Petsche / Russ / Lewis From Past to Future: Graßmann's Work in Context

Graßmann Bicentennial Conference, September 2009
1. Auflage 2010
ISBN: 978-3-0346-0405-5
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 580 Seiten, eBook

On the occasion of the 200th anniversary of the birth of Hermann Graßmann (1809-1877), an interdisciplinary conference was held in Potsdam, Germany, and in Graßmann's hometown Szczecin, Poland. The idea of the conference was to present a multi-faceted picture of Graßmann, and to uncover the complexity of the factors that were responsible for his creativity. The conference demonstrated not only the very influential reception of his work at the turn of the 20th century, but also the unexpected modernity of his ideas, and their continuing development in the 21st century.  This book contains 37 papers presented at the conference. They investigate the significance of Graßmann's work for philosophical as well as for scientific and methodological questions, for comparative philology in general and for Indology in particular, for psychology, physiology, religious studies, musicology, didactics, and, last but not least, mathematics. In addition, the book contains numerous illustrations and English translations of original sources, which are published here for the first time. These include life histories of Graßmann (written by his son Justus) and of his brother Robert (written by Robert himself), as well as the paper "On the concept and extent of pure theory of number'' by Justus Graßmann (the father).

Zielgruppe

Research

Autoren/Hrsg.

Russ, Steve

Liesen, Jörg

Petsche, Hans-Joachim

Lewis, Albert C.

Inhaltsverzeichnis

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