In this paper, we discuss a proposal for reform in the teaching of Euclidean geometry that reveals the symbiotic relationship between axiomatics and pedagogy. We examine the role of intuition in this kind of reform, as expressed by Mario Pieri, a prominent member of the Schools of Peano and Segre at the University of Turin. We are well aware of the centuries of attention paid to the notion of intuition by mathematicians, mathematics educators, philosophers, psychologists, historians, and others. To set (...) a context for Pieri’s proposal, we only seek to open a small window on views of the pedagogical role of intuition, from primary education to university study that may have informed early 20th century efforts to improve the teaching of geometry at the secondary school level. Pieri addressed the topic of intuition in many of his axiomatizations, including those in projective geometry which was his main area of concentration in foundations. We focus here primarily on his axiom systems for elementary geometry, which embraced the transformational approach of Felix Klein’s vision for the subject. Our goal is to convey Pieri’s thoughts on how to integrate two types of intuition, denoted as sensible and rational, in his endeavors to improve the teaching of the geometry of Euclid. We show how Pieri’s views on geometric intuition and pedagogical reform were either ignored or misrepresented in several notable publications at the turn of the 20th century. In particular, we give Pieri a voice in response to specific comments made in the early 1900s by Federigo Enriques, Ugo Amaldi, and Florian Cajori in widely circulated publications inspired by Klein. (shrink)

In 2000, a draft note of David Hilbert was found in his Nachlass concerning a 24th problem he had consider to include in the his famous problem list of the talk at the International Congress of Mathematicians in 1900 in Paris. This problem concerns simplicity of proofs. In this paper we review the traces of this problem which one can find in the work of Hilbert and his school, as well as modern research started on it after its publication. We (...) stress, in particular, the mathematical nature of the problem.1. (shrink)

Dependencies are identified in two recently proposed first-order axiom systems for plane hyperbolic geometry. Since the dependencies do not specifically concern hyperbolic geometry, our results yield two simpler axiom systems for absolute geometry.