Pluralism in Geometry

This article is concerned with the topics stemming from the problems with interpreting the devolopment of geometry in the nineteenth and early twentieth century. The emerging of hyperbolic and eliptic systems as alternatives to the classical Euclid’s one posed a serious challenge to the opinions about geometry of that time, particularly to the stances of Kant, who ascribed to the whole body of mathematics the status of synthetic apriori. I want to discuss to what degree the Kantian position may be defended and what should be changed in it. The relation with the development of modern logic is also hinted, particularly when dealing with the debate between Frege and Hilbert. Thus the problems of philosophy of geometry flow into broader issues about logic and mathematic knowledge and ultimately about the possibility and character of a priori knowledge in general.