Abstract
The present article first places Mach’s consideration about space and geometry into the context of the discussion of these issues in the nineteenth and early twentieth century and then proposes three interpretations of Mach’s thesis, put forward in chapter XXI of his Knowledge and Error, that the problem of measuring the volumes of material bodies is the origin of geometry. According to the first of these interpretations, Mach’s thesis is an assertion about the historical origin of the science of geometry. Alternatively, one may understand Mach as suggesting that our geometric theorizing is best understood by relating it to our handling of material bodies and our interest in their volumes. Finally, Mach’s thesis may be conceived as asserting that the most appropriate form of geometry would be a metric geometry of the volumes of solids. The article concludes with a discussion of objections raised by Brentano against both Mach’s thesis of the priority of the measurement of volumes and his conception of bodies as complexes as sensations.
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Notes
- 1.
Page numbers separated by a slash refer to the English translation (first number) of a work originally written in German and to the German original (second number).
- 2.
The wording of the German original is “den Gebilden der Metamathematiker” (plural, dative case) where the English translation just has “space of metageometry” which is more neutral and grammatically singular.
- 3.
There is no chapter numbering in the German original; it has been added in the English edition. Chapter XXIII deals exclusively with time; chapter XXIV with “Space and Time Physically Considered”. Visual space and the perception of movement are also treated in chapters VI and VII of Mach’s Analysis of Sensation (Mach 1959).
- 4.
This chapter has been originally published in 1902 as an article in the journal The Monist (vol. 12, no. 4, pp. 481–515). The article version of the text had been translated from Mach’s manuscript by Thomas J. McCormack and was included with some marginal changes into Erwin N. Hiebert’s 1976 English edition of Knowledge and Error.
- 5.
This is one of the many issues of Mach’s conception of geometry which has been severely criticized by Brentano. According to Brentano (1988, p. 126), the investigation of our spatial image of the world is the task of empirical disciplines as astronomy and geography rather than that of geometry, which science is concerned with the relationships between spatial magnitudes.
- 6.
However, the situation already changed three years later when Minkowski suggested his four-dimensional affine geometry for the special theory of relativity in 1908. Even more radical departures from the Euclidean scheme followed soon within the general theory of relativity.
- 7.
For a detailed account of the debate cf. Turner’s (1933).—In a note to Brentano’s investigations concerning space, time and continuum (Brentano 1976, p. 229, Fn. 131), Kastil ascribes the origin of the labels empiricist (“empiristisch”) and nativist to Helmholtz by referring to the 2nd edition of the latter’s Treatise on Physiological Optics; cf. von Helmholtz (1924, III, p. 10/608f). The English translation of this work is based on the posthumous three-volume third German edition from 1910. That edition, in turn, is a revised reprint of the first edition from 1867 supplemented by appendices by the editors Allvar Gullstrand, Johannes von Kries, and Willibald Nagel.
- 8.
More precisely, “matter” vanishes completely in Helmholtz’ conceptual framework since he views “our sensations as regards their qualities just as signs whose nature completely depends upon our organization”, von Helmholtz (1924, p. 586).
- 9.
Both Hering and Helmholtz dismiss the Kantian distinction between form and matter. But whereas Helmholtz eliminates the spatial features of intuitions and turns their qualitative features into formal ones (cf. Footnote 8 above), Hering—conversely—turns them into just another species of material components. Hence von Kries, in an appendix to Helmholtz’ Physiological Optics (von Helmholtz 1924, III, p. 641) is completely right when he declares it absurd “to identify the views that Helmholtz opposed, that is, so-called nativism, with the apriority of Kant.” Kant recognizes “a fundamental distinction between the spatial (and temporal) determinations of our sensations and their qualitative or intensive determinations;” ibid. p. 640. But Hering “wipes out this distinction and places the spatial determinations exactly on a par with others;” ibid. Helmholtz himself, however, conceives of Hering’s nativism as standing in the Kantian tradition; cf. von Helmholtz (1924, III, p. 36/613).
- 10.
Mach’s association between points of outer space and places of the brain reminds of the Gestalt psychologists’ principle of isomorphism according to which “[…] the organization of experience and the underlying physiological facts have the same structure,” (Köhler 1922, p. 301). For Mach, the structural similarity between physiological and geometrical space is restricted to their topology; cf. Mach (1976, p. 256/344) and he considers this insufficient for upholding (a bio-physiological version of) Kant’s thesis of the apriority of space. The thesis that only the topological properties of intuitive space are a priori is put forward by Carnap in his Ph.D. thesis; cf. Carnap (1922, pp. 62ff).
- 11.
Cf., e.g., Leibniz explanation: “Erit enim qui arbitretur corporis notionem priorem esse notione superfici et lineae, tanquam corporis terminorum, nec per se subsistentium, et has corporis sectione cognosci,” (Leibniz 1849, p. 199) (“The notion of a body will have to be considered to be prior to those of a surface and a line which as the boundaries of the body, not as something subsisting by itself, are recognized by a dissection of the body.”).
- 12.
Hjelmslev’s (1922, p. 4) position resembles that of Mach: “Die Erfahrungsgeometrie muß mit metrischen Grundlagen (Eigenschaften des festen Körpers) anfangen [Experience-based geometry has to begin with metric foundations (properties of the rigid body)].” However, in the passage just cited, Hjelmslev definitely suggests what will be called the “axiomatic thesis” below (p. 478).
- 13.
Mach (Mach 1976, p. 279f/370) ascribes this idea to Leibniz.
- 14.
Mach “glaubt […] bewiesen zu haben, daß jede geometrische Bestimmung auf eine Volumenmessung zurückgeht,” (Brentano 1988, p. 134).
- 15.
“Durch Ineinanderschiebung konnte die Gleichheit aber auch nicht konstatiert werden, da die Körper undurchdringlich sind,” (Brentano 1988, p. 134).—Here Brentano alludes to Euclid’s fourth axiom that “[t]hings which coincide with one another are equal to one another.” This axiom legitimates the method of superposition which is used twice by Euclid in order to show the congruence of two triangles by moving one of them in such a way upon the other that the two figures coincide.
- 16.
Hölder’s little book, containing his inaugural lecture held in 1899 at the University of Leipzig, is cited twice in ch. XXI of Knowledge and Error, cf. Mach (1976, p. 296, Fn. 4/360, Fn. 1, p. 298, Fn. 38/387, Fn. 1).
- 17.
“Man kann den Inhalt bei Körpern als einen Erfahrungsbegriff ansehen, der aus dem Gebrauch der Hohlmasse bei Flüssigkeiten abgezogen ist, und von dieser Seite kann man dann auch zum Inhalt ebener Flächen gelangen.” The importance of our experiences with liquids for the development of the notion of volume has also been emphasized by Lorenzen (1984, p. 21).
- 18.
There is a further problem involved here, namely: how to account for the identity of volume of non-congruent solids. The strategy applied for the corresponding problem of equal areas of two non-congruent polygons is, very roughly, to dissect non-congruent polygons of the same area into the same number of pairwise congruent triangles. It is known, however, that the three-dimensional analogue of this strategy does not work.
- 19.
This section has been omitted in John R. Aumann’s selective English translation of Hausdorff’s book from 1957.
- 20.
This echoes Mach’s explanation in The Analysis of Sensations (Mach 1959, p. 29/23): “Bodies do not produce sensations, but complexes of elements (complexes of sensations) make up bodies.”
- 21.
“[…] zweitens hat Mach […] auch gelehrt, daß jede Sensation auf innigste mit einer Raumempfindung verbunden sei.”
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Robering, K. (2019). Ernst Mach’s Geometry of Solids. In: Stadler, F. (eds) Ernst Mach – Life, Work, Influence. Vienna Circle Institute Yearbook, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-04378-0_34
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