Hermann Günther Grassmann
Hermann Günther Grassmann
|Died||26 September 1877 68) (aged|
Stettin, German Empire
|Alma mater||University of Berlin|
|Known for||Multilinear algebra, Grassmannian, Exterior algebra|
|Awards|| PhD (Hon):|
University of Tübingen (1876)
Hermann Günther Grassmann (German: Graßmann; 15 April 1809 – 26 September 1877) was a German polymath, known in his day as a linguist and now also as a mathematician. He was also a physicist, neohumanist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties.
A polymath is a person whose expertise spans a significant number of subject areas, known to draw on complex bodies of knowledge to solve specific problems.
Linguistics is the scientific study of language. It involves analysing language form, language meaning, and language in context. The earliest activities in the documentation and description of language have been attributed to the 6th-century-BC Indian grammarian Pāṇini who wrote a formal description of the Sanskrit language in his Aṣṭādhyāyī.
Mathematics includes the study of such topics as quantity, structure, space, and change.
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 is a type of school with a strong emphasis on academic learning, and providing advanced secondary education in some parts of Europe comparable to British grammar schools, sixth form colleges and US preparatory high schools. In its current meaning, it usually refers to secondary schools focused on preparing students to enter a university for advanced academic study. Before the 20th century, the system of gymnasiums was a widespread feature of educational system throughout many countries of central, north, eastern, and south Europe.
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 does not appear to have taken courses in mathematics or physics.
Prussia was a historically prominent German state that originated in 1525 with a duchy centred on the region of Prussia on the southeast coast of the Baltic Sea. It was de facto dissolved by an emergency decree transferring powers of the Prussian government to German Chancellor Franz von Papen in 1932 and de jure by an Allied decree in 1947. For centuries, the House of Hohenzollern ruled Prussia, successfully expanding its size by way of an unusually well-organised and effective army. Prussia, with its capital in Königsberg and from 1701 in Berlin, decisively shaped the history of Germany.
Classics or classical studies is the study of classical antiquity. It encompasses the study of the Greco-Roman world, particularly of its languages and literature but also of Greco-Roman philosophy, history, and archaeology. Traditionally in the West, the study of the Greek and Roman classics was considered one of the cornerstones of the humanities and a fundamental element of a rounded education. The study of classics has therefore traditionally been a cornerstone of a typical elite education.
Physics is the natural science that studies matter, its motion, and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.
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. 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 references).
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. Over the next four years, Grassmann passed examinations enabling him to teach mathematics, physics, chemistry, and mineralogy at all secondary school levels.
Chemistry is the scientific discipline involved with elements and compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they undergo during a reaction with other substances.
Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical properties of minerals and mineralized artifacts. Specific studies within mineralogy include the processes of mineral origin and formation, classification of minerals, their geographical distribution, as well as their utilization.
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.
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.
Starting during the political turmoil in Germany, 1848–49, Hermann and his brother Robert published a Stettin newspaper, Deutsche Wochenschrift für Staat, Kirche und Volksleben , calling for German unification under a constitutional monarchy. (This eventuated in 1871.) 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.
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 is now called linear algebra and the notion of a vector space. He went on to develop those methods in his A1 and A2 (see bibliography).
In 1844, Grassmann published his masterpiece, his Die Lineale 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, 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:
|“||The definition of a linear space (vector space)... became widely known around 1920, when Hermann Weyl and others published formal definitions. 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; 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. He defines the notions of subspace, linear independence, span, dimension, join and meet of subspaces, and projections of elements onto subspaces.
...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 Produktor 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 have the rule i2 = j2 = k2 = −1.) For more details, see exterior algebra.
A1 was a revolutionary text, too far ahead of its time to be appreciated. When Grassmann submitted it to apply for a professorship in 1847, the ministry asked Ernst Kummer for a report. Kummer assured that there were good ideas in it, but found the exposition deficient and advised against giving Grassmann a university position. Over the next 10-odd years, Grassmann wrote a variety of work applying his theory of extension, including his 1845 Neue Theorie der Elektrodynamikand 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). 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. Grassmann'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. Lloyd C. Kannenberg published an English translation of The Ausdehnungslehre and Other works in 1995 ( ISBN 0-8126-9275-6. -- ISBN 0-8126-9276-4).
In 1862, Grassmann 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 [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.
In 1840s, mathematicians were generally unprepared to understand Grassmann's ideas.In the 1860s and 1870s various mathematicians came to ideas similar to that of Grassmann's, but Grassmann himself was not interested in mathematics anymore.
One of the first mathematicians to appreciate Grassmann's ideas during his lifetime was Hermann Hankel, whose 1867 Theorie der complexen Zahlensysteme
In 1872 Victor Schlegel published the first part of his System der Raumlehre which used Grassmann's approach to derive ancient and modern results in plane geometry. Felix Klein wrote a negative review of Schlegel's book citing its incompleteness and lack of perspective on Grassmann. Schlegel followed in 1875 with a second part of his System according to Grassmann, this time developing higher geometry. Meanwhile, Klein was advancing his Erlangen Program which also expanded the scope of geometry.
Comprehension of Grassmann awaited the concept of vector spaces which then could express the multilinear algebra of his extension theory. To establish the priority of Grassmann over Hamilton, Josiah Willard Gibbs urged Grassmann's heirs to have the 1840 essay on tides published.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. With the rise of differential geometry the exterior algebra was applied to differential forms.
For an introduction to the role of Grassmann's work in contemporary mathematical physics see The Road to Realityby Roger Penrose.
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 that he had first developed these ideas in 1832.
Grassmann's mathematical ideas began to spread only towards the end of his life. Thirty years after the publication of A1 the publisher wrote to Grassmann: “Your book Die Ausdehnungslehre has been out of print for some time. Since your work hardly sold at all, roughly 600 copies were used in 1864 as waste paper and the remaining few odd copies have now been sold out, with the exception of the one copy in our library”.Disappointed by the reception of his work in mathematical circles, Grassmann lost his contacts with mathematicians as well as his interest in geometry. The last years of his life he turned to historical linguistics and the study of Sanskrit. He wrote books on German grammar, collected folk songs, and learned Sanskrit. He wrote a 2,000-page dictionary and a translation of the Rigveda (more than 1,000 pages) which earned him a membership of the American Orientalists' Society. In modern Rigvedic studies Grassmann's work is often cited. In 1955 the third edition of his dictionary to Rigveda was issued.
Grassmann also discovered a sound law of Indo-European languages, which was 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 an honorary doctorate from the University of Tübingen.
Extensive online bibliography, revealing substantial contemporary interest in Grassmann's life and work. References each chapter in Schubring.
|Wikiquote has quotations related to: Hermann Grassmann|
Extension, extend or extended may refer to:
Linear algebra is the branch of mathematics concerning linear equations such as
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u ∧ v, is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude of u ∧ v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. Like the cross product, the exterior product is anticommutative, meaning that u ∧ v = −(v ∧ u) for all vectors u and v, but, unlike the cross product, the exterior product is associative. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having the same area, and with the same orientation—a choice of clockwise or counterclockwise.
In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multivectors with Grassmann algebra.
William Kingdon Clifford was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular have been of ever increasing importance to mathematical physics, geometry, and computing. Clifford was the first to suggest that gravitation might be a manifestation of an underlying geometry. In his philosophical writings he coined the expression "mind-stuff".
Friedrich Wilhelm Karl Ernst Schröder was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic, by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce. He is best known for his monumental Vorlesungen über die Algebra der Logik, in three volumes, which prepared the way for the emergence of mathematical logic as a separate discipline in the twentieth century by systematizing the various systems of formal logic of the day.
David Orlin Hestenes, Ph.D. is a theoretical physicist and science educator. He is best known as chief architect of geometric algebra as a unified language for mathematics and physics, and as founder of Modelling Instruction, a research-based program to reform K–12 Science, Technology, Engineering, and Mathematics (STEM) education.
The Calculus ratiocinator is a theoretical universal logical calculation framework, a concept described in the writings of Gottfried Leibniz, usually paired with his more frequently mentioned characteristica universalis, a universal conceptual language.
Hermann Hankel was a German mathematician who was born in Halle, Germany and died in Schramberg, Imperial Germany.
In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.
In mathematics, a versor is a quaternion of norm one.
In mathematics, a Grassmann–Cayley algebra is the exterior algebra with an additional product, which may be called the shuffle product or the regressive product. It is the most general structure in which projective properties are expressed in a coordinate-free way. The technique is based on work by German mathematician Hermann Grassmann on exterior algebra, and subsequently by British mathematician Arthur Cayley's work on matrices and linear algebra. It is a form of modeling algebra for use in projective geometry.
A rotor is an object in geometric algebra that rotates any blade or general multivector about the origin. They are normally motivated by considering an even number of reflections, which generate rotations.
A scientific society, the Quaternion Society was an "International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics". At its peak it consisted of about 60 mathematicians spread throughout the academic world that were experimenting with quaternions and other hypercomplex number systems. The guiding light was Alexander Macfarlane who served as its Secretary initially, and became President in 1909. The Association published a Bibliography in 1904 and a Bulletin from 1900 to 1913.
Hermann Rothe was an Austrian mathematician.
Victor Schlegel (1843–1905) was a German mathematician. He is remembered for promoting the geometric algebra of Hermann Grassmann and for a method of visualizing polytopes called Schlegel diagrams.
Robert Hermann is an American mathematician and mathematical physicist. In the 1960s Hermann worked on elementary particle physics and quantum field theory, and published books which revealed the interconnections between vector bundles on Riemannian manifolds and gauge theory in physics, before these interconnections became "common knowledge" among physicists in the 1970s.
Homersham Cox (1857–1918) was an English mathematician.
Samuel James Patterson is a Northern Irish mathematician specializing in analytic number theory. He has been a professor at the University of Göttingen since 1981.