Judith V. Grabiner
Mathematical Association of America | 2010 |304 páginas | rar - pdf | 1,7 Mb
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Judith Grabiner, the author of A Historian Looks Back, has long been interested in investigating what mathematicians actually do, and how mathematics actually has developed. She addresses the results of her investigations not principally to other historians, but to mathematicians and teachers of mathematics. This book brings together much of what she has had to say to this audience.
The centerpiece of the book is The Calculus as Algebra: J.-L. Lagrange, 1736-1813. The book describes the achievements, setbacks, and influence of Lagrange s pioneering attempt to reduce the calculus to algebra. Nine additional articles round out the book describing the history of the derivative; the origin of delta-epsilon proofs; Descartes and problem solving; the contrast between the calculus of Newton and Maclaurin, and that of Lagrange; Maclaurin s way of doing mathematics and science and his surprisingly important influence; some widely held myths about the history of mathematics; Lagrange s attempt to prove Euclid s parallel postulate; and the central role that mathematics has played throughout the history of western civilization.
The development of mathematics cannot be programmed or predicted. Still, seeing how ideas have been formed over time and what the difficulties were can help teachers find new ways to explain mathematics. Appreciating its cultural background can humanize mathematics for students. And famous mathematicians struggles and successes should interest -- and perhaps inspire -- researchers. Readers will see not only what the mathematical past was like, but also how important parts of the mathematical present came to be.
Contents
Introduction .. . . xi
Part I. The Calculus as Algebra .. . . .1
Preface to the Garland Edition .. .3
Acknowledgement. . . .7
Introduction . . . . .9
1. The Development of Lagrange’s Ideas on the Calculus: 1754–1797 .. . 17
2. The Algebraic Background of the Theory of Analytic Functions . . . 37
3. The Contents of the Fonctions Analytiques . . . 63
4. From Proof-Technique to Definition: The Pre-History of Delta-Epsilon Methods . . 81
Conclusion . . . .101
Appendix . .103
Bibliography .. . .105
Part II. Selected Writings . .125
1. The Mathematician, the Historian, and the History of Mathematics . . . .127
2. Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus∗ . . . 135
3. The Changing Concept of Change: The Derivative from Fermat to Weierstrass† . . 147
4. The Centrality of Mathematics in the History of Western Thought† . . . 163
5. Descartes and Problem-Solving† . . . .175
6. The Calculus as Algebra, the Calculus as Geometry: Lagrange, Maclaurin, and Their Legacy . .191
7. Was Newton’s Calculus a Dead End? The Continental Influence of Maclaurin’s Treatise of Fluxions∗ . . 209
8. Newton, Maclaurin, and the Authority of Mathematics∗ .. .229
9. Why Should Historical Truth Matter to Mathematicians? Dispelling Myths while PromotinMaths . . .243
10. Why Did Lagrange “Prove” the Parallel Postulate?∗ . . 257
Index .. . 275
About the Author . . .287
