Relativizations as the Second Pattern of Change in Mathematics
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We based our description of re-codings in the history of mathematics on Frege’ interpretation of the development of arithmetic as a gradual growth of the generality of its language. Frege identified as the fundamental events in the history of mathematics the invention of the constant symbols in arithmetic, the introduction of the variable in algebra, the introduction of symbols designating functions in mathematical analysis, and finally the introduction of symbols for functions of higher orders in logic. When we complemented this interpretation of the development of the symbolic language of mathematics by an analogous interpretation of the development of the language of geometry, we obtained the first pattern of change in the development of mathematics. Nevertheless, it is important to realize, that this union of arithmetic with geometry contradicts Frege’ original intention. In arithmetic Frege endorsed the logicist view, according to which the propositions of arithmetic are analytic and can be derived from logic. On the other hand, in the field of geometry he accepted Kant’ philosophy according to which the propositions of geometry are synthetic and are based on intuition (see Frege 1884, pp. 101–102).
KeywordsEuclidean Geometry Projective Geometry Euclidean Plane Coordinative Form Klein Bottle
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