The legacy of Mario Pieri in foundations and philosophy of mathematics:
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
New York, NY
Birkhäuser
[2021]
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032690522&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | xxii, 603 Seiten Illustrationen |
| ISBN: | 9780817648220 |
Internformat
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Datensatz im Suchindex
| _version_ | 1819256982517841920 |
|---|---|
| adam_text | Contents Foreword...........................................................................................................................v Preface............................................ Illustrations............... ix xxi 1 Pieri’s Contributions to Foundationsand Philosophy of Mathematics ..............1 1.1 Pieri, the Man, the Scholar, the Teacher .................... ......... ........................ 1 1.2 Philosophy of Mathematics and Mathematical Logic ...................................... 2 1.3 Foundations of Geometry ................................................................................... 4 2 Pierl s Philosophy of Deductive Sciences............ .............................................. 6 2.1 Primitive Concepts...............................................................................................8 2.2 Definitions ...........................................................................................................8 2.3 Definitions by Abstraction...................................................................................9 2.4 Postulates, or Primitive Propositions...................................... 10 2.5 Proofs............................... 11 2.6 Abstract Deductive Science...............................................................................11 2.7 Logic and Mathematics .....................................................................................13 2.8 Pieri’s Letter to Russell.....................................................................................16 2.9
Metamathematics............................................................................................... 17 2.10 Semantics and Model Theory .......................................................................... 20 2.11 Nominalism .......................................................................................................22 3 Two 3.1 3.2 3.3 3.4 Paths to Logical Consequence: Pieri and the Peano School ..............25 Tarski’s Definition of Consequence ............................................................ 27 Aristotle’s Counterexample Method............................................................ 30 Independence of the Parallel Postulate.................................................... 32 Logical Consequence in a Model-Theoretic Context: The Peano School ...........................................................................................36 3.4.1 Peano .............................................................................................................36 3.4.2 Pieri .................................................................................................... 37 3.4.3 Padoa ............................................................................................................ 41 xvii
xviii Contents 4 Pierľs 1900 Paris Paper ......................................................................................... 46 4.1 §I..........................................................................................................................58 4.2 §II...................... 60 4.3 §III ......................................................................................................................62 4.4 §IV........................................................................................................................64 4.5 §V ........................................................................................................................66 4.6 §VI........................................................................................................................68 4.7 §ѴП......................................................................................................................72 5 Pieri and Projective Geometry........................................................................ 76 5.1 Pieri’s Studies, Research, and Teaching........................................................... 78 5.2 Evolution of Projective Ideas and Methods.......................................................82 5.3 Synthetic Projective Geometry as an Autonomous Field.................................95 5.4 Geometry as a Logical System......................................................................... 107 5.5 The Transformational Approach................................................................... Ill 5.6 Multidimensional Projective
Geometry......................................................... 120 5.7 From Duality to Plurality............................................................................... 127 6 Pieri’s 1898 Geometry of Position Memoir................................................. 136 6.1 §1 The Primitive Entities ........................................................................... 147 6.2 §2 The Alignment Relation and the Projective Line .............................. 153 6.3 §3 The Visual of a Form and Projective Planes ....................................... 156 6.4 §4 The Plane Quadrangle and the Harmonic Relationship.................... 163 6.5 §5 The Projective Segment ......................................................................... 168 6.6 §6 Further Properties of Segments............................................................. 175 6.7 §7 Natural Orderings and Senses of a ProjectiveLine ..............................181 6.8 §8 The Projective Triangle ....................................................................... 187 6.9 §9 Segmental Transformations................................................................. 194 6.10 §10 Harmonic Correspondences and Staudt’s Theorem .............................206 6.11 §11 Projective Hyperplanes of the Third Species and Ordinary Space ... 211 6.12 §12 Projective Hyperplanes of the nth Species and Absolute Projective Space....................................................................... 216 6.13 Appendix........
...................................................................................................220 7 Transformational Geometry ...............................................................................223 7.1 Motions and Transformations................................... ...................................223 7.2 Isometries and Similarities............................................................................. 226 7.3 Transformations as Tools ............................................................................... 229 7.4 Transformations in Foundational Studies ..................................................... 234 7.5 Postlude ........................................................................................................... 240
Contents xix 8 Pierľs 1900 Polnt-and-Motlon Memoir................................................................245 8.1 §1 Generalities about Point and Motion..................................................256 8.2 §2 Rotating a Line or Plane onto Itself; Midpoint·, Orthogonality..... 266 8.3 §3 Rotating One Plane onto Another; Properties of Lines, etc............ 274 8.4 §4 Points Internal or External to a Sphere; Segments, Rays, etc......... 282 8.5 §5 Relation Less or Greater between Segments or Angles; Triangles .. 292 8.6 §6 Sum of Two Segments; Continuity of a Line; Other Properties .... 300 9 Pierľs Works on Foundations and Philosophy of Mathematics ........... 307 9.1 Course Materials and a Translation........................................ 9.1.1 9.1.2 9.1.3 9.1.4 9.1.5 9.1.6 Higher Geometry Lectures by Riccardo De Poolis (Pieri 1883-1884) ... 308 Geometry of Position by G. K. C. von Staudt (translation: Pieri 1889a) . 313 Projective Geometry: Lectures at the Military Academy (Pieri 1891c) .. 324 Course Records from Catania University Archives (Pieri 1901-1908) .. 329 Projective Geometry: Lectures at Parma (Pieri 1910,1911c)......................331 Descriptive Geometry: Lectures at Parma {Pieri 1912f) ............................ 336 9.2 Foundations of Projective Geometry.............................................................. 341 Principles That Support the Geometry of Position (1895a, 1896a-b) ... 342 Postulates for Abstract Projective Geometry of Hyperspaces {1896c) ... 349 Primitive Entities of Abstract Projective Geometry (1897c)
......................352 Intermezzo (1897b)................................................... Principles of the Geometry of Position Composed into a Deductive Logical System (1898c)............................................................................... 359 9.2.6 New Method for Developing Projective Geometry Deductively (1898b) . 369 9.2.7 Principles That Support the Geometry of Lines (1901b) ............................ 376 9.2.8 Staudťs Fundamental Theorem and the Principles of Projective Geometry (1904a)..........................................................................................381 9.2.9 New Principles of Complex Projective Geometry (1905c, 1906a)............... 394 9.2.10 On the Staudtian Definition of Homography Լ1906f)................................ 412 9.2.1 9.2.2 9.2.3 9.2.4 9.2.5 9.3 Foundations of Elementary and Inversive Geometry..............................415 9.3.1 9.3.2 9.3.3 On Elementary Geometry as a Hypothetical-Deductive System: Monograph on Point and Motion (1900a) .............................................416 Elementary Geometry Based on the Notions of Point and Sphere (1908a, 1915) ................................................................ 428 New Principles of the Geometry of Inversions (1911d, 1912c) ................. 432 9.4 Arithmetic, Logic, and PhUosophy of Science............................................. 434 9.4.1 9.4.2 Geometry Envisioned as a Purely Logical System ([1900] 1901) ............ 435 On an Arithmetical Definition of the Irrationals (1906e).......................... 437 308
Contents 9.4.3 9.4.4 9.4.5 10 A Look at the New Logico-Mathematical Direction of the Deductive Sciences (1906d)..................................................................... 445 On the Consistency of the Axioms ofArithmetic (1906g)...........................455 On the Axioms of Arithmetic (1907a)......................................................... 461 Central Themes and Impact ofPleri’s Work ........................... 465 10.1 Philosophical Themes In Pieri’s Research .............................................466 10.2 Themes In Foundations of Geometry.....................................................467 10.2.1 10.2.2 10.2.3 10.2.4 10.2.5 10.2.6 Geometry as an Abstract Science............................................................... 469 Geometry from a Synthetic Perspective..................................................... 471 Geometry from a Transformational Point of View ...................................472 Geometries Constructed as Autonomous Disciplines ...............................474 Continuity and Archimedean Principles ................................................... 476 Minimizing the Number of Primitive Notions...........................................478 10.3 Pedagogical Themes ............................................................................... 481 10.4 Pleri’s Impact ............................................................................................. 485 10.4.1 Philosophy................................................................................................... 485 10.4.2 Foundations of Geometry
........................................................................... 488 10.4.3 Pedagogy ..................................................................................................... 497 10.5 Opportunities for Future Research......................................................... 499 Appendix ......................................................................................................................507 1 2 3 4 Errata and Addenda for Marchisotto and Smith 2007 .......................... 507 Two Letters from Louis Couturat............................................................... 512 Russell’s Annotations on Principles of the Geometry of Position .............516 Pieri’s 1905b Letter to Oswald Veblen....................................................... 519 Bibliography...................................................................................................... 521 Permissions and Credits............................................................................................. 575 Index of Persons ......................................................................................................... 577 Index of Subjects 590
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| indexdate | 2024-12-20T19:15:07Z |
| institution | BVB |
| isbn | 9780817648220 |
| language | English |
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| spellingShingle | Marchisotto, Elena 1945- Rodríguez-Consuegra, Francisco A. 1951- Smith, James T. 1939- The legacy of Mario Pieri in foundations and philosophy of mathematics Pieri, Mario 1860-1913 (DE-588)117687235 gnd History of Mathematical Sciences History of Science Mathematical Logic and Foundations Logic Philosophy of Science Mathematics History Mathematical logic Philosophy and science Mathematik (DE-588)4037944-9 gnd Philosophie (DE-588)4045791-6 gnd |
| subject_GND | (DE-588)117687235 (DE-588)4037944-9 (DE-588)4045791-6 |
| title | The legacy of Mario Pieri in foundations and philosophy of mathematics |
| title_auth | The legacy of Mario Pieri in foundations and philosophy of mathematics |
| title_exact_search | The legacy of Mario Pieri in foundations and philosophy of mathematics |
| title_full | The legacy of Mario Pieri in foundations and philosophy of mathematics Elena Anne Corie Marchisotto, Francisco Rodríguez-Consuegra, James T. Smith |
| title_fullStr | The legacy of Mario Pieri in foundations and philosophy of mathematics Elena Anne Corie Marchisotto, Francisco Rodríguez-Consuegra, James T. Smith |
| title_full_unstemmed | The legacy of Mario Pieri in foundations and philosophy of mathematics Elena Anne Corie Marchisotto, Francisco Rodríguez-Consuegra, James T. Smith |
| title_short | The legacy of Mario Pieri in foundations and philosophy of mathematics |
| title_sort | the legacy of mario pieri in foundations and philosophy of mathematics |
| topic | Pieri, Mario 1860-1913 (DE-588)117687235 gnd History of Mathematical Sciences History of Science Mathematical Logic and Foundations Logic Philosophy of Science Mathematics History Mathematical logic Philosophy and science Mathematik (DE-588)4037944-9 gnd Philosophie (DE-588)4045791-6 gnd |
| topic_facet | Pieri, Mario 1860-1913 History of Mathematical Sciences History of Science Mathematical Logic and Foundations Logic Philosophy of Science Mathematics History Mathematical logic Philosophy and science Mathematik Philosophie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032690522&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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