The book Mathematicians and their Gods is a collection of essays covering the relationship between mathematics and religion, from Plato to the modern day. In this blog, one of the editors of the book, Snezana Lawrence, writes about the motivation for this book and gives some examples as to when science and faith have come together…
In recent times – following political changes, the long-awaited results of the Chilcot enquiry into the Iraq war and, in the morning when I write this article, the atrocity in Nice – it is easy to slip into the mode of regarding beliefs as a dangerous invention. Stick to facts, make rational decisions, that’s what mathematics is for. Right or wrong? Well, there the problem lies. We don’t have all the facts, and so to do ‘the rational’ in such a situation is to accept the fact we don’t have all the facts. But how do we then know? Based on the facts we don’t have or only on those we do have? The medieval philosopher Gersonides, for example, tried to reconcile the two approaches: to show that reason and revelation are coextensive (Feldman, 1967). He also suggested that, in fact, you can’t make a rational decision unless you have full knowledge of all the facts. So it seems one needs to have systems through which to bridge this cognitive gap, this discomfort, and in mathematics, there are occasions when ‘alternative’ methods are employed to close this gap.
The perennial question as to what we can tell with the instrument of our mind and by using the methods, structures, and processes that mathematics gives us, is something with which the human race has struggled throughout its history. The instances of solutions to it are multiple, and they sometimes may seem to us to be more rational and sometimes less so. But one unifying characteristic they have is that, while we can study something rationally, we have to admit that there are things we don’t know (or even know we don’t know), and continue searching for the answers to those questions.
So how are theology and mathematics related and how have we presented this in our book? Well, both deal with unknowns and our responses to them. They show, with some historical examples, how mathematicians reconcile the quests for truth in different ways – the how something works and our understanding of why may not always give a coherent answer. It is in the interplay between these questions around new mathematical discoveries that beliefs sometimes have a place. In order to make our point, we looked at the different manifestations and occurrences of such developments.
Let me start from a very straightforward situation that many of you would have come across during your school days. It is the question about the purpose of mathematics. I start here from the everyday occurrence – we are bombarded daily through the media, in education, and from various initiatives – to improve the state of our competencies in mathematics. We speak about how mathematics is important for the economy and social and scientific progress in general and hence should be one of the most important aspects of modern education. The classroom consequence, despite our best intentions in such context, easily becomes formed into a question: ‘when will we need this?’. When I worked as a teacher, I tired of this question to such an extent that I bought myself a crystal ball to keep on my desk in the classroom, just in case someone would dare pose it. If there was a daring soul who would do so, I would put my hands on the ball, look at the ceiling, and very studiously wait for everyone to be quiet for a few moments, to exclaim that there will be such a moment, far enough in the future so that its occurrence could not be tested under my watch…
How does this connect with Mathematicians and their Gods? Well, this is actually a serious question – the ‘when will we need this?’. Let me try to clarify and simplify a very complex aspect of mathematics. Learning and doing mathematics gives a student tools to make up their own mind about what they know, how they know it, and how they can check the validity of such knowledge. How to investigate, check, challenge, or invent their own tools to close the gap between the known and the unknown, without a recourse to anything but a tradition in thinking and their own mind. No wonder then that many a mathematician has found in such a process an experience that transcends the ordinary when they finally find a solution to their mathematical questions – and some have linked that to a religious inclination, justification, or experience.
We didn’t try to answer how this is actually done. Instead, we gave some examples: one is Kepler, who grappled with questions about the geometry of the cosmos, made some mistakes in his search, but persevered. His strong belief that the cosmos is divine didn’t clash in any way with his mathematical description of his laws of planetary motion.
One of Kepler’s most surprising inventions came about when he taught mathematics at a school in Graz early on in his career. He pondered: what if the five Platonic solids are indeed some kind of blueprint upon which the universe is made, as Plato suggested centuries earlier? Plato discussed these solids in Timaeus ( c. 360BC) and associated them with four classical elements: earth was represented by a cube, air by an octahedron, water by an icosahedron, and fire by a tetrahedron. The fifth element (yes, just like the Fifth Element, the film by Besson, made in 1997), the dodecahedron, Plate believed must have been used for arranging the constellations of the heavens. This was the fifth element, the divine spark, the force that made all other elements come to life.
Fast forward to 1595 when Kepler worked on Platonic solids and used them to make a model of the universe, in his now famous book Mysterium cosmographicum (1596), illustrating the work with one of the most famous images of the history of science and mathematics. The image shows each Platonic solid encased in a sphere, inscribed in a further solid, encased in a sphere, which Kepler identified with the then six known planets: Mercury, Venus, Earth, Mars, Jupiter and Saturn. Kepler described that the spheres containing the solids are placed at intervals corresponding to the sizes of each planet’s paths (as they were then known), assuming that they circled around the Sun. Of course, later, while he was living in Prague, Kepler found that the orbital paths of planets of the Solar System were not circular but elliptic, but this beautiful model, even though not perfectly accurate, gave him the impetus for further research.
There are other stories that we shared in the book – one being the story of geometry as developed by and for Freemasons and the influence this had on the establishment of a modern concept of ‘sacred geometry’. This concept, in its modern form, stems from investigations of the role of geometry in the conception and structuring of the physical environment on the one hand, and in the metaphorical interpretation of geometrical methods and images as possible lessons these may have for conduct in one’s life. The parallels that some drew between the involvement in social and mathematical work, justifications they made in the descriptions that connected geometry and social engineering may seem to us, frankly, strange. But taken at the time in which they arose, the story can show us that what is now considered rational may have evolved through some irrational or strange practices. The same, of course, goes for Newton, also a prominent figure in our book, whose writings on alchemy far exceeded in volume his writings on mathematics.
So the book that I edited with my colleague Mark McCartney is about these types of questions and the way in which mathematicians have, since antiquity, found some personal answers to them. We wanted to give a broad overview but stuck with the European context mainly because we came from that background and found ourselves surrounded by people interested in this area. Of course, it would be good to look at the interactions between mathematical ideas and mathematicians from other traditions and geographical areas. We are thinking that maybe we will one day return to those questions and see what parallels and differences we may find.
Feldman, Seymour, 1967, “Gersonides’ Proofs for the Creation of the Universe,” Proceedings of the American Academy for Jewish Research, 35: 113–37.
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