The Geometry of Thought: Circling Through Concepts

Abstract

The goal of this paper is to shed light on the nature of mathematical practice, i.e., on “doing mathematics”. It explores Gödel’s perspective, which offers an approach to understanding mathematics centered on concepts, objects, and structures. The paper has two parts. In the first part, we situate Gödel’s reflections against the backdrop of formalism and Platonism. In the second part, we present the view shaped by Gödel’s ideas that resonates with contemporary discussions in the philosophy of mathematical practice, particularly in its attention to abstraction, generalization, and conceptual discovery, as essential components of mathematical reasoning. We illustrate this view through concrete examples from category theory and geometry. This approach reveals that mathematical practice, far from being merely formal, is a dynamic interplay of intuition, abstraction, structural, and conceptual reasoning. Such a focus underscores the need for developing the theory of concepts along the lines proposed by Gödel to provide a more natural framework for thinking about mathematics.