Modern Geometry, Numbers and Abstraction

GIOVANNINI, EDUARDO N.

Buenos Aires

Workshop; International Workshop on Leibniz's Principle of Identity of Indiscernibles and its Repercussion in Physics and Mathematics; 2019

Universidad CAECE,

Abstraction principles play a prominent role in several attempts to provide a philosophical foundation for modern mathematics, such as Frege?s classical logicist program in the philosophy of arithmetic as well as the contemporary position known as mathematical abstractionism. Schematically, these principles are a special class of mathematical axioms which establish criteria of identity for abstracta based on an equivalence relation over a given domain of entities (objects, concepts, n-ary relations). The aim of this talk is to discuss the use of abstraction principles in a different mathematical context, namely in the central program in modern geometry which aimed at the elimination of numbers from the foundations of geometry. As is well known, this foundational program reached a high point in the axiomatization of Euclidean geometry presented by Hilbert in Foundations of geometry (1899). We will argue that the use of abstraction principles was a fruitful mathematical technique for the program which searched for strictly synthetic foundations of geometrical theories, since it allowed the rigorous introduction of fundamental geometrical concepts avoiding the appeal to numerical assumptions. To support this contention, we will analyze the role played by the method of abstraction in important foundational debates regarding the theories of proportion and plane area, in the context of modern axiomatic geometry.