The unity of logic, pedagogy and foundations in Grassmann's mathematical work

Abstract

Hermann Grassmann's Ausdehnungslehre of 1844 and his Lehrbuch der Arithmetik of 1861 are landmark works in mathematics; the former not only developed new mathematical fields but also both contributed to the setting of modern standards of rigor. Their very modernity, however, may obscure features of Grassmann's view of the foundations of mathematics that were not adopted since. Grassmann gave a key role to the learning of mathematics that affected his method of presentation, including his emphasis on making initial assumptions explicit. In order to better understand this less well-known aspect of his work it will help to examine why some commentators have overlooked his theme of unifying logic, pedagogy and foundations, while others have recognised it.

Notes

References are made to the sections of the Ausdehnungslehre of 1844 in order to accommodate the various printings and translations.

F. Vieta in the sixteenth century linked the term ‘analytic’ to a whole branch of mathematics, namely algebra. Though Lagrange echoed this use in his Méchanique analitique of 1788, his proofs were synthetic in the sense used in Pappos's heuristics. More of this tortuous history is provided in Grattan-Guinness 1997 .

Another example of this sentiment is: ‘The conclusion is that Logic, conceived as an adequate analysis of the advance of thought, is a fake. It is a superb instrument, but it requires a background of common sense’ (Whitehead 1941 , p. 700). More examples can be found in Kline 1980 .

The author's 1977 paper was an excerpt from his University of Texas at Austin dissertation which Ivor Grattan-Guinness encouraged him to publish.

The two brothers collaborated rather closely on both the Ausdehnungslehre and the Lehrbuch der Arithmetik (1861) and, though it is not known exactly what each contributed, judging from his later work Robert, the younger brother, could well have been a major contributor within the philosophical and logical components. A brief account of Robert Grassmann's place in the history of logic is given in Grattan-Guinness 1996 and 2000 . The collaboration of the brothers is discussed in Schubring 1996a .

Because Felix Klein held Schlegel in an unfavourable light, Schlegel's views are apt not to get the historical weight they deserve. Probably due to Klein's influence, Schlegel was invited to contribute only to the bibliography in the grand Werke project Grassmann 1894 –1911. On the Schlegel–Klein relationship see Rowe 1996 .

Grassmann does not give a statement of mathematical induction but rather, in his first use of it in a proof, states that this is an example of a proof by induction. Peano's system is given in Peano 1889 , translated in Peano 1967 . An overview of the Lehrbuch's import is given in Lewis 1995 and an evaluation of its logical basis is made in Wang 1957 .

Grassmann 1878 is the basis of the 1844 version printed in volume I, part 1, of Grassmann 1894 –1911.

The representative works are: Peano 1888 , translated into English as Peano 2000 ; Whitehead 1898 , discussed in relation to Grassmann in Lowe 1985 –1990 (vol. 1, pp. 153–156); and Cassirer 1953 (pp. 96–99).

Meyer 1962 .

Schweitzer's birth date is given as 1877 in Sommerville 1970 but appears as 1878 in the Library of Congress catalogue. There is a notice in the American Mathematical Monthly, 64 (1957), p. 611, of his death on 12 June 1957.

In Schweitzer 1915 the author mentions that Schleiermacher is undoubtedly a source of a number of Grassmann's notions, in particular that of ‘leitende Idee’. Though he does not justify this observation beyond citing a few parallel passages in Grassmann and Schleiermacher, it is the closest statement I know of to the thesis in my 1977 in prior literature.

Schweitzer 1914 is reprinted in de Waal 2001 from which I first learned of this Grassmannian.

In defence of Grassmann it may be possible to argue that his more fundamental notion is ‘change’ (Änderung) and that this would be needed to distinguish whatever entities, either points or vectors (or numbers as in the Lehrbuch der Arithmetik), were being generated. Admittedly ‘change’ and ‘vector’ are naturally close concepts but if it is possible to separate them, Swimmer's own construction shows that one need not take vector as the initially generated entities within Grassmann's general framework.

On the typical German university compared with a gymnasium, such as that Grassmann taught in, see Rowe (1996, pp. 132–134). For descriptions of Grassmann as a teacher and of his other textbooks in trigonometry, Latin, and German, see the index to the proceedings Schubring 1996b .