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H. Grassmann’s 1844 Ausdehnungslehre and Schleiermacher’s Dialektik. (English) Zbl 0355.01011

MSC:
01A55 History of mathematics in the 19th century
03-03 History of mathematical logic and foundations
08-03 History of general algebraic systems
15-03 History of linear algebra
Biographic References:
Grassmann, Hermann
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References:
[1] Klein, F. 1967.Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, Vol. 1, 175–175. New York: Dover. reprint of original 1926 edition),
[2] Boyer, C. 1968.A history of mathematics, 584–584. New York: Wiley.
[3] Kline, M. 1972.Mathematical thought from ancient to modern times, 782–782. New York: Oxford University Press. · Zbl 0277.01001
[4] Hermann Grassmanns gesammelte mathematische und physikalische Werke (1894)
[5] A history of vector analysis: the evolution of the idea of a vectorial system (1967) · Zbl 0165.00303
[6] Henrici O., Encyclopaedia Britannica 11 pp 688–, 11. ed. (1910)
[7] DOI: 10.1086/368504 · doi:10.1086/368504
[8] Steiner J., Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander (1832) · JFM 28.0474.04
[9] Gauss C.F., Werke 12 pp 57– (1929)
[10] Werke 8 pp 177– (1900) · Zbl 0179.00101
[11] Körner, S. 1960.The philosophy of mathematics: an introduction, 183–183. New York: Harper.
[12] 1866.Traité des propriétés projectivesVol. 2, 357–357. Paris (The first edition of 1822 contains Cauchy’s ’Rapport’.)
[13] 1866.Traité des propriétés projectivesVol. 2, 5–5. Paris
[14] Werke 1 · Zbl 0179.00101
[15] Ohm, M. 1819.Kritische Beleuchtungen der Mathematik überhaupt und der Euklidischen Geometrie insbesondere7–7. Berlin
[16] Bolzano B., Abh. Königl. Böhm. Gesell. Wiss. 3 pp 201– (1843)
[17] Osiris 7 pp 173– (1939)
[18] Osiris 7 pp 173– (1939)
[19] Kant’s prolegomena and metaphysical foundations of natural science (1883)
[20] Encyklopädie der Elementar MathematikVol. 2, 146–146. bk. 1, sect. 2 as quoted in Cassirer (footnote 8), 106
[21] Encyklopädie der Elementar MathematikVol. 2, 145–145.
[22] 1969.The world as will and representation, Vol. 1, 73–73. New York: Dover. (translated by E. F. J. Payne
[23] Körner. 1960.The philosophy of mathematics: an introduction, 28f–28f. New York: Harper. G. Buchdahl,Metaphysics and the philosophy of science(Cambridge, Mass.: The MIT Press, 1969), 580, 611ff.
[24] Werke 3 pp viii– · Zbl 0179.00101
[25] Werke 3 pp 21– · Zbl 0179.00101
[26] Friedrich Schleiermacher’s sämmtliche Werke (1839)
[27] Werke 3 pp 91– · Zbl 0179.00101
[28] Schlegel, V. 1878.Hermann Grassmann. Sein Leben und seine Werke, 14–14. Leipzig: Brockhaus. · JFM 10.0020.05
[29] Letter to St.Venant, 18Werke1847 April 3 42 42 pt. 2
[30] Brandt R.B., The philosophy of Schleiermacher: the development of his theory of scientific and religious knowledge (1941)
[31] ’Actual, specific’ Spiegler G.The eternal covenant: Schleiermacher’s experiment in cultural theologyHarper & Row New York 1967 82 82
[32] Dialektik pp 309–
[33] Two works in English have been relied upon as guides to Schleiermacher’s work Spiegler G.The eternal covenant: Schleiermacher’s experiment in cultural theologyHarper & Row New York 1967 82 82 and R. B. Brandt (footnote 31). Also of use was W. Dilthey,Leben Schleiermachers, Zweiter Bd., compiled by M. Redeker, Bd. 14 ofWilhelm Dilthey, Gesammelte Schriften(Göttingen: 1966).
[34] Sämmtliche Werke 1 (1846)
[35] InSämmtliche WerkeG. Reimer Berlin 1846 1 333 338 div. 3
[36] InSämmtliche WerkeG. Reimer Berlin 1846 1 336 336 div. 3
[37] ’Positing particular skeptical hypotheses’ SpieglerThe eternal covenant: Schleiermacher’s experiment in cultural theologyHarper & Row New York 1967 46 46
[38] Grundlinien pp 336–
[39] Grundlinien pp 339–
[40] Spiegler, G. 1967.The eternal covenant: Schleiermacher’s experiment in cultural theology, 53–53. New York: Harper & Row.
[41] Dialektik pp 312–
[42] Dialektik pp 39–
[43] Dialektik pp 43–
[44] Dialektik pp 48–
[45] Dialektik pp 48–
[46] Dialektik pp 140–
[47] Dialektik pp 336–
[48] Dialektik pp 336–
[49] Dialektik pp 50–
[50] Dialektik pp 51–
[51] Dialektik pp 68–
[52] Dialektik pp 69–
[53] Dialektik pp 490–
[54] Dialektik pp 69–
[55] Dialektik pp 179–
[56] Dialektik pp 178–
[57] Dialektik pp 179–
[58] Dialektik pp 285–
[59] Dialektik pp 287–
[60] Dialektik pp 291–
[61] Dialektik pp 299–
[62] Dialektik pp 299–
[63] Dialektik pp 296–
[64] Dialektik pp 296–
[65] Dialektik pp 296–
[66] Dialektik pp 297–
[67] Dialektik pp 557–
[68] Immanuel Kant’s Critique of pure reason (1929)
[69] 1929.Immanuel Kant’s Critique of pure reason, 547–548. New York: St. Martin’s Press. translated by N. K. Smith 1965
[70] Dialektik pp 299–
[71] Dialektik pp 303–
[72] Dialektik pp 300–
[73] Dialektik pp 309–
[74] Dialektik pp 92–
[75] Dialektik pp 309–
[76] Dialektik,
[77] Dialektik pp 288–
[78] Dialektik pp 59–
[79] Dialektik pp 310–
[80] Dialektik pp 311–
[81] Dialektik,
[82] Dialektik pp 61–
[83] 1829.Zur physischen Krystallonomie und geometrischen Combinationslehre, erstes Heft, 172–172. Stettin: Morin.
[84] Werke 1 pp 404– · Zbl 0179.00101
[85] A122–23. The following remarks on my translations from the A1are necessary. Grassmann achieved more recognition during his lifetime as a philologist than as a mathematician, and at least a trace of Grassmann as philologist appears in the A1in his preference for simple German over Latin or other foreign terms. ’Strecke’ and ’Stufe’ are two such terms used for two of the key concepts later in the A1. I think it is important to signalize the fact that their meanings were unique to Grassmann–at least for him, it seems clear, they were new mathematical concepts. For this reason, ’stretch’ is chosen to translate ’Strecke’, since a modern reader would probably have no strong mathematical preconceptions about it and its usual meaning of a continuous extent of length suits the geometrical prototype of ’Strecke’. (This translation was suggested to me by Prof. Dr. C. Thiel.) For analogous reasons ’step’ is used to translate ’Stufe’. My translation tends to follow the German closely, and I believe that by using such words as ’stretch’ and ’step’, which have no wide currency in mathematics, Grassmann’s effect of novelty in a mathematical context is achieved.
[86] A123–23.
[87] A123–23.
[88] Grassmann R., Die Formenlehre oder Mathematik (1872)
[89] A124–25.
[90] A125–25.
[91] A125–25.
[92] A125–25.
[93] A125–26.
[94] 1817.Raumlehre für Volkschulen. Erster Theil: Ebene räumliche Verbindungslehre, x–xi. Berlin: In der Realschulbuchhandlung.
[95] 1824.Raumlehre für die untern Klassen der Gymnasien, und für Volkschulen. Zweiter Theil. Ebene räumliche Grössenlehre, 194–195. Berlin: G. Reimer.
[96] A126–27.
[97] A128–28.
[98] Grassmann, J. 1829.Zur physischen Krystallonomie und geometrischen Combinationslehre, erstes Heft, 33–34. Stettin: Morin.
[99] A128–29.
[100] Cassirer. 1832.Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander78–78. Berlin
[101] Cassirer. 1832.Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander99–99. Berlin
[102] The philosophy of symbolic forms 3 (1953)
[103] Cassirer. 1832.Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander111–111. Berlin
[104] DOI: 10.1080/00033797500200161 · Zbl 0301.01008 · doi:10.1080/00033797500200161
[105] A130–32.
[106] A146–46.
[107] Kant. 1929.Immanuel Kant’s Critique of pure reason, 577–577. New York: St. Martin’s Press. translated by N. K. Smith
[108] Kant. 1929.Immanuel Kant’s Critique of pure reason, 577–577. New York: St. Martin’s Press. translated by N. K. Smith
[109] Kant. 1929.Immanuel Kant’s Critique of pure reason, 577–577. New York: St. Martin’s Press. translated by N. K. Smith
[110] Kant. 1929.Immanuel Kant’s Critique of pure reason, 578–578. New York: St. Martin’s Press. translated by N. K. Smith
[111] Schopenhauer. 1969.The world as will and representation, Vol. 1, 70–70. New York: Dover. (translated by E. F. J. Payne
[112] Schopenhauer. 1969.The world as will and representation, Vol. 1, 52–52. New York: Dover. (translated by E. F. J. Payne
[113] Schopenhauer. 1969.The world as will and representation, Vol. 1, 71–71. New York: Dover. (translated by E. F. J. Payne
[114] Schopenhauer. 1969.The world as will and representation, Vol. 1, 53–58. New York: Dover. (translated by E. F. J. Payne
[115] A123–24.
[116] The Open Court 2 pp 1463– (1889)
[117] A163–64.
[118] A165–65.
[119] A166–66.
[120] A133–33.
[121] A128–28.
[122] A133–34.
[123] A134–35.
[124] A135–35. The edition of the A1being used here (footnote 14), has italicized the ’theorems’ (here the paragraphs beginning ’If a connection ...’) which are also set off as separate paragraphs, neither of which was done in the original 1844 edition. This is one of the consistent differences which makes theWerkeedition easier to read than the 1844 edition and which must be taken into account in using the more recent version for the text of the original. The italicizing and the use of subheadings (though from an outline by Grassmann, these also are unique to theWerkeedition) will be ignored in quoting from the A1.
[125] Grassmann’s H., Erster Theil: Lehrbuch der Arithmetik für höhere Lehranstalten (1861)
[126] Werke 3 pp 259– · Zbl 0179.00101
[127] A135–36.
[128] A136–36.
[129] A136–36.
[130] A137–38.
[131] A138–39.
[132] Cassina, U., ed. 1958.Opere scelte di Giuseppe Peano, Vol. 2, 366–366. Rome: Cremonese.
[133] A139–40.
[134] A140–41.
[135] DOI: 10.1016/0315-0860(74)90031-7 · Zbl 0288.01014 · doi:10.1016/0315-0860(74)90031-7
[136] Natucci A., Il concetto di numero, e le sue estensioni, in Areheion, archivo di storia della scienza 4 pp 382– (1923) · JFM 49.0016.01
[137] A141–41.
[138] Werke 1 pp 407– · Zbl 0179.00101
[139] A141–42.
[140] When Grassmann tries to give a more detailed and explicit account of such relations between connections of different steps (addition-subtraction, multiplication-division, and power-root) in his Lehrbuch (footnote 133) it is without much greater success. In hisLehrbuch der Arithmetik und Algebra für Lehrer und StudirendeTeubner Leipzig 1873 274 274 E. Schröder, who follows out the program of Grassmann’sLehrbuchto a great extent, refers to the rule in Grassmann’sLehrbuch, p. 313, for changing second-step theorems to third-step theorems as too general ’to the extent I understand it correctly’, and implies it could result in false theorems.
[141] A143–43.
[142] Lehrbuch (1861)
[143] A144–45.
[144] A147–47.
[145] A149–49.
[146] A149–49.
[147] A150–50.
[148] A151–51.
[149] A151–51.
[150] A152–52.
[151] A153–54.
[152] A155–55.
[153] A156–56.
[154] A159–59.
[155] Programm der Ottoschule (1839)
[156] A162–62.
[157] If the number thus produced [by counting] is made the basis for a new counting by putting it in place of the unit, then the arithmetic contact to multiplication is obtained which is therefore nothing other than a number of higher step, a number whose unit is also a number’ Grassmann J.Raumlehre für die untern Klassen der Gymnasien, und für Volkschulen. Zweiter Theil. Ebene räumliche GrössenlehreG. Reimer Berlin 1824 195 195
[158] A177–77.
[159] A177–78.
[160] A179–79.
[161] A156–56.
[162] Grassmann, J. 1829.Zur physischen Krystallonomie und geometrischen Combinationslehre, erstes Heft, 9–9. Stettin: Morin.
[163] DOI: 10.1007/BF00540144 · Zbl 0269.01006 · doi:10.1007/BF00540144
[164] Bruce, R.V. 1973.Bell: Alexander Graham Bell and the conquest of solitude, 251–252. Boston: Little Brown.
[165] A179–80.
[166] A180–81.
[167] A181–81.
[168] A182–82.
[169] A182–83.
[170] A1108–108.
[171] A1108–108.
[172] A184–84.
[173] A184–84.
[174] A1207–207.
[175] A2( =WerkeVol. 1, 10–10. pt. 2
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