Abstract
For the understanding of Riemann’s lecture and its importance, the comparison with the reasonings of the physiologist and physicist Hermann von Helmholtz is particularly important.
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Notes
- 1.
Hermann Helmholtz was born in 1821 as the son of a school teacher. For financial reasons, he initially had to work as a military surgeon, but had been able to study in Berlin with the leading anatomist and physiologist of his time, Johannes Müller (1801–1858). Having stepped forward with studies on the formation and propagation speed of nerve impulses and the paper “Über die Erhaltung der Kraft” (On the Conservation of Force) (i.e., energy conservation), in 1849, he became professor of physiology in Königsberg, then in Bonn and Heidelberg. Among his significant achievements in sensory physiology were measuring the velocity of propagation of electrical nerve stimulations and the development of the ophthalmoscope. His monographs Handbuch der Physiologischen Optik (Handbook of Physiological Optics), Leipzig, Leopold Voss, in three installments from 1856 to 1867, and Die Lehre von den Tonempfindungen als physiologische Grundlage der Musik, (The Theory of Sensations of Tone as a Physiological Basis of Music), Braunschweig, Fr. Vieweg. Sohn in 1863, laid the foundations of systematic sensory physiology. The physiological research of Helmholtz and his colleague and friend Emil du Bois-Reymond (1818–1896) (brother of the mathematician Paul du Bois-Reymond (1831–1887)), the founder of electrophysiology and successor of Müller in Berlin, led to the final overcoming of the vitalist ideas, which their teacher Müller had still vehemently defended. Helmholtz’ sensory-physiological investigations led him to an empiricist epistemology and on this basis to systematic considerations about the concept of space; these will be discussed in more detail in the text below. It is remarkable that the physiologist Helmholtz, who was only a mathematical autodidact, could penetrate so deeply into a basic question of mathematics, even if the details did not always withstand the professional criticism of the mathematician Sophus Lie (others, especially Felix Klein in his Vorlesungen,, Vol. 1, pp. 223–230, judged the contribution of Helmholtz significantly more generously than Lie, who could be unusually sharp also in disputations with other mathematicians who he took as his competitors, like Killing or Klein). Helmholtz, who in the course of his career turned more and more to issues of physics, had in fact earlier obtained an important and difficult mathematical result in hydrodynamics. He proved that vortices are conserved in a frictionless fluid. For that work, incidentally, Riemann’s theory of conformal mappings had been an important inspiration. His work, and that of his student Heinrich Hertz, contributed decisively to a general acceptance of the Faraday-Maxwell theory of electrodynamics. Helmholtz’ approach to derive the electrodynamic field equations from a principle of least action was an important precursor for development of the theory of relativity, even if Helmholtz’s own theoretical approach, although it led to the prediction of the existence of the electron, ultimately proved to be futile, because it was based on the existence of the ether. In 1871, Helmholtz became professor of physics in Berlin. He was ennobled in 1883 (and his family name was changed into von Helmholtz as part of this procedure). In 1888, he was appointed president of the newly founded Physikalisch-Technische Reichsanstalt (Physico-Technical State Institute), a pioneering large scale research institution both through its research agenda and its organizing principles. Helmholtz died in 1894. Helmholtz was the great universal scientist of the second half of the nineteenth century, and he also enjoyed the corresponding social recognition and prestige. His position in German science can perhaps be compared with that of Alexander von Humboldt in the first half of the nineteenth century. For his biography and scientific role and achievements, see Leo Koenigsberger, Hermann von Helmholtz, 3 vols., Braunschweig, Vieweg, 1902/3. A recent study is G. Schiemann, Wahrheitsgewissheitsverlust. Hermann von Helmholtz’ Mechanismus im Anbruch der Moderne. Eine Studie zum Übergang von klassischer zu moderner Naturphilosophie. Darmstadt, Wiss. Buchges., 1997. There exists an extensive literature on Helmholtz. I mention only the more recent work of Michel Meulders, Helmholtz. From Enlightenment to Neuroscience, MIT Press, 2010 (translated from the French and edited by L. Garey).
- 2.
For references see the bibliography at the end. In the sequel, I shall cite these references in abbreviated form as Axiome, Grundlagen, Geometrie and Wahrnehmung, the first and the last and also the commentaries by Hertz and Schlick with the page numbers of the edition of F. Bonk, the others from Wissenschaftlichen Abhandlungen, Vol. II.
- 3.
For example Grundlagen, p. 613, 615. This is corrected only in the supplement to this article. likewise Geometrie, pp. 637–639, where it is corrected in footnotes inserted in Wissenschaftlichen Abhandlungen.
- 4.
To what extent Helmholtz has misunderstood the Kantian notion of synthetic a priori judgment by not recognizing the difference between logical and descriptive necessity was indeed an essential aspect of the argumentation of the Kantians, but may be left open here. See also the remarks of Schlick, p. 49.
- 5.
“in welches jeder beliebige Inhalt der Erfahrung passen würde”, in Axiome, p. 16.
- 6.
Concerning this issue, modern mathematics has then gone even further in the direction taken by Helmholtz, insofar as also topological and not only metric properties of space may be contingent.
- 7.
Wahrnehmung, p. 159.
- 8.
In Axiome.
- 9.
Apparently Helmholtz was not aware that it had already been an essential postulate of Leibniz that every body needs to be thought of as movable in space without change of form, see pp. 161, 168 in Volume V of Leibnizens mathematische Schriften, ed. C. I. Gerhardt, Vols. III-VII, Halle a. d. S., 1855–1860. This is constitutive for Leibniz’s constructive approach in his geometry of position, s. Ernst Cassirer, Leibniz’ System in seinen wissenschaftlichen Grundlagen, Hamburg, Felix Meiner, 1998 (based on the edition of 1902). For Leibniz, however, what seemed clear to Helmholtz as an empirical fact, was still a mathematical and philosophical problem, s. V. De Risi, loc. cit. Leibniz carefully analyzed the difference between similarity and congruence of geometric figures. Without a direct comparison with respect to a common scale, one can determine only the similarity, i.e. the equality of the internal relations of two figures, but not their congruence, i.e. the absolute equality of their magnitudes. Leibniz does not argue with the mobility of the rigid scale, but with that of the figures to be examined, which of course also leads to the homogeneity of space. Kant is also familiar with these issues. One could now, casually speaking, think that the physicist walking around with a yardstick in the field simply ignores a pseudo-problem of the mathematician struggling with the penetration of Euclidean geometry or of the philosopher speculating in his study. However, the situation is not that simple. As will be explained in Section 5.4, Weyl later proposed to allow even a path-dependent gauge freedom in the measurement units, so that lengths can change when a body is transported in space. This idea was ultimately rejected by physicists, for example by reference to the absolute length scale of atomistics. But as explained in Section 5.4, this idea had become central for modern elementary particle physics in a somewhat different way.
- 10.
Helmholtz shows at this point a deep understanding of the geometric model of non-Euclidean space by Beltrami cited below (at that time and also by Helmholtz called pseudospherical geometry).
- 11.
“Die Sinnesempfindungen sind für unser Bewußtsein Zeichen, deren Bedeutung verstehen zu lernen unserem Verstande überlassen ist”, in Hermann von Helmholtz, Handbuch der Physiologischen Optik, Vol. III, Heidelberg, 1867; 3rd ed., Hamburg, Leipzig, Leopold Voss, 1910, p. 433 (emphasis in the original) or “Insofern die Qualität unserer Empfindung uns von der Eigentümlichkeit der äußeren Einwirkung, durch welche sie erregt ist, eine Nachricht gibt, kann sie als ein Zeichen derselben gelten, aber nicht als ein Abbild. … Ein Zeichen aber braucht gar keine Art der Ähnlichkeit mit dem zu haben, dessen Zeichen es ist. Die Beziehung zwischen beiden beschränkt sich darauf, daß das gleiche Objekt, unter gleichen Umständen zur Einwirkung kommend, das gleiche Zeichen hervorruft”, in Wahrnehmung, p. 153 (emphasis in the original).
- 12.
On the history of neuroscience, see Olaf Breidbach, Die Materialisierung des Ichs. Zur Geschichte der Hirnforschung im 19. und 20. Jahrhundert, Frankfurt/M., Suhrkamp, 1997. Here, we cannot discuss the development of sensory physiology before, by, and after Helmholtz, or the influence of Lotze’s theory of local signs (“Lokalzeichen”) or the dispute between empiricists like Helmholtz and nativists like Hering (for Helmholtz’ position, see e.g. Wahrnehmung, p. 163f.) and other such issues.
- 13.
Wahrnehmungen, p. 171f, p. 191.
- 14.
This idea was later elaborated and popularized by Edwin A. Abbott in his Flatland. A romance of many dimensions, Seeley & Co., 1884 (Reprint, with an introduction by A. Lightman, New York etc., Penguin, 1998) which he published under the pseudonym A. Square. Actually, before Helmholtz, this idea had already been mentioned by Gauss, see Sartorius von Waltershausen, Gau zum Gedächtnis, Leipzig, 1856, S. 81. But even before Gauss, the founder of psychophysics, Gustav Theodor Fechner (1801–1887), had proposed a similar idea, s. Rüdiger Thiele, Fechner und die Folgen auerhalb der Naturwissenschaften, in: Ulla Fix (Ed.), Interdisziplinäres Kolloquium zum 200. Geburtstag Gustav Theodor Fechners, Tübingen, Max Niemeyer Verlag, 2003, 67–111.
- 15.
Helmholtz also works out the monodromy principle that a body after a rotation by 360∘ again returns to its original position and shape. Lie then later criticized that this is not an independent axiom, as Helmholtz believed, but that it follows from the other axioms of Helmholtz.
- 16.
This again is part of a long line of discussion. That the curvature of space must be empirically measurable, was already known to Gauss. Whether the curvature of space actually vanishes on a cosmic scale, leads into the even today still ongoing discussion about the cosmological constant of Einstein, which recently has been revived. This came about because of some phenomena that cannot be explained by established cosmological physics. This leads to the search for so-called dark matter and dark energy.
- 17.
Concerning this issue, see the explanations of Schlick, p. 52.
- 18.
This is the philosophical direction of conventionalism (see below). Martin Carrier,Geometric facts and geometric theory: Helmholtz and 20th-century philosophy of physical geometry, in L. Krüger (ed.), Universalgenie Helmholtz. Rückblick nach 100 Jahren, Berlin, Akademie”=Verlag, 1994, 276–291, concludes that Helmholtz has thus stimulated several different directions of the philosophy of physical geometry because his views both can be interpreted in such a way that both the free mobility of rigid bodies is an empirical fact, and that it provides a useful convention, and finally, that it is the precondition of physical and geometrical measurements. A detailed description of the history of ideas of the arguments of conventionalism is found in Martin Carrier, Raum-Zeit, Berlin, de Gruyter, 2009.
- 19.
Many examples are presented and analyzed in Torretti, Philosophy of geometry, loc. cit.
- 20.
See Bertrand Russell, An essay on the foundations of geometry, Cambridge, Cambridge Univ. Press, 1897, reprinted New York, Dover, 1956, and the same, Sur les axiomes de la géometrie, Revue de Métaphysique et de Morale 7, 684–707, 1899, and the penetrating analysis of Torretti, loc. cit.
- 21.
Eugenio Beltrami, Saggio di Interpretazione della Geometria Non-euclidea, Giornale di Matematiche VI, 284–312, 1868.
- 22.
Eugenio Beltrami, Teoria fondamentale degli spazii di curvatura costante, Annali di Matematica pura ed applicata series II, Bd. II, 232–255, 1868.
- 23.
Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (Erlanger Programm), Erlangen, A. Düchert, 1872, reprinted Leipzig, Akad. Verlagsges., 1974, with additions published in Math. Annalen 43, 63–100, 1893, reprinted in K. Strubecker (ed.), Geometrie, Darmstadt, Wiss. Buchges., 1972, pp. 118–155; Klein, Über die sogenannte Nicht-Euklidische Geometrie, Mathematische Annalen 4, 573–625, 1871. For this articles and others by Klein, see also Felix Klein, Gesammelte mathematische Abhandlungen, 3 vols., Berlin, Springer, 1921–23, and the posthumously published monograph Felix Klein, Vorlesungen über nicht-euklidische Geometrie, Berlin, Springer, 1928. On the programs of Lie and Klein see also Thomas Hawkins, The Emergence of the Theory of Lie Groups. An Essay in the History of Mathematics 1869–1926. Berlin etc., Springer (here in particular Chap. 4) and Thomas Hawkins, The Erlanger Programm of Felix Klein: Reflections on its place in the history of mathematics. Historia Mathematica 11, 442–470, 1984. For the further development of Klein’s program e.g. R. Sharpe, Differential geometry. Cartan’s generalization of Klein’s Erlangen program, New York, Springer, 1997.
- 24.
Hawkins, The Erlanger Programm, however, comes to the conclusion that Klein’s manifest itself actually did not exert a significant influence, but that programmatically related ideas were developed more or less independently not only by Lie, but also by Eduard Study, Wilhelm Killing and Henri Poincaré. Anyway, like Klein’s, neither of these approaches granted a fundamental role to Riemannian metrics.
- 25.
This is due in particular to the geometer Elie Cartan, see below, p. 19.
- 26.
For a presentation of the current state, see for instance J. Jost, Riemannian Geometry and Geometric Analysis, Berlin etc., Springer, 6th ed., 2011.
- 27.
At this point, one needs to actually argue a little deeper. The issue is not only the principal indistinguishability and thus equivalence of different descriptions. Rather, the old Leibnizian idea surfaces again that the homogeneity of space is a formlessness that leads to the indifference of its parts or elements against each other. Thus, there is no rational justification for specific positionings in space (or in time). Without the assignment of physical attributes, spatial points cannot be rationally distinguished from each other. This was probably also what Riemann’s concept of a manifold was intended to express, a general term that admits different modes of determination. Any physical theory must actually be independent of the description of the underlying objects insofar as these descriptions record the same aspects and display them only in different coordinate systems. Central for a physical theory is to work out, however, through which physical properties these objects can be distinguished from each other at all. The manifold concept of Riemann incorporates both aspects, i.e. the same point in the manifold can be described and represented in different coordinates, and in a manifold, unless an additional structure enters, all points are similar and can be converted into each other by transformations of the manifold into itself (homeomorphisms in mathematical terminology). The manifold concept thus captures the variety of points, but provides no criterion for their identification or differentiation. A metric then yields distinctive relations between points, and curvature quantities can assign specific features to individual points. As Riemann has seen, this is exactly why this geometry cannot be recovered from the manifold concept alone, but requires a physical determination. This is exactly what Einstein’s theory achieves in a systematic and principled manner. In quantum theory, however, this aspect is being turned around by Heisenberg. Here the same object shows itself in different modes of appearance. Physically accessible are only these phenomena, but not the object itself.
- 28.
Hermann Minkowski, Raum und Zeit, Phys. Zeitschr. 10, 104–111, 1909, and Jahresber. Deutsche Mathematiker-Vereinigung 18, 75–88, 1909; reprinted e.g. in C. F. Gau/B. Riemann/H. Minkowski, Gausche Flächentheorie, Riemannsche Räume und Minkowskiwelt. Edited and with an appendix by J. Bohm and H. Reichardt, Leipzig, Teubner-Verlag, 1984, 100–113.
- 29.
The extensive investigations of Pierre Duhem,Le système du Monde. Histoire des doctrines cosmologiques de Platon à Copernic, 5 vols., Paris, 1914–17, have been corrected in several essential aspects by Anneliese Maier, Das Problem der intensiven Gröe in der Scholastik, Leipzig, 1939;Die Impetustheorie der Scholastik, Wien, 1940 (an extended new edition of these two works appears in: Zwei Grundprobleme der scholastischen Naturphilosophie, Roma,31968); An der Grenze von Scholastik und Naturwissenschaft, Essen, 1943, Roma,21952;Die Vorläufer Galileis im 14. Jahrhundert. Studien zur Naturphilosophie der Spätscholastik, Rom, 1949; Metaphysische Hintergründe der spätscholastischen Naturphilosophie, Roma, 1955, Zwischen Philosophie und Mechanik. Studien zur Naturphilosophie der Spätscholastik, Roma, 1958. Building upon this, see also E. J. Dijksterhuis, Die Mechanisierung des Weltbildes, Berlin etc., Springer, 1956, reprint 1983.
- 30.
In particular, Alexandre Koyré, Etudes galiléennes, Paris, Hermann, 1966, particularly p. 102. refuted Duhem’s, loc cit, claim of the continuity of the development of the medieval impetus to the Galilean momentum.
- 31.
From the extensive literature, we only mention the document collection of Alexandre Koyré, A documentary history of the problem of fall from Kepler to Newton, Philadelphia, 1955.
- 32.
The German idealist philosopher Georg Wilhelm Friedrich Hegel (1770–1831) in his Enzyklopädie der philosophischen Wissenschaften (cf. the edition byb F. Nicolin und O. Pöggeler on the basis of the version 1830, Hamburg, Felix Meiner,81991, or that of E. Moldenhauer und K. M. Michel of the second part, that is, the natural philosophy, with the oral additions from the lectures of Hegel, Frankfurt a. M., Suhrkamp, 1978; for our present purposes, §§ 262–271 are elevant) rejected the idea of a force-free body, moving without influence from other bodies, as non sensical, because in the absence of other bodies, we can neither sensibly ascribe a motion to a body nor even reasonably an existence. Between inertia as internal characterization of a body as passive and its susceptibility to external gravitational influences of other bodies, which are thereby conceived as active, he sees a contradiction, and this leads him to strong polemics against Newton while praising Kepler instead. Hegel sees this contradiction resolved in that the basic motion of a body is not the linear inertial one, rejected by him as absurd, but Kepler’s elliptical motion around a center of gravity, ultimately, the center of gravity of all masses of the universe. In the Hegelian dialectic, matter, as a principle of isolated externality and therefore not yet determinate by itself, requires other matter for its constitution and therefore reciprocally gains its inner principle through gravity. That is, it can ultimately determine itself via the detour through other matter. This might be an attractive idea, but it raises the question of its value for physics. Thus, the reflections of Hegel on inertia and gravitation have been judged very differently, in particular in retrospect after the theory of relativity. We quote here only the benevolent or positive evaluations from D. Wandschneider, Raum, Zeit, Relativität, Frankfurt, Klostermann, 1982, and depending on those, the ones of V. Hösle in Hegels System, single volume edition, Hamburg, Felix Meiner, 1988, and E. Halper, Hegel’s criticism of Newton, in: The Cambridge Companion to Hegel and nineteenth-century philosophy (ed. F. Beiser), Cambridge etc., Cambridge Univ. Press, 2008, pp. 311–343.
- 33.
See e.g. J. Jost, Geometry and Physics, Berlin etc., Springer, 2009.
- 34.
Here, I sketch the considerations of Weyl and Cartan from the historical perspective. The systematic aspect will be taken up in Section 5.4.
- 35.
See S. Helgason, Differential geometry, Lie groups, and symmetric spaces, New York etc., Academic Press, 1978.
- 36.
For details, refer to J. Jost, Riemannian Geometry and Geometric Analysis.
- 37.
We refer to the literature cited in Footnote 14 on p. 63; also Erhard Scholz, Weyl and the theory of connections, in: Jeremy Gray (ed.), The symbolic universe. Geometry and Physics 1890–1930, Oxford etc., Oxford Univ. Press, 1999, pp. 260–284.
- 38.
“we must seek the ground of its metric relations outside it, in binding forces which act upon it”, Riemann, Hypotheses, Chap. 3, p. 69.
- 39.
W. K. Clifford, On the space-theory of matter (abstract), Cambridge Philos. Soc., Proc., II, 1876, p. 157f, also in his Mathematical Papers, ed. R. Tucker, London, 1882, p. 21f.
- 40.
See, however, Schiemann, Wahrheitsgewissheitsverlust, for an analysis of the transition from an ontological to a phenomenological conception of physics also in the views of Helmholtz.
- 41.
See also above, on p. 19, the analysis of the Kantian argument of the relationship between handedness and spatial structure.
- 42.
The spiritualist medium was Henry Slade (1840–1904). Among the scientists who were taken in by him, was, for instance, Karl Friedrich Zöllner (1834–1882), the founder of astrophysics, who thereby ruined his scientific reputation. For details, we refer to Rüdiger Thiele, Fechner und die Folgen auerhalb der Naturwissenschaften, in: Ulla Flix (Ed.), Interdisziplinäres Kolloquium zum 200. Geburtstag Gustav Theodor Fechners, Max Niemeyer Verlag, Tübingen, 2003, 67–111 or Klaus Volkert, http://www.msh-lorraine.fr/fileadmin/images/preprint/ppmshl2-2012-09-axe6-volkert.pdf. Helmholtz, however, remained skeptical. A contemporary presentation can be found in F. Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, and for example a fairly free story by the theoretical physicist Michio Kaku Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension, Oxford, Oxford Univ. Press, 1994, who presents the possibility of higher space dimensions as the essential and at the time sensational discovery of Riemann. A systematic mathematical analysis of spaces of arbitrary dimension had already been conducted before Riemann in a different context by H. Grassmann, Die lineale Ausdehnungslehre, Leipzig, 1844, a work which founded linear algebra.
- 43.
L. Nelson, Bemerkungen über die Nicht-Euklidische Geometrie und den Ursprung der mathematischen Gewiheit, Abh. Friessche Schule, Neue Folge, Vol. I, 1906, 373–430; W. Meinecke, Die Bedeutung der Nicht-Euklidischen Geometrie in ihrem Verhältnis zu Kants Theorie der mathematischen Erkenntnis, Kantstudien 11, 1906, 209–232; P. Natorp, Die logischen Grundlagen der exakten Wissenschaften, Leipzig,21921, 309f.; G. Martin, Arithmetik und Kombinatorik bei Kant, Itzehoe, 1938; the same, Immanuel Kant, Berlin, 4th ed., 1969.
- 44.
S. Lie, Über die Grundlagen der Geometrie, Ber. Verh. kgl.”=sächs. Ges. Wiss. Lpz., Math.-Phys. Classe, 42. Band, Leipzig, 1890, 284–321, and S. Lie, Theorie der Transformationsgruppen, Dritter und Letzter Abschnitt, unter Mitwirkung von F. Engel, Leipzig, Teubner, 1888–1893, New York, Chelsea,21970, Abtheilung V. Lie stated himself that he had been made aware of the work of Riemann and Helmholtz already in 1869 by Klein, pointing out that in these studies the concept of a continuous group was implicitly contained, but he himself did not turn to the considerations of Riemann and Helmholtz until 1884, when he had already worked out systematically his own theory of continuous groups (S. Lie, Transformationsgruppen, p. 397). Somewhat strangely, in Hawkins, Lie groups, Helmholtz does not appear in the presentation of the mathematical development of Lie, but only in that of Killing.
- 45.
Lie, Transformationsgruppen, pp. 498–523.
- 46.
“Das Riemann-Helmholtzsche Problem … verlangt die Angabe solcher Eigenschaften, die der Schaar der Euklidischen und den beiden Schaaren von Nichteuklidischen Bewegungen gemeinsam sind und durch die sich diese drei Schaaren vor allen anderen möglichen Schaaren von Bewegungen auszeichnen.” (My translation) Lie, Transformationsgruppen, p. 471 (emphasis in the original), and a similar formulation p. 397 ibid.
- 47.
David Hilbert, Grundlagen der Geometrie, Leipzig, Teubner, 1899; 13th ed., Stuttgart, Teubner, 1987 (with 5 supplements, in which several articles of Hilbert are reprinted, as well as supplements by Paul Bernays) and 14th ed., Leipzig, Teubner, 1999, with the essay Michael Toepell, Zur Entstehung und Weiterentwicklung von David Hilberts Grundlagen der Geometrie, that treats the developments prior to and after Hilbert’s axiomatic approach to geometry; concerning the 7th ed., see also Arnold Schmidt, Zu Hilberts Grundlegung der Geometrie, in: David Hilbert, Gesammelte Abhandlungen. Vol. 2, Berlin etc., Springer,21970, pp. 404–414. Furthermore Michael Hallett, Ulrich Majer (Eds.): David Hilbert’s Lectures on the Foundations of Geometry, 1891–1902. Berlin etc., Springer, 2004, which not only reprints the original 1899 version, but also the other publications of Hilbert on the foundations of geometry. Hilbert’s original text was edited for the present series with an extensive historical commentary by Klaus Volkert, Berlin, Heidelberg, Springer Spektrum, 2015.
- 48.
Pirmin Stekeler-Weithofer, Formen der Anschauung, Berlin, de Gruyter, 2008, in contrast, analyzes the relationship between the formal logical validity and the truth of geometrical statements based on real constructibility propositions with recourse to Kant’s concept of a synthetic a priori validity. This quote must suffice here as a new example for a very extensive and controversial discussion.
- 49.
John von Neumann, Mathematische Grundlagen der Quantenmechanik, Berlin, Springer, 1932; English translation Mathematical foundations of quantum mechanics, Princeton, Princeton Univ. Press, 1955.
- 50.
See Arthur Wightman, Hilbert’s sixth problem: Mathematical treatment of the axioms of physics, Proc. Symp. Pure Math. 28, 147–240, 1976.
- 51.
Henri Poincaré, La science et l’hypothèse, Paris, Flammarion, 1902; Reprint Paris, Flammarion, 1968; English translation Science and hypothesis, Walter Scott Publ. Comp. Ltd, 1905, reprinted by Dover, 1952. See also the detailed analysis of Torretti, Philosophy of Geometry from Riemann to Poincaré.
- 52.
See the extensive discussion in Martin Carrier, Raum-Zeit. Berlin, de Gruyter, 2009.
- 53.
Hermann Weyl, Raum, Zeit, Materie, Berlin, Julius Springer, 1918; 7th ed. (ed. Jürgen Ehlers), Berlin, Springer, 1988.
- 54.
A good example can be found in Carrier, cited above. The Hollow Earth theory says that the Earth is a hollow sphere, enclosing the heavens. Geometrically, one can simply pass from the usual Euclidean geometry to such a hollow geometry by an inversion at the surface of the globe. This inversion maps the point at infinity of Euclidean space into the center of the sphere. If the laws of motion of Newtonian mechanics are transformed as well according to the rules of coordinate transformations (tensor calculus), then all the physical laws of mechanics hold as before, and no empirical difference can be found. Thus, the same physical facts have been represented in different coordinates. As we have applied a nonlinear coordinate transformation, however, in these new coordinates the laws of motion become complicated, and the Euclidean coordinates are therefore preferable. That’s all. The question of whether the hollow geometry is the real geometry, is in this context pointless, because it confuses reality with its description.
- 55.
Hans Reichenbach, Philosophie der Raum-Zeit-Lehre, Berlin and Leipzig, de Gruyter, 1928; reprinted as Vol. 2 of his Gesammelte Werke, Braunschweig, Vieweg, 1977; English translation The Philosophy of space and time, Dover, 1957.
- 56.
A reference for this section is Jürgen Jost, Mathematical concepts, Berlin etc., Springer, 2015.
- 57.
On the history of the set concept, see for example José Ferreiros, Labyrinth of Thought. A History of Set Theory and its Role in Modern Mathematics. Basel, Birkhäuser, 1999.
- 58.
The foundational issues connected with the set concept are not relevant for our purposes.
- 59.
This includes and generalizes the well-known Weierstrass \(\varepsilon -\delta\)-criterion of analysis, see below.
- 60.
Extensive material on these notions and their history can be found in the new edition of Felix Hausdorff, Grundzüge der Mengenlehre (1914) at http://www.hausdorffedition.de with detailed commentaries on the background in Walter Purkert, Historische Einführung, and a description of the evolution of the neighborhood axioms in Frank Herrlich e. a. Zum Begriff des topologischen Raumes. Section 3.2, Fundamentaleigenschaften von Umgebungssystemen, treats the relationship discussed in the text with the neighborhood axioms in \(\mathbb{R}^{n}\) historically, on the basis of Hausdorff’s own presentation in his course of the summer term 1912.
- 61.
For a detailed historical analysis see Erhard Scholz, The concept of manifold, 1850–1950. In: I. James (Hrsg.), History of Topology, Amsterdam etc., Elsevier 1999, pp. 25–64.
- 62.
For details, we refer to Scholz, Manifold.
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Jost, J. (2016). Reception and Influence of Riemann’s Text. In: Jost, J. (eds) On the Hypotheses Which Lie at the Bases of Geometry. Classic Texts in the Sciences. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-26042-6_5
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