Re-coding as the First Pattern of Change in Mathematics
The roles of geometry and of arithmetic in contemporary philosophy of mathematics are rather asymmetric. While arithmetic plays a central role in foundational approaches and therefore its logical structure is thoroughly studied and well understood (see Shapiro 2005), geometry is the central topic of the antifoundational approaches (see Boi, Flament, and Salanskis 1992). This of course does not mean that there are no foundational studies of geometry; it is sufficient to mention the work of Alfred Tarski (see Tarski 1948 and 1959). Nevertheless, in these cases geometry is just another illustration of the methods developed for the analysis of arithmetic. The visual aspect of geometry, the very fact that geometry has something to do with space and spatial intuition, is totally ignored in these studies. These studies are just exceptions, and they do not change the basic difference that the philosophy of arithmetic is dominated by the foundational approach, while the philosophy of geometry is mainly antifoundational.
KeywordsExpressive Power Analytic Geometry Integrative Power Integral Calculus Symbolic Language
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