From Kant to Hilbert

William Bragg Ewald9780198505358, 0-19-850535-3

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 important 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, Kronecker, 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 10
Copyright Permissions……Page 18
Introduction……Page 20
1. GEORGE BERKELEY (1685–1753)……Page 30
A. From the Philosophical commentaries……Page 32
B. Of infinites……Page 35
C. Letter to Samuel Molyneux……Page 38
D. From A treatise concerning the principles of human knowledge, Part One……Page 40
E. De Motu……Page 56
F. From Alciphron……Page 73
G. From Newton’s Principia mathematica……Page 77
H. The analyst……Page 79
2. COLIN MACLAURIN (1698–1746)……Page 112
A. From A treatise of fluxions……Page 114
A. Differential……Page 142
B. Infinite……Page 147
C. Limit……Page 149
4. IMMANUEL KANT (1724–1804)……Page 151
A. From Thoughts on the true estimation of active forces……Page 152
B. From the Transcendental aesthetic……Page 154
C. From the Discipline of pure reason……Page 155
D. Frege on Kant……Page 167
5. JOHANN HEINRICH LAMBERT (1728–1777)……Page 171
A. From the Theory of parallel lines……Page 177
6. BERNARD BOLZANO (1781–1848)……Page 187
A. Preface to Considerations on some objects of elementary geometry……Page 191
B. Contributions to a better-grounded presentation of mathematics……Page 193
C. Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation……Page 244
D. From Paradoxes of the infinite……Page 268
A. On the metaphysics of mathematics……Page 312
B. Gauss on non-Euclidean geometry……Page 315
C. Notice on the theory of biquadratic residues……Page 325
8. DUNCAN GREGORY (1813–1844)……Page 333
A. On the real nature of symbolical algebra……Page 342
9. AUGUSTUS DE MORGAN (1806–1871)……Page 350
A. On the foundation of algebra……Page 355
B. Trigonometry and double algebra……Page 368
10. WILLIAM ROWAN HAMILTON (1805–1865)……Page 381
A. From the Theory of conjugate functions, or algebraic couples; with a preliminary and elementary essay on algebra as the science of pure time……Page 388
B. Preface to the Lectures on quaternions……Page 394
C. From the Correspondence of Hamilton with De Morgan……Page 444
11. GEORGE BOOLE (1815–1864)……Page 461
A. The mathematical analysis of logic, being an essay towards a calculus of deductive reasoning……Page 470
12. JAMES JOSEPH SYLVESTER (1814–1897)……Page 529
A. Presidential address to Section ‘A’ of the British Association……Page 530
A. On the space theory of matter……Page 542
B. On the aims and instruments of scientific thought……Page 543
A. Presidential address to the British Association, September 1883……Page 561
15. CHARLES SANDERS PEIRCE (1839–1914)……Page 593
A. From Linear associative algebra……Page 603
B. Notes on Benjamin Peirce’s linear associative algebra……Page 613
C. On the logic of number……Page 615
D. On the algebra of logic: a contribution to the philosophy of notation……Page 627
E. The logic of mathematics in relation to education……Page 651
F. From The simplest mathematics……Page 656
References……Page 668
B……Page 690
F……Page 691
J……Page 692
N……Page 693
S……Page 694
Z……Page 695