Have you ever wished, in your daily life, that you had a simple way to find all the answers about any subject that was vexing you? Perhaps you are in a personal crisis wondering whether God exists, or maybe have a mundane issue as simple has finding your way home when lost. Well, according to 13th-century monk Ramon Llull, you're in luck. Llull devised a unique philosophical system, based on combining a set of primitive concepts, that he believed would provide the path to solving any conceivable dilemma. His primary goal was to find a way to discuss religious issues and rationally convert heathens to Christianity, without relying on unprovable statements from the Bible or other holy books. As a philosophy, his system was far from definitive or complete, and gradually faded into obscurity. But along the way he became a major contributor to mathematics, making advances in areas as diverse as algebra, combinatorics, and computer science as he tried to elaborate upon his strange philosophical methods.
Llull began by listing a set of nine attributes in each of several categories of thought, intended to represent a complete description of that category, which could be agreed upon both by Christians and non-Christians. For example, his first list was the nine attributes of God: goodness, greatness, eternity, power, wisdom, will, virtue, truth, and glory. He wanted to discuss all combinations of these virtues, but repeating them endlessly was kind of tedious in the days before word processing, so he labeled each with a letter: B, C, D, E, F, G, H, I, K. He then drew a diagram in which he connected each letter to each of the others, forming kind of a nine-pointed star with fully connected vertices; you can see a picture at one of the links in the show notes. By examining a particular connection, you could spur a discussion of the relationship of two attributes of God: for example, by observing the connection between B and C, you could discuss how God's goodness is great, and how his greatness is good. Whatever you might think of his religious views, this was actually a major advance in algebra: while the basics of algebra had existed by then, variables were commonly represented by short words rather than letters, and had been thought of as simply representing an unknown to be solved for in a single equation. For the first time, Llull was using letters to represent something more complex than numbers, and mixing and matching them in arbitrary expressions. In addition, his diagram of the relations between attributes was what we now call a graph, an important basic data structure in computer science. He also created another depiction of the possible combinations as a square half-matrix, another data structure that is common today but was unknown in Llull's time.
Llull's system got even more complicated when he introduced additional sets of attributes, and tried to find more combinations. For example, another set of his concepts consisted of relationships: difference, concordance, contrariety, beginning, middle, end, majority, equality, minority. He also had a list of subjects: God, angel, heaven, man, imaginative, sensitive, vegetative, elementative, instrumentative. Even deeper philosophical conversations could theoretically result from combining elements from several lists. This created some challenges, however. He would again label each element of these lists with letters, but keeping track of all combinations led to an explosion of possibilities: just the three lists we have so far make 9x9x9, or 729 combinations, and he had a total of 6 major lists. So to facilitate discussion of arbitrary combinations, he created a set of three nested wheels, each divided into 9 sectors, one for each letter. One would be drawn on a sheet of paper, and the other two would be progressively smaller and drawn on separate sheets that could be placed over the first one and independently rotated. Thus, he had developed a kind of primitive machine for elaborating the combinations of multiple sets: for each 9 turns of one wheel, you would turn the next larger wheel once, and by the time you returned to your starting point, you would have explored all the combinations possible on the three wheels. Several centuries later, the great mathematician Gottfried Leibniz cited Llull as a major influence when inventing the first mechanical calculating machines.
There were also several other contributions resulting from this work, which you can read about in more detail at the links in the show notes: Llull can be thought of as the first person to discuss ternary relations, or functions of more than one variable; and he anticipated some of Condorcet's contributions to election theory, which we discussed back in podcast 183. Llull, of course, was not really concerned with making contributions to mathematics, as he was concentrating on developing a comprehensive philosophical system. In his own mind, at least, he believed that he had succeeded: he claimed that "everything that exists is implied, and there is nothing that exists outside it". To help prove this point, he wrote a long treatise elaborating upon physical, conceptual, geometrical, cosmological, and social applications of his ideas. Apparently he even spent five pages showing how his system could aid the captain of a ship that was lost at sea. Personally, I would prefer to have a GPS. But even if our modern thought processes don't strictly follow Llull's guidelines, we still owe him a debt of gratitude for his contributions to mathematics along the way.
And this has been your math mutation for today.