The Legacy of Mario Pieri in Geometry and Arithmetic (eBook)

Foreword 6
Preface 8
Contents 12
Illustrations 15
Life and Works 20
1.1 Biography 23
1.1.1 Lucca 23
1.1.2 Bologna: Studies 26
1.1.3 Pisa 29
1.1.4 Turin 36
1.1.5 The Bologna Affair 44
1.1.6 Catania 51
1.1.7 Parma 63
1.1.8 Afterward 66
1.2 Overview of Pieri’s Research 69
1.2.1 Algebraic and Differential Geometry, Vector Analysis 69
1.2.2 Foundations of Geometry 73
1.2.3 Arithmetic, Logic, and Philosophy of Science 77
1.2.4 Conclusion 80
1.3 Others 81
Foundations of Geometry 142
2.1 Historical Context 144
2.2 Hypothetical- Deductive Systems 145
2.3 Projective Geometry 147
2.4 Inversive Geometry 156
2.5 Absolute and Euclidean Geometry 164
2.5.1 Point and Motion 164
2.5.2 Point and Sphere 172
3 Pieri’s Point and Sphere Memoir 176
3.10 Historical and Critical Remarks 290
3.10.1 Pieri’s Point and Motion Monograph 290
3.10.2 Hilbert’s 293
3.10.3 Veblen’s 1904 296
3.10.4 Pieri’s Point and Sphere Memoir 297
3.10.5 The Definitions 297
3.10.6 The Postulates 301
3.10.7 Building Geometry 302
3.10.8 Other Significant Features 303
3.10.9 Questions Answered 305
3.10.10 New Questions 306
Foundations of Arithmetic 308
4.1 Historical Background 309
4.1.1 The Real Number System 310
4.1.2 The Natural Numbers 313
4.1.3 Pieri’s Investigation of the Natural Number System 324
4.2 Pieri’s 1907 Axiomatization 327
4.3 Axiomatizing Natural Number Arithmetic 332
4.3.1 Dedekind 333
4.3.2 Peano 334
4.3.3 Padoa 339
4.3.4 Pieri 341
4.4 Reception of Pieri’s Axiomatization 345
Pieri’s Impact 349
5.1 Peano and Pieri 349
5.1.1 Peano’s Background 350
5.1.2 Peano’s Early Career 351
5.1.3 Peano’s Ascent 353
5.1.4 Pieri and the Peano School 356
5.1.5 Peano’s Decline 361
5.2 Pieri and Tarski 365
5.2.1 Foundations of the Geometry of Solids 367
5.2.2 Tarski’s System of Geometry 368
5.2.3 1929 –1959 369
5.2.4 What Is Elementary Geometry? 371
5.2.5 Basing Geometry on a Single Undefined Relation 375
5.3 Pieri’s Legacy 381
5.3.1 Peano and Pieri 381
5.3.2 Pieri and Tarski 385
5.3.3 In the Shadow of Giants 387
5.3.4 In the Future ... 388
Pieri’s Works 390
6.1 Differential Geometry 391
6.2 Algebraic Geometry 391
6.2.1 Beginnings 392
6.2.2 Tangents and Normals 392
6.2.3 Enumerative Geometry 393
6.2.4 Birational Transformations 394
6.3 Vector Analysis 395
6.4 Foundations of Geometry 396
6.4.1 Projective Geometry 396
6.4.2 Elementary Geometry 398
6.4.3 Inversive Geometry 398
6.5 Arithmetic, Logic, and Philosophy of Science 398
6.6 Letters 399
6.7 Further Works 409
6.7.1 Translations, Edited and Revised 410
6.7.2 Reviews 410
6.7.3 Lecture Notes 414
6.7.4 Collections 415
6.7.5 Memorials to Pieri 416
Bibliography 417
Permissions 474
Index 477

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.