Ramon Llull: Ars magna
In 1274, at the end of a long hermitage on Puig de Randa, a mountain on the island of Majorca, Ramon Llull conceives – by divine revelation, he claims – the idea for a great work that would become the core of his life and its main objective: the creation of a complex system that he calls his Ars magna: his ‘great art’. Llull’s ‘great art’ is a strange and complex system oscillating between metaphysics and logic, expressed in the form of tables, graphics, moving paper circles that may be rotated and superimposed to generate arbitrary combinations of fundamental, elementary concepts. With this system, Raymond Llull – or Lully, as he is called in the English-speaking world – intended to bring order to the world, and to convert Jews and Muslims to Christianity.
These objectives, it would be fair to say, he decidedly failed to reach. But the influence of his peculiar system has been huge. Both Giordano Bruno and Montaigne, two of the thinkers who have had most influence upon modernity, drew inspiration from Llull. But it was above all Leibniz who sought to sift the wheat of Llull’s ‘great art’ from its chaff, to cleanse it of its medieval aspects and attempt to extract a universal rational language, dubbing it ‘combinatorial art’, with the objective of translating rationality into calculation.
A direct application of this idea is the first calculating machine, designed by Leibniz and acknowledged as the progenitor of all modern computers. But the same idea is also at the root of modern developments in logic conceived as a universal grammar of rationality, from Frege to logical positivism. The art of Llull is the deep root of much modern thought and technology, and has made this great Catalan intellectual one of the most original and influential voices to come out of medieval Europe. One technical tool central to the physics that I work on, to take just one small example of that influence, is the graph: an image that codifies the way in which a certain number of elements are connected with each other. Graphs were invented by Llull.
At the root of the curious power of combinatorial art there is a simple fact. The greatest Persian poet, Ferdowsi, conveys it well in a famous legend in his epic The Book of Kings. The ingenious inventor of the game of chess, a man called Sissa ibn Dahir, gifted the game to a great Indian king. Overwhelmed by admiration and gratitude, the king asks the wise inventor how he could reward him, and he answers in the following way: ‘Give me a grain of wheat for the first square on the chessboard, two for the second, four for the third, and so on, until we have doubled the amount, using every square on the board.’ The king is amazed by such a modest demand and immediately orders that it be granted. So imagine his astonishment when his attendants return, and tell him that there is not enough grain in all the granaries in the kingdom to satisfy the wise man’s cunning demand!
The calculation is easily done: just for the last square on the board, the sixty-fourth, it is necessary to have a number of grains equal to two multiplied by itself sixty-four times, and this amounts to 18 billion billion grains. If a single grain weighs a gram, then this would be 10,000 billion tons of grain. And this is only for the last square! Dante, in canto XXVII of his Paradiso, uses precisely this legend to indicate an inordinate number: ‘And they were so many that their number was more than the doubling on the chess squares.’
What significance does it have – the fact that a number so small can give birth to one so large? It means something simple: the number of combinations is generally much larger than we instinctively imagine. By combining a few simple objects you can obtain an unexpected vastness of things, and these can be arbitrarily varied and complicated. It is not just the number of combinations that is astonishing: it is also their variety. Think of the nature that surrounds us. Physics has helped us to understand that all that we see is generated by scarcely more than four particles which interact through a few elementary forces. The few different pieces of this simple Lego set produce forests and mountains, star-studded skies and the eyes of girls.
But the extent of what could exist is even greater than the immense extent of what does exist. Think of the proteins that form the structure of all living things. A protein is more or less a sequence of a few dozen amino acids. There are twenty amino acids. Say we want to produce all the possible proteins, in order to study them: this will allow us to understand all the possible structures of living matter, and even to anticipate the evolution of life on Earth …
But there is a problem: the combinatorial calculation is easily done: the possible combinations of about twenty amino acids forming chains of a few dozen elements are so numerous that even if we managed to produce a different protein every second the entire life of the universe would not be sufficient to produce more than a small fraction of all the possible proteins … In other words, the space of the possible structures of life is still almost completely unexplored, not only by us, but by nature itself.
The first intuition of the immensity of space opened up by complexity had already been had by Democritus, twenty-four centuries ago. Democritus had understood that the whole of nature could be made up of only atoms; and that, as he puts it, it is the combinations of atoms that generates the complexity of nature, ‘just as the combinations of the letters of the alphabet can generate comedies or tragedies, epic poems and satirical plays’.
Our intuition baulks at the immense numbers and the endless variety generated by combinations. Like the king in the Persian story, it seems impossible to us that from the combination of simple things, so many other things and such complexity can be born. It is for this reason, I believe, that it seems inconceivable that things as complex as life, or our own thought, can emerge from simple things: because we instinctively underestimate simple things. We never think they are capable of much. Numbers generated by grains of wheat and a chessboard surely cannot empty all the granaries in the kingdom. And yet this is so.
Our brains contain approximately 100 billion neurons; each one of these is tied to other neurons by conjunctions, the synapses. Each neuron has several thousand synapses. Hence each of us has in our head hundreds of thousands of billions of synapses. But it is not this number that determines the potential space for our thoughts. The space occupied by our thinking is (at the very least) the space of the possible combinations in which each synapse is active or not. And this number is two multiplied by itself not sixty-four times, as in the fable of the wise Persian, but hundreds of thousands of billions of times.
The resulting number is stratospheric; to write it down you would need thousands of billions of digits, ‘so many, their number, more than the doubling on the squares’. Not even the most far-reaching cosmology deals with numbers this large. This number quantifies the immensity of thinkable space, of which we have yet to explore more than an infinitesimal corner. This is the boundless space opened up by combinations, by the art of combination, the Ars magna, the great art of Ramon Llull.