Abstract

The central claim of this thesis is that geometry is a quasi-empirical science based on the idealisation of the elementary physical operations that we actually perform with pen and paper. This conclusion is arrived at after searching for a theory of geometry that will not only explain the epistemology and ontology of mathematics, but will also fit with the best practices of working mathematicians and, more importantly, explain why geometry gives us knowledge that is relevant to physical reality. We will be considering all the major schools of thought in the philosophy of mathematics. Firstly, from the epistemological side, we will consider apriorism, empiricism and quasi-empiricism, finding a Kitcherian style of quasi-empiricism to be the most attractive. Then, from the ontological side, we will consider Platonism, formalism, Kitcherian ontology, and fictionalism. Our conclusion will be to take a Kitcherian epistemology and a fictionalist ontology. This will give us a kind of quasiempirical-fictionalist approach to mathematics. The key feature of Kitcher's thesis is that he placed importance on the operations rather than the entities of arithmetic. However, because he only dealt with arithmetic, we are left with the task of developing a theory of geometry along Kitcherian lines. I will present a theory of geometry that parallels Kitcher's theory of arithmetic using the drawing of straight lines as the most primitive operation. We will thereby develop a theory of geometry that is founded upon our operations of drawing lines. Because this theory is based on our line drawing operations carried out in physical reality, and is the idealisation of those activities, we will have a connection between mathematical geometry and physical reality that explains the predictive power of geometry in the real world. Where Kitcher uses the Peano postulates to develop his theory of arithmetic, I will use the postulates of projective geometry to form the foundations of operational geometry. The reason for choosing projective geometry is due to the fact that by taking it as the foundation, we may apply Klein's Erlanger programme and build a theory of geometry that encompasses Euclidean, hyperbolic and elliptic geometries. The final question we will consider is the problem of conventionalism. We will discover that investigations into conventionalism give us further reason to accept the Kitcherian quasi-empirical-fictionalist approach as the most appealing philosophy of geometry available.