By L. Boi, D. Flament, Jean-Michel Salanskis

Within the first half the nineteenth century geometry replaced substantially, and withina century it helped to revolutionizeboth arithmetic and physics. It additionally placed the epistemologyand the philosophy of technological know-how on a brand new footing. In thisvolume a legitimate assessment of this improvement is given byleading mathematicians, physicists, philosophers, andhistorians of technology. This interdisciplinary technique givesthis assortment a distinct personality. it may be used byscientists and scholars, however it additionally addresses a generalreadership.

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**Sample text**

Riemann's much more profound and difficult idea is the unaltered motion of infinitesimal line segments. Helrnholtz defended his view by arguing that although it restricted space to have constant curvature, it was precisely in such spaces that there could be a plausible mechanics. Indeed,'he was at pains to show that a non-Euclidean space could be experienced, and not merely imagined. Hou~l argued that "Apr~s avoir aquis l'idde de grandeur ou d'dtendue par la consid$ration du mouvement, . . nous constatons que certains corps .

In fact the sciences ought to organize themselves around central concepts (Hauptbegviffe) (Herbart 1807). Philosophy proper should then work out the connections and dissolve possible contradictions between the central concepts of the different scientific fields. By such a systematic but open communication between philosophical studies of science and philosophy proper, science and philosophy would be able to fulfil their Bildungsauflrag, their educational goal. ]. That seems to have given him a sort of mirror for the self-reflection on the task and method of mathematics.

He had used dissection of the surfaces into simply connected components, resulting in a complete classification of these by the number of boundary components and the order of connectivity ( Ordnung des Zusammenhangs) m. The latter had been introduced by him as alternating sum of the number e of cross cuts and the number f of simply connected components m = e - f = -X(F),[ x(F) the Euler characteristic of F]. ). iemann used another approach to the characterization of the the topology of surfaces, building on boundary relations of systems of closed curves inside the surface (Riemann 1857).