From Past to Future: Graßmann's Work in Context (eBook)

Contents 6
Preface 12
Abbreviations for works of Hermann Grassmann 18
On the lives of the Grassmann brothers 22
Description of the life of Hermann Grassmann by his son Justus Grassmann, probably written shortly after the death of his father, 1877 24
Life history of Robert Grassmann, written by himself (1890) 29
Historical contexts of Hermann Grassmann's creativity 36
Discovering Robert Grassmann (1815–1901) 37
An overlooked prolific polymath 37
Plan of the paper 38
Books by the score 38
Before GW: Robert's Wissenschaftslehre 40
The first planned version of GW 43
The house that Robert Grassmann built: the structure and chronology of GW 43
Some characteristics of GW 47
Robert Grassmann on the calculus and logic 49
Four final queries 51
Acknowledgements 
53 
Hermann Grassmann's theory of religion and faith 54
I 54
II 55
Why have people stopped believing in miracles? 56
Where does the knowledge of mankind come from? 57
Where do we find absolute knowledge? 58
Is the Bible the absolute word? 59
Who interprets scripture? 61
III 62
The Significance of Naturphilosophie for Justus and Hermann Grassmann 65
The philosophy of Christian Samuel Weiss 67
Emergence of matter 68
Concept of extension 70
The question of influence 74
Justus and Hermann Grassmann: philosophy and mathematics 76
Institutional development of science in Stettin in the first half of the nineteenth century in the time of Hermann Grassmann 86
Pomerania at the turn of the nineteenth century 86
The time of the Bourgeois reformers 88
Johann August Sack: governor and reformer in Pomerania 89
Stettin and its Marienstift Gymnasium 90
The Pommersche Provinzial: Blätter für Stadt und Land 1820–1825 91
The founding of the ``Society for Pomeranian History and Classical Studies'' 94
The establishment of the Stettin Provincial Archives 95
The flowering of scientific life in Stettin 96
Philosophical and methodological aspects of the work of the Grassmann brothers 99
Brief outline of a history of the genetic method in the development of the deductive sciences 100
I 100
II 101
III 102
IV 102
V 103
VI 103
VII 103
Grassmann's epistemology: multiplication and constructivism 104
Introduction 104
The product between extensive magnitudes 105
Extensive magnitudes 106
The product between extensive magnitudes 107
A comparative philosophical analysis 108
The product between vectors and multivectors 109
Domain and homogeneity 110
Conclusion 111
Axiomatics and self-reference Reflections about Hermann Grassmann's contribution to axiomatics 114
The (never ending?) debate 114
The place of axiomatics in the Lehrbuch der Arithmetik (1861): the positions of Gottlob Frege, Judson Webb, and Hao Wang 116
Hans-Joachim Petsche's interpretation 120
An alternative interpretation: axiomatics and self-reference 122
Instead of a conclusion 128
Concepts and contrasts: Hermann Grassmann and Bernard Bolzano 130
Introduction 130
Some parallels of context 131
Some divergences of working 133
The nature and classification of mathematics 134
What shall we do with geometry? 136
What makes a Presentation ``Scientific''? 137
Conclusion 139
Diversity of the influence of the Grassmann brothers 141
New forms of science and new sciences of form: On the non-mathematical reception of Grassmann's work 142
Grassmann outside mathematics 142
Grassmann in psychology and physiology 143
Basic structures and operations: relations, order and abstraction 146
New forms of science 148
Some philosophical influences of the Ausdehnungslehre 151
Grassmann as philosopher 151
Bertrand Russell 152
Ernst Cassirer 154
Paul Carus 155
Friedrich Kuntze 157
Concluding note 158
Grassmann's influence on Husserl 159
``Influence'' 159
The Grassmanns and Husserl 160
The Weierstrassian first part of the Philosophy of Arithmetic 161
The parallel structures of symbols and concepts 163
The problem and the influence of Grassmann 164
Conclusion 169
Ernst Abbe's reception of Grassmann in the light of Grassmann's reception of Schleiermacher 170
The reception of Grassmann in Göttingen and Jena 170
Mathematics, philosophy and experimentation: Abbe's scientific interests 171
Abbe's first encounter with Grassmann's Extension Theory of 1844 172
Alexander Crailsheim: Grassmann's contemporary and Abbe's inspiration 174
Hegel, Schleiermacher and Robert Grassmann's opinion 176
Schleiermacher's influence on the work of Hermann Grassmann 177
Heuristics and architectonics in the work of Schleiermacher and the Grassmanns 181
Appendix 183
Acknowledgment 183
On the early appraisals in Russia of H. and R. Grassmann's achievements 184
Hermann Grassmann's Work and the Peano School 193
Introduction 193
Peano's geometric calculus 195
Toward the minimum system 201
Conclusion 203
Did Gibbs influence Peano's ``Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann ''? 204
Introduction 204
What Polak said, and related comments 205
Polak's starting point 206
Polak on Grassmann and Peano 206
Polak's unconvincing consideration 207
Was Peano `deceived' by Gibbs? 213
Burali-Forti and Marcolongo and the Italian Vector School 214
Conclusion 215
Acknowledgements 
215 
Rudolf Mehmke, an outstanding propagator of Grassmann's vector calculus 216
Biography 216
Lectures 218
Scientific publications and instruments 219
Vector commission 220
Mehmke's main publications on vector calculus 221
Relativity theory 223
Mehmke's correspondence 223
Summary 226
Robert and Hermann Grassmann's influence on the history of formal logic 228
Introduction 228
General theory of forms 230
Logical interpretation 231
Influences 232
Acknowledgement 235
Hermann Grassmann's contribution to Whitehead's foundations of logic and mathematics 236
Introduction 236
A. N. Whitehead's Treatise on Universal Algebra 237
A picture of A. N. Whitehead by D. Emmett 238
Structure and method: From Leibniz to the Grassmanns and A. N. Whitehead 239
Leibniz's thesis 239
Divisibility 240
Combinatorics 240
What did Hermann learn from his father Justus? 241
From his Crystallonomy 241
From his philosophy 242
Parenthesis on prizes 243
The new geometries 244
Courses in Cambridge 244
Whitehead's early geometrical works 245
Present and future of Hermann Grassmann's ideas in mathematics 248
Grassmann's legacy 249
Evolution of Geometric Algebra and Calculus 250
Recent developments in Geometric Algebra 252
Products in Geometric Algebra 254
Conformal Geometric Algebra 259
The algebra of ruler and compass 260
On Grassmann's regressive product 267
A new mathematical discipline 267
An algebra of pieces of space 268
Applications to geometry and mechanics 270
The regressive product 273
Subordinate form 273
Modular lattices 275
Nonassociativity of the geometric product 275
Multiplication of flags 276
Where did this leave Grassmann? 277
Where does this leave us? 279
Giving Hermann Grassmann the final word 280
Projective geometric theorem proving with Grassmann–Cayley algebra 281
Introduction 281
Classical Grassmann–Cayley algebra 282
Theorem proving in projective incidence geometry with Grassmann–Cayley algebra 288
Conclusion 291
Grassmann, geometry and mechanics 292
Introduction 292
Grassmann, Hamilton, and Gibbs 293
Interpreted spaces 294
Points and weighted points 295
Bound vectors and bivectors 296
Sums of bound vectors and bivectors 297
The equilibrium of a rigid body 299
Momentum 300
Newton's Second Law 301
The regressive product 302
Projective geometry 303
Geometric constructions 304
Geometric theorems 304
Conclusions 307
Representations of spinor groups using Grassmann exterior algebra 308
Hermann Grassmann's theory of linear transformations 315
Introduction 315
Definition of the fraction 316
Peano's and Whitehead's takes on the fraction 319
Exchanging the denominators 321
Spectral theory 323
Concluding remarks 326
Acknowledgments 
327 
The Golden Gemini Spiral 328
Introduction 328
Notation 329
Castor and Pollux, the Gemini Twins 330
Constructing the Golden Gemini Spiral 330
The eye of the Gemini Spiral 332
Intertwining Gemini Spirals 333
A short note on Grassmann manifolds with a view to noncommutative geometry 335
Introduction 335
On Grassmann manifolds 336
A view to noncommutative geometric spaces 338
Conclusion 342
Present and future of Hermann Grassmann's ideas in philology 345
Hermann Grassmann: his contributions to historical linguistics and speech acoustics 346
Introduction 346
Grassmann's work in historical linguistics 346
Grassmann's contribution to the acoustic phonetics of vowels 350
Conclusion 352
Acknowledgements 353
Grassmann's ``Worterbuch des Rig-Veda'' (Dictionary of Rig-Veda): a milestone in the study of Vedic Sanskrit 354
Remarks on Rgveda (RV) 354
Accomplishments of the Old Indic grammarians 355
Entries in Vedic dictionaries 356
Grammatical features of Vedic Sanskrit 356
Grassmann's qualifications for such a dictionary 356
Grassmann's Dictionary of Rig-Veda 357
Exemplary comparison of Grassmann's dictionary with the Petersburg dictionary by Otto Böhtlingk and Rudolph Roth, pt. 2. (1856–1858) 360
Recognition of the linguistic accomplishments 362
The Rigveda Dictionary from a modern viewpoint 363
Lemmas, forms and meaning 364
1. Analysis of the entry 366
2. Meaning entries 368
3. Form entries 368
Metrical analysis 370
Prepositions, particles, etc. 371
Abstract language and German 373
The decisive year of 1875 374
Grassmann's contribution to lexicography and the living-on of his ideas in the Salzburg Dictionary to the Rig-Veda 376
Introduction 376
Comparing Grassmann and RIVELEX from a modern lexicographical point of view 377
Pre-Lexicography 377
Elaboration of a macrostructure 378
Working out a microstructure 380
Final remarks 385
Hermann Grassmann's impact on music, computing and education 387
Calculation and emotion: Hermann Grassmann and Gustav Jacobsthal's musicology 388
Classification of complex musical structures by Grassmann schemes 398
Global compositions 398
Classification of global compositions 401
Grassmann's technique 404
The musical meaning of Grassmann's approach 405
Varèse's interpretation 407
New views of crystal symmetry guided by profound admiration of the extraordinary works of Grassmann and Clifford 410
Introduction 410
Computer visualization of crystal symmetry 411
Appendix. Clifford geometric algebra description of space groups 415
Cartan–Dieudonné and geometric algebra 415
Two-dimensional point groups 417
Three-dimensional point groups 418
Space groups 418
Acknowledgments 419
From Grassmann's vision to geometric algebra computing 420
Introduction 420
Benefits of conformal geometric algebra 421
Unification of mathematical systems 422
Intuitive handling of geometric objects 423
Intuitive handling of geometric operations 424
Robotics application example 424
Geometric algebra computing technology 425
Compilation 427
Adaptation to different parallel processor platforms 428
Conclusion 430
Grassmann, Pauli, Dirac: special relativity in the schoolroom 431
Introduction 431
Grassmann's mathematical parenthood 432
Space and perception 433
Mathematical models of space 433
Didactical aspects of the geometric product 436
The Quantum-mechanical misconception 438
Didactical aspects of special relativity 439
Spacetime algebra 440
The quantum-mechanical misconception revisited 442
Remark about the history of the interpretation of Dirac matrices 443
Main focus at school 444
Appendix 448
On the concept and extent of pure theory of number (1827) 450
The three orders of enumeration 459
The general conjunction 463
The types of calculation 463
Survey of the types of calculation 464
Mechanical conjunction 465
Chemical conjunction 466
Dynamic conjunction 471
On the negative numbers 474
Proof that there can be no conjunction higher than exponentiation 478
Concluding remarks 481
Remarks on illustrations 483
Notes on contributors 498
References 517
Index of names and citations 545