REA's Algebra and Trigonometry large Review
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Extra info for Algebra and Trigonometry Super Review (2nd Edition) (Super Reviews Study Guides)
His concept of manifold was formed exactly to transform imagery and metaphor into strictly mathematical concepts of a generalized geometric framework, thus liberating geometrical thought from the euclidean straightjacket. He introduced the concept of a metrical manifold (Mannigfaltigkeit mit Maflbestimmung) in the well-known way by the selection of a positive determinate quadratic differential form, ds 2 = ~ gijdzidzJ i,j (1 < i,j < n), which enabled him to transfer essential constructions of Gauss' theory of surfaces to the generalized geometry of manifolds.
Riemann Nachlass, 16, 40 r) This is clear testimony that Riemann did not show much interest in detailed studies of the logical foundations of geometry, precisely because he presumed them to be fruitless from the point of view of new theorems. This position is completely understandable from his point of view, but it cannot be upheld if one is familiar with the works of Bolyai and/or Lobachevskii. Their studies of absolute geometry and of horocycle geometry in the noneuclidean case [Gray 1979], to name just two examples, is too obviously incompatible with such a strict verdict of fruitlessness.
First Glances at Other Geometrical Structures The next point I want to discuss is the surprising fine and differentiated approach to geometric thinking that was opened up by Riemann on the basis of his manifold concept. This view on geometry was in line with the most far-reaching and deep-going changes of geometric thought during the turn towards "modern mathematics" of the late 19th and the early 20th century. These changes concern both semantics and the internal structure of geometry. From the point of view of semantics the most striking feature of 19th century development is the turn away of geometric theories from predominantly or even exclusive reference to physical space (even if perhaps understood in a philosophical a priori disguise).
Algebra and Trigonometry Super Review (2nd Edition) (Super Reviews Study Guides)