1 Life and Works...... 1 1.1 Biography...... 4 1.1.1 Lucca...... 4 1.1.2 Bologna: Studies...... 7 1.1.3 Pisa ...... 10 1.1.4 Turin...... 17 1.1.5 The Bologna Affair...... 25 1.1.6 Catania ...... 32 1.1.7 Parma ...... 44 1.1.8 Afterward ...... 47 1.2 Overview of Pieri’s Research...... 50 1.2.1 Algebraic and Differential Geometry, Vector Analysis ...... 50 1.2.2 Foundations of Geometry...... 54 1.2.3 Arithmetic, Logic, and Philosophy of Science...... 58 1.2.4 Conclusion...... 61 1.3 Others...... 62

2 Foundations of Geometry ...... 123 2.1 Historical Context ...... 125 2.2 Hypothetical-Deductive Systems...... 126 2.3 Projective Geometry ...... 128 2.4 Inversive Geometry...... 137 2.5 Absolute and Euclidean Geometry ...... 145 2.5.1 Point and Motion ...... 145 2.5.2 Point and Sphere ...... 153

3Pieri’s Point and Sphere Memoir ...... 157 3.1 Point and Sphere...§I ...... 164 3.2 Orthogonality...§II ...... 178 3.3 Points Internal or External...§III...... 192 3.4 Theorems on Rotations...§IV...... 206 3.5 Relations “Smaller Than” and “Larger Than”...§V ...... 220 3.6 Parallelism...§VI...... 229 xii Contents

3.7 Products of Isometries...§VII...... 241 3.8 Ordering and Senses...§VIII...... 249 3.9 Appendix...... 265 3.10Historical and Critical Remarks...... 271 3.10.1 Pieri’s Point and Motion Monograph...... 271 3.10.2 Hilbert’s Foundations of Geometry ...... 274 3.10.3 Veblen’s 1904 System of Axioms...... 277 3.10.4 Pieri’s Point and Sphere Memoir...... 278 3.10.5 The Definitions ...... 278 3.10.6 The Postulates...... 282 3.10.7 Building Geometry ...... 283 3.10.8 Other Significant Features...... 284 3.10.9 Questions Answered...... 286 3.10.10New Questions ...... 287

4 Foundations of Arithmetic ...... 289 4.1 Historical Background...... 290 4.1.1 The Real Number System ...... 291 4.1.2 The Natural Numbers ...... 294 4.1.3 Pieri’s Investigation of the Natural Number System...... 305 4.2 Pieri’s 1907 Axiomatization ...... 308 4.3 Axiomatizing Natural Number Arithmetic ...... 313 4.3.1 Dedekind...... 314 4.3.2 Peano...... 315 4.3.3 Padoa...... 320 4.3.4 Pieri...... 322 4.4 Reception of Pieri’s Axiomatization...... 326

5 Pieri’s Impact ...... 331 5.1 Peano and Pieri ...... 331 5.1.1 Peano’s Background...... 332 5.1.2 Peano’s Early Career ...... 333 5.1.3 Peano’s Ascent ...... 335 5.1.4 Pieri and the Peano School...... 338 5.1.5 Peano’s Decline ...... 343 5.2 Pieri and Tarski...... 347 5.2.1 Foundations of the Geometry of Solids ...... 349 5.2.2 Tarski’s System of Geometry...... 350 5.2.3 1929–1959 ...... 351 5.2.4 What Is Elementary Geometry? ...... 353 5.2.5 Basing Geometry on a Single Undefined Relation ...... 357 5.3 Pieri’s Legacy...... 363 5.3.1 Peano and Pieri...... 363 5.3.2 Pieri and Tarski ...... 367 5.3.3 In the Shadow of Giants...... 369 5.3.4 In the Future...... 370 Contents xiii

6 Pieri’s Works ...... 373 6.1 Differential Geometry ...... 374 6.2 Algebraic Geometry...... 374 6.2.1 Beginnings ...... 375 6.2.2 Tangents and Normals...... 375 6.2.3 Enumerative Geometry ...... 376 6.2.4 Birational Transformations...... 377 6.3 Vector Analysis...... 378 6.4 Foundations of Geometry ...... 379 6.4.1 Projective Geometry...... 379 6.4.2 Elementary Geometry ...... 381 6.4.3 Inversive Geometry...... 381 6.5 Arithmetic, Logic, and Philosophy of Science...... 381 6.6 Letters ...... 382 6.7 Further Works ...... 392 6.7.1 Translations, Edited and Revised ...... 393 6.7.2 Reviews...... 393 6.7.3 Lecture Notes ...... 397 6.7.4 Collections...... 398 6.7.5 Memorials to Pieri ...... 399

Bibliography ...... 401

Permissions ...... 459

Index ...... 463

Illustrations

Portraits Mario Pieri ...... frontispiece, 24, 130 Augusto Righi ...... 9 Silvio Pieri and his daughter ...... 9 Enrico Betti ...... 13 Luigi Bianchi ...... 13 Ulisse Dini ...... 13 Enrico D’Ovidio ...... 21 Federigo Enriques ...... 24 Mario Pieri ...... 24 Eugenio Bertini ...... 30 Luigi Cremona ...... 30 Salvatore Pincherle ...... 30 Vito Volterra ...... 30 Virginia Pieri and Paolo Anastasio ...... 37 Pieri’s sister Gemma and her sons ...... 39 Pieri and relatives ...... 39 Angiolina Pieri ...... 45 Beppo Levi ...... 45 Corrado Segre ...... 55 Cesare Burali-Forti ...... 55 Felix Klein ...... 55 Mario Pieri ...... 130 Moritz Pasch ...... 132 Gino Fano ...... 132 Giuseppe Veronese ...... 132 G. K. C. von Staudt ...... 136 Theodor Reye ...... 136 August F. Möbius ...... 144 B. L. van der Waerden ...... 144 Gino Loria ...... 152 David Hilbert ...... 275 Oswald Veblen ...... 279 Hermann Grassmann ...... 295 Richard Dedekind ...... 295 Giuseppe Peano ...... 303, 339 Bertrand Russell ...... 339 Alfred Tarski (2 portraits) ...... 359 Adolf Lindenbaum ...... 359 xvi Illustrations

Maps Some Italian cities ...... xviii Italy’s regions ...... xix Telescopic projections of the planet Mars: Mario Pieri, 1878 ...... 9

Figures Scuola Normale Superiore, Pisa ...... 13 Pieri’s doctoral thesis ...... 14 Military Academy, Turin ...... 19 Military Academy personnel, Turin ...... 19 Pieri’s first Turin Academy of Sciences paper ...... 21 University of Catania ...... 35 Pieri became an editor of the Gioenia Atti ...... 35 Announcement of Pieri’s promotion ...... 37 Pieri’s 1922 interment in Lucca ...... 48 Pieri’s tomb ...... 48 Pieri’s definition of cyclic order ...... 130 Pieri’s definitions of collinearity ...... 152 Pieri’s definition of betweenness ...... 152 Pieri’s Point and Sphere memoir, first page ...... 159 Hilbert’s Foundations of Geometry...... 275 Veblen’s System of Axioms for Geometry ...... 279 Peano’s Principles of Arithmetic ...... 303 Pieri’s On the Axioms of Arithmetic ...... 309 Peano’s Formulario ...... 344

Table of Contents