Everything about Hermann Grassmann totally explained
Hermann Günther Grassmann (
April 15,
1809,
Stettin (
Szczecin) –
September 26,
1877,
Stettin) was a
German polymath, renowned in his day as a
linguist and now admired as a
mathematician. He was also a
physicist,
neohumanist, general scholar, and publisher. His mathematical work wasn't recognized in his lifetime.
Biography
Grassmann was the third of 12 children of Justus Günter Grassmann, an ordained minister who taught mathematics and physics at the
Stettin Gymnasium, where Hermann was educated. Hermann often collaborated with his brother Robert.
Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to
Prussian universities. Beginning in 1827, he studied
theology at the
University of Berlin, also taking classes in
classical languages,
philosophy, and literature. He doesn't appear to have taken courses in
mathematics or
physics.
Although lacking university training in mathematics, it was the field that most interested him when he returned to Stettin in 1830 after completing his studies in Berlin. After a year of preparation, he sat the examinations needed to teach mathematics in a gymnasium, but achieved a result good enough to allow him to teach only at the lower levels. In the spring of 1832, he was made an assistant at the Stettin Gymnasium. Around this time, he made his first significant mathematical discoveries, ones that led him to the important ideas he set out in his 1844 paper referred to as
A1 (see below).
In 1834 Grassmann began teaching mathematics at the Gewerbeschule in Berlin. A year later, he returned to Stettin to teach mathematics, physics, German, Latin, and religious studies at a new school, the Otto Schule. This wide range of topics reveals again that he was qualified to teach only at a low level. Over the next four years, Grassmann passed examinations enabling him to teach
mathematics,
physics,
chemistry, and
mineralogy at all secondary school levels.
Grassmann felt somewhat aggrieved that he was writing innovative mathematics, but taught only in secondary schools. Yet he did rise in rank, even while never leaving Stettin. In 1847, he was made an "Oberlehrer" or head teacher. In 1852, he was appointed to his late father's position at the Stettin Gymnasium, thereby acquiring the title of Professor. In 1847, he asked the Prussian Ministry of Education to be considered for a university position, whereupon that Ministry asked
Kummer for his opinion of Grassmann.
Kummer wrote back saying that Grassmann's 1846 prize essay (see below) contained "... commendably good material expressed in a deficient form." Kummer's report ended any chance that Grassmann might obtain a university post. This episode proved the norm; time and again, leading figures of Grassmann's day failed to recognize the value of his mathematics.
During the political turmoil in Germany, 1848-49, Hermann and Robert Grassmann published a Stettin newspaper calling for
German unification under a
constitutional monarchy. (This eventuated in 1866.) After writing a series of articles on
constitutional law, Hermann parted company with the newspaper, finding himself increasingly at odds with its political direction.
Grassmann had eleven children, seven of whom reached adulthood. A son, Hermann Ernst Grassmann, became a professor of mathematics at the
University of Giessen.
Mathematician
One of the many examinations for which Grassmann sat, required that he submit an essay on the theory of the tides. In 1840, he did so, taking the basic theory from
Laplace's
Mécanique céleste and from
Lagrange's
Mécanique analytique, but expositing this theory making use of the
vector methods he'd been mulling over since 1832. This essay, first published in the
Collected Works of 1894-1911, contains the first known appearance of what are now called
linear algebra and the notion of a
vector space. He went on to develop those methods in his
A1 and
A2.
In 1844, Grassmann published his masterpiece, his
Die Lineare Ausdehnungslehre, ein neuer Zweig der Mathematik [TheTheory of Linear Extension, a New Branch of Mathematics], hereinafter denoted
A1 and commonly referred to as the
Ausdehnungslehre, which translates as "theory of extension" or "theory of extensive magnitudes." Since
A1 proposed a new foundation for all of mathematics, the work began with quite general definitions of a philosophical nature. Grassmann then showed that once
geometry is put into the algebraic form he advocated, then the number three has no privileged role as the number of spatial
dimensions; the number of possible dimensions is in fact unbounded.
Fearnley-Sander (1979)
describes Grassmann's foundation of linear algebra as follows:
A1 also defined the
exterior product, also called "combinatorial product" (In German:
äußeres Produkt or
kombinatorisches Produkt), the key operation of an algebra now called
exterior algebra. (One should keep in mind that in Grassmann's day, the only
axiomatic theory was
Euclidean geometry, and the general notion of an
abstract algebra had yet to be defined.) In 1878,
William Kingdon Clifford joined this exterior algebra to
William Rowan Hamilton's
quaternions by replacing Grassmann's rule
epep = 0 by the rule
epep = 1.
(For quaternions, we've the rule i
2 = j
2 = k
2 = -1.) For more details, see
exterior algebra.
A1 was a revolutionary text, too far ahead of its time to be appreciated. Grassmann submitted it as a
Ph. D. thesis, but
Möbius said he was unable to evaluate it and forwarded it to
Ernst Kummer, who rejected it without giving it a careful reading. Over the next 10-odd years, Grassmann wrote a variety of work applying his theory of extension, including his 1845
Neue Theorie der Elektrodynamik and several papers on algebraic curves and surfaces, in the hope that these applications would lead others to take his theory seriously.
In 1846,
Möbius invited Grassmann to enter a competition to solve a problem first proposed by
Leibniz: to devise a geometric calculus devoid of coordinates and metric properties (what Leibniz termed
analysis situs). Grassmann's
Geometrische Analyse geknüpft an die von Leibniz erfundene geometrische Charakteristik, was the winning entry (also the only entry). Moreover, Möbius, as one of the judges, criticized the way Grassmann introduced abstract notions without giving the reader any intuition as to why those notions were of value.
In 1853, Grassmann published a theory of how colors mix; it and its three color laws are still taught, as
Grassmann's law. Grassman's work on this subject was inconsistent with that of
Helmholtz. Grassmann also wrote on
crystallography,
electromagnetism, and
mechanics.
Grassmann (1861) set out the first axiomatic presentation of arithmetic, making free use of the principle of induction.
Peano and his followers cited this work freely starting around 1890. Curiously, Grassmann (1861) has never been translated into English.
In 1862, Grassman published a thoroughly rewritten second edition of
A1, hoping to earn belated recognition for his theory of extension, and containing the definitive exposition of his
linear algebra. The result,
Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet [TheTheory of Extension, Thoroughly and Rigorously Treated], hereinafter denoted
A2, fared no better than
A1, even though
A2's manner of exposition anticipates the textbooks of the 20th century.
The only mathematician to appreciate Grassmann's ideas during his lifetime was
Hermann Hankel, whose 1867
Theorie der complexen Zahlensysteme helped make Grassmann's ideas better known. This work
Grassmann's mathematical methods were slow to be adopted but they directly influenced
Felix Klein and
Elie Cartan.
A. N. Whitehead's first monograph, the
Universal Algebra (1898), included the first systematic exposition in English of the theory of extension and the
exterior algebra. The theory of extension led to the development of
differential forms and to the application of such forms to
analysis and
geometry.
Differential geometry makes use of the
exterior algebra. For an introduction to the role of Grassmann's work in contemporary
mathematical physics, see Penrose (2004: chpts. 11, 12).
Adhémar Jean Claude Barré de Saint-Venant developed a vector calculus similar to that of Grassmann which he published in 1845. He then entered into a dispute with Grassmann about which of the two had thought of the ideas first. Grassmann had published his results in 1844, but Saint-Venant claimed (and there's little reason to doubt him) that he'd first developed these ideas in 1832.
Linguist
Disappointed at his inability to be recognized as a mathematician, Grassmann turned to historical
linguistics. He wrote books on German grammar, collected folk songs, and learned
Sanskrit. His dictionary and his translation of the
Ayurveda (still in print) were recognized among philologists. He devised a sound law of
Indo-European languages, named
Grassmann's Law in his honor.
These philological accomplishments were honored during his lifetime; he was elected to the
American Oriental Society and in 1876, he received a honorary doctorate from the
University of Tübingen.
Further Information
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