‘Axiomatics, Empiricism, and Anschauung in Hilbert’s Conception of Geometry: Between Arithmetic and General Relativity’

Abstract

Abstract To what extent the philosophy of mathematics of any individual mathematician is relevant to historically understanding his mathematical work, and to what extent his mathematical work has any bearing in understanding philosophical issues related with mathematics, are questions that have different meanings and have to be approached differently when they refer to different mathematicians. Take, for example, Descartes and Frege. These two thinkers can be considered philosophers in the strict sense of the word, with philosophical interests going well beyond the strict scope of mathematics, each of them in his own way. They devoted much of their time and efforts to develop coherent, well-elaborated philosophical systems, and their writings turned them into philosophers in the eyes of the philosophical community. Their philosophical systems are directly relevant to addressing central questions pertaining to the nature of mathematical knowledge, but they were not intended exclusively as answers to specific problems in the philosophy of mathematics. And besides their intense involvement with philosophical questions, both Descartes and Frege contributed positive mathematical results of various kinds, albeit of different overall impact on mathematics at large, and while working under quite different professional circumstances. A natural question that the historian may be easily led to ask in relation to these two thinkers concerns the mutual relationship between the philosophical systems they developed and the mathematics that each of them produced. One way to answer this question is by investigating, separately, the philosophy and the mathematics of each of them, and then trying to articulate the said relationship.