between science and the humanities as it does to bridge it. Department of History University of California at Los Angeles Los Angeles, CA 90095-1473 USA e-maih [email protected]

Mathematics and the Divine. A Historical Study edited by Teun Koetsier and Luc Bergmans AMSTERDAM, ELSEVIER, 2005, HARDBOUND, 716 PP., US $250, ISBN-$3: 978-0-444-50328-2, ISBN-IO: 0-444-50328-5 REVIEWED BY JEAN-MICHEL KANTOR

"God is like a skilful Geometrician." --Sir Thomas Browne (1605-1682), Religio Medici I, 16 "As God calculates and executes thought, the world comes into being." --Gottfried Wilhelm Leibniz (1646-1716), S~mtliche Schriflen und Briefe, 1923, ser. VI, vol. 4A, p. 22 "Par Dieu j'entends un ~tre absolument infini, c'est ft dire une substance consistant en une infinit~ d'attributs dont chacun exprime une essence ~ternelle et infinie." - - B a m c h Spinoza (1632-1677),

Ethics n recent years, science and religion have been o p p o s e d in numerous books and articles [1], but let's face it, there is one science which, since its (unknown) origins, has been closely and positively connected to religion: mathematics. To explain this proximity one notices c o m m o n features, for example, a claim for universality, for the eternity of concepts, and certainty of truths [3]. But some other explanations can be given as well. In this 700-page anthology of 35 independent chapters written by 30 authors, the editors, Luc Bergmans, a Dutch cultural historian with a special

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interest in religion, and Teun Koetsier, a historian of mathematics from Amsterdam, have gathered a loose potpourri on the theme of "religion and mathematics." Some articles address minor topics, such as numerological observations in Chinese magic squares, Michael Stiefel's biblical numerology (an ancestor of the bible code), or the significans m o v e m e n t in Holland in the late nineteenth century. 1 But some articles contain interesting analyses of parallels in the religious feelings and prescientific or scientific work of major figures, such as Kepler, Newton, Euler, Leibniz, and Cantor. The order of the chapters is essentially chronological and relates to the Western world. The historical times considered are: 9 the early pre-Greek period; 9 the Greek antiquity and its medieval and renaissance heirs; 9 the birth of modernity and of the scientific revolution. We mention only in passing the chapters connected to art, religion, and mathematics, exemplified by such material as the sempiternal divine proportion, or the "Geometry of the Divine," as described by Proclus (a neoplatonist of the Plotinian school) and represented in Constantinople's St. Sophia by the emperor Justinian. The heart of the b o o k consists of examples taken from classical figures from the middle ages and the beginning of m o d e r n times with the birth of science as w e understand it today. Historians have written on the positive influence of Christianity in the development of m o d e r n science [6], and we can test their theories here with some examples, well k n o w n (Nicolas de Cusa, Lull, Kepler) or not. C. de Pater, in a remarkable article on Newton, relies on many previous Newtonian studies to distinguish the attitude of Newton, clearly o p p o s e d to Cartesian mechanics. In short, Newton needs, more than Descartes, God's help to establish distant action. Another striking article in this book, by E. Sylla, concerns Gregory of Rimini (in the midfourteenth century) deciding h o w many angels can dance on the h e a d of a pin, a question dismissed as a scholastic stupidity in the seventeenth century. Here we have a direct contact b e t w e e n religion and mathematics in progress (premises of analysis).

One central agent of the connection between mathematics and religion is the concept of infinity (but it is not the only one!). From its first appearance under the name of "apeiron" with Anaximander of Miletus (610-546 BC), tO the recent work of Hugh Woodin [9], this is a permanent theme in mathematics--H. Weyl even wrote that mathematics is "the science of the Infinite" [8]--but the theme is also permanent in the philosophy of mathematics, and the w o r d End is not yet written. This is a fascinating sto W that has inspired philosophers and theologians, poets and mathematicians. One can follow the birth of the concept, corresponding to attributes of God (or space or time) with mathematics filling more and more space through the centuries, until the Cantorian parthenogenesis between mathematics and religion (but still with a trace of its origins with the theological Absolute to escape the paradox of the set of all sets). Religious and prescientific thoughts are intimately mixed in the view of infinity in the sefirot of kabbalistic thinkers in medieval Spain ("Is the Universe of the Divine dividable?" by M.-R. Hayoun). In a different style, the fascination of the work of Cardinal de Cusa (14011464; see the Chapter by J. M. Counet) relies on the variety of the new ideas he brings, from the geometrical representation of the Infinite in his theology to the symbolic role of mathematics in his famous "learned ignorance." A second theme running through many chapters of the b o o k is the search for a global vision uniting mathematics and religion. This can be found first in the school of Pythagoras, the object of "The Pythagoreans," an interesting study by Reviel Netz. Netz suggests, through an analysis of the mystery of the Pythagorean cult, that religion and mathematics might be able to interact, because they share some way of "rationalizing mystery" through analogies and metaphors. The global unity of mathematics with religion is central in Plato's work, and in his followers' such as Plotinus and Proclus, but also m u c h later in m o d e r n times (de Cusa, Leibniz's philosophical system). This global vision is still present in modern times, for example, in the unique vision of Father Pavel Flo.renski (see the study of S. Demidov and the detailed study of [2]),

which includes the p h i l o s o p h y of divine w i s d o m in the Orthodox tradition, the connections b e t w e e n physical a n d spiritual worlds expressed though inverted perspective, or even the use of imaginary numbers. Mathematics can also serve as an inspiration in the p r o o f of the exist e n c e of God, or as a "Staircase leading to God." Descartes for e x a m p l e p r o v e s the existence of G o d through its essence, and this relies o n the "essence" of infinity. The Cartesian "essences" will play a role in m o d ern p h i l o s o p h y of mathematics (see for e x a m p l e G6del's mathematical ontological p r o o f of the existence of God, "IF it is possible for a rational omniscient being to exist THEN necessarily a rational omniscient being exists." (see [5], and also G 6 d e l ' s unp u b l i s h e d manuscripts). The twentieth century is h a r d l y repr e s e n t e d in the book. There should have b e e n a Chapter on G 6 d e l ' s Platonism as a kind of m o d e r n religion, on the intuitionist school with B r o u w e r and H e r m a n n Weyl's philosophical views, the mystical interpretations of the Pythagorean texts by Simone Weil as s e e n in her conversations with her famous mathematician-brother Andr~ [7], a n d the extraordinary mystical views o f Alexander G r o t h e n d i e c k [4]. In conclusion, the o b v i o u s defects in the s e l e c t i o n of topics are b a l a n c e d by a v e r y rich v o l u m e , with s o m e interesting analysis of a historical character. T h e r e is a n e e d for further dev e l o p m e n t s on the subject, its r e c e n t aspects, a n d its p h i l o s o p h i c a l d i m e n sions. REFERENCES

[1] Conference, "Science and Belief," 2006, http://beyondbelief2006.org/. [2] Loren Graham, Jean-Michel Kantor, "A comparison of two cultural approaches to mathematics, France and Russia, 18901930," ISIS (March 2006) and www.math. jussieu.fr/-kantor/isis.pdf. See also Naming God, Naming Infinities, Mysticism and Mathematical Creativity. Harvard University Press, to appear. [3] Ivor Grattan-Guinness, "Christianity and mathematics: kinds of link, and the rare occurrences after 1750," Physis XXXVll (2000), Nuova ~;erie, Fasc. 2, 467-500.

[4] Alexander Grothendieck, "La clef des songes ou Dialogue avec le Ben Dieu," http://www.grothendieckcircle.org/. [5] Kurt Gd)del, Collected works Volume III, The Clarendon Press, Oxford University Press, Oxford, p. 403. [6] Alexander Koj#ve, "The Christian Origrn of Modern Science," St. John's Review Winter (1984) 22-26. [7] Simone Well, Oeuvres completes, Gallimard, Paris, 1988. [8] Hermann Weyl, The Open World (God and the Universe, Causafity, Infinity), Yale, 1933, Reprtnt Oxbow Press, 1989. [9] Hugh W. Woodin, "The continuum hypothesis. I.," Notices Amer. Math. Soc. 48 (2001) 567-576. Part II. Notices Amer. Math. Soc. 48 (2001) 681-690. Universite de Paris VII 75251 Pans Cede><75005 France e-mapl: [email protected]

Nonplussed! Mathematical Proof of Implausible ideas by Julian Havil PRINCETON, NEW JERSEY, OXFORDSHIRE, UNITED KINGDOM, PRINCETON UNIVERSITY PRESS, 2007, 208 PP. US$24.95, 14.95, ISBN: 13: 978-0-69112056-0 REVIEWED BY CHRISTINA BIRKENHAKE

Alice laughed: 'There's no use trying', she said; 'one can't believe impossible things'. 'I d a r e s a y y o u h a v e n ' t h a d m u c h practice', said the Q u e e n . ' W h e n I was y o u n g e r , I always d i d it for half an h o u r a day. Why, s o m e times I've b e l i e v e d as m a n y as six i m p o s s i b l e things b e f o r e b r e a k fast'. 'Where shall I begin', she asked. 'Begin at the beginning', said the king, 'and stop w h e n you get to an end'. (Lewis Carroll) his is Julian Havil's invitation to his intriguing collection of implausible ideas.

Altogether the b o o k consists of 14 chapters, each starting with a newspaper advertisment or a short story leading to s o m e implausible idea. In the course of the c h a p t e r the p r o b l e m is discussed and solved in full detail. Generalizations or related problems are discussed as well. For example, h a v e you ever heard of a b o d y rolling uphill? A d o u b l e cone on two inclined rails can do it (of course, the angles in the m o d e l have to be in p r o p e r relation). Or of a solid of finite volume but infinite surface area and, vice versa, o f infinite volume and finite surface area? Look at Torricelli's trumpet and Huygens's and de Sluze's drinking vessel. And as for d i m e n s i o n problems: The two-dimensional unit hypersphere is a disk of radius 1, a n d w e k n o w from geometry classes that its volume, which in this case is its area, is rr - 3.141 . . . . Similarly, the v o l u m e of the three-dimensional unit h y p e r s p h e r e is 4 ~ r ~ 3 4.188 . . . . What a b o u t dimensions n-> 3? I will not present here the respective formulae (which you'll find discussed in full detail in Havil's b o o k ) but only mention a surprising fact: The volume takes a m a x i m u m close to dimension five! That is: the five-dimensional unit h y p e r s p h e r e has v o l u m e --8 "na ~- 5.263, 15 which is greater than not only the respective volumes in dimensions 1,2,3, and 4, but also, those in n = 6,7,8, . . . , etc. Indeed, the v o l u m e as a function of the continuous variable n attains its m a x i m u m at 5 . 2 7 7 . . . a n d then tends to zero with increasing n. Some perplexing facts arising from probability are included too. The 'Birthd a y Paradox' asks for the likelihood of two individuals sharing the same birthday. It is obvious, ignoring leap years, that a m o n g 366 p e o p l e at least one repetition occurs. However, a m o n g only 23 p e o p l e there is a 50:50 chance of at least 2 coincident birthdays! Do y o u k n o w w h a t triskaidekaphobia means? It is the G r e e k comp o u n d m a d e from tris, 'three'; kai, 'and'; deka, 'ten'; a n d phobia, 'fear'. So w e learn in the c h a p t e r n a m e d 'Friday the 13th' a n d n u m b e r e d , of course, 13. Here w e find s o m e u n b e l i e v a b l e p r o p erties of o u r G r e g o r i a n calendar: the 13th of a m o n t h is m o r e likely to fall o n a Friday than o n a n y o t h e r d a y of the w e e k ; there is at least o n e Friday

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