Hermann Grassmann

Hermann Günther Grassmann (15 April 1809 – 26 September 1877) was a German thinker known for his work in many areas. During his time, he was mostly recognized as a language expert, but today he is also famous for his contributions to math.

In math, Grassmann did important work that was not fully appreciated until he was older. He ideas were ahead of their time and helped develop what we now call a vector space. He also created something called the Grassmannian, which helps describe certain patterns in math.

Besides math and language, Grassmann also studied physics and worked as a publisher. His efforts helped separate the study of language history from the study of language structure, making both fields stronger.

Biography

Hermann Grassmann was one of twelve children born to a teacher and minister in Stettin, a city in Germany. He went to school there and later studied theology in Berlin, though he did not study mathematics or physics formally.

Despite this, Grassmann grew to love mathematics and made important discoveries in his early adult years. He began teaching math in schools in Berlin and Stettin and eventually became a head teacher and a professor. Sadly, many experts of his time did not recognize the value of his mathematical ideas.

Grassmann also had a family life. He married in 1849 and had eleven children, seven of whom grew up. One of his sons also became a mathematics professor.

Mathematician

In 1840, Hermann Grassmann wrote an essay about the tides using ideas from other scientists but also using special math methods he had thought about since 1832. This essay showed the first signs of what we now call linear algebra and vector spaces.

In 1844, Grassmann published a major work called Ausdehnungslehre, which means "theory of extension." He wanted to create a new foundation for all mathematics. He showed that geometry could be built using algebra, and that space could have more than three dimensions. His work included early ideas of linear spaces, subspaces, and other concepts that are important in modern math. Unfortunately, his ideas were too new and were not appreciated during his lifetime. He kept writing more papers hoping others would see the value in his work.

Response

In the 1840s, most mathematicians were not ready to understand Grassmann's new ideas. Later, in the 1860s and 1870s, other mathematicians started to think about similar concepts, but Grassmann had lost interest in math by then.

One mathematician, Adhémar Jean Claude Barré de Saint-Venant, created a system close to Grassmann's in 1845. He and Grassmann argued about who came up with the ideas first, since Grassmann had shared his work in 1844, but Saint-Venant said he had thought of them earlier in 1832.

Hermann Hankel was one of the first to see the value in Grassmann's work. In 1867, Hankel wrote about Grassmann's algebras and quaternions, recognizing how important Grassmann's writings were.

In 1872, Victor Schlegel published a book using Grassmann's methods to explain geometry. Felix Klein reviewed it and was not impressed, saying it was incomplete. Schlegel later published more work based on Grassmann’s ideas. Meanwhile, Klein was developing his own ideas about geometry.

It wasn’t until later that mathematicians fully understood Grassmann’s work, especially after the idea of vector spaces was introduced. This helped show the importance of Grassmann’s ideas. In 1995, an English translation of some of Grassmann’s works was published. To learn more about how Grassmann’s ideas are used today, you can read The Road to Reality by Roger Penrose.

Linguist

Hermann Grassmann spent the last years of his life studying languages. After his math work wasn’t well received, he turned his attention to historical language studies and the language Sanskrit. He wrote books on German grammar, collected folk songs, and created a large dictionary and translation of the Rigveda, a very old collection of poems. Even today, his work on the Rigveda is often used by scholars.

Grassmann also discovered an important pattern in the sounds of both Sanskrit and Greek, now called Grassmann’s law. This discovery changed how people thought about language family trees, showing that German could be considered “older” in some ways than Sanskrit. For his language work, he was honored by groups like the American Oriental Society and received a special degree from the University of Tübingen.

Publications

Hermann Grassmann wrote many important books and papers. Some of his well-known works include:

• In 1844, he published Die Lineale Ausdehnungslehre in German.
• In 1847, he wrote Geometrische Analyse in German.
• In 1861, he created Lehrbuch der Mathematik für höhere Lehranstalten, a math book for schools.

His major work, Die Ausdehnungslehre, was completed in 1862. An English version called Extension Theory was published much later in 2000.

Grassmann also worked on ancient languages and published a dictionary for the Rig-Veda in 1873, followed by a translation of the Rig-Veda in 1876–1877.