The Legacy of Mario Pieri in Foundations and Philosophy of Mathematics

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 < 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