Ivor Grattan-Guinness0444508716, 9780444508713, 9786610633739
Table of contents :
front cover……Page 1
copyright……Page 6
Table of Contents……Page 7
Organisation of the articles……Page 11
Some principal limitations……Page 12
Bibliography 0……Page 18
1. René Descartes, GÉOMÉTRIE, Latin edition (1649), French edition (1637)……Page 21
Youth, from La Flèche to the REGULAE……Page 22
Descartes in Holland and Stockholm……Page 23
The `construction’ of the expressions……Page 24
Compasses, ruler-and-slide, criterion for `continuous motions’……Page 27
The problem of Pappus and an algebraic criterion……Page 28
Tangents……Page 32
Algebraic equations……Page 33
The `construction’ of equations……Page 34
From the GÉOMÉTRIE to the ENUMERATIO……Page 37
`PROLES SINE MATRE CREATA’……Page 38
Bibliography……Page 41
Background to the ARITHMETICA INFINITORUM……Page 43
Methods and results in the ARITHMETICA INFINITORUM……Page 45
The motivation to `Wallis’s product’……Page 47
Reactions to the ARITHMETICA INFINITORUM……Page 49
Bibliography 2……Page 52
3. Christiaan Huygens, book on the pendulum clock (1673)……Page 53
The three strands of Huygens’s research……Page 54
Pendulum clocks……Page 55
The five parts of HOROLOGIUM OSCILLATORIUM……Page 56
On Huygens’s mathematical style……Page 61
Unfocused reception……Page 62
Bibliography 3……Page 65
4. Gottfried Wilhelm Leibniz, first three papers on the calculus (1684, 1686, 1693)……Page 66
Leibniz’s research on the infinitesimal mathematics……Page 67
The enigmatic first publication……Page 69
The early reception of the differential calculus……Page 72
The first papers on integral calculus……Page 74
The spread of the Leibnizian calculus……Page 75
Bibliography 4……Page 77
5. Isaac Newton, philosophiae naturalis principia mathematica, first edition (1687)……Page 79
Newton’s mathematical methods……Page 80
Early studies on the motion of bodies and on planetary motion……Page 83
The PRINCIPIA (1687): definitions and laws……Page 86
The PRINCIPIA (1687): limits……Page 87
The PRINCIPIA (1687): the area law……Page 92
The PRINCIPIA (1687): central forces……Page 93
The PRINCIPIA (1687): Pappus’s problem……Page 95
The PRINCIPIA (1687): algebraic non-integrability of ovals……Page 96
The PRINCIPIA (1687): the general inverse problem of central forces……Page 97
Revisions (1690s), second (1713) and third (1726) editions……Page 98
The impact of the PRINCIPIA……Page 103
Bibliography 5……Page 105
6. Jakob Bernoulli, Ars conjectandi (1713)……Page 108
Background and story of publication……Page 109
Content and structure of the AC……Page 110
Huygens’s De rationciniis in ludo aleae with Bernoulli’s annotations……Page 112
Combinatorics as the main tool of the art of conjecturing……Page 116
The transition to a calculus of probabilities in the fourth Part……Page 119
Bibliography 6……Page 123
Background and story of the publication……Page 125
The introduction of the Doctrine of chances, and the mathematical requirements for the reader as announced in the preface……Page 127
A `new sort of algebra’ in the Doctrine of chances……Page 130
The duration of play……Page 132
The approximation of the binomial by the normal distribution……Page 134
Annuities on lives……Page 137
Impact of the DoC……Page 139
Bibliography 7……Page 140
8. George Berkeley, The analyst (1734)……Page 141
Berkeley’s life and works……Page 142
The purpose of The analyst……Page 143
The principal arguments……Page 144
Responses to Berkeley……Page 149
Bibliography 8……Page 150
9. Daniel Bernoulli, Hydrodynamica (1738)……Page 151
General remarks……Page 152
The Bernoulli equation in the Hydrodynamica……Page 154
Propulsion of ships by jet ejection, and the dynamics of systems with variable mass……Page 156
Some other topics……Page 157
Some general remarks on the style of the Hydrodynamica……Page 159
Early praise and troubles……Page 160
Bibliography 9……Page 161
Colin MacLaurin (1698-1746)……Page 163
The TREATISE ON FLUXIONS (1742): foundations……Page 164
Extrema and inflections……Page 171
Limits of series, and the Euler-MacLaurin theorem……Page 173
The method of infinitesimals, and Newton’s prime and ultimate ratios……Page 174
Book II……Page 175
Summary remarks 10……Page 176
Bibliography 10……Page 177
11 Jean Le Rond D’Alembert, Traité de dynamique (1743, 1758)……Page 179
Scientific works……Page 180
Contents of the TRAITÉ DE DYNAMIQUE……Page 181
The place of the TREATISE in the work of the author……Page 184
Posterity of the TREATISE……Page 185
Bibliography 11……Page 187
Introduction 12……Page 188
Origins and basic results……Page 189
Foundations of analysis……Page 194
Later developments: Lagrange, Euler and the calculus of variations……Page 197
Bibliography 12……Page 199
13. Leonhard Euler, `Introduction’ to analysis (1748)……Page 201
Book I, on analysis……Page 202
Book II on plane and surface geometry……Page 208
On the impact of the INTRODUCTIO……Page 209
Bibliography 13……Page 210
The second part of Euler’s trilogy on mathematical analysis……Page 211
The differential calculus and its foundations……Page 212
Applications of the differential calculus……Page 215
General remarks 14……Page 216
Bibliography 14……Page 217
15. Thomas Bayes, An essay towards solving a problem in the doctrine of chances (1764)……Page 219
Biography 15……Page 220
Bayes’s work on chances……Page 221
Price’s appendix……Page 223
Impact and influence of the work……Page 225
Bibliography 15……Page 226
16. Joseph Louis Lagrange, Méchanique analitiqueMéchanique Analitique, Lagrange’s|(, first edition (1788)……Page 228
Outline of Lagrange’s scientific biography……Page 229
Lagrange’s conception of `analytic mechanics’ as a science……Page 230
Lagrange’s fundamental developments up to the first edition of the Méchanique analitique……Page 231
Contents of the editions……Page 236
Fundamental differences between the Méchanique analitique and the Mécanique analytique and later reviews……Page 238
General assessment and reception of the work……Page 240
Bibliography 16……Page 241
17. Gaspard Monge, Géométrie descriptive, first edition (1795)……Page 245
Introduction 17……Page 246
Gaspard Monge……Page 247
The subject matter of Monge’s lectures……Page 248
The principal aims of Monge’s course……Page 255
The influence of Monge’s lectures……Page 258
Bibliography 17……Page 260
18. P.S. Laplace, Exposition du système du monde, first edition (1796); Traité de mécanique céleste (1799-1823/1827)……Page 262
Background 18……Page 263
The Exposition 18……Page 264
Mécanique céleste, the celestial volume 1……Page 266
Mécanique céleste, the planetary volume 2……Page 270
Mécanique céleste, the numerical Books 6-9……Page 271
Mécanique céleste, the miscellaneous Book 10……Page 272
Immediate influence……Page 273
Mécanique céleste, the miscellaneous volume 5……Page 274
Bibliography 18……Page 276
Introduction 19……Page 278
Algebraic analysis and the function concept……Page 281
Theorems of analysis……Page 283
Methods of approximation……Page 285
Multiplier rule……Page 289
Calculus of variations: sufficiency results……Page 293
Conclusion 19……Page 294
Bibliography 19……Page 295
Mathematical teacher and writer……Page 297
Purposes of the Traité……Page 299
Differential calculus……Page 300
Integral calculus……Page 304
Differences and series……Page 307
Impact 20……Page 310
Bibliography 20……Page 311
Biography 21……Page 312
Contents of the book……Page 314
Volume 1 (1758 and 1799)……Page 315
Volume 2 (1758 and 1799)……Page 316
Volume III (1802)……Page 317
Volume 4 (1802)……Page 320
Reception of the work 21……Page 321
Bibliography 21……Page 322
22. Carl Friedrich Gauss, Disquisitiones arithmeticae (1801)……Page 323
What is the Disquisitiones arithmeticae about?……Page 324
Content and reception of Sections 1-4……Page 327
Content and reception of Section 5……Page 330
Content and reception of Section 7……Page 332
Fame and reactions 22……Page 333
Bibliography 22……Page 334
Biographical sketch……Page 336
Inception of Gauss’s work in orbit determination……Page 337
Gauss’s early work on orbit-determination……Page 338
The determination of orbits in TM……Page 341
Themes and topics of Book I……Page 344
Sections 3 and 4 of Book II: the method of least squares; perturbations……Page 346
The impact of TM……Page 347
Bibliography 23……Page 348
24. P.S. Laplace, Théorie analytique des probabilités, first edition (1812); Essai philosophique sur les probabilités, first edition (1814)……Page 349
The Mont Blanc of mathematical analysis, and its foothills……Page 350
Laplacian probability……Page 351
Laplace’s THÉORIE and mathematical statistics……Page 353
The ESSAI PHILOSOPHIQUE……Page 357
The legacy……Page 358
Bibliography 24……Page 359
25. A.-L. Cauchy, Cours d’analyse (1821) and Résumé of the calculus (1823)……Page 361
From student to professor……Page 362
The COURS D’ANALYSE: `algebraic analysis’ and the theory of limits……Page 363
The COURS D’ANALYSE: continuous functions and infinite series……Page 364
The RÉSUMÉ: a new version of the calculus……Page 366
Reactions at the ECOLE POLYTECHNIQUE: Cauchy’s later books……Page 369
The gradual influence of Cauchy’s doctrine……Page 370
Bibliography 25……Page 372
Education and employments……Page 374
The chronology of Fourier’s researches……Page 375
Heat diffusion, internal and surface……Page 376
Fourier series and their function……Page 379
Calculating and interpreting the coefficients……Page 380
Non-harmonic series, and the Bessel function……Page 381
Later work and recognition……Page 382
On the later impact……Page 383
Bibliography 26……Page 384
Poncelet’s Traité……Page 386
Jean Victor Poncelet and the other French geometers……Page 392
Michel Chasles……Page 395
Bibliography 27……Page 396
28. A.-L. Cauchy, two memoirs on complex-variable function theory (1825, 1827)……Page 397
`Cauchy’s theorem’ foreshadowed……Page 398
An argument foreshadowing the notion of principal value……Page 401
The first hints of the residue theorem……Page 402
The background of the 1825 memoir……Page 405
Definite integrals between complex limits……Page 406
Taking singularities into account……Page 407
The use of geometrical language……Page 408
Later developments……Page 409
Bibliography 28……Page 410
Introduction: equations in general……Page 411
The problem of the quintic, 1700-1800……Page 416
Niels Henrik Abel……Page 419
Reception of the memoir……Page 420
Bibliography 29……Page 421
Green’s little-known entrée……Page 423
Poisson’s `simplifying’ theorem……Page 424
Green’s theorem in Green’s book……Page 425
Green’s applications to electricity and magnetism……Page 427
Green’s later researches……Page 428
The discovery of the book……Page 429
Bibliography 30……Page 431
Elliptic integrals……Page 432
Elliptic integrals from Fagnano to Legendre……Page 436
Genesis of the Fundamenta……Page 439
The author……Page 444
Contents of the work……Page 445
Significance of the work……Page 448
Bibliography 31……Page 450
32. Hermann G. Grassmann, Ausdehnungslehre, first edition (1844)……Page 451
Dialectics and the theory of tides……Page 452
The new branch of mathematics……Page 454
A muted reception……Page 457
Eventual recognition……Page 458
Bibliography 32……Page 459
Background and biography……Page 461
Von Staudt’s `Geometry of position’……Page 462
Impact 33……Page 465
Bibliography 33……Page 466
34. Bernhard Riemann, thesis on the theory of functionscomplex function theory|( of a complex variablefunction of a complex variable|( (1851)……Page 468
The theory of functions of a complex variable before Riemann……Page 469
Biography of Riemann……Page 471
The thesis……Page 472
Riemann’s publications from 1857……Page 474
The positive reception of Riemann’s thesis……Page 475
A complex of theories……Page 476
Bibliography 34……Page 477
From prodigy to sage……Page 480
The origin of quaternions……Page 481
The Lectures……Page 482
Reception and subsequent development……Page 486
Quaternions versus vectors: J.W. Gibbs and E.B. Wilson……Page 487
Bibliography 35……Page 489
A self-made mathematician……Page 490
Boole’s mature `Investigation’ of logic……Page 492
The algebraic methods of deduction and elimination……Page 494
The religious connotation of Boole’s logic……Page 496
Boole’s gradual influence……Page 497
Bibliography 36……Page 498
37. Johann Peter Gustav Lejeune-Dirichlet, Vorlesungen über Zahlentheorie, first edition (1863)……Page 500
A posthumous textbook……Page 501
The simplification of Gauss’s Disquisitiones arithmeticae……Page 502
Analysis and arithmetic……Page 506
Dedekind’s supplements X and XI: towards the theory of ideals……Page 507
The influence of the Vorlesungen……Page 508
Bibliography 37……Page 510
Background 38……Page 511
Riemann’s historical analysis of the integral and trigonometric series……Page 512
Riemann on the integral……Page 517
The problem of the uniqueness of the representation……Page 519
The final article: examples illustrating the diversity and complexity of trigonometric series……Page 521
Bibliography 38……Page 524
39. Bernhard Riemann, posthumous thesis `On the hypotheses which lie at the foundation of geometry’ (1867)……Page 526
The lecture……Page 527
The intellectual context……Page 531
Bolyai and Lobachevsky and the discovery of non-Euclidean geometry……Page 535
Beltrami, Poincaré, and Klein on non-Euclidean geometry……Page 537
The later reception of non-Euclidean geometry……Page 538
Bibliography 39……Page 539
40. William Thomson and Peter Guthrie Tait, Treatise on natural philosophy, first edition (1867)……Page 541
The place of T&T’ in Thomson’s work……Page 542
Collaboration with Tait……Page 544
Kinematics……Page 545
Dynamics of energy……Page 548
Extremum principles……Page 550
Abstract dynamics……Page 551
Reception 40……Page 552
Bibliography 40……Page 553
41. Stanley Jevons, The theory of political economy, first edition (1871)……Page 554
A new theory of value……Page 555
The law of exchange and the trading bodies……Page 558
Concluding remarks……Page 561
Bibliography 41……Page 562
42. Felix Klein’s Erlangen Program, `Comparative considerations of recent geometrical researches’ (1872)……Page 564
On the biography of Klein……Page 565
The Erlangen Program……Page 566
The reception of the Erlangen Program……Page 570
Bibliography 42……Page 572
Introduction: the problem of incommensurables……Page 573
The author……Page 575
Dedekind’s view of the problem of continuity……Page 576
Dedekind’s solution of the problem……Page 577
Reception of the work……Page 580
Bibliography 43……Page 583
44. James Clerk Maxwell, A treatise on electricity and magnetism, first edition (1873)……Page 584
Education and career……Page 585
Mathematical theories of electricity and magnetism in the first half of the 19th century……Page 586
Faraday and Thomson on the notion of field……Page 587
Maxwell and the theoretical reform of electromagnetism……Page 589
The publication, functions and structure of the Treatise……Page 590
Mathematical structures in the Treatise……Page 591
Electrostatics and electrokinetics……Page 595
Magnetism and electromagnetism……Page 596
The dynamical theory of electrokinetic phenomena, and the general equations of the electromagnetic field……Page 597
The electromagnetic theory of light……Page 600
The Maxwellians……Page 603
The experiments of Hertz and their impact……Page 604
Larmor and the notion of electron……Page 605
Bibliography 44……Page 606
Rayleigh’s early research on sound……Page 608
The publication of The theory of sound……Page 609
The book as compared with its predecessors……Page 610
On Rayleigh’s mathematical methods in the book……Page 612
Rayleigh on waves and vibrations……Page 613
Presenting original researches……Page 614
The influence of the book on acoustics and elsewhere……Page 616
Bibliography 45……Page 618
46. Georg Cantor, paper on the `Foundations of a general set theory’ (1883)……Page 620
Early work on trigonometric series: derived sets……Page 621
Cantor’s theory of real numbers……Page 622
The descriptive theory of point sets……Page 623
The GRUNDLAGEN: a general theory of sets and transfinite ordinal numbers……Page 624
Cantor’s nervous breakdowns……Page 627
Transfinite cardinal numbers: the alephs ()……Page 628
Transfinite mathematics and Cantor’s manic depression……Page 629
Bibliography 46……Page 631
47. Richard Dedekind (1888) and Giuseppe Peano (1889), booklets on the foundations of arithmetic……Page 633
Dedekind: biography and background……Page 634
Peano: biography and background……Page 636
Dedekind’s theory……Page 639
Peano’s theory……Page 641
Appraisal and impact……Page 643
Bibliography 47……Page 645
Introduction 48……Page 647
Origin and significance of the three-body problem……Page 648
Poincaré’s work before TBP……Page 649
The publication of TBP……Page 650
The content of TBP……Page 651
The reception of TBP……Page 655
The resolution of the three-body problem and some later developments……Page 656
Bibliography 48……Page 658
General outline of the ELECTRICAL PAPERS……Page 659
The algebra of vectors……Page 662
Heaviside’s operational calculus……Page 667
On Heaviside’s later work……Page 671
Bibliography 49……Page 672
50. Walter William Rouse Ball, Mathematical recreations and problems of past and present times, first edition (1892)……Page 673
Historical background……Page 674
The late 19th century……Page 676
Walter William Rouse Ball (1850-1925)……Page 678
Examples of new material in the book……Page 679
Bibliography 50……Page 682
51. Alexandr Mikhailovich Lyapunov, thesis on the stability of motion (1892)……Page 684
The aim and the inspiration of the Dissertation……Page 685
Lyapunov’s concept of stability……Page 686
The second method of Lyapunov……Page 688
The case of autonomous systems……Page 689
The case of periodic systems……Page 690
The influence of Poincaré’s work on Lyapunov’s Dissertation……Page 691
The early reception of the work of Lyapunov on stability……Page 693
Bibliography 51……Page 694
Education and employments……Page 697
Mechanics, a race with death……Page 698
Why mechanics?……Page 699
Images of nature……Page 700
Geometry of systems of points……Page 702
Dynamics……Page 705
Reception and impact 52……Page 707
Bibliography 52……Page 708
Background 53……Page 710
Heinrich Weber’s career……Page 711
The Introduction to the Lehrbuch……Page 712
The three volumes……Page 713
Impact 53……Page 718
Bibliography 53……Page 719
A report and almost a textbook……Page 720
The preface: number theory and arithmetisation……Page 721
Dedekind versus Kronecker, arithmetic versus algebra……Page 722
Content and structure……Page 724
Later reactions 54……Page 728
Bibliography 54……Page 729
A new direction of mathematical thought in 1899……Page 730
Fourteen editions in 100 years……Page 731
Seeing the master in his workshop: Hilbert’s manuscripts……Page 732
Foundations of projective geometry: Hilbert’s exercise-book (1879) and later……Page 733
Geometry as a system of axioms (1894)……Page 734
A vacation-course for teachers: the kernel of the Festschrift (1898)……Page 736
Lectures and an elaboration on Euclidean geometry (1898-1899)……Page 737
The further development of Hilbert’s Grundlagen der Geometrie……Page 738
A survey of the intersection theorems and the most important results……Page 741
Bibliography 55……Page 742
Education and employment……Page 744
Chronology and curve fitting……Page 745
The mathematical derivation……Page 748
Concluding remarks 56……Page 749
Bibliography 56……Page 750
57. David Hilbert, paper on `Mathematical problems’ (1901)……Page 752
The problems 57……Page 753
Concluding remarks 57……Page 762
Bibliography 57……Page 763
Biography 58……Page 768
Mathematical field theory……Page 769
The physical foundation of the Baltimore lectures……Page 770
The contents of the Baltimore lectures……Page 771
Bibliography 58……Page 776
59. Henri Lebesgue and René Baire, three books on mathematical analysis (1904-1906)……Page 777
Integrals and functions in the 19th century……Page 778
The authors 59……Page 784
Lebesgue’s Leçons sur l’intégration et la recherche des fonctions primitives (1904)……Page 785
Reception of Lebesgue’s book……Page 788
Baire’s Leçons sur les fonctions discontinues (1905)……Page 789
Reception of Baire’s book……Page 792
Lebesgue’s Leçons sur les séries trigonométriques (1906)……Page 793
Reception of Lebesgue’s second book……Page 795
Bibliography 59……Page 796
Biography 60……Page 798
Lorentz’s contributions to electromagnetism……Page 799
The lectures 60……Page 801
Bibliography 60……Page 803
The reductionist heritage……Page 804
Collaboration, and fallow years……Page 806
The writing and content of PM……Page 807
Reactions by Russell and his British followers……Page 810
German-speaking contributions……Page 812
Logic(ism) after Gödel……Page 813
Bibliography 61……Page 814
The authors 62……Page 815
The methods and the content……Page 817
The impact 62……Page 820
Bibliography 62……Page 821
63 Albert Einstein, review paper on general relativity theory (1916)……Page 822
The special theory of relativity……Page 823
The equivalence hypothesis……Page 824
The metric tensor……Page 825
Einstein’s collaboration with Marcel Grossmann……Page 826
Coming close to the solution, or so it seems……Page 827
The Entwurf theory……Page 828
The 1914 review article on the Entwurf theory……Page 830
The demise of the Entwurf and the breakthrough to general covariance……Page 831
The 1916 review paper……Page 835
Early reception of the final version of general relativity……Page 838
Going on and beyond general relativity……Page 839
Bibliography 63……Page 841
64. D’Arcy Wentworth Thompson, On growth and form, first edition (1917)……Page 843
Structure of the argument……Page 844
Thompson’s key methodological example……Page 846
How should such a work be approached today?……Page 847
Evaluation 64……Page 850
Bibliography 64……Page 851
Brief biography of the author……Page 853
Why Dickson may have written his HISTORY OF THE THEORY OF NUMBERS……Page 854
The style and content of Dickson’s HISTORY OF THE THEORY OF NUMBERS……Page 855
One salient omission……Page 859
Reception of Dickson’s HISTORY OF THE THEORY OF NUMBERS……Page 860
Bibliography 65……Page 862
Ancestry 66……Page 864
Paul Urysohn, mathematician……Page 866
Karl Menger, mathematician……Page 870
The joint contribution……Page 871
The impact of Menger and Urysohn……Page 872
Bibliography 66……Page 874
67. R.A. Fisher, Statistical methods for research workers, first edition (1925)……Page 876
The author 67……Page 877
Writing STATISTICAL METHODS……Page 879
Content of the first edition……Page 881
Subsequent editions and books……Page 885
Impact of the book……Page 888
Bibliography 67……Page 889
Introduction 68……Page 891
Celestial mechanics: the historical background……Page 892
The contents of Birkhoff’s book……Page 893
Birkhoff’s main ambition……Page 895
Birkhoff’s lecture course and its context……Page 896
On the impact and renaissance of the book……Page 897
Bibliography 68……Page 899
The discovery of quantum mechanics……Page 902
Background of Dirac’s PRINCIPLES……Page 903
The PRINCIPLES OF QUANTUM MECHANICS……Page 906
Later editions of PRINCIPLES……Page 909
Weyl, Hilbert, von Neumann and the mathematical foundations of quantum mechanics……Page 912
The MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS……Page 916
Bibliography 69……Page 919
Biographical notes 70……Page 921
Aims and contents of the book: foundations……Page 922
The consolidation of field theory and group theory: Galois theory……Page 925
Ideals and hypercomplex systems: further basic algebraic theory……Page 927
Some special topics in algebra……Page 929
Reception and historical impact of the book……Page 931
Further revision of the book 70……Page 933
Bibliography 70……Page 935
71. Kurt Gödel, paper on the incompleteness theorems (1931)……Page 937
Gödel’s life and work……Page 938
Hilbert’s program, completeness, and incompleteness……Page 940
An outline of Gödel’s results……Page 941
Importance and impact of the incompleteness theorems……Page 943
Bibliography 71……Page 945
The industrial problem of the statistical control of quality……Page 946
An approach via statistical physics……Page 947
The dialogue between mathematics and industrial practice……Page 949
Presentation of the book……Page 950
Allowable variability in quality……Page 952
Bibliography 72……Page 954
The origins of Volterra’s interest in biomathematics……Page 956
The premises and genesis of the book: research in the 1920s……Page 957
The writing and contents of the book……Page 959
The book’s reception and its influence on biomathematical research……Page 962
Bibliography 73……Page 963
The development of the theory of the Fourier integral……Page 965
New kinds of integrals……Page 968
The author 74……Page 972
Bochner’s book 74……Page 973
The aftermath: abstract harmonic analysis……Page 977
Bibliography 74……Page 979
Introduction 75……Page 980
The background of the Grundbegriffe……Page 981
Axioms for finitary probability theory……Page 983
The application of probability……Page 984
Infinite fields of probability……Page 985
The impact of the Grundbegriffe……Page 987
Bibliography 75……Page 989
Algebraic topology prior to 1934……Page 990
Seifert and Threlfall, Lehrbuch der Topologie (1934)……Page 992
Alexandroff and Hopf, Topologie (1935)……Page 995
Reception of the two books……Page 997
Bibliography 76……Page 999
Background 77……Page 1001
Naïve proof theory……Page 1002
The first volume……Page 1006
The second volume……Page 1010
Philosophical and mathematical issues……Page 1014
Bibliography 77……Page 1018
List of Authors……Page 1020
Index……Page 1024
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