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From Past to Future: Grassmann's Work in Context

Grassmann Bicentennial Conference
(1809 – 1877)
September 16 – 19, 2009  Potsdam / Szczecin  (DE / PL)


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A small-town genius

Potsdam philosopher Hans-Joachim Petsche on mathematical genius

By Mark Minnes

Hans-Joachim Petsche’s voice vibrates with enthusiasm when he brings up Hermann Grassmann. And the Potsdam professor of philosophy is perfectly capable of bringing his enthusiasm across to the listener. With every word Petsche says on the topic, it becomes clear that he is fascinated by the work of Hermann Grassmann, a mathematician from the once Prussian town of Stettin. Professor Petsche received his academic training in mathematics and philosophy. At times, Petsche’s voice will pause. He will interrupt his thoughts, just to place special emphasis on a point just made: “Modern, strange… brilliant!” This is how Prof. Petsche sees the work of this mathematician, who was born in 1809. And he should know: Hans-Joachim Petsche has published an extensive intellectual biography on the brilliant man. The book tells the story of an exceptional scholar.

Hermann Grassmann, a school teacher in the town of Stettin, which today is north-western Poland, played a key role in a philosophical and mathematical revolution which took place in the 19th century. Grassmann’s 1844 “Ausdehnungslehre” brought geometry and algebra together, creating a new mathematical discipline. Almost without using mathematical formulas, Grassmann worked out a way to calculate directly in terms of geometrical bodies. By doing this in his unique way, the mathematician crossed an ancient line of demarcation. He completed a project that had already puzzled great minds such as Descartes and Leibniz.

It took some time for people to realize that Grassmann had brought a revolution to philosophy, physics and mathematics: a new concept of spatial expansion had been born, and an ancient intellectual dogma was losing its power. This dogma had been the notion that we inevitably must conceptualize our world as a three dimensional one. By the end of the 18th century, the great Immanuel Kant had given this dogma its philosophical foundation. But with Grassmann, it was beginning to lose its grip.

Today, as in his day, only specialists know who came up with “n-dimensional vector algebra”. In his lifetime, hardly anybody reacted to Hermann Grassmann’s theories. This school teacher’s thoughts were too new and too hard to grasp. Also, he was a self-taught mathematician. He attempted in vain to become a university professor, he took a stand in the political turmoil of 1848, he taught, he thought, and he became the father of eleven children. When in 1860 his lack of success in mathematics seemed definite, he turned to linguistics. With brilliant effect: His dictionary of Sanskrit is still in print today.

So Grassmann was not just the creator of a groundbreaking theory on n-dimensional space. Hans-Joachim Petsche is convinced that the man from Stettin represents a special phase in German intellectual history. “The whole situation in Stettin was that of a small town: limited, provincial, brilliant”, as Prof. Petsche told PNN. To him, Grassmann’s great scientific achievement is linked to the German world of small towns. This was the world of school professors, of the petite bourgeoisie, and of learned societies. As Prof. Petsche points out, it was also a world of narrow-minded German nationalism and political backwardness. Certainly, curious mixture of influences.

But Grassmann’s story is also linked to the Berlin university, where he studied theology: his only venture beyond the Stettin city limits. In Berlin he was influenced by the philosopher and theologian Friedrich Schleiermacher. Schleiermacher’s thoughts, this is one of Prof. Petsches key insights, prepared the ground for Grassmann’s brilliant intellectual movements. But Grassmann needed small-town quietude to work. “A small-town genius”, as Hans-Joachim Petsche likes to call the school teacher, who had a reputation of being modest and friendly.

“At first glance, Grassmann’s scientific approach is quite simple”, says Hans-Joachim Petsche. “But then Grassmann shows an extreme will to expand and generalize his thoughts. He develops his concepts with extreme rigor, exploring their meaning to the very end.” This is how, according to Prof. Petsche, Grassmann left behind the notion of three-dimensional space. To Grassmann, the word “expansion” meant more than just expansion in space. Instead, his theory embraces a given number of directional aggregates, which his “vector calculations” handle better than any other previous approach. According to Prof. Petsche, Grassmann’s vector algebra is fundamental to complex optimization processes. It is used in satellite navigation systems, in computer technology and in aeronautics.

Hans-Joachim Petsche sees Grassmann as a case where disciplinary limitations, even academic specializations, were superseded. Grassmann stands for brilliant innovation. We should see this as a “piece of advice German history holds in store for the present”, as Petsche puts it in his biography. Hermann Grassmann moved freely between the fields of mathematics, physics, philosophy and linguistics. “A beautiful system of coordinates, full of creativity”, says Prof. Petsche. He sees this networked approach as an important bearing for scientific work today. So it seems like the perfect time for an English translation of Prof. Petsche’s book, which is underway. Next year, 200 years after Grassmann’s year of birth, Prof. Petsche hopes to bring an international conference to Potsdam. Right now, he is looking for supporters. Hans-Joachim Petsche would like to renew the intellectual spirit Grassmann brought to the small town of Stettin – in Potsdam. And with the spirit of Grassmann, Hans-Joachim Petsche would like to bring back a spirit of innovation and creativity.


Petsche, Hans-Joachim: Grassmann. Basel: Birkhäuser Verlag.


(Published in the Potsdam daily newspaper “Potsdamer Neueste Nachrichten”/ PNN on 3/26/2008; translated from German by Mark Minnes)

    Conference Patron:
    Dr. Manfred Stolpe

   Funded by:


Bundesministerium für Bildung und Forschung

Universität Potsdam

Universitätsgesellschaft Potsdam e.V.

Universytet Szczecinski

Birkhäuser Verlag