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ABSTRACT
The life and work of Hermann Günther Graßmann (1809–1877) attract not only ever again the attention of mathematicians, mathematical historians and those interested in the history of mathematics, they constitute also a challenge for the methodology of historiographical research. This challenge persists since Friedrich Engel’s biography of 1911; there, two sources were presented and interpreted in a not legitimate manner which even mislead since then various scholars. This paper faces the intricate task to unravel not only the methodological shortcomings of Engel’s biography, but also to re-assess the misinterpretations – particularly of alleged influences on Graßmann’s approaches – induced by Engel’s misleading claims. Based upon broader historically contextualised analyses of Graßmann’s innovative elements in his theory of extension and upon new primary sources for his student times at the Gymnasium and Berlin University, the paper presents a novel assessment of Graßmann’s approaches.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 https://mathshistory.st-andrews.ac.uk/Biographies/Grassmann/ accessed 25 April 2023.
2 A large number of the contributions have been published in the conference proceedings (Schubring Citation1996). In addition, further contributions have been published in a volume by the local conference organiser, Peter Schreiber (Citation1995).
3 See its review (Schubring Citation2003).
4 The extensive research and its results are reported in detail in the chapter (Schubring Citation1996a).
5 See my comment upon Engel's interpretation (Schubring Citation1996b: 59 f.).
6 See its review (Schubring Citation2007). He published the biography in English, in 2009 (Petsche Citation2009); some of the points criticised in 2007 have been revised there (Schubring Citation2011).
7 The value of the documents presented in the book is further hampered by the fact that the English translation of the documents is largely unreliable and even misleading. See the review of both volumes (Schubring Citation2011). The volume with the Röhl-Nachlass (Petsche et al. Citation2009) does not indicate its present location; many of the documents are printed without informing their location and code.
8 In a recent study on Schleiermacher's alleged influence on Graßmann, he was described as a ‘lay-mathematician’ (Achtner Citation2016: 112).
9 Carnot’s conceptions soon became known in Germany, in particular by the strong criticism published by Busse in (Citation1804); see also Fries (Citation1810).
10 Petsche dates Graßmann's discovery of vectors to the year 1832: ‘He found the geometric addition and multiplication of segments (vector addition and outer product of vectors) as early as 1832’ (Petsche Citation2006: 30). In the footnote given as evidence, he quotes a complaint from Graßmann in 1847, in which he says ‘dépuis 1832’. This ‘since’ definitely includes becoming acquainted with Drobisch and Mourey in 1834 or a little later.
11 Graßmann (Citation1839): thus, notabene, before him reading Schleiermacher’s book on Dialektik.
12 Notabene: Graßmann had already achieved essential conceptual steps for his innovations in 1840 – before studying Schleiermacher's Dialektik lecture, together with his brother.
13 His terminology was even unusual in German: ‘als gleich setzen’, resp. ‘als ungleich setzen’.
14 Likewise unusual: ‘Verknüpfung als gleich’.
15 The administration of the Protestant-Lutheran Church in the Prussian province Pomerania.
16 Engel had used then Apelt’s Nachlass.
17 Haben Sie denn Graßmanns wunderliche Ausdehnungslehre gesehen? Ich kenne sie nur aus Grunerts Archiv, aber mir scheint dem Buche eine falsche Philosophie der Mathematik zu Grunde zu liegen. Der wesentliche Charakter der mathematischen Erkenntniß, die Anschaulichkeit, scheint darin ganz verkannt zu seyn. So eine abstrakte Ausdehnungslehre, wie er sucht, könnte sich nur aus Begriffen entwickeln lassen. Aber die Quelle der mathematischen Erkenntniß liegt nicht in Begriffen sondern in der Anschauung.
18 ‘Das Geheimniß der reinen Anschauung besteht darin, daß Kant lehrt, Raum und Zeit sind keine Begriffe, keine aus der Erfahrung abstrahirten allgemeinen Vorstellungen, sondern wirkliche Anschauungen, aber nicht Sinnesanschauungen, sondern Anschauungen a priori, sie entspringen nicht, wie es Leibnitz wollte, aus dem Verhältniß der Dinge, sie sind vielmehr ein ursprüngliches und anschauliches Gesetz für das Nebeneinander und Nacheinander, welches allen Verhältnissen der Dinge vorhergeht und dieselben erst möglich macht. Die Kantische Ansicht ist für die Philosophie von unübersehbaren Folgen, aber ich glaube, sie hat auch für die Metaphysik des Calculs wie nicht minder für die Philosophie der Geometrie ihre große Bedeutung’ (Apelt-Nachlass, Nr. 7).
In the Apelt Nachlass, the copies preserved by Apelt are so-called original copies kept from his letters to Möbius. The answer-letters from Möbius are apparently no longer preserved: the Möbius archive of the Gesellschaft der Wissenschaften in Leipzig was destroyed due to the Second World War, and the partial Nachlass of Apelt in Düsseldorf does not contain the letters, which Apelt had received from Möbius.
19 The English version of 2009 maintains the same assessment, formulated somewhat differently (‘intermezzo’), (Petsche Citation2009: 146).
20 Same statement in Petsche (Citation2009: 244).
21 In section 2.7 on the arithmetic textbook, Petsche affirmed: ‘Damit aber ordnen sich die Untersuchungen zur Arithmetik organisch ein in den programmatischen Ansatz von 1844’, maintaining for this arithmetic also an influence of Schleiermacher’s Dialektik (Petsche Citation1979 I: 114).
22 The English version of the biography shows this same interpretation (Petsche Citation2009: 202 and 247).
23 See also his detailed study of the Dialektik concept: Wolff (Citation2017).
24 In his subject index, Arndt listed a large number of opposites that are addressed in the Dialektik. the opposite of the general and the particular, of concept and object, of form and matter, of possibility and reality, of truth and error, etc. (Schleiermacher 1814/15: 165 ff.).
25 The term ‘thinking collective’ introduced by Ludwik Fleck in 1935 became the basis for the concept of scientific community introduced by Thomas Kuhn in 1962.
26 And it doesn't imply any difference with Fries’s positions, who would also sign the statement (see below).
27 See my comments upon this in section 5.
28 One wonders how it could have been published to call here Paul Bernays a ‘Fries’ pupil’ (Petsche Citation2014: 539) – being born 45 years after the death of Fries.
29 The claimed influence for overcoming a ‘crisis in his life’ (Petsche Citation2014: 568) relies on the mis-interpretation of the second CV – which would have occurred, moreover, in an earlier period.
30 See below his concern for Graßmann’s ‘Lauterkeit’.
31 The English translation of this present tense puts it wrongly into past tense: ‘studied’ (Petsche et al. Citation2011: 495).
32 However, Engel erroneously stated the number of lectures by Schleiermacher attended as three (Engel Citation1911: 20), but in fact it had been four lectures: Dialektik and the Gospel according to Matthew (Engel Citation1911: 21) in the second semester, Life of Jesus in the fifth and psychology in the sixth.
33 In his fifth semester, Graßmann initially wanted to attend the geography of Palestine; the title is then crossed out.
34 Information by the Archiwum Państwowe w Szczecinie of 23 July 1985.
35 Petsche puts this abstract theory of forms, via connections, in the strange and unsuitable connection with Pestalozzi's form theory for elementary teaching of geometry (Petsche Citation2012: 214).
36 In a footnote, Petsche explains that Fries rejected ‘the dialectic philosophy of Schleiermacher’ in a review in 1842 (Petsche Citation2012: 189). In fact, Fries reviewed there parts of his Ethik published in 1841 from the Nachlass. And there Fries regretted that the Ethik und die Dialektik (1839) were published as fragments, so that one cannot get to know Schleiermacher's views as a coherent text. And he expressed his great admiration for Schleiermacher's achievements (Fries Citation1842: 159). His critical remarks about the Dialektik are primarily criticisms of Fichte and Schelling and especially of Hegel.
37 Vielmehr können wir dafür in den Operationen der Zusammensetzung immer weiter fortschreiten.
38 Strangely, even such a serious historian like Jens Høyrup accepted Petsche’s arguments as proven: ‘Schubring has also proposed that Schleiermacher’s willingness to go beyond three dimensions might be due to Fries through Hermann’s father Justus Graßmann. Petsche shows that the case in question has nothing to do with dimensions and thus can hardly have been relevant’ (Høyrup Citation2013).
39 Das besondere Sein, in diesem Sinne aufgefasst, nennen wir eine Denkform oder schlechtweg eine Form. Daher ist reine Mathematik Formenlehre.
40 Die reine Mathematik ist das vollständige System der mathematischen Formen.