Hermann Grassmann

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Hermann Günther Grassmann

Hermann Günther Grassmann (15 de Abril, 1809, Stettin – 26 de Setembro, 1877, Stettin) foi um polímata Alemão, renomado em sua época como linguista e hoje admirado como matemático. Ele também foi um físico, neohumanista, "all-round scholar" e editor.

Grassmann foi o terceiro de doze filhos de Justus Günter Grassmann, um Pastor que ensinava Matemática e Física no Liceu onde o seu filho Hermann foi educado.

Colaborou frequentemente com o seu irmão Robert, razão pela qual o seu próprio trabalho no campo da Matemática nunca lhe foi reconhecido em vida.

Grassmann foi um aluno irrelevante até obter classificações altas nos exames de admissão para as Universidades Purssas.

Começando em 1827, estudou Teologia na Universidade de Berlim e e simultanemente aulas de Línguas Clássicas, Filosofia e Literatura. Aparentemente não cursou Matemática ou Física.

Apesar da lacuna universitária no estudo da Matemática, foi o campo que mais o interessou quando voltou a Stettin em 1830, após ter completado os seus estudos em Berlim.

Após um ano de preparação, ele realizou os exames necessários para ensinar Matemática num Liceu, tendo obtido um resultado que lhe permitiu ensinar sómente a níveis de aprendizagem mais baixos. Na primavera de 1832, tornou-se assistente no Liceu de Stettin. Nesta altura ele fez as suas primeiras descobertas significantes em Matemática que o conduziram a ideias importantes por ele confirmadas em 1844.

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." 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 which reached adulthood. A son, Hermann Ernst Grassmann, became a professor of mathematics at the University of Giessen.

[editar] Mathematician

One of then many examination 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 Méchanique céleste and from Lagrange's Méchanique analytique, but expositing this theory using 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. He went on to develop those methods in his A1 and A2.

In 1844, Grassmann published his masterpiece, his Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, hereinafter denoted A1 and commonly referred to as the Audehnungslehre, 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:

"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 a_j 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, 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 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 e_pe_p = 0 by the rule e_pe_p = 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 Die Geometrische Analyse geknüpft und die von Leibniz Characteristik, was the winning entry. There was a sting in the tail, however; 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; it and its three color laws are still taught. 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, 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

"... 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 ... " (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. 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 has been applied in the study of differential forms and in 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).

[editar] 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.

[editar] Ver também

exterior algebra
Grassmann number
Grassmannian
Grassmann's law
Grassmann laws (color)

[editar] Referências

Primary:

1844. Die lineale Ausdehnungslehre. Leipzig: Wiegand. English translation, 1995, by Lloyd Kannenberg, A new branch of mathematics. Chicago: Open Court. This is A1.
1861. Lehrbuch der Mathematik fur hohere Lehrenstalten, Band 1. Berlin: Enslin.
1862. Die Ausdehnungslehre, vollstandig und in strenger Form bearbeitet. Berlin: Enslin. English translation, 2000, by Lloyd Kannenberg, Extension Theory. American Mathematical Society. This is A2. Excerpt translated by D. Fearnley-Sander.
1894-1911. Gesammelte mathematische und physikalische Werke, in 3 vols. Friedrich Engel ed. Leipzig: B.G. Teubner. Reprinted 1972, New York: Johnson.

Secondary:

Crowe, Michael, 1967. A History of Vector Analysis. Notre Dame University Press.
Fearnley-Sander, Desmond, 1979, "Hermann Grassmann and the Creation of Linear Algebra," American Mathematical Monthly 86: 809-17.
--------, 1982, "Hermann Grassmann and the Prehistory of Universal Algebra," Am. Math. Monthly 89: 161-66.
-------, and Stokes, Timothy, 1996, "Area in Grassmann Geometry ". Automated Deduction in Geometry: 141-70
Roger Penrose, 2004. The Road to Reality. Alfred A. Knopf.
Schlege, Victor, 1878. Hermann Grassmann: Sein Leben und seine Werke. Leipzig: F.A. Brockhaus.
Schubring, G., ed., 1996. Hermann Gunther Grassmann (1809-1877): visionary mathematician, scientist and neohumanist scholar. Kluwer.
Hans-Joachim Petsche: Graßmann. Basel [usw.] 2006 (Vita Mathematica 13), ISBN 3-7643-7257-5

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

[editar] Ligações externas

The MacTutor History of Mathematics archive:
- Hermann Grassmann.
- Abstract Linear Spaces. Discusses the role of Grassmann and other 19th century figures in the invention of linear algebra and vector spaces.
Fearnley-Sander's home page.