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Revision difference : point-free geometry |
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Point-free geometry is based on the idea that in geometry it is not necessary to assume as a primitive the notion of point. Instead, we can start from the notion of a region in the space and the points are defined by suitable \textit{abstraction processes}, i.e. order-reversing sequences of regions. Firstly, such a question was analized by A. N. Whitehead in the books "An Inquiry Concerning the Principles of Natural Knowledge" and "The concept of Nature" in which the inclusion relation is a primitive. Successively in "Process and Reality" Whitehead proposed a new approach in which the connection relation is considered. Whitehead'analysis is philosophic in nature. Successively several autors translated this analysis into systems of axioms for a mathematical treatment of point-free geometry (see [1]). A very interesting proposal for a point-free approach to geometry is exposed in [2].
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Point-free geometry is based on the idea that in geometry it is not necessary to assume as a primitive the notion of point. Instead, we can start from the notion of a region in the space and the points are defined by suitable \textit{abstraction processes}, i.e. order-reversing sequences of regions. Firstly, such a question was analized by A. N. Whitehead in the books "An Inquiry Concerning the Principles of Natural Knowledge" and "The concept of Nature" in which the inclusion relation is a primitive. Successively in "Process and Reality" Whitehead proposed a new approach in which the connection relation is considered. |
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| \textbf{References} |
\textbf{References} |
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1. G. Gerla, Pointless geometries, in Handbook of Incidence Geometry, F. Buekenhout and W. Kantor (eds) 1994 North-Holland.
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1. A. N. Whitehead, An Inquiry Concerning the Principles of Natural Knowledge, Cambr. Univ. Press, Cambrige 1919.
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2. H. J. Schmidt, Axiomatic Characterization of Physical Geometry, Lecture Notes in Physics, Springer-Verlag, Berlin Heidelberg 1979.
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2. A. N. Whitehead, The concept of Nature, Cambr. Univ. Press, Cambrige 1920.
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| 3. A. N. Whitehead, An Inquiry Concerning the Principles of Natural Knowledge, Cambr. Univ. Press, Cambrige 1919. |
3. A. N. Whitehead, Process and Reality, The Macmillan Co., New York 1929. |
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| 4. A. N. Whitehead, The concept of Nature, Cambr. Univ. Press, Cambrige 1920. |
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| 5. A. N. Whitehead, Process and Reality, The Macmillan Co., New York 1929. |
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