Mathematician and philologist, born in Szczecin, Poland (formerly Stettin, Germany). He studied at Berlin, and spent most of his life as a teacher there and in Stettin. His book Die lineale Ausdehnungslehre (1844, The Theory of Linear Extension) developed a general calculus for vectors. It made little impact during his life, and it is only since his death that its importance has gradually been recognized; it anticipated much later work in quaternions, vectors, tensors, matrices, and differential forms. From 1849 he studied Sanskrit and other ancient Indo-European languages and, unlike his mathematics, his work in Indo-European and Germanic philology met with immediate acceptance.
Hermann Günther Grassmann (April 15, 1809, Stettin – September 26, 1877, Stettin) was a German polymath, renowned in his day as a linguist and now admired as a mathematician.
Biography
Hermann 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.
This entry dwells on the details of Grassman's career more than usual, because his mathematical work was not recognized in his lifetime. Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to Prussian universities.
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.
In 1834 Grassmann began teaching mathematics at the Gewerbeschule in Berlin. 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. 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 "... Kummer's report ended any chance that Grassmann might obtain a university post. 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.
Grassmann had eleven children, seven of which reached adulthood. A son, Hermann Ernst Grassmann, became a professor of mathematics at the University of Giessen.
Mathematician
One of then many examination Grassmann sat required that he submit an essay on the theory of the tides.
In 1844, Grassmann published his masterpiece, his Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, hereinafter denoted A1 and commonly referred to as the Ausdehnungslehre, which translates as "theory of extension" or "theory of extensive magnitudes." 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;
Fearnley-Sander (1979) describes Grassmann's foundation of linear algebra as follows:
"The definition of a linear space (vector space)... In fact, such a definition had been given thirty years previously by Peano, who was thoroughly acquainted with Grassmann's mathematical work. Grassmann did not put down a formal definition --- the language was not available --- but there is no doubt that he had the concept."
"Beginning with a collection of 'units' e1, e2, e3, ..., he effectively defines the free linear space which they generate; "
"...few have come closer than Hermann Grassmann to creating, single-handedly, a new subject."
Following an idea of Grassmann's father, 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 Euclidian 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. Grassmann submitted it as a Ph. 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 Die Geometrische Analyse geknüpft und die von Leibniz Characteristik, was the winning entry. Grassmann's entry was the only one. 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; 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. 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 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. developed some of Hermann Grassmann's algebras and Hamilton's quaternions. Hankel was the first to recognise the significance of Grassmann's long-neglected writings ... New York: 1970-1990)
Grassmann's mathematical methods were slow to be adopted but they directly influenced Felix Klein and Elie Cartan. For an introduction to the role of Grassmann's work in contemporary mathematical physics, see Penrose (2004: chpts.
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 is little reason to doubt him) that he had first developed these ideas in 1832.
Linguist
Disappointed at his inability to be recognized as a mathematician, Grassmann turned to historical linguistics. He devised a sound law of Indo-European languages, named Grassmann's Law in his honor.
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