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Hermann Graßmann

Albert C. Lewis

Editors

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Hans-Joachim Petsche

Steve Russ

Work in ContextFrom Past to Future: Graßmann's

Graßmann Bicentennial Conference, September 2009

Jörg Liesen

EditorsHans-Joachim Petsche

14469 PotsdamGermany

Albert C. LewisEducational Advancement FoundationRio Grande St. 2303Austin, TX 78705USAalewis@edu-adv-foundation.org

Jörg LiesenTU BerlinInstitut für MathematikStraße des 17. Juni 13610623 BerlinGermany

Steve RussUniversity of WarwickDept. Computer ScienceCoventry CV4 7ALUnited Kingdomsbr@dcs.warwick.ac.uk

ISBN 978-3-0346-0404-8 e-ISBN 978-3-0346-0405-5DOI 10.1007/978-3-0346-0405-5

cThis work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting,reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permissionof the copyright owner must be obtained

Cover figure: Hermann Günther Graßmann, xylograph after a photograph from 1874 (mirror inverted).Source: Hermann Graßmann. Gesammelte mathematische und physikalische Werke. Bd. 1.1. Herausgeg.

the University of Potsdam (Communs of the New Palace at Park Sanssouci, a UNESCO World HeritageSite).

Printed on acid-free paper

Springer Based AG is part of Springer Science+Business Media

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von Fr. Engel unter Mitwirkung von E. Study, Leipzig 1894; background: The New Palace campus of

Universität PotsdamInstitut für PhilosophieAm Neuen Palais 10

petsche@uni-potsdam.de liesen@math.tu-berlin.de

© Springer Basel AG 2011

Contents

Preface........................................................................................................ xi

Abbreviations for works of Hermann Grassmann . ........................................ xvii

On the lives of the Grassmann brothers ....... ......................... ........ 1

Description of the life of Hermann Grassmann by his son JustusGrassmann, probably written shortly after the death of hisfather, 1877 ................................................................................................ 3

Life history of Robert Grassmann, written by himself (1890) . ....................... 9

Historical contexts of Hermann Grassmann’s creativity ........ ........ 17

Discovering Robert Grassmann (1815–1901) .............................................. 19Ivor Grattan-Guinness

Hermann Grassmann’s theory of religion and faith....................................... 37Johann Ev. Hafner

The Significance ofNaturphilosophie for Justus and HermannGrassmann ................................................................................................. 49Marie-Luise Heuser

Justus andHermann Grassmann: philosophy andmathematics .................... 61Michael Otte

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Institutional development of science in Stettin in the first halfof the nineteenth century in the time of Hermann Grassmann . ..................... 71Maciej Szukała

Philosophical and methodological aspects of the workof the Grassmann brothers ......................... ......................... ........ 85

Brief outline of a history of the geneticmethod in the development of the deductive sciences ................................... 87Lyubov Gavrilovna Biryukova

Grassmann’s epistemology: multiplication and constructivism ..................... 91Paola Cantù

Axiomatics and self-referenceReflections about Hermann Grassmann’s contributionto axiomatics ............................................................................................... 101Mircea Radu

Concepts and contrasts: Hermann Grassmann and BernardBolzano ...................................................................................................... 117Steve Russ

Diversity of the influence of the Grassmann brothers ............ ........129

New forms of science and new sciences of form: On thenon-mathematical reception of Grassmann’s work ....................................... 131Paul Ziche

Some philosophical influences of the Ausdehnungslehre ................................ 141Albert C. Lewis

Grassmann’s influence onHusserl ............................................................... 149Mirja Hartimo

Ernst Abbe’s reception of Grassmann in the lightof Grassmann’s reception of Schleiermacher ................................................ 161Hans-Joachim Petsche

Contents vii

On the early appraisals in Russia of H. and R. Grassmann’sachievements .............................................................................................. 175Boris Biryukov and Zinaida Kuzicheva

Hermann Grassmann’s Work and the Peano School..................................... 185Paolo Freguglia

Did Gibbs influence Peano’s “Calcolo geometrico secondol’Ausdehnungslehre di H. Grassmann : : :”? ................................................... 197Filippo Demonte-Barbera

Rudolf Mehmke, an outstanding propagator of Grassmann’svector calculus ............................................................................................ 209Karin Reich

Robert and Hermann Grassmann’s influence on the historyof formal logic ............................................................................................ 221Volker Peckhaus

Hermann Grassmann’s contribution to Whitehead’sfoundations of logic and mathematics .......................................................... 229Jacques Riche

Present and future of Hermann Grassmann’s ideasin mathematics .................. ......................... ......................... ........241

Grassmann’s legacy ..................................................................................... 243David Hestenes

OnGrassmann’s regressive product ............................................................. 261Henry Crapo

Projective geometric theorem proving with Grassmann–Cayleyalgebra. ....................................................................................................... 275Hongbo Li

Grassmann, geometry andmechanics ........................................................... 287John Browne

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Representations of spinor groups using Grassmann exterioralgebra. ....................................................................................................... 303Nikolay Marchuk

Hermann Grassmann’s theory of linear transformations ............................... 311Jörg Liesen

The Golden Gemini Spiral .......................................................................... 325A. Swimmer

A short note on Grassmann manifolds with a viewto noncommutative geometry ...................................................................... 333Jochen Pfalzgraf

Present and future of Hermann Grassmann’s ideasin philology ....................... ......................... ......................... ........343

Hermann Grassmann: his contributions to historical linguisticsand speech acoustics.................................................................................... 345John J. Ohala

Grassmann’s “Wörterbuch des Rig-Veda” (Dictionaryof Rig-Veda): a milestone in the study of Vedic Sanskrit ............................... 353Maria Kozianka

The RigvedaDictionary from a modern viewpoint ....................................... 363Toshifumi Gotō

Grassmann’s contribution to lexicography and the living-onof his ideas in the Salzburg Dictionary to the Rig-Veda ................................. 377Thomas Krisch, Christina Katsikadeli,and Stefan Niederreiter

Hermann Grassmann’s impact on music, computingand education ................... ......................... ......................... ........389

Calculation and emotion: Hermann Grassmann and GustavJacobsthal’s musicology............................................................................... 391Peter Sühring

Contents ix

Classification of complex musical structures by Grassmannschemes . ..................................................................................................... 401Guerino Mazzola

New views of crystal symmetry guided by profound admirationof the extraordinary works of Grassmann and Clifford ................................. 413Eckhard Hitzer

FromGrassmann’s vision to geometric algebra computing............................ 423Dietmar Hildenbrand

Grassmann, Pauli, Dirac: special relativity in the schoolroom ....................... 435Martin Erik Horn

Appendix .......................... ......................... ......................... ........453

On the concept and extent of pure theory of number (1827) ......................... 455Justus Günther Grassmann (transl. by Lloyd Kannenberg)

Remarks on illustrations.............................................................................. 489

Notes on contributors ................................................................................. 505

References . ..................................................................................................... 525

Index of names and citations ........................................................................... 553

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Illustration 1. Conference venue at University of Potsdam

Preface

On the occasion of the 200th birthday of Hermann Günther Grassmann1 (1809–1877), an international and inter-disciplinary conference took place from the 16th tothe 19th September 2009 in Potsdam and Szczecin. For four days more than 75 scien-tists from four continents – mathematicians, logicians, soft- and hardware specialists,physicists, engineers, historians, philosophers, scholars of religious studies, pedagogues,Indologists, musicologists, and many more – participated in an inter-disciplinary dia-logue on the work of Grassmann and his significance for current research.

Shortly before this conference, the first comprehensive English-language biogra-phy of Grassmann appeared,Hermann Grassmann – Biography. In addition, the newlypublished volume Hermann Grassmann – Roots and Traces presents many previouslyunpublished documents on the life and work of Grassmann and his family.

The present book, which includes the Proceedings of the Bicentennial Confer-ence as well as additional documents related to Grassmann, is the third and concludingvolume of this Grassmann-Trilogy.2

Point of departure

The last major conference concerned with Grassmann was devoted to the 150thanniversary of the first publication of his Lineale Ausdehnungslehre. It took place in1994 on the island of Rügen. Two publications with first-class contributions resultedfrom this conference [Schreiber 1995; Schubring 1996a]. In the following 15 years

1 Like his father Justus Günther, Hermann Graßmann used to spell his name with the German letter “ß”.Hermann’s brother Robert and his fosterfather Friedrich Heinrich Gotthilf Grassmann preferred “ss”. Inthis book, we will write Grassmann and always use “ss”, except in some titles and quotations.2 The conference proceedings will be complemented by a video-recording of the conference on four DVDs,published simultaneously by Universitätsverlag Potsdam [Petsche and Lenke 2010].

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xii Preface

the Rügen conference has stimulated numerous new publications. Some of them havea mathematical approach, some deal with philosophical aspects, and some reflectpedagogical contexts of Grassmann’s works.

A Spanish translation of Grassmann’sLineale Ausdehnungslehre of 1844 by EmilioOscar Roxin appeared in 1947. A French translation of the same work by DominiqueFlament and Bernd Bekemeier followed in 1994. In the most ambitious translationproject of Grassmann’s works yet, Lloyd Kannenberg published in 1995 and 2000English translations of the 1844 and 1862 versions of the Ausdehnungslehre and sev-eral other important papers of Grassmann. Finally, in 2008 the Russian translationsof selected papers of Hermann and Robert Grassmann – thoroughly introduced andwith extensive commentary – were published by Boris Biryukov, Lyubov GavrilovnaBiryukova, and Zinaida Kuzicheva.

Conference topics

The idea of the conference was to review and present a multi-faceted picture ofHermann Grassmann. An initial goal was to uncover the complexity of factors thathadmade Grassmann’s creativity possible. In addition, we wanted to analyze the varietyof the influential roles that his ideas have played and continue to play up to the present.The Conference Program organized the range of subjects in four focal themes:

1 Historical contexts of Grassmann’s creativity.2 The diversity of Grassmann’s influence.3 Present and future of Grassmann’s ideas.4 Grassmann and his hometown: from past to future.

These focal themes evolved into the several sections of the Contents of the currentvolume.

While at the Rügen conference the mathematical and natural scientific worksof Grassmann stood at the center of the discussions, the Bicentennial Conferencestrived for a more comprehensive access to Grassmann’s works. In this conferencewe investigated the significance of his work for philosophical as well as for scientificand methodological questions, for comparative philology in general and for Indologyin particular, for psychology, physiology, religious studies, musicology, and didactics.Special attention was given to the question which influences of Grassmann’s work arenoteworthy for current scientific research. Accordingly, thanks to the breadth of thecontributions received, the subdivision of the focal points of this book arrived at a

Preface xiii

certain arrangement that permits the relevance of Grassmann to emerge more clearly.Readers interested in the natural sciences or mathematics and those who are morehumanities oriented will both find that this makes Grassmannmore accessible to them.

The 37 contributions of 40 authors from 15 countries presented in this volumeconvincingly demonstrate that Grassmann’s legacy is more important than ever.

Structure

The main part of this volume is formed by papers presented at the Bicentennial Con-ference. In order to introduce the reader to Hermann and Robert Grassmann, and toachieve a certain independence within the framework of the Grassmann-Trilogy, westart this volume with translations of two biographies: the previously unpublished lifehistory ofHermannGrassmann (written by his son Justus Grassmann), and the life his-tory of Robert Grassmann (written by himself and concealed in a footnote of his – nowscarcely available –Gebäude des Wissens).

An English translation of the paper Ueber den Begriff und Umfang der reinenZahlenlehre by Justus Grassmann (Robert and Hermann’s father) is added as anAppendix. This paper, in a translation prepared by Lloyd Kannenberg is published herefor the first time and offers a deep insight into the world of ideas of the father andpermits the extent of his influence on his sonHermann to be traced out.

To the book is appended an index of the references of all contributions. This offersan impression of the variety of publications about Grassmann, and simplifies furtherinvestigations. The name and citation indexes aid the classification of the individualsmentioned in the contributions. Moreover, from the citation directory the referencesof the individual contributions to the different sources can be conveniently found.

Acknowledgements

Neither the arrangements for the conference, nor the publication of this book, wouldhave been possible without the active support of the following institutions:

� German Research Foundation (DFG)� German Federal Ministry of Education and Research (BMBF)� University of Potsdam� University of Szczecin

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� Universitätsgesellschaft Potsdam e. V.� Birkhäuser Publishing Ltd.

We have been able to present in this volume several previously unpublished documentson the life of Grassmann thanks to the friendly support of librarians and archivists atthe following institutions:

� ArchivWalter de Gruyter of the Staatsbibliothek Berlin PK� Greifswald University Archive� Archiwum Państwowe in Szczecin� Książnica Pomorska im. Stanisława Staszica in Szczecin

Translations of the original documents on the lives of the Grassmann brothers as wellas the paper of Justus Grassmann on the pure theory of number were generously con-tributed by Prof. Lloyd Kannenberg. In addition, we thank Prof. Kannenberg forreviewing several of the manuscripts.

Further thanks are owed to Peter Lenke, Mark Minnes, and John Ohala, whostood by our side with help in checking through the manuscripts, Gottfried Kessler,who was helpful with accessing the archival sources, as well as Matthes Uhlmann, whocontributed to the editing of the visual materials. We are also pleased to thank every-body who supported our authors and who contributed to the entire project by checkingfacts, voicing criticism, or being involved in some other way.

Finally, our warm thanks are due to Dr. Karin Neidhart, Birkhäuser’s editor of theentire project of the three Grassmann volumes:

Hermann Graßmann – BiographyHermann Graßmann – Roots and TracesHermann Graßmann – From Past to FutureShe has facilitated, guided, supported, and helpfully promoted these volumes at

every stage.

For the editors, Hans-Joachim Petsche

On16November 2010, theGraßmann-Conference received the PotsdamConference-Award from themayor of Potsdam, the capital of the federal state of Brandenburg, makingit the first-time winner in the category “Innovative and/or exceptional events”.

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Illustration 2. Opening session of the Grassmann Bicentennial Conference in Potsdam

Illustration 3. Hermann Grassmann’s great-grandson Dietrich Grassmann and the main orga-nizer of the conference Hans-Joachim Petsche in front of a picture showing the Stettin Masoniclodge garden in the nineteenth century

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Illustration 4. Inauguration of a Grassmann exhibition unveiling recently discovereddocuments in the Pomeranian Library (during the conference’s stay in Szczecin)

Illustration 5. Participants of the Grassmann Conference in front of the former Marienstifts-gymnasium in Szczecin

Abbreviations for works of HermannGrassmann

A1 Grassmann, Hermann. 1844. Die Wissenschaft der extensiven Grösse oderdie Ausdehnungslehre, eine neue mathematische Disciplin. 1. Aufl. Leipzig:Wigand (2. Aufl. Leipzig 1878). [Reprinted in: GW11, 4–312].

A1K Grassmann, Hermann. 1995b. Linear extension theory (Die Lineale Aus-dehnungslehre, 1844), transl. Lloyd C. Kannenberg. In [H. Grassmann1995a, 1–312].

A2 Grassmann, Hermann. 1862. Die Ausdehnungslehre. Vollständig und instrenger Form begründet. Berlin: Enslin. [Reprinted in: GW12, 1–383].

A2K Grassmann, Hermann. 2000. Extension theory (Die Ausdehnungslehre,1862), transl. Lloyd C. Kannenberg. American Mathematical Society,London Mathematical Society.

LA Grassmann, Hermann. 1861: Lehrbuch der Mathematik für höhereLehranstalten. Teil 1: Arithmetik, Berlin: Enslin.

RV1 Grassmann, Hermann. 1876. Rig-Veda. Übersetzt und mit kritischenund erläuternden Anmerkungen versehen von Hermann Grassmann. ErsterTheil. Die Familien-Bücher des Rig-Veda: (zweites bis achtes Buch). Leipzig:F. A. Brockhaus.

RV2 Grassmann, Hermann. 1877a. Rig-Veda. Übersetzt und mit kritischen underläuternden Anmerkungen versehen von Hermann Grassmann. ZweiterTheil. Sammelbücher des Rig-Veda: (erstes, neuntes, zehntes Buch). Leipzig:Brockhaus.

HGW11 Grassmann, Hermann. 1894. Gesammelte mathematische und physikalis-che Werke. Vol. 1.1. Ed. Friedrich Engel assisted by Eduard Study. Leipzig:Teubner.

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xviii Abbreviations for works of Hermann Grassmann

HGW12 Grassmann, Hermann. 1896. Gesammelte mathematische und physikalis-che Werke. Vol. 1.2. Ed. Friedrich Engel, assisted by Hermann Grassmann(Jun.). Leipzig: Teubner.

HGW21 Grassmann, Hermann. 1904. Gesammelte mathematische und physikalischeWerke. Vol. 2.1. Eds. Eduard Study, Georg Scheffers and Friedrich Engel.Leipzig: Teubner.

HGW22 Grassmann, Hermann. 1902. Gesammelte mathematische und physikalischeWerke. Vol. 2.2. Eds. Jakob Lüroth and Friedrich Engel. Leipzig: Teubner.

HGW31 Grassmann, Hermann. 1911. Gesammelte mathematische und physikalis-che Werke. Vol. 3.1. Eds. Justus Grassmann and Friedrich Engel. Leipzig:Teubner.

HGW32 Grassmann, Hermann. 1911. Gesammelte mathematische und physikalischeWerke. Vol. 3.2. Ed. Friedrich Engel. Leipzig: Teubner.

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Illustration 6. Pestalozzi in Stanz (lithography, 1846). An artistic representation of the manyfollowers of the pedagogue J. H. Pestalozzi (1746–1827). Among them, Hermann’s uncle,Friedrich Heinrich Gotthilf Grassmann (1784–1866). The picture documents the great influ-ence of progressive pedagogy on Hermann Grassmann’s uncle and on his father.Further information on p. 490

From Past to Future: Grabmann's Work in ContextContentsPrefaceAbbreviations for works of Hermann Grassmann

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