09082nam 2200517 450 991048418560332120230421142806.00-8176-4823-2(CKB)4100000011867314(MiAaPQ)EBC6533829(Au-PeEL)EBL6533829(OCoLC)1246576292(PPN)255291388(EXLCZ)99410000001186731420211019d2021 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierThe legacy of Mario Pieri in foundations and philosophy of mathematics /Elena Anne Corie Marchisotto, Francisco Rodríguez-Consuegra, James T. SmithNew York, New York :Birkhäuser,[2021]©20211 online resource (xxii, 603 pages) illustrations0-8176-4822-4 Intro -- Foreword -- Preface -- Style and Translation -- Evolution of the Project and Acknowledgments -- Contents -- Illustrations -- 1 Pieri's Contributions to Foundations and Philosophy of Mathematics -- 1.1 Pieri, the Man, the Scholar, the Teacher -- 1.2 Philosophy of Mathematics and Mathematical Logic -- 1.3 Foundations of Geometry -- 2 Pieri's Philosophy of Deductive Sciences -- 2.1 Primitive Concepts -- 2.2 Definitions -- 2.3 Definitions by Abstraction -- 2.4 Postulates, or Primitive Propositions -- 2.5 Proofs -- 2.6 Abstract Deductive Science -- 2.7 Logic and Mathematics -- 2.8 Pieri's Letter to Russell -- 2.9 Metamathematics -- 2.10 Semantics and Model Theory -- 2.11 Nominalism -- 3 Two Paths to Logical Consequence: Pieri and the Peano School -- 3.1 Tarski's Definition of Consequence -- 3.2 Aristotle's Counterexample Method -- 3.3 Independence of the Parallel Postulate -- 3.4 Logical Consequence in a Model-Theoretic Context: The Peano School -- 3.4.1 Peano -- 3.4.2 Pieri -- 3.4.3 Padoa -- 4 Pieri's 1900 Paris Paper -- ON GEOMETRY ENVISAGED AS A PURELY LOGICAL SYSTEM -- I -- II -- III -- IV -- V -- VI -- VII -- 5 Pieri and Projective Geometry -- 5.1 Pieri's Studies, Research, and Teaching -- 5.2 Evolution of Projective Ideas and Methods -- 5.3 Synthetic Projective Geometry as an Autonomous Field -- 5.4 Geometry as a Logical System -- 5.5 The Transformational Approach -- 5.6 Multidimensional Projective Geometry -- 5.7 From Duality to Plurality -- 6 Pieri's 1898 Geometry of Position Memoir -- THE PRINCIPLES OF THE GEOMETRY OF POSITION -- LIST OF ABBREVIATIONS -- 1 The Primitive Entities -- POSTULATE I -- POSTULATE II -- POSTULATE III -- POSTULATES IV AND V -- POSTULATE VI -- POSTULATE VII -- POSTULATE VIII -- POSTULATE IX -- POSTULATE X -- 2 The Alignment Relation and the Projective Line -- POSTULATE XI.3 The Visual of a Form and Projective Planes -- POSTULATE XII -- 4 The Plane Quadrangle and the Harmonic Relation -- POSTULATE XIII. -- POSTULATE XIV -- 5 The Projective Segment -- POSTULATE XV -- POSTULATE XVI -- POSTULATE XVII -- 6 Further Properties of Segments -- 7 Natural Orderings and Senses of a Projective Line -- 8 The Projective Triangle -- 9 Segmental Transformations -- POSTULATE XVIII -- 10 Harmonic Correspondences and STAUDT's Theorem -- 11 Projective Hyperplanes of the Third Species and Ordinary Space -- POSTULATE XIX -- 12 Projective Hyperplanes of the nth Species and Absolute Projective Space -- POSTULATE XIX' -- POSTULATE XX' -- APPENDIX -- 7 Transformational Geometry -- 7.1 Motions and Transformations -- 7.2 Isometries and Similarities -- 7.3 Transformations as Tools -- 7.4 Transformations in Foundational Studies -- 7.5 Postlude -- 8 Pieri's 1900 Point and Motion Memoir -- ON ELEMENTARY GEOMETRY -- LIST OF ABBREVIATIONS -- 1 Generalities about point and about motion. The relation of collinearity among points. Line, plane, and sphere are introduced. -- POSTULATE I -- POSTULATES II and III -- POSTULATE IV -- POSTULATE V -- POSTULATE VI -- POSTULATE VII -- POSTULATE VIII -- POSTULATE IX -- 2 Rotating a line onto itself. Midpoint of a pair of points. Rotating a plane onto itself. Orthogonality relation among three points or between two intersecting lines -- POSTULATE X -- POSTULATE XI -- POSTULATE XII -- POSTULATE XIII -- POSTULATE XIV -- 3 Rotating one plane onto another. Orthogonality of lines and planes. Various properties relating to lines, planes, and spheres. -- POSTULATE XV -- POSTULATE XVI -- 4 Points internal or external to a sphere. Segments, rays, half-planes, angles, and so on. -- POSTULATE XVII -- POSTULATE XVIII -- POSTULATE XIX.5 Relation less than or greater than between two segments or between two angles. Triangle is introduced. Congruence of triangles and other propositions of the first and third books of Euclid. -- 6 Sum of two segments. Other properties of triangles, circles, and so on. Continuity of a line. -- POSTULATE XX -- 9 Pieri's Works on Foundations and Philosophy of Mathematics -- 9.1 Course Materials and a Translation -- 9.1.1 Higher Geometry Lectures by Riccardo De Paolis -- 1882-1883 -- 1883-1884 -- 9.1.2 Geometry of Position by G. K. C. von Staudt -- Pieri's Treatment of the Fundamental Theorem -- Pieri's Handwritten Notations -- 9.1.3 Projective Geometry: Lectures at the Military Academy -- 9.1.4 Course Records from Catania University Archives -- 9.1.5 Projective Geometry: Lectures at Parma -- Pieri 1910 -- Pieri 1911c -- 9.1.6 Descriptive Geometry: Lectures at Parma -- 9.2 Foundations of Projective Geometry -- 9.2.1 Principles That Support the Geometry of Position -- 9.2.2 Postulates for Abstract Projective Geometry of Hyperspaces -- 9.2.3 Primitive Entities of Abstract Projective Geometry -- 9.2.4 Intermezzo (1897b) -- 9.2.5 Principles of the Geometry of Position Composed into a Deductive Logical System -- 9.2.6 New Method for Developing Projective Geometry Deductively -- 9.2.7 Principles That Support the Geometry of Lines -- 9.2.8 Staudt's Fundamental Theorem and the Principles of Projective Geometry -- Introduction (1) -- Proof of the Fundamental Theorem (2) -- Results Not Dependent on Continuity Principles (3-6) -- Continuity and Archimedean Principles -- Pieri's Archimedean Postulate XVIII' (1, 7-9) -- Pieri's Postulate XVIII'' (9) -- Real Projective Geometry of Degree &lt -- 2 (10-11) -- Conclusion -- 9.2.9 New Principles of Complex Projective Geometry -- Background -- Axiomatic Framework -- Incidence Postulates and Dependence (1).Chains (2-4) -- Coordinates and Cross Ratios (7) -- Further Topics, from 4-7 and 1906a -- Conclusion -- 9.2.10 On the Staudtian Definition of Homography -- 9.3 Foundations of Elementary and Inversive Geometry -- 9.3.1 On Elementary Geometry as a Hypothetical Deductive System: Monograph on Point and Motion -- 9.3.2 Elementary Geometry Based on the Notions of Point and Sphere -- 9.3.3 New Principles of the Geometry of Inversions -- 9.4 Arithmetic, Logic, and Philosophy of Science -- 9.4.1 Geometry Envisioned as a Purely Logical System -- 9.4.2 On an Arithmetical Definition of the Irrationals -- 9.4.3 A Look at the New Logico-Mathematical Direction of the Deductive Sciences -- 9.4.4 On the Consistency of the Axioms of Arithmetic -- 9.4.5 On the Axioms of Arithmetic -- 10 Central Themes and Impact of Pieri's Work -- 10.1 Philosophical Themes in Pieri's Research -- 10.2 Themes in Foundations of Geometry -- Pieri's Views on Abstract Mathematics -- 10.2.1 Geometry as an Abstract Science -- 10.2.2 Geometry from a Synthetic Perspective -- 10.2.3 Geometry from a Transformational Point of View -- 10.2.4 Geometries Constructed as Autonomous Disciplines -- 10.2.5 Continuity and Archimedean Principles -- 10.2.6 Minimizing the Number of Primitive Notions -- 10.3 Pedagogical Themes -- Projective Geometry -- Inversive and Elementary Geometry -- 10.4 Pieri's Impact -- 10.4.1 Philosophy -- 10.4.2 Foundations of Geometry -- Projective Geometry -- Inversive Geometry -- Elementary Geometry -- 10.4.3 Pedagogy -- 10.5 Opportunities for Future Research -- Foundations of Geometry -- Pedagogy -- Logic and Philosophy -- Appendix -- A.1 Errata and Addenda for Marchisotto and Smith 2007 -- A.1.1 Errors and Corrections -- A.1.2 New Items for Chapter 6: Pieri's Works -- A.2 Two Letters from Louis Couturat -- A.3 Russell's Annotations on Principles of the Geometry of Position.A.4 Pieri's 1905b letter to Oswald Veblen -- Bibliography -- Permissions and Credits -- Index of Persons -- Index of Subjects.MathematicsHistoryHistòria de la matemàticathubLlibres electrònicsthubMathematicsHistory.Història de la matemàtica510.9Marchisotto Elena252366Rodríguez-Consuegra Francisco A.1951-Smith James T.MiAaPQMiAaPQMiAaPQBOOK9910484185603321The legacy of Mario Pieri in foundations and philosophy of mathematics2830640UNINA