Foundations of Geometry - Kostenloses Hörbuch
1/42Preface, Contents, and Introduction
Zusammenfassung
The Foundations of Geometry is a mathematical work by the German mathematician David Hilbert, first published in German in 1899 as Grundlagen der Geometrie and soon translated into English. It is one of the most influential mathematics books of its era, and it helped set the pattern for the modern axiomatic method.
Euclid's Elements had served for two thousand years as the model of rigorous reasoning, but nineteenth-century mathematicians had found gaps in it, places where Euclid relied on intuition or diagrams rather than stated assumptions. Hilbert sets out to close those gaps. He begins with three kinds of undefined objects, points, lines and planes, and organizes his axioms into groups dealing with connection or incidence, order, parallels, congruence and continuity. From these he develops the familiar results of plane geometry, including the theory of proportion, areas and constructions. Throughout he also asks which axioms are independent of the others and whether the system is consistent, building models from number systems to show what follows when particular axioms are dropped.
The book is short but demanding, written in compact definitions, theorems and proofs. Listeners with some background in mathematics or logic, and anyone curious about how modern mathematics approaches certainty, will find it a direct encounter with a turning point in the subject.
Über
The German mathematician David Hilbert was one of the most influential mathematicians of the 19th/early 20th century. Hilbert's 20 axioms were first proposed by him in 1899 in his book Grundlagen der Geometrie as the foundation for a modern treatment of Euclidean geometry.
Hilbert's axiom system is constructed with six primitive notions: the three primitive terms point, line, and plane, and the three primitive relations Betweenness (a ternary relation linking points), Lies on (or Containment, three binary relations between the primitive terms), and Congruence (two binary relations, one linking line segments and one linking angles).
The original monograph in German was based on Hilbert's own lectures and was organized by himself for a memorial address given in 1899. This was quickly followed by a French translation with changes made by Hilbert; an authorized English translation was made by E.J. Townsend in 1902. This translation - from which this audiobook has been read - already incorporated the changes made in the French translation and so is considered to be a translation of the 2nd edition.
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