Friday, September 26, 2008

Hermann Grassmann: anniversary

Herman Günther Grassmann was born in 1809 and died 131 years ago, on September 26, 1877, in Stettin, Prussia, Greater Germany (now Szczecin, Poland) to a family of an ordained minister who taught maths and physics.

It's an anniversary of his death, not birth, so this story won't be quite complete. But this mathematician had an interesting CV. Unlike others, he was a lousy student and it seems that he studied no maths and physics at all. Instead, he chose classical languages, philosophy, and theology.

Because of some twists and turns, he decided that he should teach maths at a Gymnasium (a job similar to his father's). But he needed an exam. His score was so-so but he was allowed to teach at the high school. It just happened that during that time, he made some crucial steps that eventually led him to his now-famous 1844 paper referred to as "A1". Not bad for a high school teacher.

What was the paper about? Well, he defined various formal, exterior (or "combinatorial" or "wedge") products, and discussed the concept of linear independence. The only other axiomatic system known before Grassmann was the Euclidean geometry so his new abstract system was quite revolutionary. With some updates, Grassmann's work also led Hamilton to discover quaternions and it influenced Felix Klein and Élie Cartan. Linear algebra and group theory would be virtually unimaginable without Grassmann's "momentum".




The Prussian government wanted to know how valuable "A1" was. So they asked Ernst Kummer (of the K3 surface fame). He obviously didn't read the paper in detail and wrote that the paper contained "commendably good material expressed in a deficient form". That ended Grassmann's hopes for a university job. What did he mean by the deficient form? Well, the approach of Grassmann was virtually identical to the contemporary approach to linear algebra, linear independence, exterior algebras, and abstract products - it was the approach that dominates 150 years after Grassmann's paper.

Besides linear algebra, Grassmann also discovered the law how colors add, Grassmann's law. These laws were correct, unlike Helmholtz's alternatives. Add some papers about mechanics, electromagnetism, and crystallography.

There are several things named after Grassmann. A Grassmannian is the space of all linear subspaces of another linear space; it is related to projective spaces. However, the first concept that physicists connect with Hermann Grassmann are the Grassmann (anticommuting) numbers. In the present form, this specific number system was defined by David John Candlin in 1956 but it seems pretty obvious why they were named after Grassmann.

As you can see, Grassmann was 100+ years ahead of his time. It shouldn't be shocking that he had to abandon mathematics. Instead, he learned Sanskrit, translated Rigveda, and collected folk songs. With this activity, he could achieve at least some recognition: for example, he was elected to the American Oriental Society.

As a linguist, he could also give a new meaning to the term "Grassmann's law": it's the rule that the first aspirated consonant in a pair of two consecutive ones loses the aspiration. The reason, as I see it, is that it would be pretty hard to aspire so many times and so quickly. ;-) This rule holds if the two consonants belong to two different syllables and if it is Sanskrit or Ancient Greek that we talk about. ;-)

Grassmann had 11 siblings and 11 offspring, 7 of which lived up to their adulthood (it's the opposite relation than in M-theory phenomenological models where 7 dimensions stayed small). In politics, he supported the unification of Germany. With his brother Robert, they published a text supporting this scenario in the framework of constitutional monarchy. However, the newspaper that had published it was going in a different direction and Grassmann's politics became inconsequential.

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