Mathematics and the Divine (eBook)

Preface 8
List of Contributors 10
Contents 12
Introduction 16
The divine and mathematics 18
Three periods: The pre-Greek period, the Pythagorean-Platonic period and the period of the Scientific Revolution and its aftermath 19
The pre-Greek period and the ritual origin of mathematics 21
The Pythagorean-Platonic period 28
The Scientific Revolution and its aftermath 43
References 57
Chinese Number Mysticism 60
Introduction 62
The Hetu Diagram and the Luoshu Chart 62
The system of the Yijing (Book of Changes) 65
Daoist liturgy 67
Mysticism in the Chinese magic square 69
Popular beliefs in number mysticism and conclusions 72
Bibliographical comments 74
References 74
Derivation and Revelation: The Legitimacy of Mathematical Models in Indian Cosmology 76
Indian science, Indian religion: ``Orientalist'' and ``post-Orientalist'' views 78
The Puranas 80
The siddhantas 80
Contradiction and concession 81
The quest for non-contradiction 85
The status of siddhantas in the nineteenth century 86
Mathematical models in siddhantas 87
Conclusion 88
Bibliographical notes 89
References 90
The Pythagoreans 92
Introduction 94
Pythagoreanism in Plato and Aristotle 95
Pythagoreanism: some evidence from the Pythagoreans 100
Mathematics and the divine in the Pythagoreans: a suggestion 106
Notice for further reading 111
References 112
Mathematics and the Divine in Plato 114
Preliminary remarks 116
Introduction 118
The Timaeus 119
The Republic 127
Conclusion 131
Appendix A. The division of the stuff of the world soul (Timaeus 35b-36b) 132
References 135
Nicomachus of Gerasa and the Arithmetic Scale of the Divine 138
Introduction 140
Theologoumena Arithmetica 140
References 146
Geometry and the Divine in Proclus 148
Philosophy as divinisation 150
Mathematics in the divinisation of human nature 151
The nature of mathematical science 152
The metaphysics of geometry 154
St. Sophia: a geometry of the divine? 156
Bibliographical note 160
Religious Architecture and Mathematics During the Late Antiquity 162
Introduction 164
Religious architecture and heavenly measurements 164
Religious architecture and geometrical measurements 167
The building in its Earth-Heaven dialectic, or the circle above the square 170
References 174
The Sacred Geography of Islam 176
Introduction 178
The dichotomy of science in Islamic civilisation 178
The sacred geography of the legal scholars 180
The sacred geography of the scientists 181
The orientation of mosques and Islamic cities 190
Concluding remarks 192
References 193
``Number Mystique'' in Early Medieval Computus Texts 194
Introduction 196
The shape and scope of computus 198
Computus as ratio numerorum: the Irish computus of ca. 658 201
Computus as ratio temporum: Bede's revision of computistical ``number mystique'' 206
Byrhtferth's choices 210
Is the Universe of the Divine Dividable? 216
The sefirotic concept of Divinity developed in response to philosophy and rational theology 218
Eyn-sof, the perfectly unknowable origin of the sefirot 219
The struggle between mat]PlotinusPlotinus' God and the God of the Bible 220
Thought and Will in zoharic and pre-zoharic literature 220
The relations between Eyn-sof and the sefirot the One and the multiple
The origin of the term sefirot. A presentation of the nomenclature 223
Lights and colours 224
The unbreakable dynamic unity governing the sefirot and respected in prayer 226
References 226
Mathematics and the Divine: Ramon Lull 228
Introduction 230
The arbor elementalis 230
Lull's dynamic understanding of reality 231
The disciplines as productive arts 235
Geometry as an ``art'' 239
Conclusion 242
References 243
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 244
Introduction: displaying the method 246
An anatomical description of the ceiling 249
A theological imagination of odd numbers 251
A zoomorphic and soteriological arithmetic 256
The cross as cosmic sign and structure 260
The celestial cross and its function as odd-number Eon 262
Swester Katrei and Gregory of Rimini: Angels, God, and Mathematics in the Fourteenth Century 264
How many angels can dance on the point of a needle? 266
The place of angels: Gregory of Rimini 271
God and the continuum: Gregory of Rimini 275
Bibliography Swester Katrei and Gregory of Rimini 284
Mathematics and the Divine in Nicholas of Cusa 288
Introduction 290
The place of mathematics in human knowledge and its symbolic value 290
The problem of squaring the circle 301
References 304
Michael Stifel and his Numerology 306
Introduction 308
The end of the world and the Antichrist 308
The autumn of the Middle Ages 311
Stifel's numerology in 1532 312
Stifel's mathematics 318
Concluding remarks 322
References 324
Between Rosicrucians and Cabbala-Johannes Faulhaber's Mathematics of Biblical Numbers 326
Introduction 328
Pyramidal numbers and the Bible 328
Gog and Magog 332
Word calculus and signs 334
The Rosicrucian movement 336
The comet of 1618 338
Pyrgoidal numbers 343
Mathematics and the Divine: Athanasius Kircher 346
Biographical introduction 348
God and the rational foundation of music 348
The rules of the divine, combinatorial art 350
The world as God's organ 353
Universal science as an imitation of God's art 354
Mystical arithmetic 356
Mathematical theology 358
References 361
Galileo, God and Mathematics 362
Introduction 364
The mathematical sciences in early modern Europe 364
Galileo, God and mathematics 365
God and mathematics around Galileo 371
Concluding remarks 372
Selected bibliography 374
The Mathematical Model of Creation According to Kepler 376
Harmony and mathesis: the originality of Kepler 379
Mathematical ideas in God and creation 382
The limits of univocity 385
Colour Figures 390
The Mathematical Analogy in the Proof of God's Existence by Descartes 400
Introduction: Descartes' plan 402
Mathematical truth as an example 403
The ontological proof of God's existence 407
Arnauld's criticism: Descartes proof is circular 412
Truth and existence 414
Conclusion: Descartes' God and Pascal's God 417
References 418
Pascal's Views on Mathematics and the Divine 420
Introduction 422
Characteristics of God 423
God's Salvation-Plan 427
Heart and reason 428
The Wager argument (418) 429
Mathematical aspects of presentation 431
Conclusion 433
References 435
Spinoza and the Geometrical Way of Proof 438
Spinoza's time 440
The Jews in the Netherlands 441
Descent and youth 441
Influences 442
Interpretation 444
The mathematical way of reasoning 446
Social philosophy 450
Living nature 452
Summum bonum 453
References 454
John Wallis (1616-1703): Mathematician and Divine 456
The controversy with Thomas Hobbes 459
Wallis' defence of the Trinity 465
Mathematics and calendar reform 468
An Ocean of Truth 474
In search of ultimate truth 476
Natural philosophy and mathematical principles 477
Anticartesianism: passive matter and active principles 481
Prisca theologia and prisca sapientia 485
Prophecy and comets 488
Prisca geometria and fluxions 492
References 496
God and Mathematics in Leibniz's Thought 500
Introduction 502
The best of all possible worlds 504
The binary system and creation 506
A staircase leading to God 508
The existence of God 511
Concluding remark 512
References 513
Berkeley's Defence of the Infinite GodGod in Contrast to the InfiniteInfinite in Mathematics 514
Berkeley's intent 516
God and infinity 518
Mathematical exactitude 520
The astounding results of modern mathematics 521
Bibliography 523
Leonhard Euler 524
Historical background 526
Life 526
Euler on religion 530
Physico-theological arguments in Euler 534
References 535
Georg Cantor 538
The mathematician Cantor 540
Short vita of Cantor 542
Mathematics and metaphysics 543
Mathematics and religion 548
Paradoxes and truth 555
Religious denomination 558
References 562
Gerrit Mannoury and his Fellow Significians on Mathematics and Mysticism 564
The search for unifying explanations 566
Convictions and ideologies 567
Does 2+2 = 4 and is there a God to an all-levelling relativist? 569
Frederik van Eeden on the foundational status of uncertainty 571
Gerrit Mannoury on the foundational status of uncertainty 576
Beyond psychological foundations of mathematics: the unobjectifiable subject and the idea of the unreligious separation of subject and object 577
Uncertainty and docta ignorantia. The role of negation in significs, intuitionism and theology 579
Johan J. de Iongh. Shared longing as the religious meaning of mathematical dialogue 580
References 582
Arthur Schopenhauer and L.E.J. Brouwer: A Comparison 584
Introduction 586
From Kant to Schopenhauer 587
Brouwer, the prophet 589
Life, art and mysticism 591
Schopenhauer and Brouwer: A comparison 598
The turn to mathematics 602
Brouwer compared to Gödel 604
Final remarks 607
Acknowledgement 607
References 608
On the Road to a Unified World View: Priest Pavel Florensky-Theologian, Philosopher and Scientist 610
Early years 612
Moscow University 613
Theological Academy 615
The pillar and ground of the truth 616
Concrete metaphysics 618
Church and revolution 620
Religious and scientific work 621
Enemy of the people 623
Conclusion 625
References 626
Husserl and Impossible Numbers: A Sceptical Experience 628
Introduction 630
Beyond Hume 630
Prelogical use of signs 631
Husserl's philosophical quest 632
Impossible concepts 633
Natural selection 635
Structural mathematics and categorial connections 635
De facto truth and the need of a science of science 637
References 638
Symbol and Space According to René Guénon 640
Life and work of René Guénon (Blois 1886 - Cairo 1951) 642
The symbolism of space 644
References 655
Eddington, Science and the Unseen World 656
Introduction 658
Astronomer and physicist 658
Quakerism 660
The problem 660
The four-dimensional world 662
Eddington's idealism 663
Mind stuff 664
Concluding remarks 666
Acknowledgement 667
References 667
The Divined Proportion 670
Introduction 672
No logo 672
Elemental truths 674
Potentia mirabilis 676
Stupendous effects 678
The blueprint of the universe 681
Unity in variety 684
Author Index 688
Subject Index 698