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Definitions and Nondefinability in Geometry

Abstract Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid's, which handled some basic concepts awkwardly and imprecisely. They would introduce precision required for generalization and application to new, delicate problems in higher mathematics. Their work was controversial: they departed from tradition, criticized standards of rigor, and addressed fundamental questions in philosophy. This paper follows the problem, Which geometric concepts are most elementary? It describes a false start, some successful solutions, and an argument that one of those is optimal. It's about axioms, definitions, and definability, and emphasizes contributions of Mario Pieri (1860-1913) and Alfred Tarski (1901-1983). By following this thread of ideas and personalities to the present, the author hopes to kindle interest in a fascinating research area and an exciting era in the history of mathematics.