An Essay on the Foundations of Geometry

are popularizings and expansions of the article in Mind, Vol. III. The most widely read, though also, to my mind, the least valuable, of all Helmholtz's writings on Geometry, is the article in Mind, Vol. I. This contains the famous and much misunderstood analogies of Flatland and Sphereland, which will be discussed, and as far as possible defended, in answering Lotze's attack on Metageometry—an attack based, apparently, almost entirely on this one popular article. The present discussion, therefore, may be confined almost entirely to Mind, Vol. III., and the philosophical portions of the two papers quoted in Chap. I., i.e. to the articles in Wiss. Abh. Vol. II. pp. 610–660. His other works are popular, and important only because of the large public to which they appeal. [84] In the answer to Land, Mind, Vol. III. and Wiss. Abh. II. p. 640. [85] See also Die Thatsachen in der Wahrnehmung, Zusatz II., Der Raum kann transcendental sein, ohne dass es die Axiome sind. Vorträge und Reden, Vol. II. [86] See below, criticism of Erdmann, § 84. [87] See Prof. Land, in Mind, Vol. II. [88] See concluding paragraph of Helmholtz's article in Mind, Vol. III. [89] Cf. Veronese, Grundzüge der Geometrie (German translation), p. ix. Also pp. xxxiv, 304, and Note II. pp. 692–4. [90] See Chap. IV. § 197 ff. [91] Cf. the opinion of Bolyai, quoted by Erdmann, Axiome, p. 26; cf. also ib. p. 60. [92] Die Axiome der Geometrie: Eine philosophische Untersuchung der Riemann-Helmholtz'schen Raumtheorie, Leipzig, 1877. [93] On the influence of Mill, cf. Stallo, Concepts of Modern Physics, p. 216. [94] This view seems to be derived, through Riemann, from Herbart. See Psych. als Wiss. ed. Hart. Vol. V. p. 262. [95] The same irreducibility of space to mere magnitude is proved by Kant's hands and spherical triangles, in which a difference persists in spite of complete quantitative equality. [96] See §§ 146–7. [97] "Jeder Versuch, Kant's Lehre von der Apriorität als des subjectiven, von aller Erfahrung absolut unabhängigen Erkenntnissfactors, trotzdem zu halten, ist deshalb von voruherein aussichtslos." [98] Ausdehnungslehre von 1844, 2nd edition, pp. xxii. xxiii. [99] See § 129 ff. [100] Metaphysik, Book II. Chap. II. My references are to the original. [101] See Lectures and Essays, Vol. I. p. 261. [102] On the meaning of geometrical possibility, cf. Veronese, Grundzüge der Geometrie (German translation), pp. xi.-xiii. [103] Compare Calinon, "Sur l'Indétermination géométrique de l'Univers," Revue Philosophique, 1893, Vol. XXXVI. pp. 595–607. [104] Vorträge und Reden, Vol. II. p. 9: "Parallele Linien würden die Bewohner der Kugel gar nicht kennen. Sie würden behaupten, dass jede beliebige zwei geradeste Linien, gehörig verlangert, sich schliesslich nicht nur in einem, sondern in zwei Punkten schneiden müssten." (The italics are mine.) The omission of straight in such phrases is a frequent laxity of mathematicians. [105] It has been suggested to me that Lotze regards the meridians as projected on to a plane, as in a map. If this be so, there is an obviously illegitimate introduction of the third dimension. [106] This is proved by Helmholtz's remark at the end of a detailed attempt to make spherical and pseudo-spherical spaces imaginable (l.c. p. 28): "Anders ist es mit den drei Dimensionen des Raumes. Da alle unsere Mittel sinnlicher Anschauung sich nur auf einen Raum von drei Dimensionen erstrecken, und die vierte Dimension nicht bloss eine Abänderung von Vorhandenem, sondern etwas vollkommen Neues wäre, so befinden wir uns schon wegen unserer körperlichen Organisation in der absoluten Unmöglichkeit, uns eine Anschauungsweise einer vierten Dimension vorzustellen." [107] Cf. Grassmann, Ausdehnungslehre von 1844, 2nd Edition, p. xxiii. [108] See especially Stallo, Concepts of Modern Physics, International Science Series, Vol. XLII. Chaps. XIII. and XIV.; Renouvier, "Philosophie de la règle et du compas," Année Philosophique, II.; Delbœuf, "L'ancienne et les nouvelles géométries," Revue Philosophique, Vols. XXXVI.-XXXIX. [109] M. Delbœuf deserves credit for having based Euclid, already in 1860, in his "Prolégomènes Philosophiques de la Géométrie," on this axiom—certainly a better basis, at first sight, than the axiom of parallels. [110] This meaning of homogeneity must not be confounded with the sense in which I have used the word. In Delbœuf's sense, it means that figures may be similar though of different sizes; in my sense it means that figures may be similar though in different places. This property of space is called by Delbœuf isogeneity. [111] For a full proof of this proposition, see Chap. III. [112] See Chap. III., especially § 133. [113] For a criticism of this view, see the above discussions on Riemann and Erdmann. [114] Cf. Couturat, "De l'Infini Mathématique," Paris, Félix Alcan, 1896, p. 544. [115] The following is a list of the most important recent French philosophical writings on Geometry, so far as I am acquainted with them. Andrade: "Les bases expérimentales de la géométrie euclidienne"; Rev. Phil. 1890, II., and 1891, I. Bonnel: "Les hypothèses dans la géométrie"; Gauthier-Villars, 1897. L'Abbé de Broglie: "La géométrie non-euclidienne," two articles; Annales de Phil. Chrét. 1890. Calinon: "Les espaces géométriques"; Rev. Phil. 1889, I., and 1891, II. "Sur l'indétermination géométrique de l'univers"; ib. 1893, II. Couturat: "L'Année Philosophique de F. Pillon," Rev. de Mét. et de Morale, Jan. 1893. "Note sur la géométrie non-euclidienne et la relativité de l'espace"; ib., May, 1893. "Études sur l'espace et le temps," ib. Sep. 1896. Delbœuf: "L'ancienne et les nouvelles géométries," four articles; Rev. Phil. 1893–5. Lechalas: "La géométrie générale"; Crit. Phil. 1889. "La géométrie générale et les jugements synthétiques à priori" and "Les bases expérimentales de la géométrie"; Rev. Phil. 1890, II. "M. Delbœuf et Le problème des mondes semblables"; ib. 1894, I. "Note sur la géométrie non-euclidienne et le principe de similitude"; Rev. de Mét. et de Morale, March, 1893. "La courbure et la distance en géométrie générale"; ib., March, 1896. "La géométrie générale et l'intuition"; Annales de Phil. Chrét., 1890. "Etude sur l'espace et le temps"; Paris, Alcan, 1896. Liard: "Des définitions géométrie et des définitions empiriques," 2nd ed.; Paris, Alcan, 1888. Mansion: "Premiers principes de la métagéométrie"; two articles in Rev. Néo-Scholastique, 1896. Separately published, Gauthier-Villars, 1896. Milhaud: "La géométrie non-euclidienne et la théorie de la connaissance"; Rev. Phil. 1888, I. Poincaré: "Non-Euclidian Geometry"; Nature, Vol. XLV., 1891–2. "L'espace et la géométrie"; Rev. de Mét. et de Morale, Nov. 1895. "Résponse à quelques critiques," ib. Jan. 1897. Renouvier: "Philosophie de la règle et du compas"; Crit. Phil., 1889, and L'Année Phil., IIme année, 1891. Sorel: "Sur la géométrie non-euclidienne"; Rev. Phil., 1891, I. Tannery: "Théorie de la connaissance mathématique"; Rev. Phil., 1894, II.