Hermann Graßmann pp 221-255 | Cite as

# The genesis and essence of Hermann Günther Graßmann’s philosophical views in the *Extension Theory* of 1844

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## Abstract

Hermann Graßmann’s philosophical and methodological views were directly on the path his father had traced for him. His father had been influenced by Pestalozzi’s pedagogy, Leibniz’ combinatorial and synthetic approach, Kant’s constructive view of mathematics and the dialectics of Romantic philosophy of nature. Merged together, these were the elements of Justus Graßmann’s unique philosophical and mathematical position. These impulses — modified by Schleiermacher’s dialectics — also shaped Hermann Graßmann’s way of thinking.

## Keywords

Pure Mathematic Extension Theory Formal Science Philosophical Concept Philosophical Method
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## Notes

- 1.Leibniz, G. W.:
*Project of a New Encyclopedia to be written following the method of invention*(15 June 1679). In: Dascal 2008, p. 130–141 (p. 133–134).Google Scholar - 2.A1, p. 9.Google Scholar
- 4.A1, p. 9.Google Scholar
- 5.We possess no further information about this little text. — See BIO, p. 36sq.Google Scholar
- 8.A1, p. 9. See also Justus Graßmann’s teaching manual of plane and spherical geometry (1835), which deals with “The positive and negative in geometry” in § 59 (p. 26sqq) and with “The positive and negative in space” in § 163 (p. 70sqq), while also showing that geometry is the specific realm of negative magnitudes and that arithmetic only possesses this concept to the extent to which it is connected to geometry.Google Scholar
- 9.A1, p. 9.Google Scholar
- 10.See J. Graßmann 1824, p. 194. — Hermann Graßmann directly refers to this passage from his father’s treatise in the foreword to his
*Extension Theory*. — See A1, p. 9.Google Scholar - 11.A1, p. 9.Google Scholar
- 12.Ibidem.Google Scholar
- 13.Ibidem.Google Scholar
- 14.A1, p. 10.Google Scholar
- 15.Bell 1986, p. 360.Google Scholar
- 16.See ibidem.Google Scholar
- 18.A1, p. 10.Google Scholar
- 19.Ibidem.Google Scholar
- 22.Poincaré 2007, p. 79/80.Google Scholar
- 24.A1, p. 10.Google Scholar
- 25.KRY, p. 174.Google Scholar
- 27.Möbius 1827, p. xiv.Google Scholar
- 28.Möbius 1827, p. 16/17.Google Scholar
- 29.A1, p. 11.Google Scholar
- 31.A1, p. 12.Google Scholar
- 33.A1, p. 17.Google Scholar
- 35.A1, p. 15.Google Scholar
- 36.A1, p. 16.Google Scholar
- 37.See Klaus 1965, p. 102.Google Scholar
- 38.A1, p. 23. An alternate translation of this passage by Albert Lewis renders “produced by thought alone” as “posited through thought itself ”.Google Scholar
- 39.See A1, p. 23sq.Google Scholar
- 41.See Ruzavin 1977, p. 172.Google Scholar
- 43.DIAL, p. 7.Google Scholar
- 44.DIAL, p. 48.Google Scholar
- 46.DIAL, p. 53.Google Scholar
- 47.DIAL, p. 40 (note).Google Scholar
- 48.DIAL, p. 48 (note).Google Scholar
- 50.
*A1*, p. 23. Where Lloyd Kannenberg uses the concept of portrayal, we would like to suggest representation as an alternative.Google Scholar - 51.H. Graßmann 1878, p. 7.Google Scholar
- 52.Ibidem, p. 9/10.Google Scholar
- 53.Ibidem, p. 15.Google Scholar
- 54.A1, p. 23/24.Google Scholar
- 55.A1, p. 24.Google Scholar
- 56 DIAL, p. 309.Google Scholar
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- 62.Ibidem, p. 24.Google Scholar
- 63.Ibidem.Google Scholar
- 65.See Kant 2007, p. 577sq.Google Scholar
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- 67.See Šljachin 1976, p. 131.Google Scholar
- 68.Kant 2007, p. 577.Google Scholar
- 69.See Šljachin 1976, p. 131.Google Scholar
- 70.ZL, p. 3.Google Scholar
- 71.Ibidem.Google Scholar
- 72.Ibidem.Google Scholar
- 73.ZL, p. 3.Google Scholar
- 74.Ibidem, p. 4.Google Scholar
- 75.Schleiermacher 1942, p. 130.Google Scholar
- 76.DIAL, p. 386 (footnote).Google Scholar
- 77.Schleiermacher 1942, p. 33.Google Scholar
- 78.Einstein is very explicit in a remark in
*Geometry and Experience*: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” See Einstein 2004, p. 12–23.Google Scholar - 79.A1, p. 23/24. Again, in this quotation we would like to suggest “posited” as an alternate translation for “established”.Google Scholar
- 80.See Kant 2007, p. 578sq.Google Scholar
- 81.A1, p. 24.Google Scholar
- 82.See Molodschi 1977, p. 210sq.Google Scholar
- 84.Bourbaki 1950, p. 231.Google Scholar
- 85.A1, p. 46.Google Scholar
- 86.See also Molodschi 1977, p. 276/277.Google Scholar
- 87.R. Graßmann 1890b, p. 70.Google Scholar
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- 90.A1, p. 24.Google Scholar
- 91.A1, p. 24/25.Google Scholar
- 92.R. Graßmann 1890b, p. 70.Google Scholar
- 93.See, for details on Lobačevskij’s philosophical positions, Licis 1976, p. 52sqq.Google Scholar
- 94.A1, p. 25.Google Scholar
- 95.A1, p. 26.Google Scholar
- 96.A1, p. 25.Google Scholar
- 97.Ibidem. “Positing and connecting” might very well serve as alternate translations for “placement and conjunction” in these quotations.Google Scholar
- 98.A1, p. 26.Google Scholar
- 99.Ibidem.Google Scholar
- 101.A1, p. 26.Google Scholar
- 103.A1, p. 27.Google Scholar
- 104.Ibidem, p. 27.Google Scholar
- 105.See Kant 2007, p. 197sqq.Google Scholar
- 106.ZL, p. 6.Google Scholar
- 107.Enriques 1907, p. 63/64.Google Scholar
- 110.More information on Scheibert’s relation to Herbart’s philosophy can be found in Müller 1926.Google Scholar
- 111.See Torretti 1984, p. 107–109. We find a contrary argument in Laugwitz 1999, p. 223.Google Scholar
- 112.See Riemann 1876a, p. 255.Google Scholar
- 113.Riemann 1876b, p. 476.Google Scholar
- 114.Herbart 1890, p. 409sqq.Google Scholar
- 115.Ibidem, p. 415.Google Scholar
- 116.See Kant 1929, p. 12.Google Scholar
- 117.Ibidem.Google Scholar
- 118.A1, p. 27.Google Scholar
- 119.Compare Graßmann’s realization: “...although the different always somehow adheres to the equal and conversely, at the moment of consideration only the one is encompassed, while the other only appears as the requisite basis of the first.” (A1, p. 26).Google Scholar
- 120.Here the concept “metaphysical” is meant to signify “un-dialectical”, not “anti-dialectical.” See Heitsch 1976, p. 56. While the dialectical dimension prevails in the A1 of 1844, the “metaphysical” approach is dominant in the A2 of 1862.Google Scholar
- 121.See Erpenbeck /Hörz 1977, p. 133.Google Scholar
- 122.A1, p. 28.Google Scholar
- 123.Ibidem.Google Scholar
- 125.See A1, p. 28.Google Scholar
- 126.Birjukova /Birjukov 1997, p. 137.Google Scholar
- 129.See also the detailed analysis in Radu 2000, p. 173sqq. Birjukova/Birjukov 1997 also come very close.Google Scholar
- 130.Herewith Graßmann stands for a positive alternative to the “prince of mathematics”, Gauß, who almost never uncovered his scientific work-process. See Wußing 1976, p. 64sq.Google Scholar
- 131.A1, p. 30.Google Scholar
- 132.See Kant 2007, p. 577sq.Google Scholar
- 134.A1, p. 30.Google Scholar
- 135.See A1, p. 30.Google Scholar
- 136.A1, p. 31.Google Scholar
- 137.Ibidem.Google Scholar
- 138.Poincaré 2007, p. 25.Google Scholar
- 139.A1, p. 31.Google Scholar
- 140.Ibidem.Google Scholar
- 141.A1, p. 32.Google Scholar
- 142.Ibidem.Google Scholar
- 143.Ibidem.Google Scholar
- 144.In this context, see the statistical analyses of works published by Graßmann’s followers on the one hand, and Hamilton’s on the other, ranging from the second half of the 19
^{th}century to the early 20^{th}century. In: Crowe 1994, p. 109–149.Google Scholar - 149.There is a reference to Schweitzer in Lewis 2004. But when Lewis points out that Paul Carus, who was a pupil of Graßmann in Stettin, also was strongly familiar with Graßmann’s way of thinking, this is only partly true, because Carus strongly opposed the mathematical concept of
*n*-dimensional spaces as late as the 1880s (Carus 1881): “But the whole enterprise has no other significance than to show that what in fact is impossible may still be correct from a logical point of view. It is a scientific equivalent to a paradoxical piece of fiction and only confirms the undeniable fact that one cannot contradict the careful liar by his own words alone, because one will find no contradiction there.” (Carus 1881, p. 54) Carus changed his mind 25 years later (Carus 1908), when he published a foundation of a philosophy of geometry, in which he praised Graßmann enthusiastically. So Paul Carus and Victor Schlegel seem to share the same attitude. Both Carus and Schlegel admired Graßmann’s ideas only after having left Stettin.Google Scholar - 151.See also R. Graßmann 1890b, p. 83, and chapter 2, section 2.Google Scholar
- 152.Schleiermacher wrote Gaß about Bartholdy: “In Berlin, he told me about his plans for a seminar, which made me very happy, and from which I conclude that I share his view of Pestalozzi’s idea and its essential importance. Here, as well, the combination of Lutheran and Reformed higher schools is the next step.” Letter from Schleiermacher to Gaß, May 1805. In: Schleiermacher 1852, p. 23.Google Scholar
- 153.Letter from J. Chr. Gaß to Schleiermacher, 13 July 1805. In: Schleiermacher 1852, p. 25sq.Google Scholar
- 154.See Arndt 1986, p. xxiii.Google Scholar
- 155.“With much gratitude I return your copy of the
*Dialectic*..., my dear friend.” Letter from J. Chr. Gaß to Schleiermacher, 31 March 1816. In: Schleiermacher 1852, p. 125sq.Google Scholar - 156.Justus Graßmann wrote in the preface to the
*Geometry for Elementary Schools*(1817): “Enough has already been said about the justification and usefulness of this work in the foreword to Bartholdy’s ‘Versuch einer Sprachbildungslehre für Deutsche’, which has appeared recently. Its general outlines have something to do with some local schools for the poor, which I and some dear friends, now passed away, most notably councilor Bartholdy, have helped to found and establish, apart from preparing these school’s teachers for the sake of doing a good deed, voluntarily and without pay.” (Graßmann 1817, p. iii).Google Scholar - 157.See R. Graßmann 1891, p. 16sq.Google Scholar
- 158.See Veronese 1894, p. vii sqq.Google Scholar
- 159.Veronese 1894, p. 1–2.Google Scholar
- 160.See also Radu 2000, p. 166.Google Scholar
- 161.A2, p. xvii.Google Scholar

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