: Elena Anne Marchisotto, James Smith
: The Legacy of Mario Pieri in Geometry and Arithmetic
: Birkhäuser Basel
: 9780817646035
: 1
: CHF 89.50
:
: Allgemeines, Lexika
: English
: 494
: Wasserzeichen/DRM
: PC/MAC/eReader/Tablet
: PDF

This book is the first in a series of three volumes that comprehensively examine Mario Pieri's life, mathematical work and influence. The book introduces readers to Pieri's career and his studies in foundations, from both historical and modern viewpoints.

Included in this volume are the first English translations, along with analyses, of two of his most important axiomatizations - one in arithmetic and one in geometry. The book combines an engaging exposition, little-known historical notes, exhaustive references and an excellent index. And yet the book requires no specialized experience in mathematical logic or the foundations of geometry.
3 Pieri s Point and Sphere Memoir (p. 157-158)

This chapter contains an English translation of Pieri s 1908a memoir, Elementary Geometry Based on the Notions of Point and Sphere.1 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 point and the relation that holds between points a,b, c just when a,b are equidistant from c. 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 b centered at c as the set of all points a such that a and b are equidistant from c.

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 Point and Motion memoir. Finally, Pieri followed the strategy of fusionism in developing plane and solid geometry together.2

The translation is meant to be as faithful as possible to the original. Its only intentional modernizations are

punctuation,
bibliographic references, which have been altered to refer to entries in the bibliography of the present book,
rare changes in mathematical symbols, where Pieri s are inconsistent with today s mathematical practice, and
the use of a few common English mathematical terms invented more recently than Pieri s coinages, some of which were not widely adopted.3

Editorial comments [in square brackets like these] are inserted, usually as footnotes, to document changes in mathematical terms, to note or suggest corrections for occasional mathematical errors in the original, and to explain a few passages that seem particularly opaque. All [square] brackets in the translation enclose editorial comments.

The translation strategy results in a style of English mathematical exposition now regarded as old-fashioned, awkward and redundant. This may challenge a reader whose familiarity with English is limited to the styles now used in mathematical exposition. The strategy was adopted to minimize destruction of aspects of Pieri s work tied to his expository style.

Pieri employed very extensively the subjunctive mood and some other verb forms that are rare in modern English. This may have provided him shades of meaning available in today s usage only through wording that would differ considerably from his. In the translation, wording was selected that is as close to his as possible. Subjunctives and equivalent forms with auxiliary verbs are used in the translation much more than in conventional modern English, even in the translator s own writing. Readers should interpret some such instances as indications that Pieri may be shading his meaning differently from what might be conveyed by shorter, more familiar English expressions. In most cases, readers can proceed with the same caution they would use with English mathematical or philosophical prose written in Pieri s time or a decade or two earlier. But for a definitive interpretation, they should consult the original and someone more familiar than the translator with psychological nuances conveyed by Pieri s style.
Foreword6
Preface8
Contents12
Illustrations15
Life and Works20
1.1 Biography23
1.1.1 Lucca23
1.1.2 Bologna: Studies26
1.1.3 Pisa29
1.1.4 Turin36
1.1.5 The Bologna Affair44
1.1.6 Catania51
1.1.7 Parma63
1.1.8 Afterward66
1.2 Overview of Pieri s Research69
1.2.1 Algebraic and Differential Geometry, Vector Analysis69
1.2.2 Foundations of Geometry73
1.2.3 Arithmetic, Logic, and Philosophy of Science77
1.2.4 Conclusion80
1.3 Others81
Foundations of Geometry142
2.1 Historical Context144
2.2 Hypothetical- Deductive Systems145
2.3 Projective Geometry 147
2.4 Inversive Geometry156
2.5 Absolute and Euclidean Geometry164
2.5.1 Point and Motion164
2.5.2 Point and Sphere172
3 Pieri s Point and Sphere Memoir176
3.10 Historical and Critical Remarks290
3.10.1 Pieri s Point and Motion Monograph290
3.10.2 Hilbert s293
3.10.3 Veblen s 1904296
3.10.4 Pieri s Point and Sphere Memoir297
3.10.5 The Definitions297
3.10.6 The Postulates301
3.10.7 Building Geometry302
3.10.8 Other Significant Features303
3.10.9 Questions Answered305
3.10.10 New Questions306
Foundations of Arithmetic308
4.1 Historical Background309
4.1.1 The Real Number System310
4.1.2 The Natural Numbers313
4.1.3 Pieri s Investigation of the Natural Number System324
4.2 Pieri s 1907 Axiomatization327
4.3 Axiomatizing Natural Number Arithmetic332
4.3.1 Dedekind333
4.3.2 Peano334
4.3.3 Padoa339
4.3.4 Pieri341
4.4 Reception of Pieri s Axiomatization345
Pieri s Impact349
5.1 Peano and Pieri349
5.1.1 Peano s Background350
5.1.2 Peano s Early Career351
5.1.3 Peano s Ascent353
5.1.4 Pieri and the Peano School356
5.1.5 Peano s Decline361
5.2 Pieri and Tarski365
5.2.1 Foundations of the Geometry of Solids367
5.2.2 Tarski s System of Geometry368
5.2.3 1929 1959369
5.2.4 What Is Elementary Geometry?371
5.2.5 Basing Geometry on a Single Undefined Relation375
5.3 Pieri s Legacy381
5.3.1 Peano and Pieri381
5.3.2 Pieri and Tarski385
5.3.3 In the Shadow of Giants387
5.3.4 In the Future ...388
Pieri s Works390
6.1 Differential Geometry391
6.2 Algebraic Geometry391
6.2.1 Beginnings392
6.2.2 Tangents and Normals392
6.2.3 Enumerative Geometry393
6.2.4 Birational Transformations394
6.3 Vector Analysis395
6.4 Foundations of Geometry396
6.4.1 Projective Geometry396
6.4.2 Elementary Geometry398
6.4.3 Inversive Geometry398
6.5 Arithmetic, Logic, and Philosophy of Science398
6.6 Letters399
6.7 Further Works409
6.7.1 Translations, Edited and Revised410
6.7.2 Reviews410
6.7.3 Lecture Notes414
6.7.4 Collections415
6.7.5 Memorials to Pieri416
Bibliography417
Permissions474
Index477