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What does the ‘M’ in Benoit M Mandelbrot stand for?

February 25, 2014

In my last post, I set out some ideas, which largely come from the tradition of depth and archetypal psychology (the only psychology worth paying attention to) that I find helpful in understanding the encounter with La Madre Ayahuasca.

The main figures in this last postJung, Corbin and Hillman -all place soul at the heart of their understanding of what it means to be human. They do, however, understand the idea and the reality of soul in a very different way from the mainstream Christian and Eastern religious traditions.

For all of them, soul is not just individual. Soul, or psyche, offers a connection to a wider world: in the case of Corbin, to the ‘mundus imaginalis’ or imaginal world; for Hillman to the ‘anima mundi’ or soul of the world; and for Jung to the collective unconscious. Of course, indigenous people have known and practiced this for millenia.

All these thinkers are important in helping us break out of the limitations of a narrow, skin-enclosed individuality based on the ego. As James Hillman writes in the first lines of his wonderful essay, “A Psyche the Size of the Earth’, which inaugurates one of the early, key texts on eco-psychology (as opposed to ego psychology):

“There is only one core issue for all of psychology. Where is the “me”? Where does the “me” begin? Where does the “me” stop? Where does the “other” begin?”

La Madre Ayahuasca shows us that there is no sharp dividing line between the ‘me’ and the ‘other’ – they are profoundly intertwined. Dennis McKenna has described how, through drinking ayahuasca, he directly experienced the whole process of photosynthesis in plants from the perspective of a water molecule. La Madre has taken many other people to the stars and other universes. She can also connect us to past or future lives – though, in her world, the whole perspective of time in terms of a linear sequence of past-present-future is revealed as a convenient illusion.

A fundamental issue at stake here is how do we understand the relationship between the part(s) and the whole – the individual self and the wider planetary community of human and non-human beings to which we belong. Between individual and cosmic consciousness?

Conventional mechanistic science with its reductionist approach believes that the whole can be explained by reducing it to its parts. Until quantum physics came along to subvert the underlying assumptions of this mechanistic worldview, this kind of science believed it had found the basic building blocks of matter in atoms and that the study of their behavior would be able explain the functioning of every level of reality. In a similar vein, evolutionary biologists consider that complex human behavior can be explained and reduced to the functioning of genes.

mandelbrot_set

Mandelbrot Set

More recently, however, in the new sciences of chaos and complexity we see a different understanding emerging of the relationship between the whole and the parts.

A part can be described as a ‘fractal’ – a term from chaos theory, first used by the mathematician Benoit M. Mandelbrot in 1975. Technically, this describes a mathematical set, based on a family of mathematical equations called complex quadratic polynomials, that can be visually generated by computer programs.

As you zoom further and further into the image of the Mandelbrot set, you find the same pattern shown above infinitely replicated at ever finer detail, though the context in which this pattern appears is constantly changing. (For a good illustration of this, with accompanying lyrics called ‘Mandelbrot Set’ by Jonathon Coulton click here.) The Mandelbrot set, like every living system, embodies both continuity and change.

This pattern of  infinite self-replication is called self-similarity at every scale. The whole is embedded in the parts and the parts are embedded in the whole.

Hence, one of the favorite jokes of mathematicians:

Q. What does the ‘M’ in Benoit M. Mandelbrot stand for?

A. Benoit M Mandelbrot

Jackson Pollock

Jackson Pollock’s ‘Convergence’ (1952)

Fractals are not limited to computer representations of mathematical equations as in the Mandelbrot Set but can be found in phenomenon as diverse as music, heart beat rhythm, urban growth, the functioning of stock markets, soil mechanics, African textiles, the work of the artists Jackson Pollock (nicknamed ‘Jack the Dripper’) and Max Ernst, and in many patterns in nature.

Greenland coastland

Greenland coastline showing fractal form. Courtesy of http://artsonearth.com/2008/08/11-phenomenal-images-of-earth.html

A good example of fractals in nature is a coastline in which the same pattern, say of ‘ruggedness’, is found when examined in detail at every level.

Mandelbrot believed that there was an underlying order to apparently ‘rough’ chaotic natural phenomenon like coastlines, mountains, and river systems as well as organic human structures like  lungs and blood vessels. He hoped that this underlying order could be mathematically represented.

This means that the exact length of a coastline is impossible to measure as it will depend on the scale of the measuring unit. Given ever increasing small er units of measurement, the coastline becomes longer and longer. Due to the fractal nature of these phenomenon, the correct answer to the question:

“What is the length of the coastline of England”, is “Infinity”.

This work also suggests the possibility of a different approach to science – coined by the theoretical biologist Brian Goodwin as “The Science of Qualities” – which looks at the qualities of phenomena (in the case of coastlines, ‘ruggedness’) as well as their quantities. Generally, empirically-oriented science has followed Galileo’s dictum that science can only concern itself with what can be measured.

In an interview, Brian Goodwin says:

“I believe that there is a whole scientific methodology that needs to be developed on the basis of what is called the intuitive way of knowing. It’s not something that’s vaguely subjective and artistic, it’s a definite way of knowing the world. In fact, it’s absolutely essential to creative science. All the great scientists, Einstein, Feynman, you name them, would say intuition is the way they arrived at their basic insights, their new ways of putting parts together into coherent wholes.

This intuitive way of knowing is the form of knowledge practiced by indigenous cultures. Rather than separate and distance themselves from the phenomenon, they empathically identify with it, experientially enter into it, and participate with it. This is also the methodology developed by the famous poet and playright, Goethe, over two centuries ago.

So what has all this to do with La Madre Ayahuasca?

Firstly, I think that she offers us the kind of knowledge and methodology described by Goethe. Shamans typically learn about other animals by becoming them.

Secondly, apart from fractal images bearing an uncanny resemblance to some of the images experienced when drinking ayahuasca – (its noteworthy that film makers use fractal generating software to portray the visual aspects of the ayahuasca experience) – the idea of fractal suggests that the whole cosmic and planetary field of consciousness is embedded, or to use a term from the theoretical physicist David Bohm, enfolded in our individual consciousness.

We are both part and whole. Both a distinct individual identity and profoundly interconnected with the cosmos. We are fractals.

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2 Comments
  1. Kev Borman permalink

    I’ve come across this notion of fractals in relation to coastlines and river systems quite often before. Tim Robinson, for example, in his wonderful books and hand-drawn maps of Connemara and other nearby parts of western Ireland, explores this idea.

  2. This is the most interesting thing I’ve read related to mathematics for a long time, perhaps ever.

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