1 - Contents [Seite 6]
2 - Preface [Seite 12]
3 - Abbreviations for works of Hermann Grassmann [Seite 18]
4 - On the lives of the Grassmann brothers [Seite 22]
4.1 - Description of the life of Hermann Grassmann by his son Justus Grassmann, probably written shortly after the death of his father, 1877 [Seite 24]
4.2 - Life history of Robert Grassmann, written by himself (1890) [Seite 29]
5 - Historical contexts of Hermann Grassmann's creativity [Seite 36]
5.1 - Discovering Robert Grassmann (1815-1901) [Seite 37]
5.1.1 - An overlooked prolific polymath [Seite 37]
5.1.2 - Plan of the paper [Seite 38]
5.1.3 - Books by the score [Seite 38]
5.1.4 - Before GW: Robert's Wissenschaftslehre [Seite 40]
5.1.5 - The first planned version of GW [Seite 43]
5.1.6 - The house that Robert Grassmann built: the structure and chronology of GW [Seite 43]
5.1.7 - Some characteristics of GW [Seite 47]
5.1.8 - Robert Grassmann on the calculus and logic [Seite 49]
5.1.9 - Four final queries [Seite 51]
5.1.10 - Acknowledgements [Seite Acknowledgements]
- 53 [Seite 53]
5.2 - Hermann Grassmann's theory of religion and faith [Seite 54]
5.2.1 - I [Seite 54]
5.2.2 - II [Seite 55]
5.2.2.1 - Why have people stopped believing in miracles? [Seite 56]
5.2.2.2 - Where does the knowledge of mankind come from? [Seite 57]
5.2.2.3 - Where do we find absolute knowledge? [Seite 58]
5.2.2.4 - Is the Bible the absolute word? [Seite 59]
5.2.2.5 - Who interprets scripture? [Seite 61]
5.2.3 - III [Seite 62]
5.3 - The Significance of Naturphilosophie for Justus and Hermann Grassmann [Seite 65]
5.3.1 - The philosophy of Christian Samuel Weiss [Seite 67]
5.3.2 - Emergence of matter [Seite 68]
5.3.3 - Concept of extension [Seite 70]
5.3.4 - The question of influence [Seite 74]
5.4 - Justus and Hermann Grassmann: philosophy and mathematics [Seite 76]
5.5 - Institutional development of science in Stettin in the first half of the nineteenth century in the time of Hermann Grassmann [Seite 86]
5.5.1 - Pomerania at the turn of the nineteenth century [Seite 86]
5.5.2 - The time of the Bourgeois reformers [Seite 88]
5.5.3 - Johann August Sack: governor and reformer in Pomerania [Seite 89]
5.5.4 - Stettin and its Marienstift Gymnasium [Seite 90]
5.5.5 - The Pommersche Provinzial: Blätter für Stadt und Land 1820-1825 [Seite 91]
5.5.6 - The founding of the ``Society for Pomeranian History and Classical Studies'' [Seite 94]
5.5.7 - The establishment of the Stettin Provincial Archives [Seite 95]
5.5.8 - The flowering of scientific life in Stettin [Seite 96]
6 - Philosophical and methodological aspects of the work of the Grassmann brothers [Seite 99]
6.1 - Brief outline of a history of the genetic method in the development of the deductive sciences [Seite 100]
6.1.1 - I [Seite 100]
6.1.2 - II [Seite 101]
6.1.3 - III [Seite 102]
6.1.4 - IV [Seite 102]
6.1.5 - V [Seite 103]
6.1.6 - VI [Seite 103]
6.1.7 - VII [Seite 103]
6.2 - Grassmann's epistemology: multiplication and constructivism [Seite 104]
6.2.1 - Introduction [Seite 104]
6.2.2 - The product between extensive magnitudes [Seite 105]
6.2.2.1 - Extensive magnitudes [Seite 106]
6.2.2.2 - The product between extensive magnitudes [Seite 107]
6.2.3 - A comparative philosophical analysis [Seite 108]
6.2.3.1 - The product between vectors and multivectors [Seite 109]
6.2.3.2 - Domain and homogeneity [Seite 110]
6.2.4 - Conclusion [Seite 111]
6.3 - Axiomatics and self-reference Reflections about Hermann Grassmann's contribution to axiomatics [Seite 114]
6.3.1 - The (never ending?) debate [Seite 114]
6.3.2 - The place of axiomatics in the Lehrbuch der Arithmetik (1861): the positions of Gottlob Frege, Judson Webb, and Hao Wang [Seite 116]
6.3.3 - Hans-Joachim Petsche's interpretation [Seite 120]
6.3.4 - An alternative interpretation: axiomatics and self-reference [Seite 122]
6.3.5 - Instead of a conclusion [Seite 128]
6.4 - Concepts and contrasts: Hermann Grassmann and Bernard Bolzano [Seite 130]
6.4.1 - Introduction [Seite 130]
6.4.2 - Some parallels of context [Seite 131]
6.4.3 - Some divergences of working [Seite 133]
6.4.3.1 - The nature and classification of mathematics [Seite 134]
6.4.3.2 - What shall we do with geometry? [Seite 136]
6.4.3.3 - What makes a Presentation ``Scientific''? [Seite 137]
6.4.4 - Conclusion [Seite 139]
7 - Diversity of the influence of the Grassmann brothers [Seite 141]
7.1 - New forms of science and new sciences of form: On the non-mathematical reception of Grassmann's work [Seite 142]
7.1.1 - Grassmann outside mathematics [Seite 142]
7.1.2 - Grassmann in psychology and physiology [Seite 143]
7.1.3 - Basic structures and operations: relations, order and abstraction [Seite 146]
7.1.4 - New forms of science [Seite 148]
7.2 - Some philosophical influences of the Ausdehnungslehre [Seite 151]
7.2.1 - Grassmann as philosopher [Seite 151]
7.2.2 - Bertrand Russell [Seite 152]
7.2.3 - Ernst Cassirer [Seite 154]
7.2.4 - Paul Carus [Seite 155]
7.2.5 - Friedrich Kuntze [Seite 157]
7.2.6 - Concluding note [Seite 158]
7.3 - Grassmann's influence on Husserl [Seite 159]
7.3.1 - ``Influence'' [Seite 159]
7.3.2 - The Grassmanns and Husserl [Seite 160]
7.3.3 - The Weierstrassian first part of the Philosophy of Arithmetic [Seite 161]
7.3.4 - The parallel structures of symbols and concepts [Seite 163]
7.3.5 - The problem and the influence of Grassmann [Seite 164]
7.3.6 - Conclusion [Seite 169]
7.4 - Ernst Abbe's reception of Grassmann in the light of Grassmann's reception of Schleiermacher [Seite 170]
7.4.1 - The reception of Grassmann in Göttingen and Jena [Seite 170]
7.4.2 - Mathematics, philosophy and experimentation: Abbe's scientific interests [Seite 171]
7.4.3 - Abbe's first encounter with Grassmann's Extension Theory of 1844 [Seite 172]
7.4.4 - Alexander Crailsheim: Grassmann's contemporary and Abbe's inspiration [Seite 174]
7.4.5 - Hegel, Schleiermacher and Robert Grassmann's opinion [Seite 176]
7.4.6 - Schleiermacher's influence on the work of Hermann Grassmann [Seite 177]
7.4.7 - Heuristics and architectonics in the work of Schleiermacher and the Grassmanns [Seite 181]
7.4.8 - Appendix [Seite 183]
7.4.8.1 - Acknowledgment [Seite 183]
7.5 - On the early appraisals in Russia of H. and R. Grassmann's achievements [Seite 184]
7.6 - Hermann Grassmann's Work and the Peano School [Seite 193]
7.6.1 - Introduction [Seite 193]
7.6.2 - Peano's geometric calculus [Seite 195]
7.6.3 - Toward the minimum system [Seite 201]
7.6.4 - Conclusion [Seite 203]
7.7 - Did Gibbs influence Peano's ``Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann ''? [Seite 204]
7.7.1 - Introduction [Seite 204]
7.7.2 - What Polak said, and related comments [Seite 205]
7.7.2.1 - Polak's starting point [Seite 206]
7.7.2.2 - Polak on Grassmann and Peano [Seite 206]
7.7.2.3 - Polak's unconvincing consideration [Seite 207]
7.7.2.4 - Was Peano `deceived' by Gibbs? [Seite 213]
7.7.3 - Burali-Forti and Marcolongo and the Italian Vector School [Seite 214]
7.7.4 - Conclusion [Seite 215]
7.7.5 - Acknowledgements [Seite Acknowledgements]
- 215 [Seite 215]
7.8 - Rudolf Mehmke, an outstanding propagator of Grassmann's vector calculus [Seite 216]
7.8.1 - Biography [Seite 216]
7.8.2 - Lectures [Seite 218]
7.8.3 - Scientific publications and instruments [Seite 219]
7.8.4 - Vector commission [Seite 220]
7.8.5 - Mehmke's main publications on vector calculus [Seite 221]
7.8.6 - Relativity theory [Seite 223]
7.8.7 - Mehmke's correspondence [Seite 223]
7.8.8 - Summary [Seite 226]
7.9 - Robert and Hermann Grassmann's influence on the history of formal logic [Seite 228]
7.9.1 - Introduction [Seite 228]
7.9.2 - General theory of forms [Seite 230]
7.9.3 - Logical interpretation [Seite 231]
7.9.4 - Influences [Seite 232]
7.9.5 - Acknowledgement [Seite 235]
7.10 - Hermann Grassmann's contribution to Whitehead's foundations of logic and mathematics [Seite 236]
7.10.1 - Introduction [Seite 236]
7.10.2 - A. N. Whitehead's Treatise on Universal Algebra [Seite 237]
7.10.3 - A picture of A. N. Whitehead by D. Emmett [Seite 238]
7.10.4 - Structure and method: From Leibniz to the Grassmanns and A. N. Whitehead [Seite 239]
7.10.4.1 - Leibniz's thesis [Seite 239]
7.10.4.1.1 - Divisibility [Seite 240]
7.10.4.1.2 - Combinatorics [Seite 240]
7.10.4.2 - What did Hermann learn from his father Justus? [Seite 241]
7.10.4.2.1 - From his Crystallonomy [Seite 241]
7.10.4.2.2 - From his philosophy [Seite 242]
7.10.4.3 - Parenthesis on prizes [Seite 243]
7.10.4.4 - The new geometries [Seite 244]
7.10.4.5 - Courses in Cambridge [Seite 244]
7.10.4.6 - Whitehead's early geometrical works [Seite 245]
8 - Present and future of Hermann Grassmann's ideas in mathematics [Seite 248]
8.1 - Grassmann's legacy [Seite 249]
8.1.1 - Evolution of Geometric Algebra and Calculus [Seite 250]
8.1.2 - Recent developments in Geometric Algebra [Seite 252]
8.1.3 - Products in Geometric Algebra [Seite 254]
8.1.4 - Conformal Geometric Algebra [Seite 259]
8.1.5 - The algebra of ruler and compass [Seite 260]
8.2 - On Grassmann's regressive product [Seite 267]
8.2.1 - A new mathematical discipline [Seite 267]
8.2.2 - An algebra of pieces of space [Seite 268]
8.2.3 - Applications to geometry and mechanics [Seite 270]
8.2.4 - The regressive product [Seite 273]
8.2.5 - Subordinate form [Seite 273]
8.2.6 - Modular lattices [Seite 275]
8.2.7 - Nonassociativity of the geometric product [Seite 275]
8.2.8 - Multiplication of flags [Seite 276]
8.2.9 - Where did this leave Grassmann? [Seite 277]
8.2.10 - Where does this leave us? [Seite 279]
8.2.11 - Giving Hermann Grassmann the final word [Seite 280]
8.3 - Projective geometric theorem proving with Grassmann-Cayley algebra [Seite 281]
8.3.1 - Introduction [Seite 281]
8.3.2 - Classical Grassmann-Cayley algebra [Seite 282]
8.3.3 - Theorem proving in projective incidence geometry with Grassmann-Cayley algebra [Seite 288]
8.3.4 - Conclusion [Seite 291]
8.4 - Grassmann, geometry and mechanics [Seite 292]
8.4.1 - Introduction [Seite 292]
8.4.2 - Grassmann, Hamilton, and Gibbs [Seite 293]
8.4.3 - Interpreted spaces [Seite 294]
8.4.4 - Points and weighted points [Seite 295]
8.4.5 - Bound vectors and bivectors [Seite 296]
8.4.6 - Sums of bound vectors and bivectors [Seite 297]
8.4.7 - The equilibrium of a rigid body [Seite 299]
8.4.8 - Momentum [Seite 300]
8.4.9 - Newton's Second Law [Seite 301]
8.4.10 - The regressive product [Seite 302]
8.4.11 - Projective geometry [Seite 303]
8.4.12 - Geometric constructions [Seite 304]
8.4.13 - Geometric theorems [Seite 304]
8.4.14 - Conclusions [Seite 307]
8.5 - Representations of spinor groups using Grassmann exterior algebra [Seite 308]
8.6 - Hermann Grassmann's theory of linear transformations [Seite 315]
8.6.1 - Introduction [Seite 315]
8.6.2 - Definition of the fraction [Seite 316]
8.6.3 - Peano's and Whitehead's takes on the fraction [Seite 319]
8.6.4 - Exchanging the denominators [Seite 321]
8.6.5 - Spectral theory [Seite 323]
8.6.6 - Concluding remarks [Seite 326]
8.6.7 - Acknowledgments [Seite Acknowledgments]
- 327 [Seite 327]
8.7 - The Golden Gemini Spiral [Seite 328]
8.7.1 - Introduction [Seite 328]
8.7.2 - Notation [Seite 329]
8.7.3 - Castor and Pollux, the Gemini Twins [Seite 330]
8.7.4 - Constructing the Golden Gemini Spiral [Seite 330]
8.7.5 - The eye of the Gemini Spiral [Seite 332]
8.7.6 - Intertwining Gemini Spirals [Seite 333]
8.8 - A short note on Grassmann manifolds with a view to noncommutative geometry [Seite 335]
8.8.1 - Introduction [Seite 335]
8.8.2 - On Grassmann manifolds [Seite 336]
8.8.3 - A view to noncommutative geometric spaces [Seite 338]
8.8.4 - Conclusion [Seite 342]
9 - Present and future of Hermann Grassmann's ideas in philology [Seite 345]
9.1 - Hermann Grassmann: his contributions to historical linguistics and speech acoustics [Seite 346]
9.1.1 - Introduction [Seite 346]
9.1.2 - Grassmann's work in historical linguistics [Seite 346]
9.1.3 - Grassmann's contribution to the acoustic phonetics of vowels [Seite 350]
9.1.4 - Conclusion [Seite 352]
9.1.5 - Acknowledgements [Seite 353]
9.2 - Grassmann's ``Worterbuch des Rig-Veda'' (Dictionary of Rig-Veda): a milestone in the study of Vedic Sanskrit [Seite 354]
9.2.1 - Remarks on Rgveda (RV) [Seite 354]
9.2.2 - Accomplishments of the Old Indic grammarians [Seite 355]
9.2.3 - Entries in Vedic dictionaries [Seite 356]
9.2.4 - Grammatical features of Vedic Sanskrit [Seite 356]
9.2.5 - Grassmann's qualifications for such a dictionary [Seite 356]
9.2.6 - Grassmann's Dictionary of Rig-Veda [Seite 357]
9.2.7 - Exemplary comparison of Grassmann's dictionary with the Petersburg dictionary by Otto Böhtlingk and Rudolph Roth, pt. 2. (1856-1858) [Seite 360]
9.2.8 - Recognition of the linguistic accomplishments [Seite 362]
9.3 - The Rigveda Dictionary from a modern viewpoint [Seite 363]
9.3.1 - Lemmas, forms and meaning [Seite 364]
9.3.1.1 - 1. Analysis of the entry [Seite 366]
9.3.1.2 - 2. Meaning entries [Seite 368]
9.3.1.3 - 3. Form entries [Seite 368]
9.3.2 - Metrical analysis [Seite 370]
9.3.3 - Prepositions, particles, etc. [Seite 371]
9.3.4 - Abstract language and German [Seite 373]
9.3.5 - The decisive year of 1875 [Seite 374]
9.4 - Grassmann's contribution to lexicography and the living-on of his ideas in the Salzburg Dictionary to the Rig-Veda [Seite 376]
9.4.1 - Introduction [Seite 376]
9.4.2 - Comparing Grassmann and RIVELEX from a modern lexicographical point of view [Seite 377]
9.4.2.1 - Pre-Lexicography [Seite 377]
9.4.2.2 - Elaboration of a macrostructure [Seite 378]
9.4.3 - Working out a microstructure [Seite 380]
9.4.4 - Final remarks [Seite 385]
10 - Hermann Grassmann's impact on music, computing and education [Seite 387]
10.1 - Calculation and emotion: Hermann Grassmann and Gustav Jacobsthal's musicology [Seite 388]
10.2 - Classification of complex musical structures by Grassmann schemes [Seite 398]
10.2.1 - Global compositions [Seite 398]
10.2.2 - Classification of global compositions [Seite 401]
10.2.3 - Grassmann's technique [Seite 404]
10.2.4 - The musical meaning of Grassmann's approach [Seite 405]
10.2.5 - Varèse's interpretation [Seite 407]
10.3 - New views of crystal symmetry guided by profound admiration of the extraordinary works of Grassmann and Clifford [Seite 410]
10.3.1 - Introduction [Seite 410]
10.3.2 - Computer visualization of crystal symmetry [Seite 411]
10.3.3 - Appendix. Clifford geometric algebra description of space groups [Seite 415]
10.3.3.1 - Cartan-Dieudonné and geometric algebra [Seite 415]
10.3.3.2 - Two-dimensional point groups [Seite 417]
10.3.3.3 - Three-dimensional point groups [Seite 418]
10.3.3.4 - Space groups [Seite 418]
10.3.4 - Acknowledgments [Seite 419]
10.4 - From Grassmann's vision to geometric algebra computing [Seite 420]
10.4.1 - Introduction [Seite 420]
10.4.2 - Benefits of conformal geometric algebra [Seite 421]
10.4.2.1 - Unification of mathematical systems [Seite 422]
10.4.2.2 - Intuitive handling of geometric objects [Seite 423]
10.4.2.3 - Intuitive handling of geometric operations [Seite 424]
10.4.2.4 - Robotics application example [Seite 424]
10.4.3 - Geometric algebra computing technology [Seite 425]
10.4.3.1 - Compilation [Seite 427]
10.4.3.2 - Adaptation to different parallel processor platforms [Seite 428]
10.4.4 - Conclusion [Seite 430]
10.5 - Grassmann, Pauli, Dirac: special relativity in the schoolroom [Seite 431]
10.5.1 - Introduction [Seite 431]
10.5.2 - Grassmann's mathematical parenthood [Seite 432]
10.5.3 - Space and perception [Seite 433]
10.5.4 - Mathematical models of space [Seite 433]
10.5.5 - Didactical aspects of the geometric product [Seite 436]
10.5.6 - The Quantum-mechanical misconception [Seite 438]
10.5.7 - Didactical aspects of special relativity [Seite 439]
10.5.8 - Spacetime algebra [Seite 440]
10.5.9 - The quantum-mechanical misconception revisited [Seite 442]
10.5.10 - Remark about the history of the interpretation of Dirac matrices [Seite 443]
10.5.11 - Main focus at school [Seite 444]
11 - Appendix [Seite 448]
11.1 - On the concept and extent of pure theory of number (1827) [Seite 450]
11.1.1 - The three orders of enumeration [Seite 459]
11.1.2 - The general conjunction [Seite 463]
11.1.3 - The types of calculation [Seite 463]
11.1.4 - Survey of the types of calculation [Seite 464]
11.1.5 - Mechanical conjunction [Seite 465]
11.1.6 - Chemical conjunction [Seite 466]
11.1.7 - Dynamic conjunction [Seite 471]
11.1.8 - On the negative numbers [Seite 474]
11.1.9 - Proof that there can be no conjunction higher than exponentiation [Seite 478]
11.1.10 - Concluding remarks [Seite 481]
11.2 - Remarks on illustrations [Seite 483]
11.3 - Notes on contributors [Seite 498]
11.4 - References [Seite 517]
11.5 - Index of names and citations [Seite 545]
