Editing Hermann Grassmann (section)

==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 ''[[Traité de mécanique céleste]]'' and from [[Joseph-Louis Lagrange|Lagrange]]'s ''[[Mécanique analytique]]'', but expositing this theory making use of the [[vector (geometry)|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 is now called [[linear algebra]] and the notion of a [[vector space]]. He went on to develop those methods in his ''Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik'' ('''A1''') and its later revision ''Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet'' ('''A2''').

In 1844, Grassmann published his masterpiece ('''A1''') 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 [[dimension]]s; the number of possible dimensions is in fact unbounded.

Fearnley-Sander describes Grassmann's foundation of linear algebra  as follows:<ref>{{cite journal |last1=Fearnley-Sander |first1=Desmond |title=Hermann Grassmann and the Creation of Linear Algebra |journal=The American Mathematical Monthly |date=December 1979 |volume=86 |issue=10 |pages=809–817 |doi=10.2307/2320145 |url=https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/DesmondFearnleySander.pdf |ref=Fearnley-Sander |publisher=Mathematical Association of America |issn=0002-9890 |jstor=2320145}}</ref>

{{blockquote|The definition of a [[linear space]] ([[vector space]]) [...] became widely known around 1920, when [[Hermann Weyl]] and others published formal definitions. In fact, such a definition had been given thirty years previously by [[Peano]], who was thoroughly acquainted with Grassmann's mathematical work. Grassmann did not put down a formal definition – the language was not available – but there is no doubt that he had the concept.

Beginning with a collection of 'units' ''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>, ..., he effectively defines the free linear space that they generate; that is to say, he considers formal linear combinations ''a''<sub>1</sub>''e''<sub>1</sub> + ''a''<sub>2</sub>''e''<sub>2</sub> + ''a''<sub>3</sub>''e''<sub>3</sub> + ... where the ''a<sub>j</sub>'' are real numbers, defines addition and multiplication by real numbers [in what is now the usual way] and formally proves the linear space properties for these operations. ... He then develops the theory of [[linear independence]] in a way that is astonishingly similar to the presentation one finds in modern linear algebra texts. He defines the notions of [[linear subspace|subspace]], [[linear independence]], [[Linear span|span]], [[dimension]], [[join and meet]] of subspaces, and [[projection (linear algebra)|projection]]s of elements onto subspaces.

[...] few have come closer than Hermann Grassmann to creating, single-handedly, a new subject.}}

Following an idea of Grassmann's father, '''A1''' also defined the [[exterior product]], also called "combinatorial product" (in German: ''kombinatorisches Produkt'' or ''äußeres Produkt'' “outer product”), the key operation of an algebra now called [[exterior algebra]]. (One should keep in mind that in Grassmann's day, the only [[axiom]]atic theory was [[Euclidean geometry]], and the general notion of an [[Algebra over a field|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 ''e<sub>p</sub>e<sub>p</sub>'' = 0 by the rule ''e<sub>p</sub>e<sub>p</sub>'' = 1. (For [[quaternions]], we have the rule ''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = −1.) For more details, see [[Exterior algebra]].

'''A1''' was a revolutionary text, too far ahead of its time to be appreciated. When Grassmann submitted it to apply for a professorship in 1847, the ministry asked [[Ernst Kummer]] for a report. Kummer assured that there were good ideas in it, but found the exposition deficient and advised against giving Grassmann a university position. Over the next 10-odd years, Grassmann wrote a variety of work applying his theory of extension, including his 1845 ''Neue Theorie der Elektrodynamik'' and several papers on [[algebraic curve]]s and [[algebraic surface|surface]]s, in the hope that these applications would lead others to take his theory seriously.

In 1846, [[August Ferdinand Möbius|Möbius]] invited Grassmann to enter a competition to solve a problem first proposed by [[Gottfried Wilhelm Leibniz|Leibniz]]: to devise a geometric calculus devoid of coordinates and metric properties (what Leibniz termed ''analysis situs''). Grassmann's ''Geometrische Analyse geknüpft an die von Leibniz erfundene geometrische Charakteristik'', was the winning entry (also the only entry). Möbius, as one of the judges, criticized the way Grassmann introduced abstract notions without giving the reader any intuition as to why those notions were of value.

In 1853, Grassmann published a theory of how colors mix; his theory's four color laws are still taught, as [[Grassmann's laws (color science)|Grassmann's laws]]. Grassmann's work on this subject was inconsistent with that of [[Helmholtz]].<ref>{{cite book
 | last = Turner | first = R. Steven
 | contribution = The Origins of Colorimetry: What did Helmholtz and Maxwell Learn from Grassmann?
 | doi = 10.1007/978-94-015-8753-2_8
 | isbn = 9789401587532
 | pages = 71–86
 | publisher = Springer Netherlands
 | title = Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar
 | series = Boston Studies in the Philosophy of Science
 | year = 1996| volume = 187
 }} See p.74: "Helmholtz rejected almost as many of Grassmann's conclusions as he accepted."</ref> Grassmann also wrote on [[crystallography]], [[electromagnetism]], and [[mechanics]].

In 1861, Grassmann laid the groundwork for [[Peano axioms|Peano's axiomatization of arithmetic]] in his ''Lehrbuch der Arithmetik''.<ref>{{cite journal |last1=Wang |first1=Hao |author1-link=Hao Wang (academic) |title=The Axiomatization of Arithmetic |journal=[[The Journal of Symbolic Logic]] |date=June 1957 |volume=22 |issue=2 |pages=145–158 |doi=10.2307/2964176 |publisher=[[Association for Symbolic Logic]] |jstor=2964176 |s2cid=26896458 |quote=It is rather well-known, through Peano's own acknowledgement, that Peano […] made extensive use of Grassmann's work in his development of the axioms. It is not so well-known that Grassmann had essentially the characterization of the set of all integers, now customary in texts of modern algebra, that it forms an ordered [[integral domain]] in which each set of positive elements has a least member. […] [Grassmann's book] was probably the first serious and rather successful attempt to put numbers on a more or less axiomatic basis. |quote-page=145, 147}}</ref> In 1862, Grassmann published a thoroughly rewritten second edition of '''A1''', hoping to earn belated recognition for his theory of extension, and containing the definitive exposition of his [[linear algebra]]. The result, ''Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet'' ('''A2'''), fared no better than '''A1''', even though '''A2'''{{'s}} manner of exposition anticipates the textbooks of the 20th century.