Herman Grassmann

Hermann Günther Grassmann (in German Herman Günther Graßman) (Stettin, April 15, 1809-ibid., September 26, 1877) was a linguist and German mathematician. He also worked as a physicist, humanist, scholar and editor, which is why he is considered a clear example of polymathy.

Biography

He was the third of twelve children born to Justus Günter Grassmann and Johanne Luise Friederike Medenwald. His mother was the daughter of a pastor from Klein-Schönfeld. His father had also been a consecrated pastor, but he got a professorship in mathematics and physics at the Stettin Institute, and was a noted academic, author of several Physics and Mathematics school textbooks, as well as carrying out interesting research in the field of crystallography. Another of Hermann's brothers, Robert, also took up mathematics and the two worked together on many projects.

During his youth, Hermann was educated by his mother, a woman of vast culture. He then attended a private school, before entering the Stettin Institute, where his father taught. Most mathematicians stand out before his teachers from a very young age. However, despite having extraordinary opportunities from a family prone to education, Hermann did not stand out in a special way in his high school years, to the point that his father thought that he should dedicate himself to some type of manual labor., such as gardener or craftsman.

Hermann appreciated music and learned to play the piano, while continuing his studies, in which little by little he was improving, and in the final exams of secondary studies, at the age of 18, he finished the second year of his promotion. After demonstrating his academic competence at the end of his studies, Hermann decided to study Theology, and in 1827 he moved to Berlin with his older brother to study at the University. He studied Theology, classical languages, Philosophy and Literature, and it does not seem that he attended any Mathematics or Physics class.

Although it seems clear that Hermann had no formal university training in mathematics, this was the subject that interested him most when he returned to Stettin in the autumn of 1830, having completed his university studies in Berlin. Obviously, the influence of his father in this way was very important, and he could have become a professor of mathematics, but he had already decided to carry out mathematical research on his own. After spending a year doing research in mathematics and preparing for the high school teacher exam, Hermann went to Berlin in December 1831 to sit the exams.

It seems that his written exercises must not have been highly valued, since his examiners gave him the title to teach only in the first levels of secondary school. She was told that before she could teach at the higher levels, she would have to re-examine herself and demonstrate greater knowledge in the subjects she had competed for. In the spring of 1832 he obtained an assistant professorship at the Stettin Institute.

It was around this time that he made his first two significant mathematical discoveries, which were destined to lead to the important ideas he would develop years later. On the premise of his Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics, 1844), Grassmann describes how he had arrived at these ideas already around the year 1832.

In 1834 Grassmann began teaching mathematics at the Gewerbeschule (School of Arts and Crafts) in Berlin. A year later he returned to Stettin to teach mathematics, physics, German, Latin and religion at a new educational centre, the Otto Schule . This great variety of subjects to teach is proof that he was still qualified only to teach classes in schools at the lowest levels. In the four years that followed, Grassmann passed the exams that allowed him to teach mathematics, physics, chemistry, and mineralogy at all levels of secondary schools.

Grassmann was somewhat frustrated with having to teach only at the high school level, despite being able to produce innovative mathematics. In 1847 he became "Oberlehrer". In 1852 he was assigned the post that his father had previously held at the Stettin Institute, and thereby obtained the title of professor. In 1847 he applied to the Prussian Minister of Education for consideration for a post as a university professor, and the minister asked Ernst Eduard Kummer for his opinion on Grassmann. Kummer replied by saying that Grassman's essay, which had won a prize in 1846, had "(...) good material inadequately expressed". This Kummer report put an end to Grassmann's hopes of ever getting a post as a university professor. This episode further confirms the fact that the authorities Grassmann contacted never recognized the real importance of his ideas.

During the political unrest in Germany in 1848-49, Hermann and Robert Grassmann edited a newspaper in Stettin to support the unification of Germany under a constitutional monarchy. After writing a series of articles on constitutional law, Hermann, less and less in agreement with the political line of the newspaper, left it.

Grassmann had eleven children, seven of whom reached adulthood. One of his sons, Hermann Ernst Grassmann, became a professor of mathematics at the University of Giessen.

Mathematician

Among the many topics Grassmann addressed is his essay on the theory of tides. He elaborated it in 1840, based on the theory of Lagrange's Méchanique analytique and Laplace's Méchanique céleste, but exposing this theory by vectorial methods, on which he worked since 1832. This essay, first published in the Collected Works of 1894-1911, contains the first written testimony of what is now known as linear algebra and the notion of vector space. Grassmann developed these methods in Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik and Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet.

In 1844, Grassmann published his masterpiece, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, better known as Ausdehnungslehre, which can be translated as " extension theory" or "theory of extensive magnitudes". After proposing in Ausdehnungslehre new bases for all mathematics, the work begins with definitions of a rather philosophical nature. Grassmann also showed that if geometry had been expressed in algebraic form as he proposed, the number three would not have played the privileged role it has as a number that expresses the spatial dimensions; in fact, the number of possible dimensions of interest to geometry is unlimited.

Fearnley-Sander (1979) describes the creation of Grassmann's linear algebra thus:

"The definition of linear space (...) is openly recognized around 1920, when Hermann Weyl and others published the formal definition. In fact, that definition had been formulated about thirty years earlier by Peano, who had thoroughly studied the mathematical work of Grassmann. Grassmann did not formulate a formal definition - then there was no proper language - but there is no doubt that the concept was clear. "

"Starting with a collection of 'units' e₁, e₂, e₃..., he effectively defined the free linear space they generated; in other words, consider the formal linear combination a₁e₁ + to₂e₂ + to₃e₃ +... where a_j they are real numbers, define the sum and multiplication of real numbers [in the mode currently used] and formally demonstrate the linear space properties of these operations. (...) It develops the theory of linear independence in an extraordinarily similar way to the presentation we can find in modern linear algebra texts. Defines the notion of subspace, independence, length, depopulation, dimension, sum and intersection of subspaces, and projection of elements in subspaces. "

"...people were as close as Hermann Grassmann to create, working alone, a new discipline. "

Developing an idea from his father, Grassmann also defined in Ausdehnungslehre the exterior product, also called "combination product" (German: äußeres Produkt or kombinatorisches Produkt), the key operation in algebra now known as external algebra. (It should not be forgotten that in Grassmann's time the only axiomatic theory available was Euclidean Geometry, and that the general notion of abstract algebra had not yet been defined.) In 1878, William Kingdon Clifford united external algebra with William's quaternions Rowan Hamilton, substituting Grassmann's rule e_pe_p = 0 for e_pe_p = 1. For more detail see external algebra.

The Ausdehnungslehre was a revolutionary text, too advanced for its time to be appreciated. Grassmann presented it as a doctoral thesis, but Möbius did not consider himself capable of evaluating it and sent it to Ernst Kummer, who rejected it without having carried out a careful reading. Over the next 10 years, Grassmann wrote a series of papers applying his theory of extension, including an 1845 Neue Theorie der Elektrodynamik, and various papers on algebraic curves and surfaces, hoping that these applications would move others to take their theory more seriously.

In 1846, Möbius invited Grassmann to a competition to solve a problem originally posed by Leibniz: Devise a private geometric calculus of coordinates and metric properties. Geometrische Analyze geknüpft an die von Leibniz erfundene geometrische Charakteristik by Grassmann, was the winning idea. It must be said, however, that Grassmann's result was the only one presented. In any case, Möbius, who was one of the members of the jury, criticized the way in which Grassmann introduced the abstract notion without providing the reader with any insight into the validity of these notions.

In 1853, Grassmann published a theory of how colors mix; this and his three laws of colors are still taught today. Grassmann's work was in contradiction with that of Helmholtz. Grassmann also wrote on crystallography, electromagnetism, and mechanics.

In 1861 Grassmann exposed the first axiomatic formulation of arithmetic, using extensively the principle of induction. Giuseppe Peano and his followers widely cited this work beginning in 1890.

In 1862, Grassman, trying to gain recognition for his theory of extension, published the second edition of the 'Ausdehnungslehre', extensively rewritten, and the definitive exposition of his linear algebra. The result, Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet, which is known as "Teaching Dilation" was not considered any better than the original edition, despite the fact that the method of exposition of this second version of 'Ausdehnungslehre' It will anticipate what textbooks have been in the 20th century. In this work he develops a direct operational calculation for the various geometric magnitudes, which are known as Grassmann numbers.

The only mathematician who fully appreciated Grassmann's ideas during his lifetime was Hermann Hankel. In his work Theorie der complexen Zahlensysteme (1867) he helped to make Grassmann's ideas better known. This work:

"... developed a part of Hermann Grassmann's algebra and Hamilton's quaternions. Hankel was the first to recognize the importance of Grassmann's texts, which had been depreciated for a long time..." (Hankel's introduction in the Dictionary of Scientific BiographyNew York: 1970-1990)

It was slow to adopt Grassmann's mathematical methods but they directly influenced Felix Klein and Élie Cartan. A. N. Whitehead's first monograph, Universal Algebra of 1898, included the first systematic exposition in English of the theory of extension and exterior algebra. The theory of extension was applied to the study of differential forms and to the applications of these forms to analysis and geometry. Differential geometry uses exterior algebra. For an introduction to the importance of Grassmann's work in mathematical physics see Penrose (2004: chs. 11, 12).

Linguist

Upset with his inability to gain recognition as a mathematician, Grassmann turned to historical linguistics. He wrote German grammar books, compiled catalogs of popular songs, and learned Sanskrit. His dictionary and his translation of Ayurveda (which is still published today) were widely recognized among philologists. He formulated a law concerning the phonemes of the Indo-European languages, which is known today as Grassmann's law in his honor. He also produced a Dictionary on the Rig-veda (1873-1875). His philological qualities were recognized in his lifetime: he was admitted to the American Oriental Society in 1876 and was made an honorary doctor of the University of Tübingen.