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)

ArgumentUp until the French Revolution, European mathematics was an “aristocratic” activity, the intellectual pastime of a small circle of men who were convinced they were collaborating on a universal undertaking free of all space-time constraints, as they believed they were ideally in dialogue with the Greek founders and with mathematicians of all languages and eras. The nineteenth century saw its transformation into a “democratic” but also “patriotic” activity: the dominant tendency, as shown by recent research to analyze this transformation, seems (...) to be the national one, albeit accompanied by numerous analogies from the point of view of the processes of national evolution, possibly staggered in time. Nevertheless, the very homogeneity of the individual national processes leads us to view mathematics in the context of the national-universal tension that the spread of liberal democracy was subjected to over the past two centuries. In order to analyze national-universal tension in mathematics, viewed as an intellectual undertaking and a profession of the new bourgeois society, it is necessary to investigate whether the network of international communication survived the political, social, and cultural upheavals of the French Revolution and the European wars waged in the early nineteenth century, whether national passions have transformed this network, and if so, in what way. Luigi Cremona's international correspondence indicates that relationships among individuals have been restructured by the force of national membership, but that the universal nature of mathematics has actually been boosted by a vision shared by mathematicians from all countries concerning the role of their discipline in democratic and liberal society as the basis of scientific culture and technological innovation, as well as a basic component of public education. (shrink)