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Mathematics and the Divine

T. Koetsier and L. Bergmans, editors
Publication Date: 
Number of Pages: 
[Reviewed by
Calvin Jongsma
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Mathematics and religion seem a peculiar mix to most people today. What relationship could possibly exist between numbers and God, between geometric figures and religious beliefs, between abstract structures and theological doctrines? Hasn't mathematics been weaned from the primeval ties it once had to religion and metaphysics?

As the book Mathematics and the Divine demonstrates, people have made many interesting and vital connections between mathematics and religion over the years. Believers of many faiths have found significant points of contact between their religious outlooks and mathematics. Not all of these claims were made in the distant past or by certified crackpots — although, admittedly, some pretty off-beat linkages have been proposed. Religion and mathematics seems to attract eccentrics: witness the popular Bible codes of a decade or so ago, which claimed that esoteric prophetic messages could be deciphered by reading equidistant letter sequences in some sacred text.

On the other hand, reasoned arguments have also been put forward relating mathematics and religion. At least two books reviewed within the last year in MAA Reviews (Mathematics in a Postmodern Age: A Christian Perspective and The Divine Challenge) seem to fall into this latter category, as does the book now under consideration. While this doesn't constitute a movement, it does begin to address an issue that has received almost no attention by scholars interested in mathematics. In contrast, religion has been considered a legitimate research focus in history of science for quite some time.

One might be excused for thinking this topic could not support an academic book-length treatment, but Mathematics and the Divine proves otherwise. The book extends to some 700 pages, containing a long introductory essay followed by 35 chapters/articles by almost as many authors. Unfortunately, its size and Elsevier's whopping $226 price-tag make owning it prohibitive and may even make it difficult to find a copy of the book. This is regrettable, because while Mathematics and the Divine has some forgettable chapters, it also has a number of insightful and well-written contributions.

The editors weave the topics of the various chapters together in their 40-page introduction, though they do not attempt to provide a definitive schema for thinking about the topic. Their overview is helpful, for the chapters form a loosely knit chronologically ordered collection of independent case studies, focused primarily on the ideas of individual thinkers. There is little overlap or dialogue among the offerings, and at times one wishes the topic of a chapter had been expanded to position it more within a broader cultural context or relate it to some trend in the development of mathematics. But given the novelty of the topic, case studies like those presented here are probably needed before more general conclusions can be drawn about the contours of the field.

The book restricts its attention mainly to Western culture, distinguishing three periods in the relationship between mathematics and the divine: the early pre-Greek period, the classical Greek period with its medieval and Renaissance heirs, and the modern era starting with the Scientific Revolution.

The pre-Greek period is discussed briefly in the introduction and in the first two chapters, which deal with China and India respectively. Except for a later chapter on the sacred geographies developed by medieval Islam, this is the only place that treats non-Western mathematics, and it concentrates on some fairly narrow topics (Chinese magic square number mysticism, and the compatibility of Indian astronomy with Hindu sacred texts).

The classical Greek period is represented by the Pythagoreans, Plato, Nicomachus, and Proclus. Chapters on medieval mathematics take up some mathematical traditions and applications (texts devoted to calculating calendar matters, ecclesiastical architecture, and mathematical topics stimulated by theological debates) as well as ideas put forward by some well-known thinkers (Ramon Lull and Nicolas of Cusa). A chapter on the algebraist Michael Stifel's Biblical numerology is followed by ones on the mystical ideas of two later Renaissance thinkers, the rechenmeister Johannes Faulhaber and the polymath Athanasius Kircher, who lived during the early years of the Scientific Revolution. John Napier's ideas in a similar vein are mentioned but did not merit a special chapter.

The last half of the book is allocated to thinkers from the modern period. This includes some obvious choices from the seventeenth century on (Galileo, Kepler, Descartes, Pascal, Wallis, Newton, Leibniz, Berkeley, Euler, and Cantor) as well as some lesser ones (the philosophers Spinoza, Gerrit Mannoury, Husserl, and Rene Guenon; the mathematician L. E. J. Brouwer; and the scientists Pavel Florensky and Arthur Eddington). The book concludes with a chapter tracing the history of the golden ratio, pinpointing how it came to be considered a divine proportion.

The book covers many links between mathematics and religion, but it concentrates heavily on how mathematical ideas were employed for religious and metaphysical purposes: the number 6 was chosen by God for the days of creation since it is a perfect number (Augustine); numerological calculations predicted the end of the world on October 19, 1533, at 8 a.m. (Stifel); a Trinitarian view of God is bolstered apologetically by analogy with the three dimensions in a cube (Wallis); and so on. As the editors note, ever since the time of the Pythagoreans and Plato mathematics has seemed specially poised to take on this exalted role, for it alone (or in conjunction with logic) studies abstract, unchanging objects and contains truths that are deemed absolutely certain, universal, and eternally valid, things typically associated with divine realities. Numbers and shapes were exceptionally privileged, having mystical powers and divine connections.

Moreover, mathematics could rationally account for musical harmony, celestial motion, optical phenomena, and mechanical behavior. Medieval and Renaissance Christians often adopted the Pythagorean/Platonic view that mathematical structure provided the underlying constitution of reality, consecrating this outlook with a quote from the Old Testament Apocrypha, "Thou hast ordered all things in measure and number and weight" (Wisdom 11:20b). Many found it quite natural to use mathematical ideas to explain God's nature and creative activity. This tendency was still the case in the early modern period, even strongly so; in fact, mathematization of physical phenomena was one of the key driving forces behind the rise of modern natural science.

Religious concerns formed the broader motivation for natural philosophers such as Galileo, Kepler, Newton, and others, but mathematics in turn provided the language for reading the deep structure of the cosmos, for deciphering God's intentions. In taking this position, many scientists incorporated mathematics into their religious foundation: mathematics becomes (part of) the eternal wisdom used by God to create and structure the world and so has a divine character. Greek notions thus became intertwined with Biblical ideas to form an unstable religious synthesis, one that would be dissolved in the eighteenth and nineteenth centuries by the increasing secularization due to the rise of deism, agnosticism, and atheism. There are still notable cases of Western scientists and mathematicians maintaining orthodox religious beliefs (Euler is a prime example), but they become more the exception than the rule as Enlightenment views become dominant. Cantor's taking an avid interest in medieval Catholic theology because its ideas of infinity supported his maverick notion of transfinite numbers seems atypical, though case studies of other mathematicians may later change our picture of this time period.

Readers who enjoy learning how mathematical ideas have shaped religious trends and doctrines will find much of interest in this book. Those who want to learn how religious ideas have impacted the development of mathematical science over the centuries will also find relevant material. Chapter 13 by Edith Sylla is a fascinating piece of historical detective work, examining how medieval discussions later caricatured and ridiculed in the seventeenth century as being about how many angels can dance on the head of a pin actually contributed to clarifying the notions of infinity and continuity. Chapter 18 by Volker Remmert is a good discussion of the role mathematics came to play in Galileo's thinking about nature and how this transformed the relationship between science and theology. Chapter 24 by Cornelis de Pater on Newton paints a holistic picture of Newton's intellectual interests, showing how his physics (astronomy, mechanics, optics), alchemy, and theology form consistent parts of a whole, all of them motivated by Newton's deep desire to fathom the ways of God and thus show His wisdom and grandeur. Mathematics provided the tool for understanding how God governs his handiwork, while alchemy looked for the active principles mediating God's interaction with his material creation.

This highlights only three of the chapters I found very interesting, but many others are worth reading. A few chapters that might have been good are marred by poor editing and awkward phraseology: the chapter on Cantor, for instance, could have gone through another draft. Omitting certain chapters would also have strengthened the book: a couple of them are only tangentially on topic, and one or two others seem to exemplify the sort of mysticism the book analyzes. This unevenness is probably to be expected in a compilation of so many articles, but a stronger editing hand would have improved the book.

All in all, however, Mathematics and the Divine makes a valuable contribution to opening up the history of this topic. It should provide welcome encouragement and assistance to others who would like to explore this arena further for themselves.

Calvin Jongsma is Professor of Mathematics at Dordt College in Sioux Center, IA. While he teaches a wide range of undergraduate mathematics courses on all levels, his Ph.D. training is in the history of mathematics (University of Toronto) and his main interests are the history of logic and the history of mathematics education. He has also had a long standing interest in the relations, real and imagined, between religion, philosophy, and mathematics.

1. Ho Peng-Yoke, Chinese Number Mysticism
2. Kim Plofker, Derivation and Revelation: the Legitimacy of Mathematical Models in Indian Cosmology
3. Reviel Netz, The Pythagoreans
4. Ian Mueller, Mathematics and the Divine in Plato
5. Jean-Fran‡ois Mattei, Nicomachus of Gerasa and the Divine Arithmetical Ladder
6. Dominic J. O'Meara, Geometry and the Divine in Proclus
7. Marie-Pierre Terrien, Religious Architecture and Mathematics during Late Antiquity
8. David A. King, The Sacred Geography of Islam
9. Faith Wallis, 'Number Mystique' in early medieval computus texts
10. Maurice-Ruben Hayoun, Is the Divine Universe Divisible
11. Charles Lohr, Mathematics and the Divine: Ramon Lull
12. Hugue Garcia, Christian Gnosis
13. Edith Dudley Sylla, Swester Katrei and Gregory of Rimini: Angels, God and Mathematics in the Fourteenth Century
14. Jean-Michel Counet, Mathematics and the Divine in Nicholas of Cusa
15. Teun Koetsier and Karin Reich, Michael Stifel and his Numerology
16. Ivo Schneider, Between Rosicrucians and Kabbala - the Mathematics of the Biblical Numbers of Johannes Faulhaber
17. Eberhard Knobloch, Mathematics and the Divine: Athanasius Kircher
18. Volker R. Remmert, Galileo, God and Mathematics
19. Andr‚ Charrak, The Mathematical Model of Creation According to Kepler
20. Jean-Marie Nicolle, The Mathematical Analogy in the Proof of God's Existence by Descartes
21. Donald Adamson, Pascal's Views on Mathematics and the Divine
22. Ger Harmsen, Spinoza and the Geometrical Method of Proof
23. Philip Beeley and Siegmund Probst, John Wallis (1616-1703): Mathematician and Divine
24. Kees de Pater, Newton and the Ocean of Truth
25. Herbert Breger, Leibniz: Mathematics and the Divine
26. Wolfgang Breidert, Berkeley's Defence of the Infinite God in Contrast to the Infinite in Mathematics
27. Ruediger Thiele, Leonhard Euler and the Divine
28. Ruediger Thiele, Georg Cantor and the Divine
29. Luc Bergmans, Gerrit Mannoury and his Fellow Significians on Mathematics and Mysticism
30. Teun Koetsier, Arthur Schopenhauer and L. E. J. Brouwer: A Comparison
31. Sergei S. Demidov and Charles E. Ford, On the Road to a Unified View: Priest Pavel Florensky - Theologian, Philosopher and Scientist
32. Fran‡ois De Gandt, Husserl and Impossible Numbers: a Sceptical Experience
33. Bruno Pinchard, Symbol and Space According to Ren‚ Gu‚non
34. Teun Koetsier, Eddington: Science and the Unseen World
35. Albert van der Schoot, The Divined Proportion