Abstract
Plato held that specialized knowledge becomes true science only if one is aware of its foundations. To inquire into these, however, was not a task for the specialist but for the dialectician, whom we would call the philosopher. Yet philosophers from Plato to Kant have not contributed much to our awareness of the foundations of geometry. Some of them did discuss the nature of geometrical objects and the source of geometrical knowledge, but they were content to accept the principles of geometry proposed by Euclid, and they rarely went into details concerning, say, the justification of this or that particular principle or the relationship between the principles and the body of geometrical propositions. Shortly after 1800, G.W.F. Hegel claimed that mathematical axioms, insofar as they are not mere tautologies, ought to be proved in a philosophical science, prior to mathematics. Euclid, said Hegel, was right not to attempt a demonstration of Postulate 5, for such a demonstration can only be based on the concept of parallel lines and therefore pertains to philosophy, not to geometry.1 However Hegel himself did not provide the demonstration.
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Notes
Hegel, Logik; Werke, ed. Glockner, Vol.V, p.306.
Delboeuf, PPG, pp.75ff.
Since the axiom of completeness is lacking — it was added in the French translation of 1900 — the system can be modelled in the denumerable set of algebraic number triples. See p.237.
3.1 Helmhoitz’ Problem of Space
Helmhoitz (1866), p. 197.
Riemann, H, p.8. See above, p.84.
Riemann, H, p.21. See above, p. 104.
Helmhoitz (1870), G, p. 19.
Helmhoitz (1868), G, p.60.
See above, p. 100. A proof was given by Lipschitz (1870).
Helmhoitz (1866), pp.199, 201; (1868), G, pp.36, 60; (1870), G, p.29. Helmhoitz is so convinced of the necessity of this requirement that he does not even perceive that Riemann was of another mind. He simply overlooks the fact that Riemann, who shared with him the assumption that physical measurement involves superposition of physical magnitudes, inferred from this assumption, not that every body must be superposable with every other body, but that every line must be measurable by — and to this end totally or partially superposable with — every other line (p.91). We might thus say that Riemann’s physical geometry requires ideally thin, perfectly flexible and inextensible strings, but does not require, like that of Helmhoitz, absolutely rigid bodies.
Helmhoitz (1870), G, p.29.
See p. 177.
Helmhoitz, G, p.41.
Helmholtz’s reluctance to extend his axioms concerning rigid point systems to systems of any arbitrary size is motivated perhaps by a fact he learned through his investigations on the psychophysiology of vision: “The visual field exhibits a more restricted mobility of the images on the retina”. (Helmhoitz, G, p.40).
“All rotations of the system about the point r = s = t = 0” (Helmhoitz, G, p.57) really means all infinitesimal rotations about any arbitrarily fixed line element through that point.
Helmholtz (1868), G, p.57. For a reconstruction of Helmholtz’s proof, which introduces clarity and precision into it, while remaining faithful to its spirit, see Weyl, MAR, pp.29–43.
Helmholtz (1866), p.201.
Kant, Ak., Vol.IV, p.312.
Helmholtz, G, p.71f.
Helmholtz, G, p.8; PSL, p.227.
Helmholtz, G, p.25; PSL, p.241. Cf. Poincaré, FS, pp.416f.
Helmholtz, G, p.28; PSL, p.244.
Helmholtz, G, p.81.
Helmholtz (1878), p.213; G, p.62.
Schlick in Helmholtz, SE, p. 162.
“As all our means of sense-perception extend only to space of three dimensions, and a fourth is not merely a modification of what we have, but something perfectly new, we find ourselves by reason of our bodily organization quite unable to represent a fourth dimension.” Helmholtz, G, p.28; PSL, p.244.
I cannot accept, however, Schlick’s alternative suggestion that Helmholtz’s “general form of spatial intuition” exhibits space as a three-dimensional continuous manifold wherein quantitative comparisons are possible (Schlick in Helmholtz, SE, p. 161). The possibility of quantitative comparisons is not a consequence of the general form of extendedness, since it presupposes the existence of rigid bodies. A liquid mathematician living in a liquid world cannot, according to Helmholtz, develop a geometry, but he could very well share our general form of spatial intuition.
Helmholtz, G, p.29; PSL, p.244.
Helmholtz, G, p. 17; PSL, p.234. Formerly, he had tried to make this intuitively clear in the light of the two-dimensional case. In that context, he introduces a two-dimensional country inhabited by two-dimensional rational beings who “have not the power of perceiving anything outside” the surface they live on, but have, upon it, “perceptions similar to ours”! (Helmholtz, G, p.8; PSL, p.227). These beings will supposedly develop the intrinsic geometry of their country. But if they happen to live, say, upon an egg, they will be unable to build transportable rigid figures and, consequently, according to Helmholtz, they will be incapable of defining a geometry. Though Helmholtz introduces the example of an egg, he does not draw the latter consequence; had he drawn it, it would probably have shocked him out of his operationist bias, for he certainly knew that one can define with Gaussian methods the intrinsic geometry of an egg-like surface. Let us remark that the country of Flatland is used by Helmholtz merely as a didactic prop, and ought not to be taken too seriously. Greater significance must be attributed to the Helmholtzian country we mentioned on p. 165 viz. Through-the-Convex-Looking-Glass.
Helmholtz (1866), p.197; cf. Helmholtz (1868), G, pp.32f.
Helmholtz (G, pp.29f.; PSL, p.245) says “transcendental in Kant’s sense”. But the sense intended is certainly not Kant’s, for Helmholtz says in the same sentence that experience “need not exactly correspond therewith”.
Helmholtz, G, p.30; the English version in PSL, p.245 is very free.
Helmholtz, G, p.29; PSL, pp.244f.
Poincaré (1898), p.40, acknowledges that he owes much to Helmholtz.
Helmholtz, G, p.29; PSL, p.245 (quoted in Section 3.1.2, p. 160).
Helmholtz, G, p.30; PSL, p.245.
Lie (1890), p.359 n.2.
That Lie’s continuous groups are connected follows from the definition in Lie, TT, Vol.1, p.3.
Lie, TT, Vol.III, pp.385, 387, etc. Observe that by fixing once and for all the nature of the space R n on which his groups are allowed to act, Lie was able to ignore the complications that arise from considering the action of groups on manifolds of diverse global topologies.
Lie, TT, Vol.III, p.481. Two groups are similar, in Lie’s idiom, if one can be obtained from the other through a coordinate transformation of the underlying manifold (ibid., p.364). I assume that a “real point transformation” of the complex manifold R n is a coordinate transformation that maps real coordinates onto real coordinates, non-real coordinates onto non-real ones (thus preserving the division between real and imaginary points).
Lie, TT, Vol. III, p.538. The terms ‘proper subgroup’ and ‘normal subgroup’ are defined in the Appendix, p.360.
Poincaré (1887).
Killing (1892), p.129.
Killing (1892), p. 153. I understand he means transitive n-dimensional connected Lie groups.
Killing (1892), p.131.
Killing (1892), p. 137. Compare Killing’s treatment of dimension number in EGG, Vol.I (1893), Absch.3, especially pp.238–265.
Killing (1892), p. 167.
Hubert, GG, p. 181n.
3.2 Axiomatics
“Les géomètres supposent toutes ces propositions sans le dire.” I have not seen a copy of Lamy’s book; I take this quotation from P. Rossier, “Les axiomes de la géométrie et leur histoire” (1967), pp.379f., who obtained it from the 6th edition (1734).
See G. Goe (1962).
Plücker, Neue Geometrie des Raumes (1868–69); see Nagel (1939), pp.188–192.
Strictly speaking, I demand that the set of axioms be computable. For a definition of computable set see, e.g., M. Davis, Computability and Unsolvability. Informally, we may say that a set S is computable if an ordinary computer with unlimited memory can be programmed to determine whether any given object belongs to S or not.
See Oswald Veblen (1911), pp.5f.
See John Mayberry (1977).
Stewart, WW, Vol.III, p.117. See also the quotation ibid., p.123, n.1.
Stewart, WW, Vol.III, p. 114.
See, however, the passage from Buffon’s Histoire Naturelle (Vol.I, Paris 1749, pp.53f.) quoted in Tonelli (1959), p.47 n.52.
Stewart, WW, Vol.III, pp.26, 31, 32 n.1.
Grassmann, WW, I.1, p. 10.
Grassmann, WW, I.1, p.65.
See the references in Note 67 of Part 2.2.
Veronese, GG, p.674.
Plücker, Gesammelte wissenschaftliche Abhandlungen, Vol.I, p.173; quoted by Nagel (1939), pp.l91f.
Plücker, ibid., p. 149; quoted in Nagel (1939), p.191.
Bolzano (1804), p.46. (My italics.)
On Staudt, see Freudenthal (1974).
Delboeuf, PPG, p. 127.
Delboeuf, PPG, p. 128.
Delboeuf, PPG, p. 129.
Delboeuf, PPG, pp.223, 229, 222.
Hoüel, PFGE, pp. 63, 64.
Hoüel, PFGE, pp.64f.
Hoüel, PFGE, p.66. Compare Hubert’s letter to Frege, of December 1899, quoted on p.235.
Rossier (1967), p.896.
Méray (1869). Méray’s work is summarized in Mannheim, GPST, pp.80–82.
Pasch, Vorlesungen über neuere Geometrie, first edition, Leipzig 1882 (VNG); second edition, Berlin 1926 (VNG2). The second edition was revised by Pasch himself, not by Max Dehn, as we read in Kline, MT, p. 1008. Dehn wrote the valuable historical study appended to this edition.
Pasch, VNG, p. 16.
Pasch, VNG, p.4.
Pasch, VNG2, p. 18.
Pasch, VNG, p.45.
Pasch, VNG, p.6.
Pasch, VNG, p.5.
Pasch, VNG, p. 17.
Pasch, VNG, p.43.
Pasch, VNG, p.98.
Pasch (1917), p. 185.
Pasch, VNG, p.3.
Pasch, VNG, p.IV; VNG2, p.VI.
Pasch, VNG, pp.5–7; VNG2, pp.5–7. Axioms S I-S VIII are exactly the same in the first and in the second edition. S IX is added in the latter. I have been unable to trace the source of a different list of eight axioms which Paul Rossier (1967), pp.400f., says he found in VNG2. I have prefixed the letter S (for Strecke, segment), to the axioms of Pasch’s first group, in order to distinguish them from the other two groups, which I call E (for Ebene = plane) and K (Kongruenz).
Pasch, VNG, pp.20–21; VNG2, pp. 19–20. (Same text in both editions.)
Pasch, VNG, p.34.
Pasch, VNG, p.40.
Pasch, VNG, p.51.
Pasch, VNG, p.58.
Pasch, VNG, p.58.
Pasch, VNG, pp.103–110; VNG2, pp.94–101. The text of these axioms is the same in both editions. My version of them is more a paraphrase than a translation, except for K VI, which I have translated literally.
Pasch, VNG, §11, pp.83ff. See Section 2.3.9, pp.143ff.
Schur (1899) showed how to introduce homogeneous coordinates using axioms of congruence, but without having to assume the Archimedean axiom or any axiom of continuity.
Pasch, VNG, p. 120.
Pasch, VNG, pp. 125f.
Pasch, VNG, p. 127. Theorem 1.8 says that if two proper points A, B lie on a line m, you can always choose a proper point C on m which lies between A and B (Pasch, VNG, p. 10). Pasch remarks however that this theorem cannot be applied to a given line an indefinite number of times (Ibid., p. 18). Theorem 1.8 follows immediately from Axiom SII. It can also be derived from Axioms S I, S VI, S IX, EIV (Hilbert, GG, p.5, Satz 3). Consequently, one or more of these axioms have a restricted application. Such are the miseries of mathematical empiricism.
Pasch, VNG2, p. 174 (added in the 2nd edition).
Peano (1894), p.52. These ideas must have a name in every language, at least in all languages known to be suitable for geometrical discourse. Consequently, says Peano, space cannot be a basic concept of geometry, because there is no word for it in the language of Euclid and Archimedes.
Peano (1889), p.24.
Peano (1894), p.62.
Fano (1892), p. 108; quoted by Freudenthal (1957), p.112n.
Veronese, GG, p.656; my italics.
Peano (1894), p.75.
Pieri (1900), p.373.
Pieri (1900), pp.375f.
Pieri (1900), p.387. “Axiom XII” is Euclid’s Postulate 5.
Pieri (1900), p.387. See Section 3.2.10, pp.252f.
Pieri (1900), pp.387f. Cf. Freudenthal (1957), p.116.
Pieri (1900), pp.388f.
Pieri (1900), pp.373, 374. See also Pieri (1899a), p.2.
Mario Pieri (1899a), “I principii della geometria di posizione composti in sistema logico deduttivo”; (1899b) “Della geometria elementare come sistema ipotetico deduttivo. Monografia del punto e del moto.” Pieri mentions three earlier papers of his as preliminary stages of (1899a): “Sui principii che reggono la geometria di posizione” (Atti della R. Accademia di Scienze di Torino, Vols. 30, 31), “Un sistema di postulati per la geometria proiettiva astratta degl’iperspazii” (Revista di Matematica, Vol.6), and “Sugli enti primitivi della geometria proiettiva astratta” (Atti della R. Acc. di Se. di Torino, Vol.32). These papers were published in 1895–97.
Pieri (1899a), pp. 1,2.
Pieri (1899a), p.2.
Finite geometries were first introduced by G. Fano, Pieri quotes Fano’s paper “Sui postulati fondamentali della geometria proiettiva”, Giornale de Matematica, Vol.30 (1891).
Pieri (1899b), p.177. Cf. Pieri (1899a), pp.4f., Pieri (1900), pp.381f.
Peano (1900), p.279.
Burali-Forti (1900), p.290. “Definitions by abstraction” characterize a mapping f by indicating its domain and the partition of this into subsets on which f is constant. (Ibid., p.295).
Frege, Grundlagen der Arithmetik (1884).
Padoa (1900), p.318. Padoa’s phrasing is awkward. Axioms do not impose conditions on the symbols which occur in them, but on the objects (properties, etc.) which these symbols may stand for.
Padoa (1900), p.319f.
Padoa (1900), p.323.
Padoa (1900), p.321.
Padoa (1900), p.322. See Padoa (1900b).
Hubert, GG, p. 1.
Hubert, GG, p.2.
Hubert, GG, p. 1.
Hubert, GG, pp.125.
Axiom V 2 did not appear in the first edition. Its necessity was pointed out by Hilbert in a lecture “Ueber den Zahlbegriff” delivered in 1899. (Appendix VI of GG, 7th edition; suppressed in later editions.) The French translation of the Grundlagen by Laugel, published in 1900, includes a note, signed by Hilbert, which reads, in part, as follows: “Remarquons qu’aux cinq précédents groupes d’axiomes l’on peut encore ajouter l’axiome suivant qui n’est pas d’une nature purement géométrique [...]: Au système des points, droites et plans, il est impossible d’adjoindre d’autres êtres de manière que le système ainsi généralisé forme une nouvelle géométrie où les axiomes des cinq groupes (I) à (V) soient tous vérifiés; en d’autres termes; les éléments de la géométrie forment un système d’êtres qui, si l’on conserve tous les axiomes, n’est susceptible d’aucune extension. [...] La valeur de cet axiome au point de vue des principes, tient [...] à ce que l’existence de tous les points limites en est une conséquence et que, par suite, cet axiome rend possible la correspondance univoque et réversible des points d’une droite et de tous les nombres réels.” (Hilbert-Rossier, FG, pp.43f.). Editions 2–6 carry the following version of V 2: “The elements of geometry, points, lines and planes, constitute a system of objects which, if you assume the foregoing axioms, does not admit any extension.” In the 7th edition, a statement similar to this is proved as Theorem 32. A discussion in depth of the axiom of completeness is given in Bernays (1955). See also Baldus (1928).
Frege, KS, p.409.
Frege, KS, p.411.
Dehn (1900).
Huntington (1902), p.277.
Veblen (1904), p.346.
Like Peano and Veblen, Huntington regards the signs that stand for the set-theoretical predicates “x is a set”, “x is an element of set y”, as “symbols which are necessary for all logical reasoning” (Huntington (1913), p.526), and treats them as non-interpretable words.
Huntington (1913), p.530.
Huntington (1913), p.524.
Huntington (1913), p.540.
Menger (1928); Birkhoff (1935).
I follow Menger (1940), Lecture II. See also Blumenthal and Menger, Studies in Geometry (1970), Parts I and II;
Birkhoff, Lattice theory (1967), Chapter IV.
For a concise description and classification of its several currents, see P. Bernays (1959).
See Bachmann, Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd enlarged edition (1973).
Tarski (1959), p. 16.
Grassmann, WW, Vol.I.1, p.65.
See Section 4.1.4.
Hubert, GG, pp.4, 11.
Frege, KS, p.265. Cf. Frege, KS, p.282.
Frege, KS,p.411.
Frege, KS, p.411. In the first edition of Hilbert’s Grundlagen, group II comprised five axioms.
Frege, KS, p.416. Frege made this point again in the second of his two articles on Hubert’s Grundlagen (Frege, KS, p.268), and in the first of his three articles in reply to Korselt (Frege, KS, p.291). These five articles (as well as Korselt’s) bear the same title: “Ueber die Grundlagen der Geometrie”.
Cf. Freudenthal (1957), p. 116.
Hilbert, GG, p.28. In the first edition, Hilbert used indifferently the words Definition and Erklärung. After Frege suggested that these two words might express two different ideas, Hilbert changed all occurrences of Definition into Erklärung. (See Frege, KS, p.407).
Pieri (1899a), p.3 n.6. Cf. Pieri (1899b), p.173n.; (1900), p.378n.1.
Frege, KS, p.288.
Frege, KS, p.412. The kind of “invertible univocal transformation” which Hilbert has in mind has been illustrated above, on pp.81f.
Gergonne (1818), p. 13.
Gergonne (1818), p.22.
Gergonne (1818), p.23; my italics, except for the words implicit and explicit, italicized by the author.
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Torretti, R. (1984). Foundations. In: Philosophy of Geometry from Riemann to Poincaré. Episteme, vol 7. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9909-1_3
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