The Blessings of Chaos

Inside the Mind of Mathematician and Revolutionary Theorist Ralph Abraham

Sitting in his home office nestled in a redwood forest above the University of California, Santa Cruz, where he’s taught mathematics since 1968, chaos theorist Ralph Abraham expresses concern about our connection. It’s not that we’ve never met; he’s just not confident that Ooma is reliable out in the woods. “Nothing works better than a land line,” he says, though he’s dumped it in order to save money. “And I love Skype, but I have never-ending problems using it.” Surprising, since he’s a techie from way back, even writing The Web Empowerment Book, How to Get Started on the Worldwide Web in 1995 (with Frank Jas and the late Will Russell). He’s also addressed The Cybersphere as a Complex Dynamical System and has his own YouTube and Vimeo channels.

In fact, Abraham believes it was the computer revolution that allowed both chaos theory and the closely-related field of fractal geometry (pioneered by the late IBM research scientist Benoit Mandelbrot) to develop. While chaos theory suggests there is order in randomness and vice versa, fractal geometry governs the behavior of natural phenomena, from coastlines to newborn babies’ cries to brain waves. “The computer extends our intellect, which helps us create the future,” states Abraham, an avowed Mac man. “It offers a door to perceiving complex space-time realities.” And over the past 55 years, the mathematical pioneer has at times altered both his perception and his space-time reality in order to bring forth the ‘chaos revolution’ he thinks is “at least as big a deal as the wheel.”

Still, he admits that though physics now depends almost entirely on it, chaos theory is a leap, whether you’re a layperson or a mathematics professor. And though the discipline has a more formal name—dynamical systems theory (dynamics for short)—he feels ‘chaos theory’ is a good popular description. For those interested in a good primer, Abraham, who has been at the forefront of the field since 1958, says science journalist James Gleick did ‘a fairly good job’ with his book, Chaos. Abraham is himself a prolific author: he’s penned 12 books, with translations in Chinese, Japanese, Korean, Russian, Dutch, German, French, Portuguese, Spanish, and Czech. Among the works are math texts, including The Geometry of Behavior with Christopher Shaw, and philosophical books such as Chaos, Gaia, Eros, and Chaos, Creativity and Cosmic Consciousness (with cutting-edge scientist Rupert Sheldrake and the late Terence McKenna, known to some as “Mr. Mushroom”).

Which brings up the psychedelic question. At 77, Ralph Abraham still lectures and leads seminars worldwide and is unabashed about touting the benefits of psychedelic drugs, with which he experimented early in his career. “I feel I owe everything to psychedelics. They provided an opening. There were people who suggested I try meditation or other paths to enlightenment. I think that could have worked, but it would have taken much longer and the effect would have been different. I took the accelerated route.” In a field where talent peaks at a young age, there was ‘a communion’ between hippies and top mathematicians in the 1960’s—Abraham is sure of this because, as he told journalist Walter Kirn in a 1991 interview for GQ magazine, he was a purveyor of psychedelics to the mathematical community. “Today when we say the word drugs, it brings up bad drugs, those that are highly addictive and take away people’s health, like meth and crack cocaine and heroin. Psychedelic drugs are not bad; in my view they’ve had an extremely positive effect.” (Abraham is featured in the 2010 documentary, DMT: the Spirit Molecule, discussing his experiments with psilocybin, LSD, and DMT.) Still, back in the day, he kept his interest in psychedelics fairly quiet. “My friends all knew, but I didn’t speak in public about the benefits of these things.”

Benefits or not, he doesn’t do drugs anymore. Hasn’t for many years. “I stopped after five or six years because I became convinced I’d had enormous benefit and couldn’t expect more. If I was going to realize the benefit I’d gained in appreciating the Universe, going forward without drugs would give me a better chance to materialize what I’d already learned.”

Having earned his Ph.D. from the University of Michigan in 1960, Abraham has held positions at Berkeley, Columbia, and Princeton in addition to his longstanding appointment at UC Santa Cruz. He has also held visiting positions in Amsterdam, Paris, Warwick, Barcelona, Basel, and Florence. Turning to the subject of history, in which he declares an amateur interest, the math professor notes the relationship between language, music, and dance in the development in consciousness in early hominids. In 1975 he founded the Visual Mathematics Project at UC Santa Cruz, which became the Visual Math Institute ( which has been extremely popular since its inception in 1994), and in 1985 Abraham helped create the MIMI (Mathematically Illuminated Musical Instrument). Since 1992 he has staged performances in which mathematics, visual arts and music are combined into a single presentation. After a ten-year lull (Kepler’s “Music of the Spheres” was performed in 2003 at the San Francisco Art Institute), a project he’s been working on for the past few years “is approaching culmination,” and involves the presentation of computer graphic images derived from chaos theory fractals. Though very attached to MIMI, Abraham admits an affinity for his favorite conventional instrument, the pipe organ.

Raised by Jewish parents, Ralph Herman Abraham (the middle name is after his uncle) also enjoys cooking, skiing, and learning Japanese. He grew up ‘anti-religious’ and, though a trip to India ignited a belief in the Sacred, he’s remained anti-dogmatic. “Since birth I’ve been agnostic; I think it’s wonderful that life is an open book—we don’t know what’s coming on the next page.” Citing the obvious—there’s so much trouble in what people believe—Abraham goes on to say that religion and mathematics were born simultaneously in the late Paleolithic period, about 50,000 years ago. And he ties it in to the present-day belief in science as a religion. “From my earliest days in mathematics I’ve repeatedly encountered scientists who believe in a dogma; they have confidence in a mathematical model about which they know very little.”

It’s the predictability piece of mathematical modeling that irks him. “Most of the models improve your understanding, but they can’t predict outcomes,” he insists. “Magazines like Time, The Economist, etc., publish articles declaring that by using various computer modeling we can predict this or that… people who know about chaos theory know that you can’t predict outcomes.” Take climate change or the global economy, for instance. Abraham points out that both are dynamical systems with lots of actors conducting innumerable transactions and that; “Conservatives don’t want too many things changed. Progressives (whom Abraham describes as people who believe in science) on the other hand want to take the carbon dioxide out of the oceans. We actually don’t know what some of these drastic geo-engineering projects will do.”

‘The dripping faucet’ has become a popular way to describe chaos theory since, as Abraham points out, “lectures are usually given in a physics hall and they always seem to have sinks and faucets in the front.” When you crack the tap a little bit, the water drips out very regularly. If you crack the tap a bit more, the drips speed up, but are still regular. When you crack it a little more, they sound irregular, like rain dripping off a roof. “If you measure the time between drops and make a list of these numbers,” explains Abraham, “you have the paradigmatic example of a chaotic time series.” Using a method of observation now known as ‘chaoscopy,’ another leader in the field, Rob Shaw, carried out an experiment on the dripping faucet; results provided proof of the hidden order within the seeming randomness.

Chaos theory computer modeling has been extensively used in all aspects of medical and psychological science for nearly a century, and the conclusion is that some chaos is good for us. Physicians now know we shouldn’t have too regular a heartbeat, for example, and that our emotional palette should contain a healthy spectrum of nuanced ‘colors.’ But what about schizophrenia, inherited diseases, and altered DNA—or cancer—wouldn’t that be brain waves or cells running amuck, behaving chaotically? Well, yes, says Abraham, but there’s more to it. “They’re wandering off the path, changing the rules of behavior. A dynamical system has rules and the behavior of a system varies a lot, but the rules don’t. If suddenly the change is dramatic or catastrophic, it constitutes a bifurcation.” While bifurcations can be positive (the whole idea of a ‘chaos revolution’ is based on such a change), often they express negative tendencies. War, the decline of the gross national product and the spread of AIDS all express dissonance, which, according to Abraham, is a lack of mathematical understanding,

And in the world of R2Abraham (his blog handle), mathematical understanding can solve pretty much everything. He’s well ahead of the curve when it comes to the current emphasis on STEM (science, technology, engineering, and math) education. Decades ago he, along with cultural historian William Irwin Thompson, designed the curriculum for the Ross School in East Hampton, New York, where students work both locally and globally as active partners in problem solving, gaining experience as scientists, engineers, and mathematicians. “Ross School is an experiment in changing the educational system; it has been wildly successful in one school. The American educational system as a whole is conservative; its job may to be maintain stability.” In Abraham’s view, U.S. schools are not bad in general, but they are bad in mathematics. “They destroy people’s native mathematical capability,” he states emphatically. “An emotional allergy is developed. The mention of math can cause anxiety even among physicists with Nobel prizes.” This distresses him, since he feels that mathematical knowledge is part of our human heritage, something akin to learning to walk.

Thompson and Abraham posit five mathematical mentalities: aRithmetic, Geometric, Algebraic, Dynamical, and Xaotic, which they refer to as RGADX. Bolts from the Blue (2010) details their graphical, historical, and integral approach to math. “We advocate teaching geometry before algebra,” writes Thompson in the book’s introduction. “This conforms to

historical order, as G precedes A in RGADX. In the history of mathematics, Babylonian geometry evolved into Greek geometric algebra, an essential prerequisite for the Islamic development of rhetorical algebra. Breaking this sequence may be a major cause for math anxiety in our schools.”

While we often equate chaos with upheaval, many consider it the birthplace of order. “Creation came out of chaos, and the word in ancient times meant a gaping void between Heaven and Earth from which form emerged,” writes Abraham in Chaos, Creativity and Cosmic Consciousness. He says the word chaos first appeared in Greece around 800 BC. Orpheus’ Theogony depicted the creation of the gods: the three main deities were Chaos, Gaia, and Eros, understood as abstract principles. Chaos was associated with the Sky, Gaia with the Earth, and Eros with Spirit. Throughout history there have been ‘chaos societies’ based around a model of partnership rather than of patriarchy. Abraham goes so far as to suggest that mathematical models can provide strategies to escape from an arms race, since working with models increases our understanding of subtle connections. He hopes we may soon be able to map and manipulate social, cultural, economic, and geographical parameters that will help anticipate and mediate international conflict. “Nations are complex dynamical systems,” he states. “The G8 players are connected by communications, trade, migration, and emigration links. The global network might not be disturbed if a couple of nations change, but when you have a change in the strength of links in the network, mathematics can’t predict what will ensue. With sanctions on Iran, major intelligence in the links constantly changing, Syria could throw the entire region. People are afraid it’s going to throw it into chaos—the upheaval kind of chaos.”

But Ralph Abraham is a hopeful guy, and it’s his voice that resonates with optimism in the trialogues with Sheldrake and McKenna, interdisciplinary discussions of chaos, creativity, and cosmic consciousness held at the Esalen retreat in northern California over a period of years (resulting in the publication of Chaos, Creativity and Cosmic Consciousness). The idea of trialogues (rather than dialogues) introduces ‘the third thing,’ a chaos concept that intercepts the polarity created between just two points of view. The talks for a while focused on the Apocalypse of 2012. While McKenna was obsessed with that, “Rupert and I were skeptical,” says Abraham. “In early January of 2013 we would have had a great discussion. We had convinced him a major cultural transformation didn’t have to involve disaster.”

Though Abraham thinks civilization has “jumped the track” and gone into a negative, non-chaotic trajectory (“new manifestations of evil have sprung out, particularly the role of the evil drugs and their use by government agencies to control populations. The global terrorism network is another example”), he still thinks it’s possible to raise the frequency, to lighten the dark, through prayer. “I’d like to see a resurgence of magic. We need to connect the star magic of the Stonehenges and astrology, for example, with the progress of daily life and political events.” Starting, perhaps, with a third political party in the U.S.?

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By Cynthia Logan