Catálogo de publicaciones - libros

Mario Pieri was “a true bridge between the two most prestigious Italian schools of mathematics of [his] epoch: that of logic and that of algebraic geometry.” Yet his works are not as well known to today’s scholars as they should be. Pieri left a legacy of results in algebraic and differential geometry, vector analysis, foundations of mathematics, logic, and philosophy of science that are worth knowing, not just for their historical value, but as well for their mathematical and philosophical import.

Three schools of mathematical research flourished simultaneously at the University of Turin in the 1890s. The group gathered around Corrado Segre was a focal point for scholars of algebraic geometry throughout Europe. The Peano school made important contributions to analysis, logic, foundations, linguistics, and teaching. Vito Volterra and his colleagues in mathematical physics explored the dependence of the calculus of variations on functional analysis, and practical applications of integral and integro-differential equations. Pieri was active in the Segre and Peano schools; his work has been characterized as exemplifying the themes and research goals of both. This chapter provides a summary of Pieri’s results in foundations of geometry, in the context of the Peano school. His association with Segre’s group will be discussed in the third book of the present series.

This chapter contains an English translation of Pieri’s 1908a memoir, . The work had two main goals. First, it presented elementary Euclidean geometry as a hypothetical-deductive system, and showed that all its notions and postulates can be defined and formulated in terms of the notion and the relation that holds between points just when are from . As noted in section 5.2, this result gave rise, over decades, to a stream of related research that still continues. The paper’s title reflects Pieri’s extensive use of elementary set theory in developing geometry from his postulates: he defined the sphere through centered at as the set of all points such that and are equidistant from . Pieri’s second aim was to foster more extensive use of properties of spheres in presenting elementary geometry, even in school courses. In this regard, he seems to have had less impact, even though this memoir presents many useful examples. A third aim, which Pieri had already pursued for a decade, was to promote the use of transformations in elementary geometry. Pieri introduced various geometric transformations early through definitions, and employed them extensively throughout the paper, following paths already explored in his 1900a memoir. Finally, Pieri followed the strategy of in developing plane and solid geometry together.

Although primarily a researcher in geometry, Mario Pieri wrote four papers that addressed aspects of arithmetic: 1906d 1906e (completed in 1905) 1906g 1907a The first three are introduced in the following paragraphs, and will be discussed in detail in the second book of the present series. Section 4.1 presents historical background for the 1906e, 1906g, and 1907a papers. The last, , is translated in entirety in section 4.2 and discussed in detail in section 4.3.

This book has presented an overview of Mario Pieri’s life and research and a deeper study of the background of his work in foundations of geometry and arithmetic. With complete translations and detailed analyses for two of his notable papers—one in each of those areas—this book has paid particular attention to Pieri’s constructions of Euclidean geometry and of the natural number system. The second book of the present series will focus on Pieri’s research in foundations of absolute and projective geometry, and expose his views on logic and the philosophy of science. The third and final book will describe his contributions to algebraic geometry and related subjects.

This chapter catalogues all known works of Mario Pieri, published and unpublished, according to topic and type. In annotations it describes some of them in detail. The others will be annotated in subsequent books of this series. Sections 6.1 to 6.7 list Pieri’s publications in the following categories: Some of these are divided further. For example, section 6.7 has subsections for Pieri’s edited translations, reviews, lecture notes, and collected works, and one for memorial articles about Pieri. Within each section or subsection complete bibliographic citations of Pieri’s works are listed in order of the year of publication, and alphabetically within each year. Notations indicate when a work was completed in a year before the listed publication date, or when a publication spans more than one year. The citations were copied from the present book’s bibliography; their format details and conventions are described in its introductory paragraphs. The citations in the present chapter also include translations of the titles.