From Kant to Hilbert

William Bragg Ewald9780198505365, 0-19-850536-1

Immanuel Kant’s Critique of Pure Reason is widely taken to be the starting point of the modern period of mathematics while David Hilbert was the last great mainstream mathematician to pursue importatn nineteenth century ideas. This two-volume work provides an overview of this important era of mathematical research through a carefully chosen selection of articles. They provide an insight into the foundations of each of the main branches of mathematics – algebra, geometry, number theory, analysis, logic, and set theory – with narratives to show how they are linked. Classic works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare are reproduced in reliable translations and many selections from writers such as Gauss, Cantor, Kronecher, and Zermelo are here translated for the first time. The collection is an invaluable source for anyone wishing to gain an understanding of the foundation of modern mathematics.

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Table of contents :
Contents……Page 13
16. GEORG FRIEDRICH BERNHARD RIEMANN (1826–1866)……Page 18
A. On the hypotheses which lie at the foundation of geometry……Page 21
17. HERMANN VON HELMHOLTZ (1821–1894)……Page 31
A. The origin and meaning of geometrical axioms……Page 32
B. The facts in perception……Page 58
C. Numbering and measuring from an epistemological viewpoint……Page 96
18. JULIUS WILHELM RICHARD DEDEKIND (1831–1916)……Page 122
A. On the introduction of new functions in mathematics……Page 123
B. From the Tenth Supplement to Dirichlet’s Lectures on the theory of numbers……Page 131
C. Continuity and irrational numbers……Page 134
D. From On the theory of algebraic integers……Page 148
E. Was sind und was sollen die Zahlen?……Page 156
F. From the Eleventh Supplement to Dirichlet’s Lectures on the theory of numbers……Page 202
G. Letter to Heinrich Weber (24 January 1888)……Page 203
I. From the Nachlass……Page 205
19. GEORG CANTOR (1845–1918)……Page 207
A. On a property of the set of real algebraic numbers……Page 208
B. The early correspondence between Cantor and Dedekind……Page 212
C. Foundations of a general theory of manifolds: a mathematico-philosophical investigation into the theory of the infinite……Page 247
D. On an elementary question in the theory of manifolds……Page 289
E. Cantor’s late correspondence with Dedekind and Hilbert……Page 292
20. LEOPOLD KRONECKER (1823–1891)……Page 310
A. Hilbert and Kronecker……Page 311
B. Extract from Hilbert’s Göttingen lectures……Page 312
C. Two footnotes……Page 315
D. On the concept of number……Page 316
21. CHRISTIAN FELIX KLEIN (1849–1925)……Page 325
A. Klein on the schools of mathematics……Page 326
B. On the mathematical character of space-intuition and the relation of pure mathematics to the applied sciences……Page 327
C. The arithmetizing of mathematics……Page 334
A. On the nature of mathematical reasoning……Page 341
B. On the foundations of geometry……Page 351
C. Intuition and logic in mathematics……Page 381
D. Mathematics and logic: I……Page 390
E. Mathematics and logic: II……Page 407
F. Mathematics and logic: III……Page 421
G. On transfinite numbers……Page 440
23. THE FRENCH ANALYSTS……Page 444
A. Some remarks on the principles of the theory of sets……Page 445
B. Five letters on set theory……Page 446
24. DAVID HILBERT (1862–1943)……Page 456
A. On the concept of number……Page 458
B. From Mathematical problems……Page 465
C. Axiomatic thought……Page 474
D. The new grounding of mathematics First report……Page 484
E. The logical foundations of mathematics……Page 503
F. The grounding of elementary number theory……Page 517
G. Logic and the knowledge of nature……Page 526
25. LUITZEN EGBERTUS JEAN BROUWER (1881–1966)……Page 535
A. Mathematics, science, and language……Page 539
B. The structure of the continuum……Page 555
C. Historical background, principles, and methods of intuitionism……Page 566
26. ERNST ZERMELO (1871–1953)……Page 577
A. On boundary numbers and domains of sets: new investigations in the foundations of set theory……Page 588
A. Sir George Stokes and the concept of uniform convergence……Page 603
B. Mathematical proof……Page 612
28. NICOLAUS BOURBAKI……Page 633
A. The architecture of mathematics……Page 634
Bibliography……Page 646
B……Page 700
C……Page 701
F……Page 702
H……Page 703
K……Page 704
M……Page 705
P……Page 706
S……Page 707
W……Page 708
Z……Page 709