The Hilbert Challenge

by Gray, Jeremy J.

• ISBN13: 9780198506515
• ISBN10: 0198506511
• Format: Hardcover
• Copyright: 2001-01-25
• Publisher: Oxford University Press

Summary

Few problems in mathematics have had the status of those posed by David Hilbert in 1900. Mathematicians have made their reputations by solving some of them like Fermat's last theorem, but several remain unsolved including the Riemann Hypotheses, which has eluded all the great minds of this century. A hundred years later, this book takes a fresh look at the problems, the man who set them, and the reasons for their lasting impact on the mathematics of the twentieth century. In this fascinating book, the authors consider what makes this the pre-eminent collection of problems in mathematics, what they tell us about what drives mathematicians, and the nature of reputation, influence and power in the world of modern mathematics. It is written in a clear and entertaining style and will appeal to anyone with interest in mathematics or those mathematicians willing to try their hand at these problems.

Author Biography

Jeremy Gray is Senior Lecturer in Mathematics at the Open University. His research interests lie in the history of the mathematics of the 19th and 20th centuries, and in the philosophy and social significance of mathematics. He is the author or editor of nine books, most recently The Symbolic Universe: Geometry and Physics 1890-1930 (OUP 1999).

Table of Contents

The future unveiled	1	(13)
The shaping of a pioneer	14	(46)
Hilbert's student years	14	(4)
After Hilbert's graduation	18	(5)
Hilbert's breakthrough in invariant theory	23	(10)
How Hilbert came to Gottingen	33	(2)
Hilbert and number theory	35	(13)
Hilbert and geometry	48	(8)
The road to Paris	56	(4)
The beacons are lit	60	(29)
The background: mathematics in 1900	74	(4)
Roads not taken	78	(2)
Poincare's address of 1897	80	(3)
Poincare and electromagnetic theory	83	(1)
Poincare's address of 1908	84	(5)
The early response: 1900-1914/18	89	(66)
Hilbert and the viability of the Hilbert Problems	89	(6)
Hilbert's Problems on the foundations of mathematics	95	(7)
Hilbert among the philosophers	102	(2)
Axiomatising geometry in USA, responses to Grundlagen der Geometrie	104	(5)
Hilbert and Brouwer on Hilbert's Problem 5	109	(4)
Hilbert Problems on pure and applied mathematics	113	(27)
Hilbert Problems on number theory	140	(1)
Problems 9, 11, and 12--The comedy of errors	141	(3)
Hilbert's geometric Problems	144	(1)
Ragsdale and others on Hilbert's 16th Problem	145	(6)
Beiberbach on Hilbert's 18th Problem	151	(4)
Between the wars: foundations examined	155	(33)
Gottingen after the First World War; the eclipse expedition	155	(7)
Hilbert, Brouwer and mathematical logic	162	(6)
Mathematical logic after Godel	168	(5)
Soviet Union in the 1930s, Egorov	173	(1)
Gelfond's work on Hilbert Problem 7	174	(1)
Pontrjagin on Problem 5	175	(3)
The 15th Problem, on the Schubert calculus	178	(1)
Hilbert's 16th Problem on differential equations and the work of Dulac	179	(1)
Plateau problem to Douglas	180	(4)
Bieberbach on Hilbert, 1930	184	(2)
1933-45	186	(2)
After 1945	188	(45)
The Princeton Conference on Problems of Mathematics	188	(2)
Pure or applied mathematics	190	(1)
Applied mathematics after the war	191	(3)
Pure mathematics after the war	194	(3)
Nicolas Bourbaki on the international stage	197	(7)
Alexandre Grothendieck	204	(4)
Matters of taste and judgement	208	(1)
The Soviet case	209	(5)
Hilbert's 17th Problem	214	(1)
Hilbert Problems at the International Congresses of Mathematicians	215	(3)
Work on mathematical logic	218	(15)
Epilogue	233	(54)
Hilbert's lecture at the International Congress of Mathematics, Paris 1900	238	(1)
Editor's notes	238	(2)
Hilbert's lecture	240	(43)
References to the lecture	283	(4)
Appendix 1: The Hilbert Problems 1900-99	287	(3)
Appendix 2: Logical matters	290	(4)
Glossary	294	(4)
General references	298	(12)
Index	310