Document (#42902)

Author

Felgner, U.

Title

Hilberts "Grundlagen der Geometrie" und ihre Stellung in der Geschichte der Grundlagendiskussion

Source

Jahresbericht der Deutschen Mathematiker-Vereinigung. 115(2014) H.3/4, S.185-206

Year

2014

Abstract

In seinen "Grundlagen der Geometrie" hat Hilbert die Bemühungen um eine Grundlegung der Geometrie zu einem überzeugenden Abschluss gebracht. Um diese Leistung würdigen zu können, müssen wir auf die lange Geschichte der Grundlagendiskussion - von der Antike bis zur Gegenwart - jedenfalls in großen Zügen eingehen. Wir werden insbesondere über das Problem, wie die geometrischen Grundbegriffe einzuführen sind (Euklid, Heron, Descartes, Pascal, Hobbes, Tschirnhaus et al.), und über die verschiedenen Entwürfe einer axiomatischen Grundlegung (Aristoteles, Euklid, Tschirnhaus etal.) berichten und danach das von Hilbert aufgestellte Axiomensystem besprechen. Ähnlich wie Dedekind 1888 den Bereich der natürlichen Zahlen (bis auf Isomorphie) als minimales Modell eines bestimmten Axiomensystems charakterisieren konnte, gelang es Hilbert, die euklidische Geometrie als maximales Modell seines Axiomensystems zu charakterisieren.

Field

Mathematik

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