Legacy of Mario Pieri in Arithmetic and Geometry

by Marchisotto, Elena Anne; Smith, James T.

• ISBN13: 9780817632106
• ISBN10: 0817632107
• Format: Hardcover
• Copyright: 2007-05-30
• Publisher: Birkhauser

Summary

The Italian mathematician Mario Pieri (1860-1913) played an integral part in the research groups of Corrado Segre and Giuseppe Peano, and thus had a significant, yet somewhat underappreciated impact on several branches of mathematics, particularly on the development of algebraic geometry and the foundations of mathematics in the years around the turn of the 20 th century. This book is the first in a series of three volumes that all are dedicated to countering that neglect and comprehensively examining Pieria??s life, mathematical work and influence in such diverse fields as mathematical logic, algebraic geometry, number theory, inversive geometry, intersection theory, mathematical analysis, vector analysis, and differential geometry. The Legacy of Mario Pieri in Geometry and Arithmetic introduces readers to Pieria??s career and his studies in foundations, from both historical and modern viewpoints, placing his life and research in context and tracing his influence on his contemporaries as well as more recent mathematicians. Included in this volume are the first English translations, along with analyses, of two of his most important axiomatizationsa??his postulates for arithmetic, which Peano judged superior to his own; and his foundation of elementary geometry on the basis of point and sphere, which Alfred Tarski used as a basis for his own system. Combining an engaging exposition, little-known historical notes, exhaustive references and an excellent index, this text will be of interest to graduate students, researchers and historians with a general knowledge of logic and advanced mathematics, but requiring no specialized experience in mathematical logic or the foundations of geometry.

Table of Contents

Foreword	p. v
Preface	p. vii
Illustrations	p. xv
History and Geography	p. xvii
Life and Works	p. 1
Biography	p. 4
Lucca	p. 4
Bologna: Studies	p. 7
Pisa	p. 10
Turin	p. 17
The Bologna Affair	p. 25
Catania	p. 32
Parma	p. 44
Afterward	p. 47
Overview of Pieri's Research	p. 50
Algebraic and Differential Geometry, Vector Analysis	p. 50
Foundations of Geometry	p. 54
Arithmetic, Logic, and Philosophy of Science	p. 58
Conclusion	p. 61
Others	p. 62
Foundations of Geometry	p. 123
Historical Context	p. 125
Hypothetical-Deductive Systems	p. 126
Projective Geometry	p. 128
Inversive Geometry	p. 137
Absolute and Euclidean Geometry	p. 145
Point and Motion	p. 145
Point and Sphere	p. 153
Pieri's Point and Sphere Memoir	p. 157
Point and Sphere...[section]I	p. 164
Orthogonality...[section]II	p. 178
Points Internal or External...[section]III	p. 192
Theorems on Rotations...[section]IV	p. 206
Relations "Smaller Than" and "Larger Than"...[section]V	p. 220
Parallelism...[section]VI	p. 229
Products of Isometries...[section]VII	p. 241
Ordering and Senses...[section]VIII	p. 249
Appendix	p. 265
Historical and Critical Remarks	p. 271
Pieri's Point and Motion Monograph	p. 271
Hilbert's Foundations of Geometry	p. 274
Veblen's 1904 System of Axioms	p. 277
Pieri's Point and Sphere Memoir	p. 278
The Definitions	p. 278
The Postulates	p. 282
Building Geometry	p. 283
Other Significant Features	p. 284
Questions Answered	p. 286
New Questions	p. 287
Foundations of Arithmetic	p. 289
Historical Background	p. 290
The Real Number System	p. 291
The Natural Numbers	p. 294
Pieri's Investigation of the Natural Number System	p. 305
Pieri's 1907 Axiomatization	p. 308
Axiomatizing Natural Number Arithmetic	p. 313
Dedekind	p. 314
Peano	p. 315
Padoa	p. 320
Pieri	p. 322
Reception of Pieri's Axiomatization	p. 326
Pieri's Impact	p. 331
Peano and Pieri	p. 331
Peano's Background	p. 332
Peano's Early Career	p. 333
Peano's Ascent	p. 335
Pieri and the Peano School	p. 338
Peano's Decline	p. 343
Pieri and Tarski	p. 347
Foundations of the Geometry of Solids	p. 349
Tarski's System of Geometry	p. 350
1929-1959	p. 351
What Is Elementary Geometry?	p. 353
Basing Geometry on a Single Undefined Relation	p. 357
Pieri's Legacy	p. 363
Peano and Pieri	p. 363
Pieri and Tarski	p. 367
In the Shadow of Giants	p. 369
In the Future	p. 370
Pieri's Works	p. 373
Differential Geometry	p. 374
Algebraic Geometry	p. 374
Beginnings	p. 375
Tangents and Normals	p. 375
Enumerative Geometry	p. 376
Birational Transformations	p. 377
Vector Analysis	p. 378
Foundations of Geometry	p. 379
Projective Geometry	p. 379
Elementary Geometry	p. 381
Inversive Geometry	p. 381
Arithmetic, Logic, and Philosophy of Science	p. 381
Letters	p. 382
Further Works	p. 392
Translations, Edited and Revised	p. 393
Reviews	p. 393
Lecture Notes	p. 397
Collections	p. 398
Memorials to Pieri	p. 399
Bibliography	p. 401
Permissions	p. 459
Index	p. 463
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