Transcendental Curves in the Leibnizian Calculus
Seiten
2017
Academic Press Inc (Verlag)
978-0-12-813237-1 (ISBN)
Academic Press Inc (Verlag)
978-0-12-813237-1 (ISBN)
Transcendental Curves in the Leibnizian Calculus analyzes a mathematical and philosophical conflict between classical and early modern mathematics. In the late 17th century, mathematics was at the brink of an identity crisis. For millennia, mathematical meaning and ontology had been anchored in geometrical constructions, as epitomized by Euclid's ruler and compass.
As late as 1637, Descartes had placed himself squarely in this tradition when he justified his new technique of identifying curves with equations by means of certain curve-tracing instruments, thereby bringing together the ancient constructive tradition and modern algebraic methods in a satisfying marriage. But rapid advances in the new fields of infinitesimal calculus and mathematical mechanics soon ruined his grand synthesis.
Descartes's scheme left out transcendental curves, i.e. curves with no polynomial equation, but in the course of these subsequent developments such curves emerged as indispensable. It was becoming harder and harder to juggle cutting-edge mathematics and ancient conceptions of its foundations at the same time, yet leading mathematicians, such as Leibniz felt compelled to do precisely this. The new mathematics fit more naturally an analytical conception of curves than a construction-based one, yet no one wanted to betray the latter, as this was seen as virtually tantamount to stop doing mathematics altogether. The credibility and authority of mathematics depended on it.
As late as 1637, Descartes had placed himself squarely in this tradition when he justified his new technique of identifying curves with equations by means of certain curve-tracing instruments, thereby bringing together the ancient constructive tradition and modern algebraic methods in a satisfying marriage. But rapid advances in the new fields of infinitesimal calculus and mathematical mechanics soon ruined his grand synthesis.
Descartes's scheme left out transcendental curves, i.e. curves with no polynomial equation, but in the course of these subsequent developments such curves emerged as indispensable. It was becoming harder and harder to juggle cutting-edge mathematics and ancient conceptions of its foundations at the same time, yet leading mathematicians, such as Leibniz felt compelled to do precisely this. The new mathematics fit more naturally an analytical conception of curves than a construction-based one, yet no one wanted to betray the latter, as this was seen as virtually tantamount to stop doing mathematics altogether. The credibility and authority of mathematics depended on it.
Viktor Blåsjö works at the Mathematical Institute of Utrecht University. He has published widely on the history of geometry and astronomy as well as historiography. He has published award-winning papers in the American Mathematical Monthly (2006 Lester R. Ford award) and in Historia Mathematica.
1. Preliminary matters2. Introduction3. The classical basis of 17th-century philosophy of mathematics4. Mathematical context5. Transcendental curves by curve tracing6. Transcendental curves analytically: exponentials and power series7. Transcendental curves by the reduction of quadratures8. Transcendental curves in physics9. A view from the 18th century10. Concluding overview
| Erscheinungsdatum | 28.04.2017 |
|---|---|
| Reihe/Serie | Studies in the History of Mathematical Inquiry |
| Verlagsort | San Diego |
| Sprache | englisch |
| Maße | 191 x 235 mm |
| Gewicht | 540 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| ISBN-10 | 0-12-813237-X / 012813237X |
| ISBN-13 | 978-0-12-813237-1 / 9780128132371 |
| Zustand | Neuware |
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