Beauty seems either to be just there in the natural world, waiting to be recognised – think of sunsets – or to be the work of artistic inspiration. But at the end of the last century an astonishingly beautiful mathematical discovery was made – the Mandelbrot set. Nothing quite like the fantastic patterns it generates had ever been seen before 1980. The mesmerising images are made up of intricate fractal shapes – that is, irregular shapes which remain similar though never identical as more detail is revealed. Natural phenomena as diverse as snowflakes, the leaves of ferns and the form of mountain ranges can be described as Mandelbrot fractals. But they can also be seen in human constructs such as the muqarnas in Islamic architecture or in the design of Persian rugs.[1]

The term ‘fractal’ was only coined in 1975 by the man after whom the Mandelbrot set is named, the Polish-French-American mathematician and polymath Benoit Mandelbrot. The story of its discovery, and of Benoit’s role in it, is related by Arthur C. Clarke in the 1995 Netflix documentary Fractals: The Colours of Infinity (see end of article). Clarke remarks during the documentary: ‘I’m often asked, well these pictures are all very pretty but what’s their practical value?’ And it is true that while the Mandelbrot set does have practical applications – in computer graphics, for example – its usefulness is relatively limited. It does not compare with, say, Fleming’s discovery of penicillin or the uses of the laws of electromagnetism developed by Faraday and Maxwell. Or within mathematics itself, the discovery and development of calculus and differential equations which affect each of us daily: almost everything that has been engineered will have been designed using these mathematical techniques.

That Clarke was ‘often asked … what’s their practical value’ is revealing. It shows that practical value is a widely accepted yardstick. This was not always the case. Those who ask the question unwittingly live by the vision of the founding father of modern Western science, the Englishman Francis Bacon. Bacon wrote in 1597: ‘The real test of knowledge is not whether it is true but whether it empowers us.’ This was a radical and novel idea four hundred years ago. Now utility almost overwhelms all other values. But there are other purposes knowledge can serve and I want to suggest that for the Mandelbrot set, there is a very interesting and important philosophical one.

A Philosophical Role

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A core role of philosophy is to help us find answers to fundamental questions about ourselves and the reality we find ourselves in. Existential and metaphysical questions are also the ones that all the great spiritual traditions address. There are, of course, differences between the way that spiritual seekers search for answers and the methods of philosophers. One of the most essential of these is that most contemporary philosophers accept that there is just one level of reality, which is the reality we know and experience through our senses. By contrast, all spiritual traditions acknowledge at least two levels: the sensory, material world and another deeper, hidden world of spirit and consciousness.

I did not say ‘all contemporary philosophers’ – just ‘most’. It is true that most modern philosophers, as well as virtually all of Bacon’s successor scientists, investigate reality on the understanding that what is real is simply what can be observed and measured. However, there is a significant number who do not. Important examples of such heterodox thinkers in our present day include those whose work we have covered in Beshara Magazine over the last five years: Philip Goff, Iain McGilchrist, Federico Faggin and Bernardo Kastrup. They recognise that there is a higher primordial level of pure consciousness out of which ordinary reality emerges.

I also wrote ‘most contemporary philosophers’ because before the modern age, there were many major philosophers who recognised distinct levels of reality. Plato was particularly famous and influential for arguing that the phenomenal world is merely a shadow of a deeper, truer reality. That reality, he wrote, is the world of forms or archetypes. So, for example, all horses we see are manifestations of the ideal horse form according to Plato. Likewise every beautiful sunset, beautiful rose or beautiful song reveals in its own way the essential idea of beauty. Iain McGilchrist echoes Plato in describing fundamental values such as beauty as ‘ontological primitives’.

A similar understanding is brought out in Islamic philosophy. The 13th century philosopher and mystic Ibn ‘Arabi also used the analogy of shadows, maintaining that what we see in the world is really a projection – as in a shadow theatre – of events in this other, vertical dimension. This hidden reality, as Eric Winkel explained in our conversation with him last year (click here) is the domain of the ‘Divine Names’, and thus the external world is understood to be an arena where the Divine Names play out.

Interesting as the ideas of Ibn ‘Arabi and Plato are, how easy is it to accept them? Is it really plausible that what we see – this complex, rich, ever-changing world – could actually be the product of relatively few, unchanging forms or names? I argue that the Mandelbrot set provides an answer, or at least a way of opening ourselves to the answer, as it vividly demonstrates how an endless variety can emerge from something that is in itself quite simple.