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Published online by Cambridge University Press: 03 November 2016
The subject of the definition of irrational numbers is so intimately connected with questions about the existence of a limit of a sequence, which has recently formed the subject of much discussion in this Gazette, that a fuller consideration of the subject, and, in particular, of how irrational numbers are to be introduced into a course introductory to ‘higher’ mathematics, may not be undesirable.
page 201 note * October, 1905, pp. 236-237; May, 1906, p. 327; July, 1906, pp. 333-335, 349-350 October, 1906, p. 380 (all of vol. iii.).
page 202 note * See Peano in, e.g., the Formulario de mathematica of 1905, pp. 74, 83, 95, 100; Ch. Méray, , Leçons nouvelles sur l’analyse infinitésimale, 1re partie, Paris, 1894, pp. 3–10 Google Scholar; Russell, , The Principles of Mathematics, vol. i., Cambridge, 1903, pp. 149–150, 229, 374, 376-380Google Scholar; and Couturat, , Les principes des mathématiques, Paris, 1905, pp. 79–81, 138Google Scholar.
page 202 note † Not ‘expressions (or signs) for lengths’; see below, § 5.
page 202 note ‡ A sequence such that, given any positive rational ϵ, there is an integer n such that | sn – sn+m | < ϵ, for any m. That convergency is necessary for the existence of a limit is easily proved; that it is also sufficient, Bolzano and others have tried to prove (see Ostwald’s Klassiker der exakten Wissenschaften, Nr. 153, pp. 41-43, 107), but requires a prior arithmetical definition of the ‘ real numbers.’
page 202 note § For references, see Encycl. des sci. math., i. 3, pp. 147-155.
page 202 note ∥ We notice that here, if m > n, then sm ≧ sn ; but not if the condition of convergency is satisfied only for some (but an infinity of) m’s, then the sequence has many limits (Cauchy, P. du Bois-Reymond, Hadamard).
page 202 note ¶ We except the theory of Weierstrass, which has the advantages mentioned below. Méray’s theory forms an apparent exception, if the term ‘irrational number’ is thought, in this theory, to have no meaning in itself, but only the phrase ‘the variant v has a certain relation with a fictitious number V’ to have the meaning ‘there is no rational number to which v has this relation.’ Then we evidently cannot speak of ‘the fictitious numbers’ as if they were, by this, defined entities. But Méray’s theory seems (like Cantor’s) to consist in the (arbitrary, though convenient and non-contradictory) postulation of new entities {“nombres fictifs”) (cf. Encyclopédie des sci. math., i. 3, pp. 148, 149, 152 note 56).
page 203 note * Cantor said that we “coordinate to the fundamental-series (αv) a number b to be defined by it” and defined the magnitude-relations between two such b’s by relations between their corresponding series; b can then be proved to be the limit of (αv ) (Math. Ann., Bd. xxi., 1883, p. 567); see also Dedekind, Stetigkeit und irrationale Zahlen, p. 14.
page 203 note † Namely, in which a real number is defined as the ‘limit of a certain series of rationale.’ On geometrical ones, see below, § 5.
page 203 note ‡ For Méray’s statement of this, see Encyclopédie des sci. math., i. 3, p. 149; for Cantor’s and Heine’s, see Heine’s paper in the Journ. für Math., Bd. 74, 1872. For the criticism relating to this, see Russell, op, cit., pp. 270, 282, 285.
page 203 note § “Arithmetices Principia nova methodo exposita,” Turin, 1889, pp. 15-16, in the various editions of the Formulaire des mathématiques (e.g., Formulario de mathematica, 1905, p. 105), and the article “Sui numeri irrazionali,” Riv. di mat., t. 6, pp. 126-140. Peano’s logical symbolism, in which these are written, has been also described in Whitehead’s, paper in the Amer. Journ. of Math., vol. xxiv., 1902, pp. 367–394 CrossRefGoogle Scholar. As to the questions of the logical validity of this definition (“by abstraction”) and the (“nominal”) definition of real number, due to Weierstrass (see the text), Frege and Russell (op. cit., pp. 270-286; see especially the remarks on Peano on pp. 274-275: cf. Couturat, op. cit., pp. 36-43).
page 203 note ∥ Hence, in Peano’s system, we cannot introduce the idea of rational numbers being limits before studying limits in general, while the other methods have the (didactic, principally) advantage of allowing this. It is, then, not an error from Peano’s point of view to speak of “irrationals as based on limits” (Russell, op. cit., p. 274) though it is from (e.g.) Cantor’s point of view.
page 203 note ¶ The magnitude-relations of these numbers are supposed to have been defined.
page 203 note ** ‘Terminus summus’ or ‘limes summus classis a’ (Arith. Princ., p. 15).
page 203 note †† ‘Upper limit of a’ (cf. Formulario, 1905, p. 105).
page 204 note * Russell seems to me to give a wrong impression of Dedekind’s theory when he stated (op. cit., p. 280) that this theory “is designed to prove the arithmetical existence of irrationals.” It was designed to create or postulate irrationals in a definite way, and then to prove the existence of limits.
page 204 note † Namely, that to every limit of a convergent sequence belongs a line of that length.
page 204 note ‡ For the points in question may be absent; intuition is not so refined as to be able to decide on the point.
page 204 note § Especially Russell.
page 204 note ∥ According to Pringsheim (Encykl. der math. Wiss., i. A3, p. 54, note 21), Cantor (Math. Ann., Bd. 21, p. 553) had a different opinion. The passage referred to was directed against the tendency of some (like Kronecker) who regarded all extensions of the number-concepc as “marks of calculation” (Rechenmarken). It seems that Cantor did, at this time, support formalism (cf. Cantor in Math. Ann., Bd. xxi., 1883, pp. 589-590, Heine’s paper and Cantor’s remarks [“Zur Lehre vom Transfiniten,” Halle, 1890. pp. 20-21, 54], and Frege’s [“Die Grundlagen der Arithmetik,” Breslau, 1884, p. 108] note on the character of the analogous [see Cantor, ibid., pp. 34-35, 48-49] transfinite ordinal numbers), but abandoned it,—at least for whole numbers—afterwards (cf. Cantor’s criticism of Helmholtz and Kronecker, ibid., pp. 15-20).
page 204 note ¶ Die allgemeine Functionentheorie. Tübingen, 1882, p. 55.
page 204 note * Pringsheim supported the view that the “real numbers are an unlimited system of signs, which have a uniquely determined succession, and with which we can calculate according to definite rules” (p. 79 of his essay, “Über den Zahl- und Grenzbegriff im Unterricht,” Jahresber. der deutsch. Math.-Ver., Bd. vi., 1898, pp. 73-83; see also Encykl. der math. Wiss., i. A3, pp. 54-55. H. Hankel was the best-known supporter of this formalism (see his Theorie der complexen Zahlensysteme, Leipzig, 1867).
page 205 note * Cf. Dini, and Lüroth, , Grundlagen für eine Theorie ..., Leipzig, 1892, pp. 2, 6Google Scholar; and Godefroy, M., Théorie élémentaire des séries, Paris, 1903, p. 1 Google Scholar, said: “... all the other numbers can be denned as groups of integers.” This is not quite correct, as {e.g.) rationals are relations.
page 205 note † This important question of the existence of real numbers has been emphasised and solved by both Frege (op. cit., pp. 114-115) and Bussell (op. cit., pp. 270-286), and consists in that, when real numbers are defined as classes, each such class can be shown to have at least one member.
page 205 note ‡ Not ‘defines,’ with Cantor, without the necessary explanation of how it defines it, nor ‘is a sign for’ with Heine (cf. § 2).
page 205 note § b – b′n is defined to be the class (a 1 – a n, a 2 – an , ...).
page 205 note ∥ The real number zero is the class (0, 0, ..., 0).
page 205 note ¶ Gazette, July, 1906, vol. iii., pp. 349-360.
page 205 note ** Ibid., October, 1905, pp. 296-297.
page 205 note †† Ibid., May, 1906, p. 327.
page 205 note ‡‡ Ibid., July, 1906, pp. 333-335.
page 206 note * Math. Ann., Bd. xxi., 1883, p. 566.
page 206 note † See a note in § 2 above.
page 206 note ‡‡ The Connexion of Number and Magnitude; an Attempt to explain the Fifth Book of Euclid, London, 1836.
page 206 note § Thus, 
is the class of those rationale x such as x
2 < 2. Cantor’s is a class of certain of these x’s, and, though, in Cantor’s definition, there is a certain amount of arbitrariness in the choice of elements of the class, yet two such classes are defined to be ‘equivalent,’ and one name () is given o them both.
page 207 note * See Encycl. des sci. math., i. 3, pp. 146-147, 157-158.
page 207 note † Cf. § 2 above. We may add the following references to Pringsheim’s support of the “sign”-theory: Sitzungsber. der math.-phys. Cl. der Kgl. bayer. Akad. za München, Bd. xxvi., 1896, p. 606, and Bd. xxvii., 1897, pp. 321-324.
page 207 note ‡ The origin of this may be that we say habitually ‘2 is a number’ when we should say the sign for a number; but the first is usually understood.
page 208 note * Cf. G. Frege, “Was ist eine Funktion?,” Boltzmann—Festschrift, Leipzig, 1904, pp. 656-666.
page 208 note † Zur Lehre vom Transfiniten, Halle, 1890, pp. 15-20.
page 208 note ‡ I mean that the notion of class has limits of validity.
page 208 note § De Morgan, for instance, had avoided this error in his text-books.
page 209 note * Theoric des fonctions analytiques, ..., Paris, 1797.
page 209 note † Lagrange's motive was probably the economy of thought resulting from the substitution of analytical for geometrical processes.
page 209 note ‡ See Ostwald's Klassiker der exakten Wissenschaften, Nr. 153, pp. 4-7. 39.
page 209 note § Couturat's book cited above is a clearly written exposition of Russell's work.
