Abstract
In the context of 19th-century physics, geometry was quite naturally interpreted as the science of space, space itself being conceived as a self-subsisting entity, no less real than the spatial things moving across it. Paradoxically, however, the propositions of this science did not seem to be liable to empirical corroboration or refutation. Since the times of the Greeks, no geometer had ever thought of subjecting his conclusions to the verdict of experiment. And philosophers, from Plato to Kant, viewed geometry as the one unquestionable instance of non-trivial a priori knowledge, i.e. knowledge relevant to things that exist, yet not dependent on our experience of them. Even such an extreme empiricist as Hume regarded geometry as a non-empirical science, concerned not with matters of fact, but with relations of ideas. The discovery of non-Euclidean geometries shattered the unanimity of philosophers on this point. The existence of a variety of equally consistent systems of geometry was immediately thought to lend support to a different view of this science. The established Euclidean system could now be regarded as a physical theory, highly corroborated by experience, but liable to be eventually proved inexact. We have seen that Gauss and Lobachevsky, Riemann and Helmholtz took this empiricist view of geometry.
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Notes
4.1 Empiricism in Geometry
Gauss, WW, Vol.8, p. 177. Quoted above, p.55.
J.S. Mill, SL, pp.209–261, 607–621; cf. Early Draft of the Logic, ibid, pp.1083–1097. That the first principles of arithmetic are generalizations from experience is stated on p.257. Arithmetic, however, differs from geometry in so far as the former “is deduced wholly from propositions exactly true”, while the latter partly depends, as we shall see, on “hypotheses or assumptions which are only approximations to the truth”. (J.S. Mill, SL, p. 1092; cf. pp.258f.).
J.S. Mill, SL, p.218.
J.S. Mill, SL, p. 1088.
J.S. Mill, SL, p.618n. I imagine that he expects them to be coplanar as well.
J.S. Mill, SL, p.225.
J.S. Mill, SL, p.225. “Inconceivable” for Mill is apparently everything that we would repute unimaginable.
J.S. Mill, SL, p.225.
J.S. Mill, SL, p.227n. (The note from which our quotation is taken was added by Mill in 1872.)
J.S. Mill, SL, pp.228f. In the first three editions the text enclosed by the asterisks read as follows: “suppressing some of those which it has”.
J.S. Mill, SL, p.230.
J.S. Mill, SL, pp.229f. The reader will not fail to observe that the last of the two axioms is incompatible with BL geometry while the first excludes spherical geometry. That Mill, who apparently knew nothing of the new developments in mathematics, should have chosen precisely such examples shows to what extent the subject was in the air at that time.
J.S. Mill, SL, p. 1089.
J.S. Mill, SL, p.232n. (note added in 1865; my italics).
J.S. Mill, SL, p.233.
J.S. Mill, SL, p.234.
J.S. Mill, SL, p.235.
J.S. Mill, SL, pp.115f.
J.S. Mill, SL, p. 1089.
J.S. Mill, SL, p.259.
J.S. Mill, SL, p.616 (text of the first four editions).
J.S. Mill, SL, pp.616f.
Ueberweg, PG, p.314.
Ueberweg, PG, pp.312f.
Ueberweg, PG, p.268.
Ueberweg, PG, p.315.
Ueberweg, PG, p.269.
Ueberweg, PG, pp.269f.
Ueberweg, PG, p.313.
Theorem 25 (Ueberweg, PG, p.287). The ‘proof’ of the theorem involves many tacit assumptions, whereby, among other things, elliptic and spherical geometry are implicitly excluded.
Ueberweg, PG, p.291.
Theorem 47 (Ueberweg, PG, p.304).
Theorem 49b (Ibid., p.305).
Erdmann, AG, p.38. In the same year, 1877, in which Erdmann’s book was published, Georg Cantor discovered that the above expression of dimension number is inadequate, because Rn can be mapped bijectively onto R. See Cantor, GA, pp.119–133. Cantor’s proof is sketched in Kline, MT, pp.997f.
“Die Teilbarkeit ins unendliche, die jeder Raumanschauung anhaftet” (Erdmann, AG, p.38). This is, of course, highly questionable as a description of spatial perception. We must assume that Erdmann, like most empiricist philosophers, allows the bounty of imagination to make up for the stinginess of the senses.
Erdmann, AG, p.39. The alleged intuitive fact of infinite divisibility does not warrant the empjpyment of such strong analytical means for expressing it. It would suffice to let the coordinates take all rational values between their value at A and their value at B, or merely all values expressible by means of a terminating decimal fraction.
Erdmann, AG, p.40.
Erdmann, AG, pp.58f.
Erdmann, AG, pp.93f.
Erdmann, AG, p. 116.
Erdmann, AG, p. 146.
Cf. Erdmann, AG, pp.59ff., 91, 141ff.
Erdmann, AG, p.91; cf. pp.115, 135, 145, 152f. On Heimholtz, see Section 3.1.3.
Erdmann, AG, pp.144f. Erdmann refers to the dimension number of space on pp.38, 49, 83, 95, 143; nowhere do I find an argument for the above quoted statement.
Erdmann, AG, pp.H8f.
Erdmann, AG, pp.94f.; cf. p. 128.
Erdmann, AG, pp.157.; cf. p.44n.2, p.50.
Erdmann, AG, p.158; (my italics). Cf. Heimholtz (1866), p.197; quoted above, p.167, Section 3.1.3.
Erdmann, AG, p. 159. I take it that the geometric concepts of construction (die geometrische Constructionsbegriffe) are concepts such as straight line and circle, which enter into the description of geometrical constructions.
Erdmann, AG, p. 159.
Erdmann, AG, p. 160.
Erdmann, AG, p. 161.
Erdmann, AG, p. 169.
Erdmann, AG, p. 170.
Erdmann, AG, p. 170. Erdmann admits here that experimental results may show that space curvature is not constant. According to the Helmholtzian stance taken by Erdmann in the rest of the book, such experimental discovery would imply that the ideal of a perfectly rigid motion cannot be indefinitely approached, and that, as a consequence of this, there is an upper bound to the accuracy of actual geometrical measurements.
Calinon (1891), p.375.
Calinon (1891), p.375.
Calinon (1889), p.589. Cf. Calinon (1891), p.368; “La Géométrie générale est l’étude de tous les groupes de formes dont les définitions premières sont astreintes à une condition unique, qui est de ne donner lieu à aucune contradiction, lorsqu’on les soumet au raisonnement géométrique indéfiniment prolongé”.
Calinon (1891), p.375. Cf. Klein (1890), p.571, quoted above, p.149, Section 2.3.10.
Calinon (1891), p.375.
Calinon (1889), p.590. Calinon’s “general geometry” embraces Euclidean geometry, BL geometry and the so-called geometry of Riemann, that is, spherical or elliptic geometry. It is, in other words, a theory of maximally symmetric spaces (p. 184). The name “general geometry” indicates that Calinon was less open-minded than he thought.
Calinon (1891), pp.368f. Identical spaces, in this sense, were called isogenous (isogènes) by Delboeuf, a term often used in the French literature of this period. See pp.206f.
Calinon (1889), p.594.
Calinon (1889), p.595.
Calinon (1891), p.374 (my italics).
Calinon (1893), p.605. The paper of 1893 is written in what the author calls an “idealistic” language; that is, space is identified there with our perceptually based representation of space. Calinon says that his views on the matters discussed by him can also be set forth in a realistic language, a task which he proposes as an exercise to the reader. In a realistic version of Calinon’s paper we ought perhaps to ascribe a definite though unknown geometric structure to physical space. But it seems clear that in Calinon’s opinion it makes no scientific sense to state hypotheses about such a structure. The choice of a geometry in physics must be made in the light of the requirements of the problem at hand, not with a view to determining the essence of things in themselves.
Published in the Revue générale des sciences on December 15, 1891. See Part 4.4.
Calinon (1893), p.607. Such are also the criteria usually applied when choosing a coordinate system. It is amazing that the great mathematician Poincaré should have overlooked this side of the analogy drawn by him.
Calinon (1893), p.607.
Mach, EI, pp.337–448. The two longest chapters (pp.353–422) had previously been published in English in The Monist. See References.
H. Poincaré, “Le continu mathématique” (1893); “L’espace et la géométrie” (1895); “On the foundations of geometry” (1898). See Part 4.4. Federico Enriques published, shortly before Mach, a study explaining the psychological development of geometry from an empiricist point of view. (Enriques, “Sulla spiegazione psicologica dei postulati della geometria” (1901).) I do not think it would be rewarding to discuss Enriques’ views here. They are readily accessible in English in Enriques, PS, pp. 199–231.
Mach, EI, p.389; cf. pp.371, 381.
Mach, EI, p.337. Compare Mach’s illuminating description of time-sensations in EI, p.423.
Mach, EI, p.345.
Mach distinguished the geometrical and the physiological properties of a figure in space in his Beiträge zur Analyse der Empfindungen (1886), pp.44, 54, long before Poincaré introduced the concepts of espace géométrique and espace représentatif in “L’espace et la géométrie” (1895).
Mach, El, p.337.
Mach, El, p.339.
Mach, EI. p.340.
Mach, EI, p.347. I fail to see how the experiences described by Mach might justify the conclusion that space is infinite, and not just unlimited. This is one of those wild speculative jumps that empiricists must make every now and then to reconcile their philosophy with experience.
Mach, EI, p.355. The reader will probably notice that in order to transport a body from a neighbourhood FGH to a neighbourhood MNO, as required by the experiment described above, we must regard it as located in the unitary objective physical space whose very idea supposedly arises out of such experiments.
Mach, EI, p.434.
Mach, EI, p.367.
Mach, EI, p.380.
Mach, EI, p.367.
Mach, EI, p.368.
Mach, EI, pp.368f.
Mach, EI, p.380. The last remark, in fact, agrees with the opinion of authors such as Klein and Enriques, who emphasized the optical origin of our idea of straightness, for they saw in visual intuition the root of projective, not of metric geometry (see above, Section 2.3.10, p. 147; Enriques, PS, pp.205ff.).
Mach, EI, p.370.
Mach, EI, p.371.
Mach, EI, p.385.
Mach, EI, p.409.
Mach, EI, p.414.
Mach, EI, p.418.
Mach, EI, p.418.
4.2 The Uproar of Boeotians
Krause, loc. cit., p.84; quoted by Schlick in Helmholtz, SE, p.29, n.25.
“Nur ein einziger grosser und zusammenhängender Irrtum”. Lotze, M, p.234.
Lotze rejects Kant’s arguments for the “transcendental ideality” of space, but supports it with another argument of his own: space as an aggregate of equal points can only be held together by consciousness. (Lotze, M, pp.211f.)
Lotze, M, p.223.
“Eines Ordnungssystems leerer Plätze”. (Lotze, M, p.235).
Lotze, M, pp.241f.
Lotze, M, p.247. I have changed Lotze’s lettering to make the meaning clearer.
Wundt, L, Vol.I, pp.478–506; Vol.II, pp. 176–208. I quote Vol.1 after the fifth (posthumous) edition of 1924.
Wundt, L, Vol.I, p.489: “Die unmittelbare gegebene Ordnung unserer Wahrnehmungsinhalte.” Ibid., p.504: “Eine Ordnung von Empfindungen”.
Wundt, L, Vol.I, p.497.
Wundt, L, Vol.I, p.501.
Wundt, L, Vol.I, p.503.
Wundt, L, Vol.I, p.504.
Wundt, L, Vol.I, p.494.
Wundt, L, Vol.I, p.483.
Wundt, L, Vol.I, p.485.
Wundt, L, Vol.I, p.482.
Wundt, L, Vol.I, p.492. “Self-congruence” means that a figure congruent with a given figure can be constructed anywhere in space. I have quoted Wundt’s shortened definition in the second paragraph of p.492. In the first paragraph we find a more explicit definition. In it, instead of unendlich (infinite), Wundt writes unbegrenzt (unlimited). This longer definition is therefore compatible with spherical and elliptic geometry; it also covers all the hypersurfaces of elliptic and hyperbolic 4-space which are isometric to Euclidean 3-space, etc.
Kant, Ak., II, pp.404f. (quoted on p.31).
Wundt, L, I, p.488.
Wundt, L, I, p.488.
Wundt, L, I, p.484.
First published in Renouvier’s Critique philosophique in November 1889, as a reply to Léchalas’ paper on general geometry, published in the same journal in September 1889 (see ibid., p.337n.). I quote from Renouvier (1891), a considerably revised and enlarged version published in Pillon’s Année philosophique.
Renouvier (1891), p.3.
Renouvier (1891), p.64.
Renouvier (1891), p.43.
Renouvier (1891), p.4.
Renouvier (1891), p.4.
Renouvier (1891), p.20. (My paraphrase.)
“La ruine entière de l’idéation géométrique”. (Renouvier (1891), p.45.)
Renouvier (1891), p.45.
Renouvier (1891), p.66.
See pp. 153 and 206.
Delboeuf, PPG, p.67.
Delboeuf, PPG, p. 129.
Delboeuf, PPG, p.50.
Delboeuf, ANG, I, p.456. Delboeuf concludes: “L’espace géométrique ou euclidien est donc un espace imaginaire, un espace hypothétique, n’ayant de commun avec la réalité que ceci qu’elle en a fourni l’idée, parce que les solides ont l’air de s’y transporter sans altération sensible.” (Ibid., pp.464f.) As we know, this empirical fact which according to Delboeuf “provides the idea of Euclidean space” fits equally well with every space of constant curvature.
Delboeuf, ANG, II, p.372.
4.3 Russell’s Apriorism of 1897
Russell, FG, New York: Dover, 1956.
Russell, FG, p.136.
Kant, KrV, B 40.
Kant, KrV, B 246. This is one of the many passages where Kant might seem to favour a psychological conception of the a priori.
Russell, FG, p.28.
Russell, FG, p. 118.
Russell, FG, p. 132.
The reader is advised to read the argument in Russell, FG, pp. 136f.
Russell, FG, pp.139f. Russell’s rejection of infinite-dimensional spaces is not an instance of mathematical ignorance — such spaces had not yet been discovered — but of a typically unmathematical narrow-mindedness which often crops up in philosophical literature.
Russell, FG, p. 149.
Russell, FG, p.46.
Russell, FG, p. 118.
Russell, FG, p. 147.
Russell, FG, p. 119; cf. p.30.
Russell, FG, p.35.
Russell, FG, p. 164.
See above, p. 124, Section 2.3.5.
Russell, FG, p. 161.
Russell, FG, p. 163. Riemann had made the same remark; see above, Section 2.2.9, p. 104.
Russell, FG, p. 164.
Russell, FG, p. 164.
Russell, FG, p. 150.
Russell, FG, pp.78, 79.
Russell, FG, p.79.
Russell, FG, p.81.
Russell, FG, p. 152.
Erdmann, AG, p.60.
Russell, FG, p. 155.
Russell, FG, pp.153f. The deduction of the axiom of free mobility from the relativity of position in Russell, FG, p. 160, begs the question. Relativity of position implies that the form of externality is homogeneous, says Russell, and free mobility follows from homogeneity “for our form would not be homogeneous unless it allowed, in every part, shapes or systems of relations, which it allowed in any other part” (Ibid.). This argument makes sense only if metrically determined shapes have been defined in our form of externality.
Couturat (1898), p.373f.
Russell (1898), p.763.
Russell, FG, pp.158f.
Russell (1898), p.760. Russell says that the experiment was suggested to him by A.N. Whitehead.
Russell, FG, p.74.
Russell (1899), pp.703–707.
Russell, “L’idée d’ordre et la position absolue dans l’espace et le temps” (1900). Russell (1901) is the corrected and slightly expanded English version of this paper.
Russell, Principles, p.372.
4.4 Henri Poincaré
Poincaré (1898b), now in Poincaré, VS, Chapter II, especially pp.47–54.
Neumann, PGNT, p. 18.
Lange (1885), p.338 (my paraphrase). These so-called theorems are of course postulates. Lange’s “proof” of the first is really a consistency proof: given an inertial system relatively to which a particle moves in a straight line, one cannot construct a second inertial system relatively to which that same particle does not move in a straight line.
The most complete study of Poincaré’s philosophy of geometry is still Rougier, La philosophie géométrique de Henri Poincaré (1920).
The book by Mooij, La philosophie des mathématiques de Henri Poincaré (1966) devotes to it only a short rather unilluminating chapter (pp.3–29). A penetrating analysis of its foundations is given in Vuillemin (1972).
“There can be little doubt that any deductive theory is capable of translation into a ‘contrary’ deductive theory.” (Black (1942), p.345). Black does not say “into any contrary theory”, but, as we shall see, his thesis on the translatability of axiomatic theories does not lend support to geometrical conventionalism unless we understand it in this strong sense.
Black (1942), pp.341f.
See Tarski, LSM, p.306; also pp.243, 248, Section 3.2.9.
Poincaré (1891), p.774; reprinted in SH, p.77.
Kant, KrV, B146, A557/B585, A613f./B641f.
Poincaré, SH, Chapter V, “L’expérience et le géométrie”, a chapter composed of passages pieced together from Poincaré (1891), (1899), (1900), plus a short summary of some of the ideas presented in Poincaré (1898).
See Section 4.4.5.
Poincaré, SH, p.76; Poincaré (1898), pp.42f. See below, pp.351f., Section 4.4.5.
Poincaré, SH, p. 101; cf. pp.75, 95.
Poincaré, SH, p.99. Compare Einstein’s formulation of the general principle of relativity: “Natural laws are only statements about spatio-temporal coincidences; therefore their natural expression must consist in generally covariant equations”. (Einstein (1918), p.241.)
Minkowski (1908). In fact, Minkowski’s contribution concerned only the underlying manifold of special relativity, which he showed to be a semi-Riemannian 4-manifold (see Appendix, p.372). But he paved the way for the four-dimensional formulation of Newtonian mechanics by Cartan (1923, 1924).
Poincaré (1891), p.774; SH, pp.95f. (my italics). On “Riemann’s geometry” see Part 2.2, Note 52.
Poincaré foresaw the rise of “an entirely new mechanics chiefly characterized by the fact that no velocity can surpass that of light”. (Poincaré, VS, pp.138f.; the passage belongs to a lecture given in St. Louis in 1904.)
Poincaré, SH, p. 103.
Poincaré, Oeuvres, Vol.XI, p.90.
Poincaré, Oeuvres, Vol.XI, p.91.
Poincaré, Oeuvres, Vol.XI, p.91. The “operations” of the group are not what we would call thus (namely, the group product and the mapping which assigns to every element its own inverse), but simply the elements of the group; the group is viewed as “operating” through them on an underlying space.
See however Glymour (1972).
Poincaré (1898), (1903). See below, Section 4.4.5.
See pp.172f., where this is explained in connection with Lie groups.
Poincaré (1898), p.40. Lie’s concept of a Zahlenmannigfaltigkeit is discussed in Part 3.1, Note 42.
Poincaré (1900), p.81.
Poincaré (1900), p.85; SH, p.104.
A valuable recent exception is Vuillemin (1972).
Poincaré, SH, p.51.
Poincaré (1898), p. 1.
Poincaré, VS, pp.73f.; (1898), p.2.
Poincaré, (1898), p.3.
Instead of cancels, Poincaré says corrects. Instead of locomotion he says displacement. See Note 55.
Poincaré, SH, p.51.
Poincaré (1898), pp.14f. Poincaré’s argument implies that every displacement is joined to 0 by a simple physical continuum; it follows easily that any two displacements are also joined thus.
Poincaré (1898), p.23. On the form and material of a group, see above, pp.336f.
They will be found in Poincaré (1898), pp.23–32 and in a much improved version in Poincaré, VS, pp.80–93.
Poincaré (1898), p.5; cf. Poincaré (1895), now in SH, p.82. By drawing our attention to the fact that Euclidean space is not a whit more imaginable than the non-Euclidean spaces Poincaré has finally disposed of a staple Boeotian argument.
Poincaré (1898), p.42.
Poincaré, SH, p.76.
Poincaré (1898), p.42.
His main contribution is contained in his long paper “Analysis situs” (1895) and its five supplements, published between 1899 and 1904. But the “qualitative theory” of differential equations developed in his four papers “Sur les courbes définies par une équation différentielle” (1881–1886) also follows an essentially topological approach. See Kline, MT, pp.732ff., 1170ff.
Poincaré, VS, pp.59f.; DP, pp.132ff.
Poincaré, DP, p. 134. We say, today, that two figures F, G (belonging to the same or to different topological spaces) are topologically equivalent if there is a continuous bijection f: F → G whose inverse f -1 is also continuous. This concept is wider than the one proposed by Poincaré in DP. It should be noted, however, that the name homeomorphism given today to such bijections as f was introduced by Poincaré. See Georg Feigl (1928), p.274.
Poincaré, VS, p.59; cf. VS, p.55; FS, p.417; DP, pp.100, 136.
Poincaré, DP, p. 135; VS, p.60.
Poincaré (1893); reprinted in Poincaré, SH, pp.58–60.
Euclid, Book I, Definitions 3 and 6; Book XI, Definition 2. (Heath, EE, Vol.I, pp.165, 171; vol.III, p.263.) Compare Menger (1926), pp.118f.
Poincaré, SH, pp.58–60; VS, pp.61–64.
Poincaré, SH, p.60; VS, pp.64f.
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Torretti, R. (1984). Empiricism, Apriorism, Conventionalism. In: Philosophy of Geometry from Riemann to Poincaré. Episteme, vol 7. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9909-1_4
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