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2007 | 52 | 2 | 51-74

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Bernard Riemann's New Philosophy Project


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The paper makes researches of the turning point that was observed in science in the half of the 19th century and the significance of Riemann's philosophical visions, and his scientific achievements. In the 19th century the development of science faced critical situation, because a number of notions grew unreasonably, and understanding them to a high degree was based on intuition. On the other hand, there appeared new notions and mathematical structures that turned out to be very helpful in describing and explaining many mathematical issues and physical phenomena. First of all, one can number among them such notions as: class (especially infinite class), group, function and miscellany. For Riemann, a notion of miscellany was 'most favourable'. Having defined it, he tried to construct a notion of repeatedly extensive quantity with comprehensive quantity notions. It was to become a starting point for understanding relation between geometry and a notion of space. Miscellany also is to become a subject of philosophical studies - it would be a characteristic 'miscellany metaphysics'. In 1850 Riemann formulated theory, which became the central idea for his scientific searches. In his opinion, one should create a completely unsubsidized mathematical theory, which, having included fundamental laws, would describe sphere of existence filled with different influences, and would merge gravitational, electromagnetic and thermodynamical phenomena. A special role is ascribed to Riemann's Goettingen lecture (1854) in which he emphasised the flimsiness of the principles of geometry, and the need of specifying and consolidating them. In Riemann's view, a notion of space itself used in geometry seems to be unprecise. Also, status of the axioms as the first principles of geometry telling the truth of the reality (as it was widely held), and their interdependence, still remains abstruse. Riemann's task is to reveal all these hidden connexions among the axioms of geometry. In his opinion, all hitherto existing miscarriages were caused by the fact that even a comprehensive notion of repeatedly extensive quantity was not formulated. Thanks to terming such notion one could understand a notion of space (that is experienced by us), which, in fact, is a remarkable instance for thrice extensive quantity. Also, a comprehensive notion of quantity should be subjected to a thorough examination and improved. Philosophical considerations of Riemann are not only a common addition to his scientific work, but also are the significant part of his thought. Riemann is constructing the philosophy of world, which in holistic way would explain physical and spiritual influences, and would reveal their reasons. The main principle of this philosophical system is mathematics, and the keystone, filler and fulfilment – metaphysics. For the structure of theory is circular, mathematics and metaphysics are united peculiarly. Unfortunately, Riemann left his work uncompleted. He believed that its realization is possible, and it would be realized in the future thanks to understanding known laws of nature, and on the strength of laying down a law of interaction of heat, light, magnetism and electricity.










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  • W. Wojcik, Instytut Historii Nauki PAN, ul. Nowy Swiat 72, 00-330 Warszawa, Poland


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