Hermann Grassmann

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Hermann Grassmann bigraphy, stories - Mathematicians

Hermann Grassmann : biography

April 15, 1809 – September 26, 1877

Hermann Günther Grassmann ( April 15, 1809 – September 26, 1877) was a German polymath, renowned in his day as a linguist and now also admired as a mathematician. He was also a physicist, neohumanist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties.

Response

One of the first mathematicians to appreciate Grassmann’s ideas during his lifetime was Hermann Hankel, whose 1867 Theorie der complexen Zahlensysteme

… developed some of Hermann Grassmann’s algebras and Hamilton’s quaternions. Hankel was the first to recognise the significance of Grassmann’s long-neglected writings …Hankel entry in the Dictionary of Scientific Biography. New York: 1970–1990

In 1872 Victor Schlegel published the first part of his System der Raumlehre which used Grassmann’s approach to derive ancient and modern results in plane geometry. Felix Klein wrote a negative review of Schlegel’s book citing its incompleteness and lack of perspective on Grassmann. Schlegel followed in 1875 with a second part of his System according to Grassmann, this time developing higher geometry. Meanwhile Klein was advancing his Erlangen Program which also expanded the scope of geometry.Rowe 2010

Comprehension of Grassmann awaited the concept of vector spaces which then could express the multilinear algebra of his extension theory. A. N. Whitehead’s first monograph, the Universal Algebra (1898), included the first systematic exposition in English of the theory of extension and the exterior algebra. With the rise of differential geometry the exterior algebra was applied to differential forms.

For an introduction to the role of Grassmann’s work in contemporary mathematical physics see The Road to RealityPenrose The Road to Reality, chapters 11 & 2 by Roger Penrose.

Adhémar Jean Claude Barré de Saint-Venant developed a vector calculus similar to that of Grassmann which he published in 1845. He then entered into a dispute with Grassmann about which of the two had thought of the ideas first. Grassmann had published his results in 1844, but Saint-Venant claimed that he had first developed these ideas in 1832.

Mathematician

One of the many examinations for which Grassmann sat required that he submit an essay on the theory of the tides. In 1840, he did so, taking the basic theory from Laplace’s Mécanique céleste and from Lagrange’s Mécanique analytique, but expositing this theory making use of the vector methods he had been mulling over since 1832. This essay, first published in the Collected Works of 1894–1911, contains the first known appearance of what are now called linear algebra and the notion of a vector space. He went on to develop those methods in his A1 and A2 (see references).

In 1844, Grassmann published his masterpiece, his Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik [The Theory of Linear Extension, a New Branch of Mathematics], hereinafter denoted A1 and commonly referred to as the Ausdehnungslehre, which translates as "theory of extension" or "theory of extensive magnitudes." Since A1 proposed a new foundation for all of mathematics, the work began with quite general definitions of a philosophical nature. Grassmann then showed that once geometry is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial dimensions; the number of possible dimensions is in fact unbounded.

describes Grassmann’s foundation of linear algebra as follows:

Following an idea of Grassmann’s father, A1 also defined the exterior product, also called "combinatorial product" (In German: äußeres Produkt or kombinatorisches Produkt), the key operation of an algebra now called exterior algebra. (One should keep in mind that in Grassmann’s day, the only axiomatic theory was Euclidean geometry, and the general notion of an abstract algebra had yet to be defined.) In 1878, William Kingdon Clifford joined this exterior algebra to William Rowan Hamilton’s quaternions by replacing Grassmann’s rule epep = 0 by the rule epep = 1. (For quaternions, we have the rule i2 = j2 = k2 = −1.) For more details, see exterior algebra.