Table Of ContentHans-Joachim Petsche
Hermann Graßmann
Biography
Translated by Mark Minnes
Scientific Consultants:
Lloyd Kannenberg and Steve Russ
Birkhäuser
Basel · Boston · Berlin
Author:
Hans-Joachim Petsche
Hessestraße 18
D-14469 Potsdam
Germany
e-mail: [email protected]
Library of Congress Control Number: 2009929497
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Cover figure: Hermann Günther Graßmann, xylograph after a photograph from 1874.
Source: Hermann Graßmann. Gesammelte mathematische und physikalische Werke. Bd. 1.1.
Herausgeg. von Fr. Engel unter Mitwirkung von E. Study, Leipzig 1894; background: see p. 60.
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Contents
Translator’s note ....................................................................................................VII
Foreword .................................................................................................................IX
Introduction ........................................................................................................XIII
Notes .........................................................................................................................................XIX
1 Graßmann’s life .................................................................................................1
1.1 The historical context .......................................................................................................1
1.2 The family: traditions and relatives ...............................................................................9
1.3 Graßmann’s youth and university years ......................................................................15
1.4 The path to independent mathematical achievements (1830 – 1840) .................26
1.5 Mathematical productivity and the struggle for recognition (1840 – 1848) ......33
1.6 The Revolution comes to Germany (1848) ...............................................................45
1.7 Renewed struggles for recognition as a mathematician ..........................................58
1.8 Farewell to mathematics, success as a philologist and late acclaim
in mathematics .................................................................................................................77
Notes .............................................................................................................................................89
2 Graßmann’s sources of inspiration ..............................................................101
2.1 Justus Graßmann: Father and precursor of his son’s mathematical
and philosophical views ..............................................................................................101
2.2 Robert Graßmann (1815 – 1901): Brother and collaborator .............................117
2.3 Friedrich Schleiermacher’s impact on Graßmann and fundamental
thoughts from his lectures on dialectics ..................................................................130
VI Contents
Notes ..........................................................................................................................................155
3 Hermann Günther Graßmann’s contributions to the development
of mathematics and their place in the history of mathematics ....................165
3.1 Some basic aspects concerning the development of geometry
from the 17th to the 19th century ...............................................................................166
3.2 Graßmann’s examination thesis on the theory of tides.........................................171
3.3 The 1844 Extension Theory and Graßmann’s theory of algebraic curves ..........175
3.4 The prize-winning treatise on geometric analysis (1847) ....................................193
3.5 The Extension Theory of 1862 ....................................................................................196
3.6 Work on the foundations of arithmetic (1861) .....................................................198
3.7 The impact of Graßmann’s ideas on the development of mathematics .............205
Notes ..........................................................................................................................................212
4 The genesis and essence of Hermann Günther Graßmann’s
philosophical views in the Extension Theory of 1844 ..................................221
4.1 The genesis of Extension Theory’s basic principles ..................................................222
4.2 Hermann Graßmann’s basic philosophical principles concerning
his determination of the essence of mathematics ...................................................227
4.3 Hermann Graßmann’s views on restructuring mathematics and
on locating Extension Theory ......................................................................................235
4.4 Hermann Graßmann’s views on the essence of the mathematical method
and its relation to the method of philosophy .........................................................241
4.5 Graßmann’s Extension Theory and Schleiermacher’s Dialectic ............................244
Closing remarks .......................................................................................................................248
Notes ..........................................................................................................................................249
Chronology of Graßmann’s life ............................................................................257
Abbreviations........................................................................................................263
Bibliography .........................................................................................................265
List of illustrations ................................................................................................293
List of persons mentioned ....................................................................................295
Translator’s note
Prof. Hans-Joachim Petsche’s Graßmann is a book on mathematics and German cul-
tural history. As a translator, I have attempted to make the text as accessible as possible
to the English-speaking readership. Some decisions should not go unmentioned.
Hermann Graßmann’s mathematical terminology is often quite extravagant and
unusual. In many cases, the translation also gives Graßmann’s original German concepts.
With his permission and generous cooperation, I have used Dr. Lloyd Kannenberg’s ter-
minology from his translations of Graßmann, e. g. displacement (“Strecke”), magnitude
(“Größe”), conjunction (“Verknüpfung”), evolution (“Änderung”), etc.
As many German-language books in the original bibliography as possible have
been cited from their English translations. In many cases, however, quotations had to be
translated and the footnotes still refer to the German originals. The Graßmann brothers
used an early edition of Schleiermacher’s Dialectic of which there is no English transla-
tion. Of course, the same is true of many other works and letters.
All titles of books and journal articles have been translated into English. The origi-
nal titles and the corresponding bibliographical references appear in parentheses and
quotation marks. In these cases, dates refer to the year of publication of the text quoted
in the bibliography.
Finally, we should not forget that today the formerly Prussian town of Stettin is the
Polish city of Szczecin. The use of the German name Stettin should not be construed as
questioning that historical fact.
This translation was a collaborative process. I would like to thank Dr. Steve Russ
(University of Warwick) and Dr. Lloyd Kannenberg (University of Massachusetts
Lowell) for sharing their expertise and doing the hard work of proofreading the manu-
script. I thank them for many pleasant discussions and their supportive approach to the
project. The author, Hans-Joachim Petsche, kept our little research team together and
showed warm and stimulating appreciation of our efforts.
Mark Minnes
Foreword
In Wilhelm Traugott Krug’s General Handbook of the Philosophical Sciences of 1827
(“Allgemeines Handwörterbuch der philosophischen Wissenschaften”), we find the fol-
lowing entry for the terms “mathematics” and “mathematical”:
“Mathematics … only [deals with] magnitudes which appear in time and space and
which therefore can be represented, counted and measured as numbers or figures… A
philosopher should familiarize himself with mathematics and a mathematician with
philosophy, as far as their talent, interests, time and surroundings will permit. But one
should not confuse and throw into one pot what the progress of scientific knowledge
has separated, and rightly so. …mathematical philosophy and philosophical mathemat-
ics – in the commonly accepted sense of the terms, namely as a mixture of both – are
scientific or, rather, unscientific monsters. They no more satisfy and please the educated
mind than could a human body consisting of a mixture of man and woman.”1
But this view did not prevent Hermann Graßmann2, a 35 year-old secondary
school teacher from the Prussian town of Stettin, from publishing a work of mathemat-
ics which, as he later remarked, “is certain to be more pleasing to more philosophically
inclined readers”.3 Graßmann’s book also claimed to have founded a new branch of sci-
ence “which extends and intellectualizes the sensual intuitions of geometry into gen-
eral, logical concepts, and, with regard to abstract generality, is not simply one among
the other branches of mathematics, such as algebra, combination theory, and func-
tion theory, but rather far surpasses them, in that all fundamental elements are unified
under this branch, which thus as it were forms the keystone of the entire structure of
mathematics.”4
Hermann Graßmann, a novice in mathematics whose name was completely unknown
in the mathematical community of his day, did not hesitate to send his book – Linear
Extension Theory, A New Branch of Mathematics (1844) – to the most famous math-
X Foreword
ematicians of his time. But their assessment of his work remained completely within the
framework of the Kantian view of mathematics, which we find in the quotation above.
This was a disaster for Graßmann. Among German mathematicians, August Ferdinand
Möbius was closest to Graßmann’s scientific perspective. Möbius told Apelt in a letter
that he had repeatedly attempted to understand Graßmann’s book, “…but I never got
beyond the first pages … since [the book] … lacks all intuitive clarity, which is the es-
sential characteristic of mathematical insight.”5 In a letter to Gauß, Möbius wrote that
Graßmann had “strayed from the firm foundations of mathematics”6. Johann August
Grunert wrote Graßmann: “I also would have hoped that you would have refrained
from getting so involved in philosophical reflections.”7 Ernst Friedrich Apelt, a friend of
Möbius, remarked that “Graßmann’s peculiar Extension Theory … seems to be built on a
wrong understanding of the philosophy of mathematics. …The abstract extension theory
he is looking for could only be developed from concepts. Concepts are not the source of
mathematical knowledge, but intuition.”8 Finally, Richard Baltzer came to the following
conclusion: “…I begin to feel dizzy in the head and disoriented when I read it.”9 Moritz
Cantor summed up the fate of Extension Theory in one simple sentence: “The book was
published in 1844 by O. Wigand in Leipzig, nobody reviewed it, nobody bought it, and
therefore the publisher destroyed the entire first edition!”10 Half a century later, nobody
doubted the importance of Graßmann’s mathematical work. On Felix Klein’s initiative,
a six-volume collection of Graßmann’s writings in mathematics and physics was pub-
lished between 1894 and 1911.11 Thanks to mathematicians such as Hermann Hankel,
Alfred Clebsch, Felix Klein and Friedrich Engel, Graßmann’s achievements concerning
the foundations of vector and tensor calculus, the development of n-dimensional affine
and projective geometry and his fundamental work in algebra and in other areas were
recognized in retrospect. Today, Graßmann has become a familiar name in mathemat-
ics. Nevertheless, many mathematicians are quite unfamiliar with his magnum opus in
mathematics. Even though “a general feeling of respect for this mathematician from
Stettin has spread in the scientific community”, as F. Engel remarked in 1911, “this feel-
ing of respect usually does not arise from knowledge of Graßmann’s writings but, rather,
is based on hearsay.”12 Graßmann’s Extension Theory of 1844 was ignored for over a
quarter of a century. Among other reasons, the general rejection of its philosophical
approach and its philosophical mode of presentation led to this lack of recognition.
Unhappily, this anti-philosophical attitude blinded mathematicians to the true value of
Extension Theory. A closer analysis of the book will show that Graßmann’s philosophical
and, to put it more precisely, dialectical approach to mathematical problems is exactly
what gave him the inspiration he needed to create and elaborate a new mathematical
discipline, namely vector and tensor calculus. What is more: Graßmann was capable of
building an unheard-of vector-algebraic theory of n dimensions because he was familiar
with the philosophical thinking of his time and because he consciously used dialectics,
Foreword XI
the philosophy of the increasingly dominant German bourgeoisie, as a method for es-
tablishing and presenting new insights.
The present book aims to critically appreciate and explain the life and work of
Hermann Graßmann (1809 – 1877).
Notwithstanding the fact that Graßmann has entered into the history of mathe-
matics as the founder of vector algebra, he still remains a relatively unknown figure. The
hundredth anniversary of his death in 1977 passed almost completely unnoticed. A
conference held in Germany on the occasion of the 150th anniversary of the publication
of Linear Extension Theory in May 1994 was one of the last major attempts to save his
name from oblivion. In September 2009 the Graßmann Bicentennial Conference in
Potsdam will commemorate the 200th anniversary of Graßmann’s birth and attempt to
contextualize his work from a present-day perspective.
The 19th century, in which Graßmann’s scientific creativity blossomed, is still a
highly promising area for future research. Few scholars have attempted to analyze the
historical interactions between philosophy and mathematics. Very much remains to be
done.13
These are plenty of reasons to have another look at Graßmann.
Introduction
Hermann Graßmann, born 200 years before the publication of the English edition of
this book, was one of the most extraordinary personalities in 19th century science. The
circumstances of his scientific achievements are no less remarkable than the results to
which he came. Graßmann, who had originally aspired to become a theologian and
remained an autodidact in mathematics and the natural sciences, was over 30 years old
when he turned to scientific research. With the exception of three years in B erlin as
a student, he spent virtually his entire life within the walls of his Pomeranian home-
town, Stettin, where he worked as a teacher in a “Gymnasium”, or secondary school.
He had literally no contact to the leading scientists and mathematicians of his time
and lived far away from the most important centers of scientific research. His personal
library contained only a few scientific works. Nevertheless, he was extraordinarily pro-
lific in his scientific work. Graßmann has gone down in history for his discoveries in
the theory of electricity, the theory of colors and of vowels. He was also among the pi-
oneers of comparative philology and of Vedaic research. In 1996, his dictionary of the
vocabulary of the Rig-Veda, a collection of pre-Buddhist religious hymns from India
(12th – 6th century BC), was reprinted for the sixth time.14 Nonetheless, Graßmann’s
main achievements lie in the field of mathematics. His two “Ausdehnungslehren”, or
Extension Theories (A1, A2), of 1844 and 1861 make him one of the founders of vector
and tensor calculus. Remarkably, he made these discoveries without having any con-
nection to the English mathematician W. R. Hamilton. A decade before B. Riemann,
Graßmann was the first mathematician to create a theory of n-dimensional manifolds
by generalizing traditional three-dimensional geometry. Even though the results of
his mathematical projects were not officially recognized for almost 30 years, they had
an enduring scientific impact on mathematicians such as Felix Klein, Giuseppe Peano,
Alfred North Whitehead, Élie Cartan, Hermann Hankel, Walter von Dyck, Josiah
