Hermann Grassmann

Hermann Günther Grassmann ( born April 15, 1809 in Stettin, † September 26, 1877 ) was a German mathematician and linguist. He is regarded as the true founder of the vector and tensor calculus.

Life and work

The father, Justus Günther Grassmann (1779-1852) completed a three-year, actually theological, but filled with science content studies at the University of Halle and initially taught then as a private tutor, then as vice-principal of the city school in Pomerania and then at the Gymnasium in Stettin. His subjects were mathematics, physics and drawing. He has also written several textbooks on elementary mathematics for primary schools and high schools and founded in 1835 a physical society. " The principles set out in all these works, scientific and philosophical approaches were at a crucial point of departure wissenschschaftlichen developments of his son. "

Youth and studies

The young Hermann Günther did not like his contemporary and later colleague from William Rowan Hamilton as a child prodigy forth; Instead, he fell during growth initially through its limited mental tension force, forgetfulness and reverie. Up to the age of 14 could not close his extraordinary talent one. But then he became interested, and you summed up for him at first - the family was strongly influenced by pietism - a theological studies in the eye, after he had in 1827 passed the matriculation examination with the best note. In 1827 he began his studies at the University of Berlin. There he heard, among other things, the dialectic lecture and the sermons of Friedrich Schleiermacher, which influenced his thinking greatly. " Even during the university years, Hermann Grassmann independent study methods appropriated, which enabled him to his later self-paced penetration in mathematics ". Throughout his studies, he did not hear any mathematical lecture.

His tremendous additional first philologically oriented, self-imposed learning stint brought him quickly to the edge of his mental and physical powers, and he fell ill. He had in his approach initially reorient, and he eventually developed a more appropriate for him working posture. In a letter Grassmann practiced self-criticism: " The Phlegmatic ... must rather give his clarity of thought and looking in the clarity of depth. ". The basis for the extraordinary productivity, " this eerie spirit" ( Junghans ) on a variety of scientific fields was thus created. The role of Schleiermacher's dialectic as a heraldic keys to the laws of different scientific areas is particularly important. Grassmann writes that "one can learn for every science because it is less positive, as he skillfully to attack each examination from the right side and independently continue, and is in a position to find the positives themselves. " He.

Towards the theory of extension of 1844

1830 returned Grassmann back to Stettin. He took his self-study again and dealt with physics and mathematics in " close connection of geometry, arithmetic and combined doctrine." In 1831 he took a job at the Szczecin teacher training college and taught first as a teaching assistant and German doctrine of space and wrote audit work for the doctrinal examination commission in Berlin. In this work, already showing his early fundamental approach, " in which the mathematical access is always flanked by philosophical considerations or even initiated ." He received permission to teach, among other things for math classes up to the Secunda as a senior teacher. In 1834 he took his first theological examination, but had already decided on a career in science.

In 1837 he became a research professor at the Otto school in Szczecin. 1839 appeared his first designed for the teaching work on the derivatives of the Krystallgestalten, found some interest for August Ferdinand Möbius, because he had also dealt on the edge with this issue. In 1838 he completed his second theological examination, and before he even volunteered for review in mathematics and physics, certainly to improve his mathematical skills and verifiable. The new test work on the theory of the tides, in which he finished successfully apply newly developed mathematical approaches, he graduated in 1840. Grassmann was aware of the importance and efficiency of the designed by him and in this work for the first time applied vector analysis. After a change to the Friedrich- Wilhelms- school appeared in 1844 his main work, the extension theory.

"This book blew the contemporary ideas about the treatment of geometry. Extensive philosophical previews, presentation of an abstract, conceived as the basis of all mathematics theory of links, sparse formula use, rejection of the geometry as a mathematical discipline and development of a n-dimensional, metric- free theory of mathematical manifold " form the theoretical elaboration of his mathematical program. With this work, Grassmann had anticipated considerations that come into contact with the later approaches Bernhard Riemann to the theory of n-dimensional manifolds as well as the concept of Hamiltonian quaternions closely. Grassmann remained, however, a total misunderstood why he was ignored by the art, and the book did not sell. One reason was certainly the use of him as a self-taught self-created, times historically sanctioned terms. Even by its differing from the Euclidean ideal representation, he could not then win over the mathematical art.

Struggle for recognition

1846 Grassmann began with a series of papers on the theory of algebraic curves, the elaboration of an approach, which had resulted in the extension theory, of course, for further promotion of its program. Even this, however, found among his contemporaries no attention.

Then he tried his hand at editing a tendered by the Jablonowskischen company in 1844 price task. That case involved the reconstruction and further training is one of Gottfried Wilhelm Leibniz only sketchily designed geometric calculus. Successfully tackle this task was, thanks to his new method, only Grassmann able. He was awarded the prize. Encouraged by the success, he worked together with his brother Robert Grassmann with further studies and applied in 1847 to a math teaching position at a Prussian university.

He submitted his prize essay, and the theory of extension, but especially the opinion of Ernst Eduard Kummer is devastating. This writes, "that this scripture will be [ the extension theory ] also ignored by mathematicians as before; because the effort to familiarize themselves with the same, appears too large in relation to the real gain of knowledge, which is thought to derive from the same ". The application was rejected.

On the way to the second extension theory

1849 married Grassmann Marie Therese Squire, in the next few years gave birth to 11 children in a happy marriage. He has published several articles on the application of extension theory to the theory of algebraic curves. 1853 appeared a breakthrough in the field of color theory essay, which influences the colorimetry until today. It also appeared for a review of vowel theory, considered the forerunner of Helmholtz's resonance theory. Inspired by the writings of Franz Bopp, he began to study with languages ​​, mainly Sanskrit, and the still young historical linguistics.

He worked a lot on an arithmetic and on a Neuausarbeitung the extension theory, which appeared in 1861 and 1862. He had changed for these books his presentation to have certainly stimulated by the criticism of the past, and his brother seems it exercised a certain influence. He now turned to the strict, purely formulaic Euclidean representation. He fell, however, as Petsche describes, "from one extreme to another. Although the mathematicians he could no longer make the philosophical account to the accusation now; but for them was expected to get a completely unfamiliar mathematical material offered in the most inaccessible representation of the time without even having an idea of the benefits of mathematical development. A response to the publication of his work [ Ausdehnungslehre ] therefore remained completely out. " With two works he applied again to a chair at the Ministry of Culture, which was again rejected. The reason given was that you could give him no better position anyway because Grassmann had taken his place at the Szczecin High School in 1852 after the death of his father, who had been already connected to the title of professor.

He finally turned disappointed by all mathematical studies from to devote himself entirely to the science of language.

Linguistic work

In this area, brought Grassmann New and Significant forth; he won here the broad recognition of his professional colleagues. In 1863 he published the Hauchdissimilationsgesetz, for which he, however, did not claim authorship. From 1873 to his dictionary Ṛgvedasaṃhitā, which is still in Indology in use, even if many entries that are outdated appeared. Then appeared a translation of the same text. The American Oriental Society made ​​him in 1876 to its member. At the instigation of Rudolf von Roth him the University of Tübingen conferred the honorary doctorate in the same year; in the application it says: " He is one of the best linguists and Sanskritists ... Translation [ the Ṛgvedasaṃhitā ] is far superior to that of Alfred Ludwig in Prague ... started by haunting comprehension and tasteful interpretation ". Grassmann heard from today's perspective the most important Vedaforschern at the turn of the 19th to the 20th century.

Late recognition of the mathematical work

Late in life of Grassmann it did come to a general scientific recognition of his mathematical achievements. Hamilton had indeed already mentioned in his Lectures on Quaternions 1853 Grassmann's theory of extension as groundbreaking praise, however, this was, overall, remained without effect, and especially Grassmann himself had not notice anything. On November 24, 1866 he received a letter from Hermann Hankel, in which the latter his enthusiasm about the mathematics of Grassmann gave expression. Became aware he was on Grassmann by reading the quaternion. " [ I ] saw, to my great joy ," writes Hankel, "that in the same [ the two expansion teachings ] the concept of complex numbers - I call your extensive quantities - [ sic ] in a public and treated such a proper manner and is used, as I could wish it for my own education only. " Hankel had been able to understand Grassmann, and there was a regular correspondence. But Hankel had no decisive scientific weight, Grassmann ultimately to achieve a breakthrough. 1869 Felix Klein became aware of the name Grassmann by Hankel's theory of complex number systems. This in turn instructed his colleague Alfred Clebsch attention to him. In the operation of Clebsch Grassmann was finally elected in 1871 from Göttingen Society of Sciences corresponding member. 1872 published Victor Schlegel, a colleague at the Szczecin Grassmann school, the first attempt at a closed foreign statement of the views Grassmann, the system of the theory of space. The growing recognition was unstoppable. Sophus Lie even came to Stettin, to wanting to hear from Grassmann about his treatment of the Pfaffian problem.

'This way the life of a great, long misunderstood and struggling for the progress of mathematics in intellectual isolation scientist bent met its end. " Shortly before his death he lived to see a new version of the extension theory of 1844, after it had been found that at that time the entire first edition was pulped because of lack of sales almost.

Celebrate the 200th anniversary Hermann Grassmann an international scientific conference in Potsdam and Stettin was held in September 2009, the ausleuchtete the contexts and the real story of his work as well as the further development of his ideas in the present interdisciplinary.

Grassmann's vector calculus

Some basic ideas of Grassmann's vector calculus:

• Relationships between spatial variables can be described using algebraic laws linking
• Conception of the segments AB and BA is as opposite sizes ( considering the negatives in geometry ), in addition to the length of a line whose direction is important now
• In contrast to Hamilton Grassmann is interested in extending his thoughts to n dimensions
• It is AB BC = AC, even if A, B, C do not lie in a straight line
• If the same changes (now parallel shifts ) all elements of a track subjects, so the resulting path of the original is the same.
• The geometric product ( wedge product ) of two routes is the area of the parallelogram formed from them

It came with Grassmann already the concepts of linear dependence and independence, basis and dimension of all before, albeit under different names. Grassmann speaks of distances and sizes, not vectors.