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{{Short description|German polymath, linguist and mathematician (1809–1877)}}
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'''Hermann Günther Grassmann''' ({{lang-de|link=no|Graßmann}}, {{IPA-de|ˈhɛʁman ˈɡʏntɐ ˈɡʁasman|pron}}; 15 April 1809 – 26 September 1877) was a German ], known in his day as a ] and now also as a ]. He was also a ], general scholar, and publisher. His mathematical work was little noted until he was in his sixties. '''Hermann Günther Grassmann''' ({{langx|de|link=no|Graßmann}}, {{IPA|de|ˈhɛɐman ˈɡʏntʰɐ ˈɡʁasman|pron}}; 15 April 1809 – 26 September 1877) was a German ] known in his day as a ] and now also as a ]. He was also a ], general scholar, and publisher. His mathematical work was little noted until he was in his sixties. His work preceded and exceeded the concept which is now known as a ]. He introduced the ], the space which parameterizes all ] linear subspaces of an ''n''-dimensional ] ''V''. In linguistics he helped free language history and structure from each other.


==Biography== ==Biography==
{{unreferenced section|date=July 2024}}
Grassmann was the third of 12 children of Justus Günter Grassmann, an ] ] who taught mathematics and physics at the ] ], where Hermann was educated. Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an ] ] who taught mathematics and physics at the ] ], where Hermann was educated.


Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to ]n universities. Beginning in 1827, he studied theology at the ], also taking classes in ], philosophy, and literature. He does not appear to have taken courses in mathematics or ]. Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to ]n universities. Beginning in 1827, he studied theology at the ], also taking classes in ], philosophy, and literature. He does not appear to have taken courses in mathematics or ].


Although lacking university training in mathematics, it was the field that most interested him when he returned to Stettin in 1830 after completing his studies in Berlin. After a year of preparation, he sat the examinations needed to teach mathematics in a gymnasium, but achieved a result good enough to allow him to teach only at the lower levels. Around this time, he made his first significant mathematical discoveries, ones that led him to the important ideas he set out in his 1844 paper referred to as '''A1''' (see ]). Although lacking university training in mathematics, it was the field that most interested him when he returned to Stettin in 1830 after completing his studies in Berlin. After a year of preparation, he sat the examinations needed to teach mathematics in a gymnasium, but achieved a result good enough to allow him to teach only at the lower levels. Around this time, he made his first significant mathematical discoveries, ones that led him to the important ideas he set out in his 1844 paper ''Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik'', here referred to as '''A1''', later revised in 1862 as ''Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet'', here referred to as '''A2'''.


In 1834 Grassmann began teaching mathematics at the Gewerbeschule in Berlin. A year later, he returned to Stettin to teach mathematics, physics, German, Latin, and religious studies at a new school, the Otto Schule. Over the next four years, Grassmann passed examinations enabling him to teach mathematics, ], ], and ] at all secondary school levels. In 1834 Grassmann began teaching mathematics at the Gewerbeschule in Berlin. A year later, he returned to Stettin to teach mathematics, physics, German, Latin, and religious studies at a new school, the Otto Schule. Over the next four years, Grassmann passed examinations enabling him to teach mathematics, ], ], and ] at all secondary school levels.


In 1847, he was made an "Oberlehrer" or head teacher. In 1852, he was appointed to his late father's position at the Stettin Gymnasium, thereby acquiring the title of Professor. In 1847, he asked the Prussian Ministry of Education to be considered for a university position, whereupon that Ministry asked ] for his opinion of Grassmann. Kummer wrote back saying that Grassmann's 1846 prize essay (see below) contained "... commendably good material expressed in a deficient form." Kummer's report ended any chance that Grassmann might obtain a university post. This episode proved the norm; time and again, leading figures of Grassmann's day failed to recognize the value of his mathematics. In 1847, he was made an "Oberlehrer" or head teacher. In 1852, he was appointed to his late father's position at the Stettin Gymnasium, thereby acquiring the title of Professor. In 1847, he asked the Prussian Ministry of Education to be considered for a university position, whereupon that Ministry asked ] for his opinion of Grassmann. Kummer wrote back saying that Grassmann's 1846 prize essay (see below) contained "commendably good material expressed in a deficient form." Kummer's report ended any chance that Grassmann might obtain a university post. This episode proved the norm; time and again, leading figures of Grassmann's day failed to recognize the value of his mathematics.


Starting during the political turmoil in Germany, 1848–49, Hermann and his brother Robert published a Stettin newspaper, '']'', calling for ] under a ]. (This eventuated in 1871.) After writing a series of articles on ], Hermann parted company with the newspaper, finding himself increasingly at odds with its political direction. Starting during the political turmoil in Germany, 1848–49, Hermann and his brother Robert published a Stettin newspaper, '']'', calling for ] under a ]. (This eventuated in 1871.) After writing a series of articles on ], Hermann parted company with the newspaper, finding himself increasingly at odds with its political direction.
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==Mathematician== ==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 ]'s ''Mécanique céleste'' and from ]'s ''Mécanique analytique'', but expositing this theory making use of the ] 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 ] and the notion of a ]. He went on to develop those methods in his '''A1''' and '''A2''' (see ]). 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 ]'s '']'' and from ]'s '']'', but expositing this theory making use of the ] 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 ] and the notion of a ]. 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, his ''Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik''<ref>''Tr''. The linear extension theory, a new branch of mathematics</ref> , hereinafter denoted '''A1''' and commonly referred to as the ''Ausdehnungslehre'',<ref>''Tr''. Extension theory</ref> 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 ] is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial ]s; the number of possible dimensions is in fact unbounded. 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 ] is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial ]s; the number of possible dimensions is in fact unbounded.


Fearnley-Sander (1979) describes Grassmann's foundation of linear algebra as follows:<ref></ref> 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>


{{quote|The definition of a ] (])... became widely known around 1920, when ] and others published formal definitions. In fact, such a definition had been given thirty years previously by ], 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. {{blockquote|The definition of a ] (]) became widely known around 1920, when ] and others published formal definitions. In fact, such a definition had been given thirty years previously by ], 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 which 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 and formally proves the linear space properties for these operations. ... He then develops the theory of ] in a way which is astonishingly similar to the presentation one finds in modern linear algebra texts. He defines the notions of ], ], ], ], ] of subspaces, and ]s of elements onto subspaces. 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 and formally proves the linear space properties for these operations. ... He then develops the theory of ] in a way that is astonishingly similar to the presentation one finds in modern linear algebra texts. He defines the notions of ], ], ], ], ] of subspaces, and ]s of elements onto subspaces.


... few have come closer than Hermann Grassmann to creating, single-handedly, a new subject.}} 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 ], also called "combinatorial product" (in German: ''äußeres Produkt''<ref>''Tr''. outer product</ref> or ''kombinatorisches Produkt''<ref>''Tr''. combinatorial product</ref>), the key operation of an algebra now called ]. (One should keep in mind that in Grassmann's day, the only ]atic theory was ], and the general notion of an ] had yet to be defined.) In 1878, ] joined this exterior algebra to ]'s ] 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 ], we have the rule ''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = −1.) For more details, see ]. Following an idea of Grassmann's father, '''A1''' also defined the ], also called "combinatorial product" (in German: ''kombinatorisches Produkt'' or ''äußeres Produkt'' “outer product”), the key operation of an algebra now called ]. (One should keep in mind that in Grassmann's day, the only ]atic theory was ], and the general notion of an ] had yet to be defined.) In 1878, ] joined this exterior algebra to ]'s ] 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 ], we have the rule ''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = −1.) For more details, see ].


'''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 ] 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''<ref>''Tr''. New theory of electrodynamics</ref> and several papers on algebraic curves and surfaces, in the hope that these applications would lead others to take his theory seriously. '''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 ] 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 ]s and ]s, in the hope that these applications would lead others to take his theory seriously.


In 1846, ] invited Grassmann to enter a competition to solve a problem first proposed by ]: 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'',<ref>''Tr.'' Geometric analysis linked to the geometric characteristic invented by Leibniz</ref> 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 1846, ] invited Grassmann to enter a competition to solve a problem first proposed by ]: 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; it and its three color laws are still taught, as ]. Grassmann's work on this subject was inconsistent with that of ]. Grassmann also wrote on ], ], and ]. In 1853, Grassmann published a theory of how colors mix; his theory's four color laws are still taught, as ]. Grassmann's work on this subject was inconsistent with that of ].<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 ], ], and ].


In 1861, Grassmann laid the groundwork for ] in his ''Lehrbuch der Arithmetik''.<ref>{{cite journal |last1=Wang |first1=Hao |author1-link=Hao Wang (academic) |title=The Axiomatization of Arithmetic |journal=] |date=June 1957 |volume=22 |issue=2 |pages=145–158 |doi=10.2307/2964176 |publisher=] |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 ] in which each set of positive elements has a least member. 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 ]. 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.
In the early history of mathematics arithmetic with integers did not seem to need formal axioms, for facts like {{nowrap|1=''x'' + ''y'' = ''y'' + ''x''}} appeared to be self-evident. But in 1861 Grassmann showed that such facts could be deduced from more basic ones about successors and induction.<ref>{{cite book|last=Wolfram|first=Stephen|title=A New Kind of Science|publisher=Wolfram Media, Inc.|year=2002|page=1152|isbn=1-57955-008-8}}</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 ]. The result, ''Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet'' , hereinafter denoted '''A2''', fared no better than '''A1''', even though '''A2''''s manner of exposition anticipates the textbooks of the 20th century.


==Response== ==Response==
In 1840s, mathematicians were generally unprepared to understand Grassmann's ideas.{{sfn|Prasolov|1994|p=46}} In the 1860s and 1870s various mathematicians came to ideas similar to that of Grassmann's, but Grassmann himself was not interested in mathematics anymore.{{sfn|Prasolov|1994|p=46}} In the 1840s, mathematicians were generally unprepared to understand Grassmann's ideas.<ref name="Prasolov">{{cite book |last1=Prasolov |first1=Viktor V. |title=Problems and Theorems in Linear Algebra |date=1994 |publisher=] |location=Providence, RI |isbn=0-8218-0236-4 |translator1-first=Dimitry A. |translator1-last=Leites}}</ref> In the 1860s and 1870s various mathematicians came to ideas similar to that of Grassmann's, but Grassmann himself was not interested in mathematics anymore.{{r|Prasolov|p=46}}


] 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. ] 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.


One of the first mathematicians to appreciate Grassmann's ideas during his lifetime was ], whose 1867 ''Theorie der complexen Zahlensysteme'' One of the first mathematicians to appreciate Grassmann's ideas during his lifetime was ], whose 1867 ''Theorie der complexen Zahlensysteme''.<ref>{{cite encyclopedia |last=Crowe |first=Michael J. |encyclopedia=Dictionary of Scientific Biography |isbn=0-684-10114-9 |publisher=Charles Scribner's Sons |entry=Hankel, Hermann |chapter-url=https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/hankel-hermann}}</ref>
:... developed some of Hermann Grassmann's algebras and Hamilton's ]s. Hankel was the first to recognise the significance of Grassmann's long-neglected writings ...<ref>Hankel entry in the ''Dictionary of Scientific Biography''. New York: 1970–1990</ref> {{blockquote|, he developed some of Hermann Grassmann's algebras and W.R. Hamilton's ]s. Hankel was the first to recognise the significance of Grassmann's long-neglected writings and was strongly influenced by them.}}


In 1872 ] published the first part of his ''System der Raumlehre'' which used Grassmann's approach to derive ancient and modern results in plane geometry. ] 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 ] which also expanded the scope of geometry.<ref>Rowe 2010</ref> In 1872 ] published the first part of his ''System der Raumlehre'', which used Grassmann's approach to derive ancient and modern results in ]. ] 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-dimensional geometry. Meanwhile, Klein was advancing his ], which also expanded the scope of geometry.<ref name="Rowe">{{cite journal |last1=Rowe |first1=David E. |author1-link=David E. Rowe |title=Debating Grassmann's Mathematics: Schlegel Versus Klein |journal=The Mathematical Intelligencer |year=2010 |volume=32 |issue=1 |pages=41–48 |doi=10.1007/s00283-009-9094-2 |publisher=Springer }}</ref>


Comprehension of Grassmann awaited the concept of ]s which then could express the ] of his extension theory. To establish the priority of Grassmann over Hamilton, ] urged Grassmann's heirs to have the 1840 essay on tides published.<ref>] (1951), ''Josiah Willard Gibbs: The History of a Great Mind'', 1998 reprint, Woodbridge, CT: Ox Bow, pp. 113-116.</ref> ]'s first monograph, the ''Universal Algebra'' (1898), included the first systematic exposition in English of the theory of extension and the ]. With the rise of ] the exterior algebra was applied to ]s. Comprehension of Grassmann awaited the concept of ]s, which then could express the ] of his extension theory. To establish the priority of Grassmann over Hamilton, ] urged Grassmann's heirs to have the 1840 essay on tides published.<ref>] (1951), ''Josiah Willard Gibbs: The History of a Great Mind'', 1998 reprint, Woodbridge, CT: Ox Bow, pp. 113-116.</ref> ]'s first monograph, the ''Universal Algebra'' (1898), included the first systematic exposition in English of the theory of extension and the ]. With the rise of ] the exterior algebra was applied to ]s.


In 1995 Lloyd C. Kannenberg published an English translation of The Ausdehnungslehre and Other works. For an introduction to the role of Grassmann's work in contemporary ] see '']''<ref>Penrose (2004) ''The Road to Reality'', chapters 11 & 2</ref> by ]. In 1995 Lloyd C. Kannenberg published an English translation of The Ausdehnungslehre and Other works. For an introduction to the role of Grassmann's work in contemporary ] see '']'' by ].<ref>{{cite book |last1=Penrose |first1=Roger |author1-link=Roger Penrose |title=The Road to Reality: A Complete Guide to the Laws of the Universe |date=February 2005 |publisher=Alfred A. Knopf |location=New York |isbn=0-679-45443-8 |chapter=2. An Ancient Theorem and a Modern Question, 11. Hypercomplex numbers}}</ref>


==Linguist== ==Linguist==
Grassmann's mathematical ideas began to spread only towards the end of his life. Thirty years after the publication of '''A1''' the publisher wrote to Grassmann: “Your book ''Die Ausdehnungslehre'' has been out of print for some time. Since your work hardly sold at all, roughly 600 copies were used in 1864 as waste paper and the remaining few odd copies have now been sold out, with the exception of the one copy in our library”.{{sfn|Prasolov|1994|p=45}} Disappointed by the reception of his work in mathematical circles, Grassmann lost his contacts with mathematicians as well as his interest in geometry. The last years of his life he turned to historical ] and the study of ]. He wrote books on ], collected folk songs, and learned Sanskrit. He wrote a 2,000-page dictionary and a translation of the ] (more than 1,000 pages) which earned him a membership of the ]. In modern Grassmann's mathematical ideas began to spread only towards the end of his life. Thirty years after the publication of '''A1''' the publisher wrote to Grassmann: “Your book ''Die Ausdehnungslehre'' has been out of print for some time. Since your work hardly sold at all, roughly 600 copies were used in 1864 as waste paper and the remaining few odd copies have now been sold out, with the exception of the one copy in our library.”{{r|Prasolov|p=45}} Disappointed by the reception of his work in mathematical circles, Grassmann lost his contacts with mathematicians as well as his interest in geometry. In the last years of his life he turned to historical ] and the study of ]. He wrote books on ], collected folk songs, and learned Sanskrit. He wrote a 2,000-page dictionary and a translation of the '']'' (more than 1,000 pages). In modern studies of the ''Rigveda'', Grassmann's work is often cited. In 1955 a third edition of his dictionary was issued.{{r|Prasolov|p=46}}
Rigvedic studies Grassmann's work is often cited. In 1955 the third edition of his dictionary to Rigveda was issued.{{sfn|Prasolov|1994|p=46}}


Grassmann also noticed and presented a ] that exists in both ] and ]. In his honor, this phonological rule is known as ]. His discovery was revolutionary for historical linguistics at the time, as it challenged the widespread notion of Sanskrit as an older predecessor to other Indo-European languages.<ref>{{Cite web |title=A Reader in Nineteenth Century Historical Indo-European Linguistics, by Winfred P. Lehmann {{!}} The Online Books Page |url=https://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=olbp46747 |access-date=2023-10-18 |website=onlinebooks.library.upenn.edu}}</ref> This was a widespread assumption due to Sanskrit's more agglutinative structure, which languages like Latin and Greek were thought to have passed through to reach their more "modern" synthetic structure. However, Grassman's work proved that, in at least one phonological pattern, German was indeed "older" (i.e., less synthetic) than Sanskrit. This meant that genealogical and typological classifications of languages were at last correctly separated in linguistics, allowing significant progress for later linguists.<ref>{{Cite web |title=A Reader in Nineteenth Century Historical Indo-European Linguistics, by Winfred P. Lehmann {{!}} The Online Books Page |url=https://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=olbp46747 |access-date=2023-10-18 |website=onlinebooks.library.upenn.edu}}</ref>
Grassmann also discovered a sound law of ], which was named '']'' in his honor.


These philological accomplishments were honored during his lifetime; he was elected to the ] and in 1876, he received an honorary doctorate from the ]. These philological accomplishments were honored during his lifetime. He was elected to the ] and in 1876 he received an honorary doctorate from the ].


==Bibliography== ==Publications==
* '''A1''':
* '''A1''': 1844. ''''.<ref>''Tr''. "The linear extension theory"</ref> Leipzig: Wiegand. English translation, 1995, by Lloyd Kannenberg, ''A new branch of mathematics''. Chicago: Open Court.
** {{cite book |last1=Grassmann |first1=Hermann |title=Die Lineale Ausdehnungslehre |date=1844 |publisher=Otto Wigand |location=Leipzig |language=de |url=https://babel.hathitrust.org/cgi/pt?id=nyp.33433017485941 }}
* 1847. ''''.<ref>''Tr''. "Geometric analysis linked to the geometric characteristic invented by Leibniz"</ref> Available on
** {{cite book |last1=Grassmann |first1=Hermann |translator1-last=Kannenberg |translator1-first=Lloyd C. |title=A New Branch of Mathematics |date=1994 |publisher=] |isbn=9780812692761 |pages=9–297 |url=https://archive.org/details/newbranchofmathe0000gras |url-access=limited }}
* 1861. ''Lehrbuch der Mathematik für höhere Lehranstalten, Band 1''. Berlin: Enslin.
* {{cite book |last=Grassmann |first=Hermann |title=Geometrische Analyse |url=https://books.google.com/books?id=cHGrfrQVq1oC |year=1847 |publisher=] |location=Leipzig |language=de}}
⚫ * '''A2''': 1862. ''''.<ref>''Tr''. "Higher mathematics for schools, Volume 1"</ref> Berlin: Enslin. English translation, 2000, by Lloyd Kannenberg, ''Extension Theory'', ] {{ISBN|0-8126-9275-6}}, {{ISBN|0-8126-9276-4}}
* {{cite book |last=Grassmann |first=Hermann |title=Lehrbuch der Mathematik für höhere Lehranstalten |url=https://books.google.com/books?id=BeSD9fFZWDYC&pg=PR2 |volume=1: Arithmetik |year=1861 |publisher=Adolph Enslin |location=Berlin}}
⚫ * 1873. ''''.<ref>''Tr''. "Dictionary of the Rig-Veda"</ref> Leipzig: Brockhaus.
* '''A2''':
**1862. ''''. Berlin: Enslin.
⚫ ** English translation, 2000, by Lloyd Kannenberg, ''Extension Theory'', ] {{ISBN|0-8126-9275-6}}, {{ISBN|0-8126-9276-4}}
⚫ * 1873. ''''. Leipzig: Brockhaus.
* 1876–1877. ''Rig-Veda''. Leipzig: Brockhaus. Translation in two vols., published 1876, vol. 2 published 1877. * 1876–1877. ''Rig-Veda''. Leipzig: Brockhaus. Translation in two vols., published 1876, vol. 2 published 1877.
* 1894–1911. '',''<ref>''Tr''. "Collected mathematical and physical works"</ref> in 3 vols. ] ed. Leipzig: B.G. Teubner. Reprinted 1972, New York: Johnson. * 1894–1911. '','' in 3 vols. ] ed. Leipzig: B.G. Teubner.<ref>{{cite journal|doi=10.1090/S0002-9904-1907-01557-4|title=Book Review: ''Hermann Grassmanns gesammelte mathematische und physikalische Werke''|year=1907|last1=Wilson|first1=E. B.|author-link=Edwin Bidwell Wilson|journal=Bulletin of the American Mathematical Society|volume=14|pages=33–36|mr=1558534|doi-access=free}}</ref> Reprinted 1972, New York: Johnson.


==See also== ==See also==
*]
*] (Grassmann was its precursor) *] (Grassmann was its precursor)
*]
*]
*]


==Citations== ==Citations==
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==References== ==References==
* {{cite book |last1=Browne |first1=John |title=Grassmann Algebra |date=October 2012 |publisher=Barnard Publishing |location=Eltham, Australia |isbn=978-1479197637 |volume=I: Foundations}}
* Browne, John, '''', 2012, Barnard Publishing
* {{cite book |last1=Browne |first1=John |title=Multiplanes and Multispheres: Notes on a Grassmann Algebra approach with Mathematica |date=August 2020 |publisher=Barnard Publishing |location=Eltham, Australia |isbn=979-8657325379}}
* Browne, John, '''', 2020, Barnard Publishing,
* Cantù, Paola, '''' . Genoa: University of Genoa. Dissertation, 2003, s. xx+465. * {{cite thesis |last=Cantù |first=Paola |date=February 13, 2003 |title=La matematica da scienza delle grandezze a teoria delle forme: l{{'}}''Ausdehnungslehre'' di H. Grassmann |trans-title=The Mathematics of Quantities to the Science of Forms: The ''Ausdehnungslehre'' of H. Grassmann |publisher=University of Genoa |degree=PhD |url=https://air.unimi.it/retrieve/handle/2434/62423/100549/La%20matematica%20da%20scienza%20delle%20grandezze%20a%20teoria.pdf |language=it}}
* Crowe, Michael, 1967. ], Notre Dame University Press. * {{cite book |last1=Crowe |first1=Michael J. |title=A History of Vector Analysis |date=1967 |publisher=] |isbn=0-486-64955-5}}
* Fearnley-Sander, Desmond, 1979, "," ''American Mathematical Monthly 86'': 809–17. * {{cite journal |last1=Fearnley-Sander |first1=Desmond |title=Hermann Grassmann and the Prehistory of Universal Algebra |journal=The American Mathematical Monthly |date=March 1982 |volume=89 |issue=3 |pages=161–166 |doi=10.2307/2320198 |publisher=Mathematical Association of America |issn=0002-9890 |jstor=2320198}}
* {{cite conference |title=Area in Grassmann Geometry |last1=Fearnley-Sander |first1=Desmond |last2=Stokes |first2=Timothy |date=1997 |conference=International Workshop on Automated Deduction in Geometry 1996 |editor-last=Wang |editor-first=Dongming |volume=1360 |book-title=Automated Deduction in Geomtetry |publisher=Springer |location=Toulouse, France |pages=141–170 |isbn=978-3-540-69717-6 |issn=0302-9743 |doi=10.1007/BFb0022724 |series=Lecture Notes in Computer Science}}
* Fearnley-Sander, Desmond, 1982, "," ''Amer. Math. Monthly 89'': 161–66.
* {{cite book |last1=Grattan-Guinness |first1=Ivor |author1-link=Ivor Grattan-Guinness |title=The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Godel |date=2000 |publisher=] |isbn=9780691058580 |jstor=j.ctt7rp8j}}
* Fearnley-Sander, Desmond, and Stokes, Timothy, 1996, "". ''Automated Deduction in Geometry'': 141–70
* {{cite book |last1=Petsche |first1=Hans-Joachim |editor1-last=Fellmann |editor1-first=Emil A. |title=Graßmann |date=2006 |publisher=Birkhäuser |location=Basel, Switzerland |isbn=3-7643-7257-5 |language=de |volume=13 |series=Vita Mathematica}}
* ] (2000) ''The Search for Mathematical Roots 1870–1940''. Princeton Univ. Press.
* {{cite book |last1=Petsche |first1=Hans-Joachim |title=Hermann Graßmann |date=2009 |publisher=Birkhäuser |location=Basel, Switzerland |isbn=978-3-7643-8859-1 |translator-last=Minnes |translator-first=Mark |doi=10.1007/978-3-7643-8860-7 |lccn=2009929497}}
* ], 2004. ''The Road to Reality''. Alfred A. Knopf.
* {{cite book |editor1-last=Petsche |editor1-first=Hans-Joachim |editor2-last=Kannenberg |editor2-first=Lloyd C. |editor3-last=Keßler |editor3-first=Gottfried |editor4-last=Liskowacka |editor4-first=Jolanta |title=Hermann Graßmann – Roots and Traces |date=2009 |publisher=Birkhäuser |location=Basel, Switzerland |isbn=978-3-0346-0155-9 |lccn=2009930234 |doi=10.1007/978-3-0346-0155-9}}
* Petsche, Hans-Joachim, 2006. ''Graßmann'' (Text in German). (Vita Mathematica, 13). Basel: Birkhäuser.
* {{cite conference |date=September 2011 |conference=Graßmann Bicentennial Conference |conference-url=https://www.uni-potsdam.de/u/philosophie/grassmann/Grassmann-2009.htm |editor1-first=Hans-Joachim |editor1-last=Petsche |editor2-first=Jörg |editor2-last=Liesen |editor3-first=Albert C. |editor3-last=Lewis |editor4-first=Steve |editor4-last=Russ |book-title=From Past to Future: Graßmann's Work in Context |publisher=Birkhäuser |location=Potsdam-Szczecin |isbn=978-3-0346-0404-8 |doi=10.1007/978-3-0346-0405-5}}
* Petsche, Hans-Joachim, 2009. ''Hermann Graßmann – Biography''. Transl. by M Minnes. Basel: Birkhäuser.
* {{cite AV media |editor1-first=Peter C. |editor1-last=Lenke |editor2-first=Hans-Joachim |editor2-last=Petsche |date=2010 |title=International Grassmann Conference: Potsdam and Szczecin |medium=DVD |publisher=Universitätsverlag Potsdam |isbn=978-3-86956-093-9}}
* Petsche, Hans-Joachim; Kannenberg, Lloyd; Keßler, Gottfried; Liskowacka, Jolanta (eds.), 2009. ''Hermann Graßmann – Roots and Traces. Autographs and Unknown Documents. Text in German and English''. Basel: Birkhäuser.
* {{cite book |last1=Schlegel |first1=Victor |author1-link=Victor Schlegel |title=Hermann Grassmann: Sein Leben und seine Werke |date=1878 |publisher=] |location=Leipzig, Germany |url=https://archive.org/details/hermanngrassman00schlgoog |language=de}}
* Petsche, Hans-Joachim; Lewis, Albert C.; Liesen, Jörg; Russ, Steve (eds.), 2010. ''From Past to Future: Graßmann's Work in Context. The Graßmann Bicentennial Conference, September 2009''. Basel: Springer Basel AG.
* {{cite book |editor1-last=Schubring |editor1-first=Gert |title=Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar |date=1996 |publisher=Springer |isbn=978-94-015-8753-2 |doi=10.1007/978-94-015-8753-2 |series=Boston Studies in the Philosophy of Science |volume=187 |issn=0068-0346}}
* Petsche, Hans-Joachim and Peter Lenke (eds.), 2010. ''International Grassmann Conference. Hermann Grassmann Bicentennial: Potsdam and Szczecin, 16–19 September 2009; Video Recording of the Conference''. 4 DVDs, 16:59:25. Potsdam: Universitätsverlag Potsdam.
* Rowe, David E. (2010) "Debating Grassmann's Mathematics: Schlegel Versus Klein", ] 32(1):41&ndash;8.
* ] (1878) on the Internet Archive.
* Schubring, G., ed., 1996. ''Hermann Gunther Grassmann (1809–1877): visionary mathematician, scientist and neohumanist scholar''. Kluwer.
* {{citation |title=Problems and Theorems in Linear Algebra |first=Viktor |last=Prasolov |year=1994 |isbn=978-0-8218-0236-6 |series=Translations of Mathematical Monographs |publisher=] |volume=134}}


'''Note:''' Extensive , revealing substantial contemporary interest in Grassmann's life and work. References each chapter in Schubring. '''Note:''' Extensive , revealing substantial contemporary interest in Grassmann's life and work. References each chapter in Schubring.


==External links== ==External links==
*{{Commons category inline}} {{Commons}}
{{wikiquote}} {{Wikiquote}}
* The MacTutor History of Mathematics archive: * The MacTutor History of Mathematics archive:
** {{MacTutor Biography|id=Grassmann}} ** {{MacTutor Biography|id=Grassmann}}
** Discusses the role of Grassmann and other 19th century figures in the invention of linear algebra and vector spaces. ** Discusses the role of Grassmann and other 19th century figures in the invention of linear algebra and vector spaces.
* 's home page. * 's home page.
* : From Past to Future: Grassmann's Work in Context * : From Past to Future: Grassmann's Work in Context
* – A compilation of English translations of three notes by Cesare Burali-Forti on the application of Grassmann's exterior algebra to projective geometry * – A compilation of English translations of three notes by Cesare Burali-Forti on the application of Grassmann's exterior algebra to projective geometry
* (English translation of book by an early disciple of Grassmann) * (English translation of book by an early disciple of Grassmann)
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Latest revision as of 23:50, 10 November 2024

German polymath, linguist and mathematician (1809–1877) "Grassmann" redirects here. For the surname, see Grassmann (surname).
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Hermann Günther Grassmann
Hermann Günther Grassmann
Born(1809-04-15)15 April 1809
Stettin, Province of Pomerania, Kingdom of Prussia (present-day Szczecin, Poland)
Died26 September 1877(1877-09-26) (aged 68)
Stettin, German Empire
Alma materUniversity of Berlin
Known for
AwardsPhD (Hon):
University of Tübingen (1876)
Scientific career
InstitutionsStettin Gymnasium
1878 copy of Grassmann's "Die lineale Ausdehnungslehre"
1878 copy of Grassmann's "Die lineale Ausdehnungslehre"
First page of "Die lineale Ausdehnungslehre"
First page of "Die lineale Ausdehnungslehre"

Hermann Günther Grassmann (German: Graßmann, pronounced [ˈhɛɐman ˈɡʏntʰɐ ˈɡʁasman]; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties. His work preceded and exceeded the concept which is now known as a vector space. He introduced the Grassmannian, the space which parameterizes all k-dimensional linear subspaces of an n-dimensional vector space V. In linguistics he helped free language history and structure from each other.

Biography

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Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an ordained minister who taught mathematics and physics at the Stettin Gymnasium, where Hermann was educated.

Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to Prussian universities. Beginning in 1827, he studied theology at the University of Berlin, also taking classes in classical languages, philosophy, and literature. He does not appear to have taken courses in mathematics or physics.

Although lacking university training in mathematics, it was the field that most interested him when he returned to Stettin in 1830 after completing his studies in Berlin. After a year of preparation, he sat the examinations needed to teach mathematics in a gymnasium, but achieved a result good enough to allow him to teach only at the lower levels. Around this time, he made his first significant mathematical discoveries, ones that led him to the important ideas he set out in his 1844 paper Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, here referred to as A1, later revised in 1862 as Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet, here referred to as A2.

In 1834 Grassmann began teaching mathematics at the Gewerbeschule in Berlin. A year later, he returned to Stettin to teach mathematics, physics, German, Latin, and religious studies at a new school, the Otto Schule. Over the next four years, Grassmann passed examinations enabling him to teach mathematics, physics, chemistry, and mineralogy at all secondary school levels.

In 1847, he was made an "Oberlehrer" or head teacher. In 1852, he was appointed to his late father's position at the Stettin Gymnasium, thereby acquiring the title of Professor. In 1847, he asked the Prussian Ministry of Education to be considered for a university position, whereupon that Ministry asked Ernst Kummer for his opinion of Grassmann. Kummer wrote back saying that Grassmann's 1846 prize essay (see below) contained "commendably good material expressed in a deficient form." Kummer's report ended any chance that Grassmann might obtain a university post. This episode proved the norm; time and again, leading figures of Grassmann's day failed to recognize the value of his mathematics.

Starting during the political turmoil in Germany, 1848–49, Hermann and his brother Robert published a Stettin newspaper, Deutsche Wochenschrift für Staat, Kirche und Volksleben, calling for German unification under a constitutional monarchy. (This eventuated in 1871.) After writing a series of articles on constitutional law, Hermann parted company with the newspaper, finding himself increasingly at odds with its political direction.

Grassmann had eleven children, seven of whom reached adulthood. A son, Hermann Ernst Grassmann, became a professor of mathematics at the University of Giessen.

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 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 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 dimensions; the number of possible dimensions is in fact unbounded.

Fearnley-Sander describes Grassmann's foundation of linear algebra as follows:

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' e1, e2, e3, ..., he effectively defines the free linear space that they generate; that is to say, he considers formal linear combinations a1e1 + a2e2 + a3e3 + ... where the aj are real numbers, defines addition and multiplication by real numbers 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 subspace, linear independence, span, dimension, join and meet of subspaces, and projections 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 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 i = j = k = −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 curves and surfaces, in the hope that these applications would lead others to take his theory seriously.

In 1846, Möbius invited Grassmann to enter a competition to solve a problem first proposed by 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. Grassmann's work on this subject was inconsistent with that of Helmholtz. Grassmann also wrote on crystallography, electromagnetism, and mechanics.

In 1861, Grassmann laid the groundwork for Peano's axiomatization of arithmetic in his Lehrbuch der Arithmetik. 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.

Response

In the 1840s, mathematicians were generally unprepared to understand Grassmann's ideas. In the 1860s and 1870s various mathematicians came to ideas similar to that of Grassmann's, but Grassmann himself was not interested in mathematics anymore.

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.

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

, he developed some of Hermann Grassmann's algebras and W.R. Hamilton's quaternions. Hankel was the first to recognise the significance of Grassmann's long-neglected writings and was strongly influenced by them.

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-dimensional geometry. Meanwhile, Klein was advancing his Erlangen program, which also expanded the scope of geometry.

Comprehension of Grassmann awaited the concept of vector spaces, which then could express the multilinear algebra of his extension theory. To establish the priority of Grassmann over Hamilton, Josiah Willard Gibbs urged Grassmann's heirs to have the 1840 essay on tides published. 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.

In 1995 Lloyd C. Kannenberg published an English translation of The Ausdehnungslehre and Other works. For an introduction to the role of Grassmann's work in contemporary mathematical physics see The Road to Reality by Roger Penrose.

Linguist

Grassmann's mathematical ideas began to spread only towards the end of his life. Thirty years after the publication of A1 the publisher wrote to Grassmann: “Your book Die Ausdehnungslehre has been out of print for some time. Since your work hardly sold at all, roughly 600 copies were used in 1864 as waste paper and the remaining few odd copies have now been sold out, with the exception of the one copy in our library.” Disappointed by the reception of his work in mathematical circles, Grassmann lost his contacts with mathematicians as well as his interest in geometry. In the last years of his life he turned to historical linguistics and the study of Sanskrit. He wrote books on German grammar, collected folk songs, and learned Sanskrit. He wrote a 2,000-page dictionary and a translation of the Rigveda (more than 1,000 pages). In modern studies of the Rigveda, Grassmann's work is often cited. In 1955 a third edition of his dictionary was issued.

Grassmann also noticed and presented a phonological rule that exists in both Sanskrit and Greek. In his honor, this phonological rule is known as Grassmann's law. His discovery was revolutionary for historical linguistics at the time, as it challenged the widespread notion of Sanskrit as an older predecessor to other Indo-European languages. This was a widespread assumption due to Sanskrit's more agglutinative structure, which languages like Latin and Greek were thought to have passed through to reach their more "modern" synthetic structure. However, Grassman's work proved that, in at least one phonological pattern, German was indeed "older" (i.e., less synthetic) than Sanskrit. This meant that genealogical and typological classifications of languages were at last correctly separated in linguistics, allowing significant progress for later linguists.

These philological accomplishments were honored during his lifetime. He was elected to the American Oriental Society and in 1876 he received an honorary doctorate from the University of Tübingen.

Publications

See also

Citations

  1. Fearnley-Sander, Desmond (December 1979). "Hermann Grassmann and the Creation of Linear Algebra" (PDF). The American Mathematical Monthly. 86 (10). Mathematical Association of America: 809–817. doi:10.2307/2320145. ISSN 0002-9890. JSTOR 2320145.
  2. Turner, R. Steven (1996). "The Origins of Colorimetry: What did Helmholtz and Maxwell Learn from Grassmann?". Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar. Boston Studies in the Philosophy of Science. Vol. 187. Springer Netherlands. pp. 71–86. doi:10.1007/978-94-015-8753-2_8. ISBN 9789401587532. See p.74: "Helmholtz rejected almost as many of Grassmann's conclusions as he accepted."
  3. Wang, Hao (June 1957). "The Axiomatization of Arithmetic". The Journal of Symbolic Logic. 22 (2). Association for Symbolic Logic: 145–158. doi:10.2307/2964176. JSTOR 2964176. S2CID 26896458. p. 145, 147: 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. was probably the first serious and rather successful attempt to put numbers on a more or less axiomatic basis.
  4. ^ Prasolov, Viktor V. (1994). Problems and Theorems in Linear Algebra. Translated by Leites, Dimitry A. Providence, RI: American Mathematical Society. ISBN 0-8218-0236-4.
  5. Crowe, Michael J. "Hankel, Hermann". Dictionary of Scientific Biography. Charles Scribner's Sons. ISBN 0-684-10114-9.
  6. Rowe, David E. (2010). "Debating Grassmann's Mathematics: Schlegel Versus Klein". The Mathematical Intelligencer. 32 (1). Springer: 41–48. doi:10.1007/s00283-009-9094-2.
  7. Lynde Wheeler (1951), Josiah Willard Gibbs: The History of a Great Mind, 1998 reprint, Woodbridge, CT: Ox Bow, pp. 113-116.
  8. Penrose, Roger (February 2005). "2. An Ancient Theorem and a Modern Question, 11. Hypercomplex numbers". The Road to Reality: A Complete Guide to the Laws of the Universe. New York: Alfred A. Knopf. ISBN 0-679-45443-8.
  9. "A Reader in Nineteenth Century Historical Indo-European Linguistics, by Winfred P. Lehmann | The Online Books Page". onlinebooks.library.upenn.edu. Retrieved 2023-10-18.
  10. "A Reader in Nineteenth Century Historical Indo-European Linguistics, by Winfred P. Lehmann | The Online Books Page". onlinebooks.library.upenn.edu. Retrieved 2023-10-18.
  11. Wilson, E. B. (1907). "Book Review: Hermann Grassmanns gesammelte mathematische und physikalische Werke". Bulletin of the American Mathematical Society. 14: 33–36. doi:10.1090/S0002-9904-1907-01557-4. MR 1558534.

References

Note: Extensive online bibliography, revealing substantial contemporary interest in Grassmann's life and work. References each chapter in Schubring.

External links

Categories:
Hermann Grassmann: Difference between revisions Add topic