Koetsier, T.; Luc Bergmans;
Mathematics and the divine: a historical study
Elsevier, 2005, 701 pages
ISBN 0444503285, 9780444503282
topics: | science | history | mathematics | india
Derivation and revelation: The legitimacy of mathematical models in Indian
cosmology
K. Plofker, University of Utrecht chapter 2: p. 61-76
it has become commonplace to ascribe to India a uniquely pervasive
preoccupation with the divine, a special status as a land of gods and
mystics. Especially to the modern scientifically trained imagination, it may
seem incongruous to combine such a preoccupation with a simultaneous interest
in understanding the universe mathematically, that is, via self-consistent
quantitative models; certainly, the details of mathematical and scriptural
Weltanschauungen often sharply conflict. In the eighteenth and nineteenth
centuries, it became a contentious question for many Indians and
Anglo-Indians whether and how mathematics and the divine had coexisted in the
“indigenous” Indian (Sanskrit) tradition. What roles had its science offered
to divinely revealed truth (including the picture of the cosmos presented in
sacred texts such as the PurANas) and mathematically deduced fact (which
involved a very different picture of the physical universe)?
Though some colonial commentators on Sanskrit astronomy were frankly
contemptuous of the entire tradition,
[FN. Probably the most famous negative remark on the subject is that of
Thomas Macaulay to the effect that Indian astral science “would move
laughter in girls at an English boarding school” [8] T.B. Macaulay,
Minute of 2 February 1835 on Indian education. A survey
of various other opinions held by European Indologists of the eighteenth
and nineteenth centuries is given in [J. Burgess, Notes on Hindu
astronomy and the history of our knowledge of it, Journal of the Royal
Asiatic Society (1893), 717–761.]
others — especially Orientalists with some personal experience of and
enthusiasm for Sanskrit literature — inclined to a more nuanced view. This
asserted that “Hindu” sciences in earlier times accepted mathematically
rigorous models of the cosmos similar to those of the classical Greeks, but
were subsequently adulterated by various fanciful superstitions derived from
sectarian myths. Several Orientalists, particularly those with official
responsibilities concerning native education, publicly deplored what they
described as the consequent discrepancies between the classical Sanskrit
astronomical tradition and the sacred cosmology of popular belief. Lancelot
Wilkinson, a British Political Agent and educational reformer in Bhopal in
the early nineteenth century, suggested using the former pedagogically to
undermine the latter:
The followers of the Puráns [PurANas] . . . maintain that the earth is
a circular plane, having the golden mountain Merú in its centre
. . . that the moon however is at a distance from the earth double of
that of the sun; that the moon was churned out of the ocean; and is of
nectar; that the sun and moon and constellations revolve horizontally
over the plane of the earth, appearing to set when they go behind Merú,
and to rise when they emerge from behind that mountain; that eclipses are
formed by the monsters Ráhú and Ketú laying hold of the sun or moon
. . . The jyotishís [astronomer/astrologers] or followers of the
Siddhántas . . . teach the true shape and size of the earth . . . The
authors of the Siddhántas . . . have spared no pains to ridicule the
monstrous absurdities of . . . the Puráns . . .
[W]e have only to revive that knowledge of the system therein [in the
siddhAntas] taught, which notwithstanding its being by far the most
rational, and formerly the best cultivated branch of science amongst the
Hindus . . . has, from the superior address of the followers of the
Puráns . . . been allowed to fall into a state of utter oblivion
. . . Indeed, so general and entire is the ignorance of most of the
joshís [or jyotishís] of India, that you will find many of them engaged
conjointly with the Puránic bráhmans in expounding the Puráns, and
insisting on the flatness of the earth, and its magnitude . . . as
explained in them, with a virulence and boldness which shew their utter
ignorance of their proper profession, which had its existence only on the
refutation and abandonment of the Puránic system [25, pp. 504–509].
Wilkinson felt that religious and scientific worldviews in Indian thought
were more or less natural enemies, and that adherents of the former had
caused, by their “superior address,” the “oblivion” of the latter.
David Arnold, for example, identifies as “Orientalist” the notion that a
Hindu “religious” worldview ousted in medieval times a more “scientific” one,
and points out the political usefulness of this assessment to imperialists:
Although the richness and diversity of India’s ancient scientific
traditions has long been recognised, over the past two centuries it has
been the convention to see this as a history of precocious early
achievement followed by subsequent decline and degeneration. The European
Orientalist scholarship of the late eighteenth and early nineteenth
centuries represented India as having had an ancient civilisation
equalling, in some respects excelling or anticipating, those of classical
Greece and Rome. . . . In astronomy, mathematics and medicine in
particular, Hindu science was considered to have been remarkably advanced
well before the dawn of the Christian era and to have been the source of
discoveries and techniques that were only later taken up and incorporated
into Western civilisation, such as “Arabic” numerals and the use of
zero. However, according to this Orientalist interpretation, Indian
civilisation was unable to sustain its early achievements and lapsed into
decline. There followed an uncritical reliance upon earlier texts:
tradition replaced observation as surely as religion supplanted
science. . . . The history of Indian science thus served as a mere
prologue to the eventual unfolding of Western science in South Asia as
science was rescued from centuries of decline and obscurity by the advent
of British rule and the introduction of the more developed scientific and
technical knowledge of the West. This Orientalist triptych— contrasting
the achievements of ancient Hindu civilisation with the destruction and
stagnation of the Muslim Middle Ages and the enlightened rule and
scientific progress of the colonial modern age — has had a remarkably
tenacious hold over thinking about the science of the subcontinent. It
was a schema deployed not only by British scholars, officials and
polemicists but also by many Indians, for whom it formed the basis for
their own understanding of the past and the place of science in Indian
tradition and modernity [1, pp. 3–4].
[But] was there in fact a shift in Indian cosmological views away from
scientific derivation towards religious revelation, and how were these
PurANic and siddhAntic texts involved in it?
The PurANas form part of the Hindu sacred texts that are categorized as “smr.ti”, literally “what is remembered” or mediated by human authorship as opposed to “shruti”, “what is heard”, e.g., the Vedic hymns themselves and other directly revealed texts; but especially in the later medieval period, they too were venerated as a source of divine truth. Within their innumerable legends of the exploits of the gods and other beings is a common picture, assembled in the first few centuries CE from various cosmological concepts up to several centuries older, of the structure of the universe. In this picture, as Wilkinson mentioned, the earth is a flat disk five hundred million yojanas in diameter with the sacred mountain Meru standing 84 000 yojanas tall in its center. (There is no single standard value for the length of the yojana, but it is more or less on the order of ten kilometers, which makes the diameter of the PurANic earth about five billion kilometers, or approximately equal to the modern value for the size of the orbit of Neptune;Mount Meru reaches more than twice as far as the distance to the moon by our reckoning.) Its surface is covered by the concentric rings of seven continents and seven oceans, and above it the sun, moon, constellations, and planets are carried in circles around the mass of Meru, which makes them appear to rise and set. The pole-star is above Meru’s summit, upon which is the city of the gods. The moon is higher above the earth than the sun in this system, so its phases as well as solar and lunar eclipses are explained by a demon who periodically devours the luminaries, and the five visible star-planets are higher than the constellations. These worlds will endure for one “day of BrahmA” or 4 320 000 000 years, called a kalpa, which is one thousand times as long as a “mahAyuga” or “great age”. Each mahAyuga in turn is divided into four unequal yugas of which the last, least, and worst is the Kaliyuga of 432 000 years. (See [16,19].) This cosmology, unsurprisingly, is not adequate for mathematical prediction of the motions of the heavenly bodies as seen from the earth, and that was never its intended purpose. That function was performed in the last few centuries BCE by a collection of simple arithmetic rules formaintaining a relatively crude luni-solar calendar; it did not set up geometric models that challenged the PurANic cosmology.3
Shortly after the beginning of the current era, under the influence of Graeco-Babylonian and Hellenistic sources, more comprehensive astronomical treatises usually called siddhAntas— which in this contextmay be rendered by “astronomical systems”—began to appear. The remnants we still have of the earliest of these texts are devoted mostly to arithmetic schemes for predicting celestial events, comparable to those in older Babylonian texts. But as David Pingree has shown [16], by the fifth century at the latest a siddhAntic model was established that assumed a spherical earth only about 5000 yojanas in circumference, suspended in the middle of a sphere of fixed stars, around whose center the planets including the sun and moon were considered to move in tilted circular orbits with other circles included to account for their orbital anomalies. The moon was now established as nearest to the earth and the constellations most distant from it. Where possible, compromises were made with the PurANic system: for example, Mount Meru was retained as the north terrestrial pole (though greatly reduced in size), and the unfamiliar southern terrestrial hemisphere served as a convenient receptacle for exotic geographical features such as the annular continents and oceans. The PurANic divisions of time were respected, and celestial rates of motion and distances were chosen so that all the bodies could complete integer numbers of revolutions about the earth from the same starting-point in one kalpa or lifetime of the worlds. But this siddhAntic model was now committed to certain mathematical constraints in return for its increased explanatory and predictive power. For instance, to explain the varying height of the pole star as seen at different localities, the earth must be more or less uniformly spherical; to account for the unchanging appearance of the stars’ positions relative to one another, it must be tiny compared to the sphere of the heavens. Accounting quantitatively for eclipses and lunar phases by the configurations of three spherical bodies rather than by demonic agents requires that the moon’s orbit be smaller than the sun’s; and all the bodies’ motions must be predictable and geometrically constrained so that their positions can be computed trigonometrically. These assumptions were retained by most of the siddhAntas of the medieval period, of which the last to have great influence was the SiddhAntashiromaNi composed by BhAskara in the middle of the twelfth century.
Apparently from their earliest stages, siddhAntic texts began making
explicit their tensions with the existing PurANic model, although at first
not systematically. For example, the astronomical treatise of AryabhaTa
around 500 CE stressed the earth’s sphericity:
The globe of the earth [made of] earth, water, fire, and air, in the
middle of the cage of the constellations [formed of] circles, surrounded
by the orbits [of the planets], in the center of the heavens, is
everywhere circular. In the same way that the [spherical] bulb of a
kadamba-flower is entirely covered with blossoms, so is the globe of the
earth [covered] by all the beings born of the water and the land.
[vr.ttabhapañjaramadhye kakShyApariveShTitah. khamadhyagatah. |
mr.jjalashikhivAyumayo bhUgolah. sarvato vr.ttah. ||
yadvat kadambapuShpagranthih. pracitah. samantatah. kusumaih. |
tadvaddhi sarvasattvair jalajaih. sthalajaish ca bhUgolah. ||
(Gola, 6–7 [22, pp. 258–259].)]
And it flatly contradicted the PurANic magnitude of Mt. Meru: “Meru is
measured by one yojana . . . ”. [Smerur yojanamAtrah. . . . (Gola, 11 [22,
p. 261].)] In the same vein, Brahmagupta’s siddhAnta about 130 years later
explicitly challenged the PurANic assumption that the moon is farther away
than the sun: “If the moon [were] above the sun [as the PurANas indicate],
how would [its] power of increase and decrease in brightness, etc., [be
produced] from calculation [of the position of] the moon? The closer half
would always be bright”.[sitavr.ddhihAnivIryAdi shashA˙nkAj jAyate
katham. gaNitAt | upari raver indush ced arvAgardham. sadA shuklam
(BrAhmasphuTasiddhAnta 7, 1 [6, p. 100].) Brahmagupta’s contemporary
bhAskara (not to be confused with the twelfth-century author of the same
name), commenting on AryabhaTa’s work, opposed his own geographic
information to the PurANic values of terrestrial distances; he also
(according to the summary of a later commentator) explained that the
PurANic size of Meru was impossible because it would block northern stars
from sight.7 In the middle of the eighth century, an astronomer named Lalla
devoted an entire chapter (bluntly entitled “Errors”) of his own siddhAnta
to refuting various assumptions such as the causation of eclipses by a demon
and the flatness of the earth:
If your opinion is that a demon invariably devours [the moon or sun]
by means of magic, how is it [that the event can be] found by
calculation? And how [is it that there is] no eclipse except [at] new
or full moon? Eclipses, conjunctions of planets, risings, the
appearance of the lunar crescent, the rule for [computing] the shadow
at a given [time]—the solution of all five is accurate [when found]
by means of the [siddhAntic] size of the earth. So how could it be
[as] large [as the PurANas say]? Those who know calculation say
[that] a hundredth part of the circumference [of the earth] is seen
as flat. So the earth appears flat to this extent; it is just meant
in that way.
( shiShyadhIvr.ddhidatantra 20, 22, 31, 35 [3, Vol. I,
pp. 235–237]; other assertions criticized by Lalla in this chapter
include non-scriptural speculations such as ĀryabhaTa’s
hypothesis of the earth’s rotation.)
Here and in similar remarks by later authors,9 the validity of the
siddhAntic model was defended essentially on the grounds that it was
mathematically effective. The fact that “calculation” or mathematical
prediction agreed with observed result (or could be made to agree with it, in
the case of the apparent flatness of the spherical earth) was an argument in
favor of the predictive model’s reality. So a more or less open breach was
made between the PurANic model revealed by sacred texts and the
siddhAntic model derived from calculation and observation, and remarks like
these seem to make it clear which side an astronomer was supposed to be on.
But if we look at other remarks within the same texts, the picture becomes
more complicated. It turns out that instead of simply defying revealed truth
for the honor of mathematical consistency, siddhAnta authors were often
trying to have the best of both worlds— relying on their measurable and
calculable universe while still availing themselves at need of PurANic
authority. This was already indicated in the early inspiration of using
features of the PurANic universe to fill in the gaps (literally and
figuratively) in the geography or mechanics of the siddhAntic
one. Contradiction continued to be softened by compromise: for example,
AryabhaTa’s above-mentioned rejection of the traditional height of Mount
Meru did not imply any doubt as to whether the sacred mountain actually
existed. In fact, his assertion that Meru was only one yojana tall was
immediately followed by an orthodox PurANic description of its appearance,
“shining” and “covered with jewels”. Such minor concessions to scriptural
knowledge are not particularly surprising, as many ancient cosmographers
similarly incorporated tradition and legend to supply some of the vast
lacunae in the available definite knowledge about terrestrial or celestial
features.
What is more remarkable is the firmness with which siddhAntas often
explicitly insisted on the need to conform to the models of sacred
texts — though admittedly, this criterion was most often invoked by authors
criticizing rival authors for having failed to meet it. For example, the
same Brahmagupta who rejected the PurANic description of the moon’s
position made the following remark about AryabhaTa’s views on eclipses:
“How can the sun [illuminate] everything and [the demon] RAhu [be]
otherwise? Since there is variation in the [amount of] obscuration in
a solar eclipse, an eclipse of the sun or moon is not caused by
RAhu”: [what is] thus declared by Varāhamihira, S¯rISheNa,
ĀryabhaTa, and others is opposed to popular [opinion] and is not
borne out by the Vedas and smr.ti . . . “[The demon] Svarbhānu or
Āsuri has afflicted the sun with darkness”: this is the statement in
the Veda. So what is said here is in agreement with shruti and smr.ti
. . . The earth’s shadow does not obscure the moon, nor the moon the
sun, in an eclipse. RAhu, standing there equal to them in size,
obscures the moon and the sun.11
[kim. prativiShayam. sUryA rAhush cAnyo yato ravigrahaNe |
grAsAnyatvam. na tato rAhukr.tam. grahaNam arkendvoh. ||
evam. varAhamihirashrISheNaryabhaTaviShNucandrAdyaih. |
lokaviruddham abhihitam. vedasmr.tisam. hitAvAhyam || ...
svarbhAnurAsurir inam. tamasA vivyAdha vedavAkyam idam |
shrutisam. hitAsmr.tInAm. bhavati yathaikyam. taduktir itah. ||...
bhUchAyendum ato hi grahaNe chAdayati nArkam indur vA |
tatsthas tadvyAsasamo rAhush chAdayatishashisUryau ||
(BrAhmasphuTasiddhAnta 21, 38–39, 43, 48 [6,
pp. 372–374]. Here Brahmagupta has even taken the unusual step
of discarding part of the mathematically predictive model in
favor of the scriptural explanation.)
Finally, there is the question of the integrity of the siddhantas’ own
mathematical models. Although, as previously noted, the basic siddhAntic
cosmos is a quasi-Ptolemaic arrangement of spheres and circles, the
computational practices of Indian astronomy usually held a greater share of
the practitioners’ interest than the geometricmodels underlying them or the
physical consequences they implied. This almost always meant that when
computationalconvenience or creativity came up against physical or geometric
consistency, consistency took second place.
For example, a planet’s mean position in its circular orbit was geometrically
corrected by siddhAntas to its true position according to the position of
its “apex”, a point that could be considered to correspond to the apogee of
an eccentric orbit. However, it was not clearlyspecified in the correction
process whether the planet actually moved upon an eccentric circle or a
mathematically equivalent epicycle. Even Ptolemy, of course, did not always
make a confident physical assumption in favor of either of these
mathematically indistinguishable structures. But the SUryasiddhAnta, for
example, carried conventionalism much farther than Ptolemy would have dreamed
of when it posited another mathematically indistinguishable alternative, in
which the planet moves on its concentric orbit and the correction is caused
by a demon that stands at the apogee and pulls on the planet with a cord of
wind (SUryasiddhAnta 2, 1–3 [5, p. 31]). Similarly, the physical
specification of circular planetary orbits was never explicitly reconciled
with position-correcting procedures that modified the orbital radius
depending on the planet’s position, effectively turning the orbit into a
quasi-ellipse. Moreover, various coefficients were used that changed the size
of the epicycle radii, causing them to swell or shrink (sometimes
discontinuously) depending on the planet’s orbital anomaly or orientation
with respect to the observer, but no physical reason was suggested for this
phenomenon. Even the all-important sphericity of the earth and the heavens
was not handled in an entirely self-consistent manner: a commonly used
approximation for longitudinal difference effectively treated a slice of the
earth as a cylinder, and celestial spherical triangles were routinely solved
as if they were plane.15 In addition, the early siddhAntic texts also took
on, over time, a venerable status of their own, particularly when attributed
to gods or seers. (Note the distinction drawn in Yajñeshvara’s opening
comment above between the original intent of divine siddhAnta authors and
its subsequent misinterpretations in the texts of merely human astronomers.)
This meant that various approximate or erroneous formulas found in them
could, and did, continue to coexist with more exact ones in later
texts. Given the comparatively low priority allotted to strict mathematical
consistency, and the correspondingly high tolerance for physical and
geometric approximations, it seems less surprising that avirodha authors
should venture to describe the whole siddhAntic cosmos as based on
computational convenience and devoid of true physical reality.
[3] B. Chatterjee (ed.), The shiShyadhIvr.ddhidatantra of Lalla, 2 vols., New Delhi, 1981.
[4] M.D. Chaturvedi (ed.), SiddhAntashiroman.
i of BhAskarAcArya, Varanasi, 1981.
[5] K.C. DvivedI (ed.), SUryasiddhAnta, Varanasi, 1987.
[6] S. DvivedI (ed.), BrAhmasphuTasiddhAnta, Benares, 1901/1902.
[21] E. Sachau, Alberuni’s India, 2 vols., repr. New Delhi, 1992.
[22] K.S. Shukla (ed.), AryabhaTIya of AryabhaTa with the Commentary of
BhAskara I and Someshvara, New Delhi, 1976.
[23] K.S. Shukla (ed.), VaTeshvarasiddhAnta and Gola of VaTeshvara, 2
vols., New Delhi, 1986.
[24] B. SubbAji, AvirodhaprakAshaviveka, Bombay, 1837.
[1] D. Arnold, Science, Technology, and Medicine in Colonial India, The New
Cambridge History of India III, vol. 5, Cambridge, 2000.
[2] J. Burgess, Notes on Hindu astronomy and the history of our knowledge of
it, Journal of the Royal Asiatic Society (1893), 717–761.
[9] C.Z. Minkowski, NIlakaNTha’s cosmographical comments in the
BhIShmaparvan, PurANa 42 (1) (2000), 24–40.
[10] C.Z. Minkowski, The PaNDit as public intellectual: the controversy
over virodha or inconsistency in the astronomical sciences, The Pandit:
Traditional Sanskrit Scholarship in India, A. Michaels, ed., New Delhi,
2001, pp. 79–96.
[11] C.Z. Minkowski, The BhUgolavicAra: A cosmological manuscript from
Jaipur, SubhAShinI: Dr. Saroja Bhate Felicitation Volume, G.U. Thite,
ed., Pune, 2002, pp. 250–263.
[12] C.Z. Minkowski, Competing cosmologies in early modern Indian astronomy,
Studies in the History of the Exact Sciences in Honour of David Pingree,
C. Burnett et al., eds., Leiden, 2004, pp. 349–385.
[13] C.Z. Minkowski, A nineteenth century Sanskrit treatise on the Revolution
of the Earth: Govinda Deva’s BhUbhramana, to appear in SCIAMVS.
[14] D. Pingree, Census of the Exact Sciences in Sanskrit, Series A,
Vols. 1–5, Philadelphia, 1970–1994.
[15] D. Pingree, The Mesopotamian origin of Early Indian mathematical
astronomy, Journal for the History of Astronomy 4 (1973), 1–12.
[16] D. Pingree, The PurANas and Jyotih. shAstra: astronomy, Journal of
the American Oriental Society 110 (2) (1990), 274–280.
[17] D. Pingree, BIja-Corrections in Indian astronomy, Journal for the
History of Astronomy 27 (1996), 161–172.
[18] D. Pingree, PaurANic versus SiddhAntic astronomy, English Abstracts,
X World Sanskrit Conference, Bangalore, 1997, pp. 318–319.
[19] K. Plofker, Astronomy and astrology in India, to appear in the Cambridge
Encyclopedia of History of Science.
[20] G. Prakash, Another Reason: Science and the Imagination of Modern India,
Princeton, 1999.
[25] L. Wilkinson, On the use of the Siddhántas in the work of native
education, Journal of the Asiatic Society of Bengal 3 (1834), 504–519.
[26] R.F. Young, Receding from Antiquity: Indian responses to Science and
Christianity on the Margins of Empire, 1834–1844, Kokusaigaku-kenkyU 16
(1997), 241–274.
[27] R.F. Young, ‘Political’ Science: Astronomy in India, 1800–1850,
forthcoming.
Mathematics and the Divine seem to correspond to diametrically opposed tendencies of the human mind. Does the mathematician not seek what is precisely defined, and do the objects intended by the mystic and the theologian not lie beyond definition? Is mathematics not Man's search for a measure, and isn't the Divine that which is immeasurable ? The present book shows that the domains of mathematics and the Divine, which may seem so radically separated, have throughout history and across cultures, proved to be intimately related. Religious activities such as the building of temples, the telling of ritual stories or the drawing of enigmatic figures all display distinct mathematical features. Major philosophical systems dealing with the Absolute and theological speculations focussing on our knowledge of the Ultimate have been based on or inspired by mathematics. A series of chapters by an international team of experts highlighting key figures, schools and trains of thought is presented here. Chinese number mysticism, the views of Pythagoras and Plato and their followers, Nicholas of Cusa's theological geometry, Spinozism and intuitionism as a philosophy of mathematics are treated side by side among many other themes in an attempt at creating a global view on the relation of mathematics and Man's quest for the Absolute in the course of history.
Preface vii
List of Contributors ix
Introduction - T. Koetsier and L. Bergmans: 1
1. Chinese number mysticism - Ho Peng-Yoke 45
2. Derivation and revelation: The legitimacy of mathematical models
in Indian cosmology - K. Plofker 61
3. The Pythagoreans - R. Netz 77
4. Mathematics and the Divine in Plato - I. Mueller 99
5. Nicomachus of Gerasa and the arithmetic scale of the Divine -
J.-F. Mattéi 123
6. Geometry and the Divine in Proclus - D.J. O’Meara 133
7. Religious architecture and mathematics during the late antiquity
- M.-P. Terrien 147
8. The sacred geography of Islam - D.A. King 161
9. “Number Mystique” in early medieval computus texts - F. Wallis 179
10. Is the Universe of the Divine dividable? - M.-R. Hayoun 201
11. Mathematics and the Divine: Ramon Lull - C. Lohr 213
12. Odd numbers and their theological potential. Exploring and redescribing
the arithmetical poetics of the paintings on the ceiling of St. Martin’s
Church in Zillis - H. Garcia 229
13. Swester Katrei and Gregory of Rimini: Angels, God, and Mathematics in the
fourteenth century - E.D. Sylla 249
14. Mathematics and the Divine in Nicholas of Cusa - J.-M. Counet 273
15. Michael Stifel and his numerology - T. Koetsier and K. Reich 291
16. Between Rosicrucians and Cabbala — Johannes Faulhaber’s mathematics of
Biblical numbers - I. Schneider 311
17. Mathematics and the Divine: Athanasius Kircher - E. Knobloch 331
18. Galileo, God and Mathematics - V.R. Remmert 347
19. The mathematical model of Creation according to Kepler - A. Charrak 361
20. The mathematical analogy in the proof of God’s Existence by
Descartes - J.-M. Nicolle 385
21. Pascal’s views on mathematics and the Divine - D. Adamson 405
22. Spinoza and the geometrical way of proof - G. Harmsen 423
23. John Wallis (1616–1703):Mathematician and Divine - P. Beeley and
S. Probst 441
24. An ocean of truth - C. de Pater 459
25. God and Mathematics in Leibniz’s thought - H. Breger 485
26. Berkeley’s defence of the infinite God in contrast to the infinite
in mathematics - W. Breidert 499
27. Leonhard Euler (1707–1783) - R. Thiele 509
28. Georg Cantor (1845–1918) - R. Thiele 523
29. Gerrit Mannoury and his fellow significians on mathematics and mysticism
- L. Bergmans 549
30. Arthur Schopenhauer and L.E.J. Brouwer: A comparison - T. Koetsier 569
31. On the road to a unified world view: Priest Pavel Florensky — theologian,
philosopher and scientist - S.S. Demidov and C.E. Ford 595
32. Husserl and impossible numbers: A sceptical experience - F. De Gandt 613
33. Symbol and space according to René Guénon - B. Pinchard 625
34. Eddington, science and the unseen world - T. Koetsier 641
35. The Divined proportion - A. van der Schoot 655
Author Index 673
Subject Index 683