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Super-Algebra and Super-Geometry

This page collects material related to the book

which introduced for the first time basic concepts of what today is known as linear algebra (including affine spaces as torsors over vector spaces) and introduced in addition an exterior product (§37, §55) on vectors, forming what today is known as exterior algebra or Grassmann algebra, hence in fact superalgebra.

Here is Grassmann introducing the sign rule of superalgebra:

from Grassmann 1844, p. 61

from Grassmann 1844, p. 84

Grassmann advertizes his work (p. xxv) as being the theory of extensive quantity. The modern way of speaking about this is that the elements of the exterior algebra he considered are differential forms on Euclidean space.



Discussion of the book includes

  • William Lawvere, Grassmann’s Dialectics and Category Theory, in Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar, Boston Studies in the Philosophy of Science Volume 187, 1996, pp 255-264, doi:10.1007/978-94-015-8753-2_21

and the similar text

  • William Lawvere, A new branch of mathematics, “The Ausdehnungslehre of 1844,” and other works. Open Court (1995), Translated by Lloyd C. Kannenberg, with foreword by Albert C. Lewis, Historia Mathematica Volume 32, Issue 1, February 2005, Pages 99–106, doi:10.1016/

which says at one point that full appreciation of the Ausdehnungslehre requires concepts of category theory

The modern conceptual apparatus, involving levels of structure, categories of morphisms preserving given structure, forgetful reduct functors between categories, the adjoints to such functors, etc., seems to be necessary for ordinary mortals to be able to find their way through the riches of Grassmann’s geometry.

The first part of the introduction of the Ausdehnungslehre is concerned with philosophy, about which

Grassmann insists that his reason for including it is an attempt to provide an orientation to help the student form for himself the proper estimation of the relation between general and particular at every stage of the learning process (Lawvere 95).

The second part of the introduction, titled Survey of the general theory of forms considers key concepts of algebra. For instance it considers the associativity law and states its coherence law (§3).

Grassmann writes that he uses the term “form” in place of “quantity” (German: “Grösse”) (Introduction A.3, §2). It is “forms” that his algebraic operations are defined on, and which are produced by these.

The last half of that introduction is essentially one of the first expositions of the rudimentary principles of what today might be called universal algebra. The content of the first half, after considerable study of the compact formulations, appears to be a simple and clear natural scientist’s version of the basic principles of dialectical materialism, as applied to the formal sciences. (Lawvere 95)

Curiously, while Grassmann complains (on p. xv) about the “unclarity and arbitrariness” of Hegel‘s school of philosophy (German idealism, predominant in Germany at Grassmann’s time), the introduction of the Ausdehnungslehre has much the same sound as Hegel, notably it discusses “categories” such as being, becoming (p. xxii), particulars (p.xx) and the dialectic of opposites such as discrete \dashv continuous (p.xxii) and, notably, of intensive and extensive quantity (p. xxiv-xxv), which Grassmann advertizes as the very topic of his mathematical theory. That of course is the difference to Hegel, that unambiguous mathematical formalization of these otherwise vague concepts is provided (according to Lawvere 95 Grassmannn’s formalization of the pair being and becoming is via points and vectors in an affine space), and in this sense Grassmann is clearly a forerunner of Lawvere’s various proposals for formalizing Hegel’s objective logic in categorical logic/topos theory (as discussed at Science of Logic).


  • Wikipedia, Grassmann – Mathematician

  • Freeman Dyson, Missed opportunities, Bulletin of the AMS, Volume 78, Number 5 (1972) pp 635-652, doi:10.1090/S0002-9904-1972-12971-9

    In the year 1844 two remarkable events occurred, the publication by Hamilton of his discovery of quaternions, and the publication by Grassmann of his “Ausdehnungslehre.” With the advantage of hindsight we can see that Grassmann’s was the greater contribution to mathematics, containing the germ of many of the concepts of modern algebra, and including vector analysis as a special case. However, Grassmann was an obscure high-school teacher in Stettin, while Hamilton was the world-famous mathematician whose official titles occupy six lines of print after his name at the beginning of his 1844 paper. So it is regrettable, but not surprising, that quaternions were hailed as a great discovery, while Grassmann had to wait 23 years before his work received any recognition at all from professional mathematicians. When Grassmann’s work finally became known, mathematicians were divided into quaternionists and antiquaternionists, and were spending more energy in polemical arguments for and against quaternions than in trying to understand how Grassmann and Hamilton might be fitted together into a larger scheme of things.

category: reference

Last revised on August 25, 2019 at 17:28:00. See the history of this page for a list of all contributions to it.