “gay voice” (Cohler & Galatzer—Levy, 2000). One of the striking results of gay liberation has been a marked diminution in the stereotyping of men with same sex desires. One result is that it is commonly believed that it has become difficult to identify men’s sexual desires on any factor other than their stated preferences. This is not only because gay men have continued to not actually meet these stereotypes, but also because many men whose erotic interests are primarily in women have adopted styles previously associated with same-sex desire ranging from clothing styles, to interest in physique, to intense-emotional expression. From many viewpoints, including feminist, gender, transgender, queer and men’s studies, the gender binary disrupted almost everyone’s capacities to live as fully as possible. To many mental health professionals, the solution seems obvious — do your best to do away with the internal and external effects of the gender binary by affirming personal and social opposition to it when it becomes problematic. In dealing with a trans gender child, such a mental-health professional would educate family and the child to the normality of the child’s condition, help, as best the professional could, to minimize the manifestations of the gender binary at school and other institutions, and support medical interventions that help the child have a gender—syntonic body, for example by postponing puberty to allow the child time to decide on wished for secondary—sex characteristics. Similarly, in dealing with a cis straight man who was disturbed by his wish to wear dresses, a therapist might either help the patient recognize that his selfdisapproval is irrational and that this stereotypically feminine aspect of himself was worthy of embrace or, possibly, to recognize it as a neurotic symptom to be resolved. Note that both the therapeutic intervention suggested for the transgender child and the cis-straight man accept the line between masculinity and femininity, though they both somewhat move both

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it and the person’s relationship to it, and attempt to reduce the moralistic attitude with which it is maintained. More radical solutions involve attempting to do away with the meme of gender. Aside from being a large task, it would deprive us of many genderassociated pleasures, including the pleasures of transgressing gender norms. A different approach would begin with the recognition that the boundaries associated with gender are complex, and, at least in many of their aspects, fi'actal in nature. The aesthetics of fractals would suggest the need for an exploration of this fractal structure, rather than an attempt to abolish the distinction. Such an exploration might lead to far richer and more workable results in which the studied boundary is revealed to have a beauty of its own. Such an approach is evident, for example, in text’s like Garber’s Vested Interests which explores and details the complex interfaces of gendered clothing and gender (Garber, 1992); works of fiction like Virginia Woolf ’5 Orlando that probe the nature of gender boundaries and time (Woolf, 1933); or the brilliant therapeutic work with a transgender child in which Saketopoulou helped the youngster move from a despairing denial of her penis to a rich appreciation of the complexity of being a girl with a male genital (Saketopoulou, 2014).

Transpersonal psychology Transpersonal psychology can be seen, in large measure, as an effort to solve problems that arise from inserting boundaries into psychological conceptualizations, whether as an aspect of psychologists’ theorizing or persons’ attempts to organize their experience. Much of the discourse of various transpersonal psychologies focuses on the limitations of experience imposed by insisting on boundaries of various kinds, and the experiences

that can be bad if these boundaries are erased or at least treated as more, permeable than they ordinarily are. The most obvious problem arises with regard to the idea of the self. A very long history of informal folk, philosophical, and psychological traditions support the notion of something like a soul associated with any person, forming their essential being and, at the same time, distinguishing them fiom others. As Makari (2008) demonstrates, in Western thought the idea of the psychological “self” replaced that of the soul. Implicit in this idea is some notion of the bounds of the self or soul. By contrast, the Buddha’s analysis of the concept of “self” led him to the idea of anatman (no-self), the fact that the experience of an essential self is an illusion. The,

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need to formulate the concept shows how powerfully present the idea of “self” is as a psychological configuration. Views of the nature and significance of the self commonly center on conceptualizations of its boundaries. A huge psychiatric and psychodynamic

literature starting with Federn (Federn, 1926) and continuing to the present (Caligor, Kernberg, & Clarkin, 2018) describes severe-psychiatric illnesses as originating in failures of self-boundaries. This failure is manifest by phenomena such as feelings that the person may be fading out of existence, a sense of being controlled by others, having thoughts injected into one’s mind, or beliefs of being able to control things that were not ordinarily considered part of the self (e. g., being able to move objects without physical contact). Around the turn of the 20th century, phenomenological studies underlined how the bounds of the self were extended and reshaped in various nonpathological situations in mystical religious practice, group phenomena, and sexual life (Freud, 1920/2001; James, 1902; Le Bon, 1896). Another point of view, which emerged in the Chicago School of Sociology recognized the contingency of the self on the environment and framed the self as a product of the environment. The Chicago School often left obscure where the self began and the environment ended. Transpersonal psychology may be viewed as an effort to clear up the problem created by the apparent contradiction between these diverse views of the boundaries of the self. But the problem is not solved by the denial that the boundaries exist, at least as part of ordinary experience. A consideration of fi'actal and fi'actal-like boundaries provides an alternative means of explaining the complexities of separateness and continuity embodied in these various efforts to address the inappropriateness of sharp-clear boundaries as descriptions of psychological realities.

Top down causation and the individual Imagine yourself as spirals in a fractal that only becomes visible, at 10,000 times magnification (see Figure 11—14).

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Figure 11- 14. Detail, the Mandelbrot set.

Further, you are unaware of this status (i.e., you don’t recognize yourself to be part of a fiactal). However, you do know that, as the computation of the fractal is ongoing, you are changing and becoming more elaborate. In fact, you have the belief that this development is something which both occurs to you (i.e., is happening independent of your will and intentions) and there are also elements of the development that you control. Your control (i.e., your wishes) are a fimction of a set of aesthetic values that feels to you like “truth,” and which happen to correspond to the geometry of the

Mandelbrot set. At this point, you “decide” to color points in a manner which corresponds to the Mandelbrot configuration, feeling this is the most aesthetic coloring. You experience free will, though, since you are, afler all,

a tiny piece of the Mandelbrot set, in fact you could not have done otherwise. This is something like what the experience of top down causation might be for an individual. Top down causes are causes that result fiom the individual being part of a system as when people act as part of a crowd, family, historical moment, and similar configurations. While the larger configuration can be thought of as an emergent result of the interactions

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(synergies) of its individual components, being part of the configuration also shapes the behavior of individual elements. Recognition of this relationship naturally leads to the question of the relationship of the individual to the larger configuration and in particular the question of the boundaries that are implicit in the very idea of an “individua .” At minimum these boundaries must be complex, dynamic, natural, and selectively permeable and hence likely fractal.

Practical and clinical implications Fabulous amounts of discourse have been devoted to boundaries, from abstract mathematical formulations to geopolitical polemics. This dis— course tends to be simplistic. Boundaries are seen as good or bad, strong or weak, in need of reinforcement or destruction. Many descriptions regarding boundaries tend to have “common sense” qualities, in which problems from various domains are treated similarly because they involve the term

boundary 7. Recommendations for dealing with boundary-related problems tend to be unsophisticated, even in the hands of sophisticated thinkers. For example, in their carefully researched study of boundary violations (situations in which therapists engage in inappropriate activities, such as sex with patients), Gutheil and Brodsky (2008) essentially conclude that the best way to avoid such violations is to stay far from the boundary and so avoid a “slippery slope”8 which is commonly recognized only retrospectively. Similarly, although contemporary psycho-analytic practice recommends the procedure be done with delicacy and respect, the analysis of defense is often undertaken with the intent of disrupting that defense so as to allow access to underlying realities. Today’s dismantling of defenses is thought to require sophistication and to involve a technique more closely resembling the work of a careful craftsman disassembling the door of a safe than the smashing techniques advocated by early psychoanalysts (Reich, 1949). Similarly, meditative techniques designed to overcome boundaries that separate individuals from various aspects of transpersonal experience, commonly involve a subtle but nonetheless brute force technique, inviting students to 7 For a superb discussion of why common sense is a poor guide to complex systems, see Watts (2011). s The term slippery slope in this context refers to actions that are not in themselves boundary violations (e.g., touching the patient, that lead to more physically intimate acts and ultimately overt sexual behavior).

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first develop expertise in some form of concentration meditation (a capacity for sustained focus of attention) and then apply this method to various subject matters of transpersonal experience (Levey & Levey, 2015). Outside of mathematics, there is a remarkable absence of discourse about boundaries themselves. This tends to perpetuate techniques that ignore their richness. It invites unthoughtful attempts to build or destroy them. More subtle approaches involving structuring and modifying boundaries are seldom discussed. This is true whether the lack of adequate language to discuss boundaries in their richness arises from simple ignorance, an active process involving avoidance or knowledge out of anxiety, or avicious circle involving both (a state described by Bion as -K). Our situation in psychology is like that of dentistry at the end of the nineteenth century when the dentist’s options were to pull bad teeth and sometimes replace them with generally unsatisfactory false teeth. Neither of these procedures reflected the complex structure of natural teeth. Not all complex boundaries are fractal. But clearly many psychological boundaries are. This suggests the need for psychotherapeutic techniques that appreciate the richness of these configurations and approaches to them based on such richness. For example, if a therapist recognizes that an existing boundary too rigorously excludes the interactions of elements on either side of that boundary, instead of trying to disrupt the boundary, the therapist may seek to understand how the boundary configuration manages porosity and only then ask whether there are means of increasing it in its specifics. Similarly, were it recognized that a boundary was inadequate, knowing that natural boundaries that emerge from repeated actions (as a river emerges in a landscape) are more complex and more reliable, the therapist may attempt to promote experiences leading to the development of such a landscape and avoid techniques that attempt to raise rigid boundaries, which tend to be brittle and require ongoing input of energy for their maintenance.

Consider the question of a colleague asking for help because they have engaged in a sexual boundary violation or feel tempted to do so. As mentioned above, thoughtful investigators have recommended techniques designed to ensure a rigid prohibition of sexual behavior with patients ranging from avoiding any actions suggestive of sexual engagements (such as touching, conscious erotic fantasies, etc.) to the avoidance of patients who might arouse erotic feelings (e.g., for a heterosexual therapist seeing only patients of one’s own gender) (Gutheil & Brodsky, 2008). These approaches require constant rigorous attention from the therapist, usually

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supplemented by ongoing monitoring by a colleague. More subtle approaches attempt to reduce the motivation for boundary violations by recognizing their origins in personal pathology and attempting to address that pathology through therapy (Celenza, 2007). As Gabbard (2017) points out, these approaches have not been successfully applied in most communities of therapists. Subsequent developments emphasize that sexual—boundary violations occur within a psycho—cultural context and that the structure of that context must be included in any investigation of boundary violations and any attempt to effectively regulate them (Dimen, 2016). What is notably missing from almost all of this discourse is a discussion of the boundaries that ordinarily separate overt erotic behaviors from therapeutic activities, and how those separations ordinarily occur. Lacking a sufficiently rich-general theory of boundaries, there is little room for forward movement in this investigation. The enriched theory of boundaries that includes the recognition of complex fractal boundaries invites such an exploration. Similarly, a range of therapies for action prone individuals include close attention to the treatment’s frame and avoiding of crossing various boundaries between therapists (or institutions) and patient. These recommendations have two-related bases. First, adequate frames may be necessary simply to continue the therapy and protect patients and therapists from various dangers. Second, it is hoped that the experience of operating within a reasonable frame will be internalized by the patient and used in ever broader contexts. For example, the physically violent patient provided with a therapeutic frame that excludes violence from the therapeutic relationship may be able to transfer the methods used to regulate violence in the therapy to a wider arena. It is evident to anyone who has worked with violent individuals that successful treatment seldom results from attempts to create simple boundaries, for example by prohibiting unwanted behaviors. However, in the absence of an adequate theory of complex boundaries, it is difficult to conceptualize the nature of the boundaries one hopes will emerge surrounding violence, much less to think through how to aid their emergence. I would argue that the appreciation of the complexity of fractal boundaries leads the clinician to think more richly about boundaries and hence be able to explore the topic in therapy in amanner that allows patients to consider, experiment with and, if appropriate, adopt complex possibilities.

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The distinction between natural emergent boundaries, which are often fractal, and artificially imposed ones, which are seldom fractal, seems to me particularly useful since the former are generally more effective, more stable and consume far less energy. Many years ago, when I was very early in my analytic career, I had a conversation with a secretary who had worked at my psychoanalytic institute for decades and was privy to many of its secrets. We were talking about a distinguished colleague who had caused patients enormous harm and brought his career to ruin by sexually seducing several of them. I asked spontaneously, “Where did he find the time?” The secretary said, “You’re so naive!” and indicated that the sexual encounters have occurred during scheduled psychoanalytic sessions. In retrospect I realize that my question arose from a particular image of psychoanalytic sessions as filled with the work of exploring the patient’s internal world. My experience of analysis,

whether as patient or analyst, was and has been that usually there is not enough time to get as far in this endeavor as I wished. There simply is not time for sex or a variety of other tempting interactions with patients. This simple element (not having enough time for the forbidden behavior), illustrates a natural boundary with some of its emergent qualities. Note that little or no additional energy is needed to maintain it. Note also that it arises without specific intention from the setup of therapeutic practice and the larger system of which that practice is a part. Note finally, that it allows for desirable permeability in the sense that there is plenty of room for discussion of erotic feelings and wishes, which might in other contexts be associated with erotic actions, without leading to those actions. It lets through some specific elements of eroticism while excluding others.

Conclusion In the introduction to his recent book on Leonardo, Walter Isaacson writes: Afler immersing myself in Leonardo, I did the best I could to be more

observant of phenomena that I used to ignore, making a special effort to notice things the way he did. When I saw sunlight hitting drapes, I pushed myself to pause and look at the way the shadows caressed the folds. I tried to see how light that was reflected fiom one object subtly colored the shadows of another object. Inoticed how the glint of a lustrous spot on a shiny surface moved when I tilted my head. When I looked at a distant tree

and a near one, Itried to visualize the lines of perspective. When I saw an eddy of water, I compared it to a ringlet of hair. When I couldn’t understand

a math concept, I did the best Iwas able to visualize it. When I saw people at a supper, I studied the relationship of their motions to their emotions.

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When I saw the hint of a smile come across someone’s lips, I tried to fathom her inner mysteries. (Isaacson, 2017, p. 6)

The intent of this chapter is to invite you to see the world of boundaries through fractal eyes and thereby to able to consider a wider range of possibilities for addressing the issues ordinarily thought about in a more limited and constricted manner. Boundaries may be used creatively and constructively. They may also diminish freedom to see things in their richness and complexity. Two main attitudes emerged historically. One was to build systems of thought substantiated in communal practices that encourage clarification and rigorous maintenance of these boundaries. An alternative position, less popular in Western thought, but gaining strength since the Second World War, has favored disrupting and destroying boundaries which are so often oppressive and limiting of perception and possibilities. Those who have seen the value in boundaries have advocated for and actualized artificially intentional means by which boundaries are made firmer and clearer, while those who have seen the negative effects of boundaries have attempted their systematic erasure. Since boundaries consist of demarcations in the spaces of which they are part, constituting, as it were, surfaces in these spaces, the emergence of a discipline that studies the nature of surfaces in greater depth and detail would be expected to help to open our eyes to salient features of boundaries. Fractal geometry is such a discipline. The study of fi'actal geometry yields several findings that quickly enrich and expand our appreciation of boundaries. First, and most obviously, it shows that boundaries themselves can be rich and complex, and hence a worthwhile object of study in themselves, not mere footnotes to the more important investigation of what it is that they separate. It points to a deep distinction between artificial boundaries that arise from intentional efforts to create separations, and natural boundaries arising, spontaneously as it were, from ongoing processes within a system. The former are likely to have a simpler, more linear structure, while the latter are more likely to have an emergent, nonlinear structure. Fractal boundaries are likely to be permeable to transmissions across them of certain types of materials from the spaces they separate as an intrinsic element of the fi'actal boundary. Oxygen and other gases pass between the boundary separating lungs from the blood stream. Similarly, While many traditional psychoanalysts believe they should not explicitly

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speak about their feeling toward analysands, the prosody of their speech passed through this boundary and often communicates these feelings far more effectively than any explicit statement could. In contrast, artificial boundaries tend to be intrinsically impermeable to almost everything except, insofar as they fail in their boundary functions or at points of specific permeability “doors,” as it were, built into the boundary. These doors, if they are not to function simply as holes in the boundary, must have an elaborate, complex structure that differentiates what may be allowed to pass through them from what may not. These abstract ideas regarding boundaries and fractal surfaces are often difficult to recognize as they are manifest in thought and emotion because psychologists are not accustomed to recognizing and attending to them in the many other contexts in which they occur. The purpose of this chapter has been to take the reader on a brief, informal tour of the world of fractal boundaries, emphasizing the subjective experiences of them in the hope that you will recognize fractal boundaries as you encounter them in psychological work of all types. It is also hoped that the reader will see that the appreciation of the fractal nature of certain boundaries may provide an alternative to the common psychological ideas that boundaries must either be strengthened or eliminated, and instead suggest that boundaries may appropriately be of a qualitatively different sort, functioning far more effectively within a system than either the rigid boundaries so commonly conceptualized or the effective absence of boundaries which is sometimes confused with freedom. Note that the verbal walk on which I invited you to join me was not only filled with fractal objects but was, in itself, fractal with descriptions interweaving as though they were different sentences. Fractals are everywhere, beautiful and interesting, and their contemplations in their various forms can be richly informative and useful—especially when it comes to boundaries.

References Alberti, LB. (2013). On painting. Cambridge, England: Cambridge University Press.

Caligor, E., Kemberg, O.F., & Clarkin, J.F. (2018). Psychoalvnamic therapy for personality pathology: Treating self anaT interpersonal functioning. Washington, DC: American Psychiatric.

Celenza, A. (2007). Sexual boundary violations: Theraeputic, supervisory and academic Contexts. Lanham, MD: Aronson.

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Chiland, C. (2009). Some thoughts on transsexualism, transvestism, transgender and identification. In C. Ambrosio (Ed.), Transvestism,

transsexualism in the psychoanalytic dimension (pp. 41-54). London, England: Karnac.

Cohler, B., & Galatzer-Levy,R (2000). The course of gay and lesbian lives: Social and psychoanalytic perspectives. Chicago, IL: University of Chicago Press. Dimen, M. (2016). Rotten apples and ambivalence: Sexual boundary violations through a psychocultural lens. Journal of the American

PsychoanalyticAssociation, 64(2), 361-373. Einstein, A. (1945). The mathematician‘ mind, Testimonial. In J. Harad

(Ed.), The mathematician’s mind: An essay on the psychology of invention in the mathematical field (pp. 142-143). Princeton, NJ: Princeton University Press.

Federn, P. (1926). Some variations in ego—feeling. International Journal of Psychoanalysis, 7, 434-444. Fine, C. (2014). His brain, her brain? Science, 346(6212), 915—916. Freud, S. (2001). Group psychology and the analysis of the ego. In J.

Strachey (Ed.) The standard edition of the complete psychological works of Sigmund Freud (Volume XVIII) (pp. 65-144). London, England: Vintage. (Original work published 1920) Gabbard, GO. (2017). Sexual boundary violations in psychoanalysis: A 30-

year retrospective. Psychoanalytic Psychology, 34(2), 151-156. Galatzer-Levy, R. (in press). Fractal boundaries. Journal of the American Psychoanalytic Association. Garber, M. (1992). Vested interests: Cross-dressing and cultural anxiety. New York, NY: Routledge.

Gombrich, E. (1950). The story of art. New York, NY: Phaidon. Gutheil, T. & Brodsky, A. (2008). Preventing boundary violations in clinical practice. New York, NY: Guilford. Harris, A.E. (2011). Gender as a stranger attractor: Discussion of the

transgender symposium. Psychoanalytic Dialogues, 21(2) 230-238. Isaacson, W. (2017). Leonardo Da Vinci. New York, NY: Simon & Schuster.

James, W. (1902). Varieties of religious experience: A study in human nature. London, England: Routledge.

Le Bon, G. (1896). Psychologie desfoules. Paris, France: Alcan. Levey, J., & Levey, E. (2015). Mindfulness, meditation, and mind fitness. San Francisco, CA: Canari Press.

Makari, G]. (2008). Revolution in mind. New York, NY: Harper.

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Mandelbrot, B. (1977). Thefractal geometry of nature. San Francisco, CA: Freeman.

—. [MIT Video Productions] (2019, January 17). Benoit B. Mandelbrot, MIT 2001 - Fractals in Science, Engineering and Finance (Roughness and Beauty) [Video file]. Retrieved from: https:llwwwyoutube.com/watch?v=ock9Gk_aqw4

Peitgen, H. & Richter, P. (1986). The beauty offractals, images of complex dynamical systems. Berlin, Germany: Springer.

Reich, W. (1949). Character analysis. New York, NY: Farrar, Straus & Giroux.

Rovelli, C. (2018) The order of time. New York, NY: Riversice. Saketopoulou, A. (2014). Mourning the body as bedrock: Developmental considerations in treating transsexual patients analytically. Journal of

the American Psychoanalytic Asssociation, 62(5), 773—805. Schafer, R. (1976). A new language for psychoanalysis. New Haven, CT: Yale University Press.

Thom, R. (1975). Structural stability and morphogenesis: An outline of a general theory of models. Reading, MA: Benjamin. Watts, D. (2011). Everything is obvious, once you know the answer. New York, NY: Crown Business.

West, B. (2006). Where medicine went wrong: Rediscovering the path to complexity. Singapore, Malaysia: World Scientific.

Woolf, V. (1933). Orlando: A biography. London, England: Leonard & Virginia Woolf at the Hogarth Press.

Wright, R. (2017). Why Buddhism is true: The science and philosopy of meditation and englightenment. New York, NY: Simon & Schuster.

CHAPTER TWELVE HOW FRACTALS HELP Us SEE AND UNDERSTAND THE WORLD1 LARRY S. LIEBOVITCH2

Introduction George Carlin noted that we say there was hail the size of golf-balls or hail the size of baseballs, but why, he complained, is not hail ever the size of hail? We face a similar problem in describing living things and especially their cognitive functions. We do not really understand the brain or how it works, and so we don't know how to describe it. Instead we use metaphors to describe it. What is really odd is that the only metaphors that we can think to apply to biology and cognitive function are based on the machines that we build. Is that because those machines are really the only things we understand because we built them ourselves? The ancient Greeks, experienced in building machines with pipes, fluids, and steam, thought of the brain as a set of interconnected and functioning pipes. Later we thought of the brain as a telephone switching network. Then the brain was a computer. Now it is an artificial neural network. But the brain is not pipes, or a telephone network, or a computer, or an artificial neural network. It is a brain. It functions like a brain, not like pipes, or a telephone network, or a computer, or an artificial neural network.

1 A version of this chapter was published in the International Journal ofTranspersonal Studies, 38(2). 2 Professor of Physics and Psychology, Queens College, City University of New York, Physics Program, The Graduate Center, City University of New York, New York; Adjunct Senior Research Scholar, Advanced Consortium for Cooperation, Conflict; and Complexity (AC4) at The Earth Institute at Columbia University. Email: larry.][email protected]

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Fractals This struggle to find the right metaphor to capture transpersonal psychology is the subject of Terry Marks-Tarlow’s first chapter in this book. Her novel proposal is that a useful metaphor for transpersonal psychology is not a machine at all, but an area in mathematics called fractal mathematics. Fractals have some very interesting properties (Barnsley, 1998; Brown & Liebovitch, 2010; Liebovitch, 1988; Liebovitch & Scheurle, 2000; Mandelbrot, 1977). A fractal has an ever-larger number of ever-smaller pieces that resemble the whole object, like the ever-smaller branches of a tree. This property is called rely-similarity. The tree has only a few large branches, but more medium-sized branches, and a very larger number of small branches. There is no such thing as the "average" size of a branch. The closer you look, the finer the size of the branches you see. There is a relationship between the number of branches of different sizes. This property is called a scaling relationship. Because there is no average size of a branch, the statistics of fractals are very different than those bell curves

they taught you in school. A fractal distribution of measurements has a much higher probability of a value that is very far from the average than that of the bell curve. Those values are not "outliers," as they are an essential part of the fractal distribution of values. Fractal mathematics has been used to provide amore realistic description of the shapes of things in the natural world, such as rivers and mountains and clouds. It has proved useful in providing a detailed mathematical analysis and an understanding of the properties of physical, biological, and social systems. These include such things as the fluctuating-light intensity from very powerful, very distant-astronomical objects call quasars (Press, 1978), extraneous voltage noise generated by electronic chips (Milotti, 2002), the random walk of molecules moving through a material (Ben— Avraham & Havlin, 2000), the patterns formed by injecting a light fluid (such as water) into a heavy fluid (such as thick oil) (Maloy, Feder, & Iossang, 1985), the statistical distributions of the populations of cities, number of casualties in wars, frequency of earthquakes, and your delay times in replying to emails (Bak, 1996; Barabasi, 2005; Mandelbrot, 1977; Richardson, 1960).

Fractals: Proteins, prices, behaviours, heart arrhythmias In my own research, understanding and appreciating these properties of fractals has helped us to better analyse and understand phenomena in

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different systems. Ion channels are proteins in the membranes of all cells that let ions like sodium, potassium, and chloride into and out of cells. These proteins act like tiny switches, ever changing between shapes that are either open or closed to flow of those ions. Using the technique called patch clamp, we could measure the tiny electrical currents though an individual ion channel. When we measured the current at ever shorter time resolutions, we found ever shorter open-time durations. We could use the scaling relationship between those short and long durations to provide information on the dynamics of how the protein wobbles and twists itself between its conformation shapes that are open or closed to the flow of ions (Liebovitch, 1989; Liebovitch, F ischbarg, & Koniarek, 1987; Liebovitch & Krekora, 2002; Liebovitch, Scheurle, Rusek, & Zachowski, 2001; Liebovitch & Sullivan, 1987). In another project, we studied the fluctuation of commodity prices in ancient Babylon recorded in cuneiform on clay tablets. We found that there are fractal fluctuations in those prices at many different time scales, just as there are many different spatial scales in the sizes of the branches of a tree. As similar fiactal fluctuations are present in modern financial data, we could confirm the market nature of this economy over 2,000 years ago (Romero, Ma, Liebovitch, Brown, & Ivanov, 2010). We also found that fractal statistics were helpful in analysing patterns of human behaviour. In a social psychology experiment, two people with different opinions on an emotionally charged topic, such as abortion, affinnative action, death penalty, or euthanasia, were asked to talk for 20 minutes and see if they could prepare a joint statement. During these difficult conversations each person spent time trying to understand the other‘s position, which is called prosocial behaviour, or time pushing their own views, which is called proself behaviour. Typically, they spent some time in one of these behavioural states and then switched to the other, and then back again. In a way, this is analogous to the ion-channel proteins being either open or closed to the flow of ions and switching between those two states. Analysing the audio recordings of these conversations, we determined the fractal—scaling relationship between the times spent in the prosocial and proself states. That scaling relationship revealed that their previous behavioural states influenced the duration of their subsequent behavioural states. The probability to switch out of each behavioural state decreased with the time already spent in that behavioural state (Kurt, Kugler, Coleman, & Liebovitch, 2004).

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Appreciating the properties of fractal statistics also helped us in our studies of the timing of events called atrial tachyarrhythmias that can sometimes lead to heart attacks. In these atrial tachyarrhythmias, the smaller chambers of the heart beat too fast instead of more slowly and smoothly contracting to pump blood out of the heart. In those patients, a small

implanted electrical device, called an implantable cardioverter defibrillator (ICD), senses the problem and sends a strong electrical signal throughout the heart to restore its normal function. That [CD is also a small computer and has a radio transmitter which can play back information to us about the times when it was triggered to restore normal heart function. Fractal statistics, not a bell curve, were a much better fit to the data of the times between the events when the device triggered. The fractal approach provided us a better quantitative measure of how the number of short times between events compared to the number of long times between events (Liebovitch, et al., 1999; Shehadeh, Liebovitch, & Wood, 2002).

Transpersonal psychology Just as fiactal mathematics provides useful descriptions to help us better understand structures and processes in physics, chemistry, biology, and medicine, it can also help us understand the psychology of many different types of human behaviours. This is particularly true for transpersonal psychology which deals with beyond-ego perceptions such as visionary, or transcendent, or out of body experiences. We find it very hard to know how to tie together the inner and outer worlds to make sense of such experiences. Marks-Tarlow (2018) shows how many of those fractal mathematical properties are also present in the fascinating dance of the intrapersonal and interpersonal worlds in transpersonal psychology. Therefore, fractal mathematics can also provide an appropriate epistemology for transpersonal psychology to describe the structure of subj ective experience. She notes that the shadows the fire casts on the cave wall (Plato, c380 BC; see Allen, 2006) from these transpersonal phenomena have some of these properties in

common with fractal mathematics: - Fractals are objects with an infinite repetition of smaller pieces that

reflect the whole, like the interconnections between difierent levels of experience; I

There is no single value for the measurement in a fractal, the values

measured depend on the scale of the measurement, like the relativity of subjective experiences;

How Fractals Help Us See and Understand the World c

Fractal borders between different mathematical solutions of a nonlinear mathematical problem have interpenetrating boundaries, such that this mathematical problem has answers of several different values that lie jumbled so closely together that a tiny change in a parameter of the problem yields a different quantitative value for its

answer, like the connections interpersonal experiences; c

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between

intrapersonal

and

Fractal patterns can be generated by complex feedback loops, like

the complexfeedback loops between inner and outer processes, such as self and other; I

Fractal patterns are represented by statistical distributions that have a much higher likelihood of unanticipated, unlikely events, called "black swans" that happen much more often than would be expected

from the statistics of classical bell curves, like the fiequency and power of peak experiences These and many other examples that Marks-Tarlow presents make a novel and strong case for a fractal metaphor for psychology in general and especially for the extreme events that characterize transpersonal psychology. Is this the best metaphor for transpersonal psychology? Certainly, it is a much more appropriate metaphor, a much closer fit to observed human experience, than simple Euclidean forms, Newtonian one simple cause to one simple effect, and the statistics of the bell curve tightly wound around a single population mean. In its resonance with human experience, this fractal epistemology has much to contribute to how we understand, conceptualize, and reason about human experience. Is it the only metaphor for transpersonal psychology? As the psychologist Abraham Maslow noted, “Ifthe only too] you have is a hammer, you tend to treat everything as if it were a nail.” We need as many different tools as possible, and in that toolbox, Terry Marks-Tarlow has given us another fine and useful tool.

Why fractals? Stepping back from the details of specific systems, we now ask, why do fractals "work" in helping us to better understand so many different things in the physical, biological, and psychological worlds? In his essay, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Eugene Wigner (1960) considers why mathematics works so well as the basis of concepts and theories in physics. After all, in paraphrasing Bertrand

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Russell, Wigner stated mathematics is beautifully cold in its logic, sublimely pure, and capable of a stern perfection such as is found only the greatest art. But art is not science? The world is not pure. Wigner considers why this pure, but beautiful game of mathematics can actually help us understand the highly impure real world. He says that physicists fmd properties in the world that remind them of the properties that they already knew in mathematical examples and so jump to use that mathematics to represent their View of the world. He ends on the "cheerful note" that: The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither

understand nor deserve. We should be grateful for it and hope that it will remain valid in future research. (p. 14) Here, we now ask a much narrower question—what are the deep

mathematical properties of fractals that fortuitously match those same deep properties in the real world that are present in the scientific applications described above? Two properties of fractals stand out here: 1)How they are made; and 2) How they work.

Fractals: Form Fractals generate complex, and beautiful patterns, from quite simple rules. For example, chose any small piece of the fractal, and replace that piece with a smaller copy of the whole fractal. The lesson is that seemingly complex patterns can arise from the repeated application of quite simple rules. Perhaps our focus on seeing and cataloguing and measuring the properties of complex patterns in nature is misdirected. Rather, we should be focused on finding the rules that generate those patterns. The genes in our DNA do not specify the shape and structure of our bones or our heart or our eyes, rather they specify the rules for creating those structures. D‘Arcy Thompson (1961) makes this point clearly in his famous work, On Growth and F arm, where he shows that the shape of bones depends not just on genes, but on an optimization principle, on the interaction of the genes with the world. Bones need to be strong in compression to resist their ends fiom being pushed towards each other. They don't need to be strong in tension, pulling the ends apart, as those forces are well handled by the muscles around the bones. The optimization rule is to add bone where it is in compression and the additional strength is needed, and to remove bone

where it is in tension and there is more material than is needed. This interaction with the world then determines the shape of the bone. There is no blueprint of the shape of the bone hidden in the DNA with a picture of

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how the bones should look, only a rule of how the products of the genes should interact with the environment that then determines their final shape. Rules are much more parsimonious than blueprints. The many details of

all the microscopic 37.2 trillion (3.72 x 1013) cells in your body are contained in the rules specified by the actions of the proteins coded in only about 20,000 genes in your DNA (Pertea & Salzberg, 2010). Similarly, how those cells form into organs is coded by the rules of how proteins in their cell membranes interact with each other. Notice a trend here? The structure of each level of organization is determined by parsimonious rules rather than a detailed map of every piece, and then that level of organization is the starting point to build the next higher-hierarchical level of organization. Philip Morrison (1966) writes: The world is both richly strange and deeply simple . . . Neither gods nor men mold clay freely; rather they form bricks. (n.p.)

This exactly mimics fractals, where simple, iterated rules, generate hierarchical levels of increasing complexity.

Fractals: Function Some additional properties of fiactals make them valuable in solving certain problems in physical, biological, and psychological systems. The scaling relationship, described previously, tells us, how many pieces there are of different sizes. This relationship can often be characterized by a single number, called the fiactal dimension. How the pieces of the fractal are

connected to each other is called the topological dimension. For example, in a line, each point is connected to its two neighbours on either side that corresponds to a topological dimension of 1. But you can make that line so Wiggly, so very Wiggly, that it bends around so much that it nearly covers an area. Then its fractal dimension is nearly 2. Many of the essential properties of fractals, such as self-similarity and the scaling relationship, arise from the fact that the fractal dimension is larger than the topological dimension. That is how smaller pieces of the fractal can look like the Whole (self-similarity) because the larger fractal dimension has the room to include copies of the Whole object. That is how the number of smaller pieces can be related to the number of larger pieces (scaling relationship) because the amount of how much larger the fractal dimension is than the topological dimension determines how many pieces there are of each size. For example, the Koch curve has fractal dimension of about 1.2619 which is larger than its topological dimension of 1, so it is avery, very Wiggly line, much longer

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than a typical l-dimensional line, but still not so Wiggly that it covers a 2dimensional area. A fiactal object with fractal dimension 2.5 would be a surface that is so large, with so many ever smaller, tiny-bent pieces, that it almost fills a 3-dimensional volume, but is still only a surface with topological dimension of 2. What can fractals do with these properties? A fractal structure with dimension greater than 1 but less than 2 can nearly fill an entire area and yet has a very large length (approaching infinity for a mathematical fractal). An excellent design for approximately 1-dimensional roads in an approximately Z-dimensional city plan to reach as many houses as possible (Batty & Longley, 1994) or for 1-dimensional lines of capillaries in the retina in the eye to nourish as many cells as possible (Mainster, 1990) across a 2dimensional area landscape. A fractal structure with dimension greater than 2 but less than 3 fills almost an entire 3-dimensional volume, yet has an

extraordinary surface area (approaching infinity for a mathematical fractal). An excellent design for the surface of a "solid" metal catalyst to touch as many reactant molecules as possible in a chemical reaction (Avnir, 1989) or a lung to have a compact 3-dimensional volume that contains a huge 2dimensional surface to diffuse oxygen in and carbon dioxide out (Bassingthwaighte, Liebovitch, & West, 1994). A fractal has many pieces of many different sizes. Function often depends on size. A radio antenna is good at picking up signals whose wavelength is about the size of the antenna. A fractal antenna is capable of picking up a broad band of many different wavelengths. This makes it very useful in a cell phone because in a small overall size its many pieces of different sizes can receive many different radio bands (see Wikipedia: https:// en.wikipedia. or g/wiki/F ractal_antenna). Fractal structures and therefore their fractal functions span not just many scales in space, but also many scales in time. The proteins that catalyse the chemical reactions that extract energy from your food and build the structures in your organs are not shapes frozen in stone. Movements within these molecules extend fiom nanoseconds to microseconds to seconds (Dewey, 1997). We are not exactly sure why. Perhaps it is to make sure their chemical reactions go forward rather than backward, entrapping the molecules that they interact with in an ever tighter, ever longer lasting, ever stronger grip. The world swirling around (as well as inside of us) presents us with stimuli both very fast and very slow and everything in-between. Biological

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systems need to respond to all these different time scales. That may be why such fractal behaviour in time, extending fiom the very fast to the very slow, is seen in the motions of individual molecules, in the firing patterns of nerve cells sensing sound, in the timing of heartbeats, and in the delays between steps in walking (Bassingthwaighte et a1., 1994; West & Griffin, 2004). The ability to cover many different time scales sets the stage to respond appropriately, either fast or slow, in response to fast or slow stimuli or changes in the environment.

A warning Metaphors reveals some properties of a system, teaches us something new, helps us understand, and drives us to ask new questions. But each metaphor, by what it is not, by what it leaves out, by what it gets wrong, also hides from us things we want and may need to see. In "The Circus Animals” Desertion," written late in his life, William Butler Yeats (1933) laments that his metaphors “took all my love and not those things that they were emblems of” (up)

Conclus ion The most essential property of fractals, that there is an ever-larger number of ever-smaller pieces, establishes a direct link between the very small pieces of a system and the very largest pieces of a system. Similarly in time, it establishes a direct link between what is happening very fast and what is happening very slowly. In both space and time, in both form and function, these properties of fractals resonate with the form and function of physical, biological, psychological, and social systems. Hence fractals provide us a metaphor to better appreciate and understand a wide range of systems. However, we should always remain cautious of the limitations of our metaphor. Our metaphor is not the thing that our metaphor attempts to describe. Hail is actually the size of hail.

References Allen, RE. (2006). Plato: The Republic. New Haven, CT: Yale University Press.

Avnir, E. (1989). The fractal approach to heterogeneous chemistry. New York, NY: John Wiley & Sons.

Bak, P. (1996). How nature works. New York, NY: Springer-Verlag.

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Barabasi, AL. (2005). The origin of bursts and heavy tails in human dynamics. Nature, 435(7039), 207-211. Barnsley, M. (1988). Fractals everywhere. Boston, MA: Academic. Bassingthwaighte, J. B., Liebovitch, L. S., & West, B. J. (1994). Fractal physiology. Oxford, England: Oxford University Press. Batty M., & Longley, P. (1994). Fractal cities. New York, NY: Academic Press.

Ben-Avraham,D., & Havlin, S. (2000). Difl'usion and reactions in fractals and disordered systems. Cambridge, England: Cambridge University Press.

Brown, C.T., & Liebovitch, LS. (2010). Fractal analysis in the social sciences. Thousand Oaks, CA: SAGE.

Dewey, T.G. (1997). Fractals in molecular biophysics. Oxford, England: Oxford University Press. Eveleth, R. (2013). There are 37.2 trillion cells in your body. Smithsonian SmartNews. Retrieved flom: https://www.smithsonianmag.com/smart~ news/there—are— 372—trillion—cells- in—your—body— 494 1473/. Kurt, L., Kugler, K.G., Coleman, P.T., & Liebovitch, LS. (2014). Behavioral and emotional dynamics of two people struggling to reach a consensus about a topic on which they disagree. PLOS ONE, 9(1):

e84608. Liebovitch, L. S. (1989). Testing fi'actal and Markov models of ion

channel kinetics. Biophysical Journal, 55, 373-377. —. (1998). Fractals and chaos: Simplified for the life sciences. Oxford, England: Oxford University Press. Liebovitch, L.S., Fischbarg, J., & Koniarek, JP. (1987). [on channel kinetics: A model based on fractal scaling rather than multistate Markov

processes.Mathematical Biosciences, 84(1), 37-68. Liebovitch, L.S., & Krekora, P. (2002). The physical basis of ion channel kinetics: The importance of dynamics. In H.E. Layton & A.M.

Weinstein (Eds), Institute for mathematics and its applications, membrane transport and renal physiology (pp. 27-52). New York, NY: Springer-Verlag. Liebovitch, L.S., & Scheurle, D. (2000). Two lessons fiom fractals and

chaos. Complexity, 5(4), 34-43. Liebovitch, L.S., Scheurle, D., Rusek, M., & Zochowski, M. (2001). Fractal

methods to analyze ion channel kinetics. Methods, 24(4), 359-375. Liebovitch, L.S., & Sullivan, J.M. (1987). Fractal analysis of a voltage dependent potassium channel from cultured mouse hippocampal

neurons. Biophysical Journal, 52(6), 979-988.

How Fractals Help Us See and Understand the World

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Liebovitch, L.S., Todorov, A.T., Zochowski, M., Scheurle, D., Colgin, L., & Bernstein, RC. (1999). Nonlinear properties of Wood, M.A.,

cardiac rhythm abnormalities. Physical Review E, 59(3), 3312-3319. Mainster, MA. (1990). The fractal properties of retinal vessels: Embryological and clinical implications. Eye (Lond), 4(1), 235-241. Maloy, K.J., Feder, J., & Jossang, T. (1985). Viscous fingering fractals in porous media. Physical Review Letters, 55(24), 268 8-26691.

Mandelbrot, BB. (1977). The fractal geometry of nature. San Francisco, CA: W. H. Freeman. Marks-Tarlow, T. (in press). A fractal epistemology for transpersonal

psychology. International Journal of Transpersonal Studies, 38(2), n.p. Milotti, E. (2002). 1/f noise: A pedagogical review. arXiv preprint physics/0204033. Morrison, P. (1966). The modularity of knowing. In G. Kepes (Ed.),

Module, proportion, symmetry, rhythm (pp. 1-19), New York, NY: Braziller. Retrieved from http://www.noteaccess.com/RELATIONSHIPS/ModularityK.htm Pertea, M. & Salzberg, S. (2010). Between a chicken and a grape: Estimating the number of human genes. Genome Biology, 1 I (5), 206. Press, W.H. (1978). Flicker noises in astronomy and elsewhere. Comments

on Modern Physics, Part C. Comments on Astrophysics, 7, 103-119. Richardson, LP. (1960). Statistics of deadly quarrels. Pittsburgh, PA: Boxwood. Romero, N.E., Ma, Q.D.Y., Liebovitch, L.S., Brown, C.T., & Ivanov, P. (2010). Correlated walks down the Babylonian markets. Europhysics

Letters, 90(1), 18004. Shehadeh, L.A., Liebovitch, L. S ., & Wood, MA. (2002). Temporal patterns of atrial arrhythmia recurrences in patients with implantable defibrillators: Implications for assessing antiarrhythmic therapies."

Journal of Cardiac Electrophysiology, 13(4), 303-309. Thompson, D.W. (J.T. Bonner, Ed.)(1961). On growth and form. Cambridge, England: Cambridge University Press.

West, B.J., & Griffin, LA. (2004). Bioaynamics: Why the wirewalker doesn ’tfall. Hoboken, NJ: Wiley-Liss. Wigner, E. (1960). The unreasonable effectiveness of mathematics in the

natural sciences. Communications in Pure and Applied Mathematics, 13, 1-14. Yeats, W.B. (1933). The circus animals’ desertion. In R. J. Finneran (Ed),

Poems of W. B. Yeats: A new edition (n.p.). New York, NY: Macmillan. Retrieved from https://www.poetryfoundation.org/poems/43299/the— circus-animals-desertion

CHAPTER THIRTEEN

How THE CEREBELLUM AND CEREBRAL CORTEX COLLABORATE T0 COMPOSE FRACTAL PATTERNS UNDERLYING TRANSPERSONAL EXPERIENCE1 LARRY VANDERVERT2

The contention made in this chapter is as follows: A fractal epistemology for transpersonal events clearly comports with the fractal evolution of the tightly collaborative fractal relationship between the hum an cerebellum and the cerebral cortex. The chapter outlines the structure, function, connectivity, and evolution of the cerebellum before turning to fractal dynamics.

A major cognitive-neuroscience breakthrough Three decades ago, Leiner, Leiner, and Dow (1986, 1989) published two landmark articles on how evolution has made human thought processes uniquely fast, complex, and efficient. Citing the fact that the small cerebellum at the back of the brain (see Figure 13-1) had increased in size three- to four-fold in the last million years of evolution, they proposed that the connections between the cerebellum and the cerebral cortex (cerebrocerebellar connections) had evolved not only to increase the speed and skill of bodily movements but also the speed and skill of mental processes:

1 A version of this chapter was published in the International Journal ofTranspersonaI Studies, 38(2). 2 American Nonlinear Systems E-mail: [email protected]

How the C ereb ellum and Cerebral C ortex Collab orate to C amp ose Fractal Patterns Underlying Transpersonal Experience

3‘? 3

Because the cerebellum is traditionally regarded as a motor mechanism (Holmes, 193 9), these cerebrocerebellar interactions are usually thought to confer [only] a motor benefit on humans, such as increased dexterity of the hand (Tilney, 192.8). But..a detailed examination of cerebellar circuitry suggests that its phylogenetically newest parts may serve as a fast thfimatton—processmg aegi'wrct of the association cortex and could asstst this cortex in the performance of a variety ofmam'ptdattve skits, including the skill that is characteristic of anthropor'd apes and humans: the skfiifid manptdatton ofrdeas [italics added] (Leiner, et al., 1986, p. 444). Leiner, et al.’s (1986, 1989) two articles were the beginning o f a sudden and unexpected breakthrough in the cognitive neurosciences. Leiner, et al.’s watershed proposal spurred a huge amount of brain imaging research on the cognitive fimctions of the cerebellum and the cerebellum’s massive twoway connections throughout the cerebral cortex—the 40 million nerve tracts (bundles of axons) between the cerebellum and the cerebral cortex are the most numerous in the brain. This is 20 times more than the two million that connect the eyes with the visual cortex (Leiner, Leiner, 3: Dow, 1993; Ramnani et al., 2006). Moreover, the human cerebellum contains 69 billion neurons compared to a mere16 billion neurons in the cerebral cortex (see Figure 13-1)!

(:n (69 Ewillion neurons) Figure 13-1. Illustration of the cerebellum in relation. to the cerebral cortex along with their respective neuron counts. The neuron counts are based on Lent, Azev edo, Andrade-Moraes and Pinto (2012).

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The cerebellum and the rise of Homo sapiens During the last million years the lateral cognitive areas of the cerebellum seen in Figure 13—2 expanded greatly. (Note in Figure 13—2 that the cerebellum is mapped onto both the motor-sensory areas of the cerebral cortex, and its cognitive areas as well). Through the forty million nerve tracts mentioned, these cognitive areas of the cerebellum are connected richly with language, mathematics, working memory, and planning areas of the cerebral cortex (Balsters, Whelan, Robertson & Ramnani, 2013', Hayter, Langdon, & Ramnani, 2007', Marvel & Desmond, 2010a, 2010b, 2012', Vandervert, 2015, 2016a, 2017a, 2017b).

The million years of cerebro-cerebellar evolution saw the rise of Homo sapiens and the origins and rise of culture (Vandervert, 2016a). Within the context of this million years of evolution, the adaptive success of Homo sapiens is seen not so much about survival in cerebral cortex-driven pitched battles of the moment, as the product of thousands of generations of repetitive and thus cerebellum-driven, cognitive-emotional refinements toward prediction, optimization, and autom aticity (Vandervert, 2016a, 2017a, 2017b). These refinements are produced predominantly by computations in the 69 billion neurons of the cerebellum. They are experienced not only in the autom aticity of patterns of culture, as proposed by Vandervert (2016a), but also in sudden, new blendings of experience

(experienced as insight) toward optimized cognition and feeling in creative moments (Higuchi, Toda & Kawato, 2007; Ito (2008); Vandervert, 2015;

Vandervert, Schimpf & Liu, 2007). As culture develops, these refinements are progressively shared interpersonally. In this way, positive feedback loops of transpersonally inspired innovation (the sudden “flow” of insight) can leap-frog forward to be rapidly, often endlessly, further refined. There is something important to note with respect to the motor homunculi (motor body maps) in Figure 13-2. Just as these two homunculi (top andbottom) are completely mapped onto the motor-sensory homunculi in the cerebral cortex, so too are the cognitive functions of the lateral areas of the cerebellum also completely mapped onto areas of the cerebral cortex. For example, cognitive functions of the cerebellum are mapped into the

prefrontal, posterior parietal and temporal areas of the cerebral cortex (Sokolov, Miall, & Ivry, 2017; Strick, Dum & Fiez, 2009). However, they

are mapped in a much more complex fashion, depending on the experiential history of the individual.

How the Cerebellum and Cerebral Cortex Collaborate to Compose Fractal Patterns Underlying Transpersonal Experience

Anterior lobe

"Motor"

Primary

fissure

375

l

Primary fissure

i Posterior lobe

"Cognition" Lateral

_

"Cognition" Lateral

,

R ions activated by

~ 3 ”Motor": W

3

@ rigeiiit hand movements

Left hemisphere M d i m » Right hemisphere Figure 13-2. Flattened View of cerebellar surface illustrating that the anterior lobe

and intermediate parts of the posterior lobe are related to the prediction, anticipation, and streamlining of “motor and somatosensory functions,” whereas the lateral posterior cerebellum is related to the prediction, anticipation, and streamlining of “cognitive functions.”

To orient properly to the anterior/posterior axis of the

flattened view, the viewer should keep in mind that anterior/posterior refer to what is actually a substantially convex cerebellar surface (see smaller drawing to left). Arrows at (a) indicate difference between “motor” (note the maps of motor areas

related to body areas at top and bottom) and “cognition.” Arrows at (b) indicate modularity within the lateral posterior cerebellum for, for example, two different cognitive functions—many more cognitive areas have been found, for example, language and working memory. These motor and cognitive areas are often blended to produce new, creative forms of experience (Vandervert, 2015). From Imamizu, et

al., 2003, pp. 5461—5466). Copyright 2003, National Academy of Sciences, U.S.A. (Reprinted with permission)

To illustrate the analogous point-by-point parallel mapping of cognitive fimctions between the cerebellum and cerebral cortex, it is necessary to briefly discuss the cerebellum’s dentate nucleus, a little-known part of the cerebellum. The dentate nucleus provides massive output from the cerebellum to both motor and cognitive areas of the cerebral cortex. The cognitive portion of this area evolved directly fiom the motor portion (Bostan, Dum, & Strick, 2013; Leiner, et al., 1986, 1989). A huge number of nerve tracts going from the cerebellurn’s dentate nucleus to the cerebral cortex includes those going to the parietal and prefrontal areas for planning, language, and associated high-level functions of working memory (Bostan, et al., 2013; Leiner, et al., 1989; Marvel & Desmond, 2010a, 2010b, 2012). Based on extensive research studies, Bostan, et a1. (2013) put it this way: “It is likely the signal from the dentate

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to the prefrontal and posterior parietal areas of the cortex [working m em ory, executive functions and rule-based leaming] is as important to their function as the signal the nucleus sends to motor areas of the cerebral cortex” (p. 3). In sum, the cerebellum appears to play a predominant role in the refinement and blending of virtually all repeated movements, thoughts, and emotions (Adamaszek, D’Agata, Ferrucci, et al. 2016', Bostan, et al., 2013; Ito, 1997,

2008; Leiner, et al., 1986, 1989).

“Flow” made simple: How cerebral cortex—to—cerebellum— to—cerebral cortex processing produces the highest levels of skill and thought Cerebral cortex-to-cerebellum -to-cortex processing not only results in everyday skill learning and thinking, but also produces the highest levels of skill and thought. It appears that the only way we get better at highly skilled activities, such as sports, playing musical instruments, and even holding complex ideas in working In em ory, is through repeated m otor and cognitive practice that results in the creation of internal models in the cerebellum (Hayter, et al., 2007', Ito, 1997, 2008; Leiner, et al., 1986, 1989). Skills

practice and repeated thoughts must engage a healthy cerebellum, otherwise several skill, thought, language and personality-related disabilities can develop (Schmahmann, 2004', further discussed in the Conclusions section of this chapter). These refinements in skills and thought occur through the cerebellum, because cerebellar internal models unconsciously drive automaticity and error-correction toward optimization in skills and thought which is then sent to and consciously experienced in the cerebral cortex. As a result, the brain sequence that produces both higher and creative levels of skill and thought may look like this: cerebral cortex—to-cerebellum-to-cerebral cortex. The word cerebellum is italicized, because it is predominantly inthe cerebellum where refinements toward optimization and automatic flow of skill and thought execution are modeled. This occurs in what are called cerebellar internal models (internal models are models learned by the cerebellum of internal processes in the brain). I suggest that the aspect of transpersonal experience known as “flow” (Csikszentmihalyi, 1975),

whether of skill or creative insight, especially appears to be the product of cerebellar inverse dynamics models, that is cerebellar internal models that,

through practice, learn to automatically control movement and thought. Ito (1993) contrasted cerebellar dynamics models versus cerebellar inverse

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dynamics models. Both assist the cerebral cortex in related, but different, ways; each is controlled by a different part of the cerebellum: A dynamics model built into the paravermis-interpositus division of the cerebellum enables the motor cortex [or other areas of the cerebral cortex] to direct limb movement [or nonmotor functions] without peripheral feedback. By contrast, an inverse dynamics model built into the hemispheredentatus division of the cerebellum replaces the controller task of the motor cortex [and/or nonmotor areas of the cerebral cortex], rendering the control more automatic and less conscious. Hence, after repeated exercise, one

becomes able to move [or think or calculate] quickly, precisely and smoothly without conscious thought. (Ito, 1993, p. 449)

Please note that the point of Ito’s above 1993 article was to argue extension of the leaming of cerebellar dynamics and inverse dynamics models to mental functions. Ito’s arguments for unconscious automaticity in working memory have also been supported by the research of Hayter, et a1. (2007).

Examples of cerebellar inverse dynamics internal models Many flowing, creative advances in skill and thought, plus peak experiences

in sports, music, mathematics, working memory, and in child prodigies, activate areas in the cerebral cortex. This has led researchers such as Dehaene (2001) and Dehaene et al. (1999) in mathematics, and Stout & Hecht (2017) regarding the origin of culture, to be misled by assuming that such phenomena are completely the province of the cerebral cortex. But these processes are not functions of the cerebral cortex alone. A close look at cerebro-cerebellar processing indicates that flowing, high levels of skill and thought also strongly activate the cerebellum. In fact, my research (Vandervert, 2017b, 2018) indicates that flowing, creative advances in movement and thought result more from inverse dynamics models that get refined in the cerebellum and then sent back to the cerebral cortex, where they “play through” in an automatic fashion. This circular sequence allows for the unconscious orchestration of skill, high level thinking, and creativity (Hayter, et al., 2007', Ito, 2008', Vandervert, 2015, Vandervert, et al., 2007). (See Ito, 2008, for a detailed neural

explanation of how cerebral—cortex—to—cerebellum-to—cerebral cortex automaticity or “play through” takes place). Before moving on, it is important to note that the cerebro-cerebellar approach does r101r necessarily conflict with mathematics or culture models that focus on brain functions associated with the cerebral cortex. Rather, the cerebro-cerebellar approach

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brings to bear additional brain mechanisms that provide more detailed and more comprehensive explanations. Vandervert (2016a) proposed that this same cerebral cortex—tocerebellum —to—cortex unconscious automaticity and error-correction toward optimization process goes beyond individuals getting better at physical performance and complex thinking. It appears that this same sequence also drove in phylogeny (evolution) and drives in ontogeny (development) the origin and advancement of unconscious, shared, social behavior (Van Overwalle & Marién, 2016) as well as broader belief systems which undergird culture, including its technology and its rituals. Overall, it is argued that it is cerebral cortex-to—cerebellum -to-cortex processing that may have actually driven the emergence of the sapience of Homo sapiens (Vandervert, 2018).

The evidence for a fractal relationship between the cerebellum and the cerebral cortex Supportive evidence for the contention of a tightly collaborative fractal (self-similar) relationship between the hum an cerebellum and the cerebral

cortex comes from a strong research history of evidence on how fractal processing (automated self-similar sequences of movement and thought) in the cerebellum (which occurs below the level of conscious awareness) is involved in the constant optimization and automation of movement and cognitive—emotional processing. Automation is produced by the computation of constantly optimized or streamlined intemal models in the cerebellum (recall from above that cerebellar intemal models are models learned and stored in the cerebellum of what is going on inside the rest of the brain) (Hayter, et a l , 2007, Ito, 2008). Cerebellum-induced automation is in no

way robotic; rather it is an integral part of rapid and creative working memory as seen in chess masters, sports superstars (especially in their

signature moves), concert pianists, and so forth, all of whom retain the capacity to freely improvise. The whole evolutionary adaptive point of these

cerebellar intemal models is to constantly error-correct what the cerebral cortex is doing so that it gets faster, better, more automatic, and innovative at whatever it does—see Vandervert (2015) and Vandervert, et a1. (2007) for the prominent role of the cerebellum in creativity.

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The cerebellum and creativity It may seem that autom aticity and creativity would be contradictory processes, but they have been shown to be two intermingled cerebellar strategies toward goal optimization. Vandervert, Schimpf and Liu (2007) first proposed the cerebellum as the predominant source of creativity. Their proposal was based on Imam izu et al.’5 (2007) findings that the cerebellum is critically involved in the error-correction of blended internal models. Im am izu and colleagues found that during repetition (practice), trial blends of skills and thought occur in the cerebral cortex. These blends are then sent to the cerebellum where they are error-corrected and optimized for speed and appropriateness to goals. World-renowned cerebellum expert Masao Ito (2007, 2008) expressed agreement with Vandervert et al.’s proposed predominant role of the cerebellum in the unconscious eiror-correction and optimization of blends leading to sudden bursts of creative insight. I contend that the more optimized, automated and blended patterns of behavior and cognition-emotion come to underlie the development of all manner of Maslow’s (1971) “the farther reaches of human nature” and Csikszentmihalyi’s (1975) “flow.”

Strong evidence that the cerebellum optimizes and automates as a flats-tally structured sequence generator For background on the cerebellum as a fractal sequence generator of optimization-automaticity, see Anderson (2000); Pellionisz, Graham, Pellionisz, and Perez (2013); Rankin, Pink, and Large (2014); and Schm ahm ann, Anderson, Newton, and Ellis (2001). In general, Pellionisz et

al. (2013) convincingly argued that, “The cerebellum serves as the best platform for unification of neuroscience and genomics. The matrix of massively parallel neural nets of fractal Purkinje brain cells explains the sensorimotor,

multidimensional

non-Euclidean

coordination

by

the

cerebellum acting as a space-time metric tensor” (p. 1381). For a general corroborative fractal analysis of the larger, overall evolutionary branching of species, see Nottale, Chaline, and Grou, 2002).

These and additional sources on the fractal anatomy and functions of the cerebellum are described in more detail next, where I note (1) how the cerebellum learns and makes use of self-similar internal models to optimize and automate prediction of future states of cognitive-emotional affairs, and (2) how, specifically, transpersonal states of peak experiences (Maslow,

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1971) and flow (Csikszentmihalyi, 1975) can be seen as products of cerebellar optimization and automaticity. We now turn to a discussion of fractal sequence detection in the cerebellum.

How the cerebellum optimizes and automates the prediction of future cognitive—emotional states In m y own research on the mechanisms behind the development of

movement and cognitive-emotional processes, I have described how the cerebellum’s detection of sequences in movement and thought, through participation in working memory, is behind the development of high-level genius, child prodigies, culture, and mathematics (Vandervert, 2015, 2016a, 2016b, 2017a,2017b, 2018). Within these several contexts, I have proposed

that sequence detection in the cerebellum is the key brain mechanism by which the brain learns to optimize and automate the prediction and anticipation of what is coming next before it occurs. The cerebellar mechanism of sequence detection toward optimized and automated prediction (through constant error-correction) has been described in very similar ways (but independently) by leading cerebellum researchers (Akshoomoff, Courchesne, & Townsend, 1997', Ito, 1997, 2008', Leggio &

Molinari, 2015) over the last 20 years. The highly adaptive mechanism of cerebellar sequence detection and pattem form ation/recognition provides progressively faster, more consistent and more appropriate behavioral and mental prediction, anticipation, and error—corrected cerebellar internal models that are automatically sent to appropriate areas of the cerebral cortex. With extended experience or practice these cerebellar internal models acquire the form of inverse dynamics models as described earlier. Because I believe there is strong emerging evidence that this sequence detection toward prediction in the cerebellum may consist of fractal prediction trcy'ectories, I provide the earlier team’s (Akshoomoff, et al., 1997) description of cerebellar sequence detection in some detail. Before providing this evidence, it is important to understand two points: First, with each repeated iteration of any movement or thought-em otion, a new round

of cerebellar sequence detection occurs below the level of conscious awareness. Second, this cerebellar sequence detection provides the constantly error-corrected predictive power behind cerebellar internal models that are sent to the cerebral cortex toward the optimization/automation of performance. (Recall that cerebellar internal models are so named because they are models of the internal world of processes going on in the cerebral cortex).

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In brief, with each repeated iteration (or practice) toward achieving a particular goal, the cerebellum detects system sequences and uses them to predict errors in attention, working memory (thought), and movement. The cerebellum then corrects or adjusts these movem ent/mental-emotional systems toward achieving the required skill or desired performance level of that goal. Akshoomoff, et al. (1997) phrased this cerebellar sequence detection and adjustment process as follows: The cerebellum is amaster computational system that adjusts responsiveness in a variety of networks to obtain a prescribed goal. These networks include

those thought to be involved in declarative memory, working memory, attention, arousal, affect, language, speech, homeostasis, and sensory

modulation as well as motor control. This may require the cerebellum to implement a succession of precisely timed and selected changes in the pattern or level of neural activity in these diverse networks. We

hypothesized that the cerebellum does this by encoding (“learning”) temporally ordered sequences of multi-dimensional information about external and internal events (effector, sensory, affective, mental, autonomic)

[these are cerebellar “internal models”], and, as similar sequences of external and internal events unfold, they elicit a readout of the full sequence in advance of the real-time events. This readout is sent to and alters, in advance [italics added], the state of each motor, sensory, autonomic, attentional, memory, or affective system which, according to the previous “learning” of this sequence, will soon be actively involved in the current real—time events. So, in contrast to conscious, longer time—scale anticipatory processes mediated by cerebral systems, output ofthe cerebellum provides moment-to-moment,

unconscious,

very short

time-scale,

anticipatory

information [italics added] (p. 592-593).

Ito (1997, 2008) has shown how such repetitious, practice-driven processes produce adaptive, cerebellar, microcomplex, circuits which constantly error-correct toward the regulation of “the speed, consistency, and appropriateness” (1997, p. 486) of all motor and mental-emotional processes. In other words, Ito has shown how the cerebellum achieves the regulation of optimization and skillful automation toward the achievement of any and all goals! Following Akshoomoff, Courchesne and Townsend’s (1997) and Ito’s (1997, 2008) arguments, in each practice session during

learning, for example, to play the piano, shooting baskets, or even finetuning drafts of a novel or scientific manuscript, the cerebellum anticipates errors toward the overall goal(s) involved and adjusts each outcome accordingly. Subsequently, as Vandervert (2016a, 2016b) proposed in the case of child prodigies, the entire piano piece, for example, is played rapidly and Without error, as cerebellar internal models are fed forward to

motor/cognitive-emotional areas of the cerebral cortex. Meanwhile, at the

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same time, the pianist may carry on a conversation of any sort, paying little attention to what his or her fingers are.

Does cerebellar sequence detection toward optimizationfautomation occur within an overall fractal structure? It is possible that the temporally ordered sequences that the cerebellum encodes (Akshoom off, et al., 1997) have a fractal structure, although this is not definitively known presently. Toward this regard, Pellionisz, et al.’s (2013) theoretical modeling of the fractal computational structure of the cerebellum concluded in part that, “coordination by the cerebellum is to be characterized by generalized coordinates as in non-Euclidean tensor and fractal geometry” (p. 1406). Moreover, Anderson (2000) and Schmah— mann, et a1. (2001) argued that the cerebellum provided an overall integrative framework for conscious, emotion, and cognitive processes. Rankin, et a1. (2014) found the following: “[our] results demonstrate that participants use fractal temporal structure to predict tempo fluctuations and temporal structure alone is sufficient to anticipate changes in tempo” (p. 6). Since fractal temporal structure in music is modeled in the cerebellum (Rauschecker, 2014), it is reasonable to hypothesize that cerebellar sequence detection may indeed use fractal structure in modeling the

prediction and anticipation of future events in many other (or all) repetitive learning and performance regimes. Examples of such other learning regimes would include, for example, learning and excelling in the execution of skilled expert piano performance, sports skills, chess, and so forth (Vandervert, 2016b).

Are the prediction trajectories in the cerebellum fractal trajectories? It is possible that the foregoing cerebellar mechanisms of selective attention, anticipation, and prediction arefractal trajectories. John S. Torday has done considerable work over three decades on developmental physiology and evolutionary physiology. Torday (2016) proposed a detailed fractal model of evolutionary physiology, ranging from the single cell to the hum an brain. He proposed that: “The reason why physiology exhibits holistic, unitary behavior is because it is fractal. That is to say, it is self-similar at every scale, due to the underlying, integrative mechanisms of cellular ontogeny,

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phylogeny and homeostasis” (p. 5). For a directly related, broader discussion of a fractal basis of evolution see Blaisdell, Pottenger, and Torday (2013). It is important to note here that within this model, fractal integrative functions from cell to brain may have been the basis of life’s origin and its adaptive survival. The emergent functions accompanying the evolution of the cerebellum were and continue to be critical to that survival. I return to this important point in a m om ent. Directly within the context of Torday’s (2016) above quote indicating

that physiology is broadly fractal, I propose that fractal sequences are the most appropriate processing fit to the sequence detection mechanism outlined by Akshoom off, et al. (1997). Fractal sequence detection in brain processes is also strongly supported by Anderson and Mandell (1996): Consciousness evolving in a fractal world would seem parsimoniously to require the incorporation of fractal structures as well as fractal processes,

and these in turn would be integrated into sensory systems, recognition, memory, and adaptive behaviors. Fractal l/f—power spectra have been observed at fundamental levels of processing in the major auditory pathways, olfactory and visual systems and the underlying activity of ion channels which are fundamental to the function of these self-organized nonequilibriurn biological systems... Fractal, i.e., llf, power spectra imply a fundamentally new view of time in biological systems, whose processes have traditionally been interpreted in

the context of linear time. It implies that biological systems manipulate and rescale complex correlational patterns in time. For example, in the dream

state most people experience distorted time; dream events that seem to take hours in reality only require minutes of clock 1ime. Recent findings in cats suggest that fractal l-f-power spectral patterns are widespread in the brain

during REM sleep as well as orienting, arguably two obligatory brain states inhumans for the existence of mind. If these findings canbe extrapolatedto humans, they suggest that broadband I/f states may underlie shifis of attention [italics added] as well as our moment to moment perception of time [italics added]. (pp. 114-115)

According to Akshoom off, et al.’s (1997) earlier-quoted description of sequence detection, attentional control (shifts, focus, and duration) is primarily learned in the cerebellum. Regarding the perception of time, the cerebellum provides multiple timing functions for learning, perception and movement (lvry, 1997). Within this context, it has also been found that the perception of time related to such cognition and movement is a function of

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this overall cerebellar timing (Ivry, 1997; Ivry & Spencer, 2004; Spencer & Ivry, 2013). Given the overall contexts of Pellionisz, et a1. (2013), Torday (2016), and Anderson and Mandell (1996), I propose that the best way to understand the ability of cerebellar sequence detection to predict and anticipate future states is that sequence detection in fact consists of fractal iterations. In other words, both the demonstrated fractal iterations in a broad range of hum an physiology and in cerebellar sequence detection contain the most adaptive information about the future from the past on all time scales.

The origins of the transpersonal experience of “flow” The experience of “flow” in Csikszentmihalyi (1975) was described as beginning with play and as an enjoyable experience that accrues from a focused development of high levels of skill (in for example, mountain climbers, chess masters, composers of music, modern dancers, inveterate

gamblers). Flow is experienced as: ...[T]he state in which action follows upon action according to an internal logic [italics added] which seems to need no conscious intervention [italics added] on our part. We experience it as a unified flowing from one moment to the next, in which we feel in control of our actions, and in which there is little distinction between self and environment; between stimulus and

response; or between past, present, and future. (p. 43)

The “internal logic,” which needs “no conscious intervention” in the above quote describes precisely the way automaticity of cerebeliar inverse ablnamics models “plays through” the cerebral cortex as described earlier in the chapter. I propose that in addressing the origins of the transpersonal experience of “flow,” here is a relevant question: What is the essential neuro—physiological component of many positive transpersonal experiences? "Flow” in my scheme, like play and creativity (Vandervert, 2015, 2016a, 2017a, 2018) is produced through cerebellar inverse dynamics models as learned from focused, repetitive, movement and mental experience that starts in infancy and continues throughout life. These neural dynamics apply, for example, to Csikszentmihalyi’s mountain climbers, chess masters, and so forth (See Vandervert (2017a) and earlier discussion of cerebellar inverse dynamics models). Once learned through focused repetitive effort, cerebellar inverse dynamics models automatically and unconsciously flow to and though the cerebral cortex. This state of nonconscious optimization applies as much to creative movements as to

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internalized cognitive/em otional experiences. This, I argue, is how the transpersonal experience of flow is propagated (for a description of flow in the case of Albert Einstein, see: http:/fblogsbiom edcentralcom/on-biology /2017/05/25/the-cerebellum-a-new-horizon-for-brain-studies-of—thegenius-of—albert—emsteinf). Conclusion I end m y support for an inclusive fractal epistemology with a quote from

where I began: “A fractal epistemology for transpersonal events clearly comports with the fractal evolution of the tightly collaborative relationship between the 69 billion neurons of the hum an cerebellum and the cerebral cortex (which contains a mere 16 billion neurons)” Because cerebral cortex-to-cerebellum -to-cerebral cortex brain processing can be shown via brain-imaging to be behind the very existence of the highest levels of skill, thought (including mathematics, Vandervert, 2017a), and socialization into culture (including ritual), these phenomena (especially mathematics) can, within this framework, be removed from the realm of ideal forms and can be embodied in the form of their brain processes. How does this embodiment occur? Ito (1997, 2008) has convincingly shown that cerebellar internal models of mental models (of skills and thoughts) take place in, for example, the prefrontal and parietal areas of the cerebral cortex and operate through the same control system principles as voluntary bodily movement. Since transpersonal “flow” events appear to occur within this overall cerebro-cerebellar framework, I believe a valuable contribution to transpersonal epistemology is made by cerebrocerebellar modeling, rather than models involving the cerebral cortex alone. Thus, I suggest that the cerebro-cerebellar understanding of transpersonal events helps to bring together not only body, mind, and brain, but also culture. This includes transpersonal experiences provided by repetitive socialization into both religious and nonreligious ritual. Ritual, by virtue of its repetitive nature, results in the learning of cerebellar inverse dynamics models that accompany the experience of spiritual “flow.”

The transpersonal experience of “flow” appears based in the fractal anatomy and physiology of the brain (Pellionisz, et al., 2013; Torday, 2016). Transpersonal flow experiences ultimately implicate a fractal pattern that optimizes a bodily/mental sync through the earlier-mentioned motor/cognitive dentate nucleus. From there, feed-forward processes promote “flow” in the form of cerebellar inverse dynamics models. That the hypothesized fractal physiology and dynamics of the cerebellum

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predominate behind these highest of flow and creative experiences is supported by what can be seen both when things in the cerebellum go right, for example in child prodigies and highly trained experts in any field (Vandervert, 2016b, 2016c), and when things in the cerebellum go wrong. The latter case of cerebellar processing going wrong is well established in what Schmahmann (2004) refers to as the cerebellar cognitive aflective syndrome: It [the cerebellar cognitive affective syndrome] is characterized by ( l ) disturbances of executive function, which includes deficient planning, set— shifling, abstractreasoning, working memory, and decreased verbal fluency; (2) impaired spatial cognition, including visual-spatial disorganization and

impaired visual-spatial memory; (3) personality change characterized by flattening or blunting ofaffect and disinhibited or inappropriate behavior; and (4) linguistic difficulties, including dysprosodia, agrammatism and mild

anomia. The net lect of these disturbances in cognitive fimctioning was a general lowering of overall intellectual fiinction [italics added]. (p. 371)

It is apparent that within Schmahmann’s cerebellar cognitive affective syndrome, the brain with an abnormal cerebellum would be marked by the absence of transpersonal “flow.” References Adamaszek, M., D’Agata, F., Ferrucci, R., Habas, C., Keulen, S., Kirkby,

Verhoeven, J. (2016). Consensus paper: Cerebellum and K.C., emotion. Cerebellum, 16(2), 1-25. Akshoomoff,

N., Courchesne,

E., & Townsend, J. (1997). Attention

coordination and anticipatory control. In I. Schmahmann (Ed), The cerebellum and cognition (pp. 575—598). New York: Academic. Anderson, CM (2000). From molecules to mindfulness: How vertically convergent fractal time fluctuations unify cognition and emotion. Consciousness & Emotion, I (2), 193—226. Anderson, C. & Mandell, A. (1996). Fractal time and the foundations of consciousness: Vertical convergence of llf phenomena from ion channels to behavioral states. In E. Mac Corm ac & M Stam enov (Eds),

Fractals of brain, fractals of mind: In search of a symmetry bond (pp. 75-126). Philadelphia, PA: John Benjamins. Balsters, 1., Whelan, C., Robertson, I., & Ramnani, N. (2013). Cerebellum

and cognition: Evidence for the encoding of higher order rules. Cerebral Cortex, 23(6), 1433-43. Blaisdell, A.P., Pottenger, B., & Torday, IS. (2013). From heart beats to

health recipes: The role of fractal physiology in the Ancestral Health

How the Cerebellurn and Cerebral Cortex Collaborate to Compose Fractal Patterns Underlying Transpersonal Experience

387

Movement, Journal ovolution and Health, 1(1), Article 2. Retrieved from: http://jevohealth.com/joumal/voll/issl/Z. Bostan, A.C., Dum, R.P., & Strick, PL. (2013), Cerebellar networks with

the cerebral cortex and basal ganglia. Trends in Cognitive Science, 17(5), 241-54. Csikszentmihalyi, M. (1975). Play and intrinsic rewards. Journal of Humanistic Psychology, 15(3), 41-63. Dehaene, S. (2001). Precis of the number sense. Mind &Language, 16, 16—

36. Dehaene, S., Spelke, E., Stanescu, R., Pinel, P., & Tsivkin, S. (1999).

Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science, 284(5416), 970-974. Hayter,

A.L., Langdon,

D.W.,

& Ramnani,

N.

(2007).

Cerebellar

contributions to working memory. NeuroImage, 36(3), 943-954. Imamizu, H., Higuchi, S., Toda, A., & Kawato M. (2007). Reorganization

of brain activity for multiple internal models after short but intensive training. Cortex. 43(3), 338-349. Imam izu, H., Kuroda, T., Miyauchi, S., Yoshioka, T., &Kawato, M. (2003). Modular organization of internal models of tools in the cerebellum. Proceedings of the National Academy of Sciences, 100(9), 5461-5466. Ito, M. (1993). Movement and thought: Identical control mechanisms by the cerebellum. Trends in Neuroscience, 16, 448-450.

—. (1997). Cerebellar microcomplexes. In ID. Schmahmann (Ed), The cerebellum and cognition (pp. 475—487). New York: Academic. —. (2007). On “How working memory and the cerebellum collaborate to produce creativity and innovation” by L.R. Vandervert, PH. Schimpf, and H. Liu. Creativity Research Journal, 19(1), 35-38.

—. (2008). Control of mental activities by intemal models in the cerebellum. Nature Reviews, 9(4), 304-313. Ivry, R. (1997). Cerebellar timing systems. International Review of Neurobiology, 41, 555-573. Ivry, R.B., & Spencer, R.C.M. (2004). The neural representation of time. Current Opinion in Neurobiology, 14, 225-232. Lent, R., Azevedo, F., Andrade-Moraes, C., & Pinto, A. (2012). How many neurons do you have? Some dogmas of quantitative neuroscience under

revision. European Journal ofNeuroscience, 35(1), 1—9. Leggio, M, & Molinari, M. (2015). Cerebellar sequencing: A trick for predicting the future. Cerebellum, 14(1), 35—38. Leiner, H., Leiner, A., & Dow, R. (1986). Does the cerebellum contribute

to mental skills? Behavioral Neuroscience, 100(4), 443-454.

Chapter Thirteen

388

—. (1989). Reappraising the cerebellum: What does the hindbrain contribute to the forebrain? Behavioral Neuroscience, 1 03(5), 998-1008. —. (1993). Cognitive and language functions of the cerebellum. Trends in Neuroscience, 16, 444-447.

Marvel, C.L., & Desmond, J.E. (2010a). Functional topography of the cerebellum in verbal working memory. Neuropsychology Review, 20(3), 271-279. —. (2010b). The contributions of cerebro-cerebellar circuitry to executive verbalworking memory. Cortex, 46(7), 880—95.

—. (2012). From storage to manipulation: How the neural correlates of verbal working memory reflect varying demands on inner speech. Brain Language, 120(1), 42-51. Maslow, A. (1971). The farther reaches of human nature. New York, NY: Viking. Nottale L., Chaline J., & Grou P. (2002) On the fractal structure of evolutionary trees. In GA. Losa, D. Merlini, T.F. Nonnenmacher,

&

ER. Weibel (Eds), Fractals in biology and medicine. Mathematics and biosciences in interaction. Basel: Birkhauser. Pellionisz, A I , Graham R., Pellionisz, PA., & Perez, I .C. (2013). Recursive genome function of the cerebellum: Geometric unification of neuroscience and genomics. In M. Manto, D.L. Gruol, I .D. Schmahmann, N. Koibuchi, & F. Rossi (Eds), Handbook of the

cerebellum and cerebellar disorders (pp. 1381-1423). Berlin: SpringerVerlag. Ramnani, N., Behrens, T.E., Iohansen-Berg, H., Richter, M.C., Pinsk, M.

A., Andersson, & Matthews, PM. (2006). The evolution of prefrontal inputs to the cortico-pontine system: Diffusion imaging evidence from Macaque monkeys and humans. Cerebral Cortex, 16(6), 811-818. Rankin, S.K., Fink, P., & Large, EW. (2014). Fractal structure enables

temporal prediction in music. Journal of the Acoustical Society of America, 136(4): EL256. Rauschecker, IP. (2014). Is there a tape recorder in your head? How the brain stores and retrieves musical melodies. Frontiers in Systems Neuroscience, 8, 149.

Schmahmann, ID. (2004). Disorders of the cerebellum: Ataxia, dysrnetria of thought, and the cerebellar cognitive affective syndrome. Journal of Neuropsychiatry Clinical Neuroscience. 16(3), 367—378. Schmahmann, I.D., Anderson, C.M., Newton, N., & Ellis, R. (2001). The

function of the cerebellum in cognition, affect and consciousness. Consciousness & Emotion, 2(2), 273-309.

How the Cerebellum and Cerebral Cortex Collaborate to Compose Fractal Patterns Underlying Transpersonal Experience

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Sokolov, A.A., Miall, R.C., &Ivry, RB. (2017). The cerebellum: Adaptive

prediction for movement and cognition. Trends in Cognitive Sciences, 21(5), 313-332. Spencer, R.M.C., & Ivry, RB. (2013). The cerebellum and timing. In M. Manto, D. Gruol, .l. Schmahmann, N. Koibuchi, & F. Rossi (Eds)

Hanabook of the cerebellum and cerebellar disorders (pp. 1201-1219). New York, NY: Springer Press. Stout, D., & Hecht, E. (2017). The evolutionary neuroscience of cumulative

culture. PNAS, 114(30), 7861—7868. Strick, P., Dum, R., & Fiez, J. (2009). Cerebellum and nonmotor function. AnnualReview ofNeuroscience, 32, 413—34.

Torday, IS. (2016). The emergence of physiology and form: Natural selection revisited. Biology, 5(2), 15. Van Overwalle, E, & Marien, P. (2016). Functional connectivity between the cerebrum and cerebellum in social cognition: A multi-study analysis. Neuroi’mage, 124/1, 248-255.

Vandervert, L. (2013). How the cerebro-cerebellar blending of visualspatial working memory with vocalizations supports Leiner, Leiner and Dow’s explanation of the evolution of thought and language. Cerebellum, 12, 151-71. —. (2015). How music training enhances working memory: A

cerebrocerebellar blending mechanism that can lead equally to scientific discovery and therapeutic efficacy in neurological disorders. Cerebellum &Ataxias, 2(11).

—. (2016a). The prominent role of the cerebellum in the origin, advancement and individual learning of culture. Cerebellum &Ataxias, 3(10). —. (2016b). Working memory in musical prodigies: A 10,000 year old story, one million years in the making. In G.E. McPherson (Ed), Musical prodigies: Interpretations from psychology, education, musicology, and ethnomusicology (pp. 223-244). Oxford, England: Oxford University Press. —. (20160). The brain’s encoding of rule-governed domains of knowledge: A case analysis of a musical prodigy. In GE. McPherson (Ed), Musical prodigies: Interpretationsfrom psychology, education, musicology, and

ethnomusicology (pp. 245-258). Oxford, England: Oxford University Press.

—. (2017a). Vygotsky meets neuroscience: The cerebellum and the rise of culture through play. American Journal of Play, 9(2), 202-227.

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—. (2017b). The origin of m athem atics and number sense inthe cerebellum: with implications for finger counting and dyscalculia. Cerebellum & Ataxias, 4(12).

—. (2018). How prediction and anticipation in the cerebellum led to stone tools, language and culture, and, thereby to the rise of Homo sapiens. Frontiers in Cellular Neuroscience, 12, 408. Vandervert, L., Schimpf, P., & Liu, H. (2007). How working memory and the cognitive functions of the cerebellum collaborate to produce creativity and innovation. Creativity Research Journal 19(1), 1-18.

CHAPTER FOURTEEN

HIDDEN 1N PLAIN SIGHT As THE SKY HOLDS A CLOUD: FRACTALS IN ANCIENT CHINESE PHILOSOPHY

ANTHONY S. WRIGHT1

From the point of View of (organic pattern2 or If E, [author ’5 manslationD, things and the self are a unity, and there is no distinction between the internal and external.

—Zhu XI 9E; (Chan 1986, 10) Introduction Is it possible to measure, predict, and control- and thus “to know” (which is the purpose of reductive, logic-based Western science in the examination of phenomena) hum an transpersonal experience with the infinite? Is the nonlinearity of complexity science useful in this context? Can scholars be rigorous in study of that which cannot be “known” in advance? In other words, can complex deterministic non-linear experience of the infinite be “explained” (laid out on a plane) in a valid (deductive), logical way? It seems that embedded in Friedman’s (2002, 2013) distinction between “transpersonal psychology” and “transpersonal studies,” with the intention to ensure validity (or deductive proof), there is a requirement of rigor- for measure, prediction, and control— in order to be considered “psychology.”

1 Assistant Professor, English Taught Program in lntemational Business, College of Management, Shih Chien University, Taipei, Taiwan. E-mail: [email protected], PO. Box 4411, Petaluma, CA, 94955

2 In the use of the term “organic” throughout this chapter, I am referring to the patterns of the cosmos as found in Chinese and indigenous philosophies: where the cosmos is presupposed to be a single organism fiom which everything emerges, as

discussed below.

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Further, to “study” the experiential infinite is to agree to some extent to forego rigor, measure, prediction, and control. What is transpersonal human experience? Is it available only to a selected few “masters” who are considered to be “enlightened?” I propose that every human being has transpersonal experiences on a daily basis- yet we simply don’t notice or pay attention to them. I suggest that when hum an beings have transpersonal experiences, the domain of that experience is other than solely in deductive thought and reason. Transpersonal experiences have been written about by many: e.g. psychologists Abraham Maslow, and “peak experience,” (Maslow, 1964), and “flow” by Csikszentmihalyi (2009). I offer my “own” and other examples below, and close the chapter with an invitation of an iteration of the Chinese Classic of Changes or Y1‘ Jing E; éfi for participating in ongoing transpersonal experience, that parallel and point to the work of transpersonal psychologist John Welwood (2009). It is beyond the scope of this chapter to other than lightly touch upon some of the ideas and experiences that will be discussed in later offerings of a book on the topics introduced in this chapter. And while some have suggested my argument is simply a “philosophical stance,” rather than an “actual” description of “ontological reality,” this chapter begins to unfold

my participatory involvement and experience in what seems to me to be infinite, ontological reality.

Evolving paradigms and complexity science The early Greek models of how the world worked were initially organic and patterned, as experienced by musician—mathematician Pythagoras, and the artisan-stone-mason Socrates. In a laudable attempt for clarity, the then nineteen-year-old Aristotle offered his three basic “laws of thought,” in the perception of the world that became an essentially a black/white, true/'false paradigm, which has been Luiconsciously adopted as a core standard for validity and reliability in present-day scientific methodology. Fortunately, in the late 19th and early 20th centuries, mathematicians Cantor, Julia,

Serpinski, and von Koch expanded Euclidian three-dimensional geometry from its black and white parameters, into the infinite dimensions of space and time. In a m aj or developm ent in m athem atical modeling, Georg Cantor’ s work evolved into what he considered the abstract reahn of mathematics. Cantor found freedom from the limitations suggested by observations of nature in

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Newtonian natural science of the nineteenth century (Boyer 1991, 2—3), in the development of set theory, in 1875. Set theory offered potential liberation into an infinite and intuitive richness of mind, through the addition of infinite recursion. The process of recursion refers to taking the product of a mathematical formula and feeding it back into the formula, which results in continuing the sequence of new products. When this recursive production is carried out an infinite number of times, the set of products show an infinite process, rather than a single static figure or point. A simple example is the Cantor set, as will be shown below. Other recursive

formulas would prove difficult if not impossible to execute without the capability of rapid and repetitive calculations available only through the development of the electronic computer. Mandelbrot’s use of computers in the 1970s unveiled the wonders of the previously inaccessible recursive formulae of the so-called “mathematical monsters.” (Mandelbrot, 1977). In his groundbreaking work, The Fractal Geometry of Nature, Benoit Mandelbrot (1977) discusses this “mathematical crisis of the 19th century” where the mathematical structures of Cantor, Koch, Sierpinski, Menger, Peano, and others were thought to be: ..”pathological and a “gallery of monsters”...The mathematicians who created the monsters regarded them as important in showing that the world of pure mathematics contains a richness of possibilities going far beyond the

simple structures they saw in Nature. Twentieth century mathematics flowered in the belief that it has transcended completely the limitations imposed by its natural origins. (Mandelbrot, 1977, p. 3)

Thought initially to be naive, Cantor’s set theory and the paradoxes that accompanied it were mathematically contextualized (one could possibly say, validated and liberated) by Kurt Godel in 1926. One of Godel’s “incompleteness theorems” essentially demonstrated that no logical]r mathematical system of sufficient complexity could be considered selfconsistent and complete (Hofstadter 1979, p. 86). By adding a poetic and playful component of recursion— So, naturalists observe, a flea

Has smaller fleas that on him prey“, And these have smaller still to bite ’em',

And so proceed ad infinimm (Swift, 1733/2007, p. 20).

—one can transcend the limits of the system (Abraham, 1995). This can be seen physically in the form of a fem, where the basic shape of the fern is an isosceles triangle, and the compound leaves are also isosceles triangles, as

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are the leaflets that make up the leaves also isosceles triangles. The whole fem is seen as more than its parts. In evolving from the Newtonian natural science of the nineteenth century, the developm ent of set theory uncovered deeply embedded organic pattems that accurately point to self-similar structures of the cosmos-asorganism. Mandelbrot quotes Freeman Dyson: Now, as Mandelbrot points out. . .Nature has played a joke on the mathematicians. The 19';11 century mathematicians may have beenlacking in

imagination, but Nature was not. The same pathological structures that the mathematicians invented to breakloose fiom the 19th century naturalism turn out to be inherent in familiar objects all around us. (Mandelbrot, 1983, pp. 3-4)

These “pathological” structures are fractals found in nature, in the complex shapes of fems, rocks, clouds, trees, water, and more. The joke was

that the world the nineteenth century mathematicians escaped into (through the inclusion of infinity) was the actual accurate modeling of organic pattems of the natural world, rather than the previously limited Euclidian— Newtonian-Cartesian mechanistic models of the world. Curiously and gratifyingly, psychologists working with complexity science have discovered that pattems of hum an consciousness itself can be modeled by fractal

geometry and complexity science (e.g., Marks-Tarlow 1999, 2008) and that these complex models of consciousness and self are not limited by degrees of scale. Complexity science, and one of its sub-fields of fractal geometry, can therefore ofler patterns and tenets beyond “reasoned” measure, of the infinite— along with an understanding that beyond a certain point, even in deterministic systems, there is complex, non—linear information which cannot (practically) be “known” (Marks-Tarlow, 2012). Yet, as we will see, the (un—knowable) infinite can be embraced, experienced, and inhabited (Welwood, 2009).

The cosmos as organism Aristotle’s three basic “Laws of Thought,” presuppose that Westem models of the cosmos have developed a highly useful, polar, black-and—white, true or false, subject-object, falsification-based, and measurem ent-based paradigm that continue in some dimensions to be an arbitrary and exclusive standard for validity in creating models of the world. Yet nature is more than black-

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and-white, and Western paradigms have run into some difficulty (e.g. the as yet unresolved “hard problem” of separation of mind and body). In a playful commentary on the utility of both rigor and study is the academic game that philosopher-entertainer Alan Watts calls “Prickles and G00,” and how we need both. There are basically two kinds of philosophy; one is calledprickles, the other is called goo. Prickly people are precise, rigorous, logical. They like everything chopped-up and clear. Goo people like it vague. For example, in

physics, prickly people believe that the ultimate constituents of matter are particles. Goo people believe its waves. I n philosophy, prickly people are

logical positivists, and goo people are idealists. And they’re always arguing with each other. What they don’t realize is, that neither one can take his position without the other person; because you wouldn’t know you

advocated prickles unless there was somebody else advocating goo. You wouldn’t know what a prickle was unless you knew what goo was. Because

life is not either prickles or goo, its gooey prickles, and prickly goo. They go together like back and front, male and female. And that’s the answer to philosophy. See, I’m a philosopher. And I’m not going to argue very much, because if you don’t argue with me, I don’t know what I think. So, if we argue, I say thank you! Because owing to the courtesy of your taking a different point of view, I understand what I mean So, I can’t get rid of you.

(Alan Watts, 2004, recording)

In contrast and additive compliment to Western paradigms, according to Chinese, Japanese, and Indigenous understandings, the Cosmos is a single organism, from which everything emerges; or to use a fractal metaphor, iterates. Here are a few examples of arguments regarding these complimentary points. Leroy Little Bear, ID. in the foreword of Native Science (Cajete, 2000) offers the Native American (Indigenous) understanding of the Cosmos as a natural-spiritual field of organic patterns in contrast to the methods of Western approaches to science and mathematics: Western paradigmatic views of science are largely about measurement using

Westem mathematics. But nature is not mathematical. Mathematics is superimposed on nature like a grid, and then examined from that fiarnework. It is like the land survey system: a grid framework. ..supe1imposed on the

land. . . as a basis of dealing with the land, but they are not part of the nature of the land. ...Modern description leaves out so much—it leaves out the sacredness, the living-ness, the soul of the world. ...The Native American

paradigm is comprised of and includes ideas of constant motion and flux, existence consisting of energy waves, interrelationships, all things being

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animate, space/place,

renewal,

and

all

things being imbued with

spirit....Everything is interrelated...What Native Americans refer to as “spiri ” and energy waves are the same thing...Everytlung in creation

consists of a unique combination of energy waves. In other words, What appears as material objects is simply the manifestation of a unique combination of energy waves. Conversely, all energy wave combinations do

not necessarily manifest in terms of material objects. (Cajete, 2000, pp. ix-

X) Mathematician John Allen Paulos notes the futility that a singularly black—and-white approach to the multidimensional and paradoxical quality of attempting to attend to that which is beyond cognition with thought alone. It has been suggested by Zen philosophers that notions like truth and falsity, subject and object, external and internal, while essential in every day life, as

well as in scientific thought, nevertheless prevent one fiom attaining a mystic, oceanic union with the universe. The universe simply is. (Paulos, 1982, p. 52)

Sinologist FW. Mote posits, The genuine Chinese cosmogony is that of organismic process, meaning that all parts of the entire cosmos being to one organic whole and that they all interact as participants in one spontaneously self-generating life process.

(Mote, 1971, p. 19)

Chinese cosmologist, Tu Wei-ming says of Chinese cosmology, What (Joseph) Needham describes as the organismic Chinese cosmos consists of dynamic energy fields rather than static matter-like entities...The distinction between energy and matter is not made in Chinese Philosophy.

(Callicott and Ames, 1989, p. 68)

Fractal nature of the (I Ching) Yi a g 57% When the book Chaos: Making a new science by James Gleiek was published in 1987, I was captivated by the infinite beauty of fractal patterns. What particularly got m y attention, was diagram of the Cantor set (see

Figure 14-1; Gleick, 1987, p. 93). This figure looked suspiciously like a Yz‘ Jing 55,553 bi-gram. In Chinese philosophy and the Yi Jing 53%, there are two primary principles, or constituents of pattern. One is yin 3%, which literally means “the shady side of the mountain,” which is treated in the Yi J‘mg East as a

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broken line: __ _ and yang [5%, which means “the sunny side of the mountain,” which is treated in the Yr‘ a g gag as a a solid line: These are combined as below, into “bi-grams.”

Y1 Jing 53$ bi-grams (adapted from Jou, 1984, 24). New moon

First Quarter

Full Moon

Greater

Lesser

Greater

Yang

Yang

(W55?)

(Mfg)

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I I

I I

I I

—

I I

I I

I I

—

I I

(Ms)

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Yin

Last Quarter

Figure 14-1. The Cantor Set

I then wondered if the Yr‘ Ji'ng Egg was fractal. My suspicions were that the YI‘ Ji'ng 54% fr fractal were confirmed by the reports of psychologist Katya Walter, with her book Tao of Chaos: Merging East and West (Walter, 1994). Here are a few of her thoughts, discoveries,

and experiences,

regarding the Y1‘ flag Eéfl laid out in greater detail in her book. I saw that the I Ching actually works; further, that its accuracy might have a basis in the new science of patterned chaos. (Walter, 1994, P. 22)

I have slowly learned that the I Ching reveals the pattern. Not the specifics of an event, but its underlying pattern. It works through the dynamics of chaos theory, which can predict a trend without specifying its exact details. Discovering this huge hidden intelligence that rests deep in the weave of nature, even learning to communicate with it, can be disconcerting,

frightening. . .until it becomes wonderful. (Walter, 1994, p. 18) The discovery reveals a deeper truth beyond the limits of what we call normal reality. It exhibits an underlying coherent pattern in the dynamic chaos of nature itself. More eerily, it exhibits a tap-able caring that’s nestled in the very fabric of spacetime-mattergy. This huge pattern knits the cosmos

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together in physics and metaphysics. It unites the objective and subjective,

the quantitative and qualitative, the alpha and omega. Its vast dynamic shapes us, body and soul. (Walter, 1994, p. 18). How does a mere I Ching oracle answer questions? Much like a computer program. It uses a mathematical algorithm to relay its response in an analogy

from the archaic Chinese past...This analogy—or verbal analog—really does correlate in a fit far beyond chance. (Walter, 1994, p. 21)

Your hexagram answer will diagnose the situation in a specific verbal analogy that faithfully conveys the dynamic pattern of that real time event. Its analogy offers advise on how to cope with the situation—in other words,

how to go through itin Tao. (Walter, 1994, p. 22).

Alan Watts suggests that some think the archetypes of these divinatory practices are merely Rorschach ink-blots upon which the querent will project the contents of their unconscious. (Watts, 1973/2017). The Y 1‘ .1n Egg? in particular suggests that the use of the oracle in divination is primarily a training regimen, designed to train the unconscious mind of the querent to intuit the fractal patterns of the organic cosmos, without having to use the oracle (Jou, 1984).

Chinese philosophy and Li Q There are five “classics” (.1n as) in Chinese philosophy, the Classics of Rites, Poetry, History, Spring & Autumn Amaals (another history text), and the Classic of Changes (1’1‘ flag gags), all compiled by Confucius in the fifth century BCE. These were the seminal texts one was expected to understand if one was an educated person in China in the time of Confucius. Of the value of Rites or ritual (11“ $E), Confucius speaks about aligning oneself, through ceremony with the current momentary momentum of the Cosmos, thereby taking advantage of and participating in a momentum larger than oneself (Behuniak in l ones, 2008). In the Song Dynasty (960—1279 CE), a scholar named Zhu X1 (11301200 CE, see Figure 14-2) synthesized Buddhism, Daoism, and Confucianism into what was termed “Neo—Confucianism,” in emphasis on the “Four Books” (from the Classic ofRz'tes). The Four Books GEE) (Confucian Analects, Great Learning, Book of Mencius, and Doctrine of the

Mean) were the texts used to govern China from 1200 CE to 1905 CE. From these texts, Zhfi Xi developed the School of Principle, or L1“ Xué $33.33* (as

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399

translated in a Western philosophical mode). I translate L1" X ué Egg as the School of Organic Pattern.

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Figure 14—2. Zhfi Xi (1 130—1200, Song Dynasty Confucian scholar, Lushan Museum. (Public domain) The idea of L1"IE, which was initially definedby the Chinese as “patterns in jade” is fractal (see Figure 14-3). I had an insight that “patterns in jade” could be seen as “frozen turbulence.” From this awareness I was drawn to the studies of Lorenz (1963), who was studying the mathematics of

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ilflEI"

turbulence, and'was one of the first to begin working with tenets of complexity science.

Figure 14-3. Scan of marble tile with fractal patterns of frozen turbulence, like “patterns in jade.” {C ourtesyof the author) Of particular interest to me was one o f the four books, called the “Great Learning,” or D a Xué jig. The D a Xue TEE has a particular quality of being nested, which led me to its fi'actal character of seifsr'rnrirm'ijr. A number of Chinese philosophical texts have this nested quality of self-

sirnilarity. A key sentencein theDaXue 7kg, that is still being argued over to this day, suggests that if you would have the worldbe peaceful, you must “complete knowledge by investigating things.” In the discussion below, I have translated the layered character of tho se Chinese words as “to complete yuisdornyou rrmstpm‘i‘ern being,” or alignfattune oneself to organic patterns.

The great learning (De Xué fit!) If you would have the world be peaceful, you must malce the country peaceful. If the countryis to be peaceful then the town must be peaceful. If the town is to be peaceful, one ’s familymust be peaceful. Ifthe familyis to be peaceful, you must be peaceful. How does one become peaceful? One becomes peaceful by“combing-out”(rectifying) the heart-mind. How does one comb-out the heart-mind? By the completion o f wisdom. The completion of wisdom is done through the patterning of being. (to attune

with or become one with organic pattern [33' (lab- or me; see explanation

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below] .) When one’s heart-mind is thus attuned, one finds peace. When you are peaceful, your family is peacefiil. When your family is peaceful, the town is peaceful. When the town is peaceful, the country is peaceful. When the

country is peaceful the world is peaceful. Wise people conduct themselves inthis way.” (Translation and paraphrase by the author.)

Li E as fractal The Chinese character It“ GE), and its interpretation have carried a great deal of relevance as a patterned topography in the development of Chinese philosophy and Western concepts of Chinese philosophy. Rather than an indepth, granular etymological study of If (E) as has so effectively been done by Moran (1984, pp. 83—185), I explore the arguments for the legitimate yet unconventional translation of 11' (1%) as “organic pattern,” as suggested by Moran (1984, pp. 83—185), Needham and Wang (1956, 558), and Sun (1966). Chinese characters show the potential of multiple and embedded meanings, which allow for different iteratively deepened meanings of characters in a hermeneutic sense, in the same textual body, as one’s awareness of nuances of meaning and context of a text grows. This phenomenon is known as “paranomasia” (Ames, 2008, pp. 37-48).

With the unconventional and deepened translation of the Neo- Confucian use of if (E) as “organic pattern” to have legitimacy (as shown above), and the argument made for II“ (E) as “organic pattem” by Sinologist Joseph Needham (Needham & Wang, 1956, p. 558), IKE) is then (as shown below) by etym ologist Moran (Moran, 1984, p. 84) to take on a transcendent quality. The meaning of If (3%) then becomes “organic patterns” of the Cosmos, in line with the organismic quality of Chinese philosophy (Mote, 1971, p. 19). Transcendence of Li E

With a transcendent If (E) comes the idea that the transcendent field of organic pattern was the source of observed organic patterning of things in daily life (in the following discussion, “things” are called “instrum ents” [qi

138]). The [dao E], or what is above shapes, considered in itself independently from concrete things, is called [11 ii]; but considered in relation to concrete things, is called [Xing ii] (nature). In fact, nature is [H 3:33], but it is that [H E] which has fallen into [qi 1%] and falleninto it. (Sun, 1966, p. 158)

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[LI 5%] is above shapes...[Li 3E] transcends time. . .The [Li fl] existed even before the formation of the universe...the [11 5E] do not depend on the existence of non-existence of heaven and earth. Thus Chu His says: “Even if mountains, rivers, and the earth would have vanished, nevertheless the [1?

iii] are still here. . .The [H E] belong to the category of eternity. . .The [1i 333] also transcend space. . .The world of [It 5:33] is the world of patterns. (Sun, 1966, p. 163)

To sum up what Sun (and Zhfi Xi) are saying, organic patterns of 11“ (fl) transcend time and space, are eternal, and are beyond heaven and earth. With the transcendent quality of 11“ (IE), as a fimction of the dao‘ (E), the natural world emerges from [1“ ( E ) as “spirit-energy” or qi‘ (is?) into “instrumentation” (which is the physicality of the world) or at (113%). The emergence (or iteration) from the Cosmos in forms of organic patterns of If (E) is one of the found parallels with complexity science. In addition to If (E), there is one other relevant term to touch upon in this chapter that has to do with “organic pattern” which is the Chinese term

gé (Fifi), which is used in the Great Leaming (Dc‘i Xue’ jig). I suggest that

ge’ (1%) is a precursor of 11' £51. In the Great Learning (Da Xué jig), the pivotal sentence, Zhi 2111' 2611' gé wit ( fl 5311 Eifivw ), is conventionally translated as, “Extend knowledge and investigate things” (Gardner 2007, p.

136), where gé (1%) is translated as “investigate.” I found that the character ge’ (13%) could also be translated as “pattern” (Harbaugh 1998, p. 22). Thus, the pivotal sentence from the Great Learning could be legitimately translated as “the completion of wisdom and patterning being.” The character gé (1%) can be translated as “the pattern of wood that speaks” (Harbaugh 1998, p. 22). This translation is consistent with the idea of “organic pattern,” as it refers to the pattern of growth in wood that informs (speaks to) observers of such patterns. I speculate that Zhfi Xi, in his compiling of and commentary on the Four

Books and particularly the Great Learning, referenced the term ge’ (*3) for the development of the term [1“ (E), as both terms can be translated as

“pattern.” It is clear to Sinologist Joseph Needham (N eedham and Wang, 1956, p. 558) that Zhfi Xi meant “organic pattern” when using the term ii“ (E). In my opinion, the Dd X ue’ k g says that when we align our being with transcendent organic pattem, such is to align ourselves with the infinite. Wisdom is beyond thought, and when aligned with the infinite, wisdom is

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complete. Whenwe are thus alignedwith the infinite, our heart-mind is clear, we and all who are in relationship with us are at peace. In other words, when we become transpersonal, we simply become something the Cosmos is doing. I’ll touch on this point in greater detail below.

The gateless gate and epistemology of transpersonal psychology There is a Zen Buddhist idea of what is called “The Gateless Gate,” which,

as the Maine curmudgeon commented when asked for directions by a tourist, suggests “You can’t get there from here.” One cannot go beyond (trans) the personal (sound-mask of the identity) solely by thought, language, episteme-ology, or study of knowledge. In a discussion of the Gateless Gate in Zen Buddhism, Yam ada (2004) observes: How much more ridiculous to adhere to words and phrases or to try to

understand (the transpersonal) by means of the intellect. It is exactly like trying to strike the moon with a stick, or to scratch an itchy spot on the foot through the surface of the shoe. What concern do they have with reality?” The essential world cannot be grasped by intellectual contemplation or philosophical conceptualization. There is no way other than to realize it in our own living experience. It is therefore, quite foolish to try to understand it by following the meanings of words. (Yamada, 2004, p. 9)

In the story “Flatland” written by theologian Edwin Abbot in the late 19th century a similar notion of transcending limiting dimension is explored. Abbot writes of beings living in atwo-dimensional, planar world, who could not understand or ex-plane the presence of an entity, the author, in a third dimension. The experience of the flatlanders was changed irrevocably, when becoming transformed from a circle to a sphere. (Abbot, 1952). I believe this is what happens in a transpersonal experience- that our human awareness is transformed in emerging into dimensions beyond the per-sonal (Greek theater sound-mask, that we mis-take for ourselves). Alan Watts comments on this pointCertainly the revolutionary thinker must go beyond thought. He knows that almost all his best ideas come to him when his thinking has stopped. He may

have struggled and struggled to understand a problem in terms of the old ways of thinking, only to find it impossible. But when thought stops from exhaustion, the mind is open to see the problem as it is—not as it is

Verbalized and a t once it is understood (italics added). (Watts, 1951, pp. 101-102)

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Daily audience with the piano god Which is more difficult to change? Hard “stuff,” made of particles (as in the current Western static view of reality [Stanford Encyclopedia of Philosophy, 2017]), or, vibrations and waves, in a field of vibration? In the same way that to a carpenter with a hammer (or empiricist with a ruler! ), all the world is a nail (something to be measured). To me, as a professional piano technician, all the Cosmos is sound: everything is vibration. To add the understandings of the organic patterns referred to by the Chinese character 11“ E and infinite multidimensionality of complexity science, all the Cosmos is (complex) vibrating (organic) pattern (Wright, 2012). Curiously this idea that Cosmos is vibration is consonant with physicists who are advocates of string theory as mentioned below. Studies by the Wellcome Institute at Imperial College, London, confirm that there are neural patternings that develop in the hippocampus of piano technicians, for navigating a topography of vibrating patterns, that are similar to the hippocampal topographic structures also in present in the brains of the infamous “Green Badge” London cab drivers who know the complex topographic patterns of the streets of London by memory only (Teki, et. al. 2012). My first awareness of the physical manifestations of sound as reflected by “matter” was the vibrations of the circus brass band in the balloon I had held in my hands at age four. Beyond my experience as a piano technician I discovered Chladni patterns of vibrating plates (Waller, 1961), where sand

sprinkled on thin, square steel plates, and as stimulated with a violin bow (see Figure 14-4).

Figure 14-4. Chladni patterns. (\Nikimedia Commons)

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In three dimensions, vibrations are studied as Cymati cs, which patterns are shown in the books and subsequent films of physicist Hans Jenny (Jemry, 196?), (Jenny & Manners, 1936). Cymatics are where sound waves

were directed to the viscous powder of lycopodium moss spores (chosen for the capacity to flow fieely) and water (see Figure 14-5). The organic, circulatory and fractal figures that spring up in reflection of the standing wave forms at various pitches is quite astonishing.

Figure 14-5. Gymatics patterns in water. (Wikimedia Commons)

Seeing these fractal forms that had sprung up in response to stimulation of standing waves at various pitches, I wondered what vibrations and Standing waves then would give rise to ihe form, emotions, and consciousness of the human my. At present, in attending to the world around me, I see and experience everything as a fractal, cymatic response of vibrations, much as described by Leroy Little Bear earlier in this chapter (Caj ete, 2000). Theoretical physi cists Jonathan Halliwell, Imperial College, London, (\Nr‘ight, 2014), and Michio Kaku of the City University of New

York, have expressed consonance with similar metaphors that the cosmos is a “symphony of vibrating strings,” (Kaku, 2011) that can be attended to by changes in perspective. On May 13th, 1993, at 2 PM, in Minneapolis, Minnesota, Ituneda piano

for a performance by an artist who participated in the Steinway ConcertArtist program by ear (or strictly aural method). I used one tuning fork and

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established a reference octave for tuning the piano by listening to harmonic phase differences (known as harmonic “beats”) in the relationships between twelve tones of what is called the “equal temperament” (Isacoff, 2001) in the center octave of the piano. I tuned the rest of the piano to the reference octave by progressively tuning lower and lower strings and then the higher strings. So the piano would be stable in a concert setting I tuned the piano a second time, re-tuning and making now minute adjustments to the twenty tons of tension held by the 225 or so (exact string numbers differ in each

piano make and model) strings. Tuning the piano twice through for the upcoming concert took about two hours of constant focus of close listening, which was physically and psychologically effortful. I finished the tuning that day, and the piano disappeared. I had fallen into a(transpersona1) flow state (Csikszentmihalyi, 2009) where I simply could experience an emotion and hear music. I didn’t notice myself or the piano, only experienced feeling and music. The piano and I both becam efunctionally transparent. The piano and I didn’t get in the way of the music that came through the piano and through me. Jazz saxophonist Sonny Rollins has expressed a similar idea to theoretical physicist Stephon Alexander, in Alexander’s 2016 book, The Jazz ofPhysics. .. You can’t think and play at the same time. When I play, I don’t want to play the music, I want the music to play me.” (italics added, Alexander, 2016, p. 174). I returned back to myself after tuning the piano and went to have tea at a friend’s coffee shop. I thought, “if this (flow state of forgetting myself) can happen by tuning a piano, what would be needed to tune a human being (to go into flow state, of functional transparency)?” I also wondered about the “I” that is doing the tuning. I wondered to myself why it is that human beings have an ego, and what good the ego is, as the ego seems to mostly get us in trouble (by attempting to keep up the illusion of separateness from the Cosmos). The answer came to me from m y sub-conscious mind: “At its highest functioning, the ego is simply an interface, an (infinite) ‘portal’ of awareness with the infinite- (which develops through relationship, MarksTarlow, 2012), in the same way I experienced the well-tuned piano- as an infinite portal through which music comes, from vibration.” To invite this

functional transparency is to attune the portal (of both hum an awareness and musical instrument) to 3 highest functioning (Maslow’s peak experience); to open infinite relational characteristics (like the infinitely deep, but finitely bounded Koch snowflake, see Figure 14- 6) with the Cosmos, which then simply becomes something the Cosmos is doing Where the “I” happens to be (Watts, 2004).

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$133!!!!

Figure 14-6. Koch Snowflake

Forgetting the self I had seen this sort of state before. In the late 1980s, I would go to the Minnesota Zen Center on a

Wednesday night, to hear the weekly lecture by Zen priest, Dainin Katagiri, Roshi3, after a forty-minute meditation session. During the meditation sessionI noticed Katagiri’s energetic affect (a bodily sensation as discussed

in the Polyvagal Theory by Porges, 2011) as he sat zazen (Zen Buddhist meditation practice in full lotus position) with us. Then the meditation session ended, and he began his lecture. I noticed that, curiously, his affect didn’t change. Then the lecture ended, and we got up to begin talking with

one another. Katagiri mingled with the group, and I noticed that his affect had not changed! Then it came to me that Katagiri, in each of these activities, was still meditating/ He was in a continuous, ongoing flow-state. Fifth century BCE Daoist philosopher, Chuang Tzu (see Figure 14-7) develops a fictitional conversation between Confucius and Yen Hui: Yen Hui: “May I ask what fasting of the mind is?” Confucius said, “Make your will one! Don’t listen with your ears, listen with your mind (It) xin), No, don’t listen with your mind, but listen with your spirit GEE shén). Listening stops with the ears, the mind stops with recognition, but spirit is empty and waits on all things. The Way (Dao E ) gathers in emptiness alone.

Emptiness is fasting of the mind.” Yen Hui said, “Before I heard this, I was certainI was Hui. But now that I have heard it, there is no more Hui. Can

this be called emptiness?” “That’s all there is to it,” said Confucius. (Zhuangzi, 2013, pp. 57-58).

3 “Réshi” is an honorific title for a Zen Buddhist priest who has received dharrna

transmission (teachings from an established lineage of Ch’an or Zen priests). It is a Chinese word “1510 shi fififli” meaning “old teacher” as translated into Japanese.

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zen Master Dfigfll Emji in the Ganja:- Kmm m" ‘idfitmizii'rqg the flmlemd Pm‘rf’taiks about ‘forge‘mng 1112 self,” an flea fluthegan with Chinese philnsuphu' Climb; Tm,abow. Dag-:11 (1935) writes: To mflyfli Buflnwism 513113.711“ 5211'. To m m 521' i: inf-Iget 111:: 5211'. To forgetfiie self is 11:: he nmaJimedhyfiie myriad I: IlllJIZIEI fling. flan ‘weryrliiig‘filfiiiigs. Winn nmflimedb'jrfli myrhdthing.w.rhoiy ufim'ndaswellas flight-flies mimini DfEI'E'IHS drcp Haw. Ham: Ii mflimtbnrmnimmidfifis run-113m ccmtirues etflhsshr.“[p.?fljl.

I“? J'flg 5E cunsuhatinn for writing and finding this chapm hifliewfifingocffliis chapter,1fl1mighr.to consuhihe H3113 Egéi Miami questimwas:‘“-F.i1utis most Efl'ecfive infiie writing ofthe aILixle an immunology oftrai‘uspersomlpmhnlng- 'n offering readers pointers tn:fiieir ongoing taruspenmlemariame?”

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In asking a question of the Ft Jing 55%, there is an art to configuring questions. So the reader will notice certian presuppositions that are embedded within the questionI asked. My intention in this article is to offer readers pointers (indicators for attending- like sign-posts, rather than as advise, or directives, or declarations of what “ought” to occur) to an ongoing trans-personal experience. I’m presupposing that, at the most basic level, trans-personal experience is continuously available as a context of being, with our “individual” human experience as an iteration or expression of (complex) vibratory (organic) patten'is (Wright, 2012). Attending from the larger context beyond individuality is available through developing the flexibility of awareness (see below). The Y i fing 57% answered with Hexagram N°5: X11 %, “Waiting,” with changing lines in the second and fifth places.

A hexagram is a figure of six horizontal lines that are either solid or broken, that is built from the bottom line to the top. These six lines are made up of two, three line figures called “trigrams.” Each trigram represents an archetype of pattern in Chinese cosmology. Some are reflective of actual physicality of the real world, and some are representative of archetypes. The trigrams that go to make up Hexagram N°5: X11 %, “Waiting,” fer the top three lines are K ’cm, i”, or “Water”:

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And-the lower three-lines? are-aim $2, or “Heaven”:

Before writing this chapter and consulting the Yi. sling 5%, I had-not. made the following correlations with transpersonal. psychologist John Welwood’s material. Synchronously this figure of water above heaven, or “cloud in the sky” (see Figure 14-3) i s directly resonant with the following directions of J ohn Welwood for what he calls “Developing Unconditional Presence” ('Welwood, EUUQ).

Figure 14—8. Cloud in the sky, April 3, BEN, this supercell thunderstorm dropped

m inch-diameterhailowrChaparraLNewMexico. [US National Weather Service) T o develop unconditional (transpersonal) presence (according to John’ 5 instructions, as paraphrased for purposes of this chapter by the author) one first chooses to attend to and mimawieetge that one thinks, andwhat the felt.

(physiological) sense i s of that experience of thinking; how itfeeis to think, and where that felt sensafi on (IF thinking is in the Body.

1i-"iire'lwood [2009) sugests: ...to allow (the sensations or the feelings ofthirllring) to tie-present, giving (these bodily sensations} ple nty of space to be there, justas they arefHoId

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the feeling (of thinking) in the space of awareness, without reacting to it,

without judging it, Without trying to change or fix it, Without getting caught in it, without identifying with it, without making it mean something about

you, without hardening against it. Let yoims'elJf soften around it as the sky holds a cloud, without resistance, simply letting it be, or like a mother holding it baby, with gentleness and caring. What’s that like? How does it feel to allow it, and give it space to be there, just as it is? (Unpublished handout).

Thus, through this exercise, one’s thinking can become appropriately contextualized as something that is happening, something the Cosmos is doing, within afield of awareness. Thinking can be seen then for What it is, a marvelous yet other than all-encompassing tool for attending from the field of being to personal experience. There then become options for attending that are other than through thought, and beyond thought.

Text and changing lines of hexagram 5: (Jou, 1984, 134). Translation of Chinese characters by the author. 5: Xfi fi “Waiting” Cloud above, Heaven Below

The Image:

aneacnennza Cloud Mist inside center (symbol). Dense cloud no rain’s (image). The Commentary:

% ° 7%?- ° 7‘16; ° Eta” ° flifijflll ° Waiting. Have sincerity. Bright, perfect communication. Proper way, good fortune. Benefit ford great river.

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Dr. Yi Wu’s Commentary on Hexagrarn 5: “...In ancient times, the water of rain was the most important thing all people needed. This hexagram discusses what we need, and how to get what we need. . . . T o get what we

need, we go into aregion of danger [of the unknown]. We carmot rush there; we have to wait. [This hexagram] has the image to wet and benefit the low places. It thus has the meaning [of] food and drink which gives necessary

nourishment to human beings. (Vi/u, 2014, pp. 69-70). The Lines: (With two lines changed, the lower (nine in the second place)

is the minor line, and the upper (nine in the fifth place) is the major line. (Wu, 2014, p. 17). Nine in the second place.

fli°fitfi°mfifi°flfi° Waiting on sand. Small have talk [gossip]. Finish [in the end], good fortune. Dr. Yi Wu’s commentary: “Waiting on the sand. There is a little criticism; inthe end there will be good fortune.” (Wu, 2014, p. 69). Nine in the fifth place.

fli“%¥fi%°§§° Waiting on wine eat [food and drink, or nourishment]. Proper way, good fortune. Dr. Yi Wu’s commentary: “This line [the writer and reader] is the position

of one who deals with all changes [of awareness of the trans-personal], so it can cover all virtues and efforts. Beside the “proper way” [211613 a ] , there are three virtues: sincerity, constancy, and reverence.”(Wu, 2014, p. 76). Proper way: Zhén E . Aligning with Divination [cracks in tortoise shells]

[Natural organic pattern offering of value] (As money (cowrie shells), to obtain fortune).] (Harbaugh, 1998, p. 83). Sinceriiy: Fi'r 5%. Trust. Hen brooding over eggs. (Wu, 2014, p. 77). [Author’s commentary: This is trusting the process]

Constancy: Chang $3.131 banner or flag [of pattern]. (Harbaugh, 1998, 176) “...It means that on this line a ruler or leader should set up the principle [pattem', 11' fl and goal [destination] which give the spirituality to what we need.” (Wu, 2014, p. 77).

413

Hidden in Plain Sight As the Sky Holds a Cloud Reverence: “Here (gong 25%, respect) has the same meaning as reverence, so it means to practice reverence in his [the writer’s] position, and to do what he should do, which means non-action; [to wait, in a reverent attitude, without prostheletizing], then the people [readers] will conduct their affairs

naturally [according to organic patterns], and the world will be peaceful.” (Wu, 2014, p. 77).

Conclusion How are w e to successfully attend to (beyond “thinking

about”) the

experience of becoming transpersonal? And, by implication and metaphor, find utility in fractal mathematics as a bridge to epistemology? By understanding the futility of attempting to describe the transpersonal with measurement, language, or concepts, in efforts of prediction and control. There is an old story about the Buddha pointing to the moon. Many people make the error of looking at the pointing finger. (Suzuki, 2011, p. 193.) Yet, as I have argued in this chapter, parallels with the transpersonal infinite can be pointed to with the infinities offractal geometry and complexity science. The “Coastline” paradox (Mandelbrot, 1977) itself points to and offers a metaphor for transendence. In a very abbreviated form, the Coastline paradox states that when measuring (a natural organic form) the shorter the

units of measurement, the longer becomes that which is measured. When the units of measurement become zero, that which is measured becomes infinite. In other words, when we stop measuring through thought, and experience directly, we become one with that which is experienced. I speculatively suggest that when we become fimctionally transparent, infinitely bounded human portals, like the Koch Snowflake; in other words, where we “forget” ourselves in a state of “flow,” what was “our” (individual) awareness, expands beyond human limits, and beyond limits of time and space. At last, awareness which is “wisdom” beyond “knowing” then becomes what there is, just in this moment (Marks—Tarlow, 2003, 2012, in

press). As the sky holds a cloud. References

Abbot, E. (1952). FMTLAND: A romance of many dimensions (by a square). Glendale, CA: Bibliotech Press.

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Abraham, FD. (1995). Dynamics, bifurcation, self-organization, chaos, mind, conflict, insensitivity

to initial conditions, time unification,

diversity, free will, and social responsibility. In R. Robertson & A. Combs (Eds), Chaos theory in psychology and the life sciences, (pp. 155—177). Mahwah, NJ: Lawrence Erlbaum.

Alexander, S. (2016). The jazz of physics: The secret link between music and the structure of the universe. New York: Basic Books. Ames, R T (2008). Paronomasia: A Confucian way of making meaning. In DE. Jones (Ed), Confucius now: Contemporary encounters with the

analects (pp. 37—48). Chicago, Il: Open Court. Boyer, C. B. (1991). A history ofmathematics (Revised ed). New York, NY: Wiley. Cajete, G. (2000). Native science: Natural laws of interdependence. Santa Fe, NM: Clear Light Publishers. Chan, W. (1973). A sourcebook in Chinese philosophy. Princeton, NJ: Princeton University Press. —. (1986). Chu Hsi and Neo—Confucianism. Honolulu: University of Hawaii Press. Csikszentmihalyi, M. (2009). Flow: The psychology of optimal experience. New York: Harper [and] Row. Dogen (1985). Moon in a dewdrop: Writings of Zen master Dogen. (K. Tanahashi, Trans, Ed.) New York, NY: North Point Press. Friedman, HL. (2002). Transpersonal psychology as a scientific field.

International Journal of Transpersonal Studies, 21(1), 175-181. —. (2013). The role of science in transpersonal psychology: The advantages of middle-range theory. In H.L. Friedman & G. Hartelius (Eds), The Wiley-Blackwell handbook of transpersonal psychology (pp. 203-222). West Sussex, England: John Wiley & Sons, Ltd. Gardner, D.K. (2007). The four books: The basic teachings of the later Confucian tradition. Indianapolis, IN: Hackett. Gleick, J. (1987). Chaos: Making a new science. New York, NY: Viking Books. Harbaugh, R. (Ed) (1998). Chinese characters:A genealogy and dictionaiy. New Haven, CT: Yale University Press. Hofstadter, DR. (1979). Godel, Escher, Bach: An eternal golden braid. New York, NY: Basic Books.

Isacoff, S. (2001). Temperament: The idea that solved music’s greatest riddle. New York, NY: Alfred A. Knopf. Jenny, H. (1967). Cymatics: The structure and dynamics of waves and vibrations. Basel, Switzerland: Basilius.

Hidden in Plain Sight As the Sky Holds a Cloud

415

Jemiy, H., &Mam1ers, PG. (1986). Cymatics: The healing nature of sound. Newm arket, NH: MACROm edia.

Iones, D E . (2008). Confucius now: Contemporary encounters with the analects. Chicago, IL: Open Court.

Iou, T H . (1984). The tao of] Ching: Way to divination. Taipei, Taiwan: Tai Chi Foundation. Kaku, M . [Big Think] (2011, May 31). The universe is a symphony of vibrating strings [Video File]. Retrieved from:

https:llwww.youtube.com/watch?v=fW 6JF KgbAF 4 Mandelbrot, B. (1977). The fractal geometry of nature. San Francisco, CA: W.H. Freeman.

Marks—Tarlow, T . (1999). The self as a dynamical system. Nonlinear Dynamics, Psychology, and Life Sciences, 3(4), 311—45. —. (2003). The certainty of uncertainty. Psychological Perspectives, 45, 11 8-130. —. (2008). Psyche’s veil: Psychotherapy, fractals, and complexity. London, England: Routledge. —. (2012). Clinical intuition in psychotherapy: The neurobiology of embodied response. New York, N Y : W . W . Norton. Maslow, A H . (1964). Religions; values, and peak-experiences. London. Penguin Books, Limited. Moran, P E . (1984). Exploration of Chinese metaphysical concepts: The

history of some key terms from the beginnings to Chu Hsi (1130—1200). (Unpublished doctoral dissertation) University of Pennsylvania, State College, Pennsylvania. ProQuest (8326318). Mote, F W . (1971). Intellectual foundations of China. New York, NY: Knopf. Needham, J. & Wang, L. (1956). Science and Civilization in China, Volume 2: History of Scientific Thought Cambridge, England: Cambridge University Press.

Paulos, 1A. (1982). Mathematics and humor: A study of the logic of humor. Chicago, IL: University of Chicago Press. Porges, SW. (2011). The polyvagal theory: Neurophysiological foundations o f emotions, attachment, communication, and self-regulation. N e w

York, N Y : W.W. Norton.

Stanford Encyclopedia of Philosophy (2017 ) Process Philosophy. Retrieved from: https://plato.stanford.edu/entries/process-philosophy/. Sun, S. (1966). The doctrine of the ‘Li’ [E] in the philosophy of Chu Hsi.” International Philosophical Quarterly, 6(2), 155—188. Suzuki, D.T. (2011). The lankavatara sutra: A mahayana text. @stennarie: Egely Kloster.

Chapter Fourteen

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Swift, I. (2007). On poetry: A rhapsoay. Retrieved from http:lfbooksgoogle.comfbooks/about/On_Poetry.html?id=GmIIAAAA QAAJ. (Original work published 1733) Teki, S., Kumar, 8., von Kriegstein, K , Stewart, L., Lyness, C R , Moore,

B.CI., & Capleton, B. (2012) Navigating the auditory scene: An expert role for the hippocampus. Journal of Neuroscience, 32(35), 12251—

12257. Tu, W. (1989). The continuity of being: Chinese visions of nature. In I. B. Callicott & R. T. Ames (Eds), Nature in Asian traditions of thought:

Essays in environmental philosophy (pp. 67-78). Albany, NY: State University of New York Press. Waller, MD. (1961). Chladni figures: A stuay in symmetry. London, England. G. Bell and Sons. Walter, K. (1994). Tao of chaos: Merging east and west. Shaftesbury, England: Elem ent. Watts, A. (1951). The wisdom of insecurity:A messageforan age of anxiety. New York, NY: Pantheon.

—. (2004). Out ofyour mind: [essential listeningfrom theAlan WattsAudio Archives]. Louisville, CO: Sounds True. —. (2005). Do you do it, or does it do you? How to let the universe meditate you. Louisville, CO: Sounds True.

—. (2017, May 6). The methodfor transformation of consciousness. [Video File] Retrieved from: https://www.youtube.com/watch?v=Ena34ANjofU (Original work published 1973) Welwood, I. (2009). Developing unconditional presence. Unpublished workshop handout. Sausalito, CA. Wright, AS. (2012). Attunement: Deep conversations. Guest: Rupert Sheldrake. Point Reyes Station, Self—published radio program. —. (2014a). Principle and pattern: Zhu XI (5E? and complexity theory: Completion of wisdom through fathom ing pattern (Eli filfi‘élfi : Zhi Zhi Qiong Li). (Unpublished doctoral dissertation). California Institute of Integral Studies, San Francisco, California.

—. (2014b). Attunement: Deep conversations. Guest: Theoretical Physicist Jonathan Halliwell, University College, London. Point Reyes Station, Self-published radio program. Wu, Y. (2014). The I Ching: The way of dealing with changes. San Bruno, CA: Great Learning Publishing Company. Yamada, K. (2004). Gateless gate. Los Angeles, CA: Center Publications. Zhuangzi (2013). The complete works of Zhuangzi (B. Watson, Trans). New York, NY: Columbia University Press.

CHAPTER FIFTEEN

ALL THE INBENT FRACTALS OF CONNECTION1

WILLIAM J. JACKSON2

It is awonder the way the psyche can reflect, and reflect on, the universe. And the ways the psyche works with images constitute a marvel. As an historian and comparativist of religions, I continue to be amazed at the

power of religious symbols. In my works on bhakti (religious devotion) in India, I have studied the archetypes in the life-stories and lyrics of South Indian singer-saints, Tyagaraja, Purandaradasa, Annamacharya and Kanakadasa (Jackson, 2007). For example, in the case of Tyagaraj a, his lifestories include likenesses and parallels, such as a miracle of origins, initiation from a holy man, spurning invitations from a king, a divine rescue, a sign of music’s extraordinary power, and foreknowledge of his own death—self-similar themes found in other great singer-saints. Taken together, the parallels among the group of prominent culturally creative singer-saints create a narrative fractal-like resonance with kindred motifs and incidents (Jackson, 1993). Architectural fractals in Hindu temples constitute another fiactal-related aspect of Indian culture I have explored,

focusing on the shapes of Shikharas (spires) varying in scale (Jackson, 2002). My contribution to this book is not a critical analysis so much as an attempt to stretch the mind of the reader with an evocative call to open up the sensibilities to a recently discovered!invented branch of mathematics not yet fully understood, yet deeply resonant with spiritual yearnings. For some people “religion” means organized and institutional traditions; for others, religion-like dimensions can be found in events associated with yoga and meditation, the rituals of politics, sports, celebrity, etc. Spirituality may be 1 A version of this chapter was published in the International Journal ofTranspersonal Studies, 38(2). 2 Professor Emeritus, Indiana University-Purdue University at Indianapolis. Email: [email protected]

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found in many people’s lives, whether they identify themselves as religious or not: nature-loving mountain climbers, social-justice seekers, nonbelievers who serve those in need, and so forth. In our age, religion and spirituality are complex (Drescher, 2016; Manning, 2015; Zuckerman, 2014). Terry Marks-Tarlow invited me to write this chapter about transpersonal psychology’s potential to explore a fractal epistemology because I also wrote a book, Heaven ’5 Fractal Net which explores fractal shapes and fractal-like images in cultures around the world (Jackson, 2004). A primary reason I wanted to write that book was to show the ways fractals can demonstrate how seemingly paradoxical descriptions of Arman, meaning “Self” in Hinduism’s Vedanta philosophy, might be visualized. Atmcm is spiritual consciousness, shared with all others, and remains in a state of fullness even when some of it is taken away, etc. Since Atmcm is seen as the

spiritual Self in all people, and in all living creatures, the concept relates to such concepts as equality of spirit, self-similarity or degrees of sameness, cosmic consciousness, and how existence is multiple yet can be considered “all one.” Most of the fractal-related images in Heaven’s Fractal Net were meant as objects of contemplation—beautiful, functional structures of

wonder. A few words on the importance of symbols may be helpful first, maybe even necessary. As an emeritus professor of religious studies with a longtime interest in wisdom traditions, and as a researcher in Indian culture, I work with deep and attractive metaphors as transpersonal archetypes, and this essay concerns the potential for fi'actal geometry as metaphor to expand spiritual sensibilities, especially the subjective sense of awe and wonder at the cosmos. Einstein wrote: “The most beautiful experience we can have is the mysterious. It is the fundamental emotion that stands at the cradle of true art and true science” (Einstein, 1982, p. 11). With this in mind, we can appreciate the possible application of fractals to realms of the psyche (see Figure 15-1).

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The Hindu and Buddhist image of Indra’s Net is ancient, a fractal network which visualizes mutual interrelatedness. Here is a Buddhist description: Far away in the heavenly abode of the great god Indra, there is a wonderful net which has been hung by some cunning artificer in such a manner that it stretches out infinitely in all directions. In accordance with the extravagant tastes of deities, the artificer has hung a single glittering jewel in each ‘eye’ of the net, and since the net itself is infinite in dimension, the jewels are infinite in number. There hang the jewels, glittering like stars in the first magnitude, a wonderful sight to behold. If we now arbitrarily select one of these jewels for inspection and look closely at it, we will discover that in its

polished surface there are reflected all the other jewels in the net, infinite in number. Not only that, but each of the jewels reflected in this one jewel is

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In the network of pearls, light illumines, reflects, unifies and warms with mutual interactions among the pearls. Humanity and all sentient beings share consciousness, in various degrees. Wholeness is imaged well in this cosmic vision, and Stanislav Grof (2008, p. 6) observes that “In holotropic states we transcend the narrow boundaries of the body ego and encounter a rich spectrum of transpersonal experiences that help us to reclaim our full identity.” Holotropoic means moving toward wholeness, and Grot’s work seeks to access depths of the psyche through sensory, biographical, prenatal, and transpersonal levels via breathing, music, and art. Images like Indra’s Net (see Figure 15-2) help us picture a mock-up of our situation and the potential we have for fulfillment in such a context. I think it will be usefitl to explore briefly some other powerful images in history before further considering the power of fractal images.

Figure 15-2. Illustration entitled “Indra’s Pearl.” (Courtesy of Graphics Interchange)

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Images are humanity’s way of picturing and understanding complex realities. Communication with words and pictures, language and images, has brought great advances to the human race, allowing ideas to be extended in space and time. Visual images provide a deep and enduringly useful extension of vision. A coin pictured in a New Testament parable serves to show the way we can relate to seemingly contradictory realms—spiritual and worldly matters: “Give unto Caesar that which is Caesar’s. . ..” A threeleaf clover in Patrick’s hand could explain the mystery of the trinity—a simple three-in—one trick to show the reason a way around a stumbling block. We know that not all people are receptive to images in the same degree. Buddha held up a single lotus flower silently, and a teaching was transmitted to those who got it. Not all present understood or resonated with the intended meaning, but those who did spread the teaching down the ages, which endures for those able to comprehend. We can hope that many in our age will appreciate fractal symbolism, a new way of seeing patterns in a variety of levels. I will give a more modern example also. One day Professor Harvey Cox, author and faculty member at Harvard Divinity School, came into a class in which he was not the usual professor. His basic message was that having taken a psychedelic at an earlier time in his life, he wanted to tell us that, instead of calling psychedelic states “altered states of consciousness” (which suggests there is one main legitimate state of consciousness and others are alterations of that), he thought it was better to say “alternate states of consciousness,” not giving so much priority or privilege or importance to ordinary everyday consciousness. This phrase points to the possibilities of a term used by psychologist William James (18 14-1910), when he referred to “a spectrum of consciousness.” In our time, many think of scientific thought, employing logic and analysis, as the way to know the only true reality, but a great many psychologists concur that this presupposition leaves out too much. Forty years later I still recall Professor Cox’s message clearly, in part because he was wearing a top hat, a unique type of headgear which rises high from the head and represents a different status or state of mind, in comparison with say, a beanie, abowler or a baseball cap. During the years I studied at Harvard, Cox was usually hatless. Only in his chat about consciousness did he wear a hat, and a seldom-seen top hat at that. Some images are emblems or signs—for example, the flag under which a ship sails. In the history of America, there have been trial-balloon emblems to signify the citizens’ togetherness. Hands clasped in a circle; a snake cut up into 13 segments with the motto “Don’t tread on me;” and the goddess Liberty, a gift reminding the nation of its promise to be arefuge for

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the tired and poor. Religions also have their emblems—Star, Cross, Crescent and Star, Wheel, unifying social groups with an image. These symbols and others are held up to encourage people to identify with all the others in their community or country, and to serve as a method to foster loyalty, to see citizenship as different kinds of people sharing in a common good and complementing each other with different skills and backgrounds. Such metaphoric emblems serve to unite people in affirming a simple identity. Metaphors can work powerfully, especially for those who are attuned to them. Professor W. C. Smith, an historian of religion with whom I studied, told me that the metaphor of “God the Father” was something he understood more deeply only after becoming a father himself. We resonate with some images depending on memories of experiences, associations, imagination, and personal needs. I am not a mathematician; it is the visual aspects of

fractal geometry that I especially resonate with. Each person resonates according to their individual traits or subjective conditioning or personal history.

experience, their

Images are the language of the unconscious; images are the alphabet of the anima mundi, the world soul’s voice, The Voice of the Earth (Roszak, 1993). In this ancient and ecological view, the world, and all that belongs to the earth—animal, vegetable and mineral—are interconnected and have a share in consciousness. The unconscious, among other things, is a feedback process. Sometimes it tries to speak in dreams, in intriguing stories, telling us things we need to know. Our responses are not always conscious either. We are conditioned with slaps and kisses to like and dislike certain kinds of things. Idol worship and idolatry have negative connotations to many monotheists, but in India I know people to whom “idol” means “sacred image,” not “false god.” With such a wealth of sacred images, Indian thinking tends to see all the names and forms of the sacred as transparent, symbolic representations we can relate to (see Figure 15-3). Not that “sacred” and “transpersonal” are equivalent—transpersonal includes recognition of experiencing the sacred, but not exclusively, and not in a single official way.

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Figure 15-3. Kolam, drawn image of a lamp made of lamps which I saw in Tamil Nadu, south India, circumambulating Arunachala Mountain, Tiruvannamallai Village. (Courtesy of the author, who sketched the image in black ink, but the original was white lines on tan earth.)

Some images are important and powerful because they convey something that is considered sacred or capable of extraordinary showings.

The image of light in the mystical poetry of writers around the world is an example. All sighted people have experiences of the sun as revealer, clarifier, enlightener, and so forth. The physical make-up of the eye allows for experiences of light which are inspiring to the depths—they are unforgettable half a century later. Light, literally the light experienced in the excitable eye cells—tessellation Vision itself—is an experiential revelation of the light of consciousness, exploring and learning.

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Entoptic phenomena (i. e., visual images occurring or originating inside

the physical eye) can feel like a display of inherent meaningful shapes, like tessellations (deftly explored by MC. Escher, and geometric tiles in the Alhambra palace in Grenada, Spain, and in Shamanic and theogenic vision states). These fields of interlocking shapes flow like tiles with pinwheeling curlicue borders, swirling each into the next (see Figure 15-4). They may give participants the impression that they are co-creators in an energetic universe, waves of the cosmos and consciousness. Mandalas are visual focal points, circles picturing deities, worlds, deadly sins—geometrical charts and reminders of programs to focus on throughout the pilgrim’s progress of the human lifespan.

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Fractal images in their modern computerized examples are not exactly sacred, because they’re not part of a tradition with religious stories, as ancient sacred images are. But fractals do have an intrinsic cache, a drawing power, and for good reason. They are in many cases innately attractive, endowed with a charming beauty. They are aesthetically appealing in a deeper sense of symmetry and intuitive relatedness to the workings of the world at large, like a flash of insight or finally seeing a solution to a mathematical puzzle. The kinds of order they point out can have a fascinating appeal which excites curiosity and awe. Fractals are uncanny, and the uncanny has potential for becoming revered, honored, considered sacred. Nature’s visible fractal examples are plentiful and purposeful, such as branching networks of arteries, smaller and smaller. What do we mean by “uncanny”? The uncanny is paradoxical—a kind of thinking Aristotle could not accept. Uncanny phenomena seem to obey laws we are unfamiliar with, and to break laws we know. The uncanny straddles a line—hermaphrodites, eclipses, premonitions, synchronicities, omens, dreams. The uncanny is somehow a paradoxical situation of “both/and” when the usual is more Aristotelian, insisting on “one or the other.” I suppose rainbows are uncanny, sun with rain. Fractals are “not a panacea” as Benoit Mandelbrot was not shy to point out (Mandelbrot, 1977, p. 3). But they do shine a light on nature’s wisdom; they show deep natural patterns and arrangements. They work with a subtle power—think, for example, of the many functions of branching shapes—in rivers, nerves, blood vessels, and tree branches. The light they shine can be applied in various fields. As images of wholeness, fractals have a unique capacity to help envision ways of reconciling a variety of co-existing views. The paradox of fractals is seen in all the ways they show the part is whole and the whole is part.

In India, Jain philosophers wanted to consciously recognize in their view that there were seven schools of thought, and that all had a right to exist with a particular view. Syadvada is the Sanskrit name of that concept— “somehow-ism”—because somehow all these different views co-exist and can be argued and held by different schools. “Seven-valued” Jain philosophy promotes looking at other views with respect, tolerating alternate perspectives, and practicing the virtue of ahz’msa (non-violence) in the realm of thinking as well as in the physical world. It is democratic, like a jazz ensemble, which involves sharing the stage and conversing with different instruments; and it brings together noble souls in a round-table style. Jains

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hope such a View can bring a healing touch to the psyches of a war-tom world, achieving more brotherhood. What is it that the shapes of fractals can illuminate? Psyche fractals can

help illuminate the art of seeing and explore the science of wholeness (by “psyche fractals” I mean fractals selected to point out a psychological idea, or psychological relationships, or to represent a psychological type showing different proportions of introversion, extraversion, intellect, intuition, etc.).

Mandalas in Tibetan Buddhism (perhaps like “Rose windows” in cathedrals) are images which can be gazed upon as circular expressions of wholeness, the whole situation up and down, North, East, West, South, allowing one’s intuition to take it in (see Figure 15-5). In one of Jung’s

letters he referred to a confused person he knew. “The man was confused and bewildered, so something had to happen to give him clarity about the whole situation. The mandala was a sort of letter from the unconscious meant to clarify his mind... to bring order out of a state of confusion. .. it is like an amulet. . .. Amulets often have a mandala form” (Jung, 1984, pp. 114-

124). Fractals can be like that too, a geometric form to focus on, a reminder of certain principles, such as whole-part relationships. They can depict layers within layers, like Russian nesting dolls.

Figure 15-5. North Rose Window, Notre Dame, Paris. (Public domain)

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There are many aspects of nature that seem rough, unruly, chaotic. But the roughness of bark on a tree, and the cracked-rock broken terrain of mountains, the scattered stars and the jagged coastlines can be seen and discussed in their ordered principles through the lens of fractals. Marks-Tarlow’s chapter aims “to present an epistemology for the field of transpersonal psychology that helps to heal an ever—widening schism between these two positions [belonging either to the humanities or hard sciences]. To honor the call for objective rigor, I offer up the mathematics of fractal geometry as model, method, and metaphor for otherwise ambiguous and inaccessible transpersonal phenomena” (this volume, chapter one). Models or conceptual frameworks determine a lot in our thinking, and the nonlinear models which fractals offer can expand our more linear presuppositions considerably. The need for coming together—merging the disjointed disconnected parts of what could be an organic whole—as a family belonging together, is apparent. Though naturally there are varied slants of theory and practice in an intellectual community, it is like an ecosystem in which all the members need each other for the whole system to be complete. The humanities and hard sciences each have important distinct abilities, and some resources of fractal geometry can serve both. Marks-Tarlow’s plea is to consider the benefits of experimenting with fractals as model, metaphor and method for co—existing, sharing and cooperating among transpersonal psychologists, an invitation to try out the promising potential of fractals.

Prominent features of fractals The way fractals are at play in the field of vision is rich and thought provoking. They intrigue and attract us to features that draw us in and carry us beyond their first impressions. One beauty of fractals is the way the parts go together enchantingly—as all playful eyes can see. The sometimes-

spectacular visual features of fractals make their contribution unique. Their traits include varied scales, parts self-similar to the whole, concepts of wholeness, non-linearity, infinity, feedback, compositeness. They possess geometrical, orderly coherent organization, seemingly simple yet complex, with dynamic chaotic processes going on if you look for them. Beauty. Poet Robert Hass speaks of beauty as a paradox of stillness and motion. Think of a trembling aspen, tall, with fluttering-flickering leaves all over, as an example of enchantment, with beauty both still and moving. The sheer beauty of fractals can attract attention—school children who dislike,

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math are often attracted to fractals. Fractals often elicit enthusiastic appreciation, curiosity, and exploratory participation. Scales. “They reckon ill” who leave out scale, because things are going on at various sizes, on various levels of existence, all at the same time. Scale is the constant frontier of exploration in many areas. Memorable thinkers have taught some of humanity’s most extraordinary insights, conveying life’s patterns in statements using scale. Geniuses of scale include Lao Tzu, Buddha, Jesus, Chuang Tzu, Rumi and Kabir. Their parables, which use symbols playing with scale, are vividly memorable: the eye of the needle and the wealthy man’s camel; the gnat and the beam in the eye; the mustard seed; "the foot of an ant,” and so forth. Geniuses of scale show us how scale itself is a very important aspect of awareness, a place where the little can have a disproportionately large impact, and the vast and the miniscule can relate intimately to human predicaments. Fractal scales offer a resource of

flexibility. Mandelbrot voiced the idea that aesthetics which give satisfaction often employ elements in all scales (Gleick, 1987, p. 117). Infinity. Taken literally, the infinite is oddly beyond us, an abstraction outside of human dimensions. Infinity may not be accessible or useful to us as an abstraction, but embodied visually and vividly in graphic representations, it is more relatable. Fractals are said to be infinitely enfolded between dimensions, like nuances of existence in creative thinking. “Nuances are like the richness of the boundary area in the Mandelbrot set, the richness of the many scales of a fractal.. .. Nuances exist in the fractal spaces between

our categories of though ” (Briggs & Peat, 1989, pp. 194-198). Unending detail and infinite length give fractals an association of wonder. Recursiveness, non-linearity, parts self-similar to the whole, mysteries of wholeness, paradoxes—all these issues and more are related to this aspect. Compositeness. Some fiactals are built-up forms, aggregates within aggregates—elements arranged within arrangements within clusters within containments—seeds, womb, generations of a family, family resemblances— Russian dolls, kinship’s self-similarities. Literature gives examples of fractal-like compositions: India’s epic Ramayanas include the same basic storyline in different languages, different lengths, different poetries, different dramatic arts, such as puppetry, dance, drama, songs—all together they form one great fiactal. Geometric. Fractals are geometrically shaped, yet their effect is on ideas, interpretations, understanding that impact in many fields. Triangles in triangles, squares in squares, circles in circles, the “ladybug” Mandelbrot

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set, and so forth. They are geometric yet fluid, enabling us to imagine possibilities, able to open the sense of potential intuitive ideas. It is said that Einstein’s greatest gift was “the ability to transcend mathematical limitations with physical intuition” (Alexander, 2017, p. 4). Fractals teach us a way of looking at the world, enabling a consideration of organizing principles. They may help us to recognize orderly coherent organization which was hitherto unnoticed in situations. Self-similarity. This quality means recognizable patterns of parts replicate the whole, not necessarily perfectly but at least recognizably, ideally on at least three levels of size. Individually and jointly, the parts resemble the whole. When we focus on parts that resemble the whole in nature, in structures of architecture, music, poetry and literature, we are awakened to the connections/relationships between parts and in the smaller parts within parts, altogether in the overall whole. Paradox. Fractals’ ability to suggest paradox is intriguing. Bob Dylan, in his song “Early Roman Kings,” offhandedly describes such a paradox of scales when singing about gangs of leaders who are part thugs of the olden days and part modern-day urban hoodlums: “Each one bigger than all of ‘em put together”—a logical impossibility, but then again, in some ways a group of people can be stupider, or smaller-seeming than any one of them alone acting big. Functional intuition has to do with knowing something more wholly, rather than knowing only some of its parts. Fractals are both aesthetic creations and physics-law circuits for generating parts and wholes in harmonizing conjunctions. The aspects of fractals considered here are features withpotential power, clarification, and contextual understanding. Fractals make room for a practical and contemplative intellectual development in the mind. They go beyond the simplistic linear patterns of simple squares and triangles to illustrate organic paths and useful abilities in systems. They also help us intuitively to delve into the potentials which Mandelbrot suggested fi‘actals enable. In an interview he said: “(O)ne can, or should, concentrate on thinking about [geometric] shapes as live ‘wholes,’ and learn to modify an

algorithm to affect the shape it generates” (Barcellos, 1984, p. 223). The principle of fractals, when learned and incorporated in thinking can inculcate the fractal sensibility in human outlook regarding ideas explained in diagrammatic, multi-sized variations on a theme. Not every fractal is literally a geometric fractal. There are, for example, experiential fractals that can help conceptualize patterns formed in our understanding. Cascading

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bunches of willow branches making up a weeping willow form a geometric fractal of curves. And repeated acts of fixation done by one with an obsessive-compulsive personality can be fractal-like too. For example, Terry Marks-Tarlow (2008), exploring fi'actal principles in her book Psyche ’s Veil, shows how similarities, symbolic recurrences, fi'actal-like kinds of patterns can appear in people’s lives, revealing themes in their psychological experiences. The function of fractals is many—faceted. Fractals are like the abstract geometry related to soul, with distinctive shapes and mysterious depths. Nature’s fractals are a touchstone of how wholeness—in composites, in branches, in microcosms, in galaxies—seems to work. Specific ways things work out in an evolving world are often fractal-based and fi'actal-like in outcome. For example, take the human form. Consider with me how the body can be seen as a numerical fractal of fives. The physique is formed of five limbs branching from the torso: two arms, two legs, and a head. This branching figure is numerically self-similar to the branching at the ends of the limbs—hands branch with five fingers, and feet have five toes, and the head traditionally has awareness of five senses—sight, smell, hearing, taste, touch. The total number of the bones in the hand is roughly as if each finger has five bones, and the toes and their joints inside the feet have five bones each. In some cultures, the universe is composed of five elements, which the body returns to, upon death. Snowflakes, cracks in drought-dried clay, swarm patterns in space, and time units—in many ways, fractal geometry’s varied forms can serve as containers for reflecting our turbulent experiences of life. Is there anything as crystal clear and enduringly recognizable to the naked eye as the image of a fractal snowflake? Snowflakes have been around a long time, but as with many aspects of nature, we see them anew when looking with a fractal sensibility. Snowflake shapes (see Figure 15—6) can represent our inner worlds of turbulent change, fears and anxieties, hopes and dreams, and projections and creative ideas.

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Figure 15—6. Snowflakes. (Photos courtesy of Wilson A. “Snowfla'lce’1 Bentley)

With a variety of fractal vessels to reflect our issues, new possibilities can emerge. Fractal shapes used contemplatively offer scope for exploratory reflections furthering our understanding. By this, Imean fractal images can serve as visual objects of contemplation, as mandalas are used, as Rorschach images, and as diagrams within which to record one’s reflections.

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What else is so special about fractals? There is comfort in enchanting forms and artful messages. It has been said that enchantment—such as a mother’s voice humming a soothing lullaby— is the oldest kind of medicine there is (Jung, 1987, pp. 410423). Of course, Jung was referring to other kinds of enchantment than fractals—he died in 1961, before Mandelbrot made his breakthroughs in fractal research. Nevertheless, fractals represent a technology-age enchantment. They intrigue school kids who otherwise are not interested in math and evoke reverence in students of architecture and other design fields. They can provide atool for pattern recognition in nature, in literature, art and human relations. They are an imagination—spur, a reminder to think about intertwining relationships, to focus for a while on connecting dots. There are reasons to revere some things, even in a secular worldview— family, elders, selfless patriotism, masterpieces of art. We revere them for their depth of enchantment, their ability to resonate in our reflections. “All great art contains at its center contemplation, a dynamic contemplation,” Susan Sontag (2012, p. 31) wrote. We could define contemplation as consciousness focusing on a deep issue. Fractals are potentially like that too. They inspire contemplation as well as participation. Like a mandala, a fractal can be gazed upon in meditation and drawn or built in 3D; it can be danced, it can be a chart (Jung, 1987). Just as a rainbow or a tree can be a spiritual sign and a visual parable, fractals serve a similar purpose by displaying the beauties of such features as self-similarity, which is a great mystery as well as an obvious echoing, an organic wholeness, a sense of oneness and naturally inter-fitting parts. And fractals visually suggest infinity. At times in the past the concept and manifestations of infinity caused fear among those with a mathematician’s view because it seemed uncanny, a freakish monster-like aberration of finite norms. The unfamiliar and endlessly expandable can at first seem outlandish, maybe even threatening; the finite engenders more control. But visual suggestions of infinity can also represent consciousness. The Hindu view of the ultimate is that a subtle spiritual consciousness which is the basis of existence—Brahman, the formless sat-cit—ananda (literally “Being, awareness, bliss”) is shared by all. Intellectual and other kinds of consciousness (such as visionary, intuitive, and awareness of feelings) share some of the qualities of infinity, adapting to endless possibilities, lighting the way to understanding. Fractals help us envision the dynamics of this kind of view.

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In some cultures, fractals are prominent motifs in structures and

organizational schemata; in others they are present but less obvious. For example, Joseph Campbell’s (1949/2008) Hero with a Thousand Faces

seems to show that there are multitudes of hero stories with heroes large and small, and there are shared similarities in their venturing forth, overcoming

a challenge, and returning home older and Wiser. Another example includes many mountain-like temple silhouettes in India featuring fractal-like shapes—little zigzags inside larger zigzag contours making up the overall zigzagging structure (see Figure 15-7). For

more on this, see my Hindu Temple Fractals post at https://WWW.academia. edu/347639/Hindu_Temple_Fractals (Jackson, 2002).

Figure 15-7. Hindu Temple Fractals Curve, using outline of Kandariya Mahadeva temple in Khajuraho. (Collage courtesy of the author)

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Ron Eglash’s (1999} book African Fractals illustrales. and discusses fi'actal ways of designing and organizing in some African cultures, favoring self-similar shapes in composing the lay-outs of villages, and individual family compounds (see Figure 15-8).

Figure [5-8. An African fractal showing the arrangement of compound huts.

(Courtesyr of Ron Eglash) These traditions select simple memorable functional fractals as a principle for organizing space and expressing harmonious order. They make good sense as a convenient method, repeatable for generations, selfregenerating. Instinct or traditional wisdom—both are memory sources of what worked .in the past—to organize matters according to self-similar favored shapes. What are .fractals that we should be s o mindful of them?

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What makes them important? For one thing, as mentioned above, fractal

structures are “useful” to nature, suiting form to function in a variety of ways. I give a couple of examples. For streng1h, one kind of fractal structure is composed of bundles made of bundles, made into a single bundle—as the cross—section of a palm tree trunk is made of bundles of bundles of fibers. In the ancient Hindu text Hitopadesha (“Book of Good Counsels” , there is a verse translated by Sir Edwin Arnold which describes this kind of structure: “Small things wax exceeding mighty, being cunningly combined:/ Furious elephants are

fastened with a rope of grass-blades twined” (Wilson, 1900, p. 11). Airplane cables (see Figure 15-9) and the steel cables for suspension bridges are constructed in this bundled manner as well. Fiberoptic cables and nerve structures in the human body, all these structures are tightly packed strandbundles, making a powerful braided strand out of many small interwoven strands, a mighty form made of tiny fibers, bound and bound and bound again. Cross—sections of cables and blood vessels clearly show such fractal structures (see also Jackson, 2004, p. 191, for illustrations of this structure.)

Figure 15-9. Cross-section of cable used for aircraft. Cables within cables within a cable.

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Other fractal shapes in nature are for distribution and circulation, branches spread out filling an area—rivers, trees, blood vessels, lungs. Fronds of ferns have frond-shape leaves, each one having fern-shaped

divisions (see Figure 15-10).

Figure 15-10. Fern fronds with self-similar frond-shaped leaves, and sub-leaves.

Some clouds have fractal shapes, and the wind has a fractal dimension— it comes into existence as waves of energy released from the burning sun

spread out into the atmosphere. The Wind’s fractal aspects have been researched. For example, Syu and Kirchoff (1993) state: “The fractal dimension is related to the universal estimate of the decay rate of turbulent

kinetic energy in the wind

wind parameters have been described as ‘a

fractal and its fractal dimension’” (p. 151). The sun is the generator of

winds, and a model of behaviors, too, in a manner of speaking. Anyone can experience fractals in the world around us. For example,

circle patterns are a sign and symbol, up in the sky, and down on earth. Emerson cited ideas which Plato philosophically wrote:

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By us it is asserted that God invented and bestowed sight on us for this purpose that on surveying the circles of intelligence in the heavens [sun, moon, stars] we might properly employ those of our own minds which

though disturbed when compared with the others that are uniform, are still allied to their circulation; and that having thus learned, and being naturally possessed of a correct reasoning faculty, we might by imitating the uniform revolutions of divinity, set right our Own wanderings and blunders. (Emerson, 1940,13. 485)

Consider how over the centuries philosophers like Plato have understood our relation to natural occurrences around us—thinking of self—similarities on various scales. The humanities offer many meditations on how human life relates to the universe and shares some self-similar aspects. Emerson

cited Plato ’5 Republic: “By each of these disciplines a certain organ of the soul is both purified and reanimated, which blinded and buried by studies of another kind, an organ better worth saving than 10,000 eyes, since truth is perceived by this alone” (Emerson, 1940 p. 485). Consciousness determines fate in this view—what you see is what you get, no more, no less, in Plato’s understanding of the crucialness of seeing: “The soul which has never perceived the truth, cannot pass into the human form” (Emerson, 1940, p. 484). Perhaps this is a way of saying that familiarity with reality (inner and outer) and veracity is a norm for human life, in the philosopher ’s worldview. Emerson wrote in his essay “Circles”: “The eye is the first circle; the horizon which it forms is the second; and throughout nature this primary figure is repeated without end. It is the highest emblem in the cipher of the world” (Emerson, 1940 p. 279). A series of roundness experiences— circular-motif fractals (see Figure 15-1) potentially extend this arrangement, take it to further levels of awakening. Jung elaborates on this topic too, reflecting on how the sun is a great life lesson, an exemplary image of energy and generative power played out in so many similar—smaller existences: Out of the unfolding embrace, the enveloping womb of the sea, the sun tears itself fiee and rises victoriously, and then, leaving the heights of noonday and all its glorious works behind it, sinks back into the maternal sea, into the night which hides all and gives new birth to all. This image was the first to become and with the most profound justification the symbolic bearer of human destiny: in the morning of life, man painfully tears himself away from the mother, fiom the home-hearth, and fights his way up to his full heights. . . . And once he has reached the noonday heights, he must sacrifice his love for his Own achievement, for there can be no standing still. The sun also

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of things, which in their turn begin and complete the sun’s course over again.” (Jung, 1961, pp. 294-295)

Life on earth needs sun, and in our lives, we reiterate in multiple ways the cycles of growing and decline. Mandelbrot noted that fractals were not invented by him but rather discovered. This includes the fractals of the Mandelbrot set, which he illustrated with the help of computer-generated images. They seemed timeless and already there, following their system’s own rules, like the ways of nature. They are summations of inevitabilities in physics, like water rolling downhill in a watershed obeying the law of gravitation to make a branching network of flowing channels—a ridge of land where creeks, brooks, rivers, deltas, take their shapes—a fractal structure. If you speak beneath a bridge, there is likely an echo; echoes are fractals, little repeats of the main sound, little by little disappearing. Many natural fractals, meaning the fractal shapes we notice around us in nature, seem like summations of natural wisdom and temporal changes. These include the rough textures of tree bark, buds and seed clusters, structures of lungs and brains, and clouds, made of small puff-shapes making up larger puffs, and yet larger puffs. This “already there” or “order for free” quality (to use Stuart Kauffman’s [1993] phrase) is like Fibonacci numbers, or great music in the hands of composers, jazz artists, and the raga originators of ancient India. The patterns seem to be already there in the logic of the system; so the potential is uncovered, not invented. Logic or habit may say it is anthropomorphism to say nature has wisdom, but perhaps that is too literalistic a view. Probably human ideals of bicameral or bilateral symmetry for checks and balances (as in the deliberative bodies known as the American Congress and Senate) are derived from seeing nature at work. We have two eyes, two lungs, two hands, two halves of the brain. Nature’s wisdom is non-linear and follows paths of least resistance like water following the forces of gravity. Is there wisdom in nature? Skills of resilience, adapting, surviving are kinds of wisdom. Lao Tzu’s and Chuang Tzu’s ancient philosophy of Tao suggests that humans should respect and emulate wisdom of nature (Tzu, 1963). I define wisdom as a life—supporting, justice-proportionate awareness inherent in psyche. Wisdom shows depth of foresight to anticipate outcomes and guide lives to fullness. Our unconscious is automatic, like nature. We can learn from natural wisdom. (Consider the Nature Channel’s “Wisdom of the

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Wild” TV series for examples of reverse engineering—where humans study nature’s ways of structuring, evolving, adapting to understand how things are done in nature, in order to model technological designs). “Wise” means being forewarned and thus forearmed—with provisions for improvisations of struggle and adaptation. The migration of monarch butterflies is a wonder—such a long migration of many fi'agile insects to such specific locations—and shows a wisdom of survival. To return to Mandelbrot’s intriguing point that fractals were not invented by him but discovered, let us explore that a little. This idea echoes the experiences of different people in different ages and places. It seems to carry a significant sense that some things are intrinsic, pre-existent to our individual experience. We live in an era of novelties and value the innovators of originality, so it may require some context, recalling examples of uncovering the already-existing to consider this. When Mandelbrot speaks of his experience discovering the formula for depicting the fractal geometry of nature in a similar way, not as innovation

but as uncovering what was always there, this may seem impossible, but it may be an honest way of saying “this is what the experience was like.” Michelangelo spoke of finding the form of the pieta (his masterpiece

sculpture of the archetypal mother and her slain son) alreaay there in the marble. It is like the Taoist craftsmen finding and valuing the grain in the

wood alreaaiv there potentially. In Overbye’s (2002) words: “Mathematicians often say that they feel as if their theorems and laws have an objective reality, like Plato’s perfect realm of ideas, which they do not create or construct as much as simply discover” (p. D5). Experientially, it feels like a timeless hidden pattern already in existence. Of course, interpreting what is uncovered and elaborating it involves new words and human means to apply what is found.

Mandelbrot explored and brought to light the nature-secret image of fractals, explored it for mathematical aspects and various applications. He would agree that there are multiple uses for fractals, many as-yet unknown. If you use a cell phone, you should know that each phone uses an antenna which employs a fractal shape: squares within squares within squares. Some poets, musicians and writers have already found fiactals to be very useful. Listen to the Oscar-winning song lyric in Disney’s movie “Frozen” for an evocative mention of the term “fractal.” Writer David Foster Wallace employed a fractal model (the Sierpinski Gasket) for his novel The Pale King and used the concept of fiactals to great effect in a story “Good Old

Neon” in the final paragrap : “.. . this is what it ’5 like. That it’s what makes

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room for the universes inside you, all the endless inbent fractals of connection and symphonies of different voices, the infinities you can never show another soul” (italics mine, Wallace, 2004, p. 179). This is like Heraclitus’ remark that soul is what you never find the end of, no matter how far you go. As fractals are being explored in various other fields, with research findings constantly growing, Marks—Tarlow is voicing the need to explore possible uses of fractal patterns specifically in transpersonal psychology. For visualizing the relations of humanities and hard sciences among transpersonal-psychology professionals, Marks-Tarlow suggests trying out fractals as potentially able to help us envision the overarching harmony. The visual example of fractals helps us think in more holistic, nonlinear ways, with more freedom to realize relationships of wholes within parts, mindful

of necessity.

Seeking goals of a plea: Reflections in a network of pearls Marks-Tarlow’s plea for a fractal epistemology is an attempt to envision more sense of wholeness in the community of transpersonal psychology. Divergences among types of psychologists include: more scientific rigor or less, more theory or more practice, more introverted or more extrovertcd, more prosaic analytical research or more Shamanic revelations, more humble and practical or more big—picture oriented, humanities or hard sciences, more brain neurology or more myth, literature and art.. .. The schools of transpersonal psychology addressed in Marks-Tarlow’s plea are characterized as valuing multiple approaches—using as many as three other schools or therapies in defining their approach. Because transpersonal psychology is diverse and eclectic in its various practitioners’ methods, visualization of their wholeness/partness could help provide a sense of overall coherence and cohesion. Fractals could offer conceptual images to accommodate these needs. Because epistemology studies the valid proofs and concern with the theory of knowledge—methodology, scope and boundaries of a field, it is foundational even if it develops over the years, catching up to actual practice and experience. Epistemology also involves researching and critically thinking about factors that distinguish justified belief in the factual from unproven opinion. What are the features of a fractal epistemology? Fractals are mathematical, and so abide in neutral, non-judgmental ways. Epistemological pluralism is a conscious choice, with multiple—valuable sides honored as

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useful and necessary. Both hard sciences and humanities can co-exist, enrich each other, while also respecting the differences of their own spheres. How might the non-linear, aesthetic, geometric, fi'actal sensibility support an epistemology valuing the limits and defined aspects of hard sciences’ knowledge, while respecting the creativity and rich multiplicity of the humanities’ side of transpersonal psychology? A tolerant “Somehowism” of accepting that many perspectives exist, as parts of a larger whole, harmonized constantly by acknowledging the other’s existence. A geometry which is organic, supporting both the humanities and sciences, and finding ways to appreciate and revel in multidisciplinary, holistic approaches. Austere formulas which allow for Pandora’s box of hopeful images, fantasies, meaningful riffs. Respect for the reductionism of Occarn’s razor while organizing in a fiactal—like manner a therapeutic toolbox to accommodate both. Marks-Tarlow’s proposal is a call to explore ways of fitting together rather than being split apart—a consortium, with family resemblances. Fractals can help this transpersonal psychology community contemplate its self-organization, visualizing understanding of the whole, refining interrelationships, increasing contextual coherence in community terms. The necessary resources to re-vision the relationships among differing styles of transpersonal psychology include fiactal designs used by timeless nature along with tools specific to our age—computers, statistics, and so forth. We seek a harmony, and some fractal images show how contrasting aspects are interrelated or even complementary, how in some cases each is inside the other—interrelated by sharing mutuality. We hope to visualize harmony in the fractal forms of templates of organic composition, beautiful like Gaudi’s architecture (see Figure 15-11), vivid and graceful with an economy of function. An elegant fractal functions well in doing this—we recognize graceful coexistence when we see it, we instinctively know it is not a grotesque Rube Goldberg Machine which just grew any which way by happenstance.

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Figure 15-11. Gaudi’s Sagrada Familia in Barcelona, Spain. (Courtesy of Bernard Gagnon)

Some fractals can help expand the canvas of our imaginal possibilities. Marks—Tarlow’s insightful writings about the “fractal self ’ explore how fractal geometry helps us think about relativity and can lead to a non-linear dynamical theory of self (Marks—Tarlow, 1999). Her article on “Fractal Self

at Play” is a helpful exploration of rich observations about development of the self. “A fractal model suggests that the whole of the self, intact during early play, exhibits self-similar resonances in the content and forms of selfexpression throughout life” (Marks-Tarlow, 2010, p. 31). Fractals can add a new element or dimension to charting our understanding of human interrelations. Used this way, a fractal is a model, metaphor, schematic template of wholeness. To chart psychological

conditions, to help provide common cause or sense of belonging to those who are alienated, fractals are a worthy candidate, a potentially helpful conceptual tool for transpersonal psychology to employ. How are

professional decisions like using a fractal epistemology in transpersonal psychology made, how do professional communities adopt a valuable tool? Usually through study, debates and conference meetings and votes and usage. How do things become part of the culture in America? An embrace of the pragmatic is a traditional value in American history. If it works, it is good, Americans of the past have philosophized. But at first new ideas are

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often not accepted. It takes time and effort to focus on them, understand them, try them out. General/clinical psychology may steer its procedures one way, and transpersonal psychology, which includes the study of expanded or altered states of consciousness, has its own ways. For both, the term “fractals” should be included in everyone’s vocabulary, and in the lexicon of every educated person. They enhance our awareness of the world and of ourselves. As noted, Mandelbrot himself famously said fractals are not a panacea, so a cautious optimism is required. In a sense, I believe Mandelbrot is echoing the saying: “Be bold, be bold—be not too bold! ” Fractals will prove their own value by being useful when tried out in actual situations. In other words, proceed with enthusiasm, yet with care. Fractals are being explored in an astounding number of possible applications in science. They can offer some freedom of speculation and scope for imagination on both pleasant and thorny issues. Fractals will become integrated as a part of psychology when fractal orders are registered as impacting our psyches and our lives.

Tapping hidden fractal wisdom like the swan Much depends on the odd structure of the psyche which possesses a dualpotentiality, “Everything of psychic origin has a double face. One face looks forward, the other back. It is ambivalent and therefore symbolic, like all living reality” (Jung, 1961, pp. 115-116). As the ancient Romans honored Janus as double—face god of doorways, drama has tragedy-and—comedy masks, some fractals have intricate borders of mirror-like contours, yin and yang polarities contrasting, bargaining, and dancing with each other every step of the spiral way. Transpersonal psychology depends on conscious analysis as well as openness to learning from the unconscious. One’s honest unconscious illuminates what one does not consciously know by reminding one—it

wakes me up in the middle of the night and I think “Oh yeah!—there’s a blind spot, a background dimension to explore.” The wisdom of uncertainty

is reliant on the unconscious sense. “If you are too unconscious it is a great relief to know a bit of the collective unconscious. But it soon becomes dangerous to know more, because one does not learn at the same time how to balance it through a conscious equivalent...” (Jung, 1976, p. 173). In the origin story of major Western traditions, Adam and Eve ate from the tree of knowledge of good and evil, and it was complex. Sometimes a question leads to uncertainties,

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sometimes discoveries bring ambivalence and dilemmas, but in time wisdom fi'om one’s psyche helps. Knowledge from personal experiences of the unconscious can offer fresh perspectives and challenges. Aware of the unconscious we explore possibilities to work with it, and “In dreams begins responsibility” (Yeats, 1914, up). Being “woke” in current vocabulary is a state related to this— awakened awareness of the ignorance inherited fiom past societal injustices, like institutional racism. Cooperating with the unconscious is rewarding in understanding if not always comforting—there are frustrations involved in “being conscious in an unconscious society” (Davis, 2017, n.p.). The shadow side of the culture, and one’s own shadow are personal challenges. Some truths may be inconvenient, encountering complexities and depths raises questions and debates. James Hillrnan’s

insights into the unconscious and archetypes in

literature are profound resources (Hillrnan, 1979, 1997). And Stanislav Grof has researched the realms of the human unconscious with appropriate care and found valuable and enduring insights into human nature (Grof, 1975). Grof’s explorations of healing holotropic experiences and related topics integrate Freudian, Jungian, Shamanic, and wisdom traditions and are fearlessly non-reductive. Grof personally experienced geometrical imagery in his experimental therapies with sound and light. This imagery included shapes he identified as visual fractals during a dramatic-crucial moment of his life and research (Van Nise, 2011). Boundaries are fluid in the experiences of healing holotropic states, and fractals can depict boundaries with precision and the vitality of playfulness. The non-reductive nature of fractals and consciousness require due attention, and reveal depths requiring time, thought and maturation. Fractals are a nonlinear, non-Euclidian resource, which engender a sensibility, a way of thinking. Like the genetic structure of DNA, and like the meridians referred to in acupuncture, they offer a set of possible configurations in a system with which we are not fully familiar. New thinking begins with observation, subtle hints and hunches, possible paths to pursue—threads of thought emerging from the unconscious. Fractals are timely, they voice an emerging realization in our time, articulating the spirit of this age, with its new studies in chaos theory, dynamical non-linear systems theory, quantum theory, and so forth—and with our needs of visualizing harmonies of parts and wholes, a sense of mutuality in self-similarities. There is a fiactal image which is entitled

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“Peaceful Borders between Three Countries” (see Figure 15-12). Because each contains some of the other, and so is not entirely other.

Figure 15-12. "Peacefirl Borders between Three Countries.” Each contains some of the other. We delve into and sort out the issues w e are confronted with, and the

process of finding clarity in chaotic situations, of finding knowledge inthe midst of ignorance, is a heroic struggle. Discerning the perennial philosophy’s timeless truths or spiritual meanings is at the heart of being a seeker. For an excellent review, see “From Philosophy to Phenomenology :. The Argument for a ‘Soft’ Perennialism,” by Steve Taylor (2016), for those who are interested in the Perennial Philosophy. In South India there is a memorable image for this practice of seeking: the swan is a mythological creature who is ostensibly able to separate milk from water when drinking a mixture, taking the milk and leaving the water. The swan is symbolized by hands held palms together as in prayer, with fingers like wings and thumbs nearest the heart, like a swan tasting truth there. The milk represents the timeless truth or eternal reality found mixed in the changing world of

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samsara, focusing on the enduring spiritual value. Similarly, the Buddhist

mantra “The jewel is in the lotus” (“0m mam' padme hum”) means finding the ultimate in the world of growing and declining. We humans seek clarity and coherence in our world of turmoil. Large or small, coherence is recognizable and appreciated as a value to appraise and evaluate phenomena. At their open-ended worldview meeting places, science and perennial philosophy are compatible. Both guide and help us through the ups and downs of life. Science can appreciate wisdom, and wisdom can appreciate science. This can especially be seen in a scientist like Einstein, who honors the mystery which human egos cannot fathom. The cosmos itself is a great mystery. Are there any known gravitationallybound structures anywhere in existence larger than galaxy super-star clusters? Star clusters are structures of stars within galaxies (see Figure 15— 13). Significantly, the more massive galaxy clusters are scaled-up versions of smaller ones—meaning, they are fractal shapes. Such vast distances, given shape and dimension by super-cluster galaxies may bring to some people a mood of feeling that the individual human being is inconsequential Transpersonal psychology does not leave out cosmic mystery, nor does it ignore suffering. Is there any reason to think that fractal geometry might be too abstract to deal with chronic existential experiences such as suffering? Can fractals play apart in assuaging pain, anxiety, bewilderment? Can they be useful in psychotherapy? If not, how useful are they in therapy? I believe they can be a tool for those issues, and I don’t think the same thing can be said for algebra or Euclid’s geometry. The inspired, who see the potential of the new images of fractal geometry, will be trying it out. When I search for a touch of kindness fi'om fractal studies, I find this aspect of fractals is already happening. Allow me to mention at least a couple examples showing how the presence of fractal sensibilities lends itself to help with suffering and pain.

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Figure 15-13. Hubble space telescope image of the Antennae Galaxies. m ASA)

One example I immediately found was in a book review entitled “Faith,

Fear, and Fractals,” by Tarn Wilson (2016). In his review of the book Fractals by William Bradley, Wilson had this to say about the author who wrote about overcoming a serious illness: I believe Bradley hopes we will see something of ourselves in his reflections and little snippets of memory, and these glimpses will represent more than the moments of one small life. In the essay “Self-Similar,” Bradley explains his fascination with fractals. He muses, “I can’t help but feel like we’d be better as a species if we tried to imagine each individual as a self-similar component of the larger human race . . . fundamentally identical parts which

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Chapter Fifteen represent a similarly identical whole.” Although he writes in the same essay that “it’s dangerous to make sweeping statements about universal truths,” he asserts that, because we all are part of the whole, any story, honestly told, will transcend our own experience.” (Wilson, 2016, n.p.)

Another piece of writing that turned up was “Pain—what do fractals have to do with it?” This is the title of a website for Kory Zimney, titled “Modern manual therapy—the eclectic approach” (see https://www.evidencein motion.com/blog/2017/08/l 8/pain-what— do-fractals-have—to-do-with—it/). It takes the image of fractals, about which many already have a general idea from documentaries like science writer and inventor Arthur C. Clarke’s

(Lesmoirs-Gordon & Sinclair, 1995) “Colors of Infinity” and relates it to pain by discussing those features. It is the intuition of those descriptive aspects of fractals which inspired Dr. Zimney, the physical therapist writing about this topic, to choose the fractal idea or ideal to stand for the new paradigm for the healing work he is engaged in. It is heartening to know that in our era we can witness such interesting advances in various sciences, many related to complexity and chaos theory, nonlinear systems theory and other emerging studies related to fractals. Look back at previously unrelated fields, separate and isolated; then look at the research pouring out. We live in a time of fruitful conjunctions, new ways of seeing systems and processes. Our sciences are coming together— geometry, physics, physiology, zoology, botany, ecology, medicine, astrophysics, climatology, electronic technology, and so forth. Consider E. O. Wilson’s (1999) book Consii'z'ence for example. It seems that in every field there are fractal studies exploring aspects of the unknown universe around us.

Conclusion As we incorporate new knowledge, transformations both inner and outer occur. Epistemology keeps us honest about our knowledge. In Jung’s words, “To have a Weltanschauung [worldview] means to make an image of the world and of oneself, to know what the world is and who I am. Taken literally this would be too much. No one can know what the world is, and as little also can he know himself; but cum grano salts, it means the best possible knowledge—a knowledge that requires wisdom and the avoidance of unfounded assumptions, arbitrary assertions, and didactic opinions” (1961, p. 247). With our modesty tempered by this knowledge of the scale of the universe and the depth of our psyches, fractal research promises, still

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at the beginning of its career, to continue uncovering life-changing perspectives and applications. The fractals of the future, their uses and beauties, are waiting in the wings. They will appear and perform their functions, with the wisdom of interconnected wholenesses. Arthur C. Clarke pointed to the image of the map described in old stories, a diagram on parchment showing the location of a buried treasure, and he observed that, in the case of fractals, the map— the idea of fractals itself— is the treasure (Lesmoirs-Gordon & Sinclair, 1995). The fractal maps nurture in our intuitive intelligence a fractal sensibility, endowing upon us a familiarity with organic wholes and parts, self-similarities echoing variations of the theme, opposites joined in nonlinear dynamics. The fi‘actal sensibility shows an aptitude to serve as a helpful window and background to enable “a both/and kind of thinking” in the world of transpersonal psychology. I mean here not the mathematical formulas of fractals, which are beyond most of us, but the basic ideas and beneficial potentials associated with fractals which I have been discussing. Fractals—uniquely useful when you are trying to be specific, yet infinitely deep about the personal and the cosmic.

References Alexander, S. (2017). The jazz of physics: The secret link between music and the structure of the universe. New York, NY: Basic. Barcellos, A. (1984). Benoit Mandlebrot. In D. Albers & G. Alexanderson

(Eds), Mathematical people (pp. 213-234). Cambridge, MA: Birkhauser Boston.

Briggs, J. & Peat, F. (1989). Turbulent mirror: An illustrated guide to chaos theory and the science of wholeness. New York, NY: Harper & Row. Campbell, I. (2008). Hero with a thousand faces. Novato, CA: New World Library. (Original work published 1949)

Cook, F.H. (1977). H ua-Yen Buddhism: The jewel net of Indra. University Park, PA: Penn State Press.

Drescher, E. (2016). Choosing our religion: The spiritual lives ofAmerica ’s nones. New York, NY: Oxford University Press. Davis, B. (2017, October 9). People think it’s fun and cool to be ‘woke’ 101 it’s actually. . .pretty shitty and frustrating being conscious in an unconscious society. [Tweet]

Eglash, R. (1999). African fractals: Modern computing and indigenous design. New Brunswick, NJ: Rutgers University Press. Einstein, A. (1982). The world as] see it. New York, NY: Crown Publishers.

Chapter Fifteen

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Emerson, R.W. (1940). The selected writings of Ralph Waldo Emerson (B. Atkinson, Ed.). New York, NY: The Modern Library. Gleick, J. (1987) Chaos: Making a new science. New York, NY: Penguin.

Grof, S. (1975). Realms of the human unconscious. London, UK: Souvenir Press. —. (2008). A brief history of transpersonal psychology. Retrived fiom: http://theeggshelllanding.comfresources/A_Brief_History_of_Transper sonal_Psychology_Grof.pdf.

Hillman, J. (1979). The dream and the underworld. New York, NY: Harper and Row.

—. (1997). Re-visioning psychology. New York, NY: William Morrow. —. (2005). Senex & puer (Uniform Edition of the Writings of James Hillman, Volume 3). Putnam, CT: Spring.

Jackson, W.J. (1993). T yagaraja: Life and lyrics. New Delhi, India: Oxford University Press. —. (2002). Hindu temple fractals. Retrieved fiom Academia website https:l/www.academia.edu/347639/Hindu_Temple_Fractals.

—. (2004). Heaven ’5 fractal net: Retrieving lost visions in the humanities. Bloomington, IN: Indiana University Press.

—. (2007). Vijayanagara visions: Religious experience and cultural creativity in a south Indian empire. New Delhi, India: Oxford University Press.

Jung, C.G. (1933). Modern man in search o f a soul (W.S. Dell & C.F. Baynes, Trans.) New York, NY: Harcourt Brace and World. —. (1961). Psychological reflections (J. Jacobi, Ed.). New York, NY: Harper Torchbook. —. (1976). Letters Vol. II 1951-1961. (G. Adler, Ed, J. Hulen, Trans). Princeton, NJ: Princeton University Press. —. (1984). Dream analysis seminar (William McGuire, Ed.,Trans.) Princeton, NJ: Princeton University Press. Retrieved from https:l/carljungdepthpsychologysite.blog/2017/08/02/carl-jungsvisions-seminar-lecture—iv-l3-february-1929/ —. (1987). C.G. Jung speaking: Interviews and encounters. ( R F C . Hull, Ed., Trans). Princeton, NJ: Princeton University Press.

Kauffinan, S. (1993). Origins of order: Self-organization and selection in evolution. Oxford, England: Oxford University Press. Lesmoir-Gordon, N . , & Sinclair, P. (Producers) [with Lesmoir-Gordon, M.

(Director)]. (1995). The colors of infinity [TV movie]. United States: Gordon Films, Iterated Systems, New Moon Pictures.

Mandelbrot, B. (1977). The fractal geometry of nature. San Francisco, CA: W . H . Freeman.

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Manning, C. (2015). Losing our religion: How unafi‘iliated parents are raising their children. New York, NY: New York University Press Marks-Tarlow, T. (1999). The self as a dynamical system. Nonlinear Dynamics, Psychology, and Life Sciences, 3(4), 311—45. —. (2008). Psyche ’s veil: Psychotherapy, fractals and complexity. New York, NY: Routledge.

—. (2010). Fractal self at play. American Journal of Play, 3(1), 31-62. Overbye, D. (2002) The most seductive equation in science: Beauty equals truth. New York Times, March 26.

Roszak, T. (1993). The voice of the earth: An exploration of ecopsychology. New York, NY: Touchstone.

Sontag, S. (2012). As consciousness is harnessed to flesh: Journals and notebooks, 1964-1980 (P. Rieff, Ed.). New York, NY: Farrar, Straus and Giroux. Syu, C.Y., & Kirchhoff, RH. (1993). The fractal dimension of the wind.

Journal of Solar Energy Engineering, I I 5(3), 151-154. Taylor. S. (2016). From philosophy to phenomenology: The argument for a

‘soft’ perennialism. International Journal of Transpersonal Studies, 35(2), 17-41. Tzu, L. (1963). Tao te ching. (D. C. Lau, Trans.) Retrieved fi'om: https://www.centertao.org/essays/tao—te—ching/dc-lau/chapter-56commentary/.

Van Nuys, D. (Producer) (2011, March, 31). Stanislav Grofi MD on transpersonal psychology and the meaning of psychedelic experience [Audio podcast]. Retrieved from: https:l/podcasts.apple.com/us/podcast/stanislav-grof—md-on— transpersonal—psychology-meaning/id2 1 8827921?i=1000092637609 Wallace, DP. (2004). Oblivion: Stories. New York, NY: Little, Brown & Co.

Wilson, E. (1900). The literature of India, Oriental literature, Volume III, H itopadesha (S. E. Arnold, Trans). New York, NY: Colonial.

Wilson, ED. (1999). Consilience: The unity of knowledge. New York, NY: Vintage. Wilson, T. (2016, April 6). Faith, fear, and fractals, areview by Tam Wilson of Fractals by William Bradley. Retrieved from: http://wwwriverteethj ourna1.com/’blo g/20 1 6/04/08/faith—fear-and— fi'actals

Yeats, W.B. (1914). Responsibilities anaT other poems. Dublin, Ireland: Cuala.

Zuckerman, P. (2014) Living the secular life: New answers to olaT questions. New York, NY: Penguin.

CHAPTER SIXTEEN

THE FRACTAL QUALITIES OF HALLUCINATORY PHENOMENA: ON GEOMETRIC FORM CONSTANTS AND THEIR IMPLICATIONS FOR THE PSYCHE J ESUS-lVlARIO SERNA‘

To introduce fractal geometry and illustrate the need for a more accurate description of natural phenomena, its founder famously stated: “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line”(Mandelbrot, 1982, 13.1). Classical Euclidean geometry primarily studied ordered and smooth forms seldom found in nature, like very regular curves or plane figures bounded by lines. Mandelbrot developed his “theory of roughness” to describe natural forms with mathematical tools. He also studied mathematical forms such as the Cantor set or the Koch curve, both of which contain forms of infinity—infinitely small for the first and infinitely long for the second. Before the advent of fractal geometry, these forms were called mathematical “monsters” and “pathological” constructs devoid of concrete interest because of paradoxes or difficulties analysing them. Nonetheless, the concrete application of these examples led the father of fractal geometry to propose a novel measurement method for fractured natural shapes such as coastlines, by taking into account their irregularities and fractional dimensions (Mandelbrot, 1967), as well as tackling realworld problems like the errors in electronic transmission lines. Phenomena 1 Clinical Psychologist; PhD. candidate, Research in Psychoanalysis and Psychopathology, University of Paris VII; Center for Research in Psychoanalysis, Medicine and Society, CRPMS, F-75013. Research and Teaching assistant at the Institute of Psychology, Université de Paris, PCPP, F-92100. Email: [email protected]

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are often considered “irrational” or “meaningless” when they cannot be explained within the framework of a prevalent paradigm, and thus tend to be ignored. Nevertheless, efforts to integrate such outliers as a valid source of knowledge can also lead to new disciplines that offer previously unavailable explanations. Likewise, one can say that this kind of approach led Freud to create a new method for understanding and treating hysterical phenomena (Freud & Breuer, 1893/2001). Along with Breuer, Freud proposed deeper causes and underlying relations that deviated from mainstream medical views of the time. Unable to find linear causes in the scope of their discipline (i.e., physical paralysis without any damage to bodily tissue), physicians of the day tended to reduce hysterical aberrations to the uterus, capriciousness, or hereditary factors. Freud’s close attention to contradictions led him to the

systematic study of unconscious processes that eventually founded psychoanalysis. The effort to understand phenomena left outside existing scientific paradigms can also be found in transpersonal psychology, including its study of phenomena beyond the usual boundaries of the ego, such as nonordinary states of consciousness (Grof, 2008). In this regard, what I find most remarkable about Terry Marks-Tarlow’s proposition of a fractal epistemology for transpersonal psychology (this volume, chapter one) is the therapeutic edge to this research approach, which carries the potential to integrate conflicting views in the field. This novel exploration opens the opportunity for active interdisciplinary dialogue, which is necessary to work through research issues. In any thought system, to consider unfamiliar views can generate a temporary disorganization, but this is a necessary step for empirical evaluation and reorganization into a higher dimension of complexity. I have conducted research on repetition and self-similarity in the field of psychoanalysis and so am acquainted with what is proposed here for transpersonal psychology. In a recent study (S ema, unpublished manuscript), I carried out a bibliographical exploration of the largest database for

psychoanalytic publications, Psychoanalytic Electronic Publishing (PEP). I searched for documents that mention fractals in order to track their use and spread over time. Preliminary results fiom this literature review show an undeniable increase in the relative frequency of fractal citations in the database, extending from their first publication in 1988 until the present. To explore how this might translate to the field of transpersonal psychology, I next studied the database of the California Institute of Integral Studies’

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(CIIS) Digital Commons repositoryz. Results3 over the last twenty years indicate that the relative frequency of the mention of fractals is approximately twelve times higher in the CIIS Digital Commons repository than in the whole of the CIIS repositories, which comprises 478 institutions and a myriad of disciplines“, and about five times higher than the ratio in PEP5. Although these numbers reflect different scales of magnitudeé, and the PEP and CIIS repositories do not function equally, results suggest that transpersonal researchers7 are more aware of fractals than are psychoanalysts, all schools mixed. Given the foundational interest in transpersonal psychology for nonordinary states of consciousness (Hartelius, Caplan, & Rardin, 2007), peak experiences, and experiential therapy, this chapter focuses on the striking fractal manifestations in the psyche that are linked to hallucinatory phenomena. Such phenomena often come along with the common denominators of transpersonal experiences, such as “the individual's feeling that his or her consciousness has expanded beyond the ego boundaries and has transcended the limitations of time and space” (Grof, 1985, p. 41). Nonordinary states of consciousness and their concomitant hallucinatory phenomena undeniably demonstrate fractal qualities, and thus require a more accurate descriptive framework. This need gets clearer when passing from the basic geometric-form constants in the first stage of hallucinations

2 https://digita1commons.ciis.edu/ 3 Out of 887 documents in the CIIS Digital Commons repository, there were 23 fractal hits versus 5601 fractal hits out of 2786492 documents in the larger CIIS

database. This translates to approximately 2.59% fractal hits in the CIIS Digital Commons Repository versus 0.2% in the general platform. 4 The full list of institutions using the Bepress Digital Commons software is available here: httpszllwww.bepress.com/categories_wdc/all—institutions/ 5 In the same timeframe, from 01/01/1999 to 12/21/2018, in the PEP database there are 39038 total documents, out of which 206 have fractal references, for a ratio of 0.52%, making the CIIS fractal hits ratio around 4.98 times that of PEP.

5 As the CIIS Digital Commons repository is smaller and less robust than the PEP Database (with publications in the hundreds, not in the thousands), the scales of magnitude are quantitatively different. There is a recent increase in fiactal citations in CIIS Digital Commons, but it is also more sensible to singularities that can change

the proportions more readily. To exemplify this, we can see that the fractal references in the C118 Digital Commons are mostly stable from 1999 to 2015, with up to two hits per year, suddenly spiking to six in 2016 and eight citations in 2018,

all of which stem from the Journal of Conscious Evolution. 7 Most of the 23 fractal hits in the CH3 Digital Commons repository derive fiom sources such as the Inremarional Journal of T rampersonal Studies.

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to higher-ordered complexity and more elaborate constants, such as deathrebirth experiences. In order to exemplify how a fractal epistemology could further illuminate our knowledge of transpersonal phenomena and offer useful clinical applications, I first review research from the neuroscience of hallucinations, including a close examination of Heinrich Klfiver’s form constants (recurring geometric patterns in hallucinations). I also summarize some of the work by Jack Cowan and his team, who propose a neuromathematical model for the functioning of the visual cortex that can explain the emergence of form constants. These findings shed light on the underlying mechanisms at work in different levels of the psyche that might be linked to self-similar reverberations at difierent scales of observation.S Next, I articulate these discoveries with insights accumulated from years of experiential therapies in transpersonal psychology, stemming primarily from the work of Stanislav Grof and Richard Yensen. Finally, I briefly mention insights from my own clinical experience in the field of psychosis, presenting an excelpt from a schizophrenic patient that exemplifies how fractal geometry can orient clinical work. I further illustrate this by identifying phenomenological similarities to Stanislav Grof’s (1975) cartography of inner space. Grof’s cartography extends from abstract and aesthetic levels to psychodynamic, perinatal, and transpersonal domains, which can share similar elements and processes separated by fuzzy boundaries. My hypothesis is that, if one can identify elements in the psyche that repeat at different scales, exhibiting the hallmark quality of fi'actals— self-similarity, then fractal geometry can be a viable fi'amework to reveal aspects that are otherwise hard to appreciate. In this way, the understanding of basic self-similar processes at work might elucidate underpinning mechanisms and their links to various levels of psyche. This could bring about a better understanding of higher-ordered phenomena, including transpersonal experiences.

5 One should be careful not to reduce various phenomena at different levels of observation to the same dimension. For example, mirror neurons are not equal to

empathy, but rather they might correlate with empathic experience. Thus we cannot reduce and dissolve the empathic experience observed at a psychodynamic level to the neural correlates that may be linked to it. One could say that we encounter two different phenomena stemming fiom different fields of observation that might intersect at some points. Nonetheless, identifying self-similar processes can help us understand parallels at different levels, for example between the neural and biological substrates to the psychodynamic levels of the psyche.

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Background on a fractal epistemology for transpersonal psychology A Non-linear Dynamical Systems (NDS) fi'amework considers the system as a whole and describes how complex phenomena change over time—their possible patterns and interrelationships. Nonlinear systems respond in different ways to the same input, depending upon their state and context, in other words, their “set and setting.” Effects are not always directly proportionate to causes, and extreme sensitivity to initial conditions abounds. Fractal geometry is “a member” of NDS, offering closer and more harmonious descriptions of natural forms than Euclidean geometry, with its classical, deterministic, and linear conceptions that governed scientific thinking for over three centuries (Prigogine & Stengers, 1979). Terry Marks-Tarlow proposes fractal geometry as a “model, method, and metaphor for otherwise ambiguous and inaccessible transpersonal phenomena” (this volume, chapter one, p. 2). She describes self-similarity as the hallmark property of a fractal, such that the large-scale pattern of the whole gets repeated on multiple size or time scales within the parts, as involves recursive or self-reflexive symmetry. Fractals also provide tools for grasping the elusive concept of infinity, as found in mathematical objects like the Cantor and Mandelbrot sets, as well as in accounts of hallucinations. These concepts are useful for considering ego boundaries. As Marks-Tarlow states (this volume, chapter one):

This very sense of boundary-less interconnection and complete interpenetration of inside and outside realms corresponds to mystical experiences and peak states like “nondual” awareness, whether facilitated by psychedelic substances or occurring naturalistically (p. 13);

Because of the properties of being multi-scaled and infinitely deep, fractals do not have clear boundaries. Their measurement is not fixed but is fuzzy and dynamical instead (p. 14);

Computer modeling of interpersonal dynamics demonstrates yet again how fractal bOundaries arise out of complex feedback loops between inner and outer processes, such as self and other (p. 27); and (F)ractals can illuminate complex interrelationships, such as the interpenetration between brain and mind, self and other, or inner versus outer realms (p. 4).

In this chapter, I illustrate how we might pursue this enterprise by Considering self—similar links between neural correlates of specific hallucinatory phenomena and their reverberations in the psyche throughout

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distinct realms of experience within Grof’s cartography. This complex interrelation would operate in nonlinear fashion, involving iteration and feedback loops between levels, without employing an exclusively top-down or bottom-up causal progression. In psychoanalysis, awareness of NDS, chaos, and complexity theory dates back to at least 40 years (Galatzer-Levy, 2017). A growing interest in fractals has crossed the often-rigid boundaries of psychoanalytic orientations, transcending geographical boundaries, as well as language barriers (Serna, unpublished manuscript). Psychoanalyst Robert Galatzer— Levy concludes that: This group of explanations, which has globally been referred to as chaos theory, involves a qualitative as opposed to quantitative view of nature, and addresses questions in a wholly different fashion fi'om a traditional physical science investigation. Its capacity to deal with the intricacies of nature and the complexities of real systems, albeit in a different way from classical physics, makes it a promising candidate for a different form of theorizing than has previously been used in psychoanalysis. (Galatzer-Levy, 1995, p.

1110) Even though Freud did not have such models at hand, he believed in “a strict and universal application of determinism to mental life” (Freud, 1910/2001, p. 52). A closer look at the determinism applied in Freud’s model of the unconscious reveals that it deviates from simple-linear models by delving into complex interrelations readily found in unconscious processes, which I call “the Freudian leap into complexity.” One clear example is the Freudian notion of overdeterrnination or multiple determination, which is used extensively in his foundational work on dreams and is a central notion underpinning his entire body of work9. In the field of neuroscience, NDS, chaos theory, and network theory have proven to be useful frameworks for studying the nervous system’s functions and complex interactions. Notable work in this line of research includes the fractal model of neurons (Pellionisz, 1989), chaotic neurocortical dynamics (Freeman, 2013), nonlinear dynamical analysis of sleep electroencephalography using fractal approaches (Ma, Shi, Peng, & Yang, 2018), and neural network models (Bassett, Meyer-Lindenberg,

9 I presented a thorough investigation of this topic for the grade of Masters in Research in Psychoanalysis, specialized in clinics of the body and society,

completed on July 2013 at the University of Paris VII, under the supervision of Pr. Paul-Laurent Assoun, with jury by Markos Zafiropoulous.

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Achard, Duke, & Bullmore, 2006). Jaak Panksepp, an essential figure in the field of neuropsychoanalysis, commented briefly on the subject: During development, especially early childhood, the basic emotional structures, when aroused intensely, may flare with the light of affective consciousness, which may serve as global attractors for other brain activities as the passions fractally spread through reverberating, hierarchically arranged ensembles of neurons through many areas of the brain. As the higher brain areas become engraved with the fi'actal signatures of the various substrates of the SELF (e.g., emotional temperaments), they may develop a hegemony that promotes specific types of cognitive some independence tendencies capable of elaborating various affectively biased views of the

world. (Panksepp, 2000, p. 31) In humanistic psychology, complex systems have been proposed as a useful means to study the transcendental potentials of the psyche at various levels of complexity (Krippner, Ruttenber, Engelman, & Granger, 1985). It has also been noted that “the application of chaos theory to human behavior and experience demonstrates an affinity with humanistic psychology that deserves attention” (Krippner, 1994, p. 48). In the field of transpersonal psychology, the importance of being open to interdisciplinary inspiration from discoveries by other scientific fields as they converge with transpersonal phenomena has long been highlighted by some of its founders. In Beyond the Brain, Stanislav Grof (1985) mentioned a plethora of scientific references that influenced his thinking. Among these Grof cited his personal relationship with cybernetician Gregory Bateson while at the Esalen Institute in California. Bateson was a special friend who enriched Grof’s understanding of information and systems theory, offering a key critique of mechanistic thinking in science. Grof also mentioned that “certain universal principles can be discovered that will be applicable in different domains” (Grof, 1985, p. 88), notably Prigogine‘s order through fluctuation (Prigogine, 1980) and René Thom's catastrophe theory (Thom, 1975). He further emphasized: It can be demonstrated without much effort that most of the material fi'om LSD psychotherapy, although quite puzzling and incomprehensible fiom the point of view of mechanistic science, presents far less difficulty when

approached in the spirit of quantum-relativistic physics, information and systems theory, cybemetics, or recent discoveries in neurophysiology and biology. (Grof, 1985, p. 51)

Grof also brought up Arthur Young’s (1976) theory of process, which proposed abasic pattern of the universal process that repeats itself—highly reminiscent of the fractal property of self-similarity. Although associated

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with some of the aforementioned fields, fractal geometry was not explicitly

brought up by Grof at the time”, but strange attractors“, most of which are fractal, did made a later appearance in his joint work on holotropic breathwork (Grof & Grof, 2010). Furthermore, the vast amount of clinical material that Stanislav Grof accumulated from his work with experiential therapies12 led him to suggest a new cartography of the psyche, where he identified four main levels of distinct experience. This cartography originally stems from Grof’s therapeutic work with LSD, but it could also apply to other kinds of clinical work that do not involve the use of substances, such as his trademark approach to holotropic breathwork. In my opinion, Grof’s cartography clearly reflects phenomena observed in clinical work with psychoses and

autistic states; and although less obvious in neurotic structures, it is still relevant. Grof refined his cartography several times, but its main elements remained quite stable throughout his body of work. The first level encompasses abstract or aesthetic experiences, mainly explained in terms of Western medicine’s understanding of the anatomy and physiology of the sensory organs. Richard Yensen (1998) explicitly recognized the value of fractal geometry for grasping phenomena often found at this level, like the perceived infinity of psychedelic visions. Such abstract or aesthetic experiences may encompass any of the senses. They often include geometric distortion of objects in the environment, the appearance of architectural patterns like temples, mosques, or cathedrals, or surreal images reminiscent of the art of Salvador Dali, Braque, or Picasso.

The second realm concerns psychoobinamic, biographical, or recollective levels. This level is in accordance with Freudian theory but may have an added “experiential” quality. Here, Grof points out elements I argue possess

a self—similar quality:

1° Although Mandelbrot’s theory was drafted as early as the mid-seventies, his ideas spread outside specialized mathematical circles mainly after the English publication of The Fractal Geometry of Nature in 1977. 11 In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system (Boeing, 2016); strange attractors often have fractal qualities while exhibiting chaotic dynamics.

12 Since 1954, spanning nearly three decades, Grof personally guided more than 3,000 sessions with LSD (Grof, 1985).

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In deep experiential psychotherapy, biographical material is not remembered or reconstructed; it can be actually fully relived. This involves not only

emotions but also physical sensations, pictorial elements of the material involved, as well as data from other senses. This happens typically in complete age regression to the stage of development when the event

happened

Another important distinction is that the relevant memories

and other biographical elements do not emerge separately, but form distinct dynamic constellations, for which I have coined the term COEX systems, or systems of condensed experience. A COEX system is a dynamic constellation of memories (and associated fantasy material) from different

periods of the individual's life, with the common denominator of a strong emotional charge of the same quality, intense physical sensation of the same kind, or the fact that they share some other important elements. Ifirstbecarne aware of COEX Systems as principles governing the dynamics of the individual unconscious and realized that knowledge of them was essential for understanding the inner process on this level. However, later it became obvious that the systems of condensed experience represent a general

principle operating on all the levels of the psyche, rather than being limited to the biographical domain. (Grof, 1985, pp. 96-97) The third level is coined the perinatal realm, whose core involves the twin experiences of birth and death, with the concomitant rebirth processes that seem to open “intrinsic spiritual areas in the human mind that are independent of the individual’s racial, cultural, and educational background” (Grof, 1985, p. 3 8). This level is further broken down into four basic perinatal matrices (BPMs) that bear similarities to various stages of biological delivery. This opens a passage to the last level that constitutes the whole spectrum of transpersonal experiences, which pose a serious challenge to the Newtonian-Cartesian mechanistic model of the universe. Grof pointed out that these four levels interact with each other with “experiential similarity,” yet they are distinct enough to justify their differentiation. One example is “the deep experiential similarity between the pattern of biological birth, sexual orgasm, and the physiological activities in the individual erogenous zones” (Grof, 1985, pp. 155-156). Grof concluded that, in the unconscious, “birth, sex and death form an inextricable triad and are intimately related to ego death” (p. 157). In my opinion this triad is not far from Freudian drive theory (Freud, 2001 g), with the psychoanalytic realization that Bros and Thanatos are just two sides of the same drive coin. Furthermore, the interpenetration and differentiation that Grof described is reminiscent of the topology-inspired conception of the Real, the Symbolic, and the Imaginary orders as inextricably bounded

in the form of a Borromean lcnot (see Figure 16—1), as suggested by French psychoanalyst Jacques Lacan (1975).

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Figure 16—1. Emblem of Lorenzo de Medici in San Pancrazio, Florence, displaying the Borromean rings. An interesting particularity of such a “Borromean knot” is that the three bound rings are linked in such a way that, if any of them is cut, all three become detached. (Courtesy of the photographer, Sailko, 2009)

Some dimensions of Grof s cartography concerning archaic unconscious processes are also reminiscent of psychoanalytic observations by Melanie Klein (1935, 1946) and Frances Tustin (1990). Therefore, this cartography is a useful reference for clinical work with children, as well as for understanding psychotic and autistic phenomena. In relation to corporeal phenomena, 3 significant reference is Paul-Laurent Assoun’s work Corps et

sympto‘me (2004). Assoun further clarified the metapsychology of somatic symptoms, following Freud’s intuition that the unconscious could be the missing link between psyche and soma, as well as the relation of hysteric

symptoms to real organs, excitation and libido fixation—like the biomineral

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analogy of a “grain of sand around which an oyster forms its pearl” (Freud, 2001a). In a nutshell, he studied complex feedback interactions between the unconscious, the subject, different levels of the body (for example differentiating the organic body from the cams pulsionnel”), language, symptoms and culture. Finally, concepts like Lacan’s big Other14 are also useful to shed light on intricate phenomena that could otherwise not be understood by restricting the understanding of the subject to the ego. Fractal geometry proposes principles that have been applied to a plethora of phenomena across awide spectrum of disciplines, and it can be instrumental to elucidate links between the brain and cognition. As already mentioned, the problem of identifying difference, similarity, and fuzzy boundaries is a theme central to fractal geometry. I suggest that these conceptual tools are useful to identify the differentiation and the interrelationships between the realms of Grof’s extended cartography. Ithus propose that self-similarity operates on different levels of the psyche as a nonlinear reverberating process, with elements and processes that repeatedly appear within and between distinctive levels. This includes qualitative differences that arise as the scale changes from neuronal substrates to perception to psychodynamic, perinatal, and transpersonal levels. This might be akin to stochastic fractals that are readily found in nature. This process begets repetition of form with slight modifications at each iteration, which stands in contrast to linear fractals that show exact scale-invariance, such that the identical form is stereotypically repeated at each iteration, with no variation nor possibility for change. How do we begin to pinpoint such fractal reverberations in the psyche more concretely? Geometric-visual hallucinations may provide a solid starting point to tackle this problem.

13 Translated literally from French as the “body of the drive” or the “erogenous

body”. 14 As Dylan Evans puts it, the big Other (A) is “perhaps the most complex term in designates radical alterity, an other-ne ss that transcends the illusory Lacan’s work Lacan equates this radical alterity with language and otherness of the imaginary the big Other is the symbolic insofar as it is particularized for each the law subject. The Other is thus both another subject, in his radical alterity and unassimilable uniqueness, and also the symbolic order which mediates the In arguing that speech originates not in the relationship with that other subject ego, nor even in the subject, but in the Other, Lacan is stressing that speech and language are beyond one’s conscious control: they come from another place, outside

consciousness” (Evans, 2006, pp. 132-133).

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On hallucinations and their fractal qualities: Heinrich Kliiver’s geometric form constants The subject of hallucinogenic phenomena has vast ramifications, accompanied by extensive data plus differing conceptions (the full description lies beyond the scope of this chapter). The history in Western thought regarding what is now called hallucinations shows substantial changes over the last 300 years. This history extends from the 17th century, where hallucinations were valued within a cultural context of bringing meaning to the subject or the

world to the mid-17“h and 18th centuries, where hallucinations acquired a medical quality in mental and organic illnesses, until their full integration

within psychiatry by Esquirol in the 18th and 19th centuries (Telles-Correia, Moreira, & Goncalves, 2015). The prevailing present-day conception of an hallucination is generally a perception without an object. Consider William James’ (1890) definition: “An hallucination is a strictly sensational form of consciousness, as good and true a sensation as if there were a real object there. The object happens to be not there, that is all” (p. 116). As the neurologist Oliver Sacks (2012) pointed out, precise definitions can vary considerably, and there are fuzzy boundaries between hallucinations, misperceptions, and illusions. Nonetheless, cues fiom neurological data help to make some clear distinctions. The vivid experiential quality usually reported in visual hallucinations might be explained by their close resemblance to perception, as the two share most of the same visual areas and pathways in the brain. Moreover, different cortical activity can be observed in the brain between normal visual imagination and actual hallucinations (Sacks, 2012). Functional MRI findings in sensory deprivation experiments exemplify this point, like the work of Sireteanu et

a1. (2008): Neural activity related to hallucinations was found in extrastriate occipital, posterior parietal, and several prefrontal regions. In contrast, mental imagery of the same percepts led to activation in prefi'ontal, but not in posterior, parietal, and occipital regions. These results suggest that deprivationinduced hallucinations result from increased excitability of extrastriate visual areas, while mentally induced imagery involves active read-out under the volitional control of prefiontal structures. This agrees with the subject's

report that visual hallucinations were more vivid than mental imagery. (p. 1805) Useful as they are, these kinds of explanations have their limits. As Sacks pointed out, we cannot fully attribute hallucinations, nor any cerebral function for that matter, to specific brain regions, as equal attention is

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needed for the connections at work between different areas (Sacks, 2012). Moreover, we must add personal and cultural elements, psychodynarnic models, and transpersonal dimensions to the equation for a more comprehensive understanding. In this chapter I abstain fi'om a philosophical discussion on the matter, maintaining the basic principle that these phenomena seem very real to the person experiencing them, but their perception is not readily shareable by “outside” observers. I next focus on one of the most basic phenomena for which there is general consensus—geometric—visual hallucinations. I begin my inquiry with the following question: why do basic hallucinogenic phenomena like visual-geometric hallucinations seem to show fi'actal qualities, and what does this imply for the rest of the psyche? My inquiry sprouts from readily available clinical data, from case studies in psychoanalytic and experiential

therapies, contemporary surveys and classical literature, to which I add my own clinical experience in the field of psychosis. To exemplify the assumptions raised by these questions, I embark on a close examination of the work by perceptual psychologist Heinrich Kluver, mostly known for the syndrome that bears his name”, as well as for ties to the cybernetic movement“. I begin by examining accounts of visual hallucinations induced by the intake of mescaline, as recollected in Kluver ’s

(1928) book, Mescal, the ‘Divine ’ Plant and I is Psychological Efleets: Circles... concentrically arranged, the innermost being infinitely small The colour is intensely beautiful, rich, deep, deep, deep, wonderfully deep blue... circles are now developing upon it; the circles are becoming sharp now they are rhomboids, now oblongs, and now all sorts and elongated mathematical figures are chasing one of curious angles are forming another wildly across the roof. The colours are changing rapidly... passing a steel veil successively through an mfinite variety of transitional shades transparent the meshes of which are constantly changing in size and form the chessboard motive repeated itself oriental rugs, but infinitely small

the design became increasingly delicate, the details assumed the character a kaleidoscopic play of modern cubistic patterns of ornaments ornaments, patterns, crystals, and prisms which creates the impression of a

never-ending uniformity

Coloured threads running together in a

15 The Kltiver-Bucy syndrome is observed in humans after bilateral lesions in the temporal lobe that involve the amygdala, hippocampal formation, and adjacent neural structures (Afifi & Bergman, 1998). It has a very peculiar symptomatology, like hypersexuality, docility, and ‘putting strange objects in the mouth’.

15 Charter member of the cybemetics group (Kline, 2015) and active participant in the Macy Conferences of the 19405 and 1950s (Oliva, 2016).

The Fractal Qualities of Hallucinatory Phenomena revolving center,

perspectives

the whole similar to a cobweb

growing into the infinite

465

deep beauti'fhl

vision of irregular fragments.

(Kltiver, 1928, excerpted fiom pp. 19, 21, 34, 35, 37-39, 45, italics mine)

Even from a casual inspection of these excerpts, one can readily identify fractal qualities like the self-similar arrangement of nested circles, or the mention of iterative growth to infinity, reminiscent of the infinitely small in the Cantor set or the zoom into progressive detail within the Mandelbrot set. Indeed, there are repeating forms across different scales, most notably size. Repetition across different senses is also commonly reported, a phenomenon known as synesthesia, with audio—visual crossover seemingly the most frequent occurrence in mescaline—induced states. It is noteworthy that language descriptions echo these repetitions, as in “Deep, deep, deep

wonderfully deep?” Klfiver also mentions that the succession of visionary phenomena may turn into an “indescribably overwhelming complexity” of a “stationary presence.” Finally, accounts abound that echo Grofs findings on cosmic themes: “Some subjects refer to ‘cosmic emotions’ and to ecstatic states in which ‘our exclamations of enjoyment become involuntary” (Klfiver, 1928, p. 101). The mention of Haeckel’s Radiolarian proves to be a point of special importance: Regular and irregular forms in iridescent colours reminding of radiolaria, sea urchins and shells, etc., in symmetrical or asymmetrical arrangement.Shells illuminated from within radiating in different colours, moving t0wards the right, turned about 45° towards the right and somewhat towards me. A little piece in every shell is broken out; objets d'art similar to Haeckel's

radiolarian. . .” (Kltiver, 1928, p. 28) Likewise reported by Grof, Haeckel’s imagery has been recurrently referenced in the observations of hallucinatory phenomena. As far back as

1862, Haeckel’s Monographs on Radiolarians strikingly revealed some of the fractal features that Mandelbrot would describe more than a century later (see Figure 16.2). Perhaps this is not so surprising, as both Mandelbrot’s and Haeckel’s work focus strongly on symmetry and pattern formations within nature. A bit more mysterious is the connection of these patterns to visions in non-ordinary states of consciousness. Yet, it is also common knowledge that in many cultures, geometric and/or repetitive fractal-like forms are

17 This repetition could be linked to reiterated discharge in language phase space,

similar to Ariana Bazan’s “phonemic phantom” neuropsychoanalytical hypothesis (Bazan, 2009)

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indicative of the divine, such as within Islamic art or cathedral architecture, as well as within mandalas, specifically designed to evoke transcendental mental states (Ash, 2008).

Figure 16-2. Illustration from Haeckel’s “Artfomis in Nature” depicting radiolaria, marine protozoa. (Haeckel, 1904)

I now take a small detour to review another notable facet of Haeckel’s work, which draws upon embryology and Darwin’s evolutionary theory: recapitulatz‘on theory or biogenetz‘c law. Built upon the assumption of a oneto-one correspondence between phylogeny and ontogeny, Haeckel’s proposal is well summarized by the phrase, “ontogeny recapitulates phylogeny,” which alongside his art, demonstrates an intuition for selfsimilarity. Although the validity of this theory has been deemed inaccurate by experimental morphologists and biologists, other contemporary scientists now explore how these two processes are indeed intertwined”. Haeckel’s proposal may have served like a linear toy-model, by ofi‘ering an oversimplification later to be resurrected and refined. Haeckel’s notion that ontogeny recapitulates phylogeny might ring a bell to anyone familiar with the work of Sigmund Freud. Indeed, these ideas were quite popular during Freud’s time, and it was here that the founder of psychoanalysis found a well of inspiration to water arid regions of the day’s "5 See http://www.ucmp.berkeley.edu/history/haeckelhtml

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most challenging theoretical questions. Several of Freud’s classic works refer to these ideas, including Totem and Taboo (2001d), the preface for

Three Essays for a Theory of Sexuality (2001b), and Introductory Lectures on Psychoanalysis (2001f). After carefully studying Freud’s work, Bolens (2001) affirmed that it was the close connection established by Haeckel between the ontogenetic and phylogenetic axes that allowed Freud to explain his strong intuition that the individual psyche and collective thought obey similar laws. What is more, in Beyond the Pleasure Principle (2001 g), Freud famously mentioned his intention to consider Haeckel’s views for speculating on a prickly subj ect—the death drive. According to Kirsch and Hogenson (2017), Haeckel promoted and popularized Charles Darwin’s writing in Germany, while his art and ideas influenced many other psychologists, biologists, and even artists of the time. Traces of Haeckel’s influence can also be found in the work of Carl Gustav Jung, although less explicitly. Cambray and Sawin (2018) link some of the

drawings from Jung’s Red Book to Haeckel’s radiolarian, also mentioning the fractal qualities of both sets of images. Of more immediate relevance to this chapter, Grof also reports observations of images like Haeckel’s alongside Klfiver, to whose work I turn next. Returning to Kluver’s mescal work, subsequent research confirms what

he initially (Klfiver, 1928, pp. 36, 108) described as super-individual form constants—geometric patterns widely observed in mescaline—induced states regardless of “inter- and intra-individual” variations, which appeared not to be constructed from memory of past experiences in the subject’s history. Klfiver (1942) would later refine his conclusions about hallucinations by seeking a common denominator and identifying three main levels: form constants; alterations in number, size, and shape; and changes in spatiotemporal relations. Moreover, the basic form constants are not exclusive to mescalineinduced hallucinations, but also appear following very different triggers: neurological conditions, migraine auras, fever states, substance intake or abstinence, psychotic states, sensory deprivation, near-death experiences, and intense psychological stress, among others. Consequently, these patterns may represent general characteristics of underlying physiological processes. This hypothesis has been taken seriously, and Klfiver’s work on form constants has sprouted interest over the years. It has been proven useful for understanding brain processes underlying perception, as well as the architecture of the visual cortex. The following remark, stemming from

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Klfiver’s work with mescaline hallucinations, resonated in the scientific community, as addressed in the next section: Other aspects of hallucinations and the hallucinatory process itself are often characterized by instability and fluctuations, and it is the task of future research to deduce the occurrence of these fluctuations on the nature of the underlying mechanisms. (Kluver, 1942, p. 18)

Neurocomputational

models for the visual cortex

While it is evident that people perceive fractal-like phenomena in nonordinary states of consciousness (see Figure 16-3), a remarkable fact is that up to this day, no widely accepted consensus has been established as to why this might be the case. How can this phenomenon be explained beyond simple observation? What implications might this have for consciousness on a broader scale? Recent findings in computational neuroscience may

shed new light on these issues, while opening viable areas of inquiry for the harder problem of how these fractal qualities relate to the rest of the psyche. Hubel and Wiesel (1959) studied how neurons in the brain are organized to produce visual perception. They demonstrated how orientation processing depends on vertical columns of neurons, plus specific neuron types within the striate cortex as arranged hierarchically from retina to cortex (see Wurtz, 2009). Within computational neuroscience, Jack D. Cowan and colleagues took these findings 3. step further by formulating a model of how the geometry of many of Klfiver’s basic form constants— notably tunnels/ funnels, spirals, cobwebs, and lattices, including honeycombs and triangles—relates to functional neuroanatomy; this research established a direct correspondence between the geometrical structure of these basic visual hallucinations and the spatial organization of the visual cortex (see Mauro, Raffone, & VanRullen, 2015). Findings support the hypothesis that the cortical mechanisms that generate geometric visual hallucinations closely relate to neural mechanisms underlying the processing of edges, contours, surfaces, and textures (Bressloff, Cowan, Golubitsky, Thomas, & Wiener, 2001).

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Figure 16-3. Image resembling Klfiver’s funnel form constant. “Modified binary decomposition of whole dynamical plane with circle .Tulia set.” Image with snc code. Courtesy of Majewski (2008)

In essence”, Cowan and his team provided a plausible explanation of how geometric visual hallucinations might be generated in the brain with mathematical investigation and computer modeling. They analyzed spatial pattern formation associated with geometric visual hallucinations in a 19 Cowan presented his research on October 20, 2014 in the seminar of ReactionDiffusion Equations, Propagation and Modelling (ReaDi) at the Ecole der Hauter Etudes en Sciences Sociales (EHESS) in Paris. His presentation was entitled

Geometric Visual hallucinations: Past Present and Future. Unless otherwise stated, the following remarks stem from my personal notes. In order to go beyond my simplified comments, I refer the reader to the original works.

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simplified model of the visual cortex (ReaDi—EHESS, 2014), putting forward the idea that the visual cortex possesses a particular Euclidean symmetry. Taking into account the stimulus fi'om images—for example artists’ drawings that depicted hallucinatory forms akin to Klfiver’s from constants, as well as some cave art that is speculated to be inspired from hallucinatory states—they calculated how this would look in brain coordinates. For this they used the retinocortical map—a topographical map extending from the projection of the retina to the striate cortex. In this way, form constants found in the first stage of hallucinations would be related to intrinsic brain activity, like “the brain looking at itself.” In the talk I attended in 2014, Cowan recalled an exchange he had around 1962 with Norbert Weiner, a main figure in cybemetics, a field that was an important precursor to the complexity sciences, including chaos theory and fractal geometry. Cowan asked this influential thinker what kind of mathematics would be most appropriate for biological networks. Weiner answered that using the methodology of stochastic processes could serve as the right theoretical framework for approaching the subject. Despite Weiner’s early prescience, it took Cowan and his collaborators some 40 more years to fully develop a working model for the problem of stochastic neurodynamics. Here is where Turing’s (1952) ideas about a chemical basis for morphogenesis enters the scene. The now famous mathematician constructed a plausible theory for the origin of repeating patterns in biology, such as in the skin of zebras or leopards. Turing’s reaction-diffusion model centers on two interacting chemical reactions—an activator and an inhibitor. When the diffusion constants of these two processes differ, one might either see stripes or blob patterns, depending upon different aspects of the underlying bifurcation structure. Because the connectivity of the visual cortex exhibits similar activation]inhibition processes, the application of Turing’s model to neural networks was deemed adequate, even though in its original form, the toy model was too simplistic to apply to complex and “noisy” systems like brain dynamics. Cowan’s team took inspiration from Turing’s model, but changed diffusion to short or long axonal connections with local excitation and lateral inhibition, which incorporates physiologically realistic connections between neurons. The mathematics of statistical neural networks allowed for the factoring in of noise and criticality. Instability leads to pattern formation, and it seems that the visual cortex has evolved a specific network structure that can tolerate internal perturbations that could interfere with the

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perception of outside input By having a limited number of long range inhibitory connections between hypercolumns, the visual system would be less prone to the intrinsic noise that can lead to hallucinations, as the dominant signals will come from external stimuli. In a model with a generic network that has random connections, the normal-visual function would be substantially degraded because the random firing of neurons amplifies the stochastic Turing patterns and the perception funnels, cobwebs, spirals and so forth (Ouellette, 2018). According to Cowan: If the coupling strength between excitation and inhibition goes beyond a certain critical strength, where the excitatory neurons are not controlled anymore by the inhibitory ones, statistically there is still inhibition. But it is too weak to influence a decrease in excitation. The system goes beyond this critical point, and we basically encounter a phenomenon that is very similar

to a grand mal seizure, where ‘everything is on.’20 Similar neurodynamics also appear in near death experiences.21 This is especially interesting for the transpersonal study of the phenomenon of “going into the tunnel of light.” Although this topic has been hotly debated by neuroscience as well as transpersonal researchers, I believe both sides could benefit fiom the other’s perspective. Moreover, one could speculate that such a generalized activity in the brain might link to what people commonly describe as “seeing their whole life in an instant.” In any case, these details explain one level of the phenomenon, but how this level relates to psychodynamic and transpersonal realms, their influence and interpenetration, requires further study. Freelance writer Sam Woolfe reports a similar account from another lecture delivered by Cowan, adding the thoughts of Carhart—Harris, known for his recent research on psilocybin in treatment-resistant depression: Cowan says it's all to do with the physical structure of the brain. With eyes closed, since there is no external input, the geometric hallucination should reflect the architecture of the brain; more specifically, the architecture of the That architecture is fractal by nature the same patterns visual cortex are repeated at different scales of size. It should come as no surprise then that fiactal hallucinations are reported by those who take psychedelic drugs. Robin Carhart—Harris has commented on this, saying that, “Like tree branches, the brain recapitulates. You are not seeing the cells themselves,

2° Conclusion from the aforementioned conference in Paris, on October 20, 2014. 21 Cowan referred to measurements made with dying mice; other neurobiological studies, such as Saavedra—Aguilar (1989), reveal striking similarities between near

death experiences and temporal lobe epilepsy.

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but the way they're organized — as if the brain is revealing itself to itself” . . ..

[H]allucinating geometry is the experience of seeing the structure of your brain. (Woolfe, 2003, n.p.)

All in all, we can see how form constants present during the initial phase of hallucinations are better understood with the help of mathematical modeling. Cowan and his colleagues effectively modeled simple form constants and their topological correspondence in the Visual cortex, but not all of Kiivler’s constants fit in their model. They are notably aware of the difficulty to find a mathematical angle for modeling hallucinations that might involve memory and context, and hence symbolic processes. Thus, beyond fundamental sensory underpinnings, how could we conceptualize the more-complex phases of hallucinations? As the brain has similar wiring, what works for Vision might explain

similar mechanisms in other senses (Ouellette, 2018). I argue that taking into account the seemingly trivial self-similar qualities of simple form constants—repetitive patterns that appear on different levels, from retinal to cortical, manifesting qualitative changes that relate to shifts in processes specific to each level— could give some insight into what might be going

on in even more complex phenomena that could share similar processing. As fractal geometry has proven useful to describe the morphogenesis of

other natural phenomena, it might also prove a useful framework to tackle these harder questions that involve the rest of the psyche (see figure 16—4).

Figure 16-4. A Hilbert curve, a continuous, space-filling fractal, on the surface of a

cube, which is then deformed into a sphere. This image by Philip Rideout (2012), resembles the contours of some tunnel hallucinations generated by LSD, as well as more complex hallucinations redrawn by Oster (1970) that do not readily fit Cowan and colleagues’ current model (Bressloff et al., 2001).

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Clinical integration Within a fractal framework In order to extract valuable knowledge from the preceding sections and open an interesting line of inquiry that may have therapeutic outcomes, a confrontation with clinical facts is necessary. I thus make links to transpersonal phenomena by taking into consideration clinical evidence from experiential therapies that stem from the work of Richard Yensen and Stanislav Grof, as well as my own clinical practice in the field of psychosis. Far from offering the reader definite conclusions, I instead suggest possibilities for future interdisciplinary research, including some potentially valuable insights for transpersonal studies. I focus on two possible “fractal operations”: self-similar reverberations within different levels of the psyche, and the perception of infinite boundaries within ego dissolution. I also exemplify how strange—attractor dynamics can be useful to orient ourselves in apatients’ chaotic discourse. I start by restating that the fractal quality of self-similarity is observable in simple hallucinatory—fonn constants linked to basic brain functions from sensory processing and neural architecture. These phenomena can reverberate among several senses, with repercussions in other levels of the psyche, albeit with different qualitative “processing.” From Klfiver to Grof, and Yensen, as well as others like Masters and Houston (1966), there is widespread agreement that visual hallucinations start “simple” before becoming more complex. This is not only a logical assumption—initial increase in entropy that leads to re-organization at a higher degree of complexity—but is in fact a well-established observation. One might imagine a bifurcation point between simple form constants and complex hallucinations that include intricate feedback loops between several levels of the psyche. If this is the case, more complex hallucinatory phenomena would profit from descriptions that enfold concepts like self-organization, dissipative structures and fractal geometry. It is evident that most subjective phenomena are more easily described in qualitative terms. As basic statistics reveal, while we may not always be able to grasp experiential phenomena in an objectively quantifiable way, this does not necessarily imply the absence of observable facts. Instead, it may indicate difficulty quantifying the variables of importance in exact mathematical intervals. Fortunately, qualitative assessments are still available, such as approximate magnitude”. Also, in psychology we, 22 For example, consider a basic ordinal measuring scale for a survey, where we might assign an order of magnitude, including options such as, “I completely agree,”

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frequently identify structures and forms”, as well as trajectories and oscillations between psychopathological states“. The qualitative “nature” of some phenomena may not prevent them fi'om mimicking processes or presenting structural similarities to other phenomena that can be readily measured quantitatively. Fractal geometry tries to bridge this gap by quantitatively describing qualitative aspects (e. g. form and fractal dimension). These are some ways that mathematical models may provide valuable insights for understanding psychodynamic processes and transpersonal phenomena. One must, however, keep in mind the epistemological heterogeneity of different fields, avoiding direct transpositions that conflate specifics. Grof did not ascribe much importance to the first level of the LSD experience, related to the aesthetic/abstract realm and corresponding to the emergence of form constants. He willingly omitted close inspection of this level according to his idea (now substantiated) that “the geometrical visions seem to reflect the inner architecture of the retina and other parts of the optical system” (Grof, 1975, p. 95). His focus was on investigating more complex levels, represented by experiential realms of psychodynamic, perinatal, and transpersonal phenomena. However, Grof did realize that this first level is not entirely isolated. I would argue that in order to acquire an integral view of experiential processes, we must consider this first, very basic level and reexamine its connection to the rest of the psyche. In this way, it might be plausible that self-similar processes—even very archaic ones, like a shift-twist, a magnification or a replication—present at this first level repeat and exert their influence in other areas of the psyche. An excerpt from Grof could exemplify such a phenomenon: [ w a s recently contacted by Arthur, a 46-year-old mathematician who had

had LSD experiences in the past for didactic purposes and as a means of finding the roots of his neurotic symptoms. Much of the work he had done

“I somewhat disagree,” “I am neutral to the statement” or “I completely disagree.” Using a Likert measuring scale, one can assign numbers to these values, thereby coding the expression of a subjective order of magnitude. Nonetheless, these numbers do not have a strict mathematical value in that one cannot ascertain exact numerical intervals between these different categories.

23 Examples within the psychoanalytic model include structures like neurosis, psychosis, and perversion, or different kinds of anxiety.

24 For example, the shift from a manic to a depressive phase in psychiatry, or a shift from the Kleinian depressive to paranoid-schizoid positions.

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in his sessions focused on problems of embryonal development and birth He had to face special complications in these areas owing to the fact that he had a twin sister. In many of his LSD sessions he had visions and

experiences of creatures with complex geometrical organization. He felt very emotionally involved in these experiences, although they were strange and did not make any sense. He could not understand why he spent so much

time on these bizarre and incomprehensible forms ....During the convalescence period following a heart attack, he acquired Ernst Haeckel's bookArt Forms in Nature. . .. He was astounded when, looking through the book, he recognized many of the forms that had represented such an important part of his LSD sessions. In an instant he received insights into the nature of the process that he never completed. As a twin, he had to face special problems related to symmetry during his embryological development. His experiences of different stages of his embryological development were associated in the LSD sessions with corresponding animal forms in accordance with Haeckel's biogenetic law. In this context, he recognized that the heart as an asymmetrical organ presents special problems during embryogenesis. It was on this level, in the realm of the basic

geometry of nature, that Arthur found the deepest roots of his liflz—long interest in mathematics symmetry, and geometrical forms. (Grof, 1977, p. 290-291, my italics highlighting the need for a fractal framework)

The “deeper meaning” ascribed to Haeckel’s imagery that, as previously seen, deals with self-similar aspects and fi'actal-like organization, can be understood as triggering an nature‘s-coup25 process, in the Freudian sense of deferred action. This reveals that basic levels can be integrated in higher levels of experience, even if at a later point in time. One could simply conclude that the form constants may be used as background imagery to project unconscious material, much like in a Rorschach test, but it is doubtful that these connections are merely arbitrary, and here one must take into account the importance that was given to them by the subject. Closer inspection could show the specific determinations that led to this particular link, and Grof accurately points to an echoing of problems with symmetry. I propose an analysis at the level of self-similar processes at work in different realms of the psyche. At first, there would seem to be a missing link, possibly because the very early perinatal phenomena described by

25 In the Freudian sense, Nachtriiglichkeit, translated as deferred action or retroaction, apres-coup is “used in connection with psychical temporality and causality: experiences, impressions and memory-traces may be revised at a later date to fit in with fresh experiences or with the attainment of a new stage of development. They may in that event be endowed not only with a new meaning but also with

psychical effectiveness.” (Laplanche & Pontalis, 1973, p. 111)

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Grof, as well as psychedelic visions, are closer to “thing-presentations” in primary processes, or to the un-representable “rea ” of the experience in lacanian terms.26 This poses difficulty for their conscious translation by connection to “word-presentations” in the Freudian sense,27 thus delaying the integration of the experience that would later be catalyzed by Haeckel’s imagery. In this account, it seems that a symmetrical process was a key element that reverberated in the psyche. In a nutshell, I suggest that recognizing self-similarity of processes at different levels can lead to a better integration of the experiential quality of non-ordinary states of consciousness, psychodynamic material, perinatal realms, and the effort towards holotropic realization. I now tum to the studies of Richard Yensen (1998). In his recollection of works, Towards a Psychedelic Medicine, he proposed that a model for psychedelic medicine can only be studied with methods that incorporate the basic unity of the studied processes. This brings us back to the value of the NDS framework, as well as to consideration of the system as a whole. Such an approach could enable integration of the complex interrelations between the set and setting, personal history, neurological levels, patient and therapist, and pharmacological factors. It could also allow for the possible emergence of novel configurations in far from equilibrium conditions. For one, such a framework enables us to consider factors like set and setting in less mysterious or “magical” ways. Here is one of Yensen’s (1998) insights: The phantastica (LSD, mescaline, psilocybin, etc.) seem to be the only substances, amongst the psychoactive compounds, with extreme sensibility

to extra-pharmacological factors, which has created a great problem for the researchers that try to evaluate the effects of these substances. (p. 52, translated by the author from the original in Spanish)

25 As Evans (1996) exemplifies, one lacanian conception of the order of the “real”

is “that which resists symbolization,” thus “when something cannot be integrated in the symbolic order, as in psychosis, it may return in the real in the form of a

hallucination” (pp. 160-161). 27 In The Unconscious, Freud (20016/1915) writes that “the conscious presentation comprise the presentation of the thing plus the presentation of the word belonging

to it, while the unconscious presentation is the presentation of the thing alone. The system c . contains the thing-cathexes of the objects, the first and true objectcathexes; the system Pcs. comes about by this thing-presentation being

hypercathected through being linked with the word-presentations corresponding to it. It is these hypercathexes, we may suppose, that bring about a higher psychical organization and make it possible for the primary process to be succeeded by the secondary process which is dominant in the Pcs.” (p. 201-202)

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Yensen’s observation could be reformulated as extreme sensitivity to initial conditions typically found in nonlinear systems. Therefore, psychedelic experience would go hand in hand with deterministic chaos. This insight renders a solely linear fi'amework inadequate for evaluating their effects, pointing to the need for a framework that includes chaos theory when studying phantastica—related phenomena. Yensen indeed suggested that psychedelics seem to introduce a chaotic quality in the ordered processes of thought and perception, by deconstructing set beliefs in neurotic and psychotic systems, while opening the concomitant possibility for new creations. It is interesting to note that recent research based on neuroirnaging data also suggests elevated entropy in certain aspects of brain function after the intake of psilocybin (Carhart-Harris et al., 2014). ~What is it that drives experiential phenomena into transpersonal realms in therapeutic sessions with psychedelics? Yensen affirms that it is the alterations in the boundaries of the ego over time, through space, and in body image, and their repetition in a series, that push the subject into transpersonal levels. One might frame these dynamics in a more specific way as self-similar repetitions in phase space, within the chaotic trajectory of a strange attractor which displays fractal properties. Concerning the therapeutic possibilities brought about by alterations of the ego, the ego-dissolution experience is said to be valuable if fruitfully reintegrated, which can be aided with an appropriate setting and clinical guidance. The subjective aspects linked to such states, notably the deathrebirth experiences, have been widely studied. They may comprise a higher order constant found not only in psychedelic— induced experiences, but also in near-death experiences, archetypical stories, art, religion and mythology. Analyses of certain neural substrates provide hints to subsequent processes at psychodynamic, perinatal, and transpersonal levels. However, reductive and linear explanations that diminish the phenomenon to a “chicken or egg” question (another birth metaphor) should be avoided. Psychodynarnically speaking, one can envisage the ego-dissolution state as a logical consequence of the disintegration of clear boundaries between the self and the universe at large, identified as “non-self.” This state could be experienced as the feeling of “boundless infinity,” as inextricably linked to the mystical revelation that “all is one.” In a clinical setting with humans having access to language, this kind of internal phenomena can be recognized in externalizations like “you are me,” “I am the table,” or “I am the walrus,” for that matter. From a psychopathological perspective, this is a basic point that leads to psychedelics being considered as ‘psychotomirnetic,”

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denoting substances capable of producing an effect on the mind that resembles a psychotic state. A crucial difference is that in a consolidated psychotic structure, the persistence of this “chaotic” state is actually the “ordinary state,” versus its presence as a transient state during psychedelicassisted therapy. In the experiences of ego dissolution, the feeling of dying logically correlates to an awareness of the ego’s usual boundaries and constitution melting away, thus perceived as its—temporary—death. This might lead to the perplexing realization that there remains life beyond the vanishing ego—that beyond the usual ego boundaries there is still more to experience, even if “lived” or ascribed non-locally. Then, with an oscillation back to baseline, the ego slowly regains its configuration. People report therapeutic effects from this “rebooting of the system,” often accompanied by a feeling of relief (Pollan, 2018). This dimension of integrating part of the experience back into the ego, including re-working or elaborating certain issues that had been rigidified, might add complexity in the form of additional degrees of freedom to the self-system.

In other words, one might say “Es war, so]! [ch werden,” the famous Freudian quote meaning literally “Where it (id), there I (ego) become.” This phrase epitomizes Freud’s explanation of dynamic structural boundaries, nonlinear and sometimes fuzzy, as well as some therapeutic efforts in psychoanalysis that have a similar line of approach to mystical practices that, in regard to the ego, might broaden its perceptual field and its organization to appropriate new portions of id. As translated by Strachey, Freud writes: In thinking of this division of the personality into an ego, a super-ego and an id, you will not, of course, have pictured sharp frontiers like the artificial ones drawn in political geography. W e cannot do justice to the characteristics of the mind by linear outlines like those in a drawing or in primitive painting, but rather by areas of colour melting into one another as

they are presented by modern artists. After making the separation we must allow what we have separated to merge together once more. You must not judge too harshly a first attempt at giving a pictorial representation of something so intangible as psychical processes. It is highly probable that the development of these divisions is subject to great variations in different individuals; it is possible that in the course of actual functioning they may change and go through a temporary phase of involution. Particularly in the case of what is phylogenetically the last and most delicate of these divisions the differentiation between the ego and the super-ego

something of the sort seems to be true. There is no question but that the same thing results from psychical illness. It is easy to imagine, too, that certain

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mystical practices may succeed in upsetting the normal relations between

the dijfirent regions of the mind, so that, for instance, perception may be able to grasp happenings in the depths o f the ego and in the id which were

Where id was, there ego shall be. (Freud, otherwise inaccessible to it. 2001i, pp. 79-80, italics mine) W e could also think of ego-dissolution in terms of a dynamical system pushed far from equilibrium (see Wilcox & Combs, this volume), including the possible emergence of dissipative structures (Prigogine, 1994). I n this regard, a similar framework that uses the concept of entropy has been proposed within neuroimaging studies, where waking consciousness is formulated as a middle state between high entropy (disorganization) and low entropy (less fluidity/more rigidity) states, with psychedelic states pushing the system towards higher entropy (Carhart—Harris et al., 2014). As has been previously stated, a similar line of thought brought Yensen to the conclusion that psychedelics nudge the system’s organization towards chaos, which also seems to tie in with the neuro-mathematical model that explains how geometric-visual hallucinations emerge when the visual cortex becomes unstable. Another psychodynamic process might play a role in the ascribed therapeutic benefits of ego-dissolution and “resolution”—its resemblance to elaboration through the change in passive and active positions during children’s play. Freud pointed out that part of the work of mastering the external world includes supplementing a passive experience with an active piece of behavior (Freud, 2001h). For example, when playing house with her doll, a little girl might reenact a past situation previously lived in a passive, dependent position with respect to her mother. This form of play, unconsciously working over the situation b y taking an active position, functions as a self-similar repetition leading to elaboration and mastery of a conflictual situation. Likewise, Terry Marks-Tarlow (2010) has also studied the dynamics of play and self-similar development, suggesting that feedback provided through early play calibrates internal systems. She has also exemplified the “universal desire in children to play with the same shape on different size scales,” an important reason being the dynamics of the visual field and our eyes abilities’ to “intuitively understand the multiscaled quality of fractal dynamics” (this volume, chapter 1 , p. 5-6). Thus, experiencing self-similar processes28 might play a role in the long” For instance, a terminal patient experiencing ego-death in a psychedelic assisted therapy could be viewed as pre-emptively taking an active stance in facing their own death, thus striving towards mastering a similar process in an otherwise passive

situation.

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lasting relief that most terminal patients report after working through egodissolution states in experiential therapies, like the recently re—introduced psychedelic-assisted therapies. Notwithstanding, I must reiterate that these are not the only mechanisms at work, though such dynamics might contribute on one level that may not be completely isolated from all others. In experiential therapies that use psychoactive substances, the trigger to bring about non-ordinary states of consciousness primarily comes fiom a pharmacological action. As the effect eventually subsides, the experience can be integrated once the subject returns to baseline, all the while being safely contained in a “good enough” therapeutic setting. But what about psychotic breakdowns where the triggers may not necessarily involve pharmacological agents and are difficult to pinpoint? How could one conceive of a subject’s “return to baseline”? For example, what about cases triggered by traumatic experiences, such as a violent rupture or separation, or the case of Japanese tourists’ extreme shock and disillusion upon visiting Paris, described as “the Paris syndrome” by Hiroaki Ota (see PalaciosSénchez et al., 2018)? There is still a lot of work to do to improve hospital models for dealing with psychiatric emergencies. Lifting the repression on psychedelic research, and taking into account the insights from experiential therapies, might bring about effective approaches for accompanying “non-ordinary states of consciousness” characterizing psychotic breaks, including “spiritual emergencies.” If the trigger arises from an unintegrated traumatic experience, one might assume that elaboration of the main trigger must be accomplished before the subject can start “coming back.” This might be easier in the case of non-ordinary states of consciousness induced chemically. When the effect subsides, the experience integrates itself almost automatically within an appropriate set and setting. Grof pointed out that stopping a difficult experience cold with tranquilizers, instead of reaching the core unconscious issues, makes for an inefficient treatment that can lead to hospitalization, flashbacks and years of struggle. In my own clinical experience, I have been confronted with cases where biographical accounts indicate amore or less “nonnal” life up to a psychotic break, a bifurcation that degenerated into a life-long struggle with schizophrenia. Several years ago, I was hired as an external psychologist by the parents of R29, a man in his forties, with the hope of “instilling in him

29 To preserve confidentiality, names are changed and the material remains general,

only evoking the essential points that are relevant to our present discussion.

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some joie de vivre” while he was hospitalized in a psychiatric ward. R often manifested violent outbursts, and he was undergoing further psychiatric and medical evaluation that lasted several months due to a mixed diagnosis of bipolar-disorder and schizophrenia, as well as several bodily health issues. His parents stated that, up until a school trip to Florence in his teens, the patient was “a normal teenager,” but that on this trip “something happened” that triggered a psychotic break”. Upon R’s return “they did not recognize him.” Since that point in time, R underwent several hospitalizations and pharmacological treatments, but he only seemed to plunge into deeper states of disorganization. When I visited him in his hospital cell, R constantly paced about and spewed what at first glance seemed like nonsensical logorrhea. Nonetheless, as the days went by, and our familiarity with each other grew, I began to identify some overarching signifiers in R’s apparently

random ramblings. My understanding of NDS and fractal geometry aided in my pinpointing something I could describe as “strange points of attraction” in his discourse, with two main attractors: “death” and “opening.” R would, for example, talk about the president, “who is dead,” or he would begin to read the newspaper before commenting that “Francois Mitterrand is dead.” I would ask him to tell me the story behind his assertions, to which he would associate amass of political elements, sometimes in a disjointed way:

“Le Pen, the father of the (political) party, Chocolate, the dog Macron is dead... Manuel Valsyou know him, Darth Vader.. ..” Then he would go on to say that his father was dead—which was not the case in consensual, up there. ” Is he in material reality— “He has my cancer that’s mine heaven then? “God? It ’s not possible, it’s over, I can’t anymore, he ’s not

alive, he did not come, dead... ” He would then state that he himself was dead and often utter “C ’est mart, ” which in colloquial French describes a situation where there is no more hope, and one must give up, literally signifying “it’s dead.”

3° This case history suggests resemblance to the Stendha] syndrome. It was named after the 19th—century French author who, upon visiting the Basilica of Santa Croce in Florence, found himself overcome with emotion, ecstasy, and complete absorption in the contemplation of sublime beauty that reached “the point where one encounters celestial sensations.” It is noteworthy that its facade bears resemblance to an iterated system fimction, having similar fractal qualities to the Duomo in Milano (Sala, 2006). Such experiences might lead to something like a spiritual emergency, as described by Grof, or to a psychotic decompensation in predisposed

cases.

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Through the oscillations in R’ s discourse, the subject of death would pop up quite often, and associated themes did not seem entirely random. R often evoked hopelessness and the death of the symbolic father, infinitely dying over and over again, but never actually arriving, like a hole/void that never gets filled or a loop that does not get cut. His constant pacing, moving back and forth, was symbolically self-similar: his discourse evolved like a strange attractor, passing close by similar points repeatedly, but never in the same exact trajectory. For the sake of brevity, I cannot present here the full scope of the material, nor is this my purpose. Rather, my intention is to provide the reader with a feel for the overall situation, which in looking back bears striking resemblances to some uncanny realms within Grof’s cartography. I later pieced together and found out more of R’s biographical history. For instance, his dog had died, and in his room he had a life-sized stuffed animal that he called by the same name, Chocolate. The father of R’s fiancee had died, after which his marriage was called off. Both families distanced themselves from one another afterwards. R’s parents refused to talk about any of this in R’s presence, thus ripping a further “hole” in his already compromised state. After that incident, R had a further breakdown, progressing into the state in which I initially found him— dangerously thin, constantly pacing, and with frequent violent outbreaks, even under heavy medication. The second signifier reminding me of a strange attractor in R’s discourse was it faut ouvrz'r, whose literal meaning from French can be “one must open.” I do not know how much time this attractor had been oscillating in R’s discourse, but it seemed directly applicable to his current state, where he would scream over and over that “one must open,” as he tapped on the walls repeatedly. Sometimes the medical nurses would come “too see if everything was alrigh .” I would ask them to open the locked window, which had heavy protective bars. R would find some relief peering out the window, whose panorama included a nearby church. However, the repetition of “the opening” seemed to link to more metaphysical themes of opening a view, or maybe opening his stereotypical discourse away from a deadening repetition cycle. In hindsight, R’s experiences bear remarkable resemblances to Grof’s cartography, notably COEX systems and his description of the perinatal domain of the unconscious, particularly BPM’s II and III. This realm exhibits symbolic and experiential self—similarity with the onset of biological delivery, when the cervix is still closed, thus constricting the fetus before its release. In this regard, Grof (1985) elaborates:

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The symbolic counterpart of a fully developed first clinical stage of delivery

is the experience of no exit or hell. It involves a sense of being stuck, caged, or trapped in a claustrophobic, nightmarish world and experiencing incredible psychological and physical tortures. The situation is usually As far as absolutely unbearable and appears to be endless and hopeless the organizing function of BPM II is concerned, it attracts COEX systems with memories of situations in which the passive and helpless individual is subjected to, andvictimized by, an overwhelming destructive force with no

chance of escaping. It also shows affinity to transpersonal themes with similar qualities. (pp. 112-113) In working with R, I tried progressively to “open up” his “space.” I would visit him as scheduled in his hospital room, unless the hospital would not allow outside visits based on daily changes in his aggressive behavior, described as chaotic and unpredictable. We started to take walks outside of his living quarters, by gradually circling inside the hospital grounds and garden as well as visiting the nearby church“. We finally went further outside the hospital grounds to a nearby park”, and even to a restaurant in front of the Salpétn'ére, where R liked to order a crepe and an Orangina with the pocket money his parents provided for the day. Hopefully, this instilled some life into his entrapped routine in the hospital. After R’s evaluation period at the hospital was over, he was transferred to another institution outside of Paris, and our work together came to an end. In encountering cases such as this, I cannot help but wonder if an appropriate setting to accompany the first psychotic break, akin to what Grof (2005) proposed for containing psychedelic crises, could help some people in sensitive states of bifurcation before they “solidify” into fullfledged psychotic disorganization. Such an approach might facilitate resolving unconscious material during intensive sessions within the right setting, versus the alternative of brief medical assessment and immediate pharmacological dampening.

Frances Tustin (1990) points to similar phenomena in her work with autistic states, particularly in certain cases of high sensitivity when coming

out of a “protective shell.” In her classification of “encapsulated secondary 31 R had deep, but conflicted, ties to religion; as we passed by the church he would say, “The Christ in the church over there is fake... He is dead.” Meanwhile, the empty church would just echo our footsteps. 32 Upon looking at the statues along the way, R would point out that these too were dead. “Of course,” I would say, “they represent important people that lived long ago.”

“Yes,” he would answer in a whirling pace while reading the plaque underneath the statue of Pine], “Dead in 1826!”

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autism,” Resnjk (2011) explains that in extreme cases, this autistic encapsulation may appear as an “infinite enclosure,” whose opening or rupture might confront a subject with a “catastrophic renaissance and the dissolution of its own limits in the face of daily reality.”33 Such a process might lead to further integration if contained in the right setting, but also risks further disorganization or deeper encapsulation if not managed appropriately. If more clinical personnel could consider these insights for the early treatment of psychotic outbreaks, they might find fruitful results in methods such as promoted by Grof. Not only might such an approach allow for a gain in taxpayers’ dollars within public healthcare systems, but it could also provide opportunities for people to live more productive lives outside of reclusion. With respect to processes underlying these realms, what do recent scientific developments offer? First, interesting perspectives regarding network theory, alpha synchronization in the brain, the default mode network (DMN), and its relation to inhibition (Greicius, Krasnow, Reiss, & Menon, 2003; Mayhew, Ostwald, Porcaro, & Bagshaw, 2013) are relevant to the study of schizophrenia, early psychosis, meditation, and the overall understanding of consciousness. In my opinion, these lines of research also provide new foundations for revisiting Freudian theory (e.g. the ego’s inhibitory function, which might link to activation/inhibition in the DMN, as well as bearing striking resemblance to the stochastic Turing mechanisms proposed for geometric visual hallucinations). Moreover, the recent comeback of research on psychedelics (Schenberg, 2018) sheds new light on their neural correlates, by describing changes in brain activity and network communication (Carhart—Harris et al., 2016). Such research offers further insight into numerous phenomena of the utmost interest to transpersonal studies, such as ego-dissolution (Nour, Evans, Nutt, & Carhart—Harris, 2016).

Conclusion I now briefly recapitulate some of the main elements mentioned in this chapter that allow for the general conclusion that a self-similar process might be repeated throughout the psyche as follows: decrease of inhibition,

increase of entropy, and emergence of patterns that are themselves determined by the primary fimctions and architecture of the same system, thus repeating or iterating qualities of itself in the process.

37’ Translated here from the French version

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First of all, let us take into account the general statement that images can be generated in the visual cortex when its inhibition/activation state becomes unstable, much like dissipative structures emerging in systems that are far-from equilibrium. They can be modeled mathematically, resembling stochastic Turing patterns, and are connected to the brain’s functions and the visual-cortex architecture. These patterns match most of Klfiver’ s simple form constants that often arise at the beginning of the first level of hallucinatory states, as has been also described in Grof’ s cartography of the psyche. It has been suggested that basic hallucinations in other perception systems might work in a similar way. Hallucinations are often reported to have fractal qualities in their form, repetition and “sense” of infinity. If these phenomena were to repeat in an exact or similar form in various scales of size, time, or even other levels of the psyche, then further research that explores their self-similar qualities could increase our insight as to what might be at work. Due to the scope of this chapter I only briefly mentioned the assumed role of the DMN, which is associated to higher—order ego functions. Moreover, recent research has revealed that psilocybin administration correlates to a decrease in connectivity in some areas of the brain’s network, notably those linked to the DMN, while at the same time correlating with the subjective experience of ego dissolution (CarhartHarris, 2014). If one were to tie this together it would ensue that psychedelics decrease connectivity and activity in the DMN, leading to higher-entropy states where far-from equilibrium conditions pave the way for the emergence of hallucinatory-perceptual patterns that iterate the systems’ own structure and functions. But how could the aforementioned self-similar process be at work specifically at the psychodynamic level? This question is particularly interesting in the light of clinical observation, as psychoanalytic metapsychology has long proposed that the ego has inhibitory functions in regards to perceptions and primary processes. Thus, a sharp decrease of inhibition favors the propagation of an excitatory flux, leading to a higher chance of instability and of primary processes taking over. This instability paves the way for the emergence of hallucinations. Following the principles of the self-similarity hypothesis, these emergent phenomena would be built with “primary psychodynamic architecture,” thus unconscious processes and material (e. g. early identifications and repressed material) would play a major role in its pattern formation. A telling-clinical example can be the accusatory voices in a paranoid state, verbal injunctions from the super ego that might themselves repeat in an exact 01: stochastically similar manner.

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I hope that, in this chapter, the reader can grasp the utility of a fractal framework for hallucinatory phenomena linked to holotropic states. In my view, after this analysis, we are in a better place to fathom the distinction, but also the interpenetration and complex feedback interactions of several levels of the psyche beyond a linear cause-effect or a reductionist bottomup logic. I find this a particularly valuable compass for navigating the expanded cartography of the psyche, upon which transpersonal psychology has already embarked. At the very least, a fractal framework can help us to understand iterative processes behind phenomena that otherwise might seem too complex or outlandish at first sight. At the very most, recognizing self-similarity of processes at different levels can lead to a better integration of the experiential quality of non—ordinary states of consciousness, psychodynarnic material, perinatal realms, and the effort towards holotropic realization. As a final remark, I end this chapter by pointing to a subject that is likely to gain importance in the following years, namely what could be called “the return of the repressed psychedelic research,” highlighting the need to revisit early tenets of transpersonal psychology and its original experiential therapies with psychedelics. In 1998, Yensen wrote that there were only two LSD psychotherapy trials going on in the United States, compared to over 200 ongoing research projects using LSD and other “phantastica” in 1965, a time before dwindling funds and legislative difficulties came to into effect. Now, 20 years later, the global situation has evolved, with at least 95 ongoing trials that include LSD, psilocybin, MDMA, and ketamine (Schenberg, 2018). Some notable actors include organizations such as the Multidisciplinmy Association for Psychedelic Studies (MAPS) (Emerson, Ponte, Jerome, & Dobliri, 2014), the H efi‘ter Institute (Belser et al., 2017)

and the Psychedelic Research Group at the Imperial College London (Carhart—Harris et al., 2016). This research reveals promising results, not only for the overall understanding of the brain but also for therapeutic application of psychedelic agents (Carhart—Harris et al., 2012). One remarkable example is MAPS’ lVlDMA—assisted psychotherapy for PTSD. Its protocol was awarded “breakthrough therapy designation”34 by the FDA in 2017. Phase II results indicate that over 70% of the participants no longer qualified for the diagnosis of PTSD after 12 months. Meanwhile, the remainder of participants had less intense symptoms and 34 “Given to studies whose preliminary clinical evidence suggests that it may provide a “substantial improvement over existing therapies.”

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improvements that lasted up to 4 years. These outcomes were accomplished primarily without additional treatments or inducing drug abuse or dependence (Schenberg, 2018). This study is currently reaching phase III trials. It is expected that MDMA—assisted psychotherapy could be more widely available as soon as 2021 (Theberge, 2018). Other studies conducted at John Hopkins (Griffiths et a1., 2016) and the New York University (Belser et a1., 2017) have supported the use of psilocybin for cancer-related anxiety and depression. Psychedelic research remains difficult to carry out; only time will tell when the tide will change towards its full integration into mainstream science. Nonetheless, if this rebirth were to continue its current pace, the

contribution to clinical research from transpersonal studies could be of inestimable value, given the historical connection of the field with psychedelic-assisted therapies. Not only does the need exist to pick up the research where it left off, but also to consider knowledge that has since accumulated in other fields, including NDS, neuroscience, and panorama, a fractal psychoanalysis. Within this multidisciplinary framework could likely play an important role. Thus, I could not agree more with Stanley Krippner and Harris Friedman: There is an urgent need in today's fractious world for integrative transpersona] perspectives, especially if presented in ways that are selfcritical and able to be linked in contemporary scientific and practical concerns (Krippner, 1998). Returning to its scientific roots is the only path for transpersonal psychology to take in order to make such needed contributions. Furthermore, accelerating advances within science, such as sophisticated new neurotechnologies applicable to studying consciousness, are increasingly opening innovative and exciting scientific avenues for

exploring transpersona] psychology. (Friedman, 2002, p. 185)

References Afifi, A.K., & Bergman, RA. (1998). Functional neuroanatomy: Text and atlas. New York, NY: McGraw—Hill.

Ash. (2008, March 19). Form constant. Retrieved from http:l/greathmkings.blogspot.com/2008/03/fonn-constanthtm1 Assoun, PL. (2004). Corps et symptéme. Paris: France: Anthropos. Bassett, D.S., Meyer-Lindenberg, A., Achard, 8., Duke, T., & Bullmore, E. (2006). Adaptive reconfiguration of fractal small-world human brain

488

Chapter Sixteen

functional networks. Proceedings of the National Academy of Sciences ofthe United States ofAmerica, 103(51), 19518-19523. Bazan, A. (2009). Des fantémes dans la voix.‘ Une hypothe‘se neuropsychanalytique sur la structure de l’inconscient (Unpublished doctoral dissertation). Université Lumiere-Lyon II, Lyon, France. Belser, A.B., Agin-Liebes, G., Swift, T.C., Terrana, S., Devenot, N., Ross, S. (2017). Patient experiences of psilocybinFriedman, H.L., assisted psychotherapy: An interpretative phenomenological analysis.

Journal ofHumanistic Psychology, 57(4), 354-388. Boeing, G. (2016). Visual analysis of nonlinear dynamical systems: Chaos, fractals, self-similarity and the limits of prediction. Systems, 4(4), 37. Bolens, JD. (2001). La théorie de la récapitulation de Haeckel a Freud, Topique, 75(2), 13-34. Bressloff, P.C., Cowan, J .D., Golubitsky, M., Thomas, P.J., & Wiener, M.C. (2001). Geometric visual hallucinations, Euclidean symmetry and the

functional architecture of striate cortex. Philosophical Transactions of the Royal Society of London. Series B, 356(1407), 299-330. Cambray, J., & Sawin, L. (2018). Research in analytical psychology: Applications from scientific, historical, and cross-cultural research. New York, NY: Routledge. Carhart—Harris, R.L., Leech, R., Hellyer, P.J., Shanahan, M., Feilding, A., Nutt, D. (2014). The entropic brain: A theory of Tagliazucchi, E., conscious states informed by neuroimaging research with psychedelic drugs. Frontiers in Human Neuroscience, 8. Carhart—Harris, R.L., Leech, R., Williams, T.M., Erritzoe, D., Abbasi, N., & Nutt, D. J. (2012). Implications for psychedelicBargiotas, T., assisted psychotherapy: Functional magnetic resonance imaging study with psilocybin. British Journal of Psychiatry, 200(03), 23 8-244. Carhart—Harris, R, L., Muthukumaraswamy, S., Roseman, L., Kaelen, M., & Nutt, DJ. (2016). Neural correlates of the Droog, W., Murphy, K., LSD experience revealed by multimodal neuroimaging. Proceedings of

the National Academy of Sciences of the United States of America, 113(17), 4853-4858. Emerson, A., Ponte, L., Jerome, L., & Doblin, R (2014). History and future of the multidisciplinary association for psychedelic studies (MAPS).

Journal of Psychoactive Drugs, 46(1), 27-36. Evans, D. (1996). An introductory dictionary of Lacanian psychoanalysis. New York, NY: Routledge. Freeman, W.J. (2013). Chaotic neocortical dynamics. In A. Adamatzky &

G. Chen (Eds), Chaos, CNN, memristors and beyond (pp. 271-284). Singapore, Malaysia: World Scientific.

The Fractal Qualifies of Hallucinatory Phenomena

489

Freud, S. (2001a). Fragment of an analysis of a case of hysteria. In J.

Strachey (Ed), The standard edition of the complete psychological works of Sigmund Freud (Vol. VII) (pp. 1-122). London, England: Vintage. Original work published 1905) —.. (2001b). Three essays on the theory of sexuality. In J. Strachey (Ed),

The standard edition of the complete psychological works of Sigmund Freud (V 01. VII) (pp. 123-246). London, England: Vintage. Original work published 1905) —. (2001c). Five lectures on psycho-analysis. In J. Strachey (Ed), The

standard edition of the complete psychological works of Sigmund Freud (V 01. XI) (pp. 1-56). London, England: published 1910)

Vintage. (Original work

—. (2001d). Totem and taboo. In J. Strachey (Ed), The standard edition of the complete psychological works of Sigmund Freud (V01. XIII) (pp. vii162). London, England: Vintage. Original work published 1913)

—. (2001e). The unconscious. In J. Strachey (Ed), The standard edition of the complete psychological works of Sigmund Freud (Vol. XIV) (pp. 159-215). London, England: Vintage. (Original work published 1915) —. (2001f). Introductory lectures on psycho—analysis. In J. Strachey (Ed),

The standard edition of the complete psychological works of Sigmund Freud (V 01. XV) (pp. 1-240). London, England: Vintage. (Original work published 1916) —. (2001g). Beyond the pleasure principle. In J. Strachey (Ed), The

standard edition of the complete psychological works of Sigmund Freud (V 01. XVIII) (pp.1-64). London, England: Vintage. (Original work published 1920) —. (2001b). Female sexuality. In J. Strachey (Ed), The standard edition of the complete psychological works of Sigmund Freud (V01. XXI) (pp. 221-244). London, England: Vintage. (Original work published 1931) —. (2001i). New introductory lectures on psycho-analysis. In J. Strachey (Ed), The standard edition of the complete psychological works of Sigmund Freud (V01. XXII) (pp. 1-182). London, England: Vintage. (Original work published 1933) Freud, S., & Breuer, J. (2001). On the psychical mechanism of hysterical phenomena: Preliminary communication. In J. Strachey (Ed), The

standard edition of the complete psychological works of Sigmund Freud (V 01. II) (pp. l-17). London, England: Vintage. (Original work published 1893) Friedman, H.L. (2002). Transpersonal psychology as a scientific field.

International Journal of Transpersonal Studies, 21(1), 175-187.

Chapter Sixteen

490

Galatzer-Levy, RM. (1995). Psychoanalysis and dynamical systems theory:

Prediction and self similarity. Journal of the American Psychoanalytic Association, 43, 1085-1113.

—. (2017). Nonlinear psychoanalysis: Notesfrom forty years of chaos and complexity theory. New York, NY: Routledge. Greicius, M.D., Krasnow, B., Reiss, A.L., & Menon, V. (2003). Functional connectivity in the resting brain: A network analysis of the default mode

hypothesis. Proceedings of the National Academy of Sciences, 100(1), 253-258. Griffiths, R.R., Johnson, M.W., Carducci, M.A., Umbricht, A., Richards, & Klinedinst, M.A. (2016). Psilocybin W.A., Richards, B.D., produces substantial and sustained decreases in depression and anxiety in

patients with life-threatening cancer. Journal of Psychophannacology, 30(12),1181-1197. Grof, S. (1975). Realms of the human unconscious: Observations from LSD psychotherapy. New York, NY: Viking. —. (1980). LSD Psychotherapy. Nashville, TN: Hunter House —. (1985). Beyond the brain: Birth, death, and transcendence in psychotherapy. Albany, NY: State University of New York Press.

—. (2005). Psicoterapia con LSD: El potencial curativo de la medicina psiquedélica. Barcelona, Spain: La Liebre de Marzo. —. (2008). Brief history of transpersonal psychology. International Journal

of Transpersonal Studies, 27(1), 46-54. Grof, S., & Grof, C. (2010). Holotropic breathwork: A new approach to selfexploration and therapy. Albany, NY: State University of New York Press.

Haeckel, E., (1904). Kunstformen derNatur. Leipzig, Germany: Verlag des Bibliographischen Instituts. Hartelius, G., Caplan, M., & Rardin, MA. (2007). Transpersonal psychology: Defining the past, divining the future. The Humanistic

Psychologist, 35(2), 135—160. Hubel, D.H., & Wiesel, TN. (1959). Receptive fields of single neurones in

the cat‘s striate cortex. Journal ofPhysiology, l 48, 574-591. James, W. (1890). The principles of psychology. New York, NY: H. Holt. Kirsch, T., & Hogenson, G. (2017). The Red Book: Reflections on CG. Jung ’s Liber Novas. New York, NY: Routledge. Klein, M. (1935). A contribution to the psychogenesis of manic-depressive

states. International Journal ofPsychoanalysis, I 6(1), 145—174. —. (1946). Notes on some schizoid mechanisms. The International Joumal of Psychoanalysis, 27, 99.

The Fractal Qualifies of Hallucinatory Phenomena

491

Kline, RR. (2015). The cybernetics moment: Or why we call our age the information age. Baltimore, MD: JHU Press.

Klfiver, H. (1928). Mescal: The divine plant and its psychological efiects. Oxford, England: Kegan Paul. —. (1942). Mechanisms of hallucinations. In Q. McNemar & M. A. Merrill, Studies in personality (pp. 175-207). New York, NY: McGraw—Hill. Krippner, S. (1994). Humanistic psychology and chaos theory: The third revolution and the third force. Journal of Humanistic Psychology, 34(3),

48-61. —. (1998). Foreword. In D. Rothberg & S. Kelly (Eds), Ken Wilber in

dialogue (pp. ix—xi). Wheaton, IL: Quest. Krippner, S., Ruttenberg, J ., Engelman, S., & Granger, D. (1985). Toward the application of general systems theory in humanistic psychology. Systems Research, 2(2), 165-115.

Lacan, J. (1975). Le seminaire. Omicar? Bulletin Periodique du Champ Freudien, (2-5). Laplanche, J., & Pontalis, J.B. (1973). The language of psycho—analysis. (D. Nicholson- Smith, Trans). Oxford, England: W. W. Norton. Ma, Y., Shi, W., Peng, C.K., & Yang, A.C. (2018). Nonlinear dynamical analysis of sleep electroencephalography using fi'actal and entropy

approaches. Sleep Medicine Reviews, 37, 85-93. Majewski, A. Modified binary decomposition of aynamical plane for fc(z) =2 *2. Retrieved from: Julia_IIlVI_6_circle.png https://commons.wikimedia.org/wiki/File: Mandelbrot, B. (1967). How long is the coast of Britain? Statistical self— similarity and fractional dimension. Science, 156(3775), 636-638.

—. (1977). The fractal geometry of nature. San Francisco, CA: W.H. Freeman. Marks-Tarlow, T. (2010). The fi'actal self at play. American Journal ofPlay,

3(1), 31-62. Masters, R.E., & Houston, I. (1966). The varieties o f psychedelic experience. Rochester, VT: Inner Traditions/Bear. Mauro, F., Raffone, A., & VanRullen, R. (2015). A bidirectional link

between brain oscillations and geometric patterns. Journal of Neuroscience, 35(20), 7921—7926. Mayhew, S.D., Ostwald, D., Porcaro, C., & Bagshaw, A.P. (2013). Spontaneous EEG alpha oscillation interacts with positive and negative BOLD responses in the visual—auditory cortices and default-mode network. NeuroImage, 76, 362-372.

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Nour, M.M., Evans, L., Nutt, D., & Carhart-Harris, R.L. (2016). Egodissolution and psychedelics: Validation of the ego-dissolution inventory (EDI). Frontiers in Human Neuroscience, I 0, up. Oliva, G. (2016). The road to servomecham'sms: The influence of cybernetics on Hayek from the sensory order to the social order. In L.

Fiorito, S. Scheall, & C.E. Suprinyak (Eds), Research in the history of economic thought and methodology (pp. 161—198). Bingley, England: Emerald Group. Ouellette, B. (2018, July 30). A math theory for why people hallucinate. Quanta Magazine. Retrieved from: https://www.quantamagazine.org/a— math—theory-for-why-people-hallucinate—20 1 80730 Palacios- Sénchez, L., Botero-Meneses, J .S., Pachon, R.P., Hernandez, LE. P., Triana—Melo, J. del P., & Ramirez-Rodriguez, S. (2018). Stendhal syndrome: A clinical and historical overview. Arquivos de Neuro—

Psiquiatria, 76(2), 120—123. Panksepp, J. (2000). The cradle of consciousness: A periconscious Panksepp. Jaak by Commentary homunculus?: emotional

Neuropsychoanalysis, 2(1), 24—32. Pellionisz, A.J. (1989). Neural geometry: Towards a fractal model of

neurons. In C. Rodney (Ed.),Models of brain function (pp. 453—464). Cambridge, England: Cambridge University Press. Pollan, M. (2018). How to change your mind: What the new science of psychedelics teaches us about consciousness, dying, addiction, depression and transcendence. New York, NY: Penguin.

Prigogine, I. (1980). From being to becoming: Time and complexity in the physical sciences. New York, NY: W. H. Freeman. —. (1994). Les lois du chaos. Paris, France: Flammarion. Prigogine, I., & Stengers, I. (1979). La nouvelle alliance: Me'tamorphose de la science. Paris, France: Gallimard. (2014, October 20). Retrieved from: httpszllreadiReaDi-EHESS, project.weebly.com/october-20th--2lth-2014.html Resnik, S. (2011). Reflexion sur autisme et psychose: Hommage a Frances Tustin. Journal de la Psychanalyse de L ’enfant, 1(2), 209-230. Rideout, P. (2012). Hilbert cubed sphere. Retrieved from: httpszliIcommons.wikimedia.org/wiki/File:Hilbert_Cubed_Sphere.png

Saavedra—Aguilar, J.C., & Gomez-Jeria, J. (1989). A neurobiologicalmodel for near-death experiences. Journal of Near-Death Studies, 7(4), 205222. Sacks, O. (2012). Hallucinations. New York, NY: Alfred A. Knopf.

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Sailko. (2009). A classic visual representation of the Borromean rings, used as an emblem of Lorenzo de Medici in San Pancrazio, Florence. Retrieved from: https://commons.wikimedia.org/wiki/File:Tempietto,_formelle_18_im presa_persona1e_di_lorenzo_de%27_medici.JPG Sala, N. (2006). Fractal geometry and architecture: Some interesting connections. In C. A. Brebbia & R. Pulselli (Eds), Eco-architecture: harmonisation between architecture and nature (Volume 1) (pp. 163—173). The New Forest, England: WIT. Schenberg, BE. (2018). Psychedelic-assisted psychotherapy: A paradigm shift in psychiatric research and development. Frontiers in Pharmacology, 9, 733, n.p. Serna, 1M. (unpublished manuscript). >30 years of fractals in psychoanalysis: State of the art and further implications. Sireteanu, R., Oertel, V., Mohr, H., Linden, D., & Singer, W. (2008). Graphical illustration and functional neuroimaging of visual hallucinations during prolonged blindfolding: A comparison to visual imagery. Perception, 37(12), 1805-1821. Telles-Correia, D., Moreira, A.L., & Goncalves, IS. (2015). Hallucinations and related concepts—their conceptual background. Frontiers in

Psychology, 6, 991, n.p. Theberge, A. (2018). Psychedelic therapy: The state of the art in 2018.

Transcendent Counsel. Retrieved from: https: // WWW.transcendentcounsel. com/news/20 1 8/ 3/ 9/psychedelic— therapies-flie-state- of—the— art-in-20 l 8

Thom, R. (1975). Structural stability and morphogenesis (D. Fowler, Trans). Reading, MA: Benjamin. Turing, A.M. (1952). The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237(641), 37-72.

Tustin, F. (1990). The protective shell in children and adults. London, England: Karnac. Woolfe, S. (2013). How geometric hallucinations are generated in the brain. Retrieved from https://www.samwoolfe.com/2013/06/a—model-of—how— geometric-hallucinations.html

Wurtz, RH. (2009). Recounting the impact of Hubel and Wiesel. The Journal of Physiology, 587(12), 2817-2823. Yensen, R. (1998). Hacia una medicina psiquede’lica: Reflexiones sobre el uso de enteogenos en psicoterapia. Barcelona, Spain: Los Libros de la Liebre de Marzo.

Young, A.M. (1976). The geometiy of meaning. New York, NY: Delacorte.

CHAPTER SEVENTEEN CRACKED ORLANDO: DRAMMA PER MUSICA E FRACTALS,

DIMENSIONS OF A FRACTAL BAROQUE OPERA JONATHAN DAWE1

Music and fractal geometry Although Mandelbrot’s fractal focus was largely directed toward the natural world, emerging technologies, and financial markets, he clearly took delight in and was often happily surprised by the broad application of this new field of mathematics among a host of unforeseen and diverse areas. This is not surprising, especially regarding the visual arts, as Mandelbrot asserted that his earliest breakthroughs developed when rethinking aspects of pure math as visual objects, “that part of geometry in which mathematics and the eye meet” (TheBITK, 2016, 00:23). He did caution against assuming relationships between math, music, painting, and architecture yet still referred to this melting pot as a “paradox,” acknowledging perhaps the elusive yet fullyfelt powerful overlaps existing between these different domains (Mandelbrot, 1977/ 1983). For Mandelbrot fractals seemed a potential underlying force manifest within these fields. Among these inklings, he seemed quite comfortable with the thought that music did indeed possess fractal aspects (Mandelbrot, 2012). Of the principal features of fractals: iteration, sefisimilariiy among levels of scaling of size, and the measurement of “roughness” as dimension, all have meaningful existence in Western music. First, repetition is a common, if not pervasive, aspect of musical construction and logic. Second, the presentation of similar musical events on varying levels of temporal size

1 Graduate Studies Faculty, Theory and Analysis Faculty, The Juilliard School. Email: [email protected]

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is a significant feature of cohesion and hierarchy in musical structure.2 And third, the varying density of musical events and/or the contour of musical lines certainly can be conceptualized as a type of topology, measurable and lying somewhere between what would be a block mass of sound and a single flat line of music. Upon closer measurements, music may reveal greater levels of detail and thus a fractal dimension may be determined.3 The sense that music possesses a fractal quality has received increasing attention. This chapter is less concerned with the search and discoveries of fractal attributes in music, in general, as tools for analytic applications. Rather, the following discussion presents particular fractal applications used as compositional strategies that were the generative force behind an Opera’s creative construction. But first, some general considerations are beneficial.

Music as a metaphor on the Cartesian plane It might seem tidy to keep the discipline of music abstract and unfettered when involving the application of fractal thinking; as a pursuit, this is possible by applying pure math to pure music. However, our very construal of music both creatively and perceptually is bound up with metaphors whose tactility lies squarely in the visual domain. If the common playing field of fractals is in geometry (and its visual models), then this is perhaps the most intuitive bridge to music. There are many types of visual metaphors are constantly at work affecting the various ways we construe the domains of music. Consider how a pitch may sound higher or lower (verticality), or that music unfolds in time (from lefi to right). These are just two of the pervasive symbolic dimensions that map the body’s perspective onto music. Western notation of music itself is deeply metaphoric in linking the, phenomena of organized sound to optical imagery.4

2 These ideas resonate with those of the theorist Heinrich Schenker (1868-1935) in which the technique of prolongation is operative on all structural levels and at differing time scales. 3 The fiactal dimension of the topology or texture/density of music may be measured

in several ways (see Madden, 1999, pp. 119-137). Madden maps musical data onto what he terms a “scatter diagram.” This suggests in turn a provocative idea that all

music possesses a topology such that infinite dimensions reside between each of the sounding notes, as overtones, or perhaps an imaginary music of implication. 4 Other forms of “notation” with strong musical metaphoric mappings exits, such as MIDI “piano role” and spectrogram imaging.

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A starting point then is accepting these analogies as a viable portal

between visual and musical structures. A basic phenomenological stance is the denotation of pitch (frequency) and duration (time) as intrinsic to visual symbols of location and direction, vertically and horizontally, respectively. Musical design thus may be mapped onto the complex plane where discrete pitches are indexed by the Y axis, and discrete units of time are plotted on the X axis (see Figure 17-1). An additional significant metaphor, especially

as music may relate to fractal geometry, involves the awareness of the shape

9

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of a musical line. It is in this contour, as it ranges from smoothness to roughness, that strongly presents itself for fractal interpretation.

EEULLEILEUEQ Figure 17-1. The opening of Johann Sebastian Bach’s F uga 1, Well-Tempered Clavier Book II (left) mapped onto the complex plane (right). X = pitch, Y = time stream, in this case 16th notes are the basic unit.

A musical fractal

Engaging the fractal tryptic, iteration with self-similar scaling, a simple musical fractal is constructed; its evolving topology possessing a fractal dimension. Here a motive is composed with three pitches; this starting musical “shape” is analogous to a fractal motif or initiator. The figure is then transposed and “nested” onto each of the three pitches of the original motive creating an outgrowth of system branching, that potentially can be repeated over and over again, creating a musical texture of ever—increasing detail (or roughness) (see Figure 17—2).

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fl / Q fl :W "' $.51: A

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Figure 17-2. A three-note motive is nested Within itself, the process then repeated.

Baroque “affect” and recursive growth A unique musical idea will grow, magnifying its own unique qualities. If a self-referential musical event is amplified, its highly contextual vocabulary is maintained and advanced, and this is fine. However, music with its ability

to reference previous music and/or draw upon more universal-musical syntaxes allows for fractal elaboration possessing a greater archetypal sweep. Further, musical material that embodies extra-musical references may produce upon expansion a far-richer pallet of associations for a listener; as it is not simply a growth of the pitches but also the associations and/or

emotions they attempt to evoke. Eighteenth-century musical rhetoric makes for a fascinating area for fractal growing. In Baroque rhetorical musical thinking, compositional

devices parallel or even mimic the figures of speech and mannerisms formalized in the art of persuasive speaking. This can involve a wide range of applications from more simple physical gestures (e.g., raising volume or speaking more quickly at a key moment to emphasize a point) to particular

figures of speech (e.g., the repetition or omission of a word [or idea]). Rhetoric techniques may even engage ways of depicting actual emotional states (e. g., sorrow, rage, joy). In music, emblematic codes represented by specific musical gestures would symbolize particular ideas or emotions.

These figures were understood to carry a degree of universality, often

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immediately recognizable by an audience. The high potency of commonality of these musical devices would allow for their existence in many different compositions and musical contexts. As such, they embody a paradigmatic presence. Connections between verbal- and musical-rhetoric figures were often quite direct. Ellipsis, for example, denotes a sudden interruption of an idea; musically it denotes a sudden shift of haimony or an omission of an expected resolution. Other devices are more depictive of an overarching emotional affect. For example, downward slides of a pitches by step may represent moments of sorrow and deep lament. A musical fractal based upon a musical moment that simply stands on its own merits (a figure with no outside reference) will create in its recursive growth reference only to itself. For example, a random succession of pitches presents its own identity but does not necessarily depict any universal gesture or emotion. As a musical structure, this is meaningful, as repetition of any kind advances wholeness and structural integrity. However, it is compelling to magnify a musical event that implies more than simply the sum of its pitches, but rather that suggests an emotive state or mood. If fractals are understood (in part) as revealing the organic-recursive building blocks of the natural world, then a powerful alchemy may reside in extending this to music and the patterns of emotional responsiveness.

Cracked Orlando: dramma per musica e fractals Cracked Orlando: dramma per musica e fractals (2010)5 is an eighty— minute mini-opera that recast the sounds and energies of Baroque music using compositional workings based upon fractal processes. Musical fragments were drawn from three Orlando F unioso operas of the eighteenth century, and then expanded and transformed.6 In places throughout this dramatic work, the original Baroque borrowings are readily perceivable; in other passages the fractal transformations are more engaged, obscuring the original artifacts. Thus, a respiration ebbs and flows between passages that are highly reflective of music of the past and musical moments that are decisively modern. This conj oining of older expressions with the atonal mix that results from the fractal manipulations sets this work firmly within a category of musical post-modernism. On an initial level, the 5 This work premiered at The Italian Academy of Columbia University in 2010 and was later produced by The Iuilliard School in 2017. 5 The Baroque composers from which fragments were drawn are George Frideric

Handel (1685-1759, Nicola Porpora (1686-1768), and Antonio Vivaldi (1678-1741).

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opera follows the conventions of eighteenth-century opera seria, where an alternation unfolds between recitatives (more akin to spoken dialogue) and arias (more songlike, where emotional states are expanded and explored).

The story The opera presents a dramatic episode from Ludovico Ariosto’s epic sixteenth-century Italian poem Orlando Fur/rose, an expansive tale of fantasy, love, and heroic deeds. Set upon the sorceress Alcina’s enchanted island of the sorcerers, Angelica, a princess from the east, has arrived seeking sanctuary. She flees from the aggressive unwanted love advances of headstrong Orlando. Meanwhile Medoro, Angelica’s true love, arrives upon the same island in dire search of her. Orlando also appears in hot pursuit. Bewildered by Alcina’s magic and Angelica’s trickery, Orlando finds himself imprisoned in a dark sooty cave and tormented by demons. His gradual realization of betrayal manifests and mounts. Upon his escape from the cavernous dungeon, the growing realization of the romantic union of Angelica and Medoro leads him to despair, jealousy, and rage. Amidst latenight wanderings deep in the forest, he stumbles upon a secret sensual wedding party of the two lovers. Such vision leads Orlando to madness. At morning, a magical zephyr, representative of a natural (if not divine) force, whispers the means of a cure for Orlando’s mental implosion. A seemingly simple solution, a single candle that burns on Alcina’s island need be extinguished. Upon hearing of the solution, Angelica puts out the candle, and all of the lush fantastic surroundings of Alcina’s island vanish. All was an illusion; only a barren empty landscape remains. This leads to the cure, as now Orlando is able to see clearly and examine himself deeply for the first time. In this inner investigation of his own levels of selfsimilarity, he is able to release himself, to ‘crack open’ in order to see his way to acceptance, forgiveness, and wholeness.

The libretto The libretto is constructed from excerpts of the libretto by Grazio Braccioli for Vivaldi’s Orlando Furiosio (1727). The driving idea behind the extraction of words to form the new libretto was to select the fewest lines necessary that would be needed to advance the plot. The resulting verse therefore reveals a lean, at times fragmented quality, but nevertheless

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maintains a clear single thread of the basic narrative. These Italian words are set in the conventions of eighteenth-century opera sen'a, highlighted by an alternation between recitatives and arias. Brought into the mix of this skeletal libretto in Italian, however, is a significant new dimension involving the interpolation of lines in English. The English verse, newly composed by Terry Marks—Tarlow, creates a psychological drama that runs concurrently throughout the work. Four discrete emotive/psychological states unfold and are interwoven into the Italian-set story line. These themes, revealed in Figure 17-3, reach into the deeper levels and raw emotional substance of the drama. Thread 1 WHOLENESS

Thread 2 POLARITY: THE FIGHT Thread 3 DYNAMICS, CHANGE, PROCESS Thread 4 MANIFESTATION, RESOLUTION, TRANSFORMATION Figure 17-3. The four archetypal, interpolated themes of Cracked Orlando Each thread/theme undergoes its own expansion based upon Fibonacci growth patterns. The Fibonacci numerical sequence is attributed to the 12century mathematician, Leonardo (of Pisa), later named Fibonacci. A Fibonacci sequence is quite simple in concept, constructed such that each subsequent number results from the sum of the previous two numbers in the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34.... The larger the numbers grow, the closer the ratio of adjacent numbers creeps towards the golden mean (161803.. .), a ratio revered for millennia for its properties in nature and utility in art and architecture (Olsen, 2006). Each word represents the unit of measurement for that growth. The first fractal thread, WHOLENESS, follows the most standard growth structure—1, 2, 3, 5, 8, 13, 21, 34—where each number represents the number of words in the stanza (see Figure 17-4 below).

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Thread 1: WHOLENESS

ALCINA:

0h! 1

ORLANDO:

0h tigress 2

MEDORO:

Oh God help! 3

(continued) ALCTNA:

Bring me peace through love 5

ORLANDO:

Seeking wholeness, darkness finds its place in me 8

MEDORO:

Blow gentle zephyr, sigh you whisper, lovestruckI sway in the beloved breeze 13

ALCINA:

Alas for me! I oflered you precious gifts. Now stn'pped of all, I refuse this fate. You must pay, you must pay! 21

ALL:

The ribbon binding love sufi‘ers perilous journeys Loneliness crushes its fabric; jealousy frays its threads. The ribbon falls to earth; sullies beneath our feet YetHeaven ’s eternal bonaT forever restores even the dustiest threads. 34

Figure 17-4. Fibonacci word-growth for Thread 1.

The three remaining threads each begin with different starting conditions (numbers of words). This renders poems of varying size, all self—similar as they move toward golden ratio proportions (see Figure 17-5).

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Thread 2: POLARITY/THE FIGHT

2, 4, 6, 10, 16, 26, 42 Thread 32 DYNAMICS, CHANGE, PROCESS 3, 6, 9, 15, 24, 39, 63

Thread 4: MANIFESTATION, RESOLUTION, TRANSFORMATION 4, s, 12, 20, 32, 52, 84 Figure 17-5. Fibonacci growth for threads 2, 3, and 4. Numbers refer to thenurnber Of words in each ‘stanza.’

The placement of the individual stanzas (derived from each of the four threads) are braided throughout the length of the opera, appearing in key dramatic moments. As they each grow, a dramatic transformation emerges as the passages in English expand. Each of the three Acts therefore grows in length and amount of purer emotional content intensified for the audience by the immediacy of the English words7. The effect creates a dynamic, yet subtle growth as fleeting moments of emotional asides (stanzas with one or several words), later transform into outbursts and reflections, finally emerging as full soliloquies of sentimental states by the Opera’s end. As such, the more stylized Italian landscape recedes into the background while the organic, emotional English domain becomes pervasive. At the Opera’s conclusion, the psychological reahn, hidden at the drama’s start, turns inside out, exposing itself fully for all too see and hear; the inner drama now becoming the outer one. The more emotional-evoking arias of the Italian opera seria layer are now relegated to the stylized and stiff world of the recitaiives through the recursion that occurs with their own relationship to the evolving hyper-emotional threads (see Figure 17-6).

7 For an English—speaking audience.

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START —

EMOTION

CTl

A

0N

> FINALE

THREADS

ARIAS

REaTWES

Figure 17-6. Depiction of Thread growth and Aria/Recitative decline during the course of the opera.

Although each of the threads is linked to aparticular emotional state/theme,

they nevertheless migrate through different characters of the opera (see Figure 17—7). Scene i

MEDORO:

Qui dove dolce Zefl’iretto spira E per l ’amata aurea innamorato (Fragment from Grazio Braccioli libretto)

Blow gentle zephyr Sigh your whisper Lovestruck] sway in the beloved breeze (Stanza 6, Thread 1, Marks-Tarlow) Scene ii

ORLANDO:

Ah sleale, ah spergiura Donna ingrata infedele, con traditor! “ll/fedora qui. d ’Angelicafit sposo! ” (Fragment from Grazio Braecioli libretto)

So tormented be her deceit. The wilting of my wings; the piercing of my hopes. Where hides my ungratqpul love now? My soul writhes and rages. Escape brings no refuge at all. (Stanza 5, Thread 4, Marks-Tarlow)

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I cannot contain the sorrow

I am flooded And carried away by torrents o f grief Damn my helplessness! Nowhere to go The fight cinders my soul

Shatters my rage Cracked to pieces Where is Orlando? Orlando exists no more... . (Stanza 6, Thread 3, Marks-Tarlow) Figure 17-7. Cracked Orlando libretto, opening of Act 3

Fractal musical workings Each of the three acts of this miniature opera embodies a different fractal compositional strategy, which highly differentiates the quality and sound

world of each act. Three primary compositional fractal procedures were employed; all involve iteration and layers of scaling.

ACT I: Rotational Arrays ACT II: Cellular Automata

ACT III: Overtone Projections Each of these three fractal procedures is defined and illustrated in sections to follow.

Rotational Arrays Rotational arrays are two-dimensional matrices generated b y a starting series of pitches, such as these three-notes:

E In the creation of the array, each “new” row is generated b y a simple rotation, such that the row below begins with the second pitch of the row above. In addition, the pitch that “came before” gets wrapped to the end of

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the newly-rotated row (see Figure *-7, for further discussion, see Straus, 2005, pp. 231-234).

//

fiE/ fl fi fi Figure 17-8. A three-note “tune”/shape becomes nine notes, as each of the 3 rows of the array on the left are made into one single linear unfolding of pitches.

While the array in Figure 17-8 does create an iteration of the original pitch row (with rotations), it does not embody an intrinsic sense of scaling. However, a degree of scaling is a feature of a transposed rotational array. Here, each of the rows is transposed (after the rotation process) to begin on the same initial pitch as the original row (see Figure 17-9).

[

\

1

Figure 17-9. A transposed rotational array, with its musical realization.

A linear sequence of the three rows consists of musical “shapes” that are nested in the contour of the same shape as projected in the large. Please note

that the larger shape is presented in inversion (with its contour direction, up/down reversed.) Saturation of the original motif (row) embedded Within a larger scaling of itself leads to further fractal recursions. The full structure (all of the

pitches) of a rotational array can be extended by reinterpreting it as an original row. Through this process, an “array of an array” can be constructed,

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which now engages four levels of scaling (see Figure 17-10). In the figure below, arrows indicate the original and now largest structure. In this example, a single motif consisting of a three-pitch row, first transforms into

nine (3*3) and then into 81 (9*9).

T Figure 17-10. A fractal array sketch presenting multiple levels of an original motif. Integers represent all pitches C = 0, C# = 1, D = 2, ...B = E (eleven); mod 12.

These compositional workings contribute to the lyrical, at times serpentine, music ofthe first Act (See Figure 17-11).

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oleimx I

E

Figure 17-11. Composer’s sketch of music for Act 1, scene 1.

Cellular A utamata Cellular Automata utilize feedback algorithms to determine a series of cell configurations on a grid of a specified shape. The design growth upon the grid functions as a dynamical system that evolves through a series of discrete time steps as determined by a simple set of rules (Wolfram, 2002). Each cell in the grid can be in one of only two states—either on or ofi‘: In each new generation, Whether a cell remains on, turns off, or if a previously off cell comes to life, is determined by the cell structure of the previous generation. Starting with an initial cell configuration, the same algorithm is applied again and again. New configurations emerge that grow, transform, and decay over time.8 There are many different types of cellular automata (CA), which all engage unique sets of “rules,” resulting in various types of growth behavior. Wolfram (2002) demonstrates how many complex patterns and seemingly chaotic textures of nature may be replicated by these

simple codes of cell growth. The Fredkin’s CA, which is applied to the music of the second Act, is of particular interest (see Poundstone 1985, pp. 136-137). Named after the mathematician Edward Fredkin (b. 1934) this CA involves the following rule: Cells remain on (and empty cells come to life) in the next generation if and only if 1 or 3 if its 4 neighbors (N, S, E, W) are on in the current generation. If this condition is not met for a given cell, it will die ofi’ in the next generation. After a number of generations, four presentations (and later multiples of that) of the original structure emerge (see Figure 17-12). The

3 Perhaps the best-known example of Cellular Automata is The Game of Life by John Horton Conway (b. 1937).

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l l

[r‘

number of generations required to yield replications depends upon the complexity of the original configuration (see Figure 17-13).

Figure 17—12. A starting configuration of Fredkin’s CA that replicates four copies of the original in the fourth generation

Figure 17-13. Transformation of Fredkin’s CA from the 11th generation (left) to the

12th generation (right), revealing l 6 replicas of the original.

In Cracked Orlando, fragments of Baroque music are used as the initial cell structures with pitch and duration mapped upon the grid. The intention was to grow a fragment of Baroque music, including its emotion punch, based on the same simple laws that appear to be at the heart of the complexity and organic patterning within nature—a compelling thought (see Figure 17-14).

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51 0

Chapter Seventeen

The resulting music of the multiple generations of cell growth is ofien presented simultaneously, creating contrapuntal textures. The rich, at times

dense, polyphony creates a percussive—driving effect that underscores the volatile and dramatic action of the second Act (see Figure 17-15).

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0vertane Projections The third and final act of Cracked Orlando involves fractals based on compositional projections onto the overtone series. This structure, also called the harmonic series, consists of a rising sequence of tones, where the frequency of each sound is an integer multiple of the fundamental lowest tone. In practice, when a fundamental tone generates upper frequencies, the resulting overtones tend to be faint to non-existent for human perception;

51 1

Cracked Orlando

this especially holds for the higher frequencies. However, it is the emphasis of particular partials (harmonics) over a given fundamental frequency that

partially determines a pitch’s timbre (color/texture). Every tone can serve as a fundamental that generates its respective overtone spectrum? The overtone series also contains internal projections of itself that spread over increasingly large scales of self-similarity (see Figure 17-16).

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The music of this final Act projects musical fragments onto this structural design in a variety of ways. In Figure 17-1, an 18th century fragment (top line/glockenspiel) is transposed to the 16th partial above the fundamental pitches. The same music is projected onto the 15‘1‘, 1411‘, and 13th partials, such that their respective rhythms reflects these ratios, and a

spectral polyphony results. mm

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9 Exception to this general rule can be found in the sine-wave tone, which as a pure tone results in no overtones.

51 2

Chapter Seventeen

overtones represent the upper partial of thatparticular series (see Figure 17—

18).

Figure 17-18. Another application of overtones inAct 3 of Cracked Orlando. The application of this compositional process in the third and final Act creates a music that is curiously harmonious, resulting from the presentation of the lower eight partials of the overtone series. The upper partials move into a more dissonant pallet, resulting from the pervasive appearance of

Cracked Orlando

513

microtones, or pitches in between the twelve pitches of Western music, creating a sparkling and shimmering sound.

Threads and their musical fractals Returning to the psychological threads intercalating the opera, each of the four themes is supported musically by the growth of its own fractal design. The Fibonacci expansion of words within each thread’s stanzas coincides with the scaling magnification of the musical fi'actals. Each musical fractal underscoring a thread is based upon a small musical passage that was drawn from one of three Baroque composers and their respective Orlando operas. Two of the threads ascend in pitch direction while the other two threads descend. Additionally, one of the rising lines is chromatic, while the other is diatonic; likewise, one of the falling lines is chromatic, while the other is diatonic. A general sense of direction (up or down) is significant because its continued expansion augments this readily perceivable contour. Sequential shapes are frequently found in the catalog of 18‘11'century rhetorical devices. They serve to advance one’s sense of a growing emotional force, a gestalt infused with the mounting impact of the psychological import of the growing verse. As rhetorical devices, these may be understood as follows: Anabasis: a rising figure that depicts heightened feelings of exaltation; and Catabasis: a falling line that depicts feelings of deflated, lamenting moods. Each of the fractal processes for these evolving structures involves the feedback of their basic rules onto themselves. This looping process creates a hyper-fractal, or higher-order fractal displaying different algorithms on different scales. In Figure 17-19, the fragment 2) by Vivaldi undergoes Cellular Automata (Fredkin’s game) procedures, while 3) by Vivaldi incorporates a Rotational Array, and 4) by Porpora utilizes Overtone mappings. (The borrowed fragment in 1) is be discussed later.)

Chapter Seventeen

5 14

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516

Chapter Seventeen

In Figure 17-20, the original descending line (D, C#, C, B, B—flat, A) of Vivaldi’s Orlando is transformed into a rotational array that becomes the starting line for a new array, and so on. Each generation of a larger array serves as a new underpinning for the word thread stanzas. In the process, an expanding iteration of the original Baroque affect mounts in intensity, establishing greater levels of emotional impact. Since there are four musical threads and only three fractal methods (applied to each of three Acts and three of the four threads), a fourth fractal design is brought into play for the first fractal thread and motif by Handel (see 1) in Figure 17-19). This cellular automaton works from a single cell to grow upwards. Its discrete one-dimensional growth creates Pascal’s triangle, which is rich with levels of self-similarity (see Figure 17-21).10

Figure 17-21. Wolfram Rule 90, Cellular Automata of Pascal ’s triangle.

Here, each of the lines (from a single box and up) are respectively applied

to the expansion of Handel’s Orlando passage (see Figure 17-22).

10 Stephen Wolfram (b.1959), a pioneer in cellular automata, has named this particular CA as Rule 90. Here, a new cell will turn on in the subsequent generation (level above) if its cell space touches its neighbors at the angles, and only the angles.

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Figure 17-23. Multiple polyphonic entries of the Handel motive results from a slice of the 5th-9th generation of the CA structure.

522

Chapter Seventeen

At the work’s climax, a rich contrapuntal texture built from Handel’s three-note figure saturates the final scene (see Figure 17-23), which is rhythmically cast to reflect the patterns fi'om higher portions of the automaton structure that permeates the final act. As for the culmination of the fourth fractal thread, WHOLENESS, Orlando’s own realization of his self, his worth, and the resonance of love are accompanied by the original passage of Porpora’s fragment that pushes higher and higher up into the overtone series until the resulting resonation of high upper partials reveal the very essence of the original music.

What is this? With Alcina ’s torch extinguished I now can see... How could I be so blind? Why chase a love that only blossomed within myself? For a love is not a love unless both hearts crack open. I was chasing only illusion; Seduced by its beauty; grabbingfor thin air; falling into delusion. I lost only myself falling into madness. Cracked open into hell. What crawled through the darkness, what I found were my knees. Now I pray that your love has no end. Postlude I met Benoit Mandelbrot in Boston, in January of 2006. The Boston Symphony was premiering a new orchestral work of mine, The Flowing Arts. A featured article in The Boston Globe entitled ‘A Symphony of Flowers and Fractal’ had caught his attention. Mandelbrot contacted me from Cambridge, where he was living. We decided to meet before one of the performances along with his wife, Aliette, at the elegant bar of the Copley Hotel. During our brief exchange I showed him some of the sketches and fractal workings of the composition to be performed later that night. He understood with smiles and familiarity the graphics and charts, but the more musical translations seemed more evasive to him. We both coyly maintained an inherent sheepishness, due to what seemed to be a mutual awareness of the respective disadvantages we both shared regarding each other’s field of expertise—music versus math/science. Nevertheless, the common ground of our discourse was fanciful, if not fun. Aliette was rather intrigued by what I termed fractal—Baroque opera.

Cracked Orlando

523

After the performance, we met briefly in the lobby of Symphony Hall. Mandelbrot impishly stated, “I did enjoy the piece very much, but I am not entirely certain that I heard any fi'actals.” In providing an on—the-spot disclaimer, I mentioned that whether one hears them or not, it is the fractal process of the work’s creation that built it into what it is in the end. But more.. . (and sometime later) I wondered if musicians, for whom music is a fluent second language, are much more likely to make the metaphoric shift from the purely audible into the imagined realm of visual geometry when hearing such sounds. Mandelbrot died on October 9th, 2010, on the eve of Cracked Orlando ’s premiere at The Italian Academy in New York City. In his memory, each of the performances was dedicated to his life of dynamic discoveries. It is my hope that this opera offered a small contribution to Mandelbrot’s powerful

mathematics and the diverse areas they still inspire. References Madden, C. (1999). Fractals in music: Introductwy mathematics for musical analysis. Salt Lake City, UT: High Art. Mandelbrot, B. (1983). The fractal geomet7y of nature. San Francisco, CA: WH. Freeman. (Original work published 1977)

—. (2012). Thefractalist: Memoir of a scientific maverick. New York, NY: Pantheon/Random House.

Olsen, S. (2006). The golden section: Nature’s greatest secret. New York, NY: Walker.

Poundstone, W. (1985). The recursive universe: Cosmic complexity and the limits of scientific knowledge. Ontario, CA: Contemporary. Straus, J. (2005). Introduction to post tonal theory (3rd Edition). Upper Saddle River, NJ: Pearson/Prentice Hall.

TheBITK. [Producer]. (2016, June 12). The Mandelbrot set: The only video you need to see [Video File]. Retrieved from: https:l/wwwyoutube.com/watch?v=56n00d6DU.

Wolfram, S. (2002). A new kind of science. Champaign, IL: Wolfram Media.

AFTERWORD

TOWARDS A DIALOGUE BETWEEN TRANSPERSONAL AND NATURAL SCIENCES

YAKOV SHAPIRO TERRY MARKS-TARLOW KATTHE P. WOLF HARRIS L. FRIEDMAN

What can be the impact of fractal thinking on transpersonal psychology and the natural sciences as a whole? When we ponder the diverse contributions in this volume, several emerging trends become apparent.

First, we observe self—similar patterns among multiple domains of physical, biological and psychosocial phenomena. These patterns emerge in the process of self-organizing complexity in nature, including inorganic, biological, and sociocultural evolution. Parallel patterns of self-similarity are also apparent between the ontology of what exists and the epistemology of our evolving knowledge of it. Just as cosmological, biological, and cultural evolution demonstrates a progression of self-organizing complexity, the evolution of knowledge can be visualized as a spiral of prevalent epistemological paradigms that emerge, solidify, and are eventually replaced by more encompassing and accurate representations of the world. Culture-specific creation myths and supernatural conceptions of nature are superseded by universal principles that can be subjectively discovered and objectively corroborated, forming an intersubjective tapestry of scientific knowledge that transcends cultureinformed imagination. Insofar as the main tasks of the natural sciences are to understand the principles that drive the evolution of nature, and to separate scientific facts from mythical narratives, then the scientific process itself is a necessary synthesis of objective, subjective, and intersubj ective perspectives that may be in fiactal correspondence with each other.

Towards a Dialogue between Transpersonal and Natural Sciences

525

Second, the fractal perspective points to the property of scale invariance

at dzfierent levels of evolutionary complexity. Scale invariance holds both as a cross-section of hierarchical organization of what exists, fi'om quantum to physicochemical, biological, psychological, sociocultural, and technological domains, and across microlevel to cultural and cosmological time scales.

The validity of fractal evolution would necessarily imply that the seeming separateness and dissimilarity between various fields of scientific inquiry, such as in physical versus psychological sciences, is only superficial. Scaleinvariant and self-similar patterns evident within each domain, point to common-informational foundations for all reality. In these terms, the fractal perspective is inherently holistic, allowing for the consilience among the diverse disciplines and perspectives, from hard sciences to humanities. This argument brings us to the third aspect of the fractal perspective, that

offractal boundaries and interdimensionaliiy. Within fractal epistemology, the very concept of a boundary shifts from being a deceptively linear construct, separating phenomena in question, to a complex, semi-permeable matrix that simultaneously divides and bridges distinct but interconnected domains, just as a cell membrane maintains internal homeostasis through a complex interchange between its inner and outer environments. Within fractal epistemology, the seemingly dichotomous distinction between “objective reality” and our subjective experience of it assumes a unified, interdimensional quality, where novel constructs may emerge and span both domains. Such objective/subjective unity is best seen in the discipline of mathematics itself, pointing toward self-similar correspondence between “constructs of the mind” and physical processes in ontological reality, the paradox Eugene Wigner (1960) described as the “unreasonable effectiveness of mathematics in natural sciences.”

Fractal epistemology manifests the reality of interdimensional phenomena that are bot “created” and “discovered” by the human mind, such as natural morality, esthetics, art, science, and culture as a whole. We can discover what exists and create novel realities in the process of its discovery, which recursively shape the very process of knowing what exists. Thus, creation of symbolic language as a means of intersubj ective discourse eventually enables Shakespeare to create/discover Hamlet (think of Jorge Borges’

(1941/2000) apocryphal story of The Library of Babel, which contains a complete compilation of books with every possible combination of 25 letters and signs, thus containing everything that has ever been or will ever be written). The creation of musical notations enables Mozart to create/ discover his last J upi ter symphony, just as the creation of sculpture as a form of art enables Michelangelo to create/discover David in a block of marble,

526

Afterward

famously saying that he only had to chisel away the extra stone that prevented others fi'om seeing it

Similarly, the matter/mind distinction inherent in separating physical from psychological sciences can be conceptualized as a fractal boundary. Fractal epistemology strongly suggests that Cartesian stratification of reality adopted by the reductive science is superficial and limiting. Insofar as “hard” sciences are based on mathematical language and intersubjective discourse between conscious observers, objective knowledge cannot exist without its subjective and intersubj ective dimensions, just as subjective and cultural phenomena cannot be separated from their biophysical foundations. This opens the possibility of a meta-reductive scientific paradigm (MRP Scott, this volume, chapter four) that extends the naturalistic perspective beyond conventional materialist definitions and allows for emergent phenomena of arbitrary complexity to be integrated in the dual reductiveholistic model. Other examples include neurological positivism (V andervert, this volume, chapter thirteen), Bohmian informational dynamics (Shapiro, this volume, chapter three), and the fractal epistemology model itself (Marks-Tarlow, this volume, chapter one). In envisioning evolution as a fractal process, we begin to glimpse recursive patterns of self-similarity that underlie physical, biological, and psychocultural phenomena, which may go back to a universal property of reiationaiity among both classical (inorganic and living) and quantum systems to “sense” their environment and maintain informational connections with it (see Kauffinan, this volume, chapter five). The fi'actal approach powerfully argues that trying to separate “objective” fi'om “subjective” sciences is a fool’s errand. A more encompassing MRP must involve systematic examination of the essential objective/subjective balance that defines the scientific process and ontological reality as awhole. In this light, fractal epistemology radically shifts the question of whether we should privilege an “objective” or “subjective” perspective in scientific discourse, suggesting that any science, defined as a systematic set of verifiable knowledge about the world and us in it, needs to incorporate and acknowledge both perspectives.

Can fractal epistemology throw some light on the persont‘ranspersonal boundary inherent in mystical, out-of-body, non-dual and psi-based experiences? Just as denying the existence and causal impact of ordinary consciousness and free will leaves natural sciences trapped in the determineistic clockwork Universe and Cartesian reductive epiphenomenalism, so denying the transformative power of alternate states of consciousness

Towards a Dialogue between Transpersonal and Natural Sciences

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perpetuates the artificial separation of rational and intuitive modes of knowing and shortchanges the scientific process itself. Creative imagination is the engine of science; fi'actal self-similarity between processes of the mind, processes of the brain, and quantum/classical processes in wider reality may serve as an indispensable source of knowledge about the world. The interdimensional qualities of the personal/transpersonal boundary open up wider horizons of the unified psychobiological reality that transcends the confines of the “dual self” worldview, just as rigorous elaborations of “quantum weirdness” helped us transcend the limitations of the classical model in understanding the physical world. Rigorous transpersonal science carries the potential to elaborate a “mathematics of unitary experience,” building on self-similar patterns beyond the confines of ordinary dual Self. It may expand the reach of the rational mind to wider psychophysical domains, just as recent technologies allowed us to probe the microworld and reach back in time in probing the cosmological horizons not directly accessible to our senses. By way of an illustration, we could picture psychobiological reality as a vast Mandelbrot set, where conscious observers form an inseparable part of the reality they observe, and the iterations of matter/mind informational processes unfold in self-similar ways across spatial and temporal domains. As a microcosm of such a meta—set, our subjective psychological reality, including analytical, intuitive, and mystical experiences, can inform us of the ontological patterns in the Universe as a whole, much as abstract mathematical constructions often do. Fractal epistemology thus provides an elegant solution to the Wigner’s paradox: if physical and psychological domains are in fact self-similar, then mental constructions, such as in the field of mathematics, will show meaningful correlations with “reality out there.” In turn, the fact that we consistently observe such correlations provides an additional source of support for the fractal nature of the unified psychobiological reality. The rigorous elaboration of transpersonal science within a metareductive fractal epistemology framework leads us to as yet unexplored vistas of interdimensional science. For example, within the current reductive paradigm, we tend to approach a problem analytically, such as calculating the wavelength of a color that results fi'om mixing green and red spectrum light. But we could also imagine experiencing the solution directly, such as in “seeing” bright yellow at the intersection of green and red laser beams in the mind’s eye and then substantiating our experience analytically. We could say that the two approaches are interchangeable, just like the seemingly separate surfaces of the Moebius band. However, whether we

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arrive at a solution in analytical or experiential terms, we have to employ the empirical method to separate facts from imagination—posing a question and verifying the information we obtain through independent means. It is not enough to have an experience or to have faith in whatever construct we postulate—we can imagine any number of things that have no ontological validity whatsoever. Science does not equate to imagination; it involves ongoing rigorous analysis of our assumptions and relevant data in order for our imaginations to become valid scientific theories, and for scientific theories to become confirmed as scientific facts. Perhaps the most important message that the fractal epistemology perspective imparts to scientific process as awhole is that of bridging, rather than separating, the domains under scientific study. It may function as an antidote to the binary logic that all too often serves as a default mode of either-0r mentality of the rational mind: brain vs. mind, objective vs. subjective, rational vs. intuitive, personal vs. transpersonal. The current reductive paradigm in natural sciences can well be described by the Hopi term Koyaanisqatsz', life out of balance, as documented in Reggio’s (1982) powerful film. The fiactal self-similarity perspective concerning the ontological and subjective dimensions, and infinitely deep, semi-permeable boundaries bridging them, strongly suggest the need for a systemic worldview as a counterbalance to reductive thinking alone. The evolving symmetry of fiactal dynamics can be illustrated by considering the inherent unity of the basic-building blocks of all reality: matter, energy, and information. While matter and energy are interchangeable based on Einstein’s famous equation E=mcz, both matter and energy can also be derived from information (John Wheeler’s famous formulation of “it from bit”). Each one can thus provide a link to the others, matter bridging information and energy (consider the brain converting the energy of the body into the information of the mind); energy bridging matter and information (information needs energy to inform material processes, and the brain needs biochemical energy to “produce” consciousness); and information bridging energy and matter, as in a common substrate between the processes of the brain and mind. The Dutch mathematician and painter M.C. Escher provided a beautiful illustration of this unity in his “Bond of Union.” Is fractal epistemology sufficient to resolve the current scientific irnpasses related to consciousness, such as “the hard problem” of brain/mind or the reality of spiritual/mystical experiences? Are there transcendent or supernatural dimensions out there that will forever remain beyond the reach

Towards a Dialogue between Transpersonal and Natural Sciences

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of empirical science, logical systems such as mathematics, or indeed any artificial contrivance? We hope that this volume stimulates enough curiosity for the readers to ponder these questions. As a metaphor for psychobiological reality, the Mandelbrot set is a mapping of the complex number plane onto itself. Such mapping can never be complete: an eye can never get a complete view of itself, and a mind will never completely apprehend its own entirety. We weave the tapestry of knowledge out of a recursive interplay of intuitive and analytical threads, where subjective, objective, and intersubj ective aspects of reality are inseparably intertwined. The transpersonal dimension itself may arise as a manifestation of the fractal nature of wider reality, the self-similar ripples that link the personal and ontological dimensions via the transpersonal boundary of mystical knowing. Yet, any system of thought, including fractal epistemology, will necessarily carry its own inconsistencies that can only be resolved in a more encompassing system—ad infinitum.

References Wigner, ER

(1960). The unreasonable effectiveness of mathematics in

natural sciences. Communications on Pure and Applied Mathematics, 13, 1—14. Borges, J. (2000). The library of Babel (E. Desmazieres & A. Hurley, Trans). San Francisco, CA: GoodReads. (Original work published

1941) Reggio, G. (1982). Koyaanisqatsi [film]. Retrieved fi'orn httpszllwrwkoyaanisqatsi.org/films/koyaanisqatsiphp

INDEX

0A0

abstract’analytical mode of knowledge, 68—70, 85—89, 97 acausal connection, 52, 276, 291— 293 Action-Perception Cycle, 178, 195, 197—198 Adam and Eve, 443

aesthetics (of fractals), 193, 194, 350, 428 Adaptive Resonance Theory of

strange: 186, 281, 284, 459, 473, 477, 482 autism, 484 automaticity, 374—384 awareness conscious, 13—14, 22, 71, 1,53, 281, 380 implicit, 5 of awareness, 73, 127 0B.

Perception, 211 Agent-based modeling, 27

bi-directional processing pathways,

ahimsa (non-violence), 425

bifurcation diagram, 223—224, 257

algorithm

bilateral symmetry, 438

decay, 29 growth, 8 ALI (Analytical Local—Interactive) channel, 94—95 anabasis, 513 anima, 190 anxiety, 152, 160, 209, 286—288, 297, 354, 446, 474, 487 archetype, XX, 188, 219, 261—264, 268, 284, 398, 409, 417—418,

444

Aristotle, 334, 392, 394, 425 atnum, 418

attachment, 46, 106, 164, 275, 278, 279, 282, 285, 290 attentional control, 383 attractor, xviii, 197, 199, 203, 223,

255, 292, 294, 481 chaotic, 193, 198, 200, 202 dynamic, 160, 165 fractal, 260, 294

periodic, 27, 202 static, 202

148

binarylogic, 149, 158,167—169, 528 body dysmorphism, 209 body/brain, 80, 226, 228—229, 232 body/mind, 37, 226 Bohm, David, xxii, 76, 80, 92, 94, 127, 172, 243, 246, 292, 526 Bohr, Neils, 161, 165—166, 170 Borderline Personality Disorder, 2 4 bottom—up causation, XXV, 46, 48, 75, 76, 95, 114—115, 227, 457 physical regulatory processes, 145, 147 reductive logic, 106—108, 110— 111,114—115,121,123,

127,132,135,148 self—regulatory signals, 152

signaling, 151, 158 boundary chaotic, 2 4 conditions, 50, 151, 274, 303,

305, 313—314

A Fractal Epistemology for a Scientific Psychology crossing, xxii fractal, 22, 24, 28, 42, 52, 56,

161, 173, 274, 292, 295, 316, 324, 332, 348, 357— 358, 456, 525—526

fuzzy, 14—15, 23, 29, 55, 89, 92‘, 111,161,177, 328, 455—

456, 462—463, 478 interpenetrating, 22—24, 51, 161, 164, 170

natural, 50, 303, 336—337, 344— 346, 354, 356—357

subjective, 22. zone, 22, 24, 29, 258 Bradley, William, 447 brain architecture, 471, 473—474, 485 brain/mind dichotomy, 69, 73—76, 80, 83, 85, 94—98, 124—127, 528 Brownian motion, 235, 222, 246—

247 Bnmelleschi, Fillippo, 36 Bucke, Richard Maurice, 307 Buddha, 209, 350, 408, 421 Buddhism, Buddhist psychology,162, 210, 229, 232 Tibetan, 426

531

causal operator, 67—68, 88, 95—96. cell membranes, 332, 345—346, 367 cellular automata, 225, 504, 507, 51 3, 51 6 Central Limit Theorem, 221, 236

Cerebellar Cognitive Affective Syndrome, 386 cerebellum, xxvi, 47, 152—153, 372— 386 cerebral cortex, 14, 372—386 chance, randomness, 8, 25, 29, 49, 232, 291, 293, 296 chaos game, 293—294 theory, 41, 109, 150, 397, 444, 457—458

Chopra, Deepak, 292 Chuang Tzu, 407—408, 438 clairvoyance, 71, 92, 95—96, 115—

116 Clarke, Arthur C., 448—449 classical physics, 90, 170, 172, 175,

246—247, 457 clinical intuition, 52, 74, 86, 222, 274 cogito ergo sum, 70, 106, 154 cognition, 88, 156—157, 195, 211,

227, 308, 374, 379, 383, 396, 0C.

calculus, xx, 9—11, 268

Campbell, Joseph: 433 Cantor dust, 9—1 1, 248

set, 41, 222—223, 248—250, 252— 253, 256, 260, 271, 393, 396—397, 452, 465 Carhart—Harn's, Robin, 89, 471, 477, 479, 484—486 Cartesian dualism, epistemology, 35, 40, 45, 78—79, 105, 394, 460,

526 plane, 495

split, 106, 289 catabasis, 513

462 cognitive-emotional states, processing, 374, 380—381 collective unconscious, xvi, 176, 188, 275, 284, 307, 443

Complex Adaptive Systems (CAS), 69, 85, 90, 94, 114, 120—123, 145, 148, 165

complex numbers, complex number plane, 15, 224, 255, 257, 259, 529 complexity sciences, 27, 145, 470 compositeness, 427, 428 computer calculating power, iterations, 16, 27, 274, 313, 397—398 generated images, 215, 331, 438 modeling, 27—28, 456, 469

532

Index

used as a microscope, 16 Confucius, 162, 263, 398, 407 conjugate variables, 170—171 consciousness altered, alternate states of, nonordinary states of, 2, 53—55, 71, 85, 105,129, 135, 308,

421, 443, 465, 476, 480, 486, 526 cosmic, 307, 309, 418 hard problem of, 13, 59, 70, 78, 94, 153, 169 nondual awareness, 14, 29, 43, 71,177, 304, 305, 317—319 ordinary states of, 68, 111, 135 consilience, 66, 113, 120, 136, 309, 448, 525 contemplation, 432

Descartes, Rene, 70, 106—108, 123,

173 dimension, dimensionality box, 235, 250, 253 Euclidean, 5, 15, 17, 29, 37 fractal, 4, 17, 19, 26, 65, 72, 150,193, 194, 195, 200, 203, 218, 245, 250—252, 260, 280, 316, 367, 368, 436, 474, 495, 496 Hausdorff, 235, 250, 252, 256,

260 Lyapunov, 250

similarity, 253—254 dissolution (of sense of self, ego dissolution), 51, 56, 57, 177,

305, 312, 317, 319, 473, 477— 480, 484, 485

contextual probability theory, 21 1— 213.

distant healing, 71, 95—96

coordination dynamics, 23—24, 74 countertransference, 87 Cozolino, Louis, 289 Cox, Harvey, 421 Creativity, xvi, 47, 71, 95, 202, 244,

double-bind, 163 dreams, xxvi, 22, 52, 82, 88, 271,

311, 343, 377—379, 384 Csikszentimihalyi, Mihaly, 152, 167, 376, 379, 380, 384, 392, 406 cultures, collectivist, 289 cybernetic feedback, 149, 178 loop, 158

divination, 71, 116, 264, 398, 412

274, 275—287, 295—299, 422, 425, 430, 444, 457 dualism, xxii, 35, 44, 46, 51, 78—79, 105, 123 dynamical system, 26, 47, 134, 193,

256, 260, 479, 507 dynamics dynamical systems, xviii, xxiii,

xxv, 6,14, 38, 225, 254— 261, 328, 449, 528

cyrnatics, 405

attachment, 275, 278—279 coordination, 23, 24, 74

0D.

fractal, 39, 49, 50, 53, 72, 111, 309, 372, 528 non—linear, 27, 50, 74, 90, 145,

150, 311, 314, 316, 319

da Vinci, Leonardo, xvi, 337—341 Dalai Lama, 162, 208 decoherence, 83, 173—174 deep ecology, 98 default mode network, 197, 229,

inverse models, 383—385 interpersonal, 28, 67, 456 informational, 35, 73, 81, 526 neural, 47, 85, 157

484 dentate nucleus, 375, 385 depression, 26, 209, 471, 487

0E.

earthquakes, 25, 26, 310, 318, 362

A Fractal Epistemology for a Scientific Psychology 418, 440, 442, 453, 456, 525—529

edge-of—chaos, 47, 50, 148, 149,

151,152,161, 227, 233, 311, 314 EEG (electroencephalography), 40, 54, 71, 78, 96, 196, 198, 199

meta-reductive, 111 equilibrium, 152, 219, 305, 319 Escher, Mauritz Cornelius, xvi, 424,

528

ELI (Experiential Local—Interactive)

channel, 85—86, 88, 94—95

Euclidean geometry, numbers, 36,

216, 218, 220, 246, 452, 456

emergence, xviii, 44, 72, 76, 109,

111,114,116,123,133,134, 136, 150,157, 319, 402

533

evidence-based, 209 evolution, evolutionary theory, vii,

ego

xii, xiii, xx, xxii, xxvii, 60, 64— defenses, 167 sense of self dissolution, 51,

67, 72, 73, 76—77, 97, 99, 104, 105, 107, 109—123, 127—141, 143, 145, 146, 148, 153—157, 159, 169, 178, 180, 183, 184, 188, 200, 203, 204, 276, 306,

177, 305, 313, 317, 319, 473, 477—479, 485 Emerson, Ralph Waldo, 436—437 Emo—Etho—Eco—Evo—Devo model,

314, 320, 322, 345, 346, 372, 374, 378, 383, 385, 387—389,

178 emotion, emotional biology, 145, 147, 148, 160,

162, 167 experience, processing, xxv, 67, 144, 147,149, 152, 158, 163—169, 227, 270, 287, 291, 295, 304, 406, 418, 460, 465, 497—503, 508, 516 qualia, 146, 152, 154—155, 160— 161 science, 156, 162 empathy, 288, 455, entanglement, 73, 77—79, 81, 83,

92—93, 94, 112, 115,123 macroscopic, 77—78

yin/yang, 36, 187 epigenetic, epigenetics, 38, 153,

166, 294 phenotype, 156

403, 450, 454, 466, 524—526 evolving symmetry, 69, 528 executive network, 230

experience-ne at, 330 experiential/intuitive mode of knowledge, 70, 83, 85—86, 88, 95 experimenter effect, 97 explicate domain, 75—77, 81, 85, 92—94, 127 extraordinary knowing, 82, 85$6, 88—89, 96 Extra-Sensory Perception (ESP), 34, 71, 89, 96 Euclid, Euclidean geometry, xv, 5, 7, 15,17, 19, 29, 36—38, 216, 218—220, 244, 246, 253, 254, 257, 313, 332, 365, 392, 394, 446, 452, 456, 470

regulatory processes, 147, 158,

175

OF.

epistemology, 187, 303, 307, 403,

440, 448, 524

faith, 86,162,181, 447, 451, 528

and ontology, 36, 118, 187 fractal, xix, xxvi, 2, 35, 39, 46—

far-from-equilibrium, xxvi, 50, 85,

59, 82, 85, 97, 104, 117, 119—120, 128, 136, 275, 288, 364, 365, 372, 385,

304—305, 309—310, 317—318 feedback dynamics, 148 feminist, 37, 44, 349

534

Index

dynamic, 39, 49, 50, 53, 72,

Ferrer, Jorge, 3, 30, 38, 42, 45, 48, 60, 105, 138 Fibonacci numbers, sequence, 7, 438, 500 Fickian diffiision, 221, 236

111, 309, 372, 528

enfolding, 5, 29, 119, 149, 294, 316, 319, 428 epistemology, 29, 45—48, 97,

flow, 31, 42, 44, 71, 73—74, 81, 92,

104,119-121, 124, 274—

94, 95, 97, 151—153, 162, 165, 170

275, 288—289 geometry, xv, xxiii, 46, 10—17,

172, 178, 180, 188, 263, 294, 299, 301, 346, 380, 407,

28—29, 69, 117—118, 151,

314, 316, 322, 332—335, 363, 374, 376—377, 379, 384—386, 392—393, 405— 413—414, 421, 424, 438,

171,178, 215—218, 245— 247, 250, 260, 312—313, 357, 393—394, 452, 459, 462, 474, 496 mathematics, xviii, 69, 109, 362 natural, 261 pattern, 119, 281, 296, 385 physiology, 276, 385 process, 76, 378, 383, 498, 513, 523 statistics, 363—364

522 fluctuations, 25, 26, 219, 231, 235, 282, 305, 363, 382 fMRI, 40, 78, 96, 99, 170, 201, 291 form constants, 452, 454, 455, 463, 467—470, 472—475, 485 formal system thinking, 133 Fowler's model of spiritual

structure, 41, 50, 135, 150—151,

development, 167 Fracta1(s), xv—xvii, xxii, 4—8, 14—17, 21—22, 25—27, 29, 39—42, 69, 118, 128, 149—150, 215—222,

158, 221—223, 234, 332,

346, 368, 382, 435 Freud, Sigmund, 44, 82, 88, 100, 191, 275, 351, 359, 444, 453,

457, 459—461, 466—467, 475-

226—227, 231-234, 246—250, 254—263, 293, 297—298, 311— 316, 330—332, 348, 362—368, 425—432, 437—448, 455—457, 494, 513 African, 434 architecture, architectural, 417 Gaudi’s architecture, 441 Hindu temple, 433

boundaries, 22, 28—29, 56, 118, 173, 274, 330—331, 348, 456, 525—526 complex cycles of existence, 130—131 complex natural histories, 129—

131, 135 consciousness, 274—275 dimensionality, 17, 19, 193— 195, 200, 250—252, 367— 368, 436, 495—496 distribution 26, 362

476, 478—479, 484, 489, 491

free wi]l, 47, 48, 66, 70, 84, 94, 175, 179, 352, 414, 526 0G.

galaxy, super—clusters, 4, 446 gamma oscillations,14, 199 Gell-Mann, Murray, 111, 120, 139, 314, 320 gender, 44—45, 51, 60, 62—63, 336,

348—350, 354, 359 gender binary: 45,51, 62, 336, 348,

349. General Systems Theory, 491 genes, genetic code, 8, 74, 154, 294,

366—367, 371 geometry, 213, 233, 330, 430, 448 see Euclid, Euclidean

A Fractal Epistemology for a Scientific Psychology fractal, xv, xviii, xix, xxiii, xxv, 3, 4, 5, 14—29, 38—39, 69,

117,145,148,150,151, 160,171,178, 215, 217, 218, 243, 245, 246—247,

250, 260, 270, 293, 312— 313, 357, 382, 393—394,

413, 418, 422, 427, 430, 439, 442, 463

see also fractal(s), geometry

535

Hegel, Georg Wilhelm Friedrich, 308 Heisenberg uncertainty, 170, 172 Hillman, James, 444

hippocampus, 282, 404 holistic operator, holism, 67—68, 87, 89, 95—96 holotropic breathwork, 42, 55, 459 homo sapiens, 116—117, 374 hyperscanning, 24, 52, 88, 290

of integrated information, 73

geopolitical boundaries, 336 Gleick, James, 23, 41, 61, 113, 139, 306, 316, 321, 396, 414, 428, 450 Godel, Kurt, xx, xxiii, xv, 107—

109,112,113,138,139,140— 142, 393, 414 golden mean, ratio, 500

.1.

I Ching, 246, 265, 268, 270—271, 396—398 imagination, 13, 15, 19, 29, 118, 245, 343, 463, 527, 528 implantable cardioverter

defibrillator (1CD), 364

good and evil dichotomy, 163—164

implicate domain, 75—78, 80—82, 85,

gradient interfaces, 305, 314 Grof, Stanislav, 12, 31, 42, 61, 116,

139, 179,181, 309, 321, 420,

92—94, 98 implicit processing, 56, 81—82, 87— 88, 92, 95, 97, 265, 288

444, 450, 451,453—455, 457— 462, 465, 467, 473—476, 480—

indigenous, 33—34, 71, 179, 289, 309, 395

485, 490

India’s Net, 419— 420

Grofs cartography, 455, 457,

infinity, infinite depth, xx, xxiii,

459, 461, 482, 485 group theory, 117, 134 01-10

xxiv, 9,10,15,16, 21, 22, 38, 51, 55, 91,118,169, 172, 224, 249, 251, 252, 255, 256, 258, 259, 274, 317, 368, 392, 427,

428,432, 452, 456, 465, 477, hallucinations, 454, 455—456, 462,— 464, 467—473, 479, 484—485, 488, 491, 492—493 happiness, 162, 166, 182, 184 harmonic series, 510—511

harmony, 162, 243, 264, 440, 441, 498

heart arrhythmias, 362 hedonic behavior, 146, 159, 164

qualia, 145, 146, 147, 154, 156, 158,159, 169, 170, 175, 178 tendencies, 191

485, 528 information, informational, 72, 76, 78, 80—82, 85, 88, 92—96, 146, 148—149, 157, 163, 166, 169— 171, 174—175, 178, 210, 219,

225, 250, 290, 307, 314, 316— 319, 381, 384, 394, 458, 528 active, xxii, 75, 80, 94, 98 channels, 67

enfolding, 81 flow, 73 functional, 74 insertion, 95—96 integrated, 73

536

Index

nonlocal, 90, 93 processing, 13, 40, 67—68, 73,

83, 87, 90, 95, 147, 151, 305, 373 quantum, 93, 115, 173

relationships, 73 system, 134 inner speech, 201 instincts, approach and avoidance, 159

integral grid (Wilber), 24 intentional stance, 68 interconnection, interconnectedness, interconnectivity, 196, 292, 305‘, 31 9 inter-dimensional, inter-

djrnensionality, xxii, xxiii, 525, 527 interobjectivity, 292

0J0

James, William, 33, 145, 188, 189, 190, 209, 303, 421, 463 Janus, 230, 443 Josephson, Brian, 83 Julia set, 41, 258 —259, 313, 469 Jung, Carl, xvi, xix, 33, 88, 176, 188, 190, 191, 275, 284, 291, 305, 306, 307, 313, 426, 432, 438, 443, 444, 448, 467 OK.

Kant, Immanuel, 308, 311 Kauffrnan, Stuart, 173, 313, 438 ketamine, 486 Klein bottle, xxi, xxii, xxiii, 306 Kluver, Heinrich, 455, 463—465,

467—470, 473, 485

interpenetrating, interpenetration,

xxi, 4, 14, 22, 23, 24, 27, 29, 51,

Koch curve, snowflake, xxii, 7, 8,

56,161,164,170,173,177,

10, 11, 12, 14, 22,250, 251,

365, 456, 460, 471, 486

315—316, 347, 367, 406, 407,

interpersonal, 24, 28, 46, 52, 75, 76, 88, 104, 106, 123, 124, 127,

128, 132,169, 276, 289, 290, 364, 365, 456 interpersonal neurobiology, 289 intersubjective space, 87 intersubjectivity, 210, 292 introspection, 191, 299 intuition, 24, 34, 52, 74, 86, 244, 274, 426, 429 inverse dynamics models, 376, 377, 380, 384—385 ion charmels, 363, 383

isomorphism, 306 iteration, 7, 10, 15, 42, 73, 136, 171, 178, 223, 237, 248, 249, 265,

271, 283, 297, 299, 304, 314, 347, 380, 381, 384, 392, 402,

413, 452

Kolmogorov additivity law, 211 Krippner, Stanley, 487 0L.

Lao Tzrr 428, 438 Law(s) of Large Numbers, 221, 236 of nature, 109 Laszlo, Ervin, 313.

Learning,146,156,161,166,191, 196, 197, 199, 227, 282, 319, 376, 377, 381, 382, 383, 385,

400, 402 Left-Hemisphere Interpreter (LHI), 67, 97

409, 457, 462, 494, 496, 504,

liar's paradox, 27—28 logarithm, logarithmic, 25, 134

505, 516, 527

logic Aristotelian, binary, 149, 158, 167,169, 172, 528 evaluative, 149, 153, 161, 163

A Fractal Epistemology for a Scientific Psychology evolutionary, 111, 120, 146, 148, 163, 165, 178

fiizzy, 28, 171 paradoxical, 28, 149 reductive, bottom—u , 104, 106,

107, 108—111,112, 114,115, 121,123,125—127,132,

135, 136, 152, 486 self-regulatory, 147, 149, 160, 161 Yes/no evaluative, 161—163

Yes/no evolutionary, 148 logical positivism, 188 LSD (lysergic acid diethylamide), 458, 459, 474—475, 486

537

unreasonable effectiveness of, 13, 82,171, 179, 365, 525, see also Wigner, Eugene

MDMA—assisted psychotherapy, 486—487 measurement, 211—212, 219, 394— 395 ofbrain activity, 196, 198, 201, 290 fractals and, xx, 4, 9, 14—15, 21— 22, 29, 170, 176, 362, 364, 413, 452, 494, 500 reaction time, 49, 195 quantum, 84, 9 4 medical model, 53, 336 memory, 149, 1,57, 158, 175, 176,

198, 200, 227, 232, 282, 284,

0M0

374—378, 380, 381, 383, 386, magical thinking, 12, 116 mandala, 424, 426, 431, 432, 466 Mandelbrot Benoit, xv, xvii, 4, 21—22, 36—

37, 69. 73, 117—118,160, 215—218, 235, 248, 250— 252, 311, 330—331, 348,

393—394, 428, 438-439, 452, 494, 522—523 equation, 174, 176 set, xxiv—xxv, 15—16, 41, 118, 216—216, 224, 244, 258— 263, 294, 316, 331—332, 352, 428, 438, 465, 527, 529 zoom, 41—42, 331

MAPS (Multi—disciplinary Association for Psychedelic

Studies), 486 Maslow, Abraham, 2, 33, 153, 165,

187, 309, 365, 379, 392 materialism, materialist, xix, 12, 35,

38, 53, 73, 526 mathematical, mathematics constructs, 83 monsters, 12, 41, 53, 215, 393— 394, 452 non-Euclidian, 25, 379, 382

404, 434, 467, 472 mescaline, 464—468 meta-reductive paradigm, 72, 98, 123, 124—126, 130, 135—137 meta, trans-binary, 43—45

metaphor, metaphorical music and, 495—496, 523 use of mathematics and fractals as, xix, xxv, xxvi, 3, 4, 33, 40, 59,114, 122,131,151, 153,156, 170, 202, 213— 214, 230, 233, 234, 245, 362, 365, 369, 395, 413,

427, 442, 456 limits of, 365, 369, 421—422, 529 metaphysics, 40, 105 Michelangelo, 439, 525 microtubule, 92—93 mindfulness, 209, 230

mind 5 E model of, 156—158 mind-matter interactions, 71, 96 mirror neuron, 87—88, 291 Modular Model of Mind-Matter

Manifestations (M5), 91 Moebius band, strip, xx—xxiii, 306,

527

538

Index

mother, mothering, 87, 268, 278, 283, 285, 287—289, 432, 439, 479 music, 89, 330, 332, 377, 382, 406, 417, 494—522

nondual awareness, consciousness, 14, 29, 43, 71, 177, 304, 305,

317—319 nonlinear systems, 46—47, 56—58, 74, 111, 115,134, 225, 271,

308, 444, 456

mystical experiences, 14, 29, 55, 83, 105, 307, 309, 319, 527 knowing, 72, 88, 94, 529 ON.

narcissism, 209 nature, natural objects, xv—xvii, 4—7,

12—13, 19, 25, 30, 35—37, 40— 44, 55, 72, 93,109,111,114— 121,128,132,135,137,145, 148—150,161—162, 191, 210, 212, 216, 221, 244, 260, 264, 268—269, 292, 294, 303, 310, 311, 314—316, 344, 366, 392— 395, 425, 427—430, 432, 435— 439, 452, 462, 465, 500, 507, 508, 524

nonlinearity, 151, 225, 305, 391,

427—428 nonlocal correlations, 78, 93 realm, 173 neurodynamics, 93, 97 nonlocal-participatorymode of knowledge, 76, 80—82, 86, 93, 97

NPC (Nonlocal—Participatory Channel) of knowing, 76,

80—82, 85—86, 95, 97 numbers complex, 224, 237, 255, 259,

529 imaginary, 171 irrational, 1 0 Law ofLarge, 221, 236 real, 220, 249, 251, 253, 257

naturalistic, naturalism, 14, 38,

42, 43, 65, 71, 72, 85, 86, 90, 97, 98, 105, 147, 526

symbolism of, 268 theory of, xix—xx

Near Death Experiences (N DE), 55,

whole, xxiv, 15, 113

71, 80, 467, 471, 477 Neolithic period, 116 neurophenomenology, 40, 72, 203 neuroscience, xxvii, 47, 54, 71, 89,

106,116,119,126,128,186, 379, 455, 457, 468, 471, 487 cognitive, 59, 186, 372—373 social, 290 neurotransmitter, 227 New Testament, 421 Newton, Isaac, Newtonian, 9, 22—

.0.

object constancy, 283 objective, objectivity, xix, xxv, 3,

13, 14, 24, 37, 51, 58—59, 73, 81, 85, 86, 89, 90, 93, 94, 96, 105,106,109,110,131,146, 186, 187, 188, 189, 190, 193, 194, 201—203, 210—213, 228— 229, 234, 269, 292, 398, 427, 439, 473, 524—529

24, 70, 79, 83, 98,111, 113, 120, 173, 243, 365, 393, 394, 460

observer dependence, 21, 29, 52,

Newton’s method of

ontology, 34, 36, 54, 56, 111, 118,

approximation, 22—24 nocebo responses, 148

149,170,176, 275, 297 120,121, 128,145,187,188, 189, 524 operationalisrn, 1 88—190

A Fractal Epistemology for a Scientific Psychology optimization, 152, 366, 374, 376, 378—382, 384 ordinary states of consciousness, 68, 111, 135 original sin, 154, 155, 163

539

of science, 106, 188 physics, 25, 40, 49, 83, 84, 90, 109,

113,122,128,131,161,170, 172,175, 211, 212, 214, 217, 246—247, 290, 292, 309, 317,

319, 364, 438 448 0P0

Piaget, Jean, 48, 167

Plato, 28, 263, 436—437 P-adic system, 220 panpsychism, 54, 176 paradigm shift, 35, 172 paradigm meta-reductive scientific

(MRP), 70—72, 79, 88, 98, 104, 108, 120—126, 130—137 reductive scientific, 70, 104,

113, 117

Platonic forms, 261 realm, 173, 439 solids, 5, 36 Pollack, Jackson, 40 polyvagal theory, 407 Popper, Karl, 111, 136 post-modernism, 188, 498

post-traumatic stress, 232

paranormal, 24, 148, 188 parapsychology, 71, 97, 116, 290 Parmenides, 243, 333 participant observer, 86, 149

power laws, 25, 49, 134, 282

participatory

prefrontal cortex, 152, 198, 229, 230 premonition, 96, 115, 116, 425

process, 76, 81, 156

pragmatism, 188 precognition, 34, 71, 92, 96, 115—

116

self—actualizing universe, 179 Pascal's triangle, 516

prisoner’s dilemma, 211 proself, prosocial behavior, 363

peak experiences, 2, 71, 104, 187,

proto—self awareness, 146 psi phenomena, xxv, 34, 55, 71—72, 78, 80, 82, 85, 94—97, 526 psychedelic experience, 14, 34, 47, 421, 459, 476—478, 479 research, 89, 480, 484—487

309, 365, 377, 379, 454 Peano, Giuseppe, 11, 12, 393, 419 Peano curve, 11 Peirce, Charles Sanders, 189, 202 Perception-Action Cycle, 195, 198 phenomenology, phenomenological/experiential approaches, 3, 39, 85, 153, 186,

195, 201, 234, 308, 351, 445, 495 philosophy, 13, 23, 35, 37, 106, 179,

202 philosophy analytic, 188

Chinese, 391, 396, 398, 401 Jain, 425 Perennial, 445 Platonic, 263 Vedanta, 418

of mind, 106—107

psychiatry, 71, 209, 219, 221, 463 psychoanalysts, xxiv, 34, 82, 284, 287, 327, 329, 335, 353, 357, 454, 457, 460

psychobiological reality, 46, 75, 82, 527 systems, 46, 75 psychokinesis, 71, 92, 96 psychology

analytic, 188, 191 Buddhist, 210, 229, 232 Humanistic, 209, 458 mainstream, 12, 34, 53, 55, 167,

209, 213, 234

540

Index

positive, 167

IR.

scientific, 58, 104, 210, 230 transpersonal, 2—5, 12, 24, 29— 30, 33—59, 97, 104—112, 116—119,128,136,145,

random walk, 222, 236, 362 randomness

148, 158,178, 186, 187, 208, 209, 210, 213, 215,

226, 228, 230, 234, 275, 288, 303, 350, 351, 362, 364—365, 391, 403, 408, 418, 427, 440—446, 449, 453—454, 456, 458, 486, 524 psychotherapy, 22, 52, 86, 202, 230—231, 275—297, 336, 446, 460, 486—487

mild, 221 slow 221, 233 Wild, 221, 228, 230, 233 recoherence, 173—174 recursion, 177, 393, 502, 505 reductive

epiphenomenalism of consciousness (REC), 70, 78, 107, 123—124, 526 method, 70, 106, 124 science, reductionism, 48, 104, 106,108—109, 112—119,

123,125, 132, 134, 329,

0Q.

441, 526 qualia hedonic, 145—147, 178, 154— 156, 159, 169—170, 175 emotional, 146, 152, 158—161 of experience, subjective, 14,

59, 73, 85, 199 quantum biology, 84, 174 entanglement, 51, 79, 93, 112,

115 epiphenomena, 107, 124 mechanics, 79, 113—115, 118,

149,170—172, 174, 212 neurobiology, 72, 78, 83, 9 2 nonlocality, 79, 81, 83, 92, 170,

304 phenomena, 79 physics, 84, 113, 128, 131 quantum-classical limit, 75—76,

84—85, 90, 94, 97, 527 realm, domain, 4, 79, 92, 127,

172, 174, 176

relational unconscious, 282, 284,

287 relativity theory, 113—114, 131 reliability, 190, 392 REM sleep: 281—282 repellors, 160, 165 res cogt'tans, 70, 73, 106, 173 res extensa, 70, 106, 173 res potential, 173 Richter scale, 25 right hemispheric network, 67 Rorschach, 19, 280, 398, 431, 475 roughness, 217, 219, 427, 452, 494, 496 as.

salience network, 230. sai—cii—ananda (bliss), 432 scale invariance, xix, 6, 14, 17, 47, 69, 97, 217, 462, 525

scaling relationship, 362—363, 367

systems, 92, 109, 127, 526

Schrodinger equation, 79, 174

theory, 444

Self (experiential) dual, 98, 165, 527

weirdness, 527 queer, 44, 349

quorum sensing, 164

self-expansiveness, 26, 34—36, 38, 55—58

A Fractal Epistemology for a Scientific Psychology Self Expansiveness Level Form

(SELF) 55

spiritual, spirituality, 12, 33, 43, 145, 162, 190, 417—418 emergencies, 480

self-preservation, 165, 166, 177 self-regulation, 147—148, 159,

162, 167—168 self-system, 147, 154, 478

sense of, 55, 89, 104, 154, 232, 305, 309, 313, 316, 319 spiritual, 418 theory of, 442 self-affinity, 218, 235 self-organization, self-organizing system(s), xviii, 23, 26, 50, 73,

74, 76, 77, 80, 114, 145, 148— 151,154,160—161,165—168, 175—178, 188, 189, 196, 201, 202, 441, 473, 524

541

epiphanies, 71 statistical, statistics, bell curve, 156 nonlinear, 363 power laws, 25, 49, 50, 134, 282 normative, Gaussian, 25—26, 29,

151, 219, 365 stock market, xvii, 25, 26, 49, 219—

221, 235, 254—255, 348 Stone Age, 117 strange attractor, see attractor,

strange subjective, subjectivity, 59, 71, 107,

156,160, 176, 191, 210, 234, 292

self—similarity, xix, 6, 14, 16, 25,

69, 82, 92, 120, 121, 134, 135, 170, 202, 218, 233, 234, 247, 249, 253, 304, 311, 313, 331, 332, 362, 367, 400, 418, 429, 432, 453, 455, 456, 458, 462, 473, 476, 482, 485, 494, 511, 516, 524, 526, 527, 528

consciousness, 146 experience, xix, 12—14, 29, 37,

59, 70, 73, 78, 86, 90, 105, 122,144, 146, 149, 153, 154,157, 175, 178, 186, 188, 209, 210, 213, 215, 217, 228—234, 285, 358, 364, 422, 485, 525

selfishness, 154, 164

semiotic triangle, 189

supernatural, 38, 43, 44, 46, 54, 56,

sensory receptors, 210 sensory-motor control, 145, 147,

support theory, 211

161, 164

57, 59, 72, 165, 358, 524 syadvada (somehow—ism), 425

sentience, xxv, 73, 80, 146, 147,

synchronicity, xix, 34, 52, 88, 274,

153—156, 168—176 sequence detection, 380—384

synchrony, xxvi, 52, 274, 276, 287—

sequence generator, 47, 379 Shamanic States of Consciousness

(SSC), 53—54, 58, 72, 90, 96, 115, 117, 289, 424

276, 291—299 291 biobehavioral, 288—290 synesthetic experiences, 89 systems, complex, 27, 49, 109, 114,

116, 148, 169, 219, 225, 234, 336, 458

Sierpinski carpet, pyramid, triangle,

17, 222—223, 248, 252—253, 293, 294 Sierpinski gasket, 439 Smith, Wilfred Cantvvell, 422

IT.

South Indian singer-saints, 417

Tao, Tao Te Ching, 45, 162, 171, 264—268, 438

space-time, 77—79, 85, 93—94, 118,

309, 379

telepathy, 71, 78, 92, 95—96 tensegrity, 307, 309

542

Index

top-down causation, 46—48, 75—76, 95, 114—115, 122—124, 127,

133, 151—153, 157—158, 175, 227, 457 topology, 57, 303, 305—306, 308,

unity consciousness, 309 universal needs, 169 universality, 167, 179, 497

unpredictability: 26, 49, 296 Unus Mundus, xix, 291

319, 460, 495, 498 topological dimension, 247,

367—368 trance, 58, 71, 72, 89, 95, 105 transcendent experiences, 303—307, 309, 313, 316—319 filnction, 305, 306—307, 313 transcranial stimulation, 96 tIansgender, 44, 349—350

transpersonal domain, xxvii, 57, 156, 455 experiences, xxvi, 12, 49, 50, 53, 54, 55, 59, 71, 104, 117,

123, 145, 150, 209, 228, 234, 384, 385, 392, 420, 454, 455, 460 phenomena, xxvi, 3, 5, 12, 25, 26, 29, 39, 40, 43, 46, 49, 52—54, 58—59, 88, 104—105,

108—112,115—118, 132, 136, 179, 228, 364, 427, 455, 458, 473, 474 psychology, see psychology, transpersonal trauma, 116, 277, 285, 286 traumatic injury, 230 trigram, 268—270, 409 triune structure of vertebrate brain, 169 turbulence, 37, 221, 310, 338, 399— 400

0V0

Varela, Francisco, 40, 80, 146, 156 visual cortex, 373, 455, 467—468, 470— 472, 479, 485 field, 6, 196, 479

Vygotsky, Lev, 201 CW.

Wallace, David Foster, 439 ways of knowing, 148, 172, 187—

188 Weierstrass function, 217—218, 235

Weltanshauung, 448 Welwood, John, 392, 410 Wigner, Eugene, 13, 82, 171, 179, 365—366, 525, 527 Wilber, Ken, 3, 24, 35 Wilson, Edward Osborne, 448 Wilson, Tarn, 447 wisdom of the heart, 162 0Y0

Yeats, William Butler, 369, 444 Yensen, Richard, 455, 459, 473, 476—477, 479, 486 Yin/Yang, 36, 51, 61,149,162,171, 177—178, 187, 265—268, 397,

443 0U. oz.

undecidable reductive proposition, 108—11 2 unified field of consciousness, 14, 38

Zen, 403, 407—408 Zeno effect, 84, 94 Zimney, Kory, 448