Citizendia
Your Ad Here

Hermann Günther Grassmann
Hermann Günther Grassmann
Hermann Günther Grassmann
Born April 15, 1809
Stettin (Szczecin)
Died September 26, 1877
Stettin
Residence German

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. Events 1450 - Battle of Formigny: Toward the end of the Hundred Years' War, the French attack and nearly annihilate English Year 1809 ( MDCCCIX) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Common year Events 46 BC - Julius Caesar dedicates a Year 1877 ( MDCCCLXXVII) was a Common year starting on Monday (link will display the full calendar of the Gregorian calendar (or a Common Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. Events 1450 - Battle of Formigny: Toward the end of the Hundred Years' War, the French attack and nearly annihilate English Year 1809 ( MDCCCIX) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Common year Events 46 BC - Julius Caesar dedicates a Year 1877 ( MDCCCLXXVII) was a Common year starting on Monday (link will display the full calendar of the Gregorian calendar (or a Common Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. A polymath ( Greek polymathēs, πολυμαθής "having learned much" is a person whose knowledge is not restricted to one subject area Linguistics is the scientific study of Language, encompassing a number of sub-fields Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and He was also a physicist, neohumanist, general scholar, and publisher. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Humanism is a broad category of ethical philosophies that affirm the dignity and worth of all people based on the ability to determine right and wrong by appealing to universal His mathematical work was not recognized in his lifetime.

Contents

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. A gymnasium (pronounced with ɡ- in several languages is a type of school providing Secondary education in some parts of Europe, comparable to English grammar 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. Prussia ( Latin: Borussia, Prutenia; Prūsija Prūsija Prusy Old Prussian: Prūsa) was most recently a historic state Beginning in 1827, he studied theology at the University of Berlin, also taking classes in classical languages, philosophy, and literature. Theology is the study of a god or the gods from a religious perspective "Classical literature" redirects here For literature in Classical languages outside the Graeco-Roman sphere see Ancient literature. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language He does not appear to have taken courses in mathematics or physics. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Mineralogy is an Earth Science focused around the Chemistry, Crystal structure, and physical (including optical) properties of Minerals

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. The unification of Germany took place on January 18, 1871, when Prussian Chief Minister Otto von Bismarck managed to unify a number of independent A constitutional monarchy, or a limited monarchy, is a form of Constitutional Government, wherein either an elected or hereditary Monarch is (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. Constitutional law is the study of foundational or basic Laws of nation states and other political organizations

Grassmann had eleven children, seven of whom reached adulthood. A son, Hermann Ernst Grassmann, became a professor of mathematics at the University of Giessen. The University of Gießen (German Universität Gießen) is officially called Justus Liebig-Universität Gießen after its most famous member

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 had 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. Linear algebra is the branch of Mathematics concerned with In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added 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 [The Theory 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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

Fearnley-Sander (1979) describes Grassmann's foundation of linear algebra as follows:

“ The definition of a linear space (vector space). In Mathematics a linear space can mean one of two things In Linear algebra or Mathematical analysis, a Vector space In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added . . became widely known around 1920, when Hermann Weyl and others published formal definitions. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. In fact, such a definition had been given thirty years previously by Peano, who was thoroughly acquainted with Grassmann's mathematical work. Giuseppe Peano ( August 27, 1858 &ndash April 20, 1932) was an Italian Mathematician, whose work was of exceptional 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; that is to say, he considers formal linear combinations a1e1 + a2e2 + a3e3 + . . . where the aj are real numbers, defines addition and multiplication by real numbers [in what is now the usual way] and formally proves the linear space properties for these operations. . . . He then develops the theory of linear independence in a way which is astonishingly similar to the presentation one finds in modern linear algebra texts. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors He defines the notions of subspace, independence, span, dimension, join and meet of subspaces, and projections of elements onto subspaces. Subspace may refer to;Mathematics Euclidean subspace, in linear algebra a set of vectors in n -dimensional Euclidean space that is closed under addition Independence is the Self-government of a Nation, Country, or State by its residents and population or some portion thereof generally exercising In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

. . . 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 Euclidean geometry, and the general notion of an abstract algebra had yet to be defined. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules ) 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. William Kingdon Clifford FRS ( May 4, 1845 &ndash March 3, 1879) was an English Mathematician and Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician (For quaternions, we have the rule i2 = j2 = k2 = -1. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician ) 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. "PhD" redirects here for other uses see PhD (disambiguation. August Ferdinand Möbius ( November 17, 1790 &ndash September 26, 1868, (ˈmøbiʊs was a German Mathematician and Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. 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). August Ferdinand Möbius ( November 17, 1790 &ndash September 26, 1868, (ˈmøbiʊs was a German Mathematician and 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. In Optics, Grassmann's law is an empirical result about human color perception chromatic response is (approximately Linear. Grassman's work on this subject was inconsistent with that of Helmholtz. Grassmann also wrote on crystallography, electromagnetism, and mechanics. Crystallography is the experimental science of determining the arrangement of Atoms in Solids In older usage it is the scientific study of Crystals The Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of Mechanics ( Greek) is the branch of Physics concerned with the behaviour of physical bodies when subjected to Forces or displacements

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. Giuseppe Peano ( August 27, 1858 &ndash April 20, 1932) was an Italian Mathematician, whose work was of exceptional 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. Linear algebra is the branch of Mathematics concerned with The result, Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet [The Theory 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. Hermann Hankel ( February 14, 1839 - August 29, 1873) was a German Mathematician who was born in Halle, This work

“ . . . developed some of Hermann Grassmann's algebras and Hamilton's quaternions. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Hankel was the first to recognise the significance of Grassmann's long-neglected writings . . . " (Hankel entry in the Dictionary of Scientific Biography. New York: 1970-1990) ”

Grassmann's mathematical methods were slow to be adopted but they directly influenced Felix Klein and Elie Cartan. Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group Élie Joseph Cartan ( 9 April 1869 &ndash 6 May 1951) was an influential French Mathematician, who did fundamental 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. Alfred North Whitehead, OM ( February 15 1861, Ramsgate, Kent, England &ndash December 30 1947, The theory of extension led to the development of differential forms and to the application of such forms to analysis and geometry. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is Analysis has its beginnings in the rigorous formulation of Calculus. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Differential geometry makes use of the exterior algebra. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry For an introduction to the role of Grassmann's work in contemporary mathematical physics, see Penrose (2004: chpts. Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. 11, 12).

Adhémar Jean Claude Barré de Saint-Venant developed a vector calculus similar to that of Grassmann which he published in 1845. Adhémar Jean Claude Barré de Saint-Venant ( August 23, 1797 - January 1886 was a Mechanician and mathematician who contributed to early Stress analysis 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. Linguistics is the scientific study of Language, encompassing a number of sub-fields He wrote books on German grammar, collected folk songs, and learned Sanskrit. Sanskrit (sa संस्कृता वाक् saṃskṛtā vāk, for short sa संस्कृतम् saṃskṛtam) is a historical His dictionary and his translation of the Ayurveda (still in print) were recognized among philologists. Ayurveda ( Devanāgarī: आयुर्वॆद the 'science of life' is a system of Traditional medicine native to India, and practiced in other He devised a sound law of Indo-European languages, named Grassmann's Law in his honor. Grassmann's law, named after its discoverer Hermann Grassmann, is a Dissimilatory Phonological process in Ancient Greek and Sanskrit 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. The American Oriental Society was chartered under the laws of Massachusetts on September 7, 1842. Eberhard Karls University of Tübingen ( German: Eberhard Karls Universität Tübingen, sometimes called the "Eberhardina Carolina" is a public university

See also

References

Primary:

Secondary:

Extensive online bibliography, revealing substantial contemporary interest in Grassmann's life and work. References each chapter in Schubring.

External links

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
 Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic